Complex Vector Functional Equations

Ice Risteski Valery Covachev World Scientific (complex Vector Functional Equations Complex Vector Functional Equati...

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Ice Risteski Valery Covachev

World Scientific

(complex Vector Functional Equations

Complex Vector Functional Equations

Ice Risteski Valery Covachev Bulgarian Academy of Sciences

V^fe World Scientific « •

New Jersey • London »Sint • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

COMPLEX VECTOR FUNCTIONAL EQUATIONS Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4683-8

Printed in Singapore by Uto-Print

To the authors' sons Branko Risteski and Svetoslav Covachev

Preface

Complex vector functional equations are a new field in the theory of functional equations. Their development up to now has not been particularly dynamical. As every new mathematical discipline, complex vector functional equations call for their more detailed investigation. From this standpoint, in the last few years we dedicated a series of papers to these equations. Our multi-annual experience and the results obtained by the research in this new area show us the necessity to write a strictly scientific systematized monograph. The object of this monograph is to give an overview to the recently obtained results of the authors from all aspects of utilization, which is obviously not possible with partial results in papers published in distinct mathematical journals. We have divided our exposition on the field of complex vector functional equations in two parts which are devoted respectively to linear and nonlinear equations. Part One consists of five chapters. In the first chapter the general classes of cyclic functional equations in a complex vector form are considered. Among these are: the basic cyclic functional equation, the derived cyclic functional equation, the semicyclic functional equation, the special cyclic functional equation and the condensed cyclic functional equation. In the second chapter the complex vector functional equations with operations between arguments are considered. First an operator functional equation is solved. Also, in this chapter the generalized equation of operator vii

Vlll

Preface

type is solved. Next, some simple functional equations and functional equations with distinct functions are solved. Of this type of functional equations we can mention Frechet's functional equations, to which an adequate place in this chapter is given. Finally, some complex vector functional equations with two operations between arguments are also solved. T h e functional equations with constant parameters are considered in the third chapter. T h e equations of this type which are solved here are: the general parametric functional equation, the special parametric functional equation, the expanded parametric functional equation and the general expanded parametric functional equation. T h e linear complex vector functional equations with constant coefficients are considered in the fourth chapter by a new m a t r i x approach. In particular, we pay attention b o t h to homogeneous and nonhomogeneous functional equations with constant coefficients. Finally, a paracyclic functional equation with complex constant coefficients is solved. In the last fifth chapter of P a r t One two types of systems of functional equations are studied. First some systems in which each equation contains all unknown functions are solved, and then some systems in which not all equations contain all unknown functions. P a r t Two is also divided in five chapters which are described below quite briefly. C h a p t e r s 6, 7 and 8 are devoted respectively to quadratic, modified quadratic and expanded quadratic equations. Higher order complex vector functional equations are considered in the ninth chapter. Finally, in C h a p ter 10 systems of nonlinear (quadratic and higher order) complex vector functional equations are studied. T h e present monograph is intended for mathematicians, physicists and engineers who use functional equations in their investigations. T h e authors are well aware t h a t all aspects of the complex vector functional equations cannot be included in an adequate way in a newly written book, a n d they will be grateful for all remarks and suggestions which will be valuable for the correction and completion of a subsequent edition. T h e second author acknowledges partial support by G r a n t M M - 7 0 6 with the Bulgarian Science Fund. A part of the book was written during the work of the second author at Fatih University, Istanbul, Turkey.

Contents

Preface PART 1

vii Linear Complex Vector Functional Equations

Chapter 1 General Classes of Cyclic Functional Equations 1 Basic Cyclic Functional Equation 2 Derived Cyclic Functional Equation 3 Paracyclic Functional Equation 4 Semicyclic Functional Equation 5 Special Cyclic Functional Equation 6 Condensed Cyclic Functional Equation

1 3 3 6 29 65 69 72

Chapter 2 Functional Equations with Operations between Arguments 75 7 Operator Functional Equation 75 8 Generalized Functional Equation 81 9 Simple Functional Equations 94 10 Functional Equations with Several Unknown Functions 99 11 Frechet's Functional Equations 106 12 Functional Equations with Two Operations Ill Chapter 3 Functional Equations with Constant Parameters 13 General Parametric Functional Equation 14 Special Parametric Functional Equation 15 Expanded Parametric Functional Equation 16 General Expanded Parametric Functional Equation ix

121 121 135 142 161

x

Contents

Chapter 4 Functional Equations with Constant Coefficients 173 17 Matrix Equations 173 18 Homogeneous Functional Equations with Constant Coefficients 175 19 Nonhomogeneous Functional Equations with Constant Coefficients 190 20 Paracyclic Functional Equations with Constant Coefficients . . 197 Chapter 5 Systems of Linear Functional Equations 21 Systems in Which Each Equation Contains All Unknown Functions 22 Systems in Which Not All Equations Contain All Unknown Functions

205

PART 2

217

Nonlinear Complex Vector Functional Equations

205 214

Chapter 6 Quadratic Functional Equations 23 Simple Quadratic Functional Equation 24 Special Quadratic Functional Equation

219 219 227

Chapter 7 Modified Quadratic Functional Equations 25 Basic Quadratic Functional Equation 26 Permuted Quadratic Functional Equation 27 First Modified Quadratic Functional Equation 28 Second Modified Quadratic Functional Equation 29 Third Modified Quadratic Functional Equation

231 231 234 238 241 244

Chapter 8 Expanded Quadratic Functional Equations 247 30 Expanded Quadratic Functional Equations with Functional Arguments 247 31 Expanded Quadratic Functional Equations with the Same Signs between the Functions 251 32 Expanded Quadratic Functional Equation with Alternating Signs between the Functions 259 33 Generalized Quadratic Functional Equation 262 Chapter 9 Higher Order Functional Equations 34 Higher Order Functional Equation without Parameters 35 Higher Order Functional Equation with Complex Parameters 36 Nonlinear Operator Functional Equation

265 265 267 273

Contents

xi

Chapter 10 Systems of Nonlinear Functional Equations 37 Systems of Quadratic Functional Equations 38 Systems of Higher Order Functional Equations

285 285 311

Bibliography

319

Index

323

PART 1

Linear Complex Vector Functional Equations

Chapter 1

General Classes of Cyclic Functional Equations

In this chapter six general classes of cyclic partial linear complex vector functional equations are solved, namely the basic cyclic functional equation, the derived cyclic functional equation, the paracyclic functional equation, the semicyclic functional equation, the special cyclic functional equation and the condensed cyclic functional equation. The results given here were obtained in [I. B. Risteski (to appear A); I. B. Risteski and V. C. Covachev (2000); I. B. Risteski et al. (2000A)].

1

Basic Cyclic Functional Equation

First we will introduce the following notations. Let V be a finite dimensional complex vector space and let there exist a mapping / : Vn i-> V, Zj (1 < i < n) are vectors in V and C is a constant complex vector in the same space. We assume that Zj = (za(t),--- ,Zin(t))T, where zy(t) (1 < i < n) are complex functions and O = (0,0, • • • , 0) T is the zero vector in V. Now we will give the following results which were obtained in [I. B. Risteski et al. (2000A)]. Theorem 1.1 The general solution of the basic cyclic complex vector functional equation n

2__/f(Zi,Zi+i,---,Zi+n-i)

= o

4=1

3

( z n + j = Zj)

(i.i)

4

General Classes of Cyclic Functional

Equations

is given by / ( Z 1 , Z 2 , - - . , Z „ ) = F ( Z i , Z 2 ) - " ,Zn)-F(Z2,Z3,---,Zn,Z1) where F is an arbitrary Proof.

complex vector function

(1.2)

with values in V.

Prom Eq. (1.1) it immediately follows t h a t / ( Z i , Z 2 , • • • ,Z„) = —/(Z2,Z3,- • • ,Z„,Zi) —

/(Z3,Z4)- • • ,Zi,Z2) — • • • — /(Zra,Zi, • • • ,Z„_i)

and n

f(Zi,

Z 2 , • • • , Z „ ) = / ( Z i , Z 2 , • • • , Z „ _ ! , Zn) — / ( Z 2 , Z 3 , • • • , Zn, Z i ) +/(Zi,Z2, • • • ,Z„_i,Zn) - /(Z3,Z4,- • • ,Zi,Z2) + • • • +/(Zi,Z2, •• • ,Zn_i,Zn) - /(Z„,Zi, • •• ,Zn_2,Z„_i) = /(Zi,Z2,--- ,Z„_i,Z„) - /(Z2,Z3,--- ,Z„,Zi) +/(Zi,Z2,--- ,Zn_i,Z„) — /(Z2,Z3, • •• ,Z„,Zi) + / ( Z 2 , Z 3 , • • • , Z „ , Zj) - / ( Z 3 , Z 4 , • • • , Z i , Z 2 ) + / ( Z i , Z 2 , • • • , Z „ _ i , Z n ) — / ( Z 2 , Z 3 , • • • , Zn,

Z{)

+ / ( Z 2 , Z 3 , • • • , Z „ , Z i ) - / ( Z 3 , Z4, • • • , Z i , Z 2 ) +/(Z3,Z4,-- • , Zi,Z2) - /(Z4,Z5,-- • ,Z2,Z3) H +/(Zi,Z2, • •• ,Zn_i,Z„) — /(Z2,Z3, • •• ,Z„,Zi)

+/(z 2 ,z 3 , • • • ,z„,Zi) — /(z 3 ,z 4 , • • • ,Zi,z 2 ) + • • • + / ( Z „ - i , Z n , • • • , Z n _ 3 , Z n _ 2 ) — f(Zn,

Zi, • • • , Z„_2, Z„_i).

By denoting t h e sum of members with positive sign by nF(Zi, we obtain Eq. (1.2).

Z2, • • • , Zn)

Basic Cyclic Functional

Equation

5

On the other hand, every function of the form Eq. (1.2) satisfies the complex vector functional equation (1.1). • Theorem 1.2 The general solution of the basic cyclic complex vector functional equation n

^ / ( Z i . Z i + i , - - - ,Zi+p_i) = 0

(n>2p~l;Zn+i

= Zi)

(1.3)

i=\

is given by / ( Z i , Z 2 , • • • , Z„) = F(Z1,Z2,

••• , Z p _ x ) - F ( Z 2 , Z 3 , • • • , Z p )

(1.4)

tu/tere F is an arbitrary complex vector function with values in V. Proof. A direct calculation shows that every function F of the form Eq. (1.4) satisfies the functional equation (1.3). Now we will prove the converse, i.e. that / has the form Eq. (1.4). For Zj = C (1 < i < n) where C is a constant complex vector from V, the equation (1.3) gives nf(C,C,--,C) = 0,i.e. f{C,C,--- ,C) = 0. By putting Zj = C (p + 1 < i < n) into Eq. (1.3) we obtain / ( Z 1 , Z 2 , - - - , Z P ) + / ( Z 2 , Z 3 ) - - - , Z P , C ) + ---

(1.5)

+/(Zp,C,C,..-,C) + /(C,C,-.-,C,Z1) +f(C,C,---

,C,Z1,Z2)

+ --- + f(C,Zl,Z2,---,Zp-1)

= 0.

If we substitute Zp = C into the above equality, we obtain / ( Z 1 , Z 2 , - - - , Z P _ 1 , C ) + / ( Z 2 , Z 3 , - - - , Z P _ 1 , C , C ) + --+f(Zp-1,C,C,---,C) +f(C,C,---

+

(1.6)

f(C,C,---,C,Z1)

, C , Z i , Z 2 ) + --- + / ( C ) Z 1 , Z 2 > " - ,Z p _i) = 0 .

By subtracting Eq. (1.6) from Eq. (1.5), we get / ( Z i , Z 2 , • • • ,Z P ) = / ( Z i , Z 2 , • • • , Z p _ i , C ) — / ( Z 2 , Z 3 , • • • , Z P , C ) +

/ ( Z 2 , Z 3 , . . . , Z P _ 1 , C , C ) - / ( Z 3 , Z 4 , . . - , Z P , C , C ) + ---

+

/ ( Z P - i , C , C , • • • ,C) — f(Zp,C,C,

• • • ,C).

(1.7)

6

General Classes of Cyclic Functional Equations

By putting F ( Z i , Z 2 , - - - ,Z p _i) = f(Z1,Z2>+

f(Z2,Z3,---

,Zp_1,C,C)

•• , Z p _ i , C )

+ --- + f(Zp_1,C,C,---

,C),

the equality (1.7) takes the form Eq. (1.4).

•

Some particular cases of the functional equation (1.3) are considered in [J. Aczel et al. (1960); M. Ghermanescu (1940); M. Hosszii (1961)] under the hypothesis that the functions and the independent variables are real.

2

Derived Cyclic Functional Equation

Let V be a finite dimensional complex vector space and let there exist mappings ft : Vp >-» V (1 < i < k). Throughout this section Zj (1 < i < n) are vectors in V, and d are constant vectors in the same space. We may assume that Zj = (zji(t),--- ,zin(t))T, where the components Zij(t) (1 < i < n; 1 < j < n) are complex functions, and that O = (0,0, • • • , 0) T is the zero vector in V. Next we will give the following results obtained in [I. B. Risteski and V. C. Covachev (2000)]. T h e o r e m 2.1 The general solution of the derived cyclic complex vector functional equation n

2_jfi\^ii

Z*+i> • • • ; Zj + n _j) = O

(Zn+i = Zi, fn+i =

fa)

(2.1)

i=l

is given by fi(ZuZ2,..-,Zn) =

Fi(Z1,Z2,---

(2.2) ,Zn)-Fi+1(Z2,Z3,---

,Zn,Zi)

(l
where Fn+i = i*1* (1 < i < n) are arbitrary complex vector functions with values in V. Proof.

Using the conventions fi = fn+i

and Zj = Z ; + n , the equa-

Derived Cyclic Functional Equation

7

tion (2.1) can be written in the following form / i ( Z j , Z j + i , • • • , Z j + „ _ i ) + / j + i ( Z j + i , Zj+2, • • • , Z j + n _ i , Zj) + • • • ) +

(2-3)

/ i + n - l ( Z i + n - l , Z j , • • • ,Zj+n_2) = 0 .

Then /j(Zt,Zi_)_i, • • • , Z j + n _ i ) = — / j + i ( Z j + i , Z j + 2 , • • • , Z j + n _ i , Z j ) — • • • —

/ i + n - 2 ( Z i + n - 2 , Z j + n _ i , • • • , Zj+„_3) — / i + n _ i ( Z j + n _ i , Z j , • • • , Zj-|_n_2)

and n

/ t ( Z i , Z j + i , • • • ,Zj.(- n _i)

= /t(Zj,Zj+i, • • • ,Zj+n_i) — /j+i(Zi+i,Zt+2, • • • ,Zj+„_i,Zj) + / j ( Z j , Z j + i , - • • , Z j + n _ i ) — /j+2(Zi+2,Zi+3,- • • , Z j , Z j + i ) + • • • + / t ( Z j , Z t + i , • • • , Z j + n _ i ) — / i + n _ i ( Z j - | - n _ i , Z i , - • • ,Zj+ r a _2) = / i ( Z j , Z j + i , • • • , Z j + n _ i ) — /i-|_i(Zj-|-i, Zj+2, • • • , Z j + „ _ i , Zj) + / i ( Z i , Z j + i , • • • , Z j + n _ i ) — /j-)_i(Zj+i, Zj+2, • • • , Z j + n _ i , Z j ) + / i + l ( Z j + l , Zj+2, • • • , Z j + „ _ i , Z i ) — /j+2(Zj+2, Zj+3, • • • , Z j , Z j + i ) + / i ( Z j , Z j + 1 , • • • , Zj+n_i) —/j+i(Zi+i,Zj+2) • • • , Z j + n _ i , Z j ) + / i + l ( Z j + l , Z t + 2 , • • • , Z j + n _ i , Z j ) — /j+2(Zj+2,Zj+3, • • • , Z j , Z j + i ) + /i+2(Zj+2,Zj+3, • • • , Z ; , Z j + i ) — /i+3(Zi+3,Zj+4,- • • ,Zj+i,Zj+2) + • • • +/i(Zj,Zj+i,-• • ,Zj+n_i) — /j+i(Zj+i,Zj+2,- • • ,Zj+n_i,Zj) + /j+l(Zi+l,Zj+2,- • • , Z j + n _ i , Z j ) — /i+2(Zj+2,Zj+3, • • • , Z j , Z i + i ) + • • •

8

General Classes of Cyclic Functional

Equations

+fi+n-2\Z'i+n-2,

Z j + „ _ i , • • • , Z j + „ _ 3 ) — /i-f-n—1 ( Z j + n _ i , Z j , • • • , Z j + n _ 2 ) .

By denoting the sum of the members with positive sign by nFi(Zi, Zi+i, • • • , Z i + n _ i ) we obtain Eq. (2.2). On the other hand, every function of the form Eq. (2.2) satisfies the equation (2.1). • T h e o r e m 2.2 The general solution functional equation

of the derived cyclic complex

^/i(Zi,Z4+i,---,Zi+J,_1)=0

vector

(2.4)

f=i

(n>2p-

1; Zn+i

= Z j , fn+i

= f{)

is given by /i(Z1)Z3,--.,Zp) =

Fi(Z1,Z2,---

(2.5)

,Zp-1)-Fi+1(Z2,Z3,---

where Fn+i = Fi (1 < i < n) are arbitrary values in V.

,ZP)

(l<*
complex vector functions

with

Proof. If we take into account t h e conventions fi = fi+n a n d Zj = Zi+n, then we may rewrite the functional equation (2.4) in the following form /i(Zj,

) + /i+i(Zj+i,Zj+2, • • • ,Zj+p) + • • • i Ji+n—p\^i+n—p;

(2.6)

^ i + n — p + l i ' ' ' > ^"i+n —1 j

+/j+n-p+l(Zj+n_p+l,Zj+n_p+2,'- • • ,Zj+n_i,Zj) + • • • +fi+n-l(Zi+n-l,

Z j , • • • , Zj+ p _2) = O .

By p u t t i n g Zi+P = Z j + p + 1 = • • • = Z i + n _ i = C in Eq. (2.6), where C is a fixed vector from V, we obtain /j(Zj,Zj+i,-• • ,Zj+p_i) + / j + i ( Z j + i , Z j + 2 , • • • ,Zj+p_i,C) + / t + 2 ( Z j + 2 , Z j + 3 , • • • , Z j + p _ i , C , C) + • • • + / i + p - i ( Z j + p _ i , C, C, • • • , C) 4- fi+p(C,

C, • • • ,C)

(2.7)

Derived Cyclic Functional Equation

+ft+p+i (C> C, • • • , C) + • • • + fi+n-p(C,

9

C, • • • ,C)

+fi+n-p+l (C, C, • • • ,C, Zj) + /j+„_p_|_2 (C, C, • • • , C, Zj,Zj+i) + • • •

+ /i+n-l(C, Zj,Zj + i, • • • ,Zj + p_ 2 ) — O. The substitution Zj+p_i = C in Eq. (2.7) yields i+1 j Z j + 2 , • ' • , Z j + p _ 2 , C,C)

+ -..

+/i+p-2(Zj+ p _2,C,C, • • • , C) + fi+p-i(C, C, • • • ,C) +fi+p(C,C,

• • • ,C) + • • • + fi+n-p(C,C,

iji+n—p+l

(2.8)

• • • ,C)

P+2(C,C, • • • ,C, Zj, Zj + i) + • • •

+fi+n-l{C ,?'i,'Z'i+l, • • • ,Zj + p_ 2 ) = O. By subtracting Eq. (2.8) from Eq. (2.7), we get the formula /j(Zi,Zj + 1 ,- • • ,Zj + p _i)

(2.9)

= /j(Zj,Z; + i, • • • ,Zj + p_2,C) — fi+l(Zj+i, Z i+ 2, • • • ,Z; + p_i,C) + /t+l(Zi+l,Zj+2, • ' • , Z; + p_2,C, C)-/i+2(Zj + 2,Zj + 3,- • • , Z,+ p _i, C, C) + + fi+p-2 (Zi+p-2 ,C,C,- • • , C) — fi+p-1 (Zj+p_i ,C,C,y

+/ i + p_ 1 (C,C,---,C)

• ,C)

(l
Let us put now G r i(Zi,Z 2 ,-• • ,Zp_i) +/i+i(Z2,Z3, • • • ,Zp_i,C,C) At

= + =

/i(Zi,Z 2 , • • • ,Zp_!,C)

(2.10)

• • • + /i + p_ 2 (Zp_i,C, C, • • • ,C), /,+„_! ( C C - . - . C ) .

(2.11)

10

General Classes of Cyclic Functional

Equations

Since fi = fi+n, we conclude that G{ = Gi+n- By putting Zj = C (1 < i < n) in Eq. (2.4) we obtain n

Y,Ai = 0.

(2.12)

According to Eq. (2.10) and Eq. (2.11), the formula Eq. (2.9) can be written in the following form / i ( Z i , Z 2 ) - • • ,Z P ) = Gj(Zi,Z 2 , • • • ,Z P ) — G i + i ( Z 2 , Z 3 , • • • ,Z P ) + Ai, (2.13) (1 < i < n). Finally, with the notations Fx

-

i*2

=

F3 = Fn

=

Gu Gi — A\,

Gz-A1-A2, ^ n — Ai — A2 — • • • — A n _i

the expression Eq. (2.13) can be written in the form Eq. (2.5). Conversely, the direct calculation shows that the function Eq. (2.5) satisfies the derived functional equation Eq. (2.4) for arbitrary Fi. • T h e o r e m 2.3 The general solution of the derived cyclic complex vector functional equation k

^/i(Zi,Zi+1,...,Zi+p_1) = 0 (p < n < 2p - 1; k
(2.14)

Z n + j = Zj)

is given by the formulae min(/s—r,n—p)

/ r ( Z 1 , Z 2 , . . . ,Z p )

=

2_,

("I)*

Fri(Zi+i,Zi+2,

. . . ,Z p )

i=l n—p

+

J2 i=n—r+1

(-ir 1 ^ri(Z i + i,Z i + 2 ,...,Zp)

(2.15)

Derived Cyclic Functional Equation

11

min(fc — r,p—1)

+

2_^

(-1)'

Fri(Zi+i,Zi+2,.

. . ,Zp,Zi,Z2, . . . ,Zp+i)

i=n—p+1

P-l

+

( — I)" l-Fi+T-,n-i(Zl,Z2, . . . ,Z p + i,Zj+i,Zj + 2, . . . , Z p )

2^

i=raax(n- r+l,n—p+1)

k—r

+ ^ ( ~ 1 ) " '•' 7 i+r,n-t(Zi, Z 2 , . . . , Z p + j) i—X>

n-1

+

(-l)n"<^+r.n-i(Zi»Z2,...,ZjH.i)(l
Z)

fc
i=max(n-r+l,p)

where s =

/

J

"

[fl > S j ;

1*111+11,1

==

•''mij

•''ijm+n

=

•''tm

a

and Fy are arbitrary vector functions from V. Proof.

The proof of this theorem will be given by induction.

For k = 2, Eq. (2.14) becomes / i ( Z i , Z 2 , . . . , Z p ) + / 2 ( Z 2 , Z 3 , . . . , ZJH-I) = O.

(2.16)

Putting Zi = Ci into Eq. (2.16), we obtain / 2 ( Z 2 , Z 3 ) . . . ,Zp + i) = -Fii(Z2,Z3,...,

Z p ),

(2.17)

where the notation •Fii(Z 2 ,Z 3 ,... ,Z P ) = / i ( C i , Z 2 , Z 3 . . . ,Z P ) is introduced. If we put Eq. (2.17) into Eq. (2.16), we have /1 (Zi, Z 2 , . . . , Zp) = F n ( Z 2 , Z 3 , . . . , Zp).

(2.18)

General Classes of Cyclic Functional

12

Equations

Since min(l,n—p) = 1, max (n,n— p+1) = n, min(0, n-p) — 0, m a x ( n - l , n — p+l) = n — 1,

min(l,p—1) = 1, max (n,p) = n, m i n ( 0 , p - l ) = 0, max(n—l,p) — n — 1,

from Eq. (2.15) we obtain Eqs. (2.18) and (2.17) respectively for k = 2, r = 1 and for k = 2, r = 2. Therefore, the theorem holds for k = 2. Now we will suppose that the theorem holds for some fixed k < n, i.e. let the general solution of the functional equation k

]>3<7 i (Z i ,Z i+ i,...,Z i+p _ 1 ) = 0

(k
(2.19)

i=i

be given by Eq. (2.15), where in the place of fr we have put gr (1 < r < k). Let us consider the following functional equation k+\

£/i(Zi>Zi+i>...,Z4+1,_i) = 0

(* + l < n ) .

(2.20)

j=i

We will distinguish the following three cases: 1° Let 2 < k < n-p+1. If we put Z* = C{ fori ^ &+l,fc+2,... ,k+p (mod n), then Eq. (2.20) becomes /jfc+i(Zfc+i,Zfc+2,... ,Zjfc+p)

(2-21)

n-l =

^

("I)"

'-fi+A+l.n-^Zft+ljZfc^, . . .

,Zi+p+k),

i=n—k

where we introduced the notation ( — 1)™

%

Fi+k+l,n-i('Z'k+l,'Z'k+2,

/ i + f c + l - n ( Z t + f c + i _ n , Zj+fc+2-n, . . . , Zj-)_fc+p_ n )

• • • ,Zj + p + jfc)

Z; = C; for t ^ fc + 1, A: + 2,. .. , k + p (mod n)

Derived Cyclic Functional Equation

13

From Eq. (2.21) it follows that n-l / f c + l ( Z i , Z 2 , . . . , Z p ) = 2_j ( _ l ) " ~ ? r i + i k + l , n - i ( Z i , Z 2 , . . . , Z j + p ) . (2.22) i=n—k

If we substitute Eq. (2.21) into Eq. (2.20) and introduce new notations by the equalities <7i(Zi,Zj + i,... ,Zj+ p _i) = / ; ( Z i , Z j + i , . . . , Z , + P _ i ) + (-1)

~'Fiti+k-i(Zk+i,Zk+2,---,'Zi+p-i)

(2.23) (i =

l,...,k),

we obtain the equation (2.19). For 2 < k < n — p, since min(k — r,n — p) = min(k — r,p — 1) = k — r,

n — p+1 > k — r,

on the basis of the expressions Eq. (2.15) the general solution of Eq. (2.19) is k—r

<7r(Zi,Z2, . . . ,Z p )

=

2 ^ ( —1)'~ i r rt(Z»+l,Zj+2, • • • ,Z p )

(2.24)

«=1

n—p

+

2_^

(~1)* _ Fri(Zi+i,Zi+2,.

. . , Zp)

i=n—r+1 p-1 + 2^j ( " I ) " '-Pi+r,n-i(Zl,Z2, . . . , Z p - | _ j , Z j + 1 , Z i + 2 , . . . , Z p ) i=max(n—r+l,n—p+1) n-l

+

£

(-l)"-\F i + r , n _ i (Z 1 ,Z 2 ,...,Zp + i )

(l
i=max(n- r+l,p)

On the basis of Eqs. (2.22), (2.23) and (2.24), we obtain / r ( Z i , Z 2 , . . . ,Zp)

=

+

fc+l-r 2_, (~I)*" i=l n—p

£J2

i=n—r+1

Fri{Zi+i,Zi+2,

. . . ,Zp)

(-l)i_1^ri(Z t+ljZj+2,

. . . ,Zp)

(2.25)

14

General Classes of Cyclic Functional

Equations

p-1

+ 2_^i (— 1 ) i = m a x ( n - r+l,n—p+1)

Fi+T,n-i

( Z i , Z2, . . . , Z p + j , Z i + i , Zj+2, . . . , Z p )

n-1

+

£

(-l)"-iFi+r,n_i(Z1,Z2,...,Zp+i)

( l < r < A + l).

i = m a x ( n - r+l,p)

Therefore, for 2 < A; < n—p+1 the theorem holds for A; + 1 if it is true for k. This means that the theorem holds for all such k, and also for k = n —p +1. 2°. Let n-p+1 < k < p. If we put Zt = d fori ^ AH-l,/M-2,... ,k+p (mod n), then Eq. (2.20) becomes /*+i(Zfc+i,Z/fc + 2,... ,Zj. + p) —

(2.26)

p-i / „ ( — I)™ * f i + k + l , n - i ( Z f c + l , Z f c + 2 , . . . , Z p + / . + j , Z j + f c + i , Z j + f c + 2 , . . . , Z p + f c ) i=n—k n-1 + ^ ( — 1 ) " *-Pi+fc+l,n-t(Z*i+l, Z / . + 2 , . . . , Zp+fc+j), i=p

where we have introduced the notation /j+Jfc+l-n(Zi+fc+l-n> Z i + f c + 2 - n , • • • i Z j + f c + p - n )

( —1)™ _ = < K

Zj = C; for i £ fc + 1, It + 2 , . . . , k + p (mod n)

-Fi+fc+l,n-i(Zfc+l, Zfc + 2, . . . , Z p+ fc_|_j,

Zi+fc+1,Zi+fc+2,...,Zp+t)

(-l)n-i+1Fi+k+ltn-i(Zk+1,

(i = n - k , n - k

+

Z f c + 2 , . . . ,Zp+k+i)

l,...,p-l),

(i=p,p+l,...,n-l).

Substituting Eq. (2.26) into Eq. (2.20) and introducing new functions by 9i = fi + (-l)1+k-iFi,1+k-i

(i = l,...,fc),

(2.27)

we obtain Eq. (2.19). On the basis of the inductive hypothesis and Eq. (2.27), the general solution of Eq. (2.20) is n—p

/ r ( Z i , Z 2 , . . . ,Zp) = 2 _ , ( - l ) * t=l

^rt(Zj+i,Zj+2, . . . ,Z p )

Derived Cyclic Functional

Equation

15

k—r +

(— 1 ) * _ • P ' r i ( Z t + l , Z i + 2 , . . . , Z p , Z i , Z 2 , • . . , Z p + i )

2^ i=n—p+1

+ ( — 1)

_r

-Fr,fc+l-r(Zfc+2-r, Zfc+3_r, . . . , Z p , Z i , Z2, . . . ,

Zp+k-r+l)

n—p +

(— l ) t _ -Fri(Zi+l,Zj+2, . . . , Z p )

2_j

i=n—r+1 p-1 +

(~l)"-t-Fi+r,n-i(Zl,Z2, . . . ,Zp+i,Zi+i,Zi+2, . . . ,Zp)

2_^

i=max(n—r+l,n—p+1) n-1

+

(-l) n _ i i ; i+r,n-i(Zi,Z2,...,Zp + i )

J2

(r = l , 2 , . . . , p + * - n ) ;

i=max(r»-r+l,p)

(2.28) k-r /r(Zi,Z2,...,Zp) = ^ ( - 1 ) * i=l

+ ( — 1)

r ;l

i

_

Fri(Zi+l,Zi+2,.

r)fc+i_r(Zfc+2-r,

. . , Zp)

Zfc+3_r, . . . , Zp)

n—p +

2 ^ (— 1 ) * ~ i=n—r+1

•Fri(Zi+l,Zj+2,...,Zp)

p-1 +

(-l)"_t-fi+r,n-i(Zi,Z2,. . . ,Zp+j,Zi+i,Zi+2, . . .,Zp)

^ ,

i=max(n—r+l,n—p+1) n-1

+

13

(-l) n ~ i i r i+r,n-i(Z i + 1 ,Z i + 2 ,...,Zp)(r=p+fc-n + l,...,A;).

i=max(n—r+l,p)

On the basis of Eqs. (2.26) and (2.28), the general solution of Eq. (2.20) is determined by Eq. (2.15), where fc must be replaced by k + 1.

General Classes of Cyclic Functional

16

Equations

Therefore, forn— p+1< k < p the theorem holds for k + 1 if it holds for k, i.e. the theorem is true for all such k, and also for k — p. 3° Let p < k < n - 1. For Zt = d when i ^ k+l,k (mod n) Eq. (2.20) becomes

+ 2,

...,k+p

n—p

/*+i(Z*+i, Zk+2, • •., Zfc+p) = 2_^(—l)1

Fk+iti(Zi+i, Z j + 2 , . . . , Z p ) (2.29)

i=n—k p-1 + 2 ^ (_l)n_lfi+fc+l,n-i(Zfc+l,Zfc+2, . . .,Zp+fc+i,Zj+fc+1,Zj+fc+2, . . i=n—p+1

.,Zp+k)

n-1 + / ^,( — 1 ) " *-Pi+fc+l,n-t(ZAi + l , Z f c + 2 , . . . , Zp+fc+j), i=p

where we have introduced the notation fi+k+l-nC^i+k+l-n,

Zj+fc+2-rai • • • > Z j + f c + p _ n )

Z; = C, for t ^ fc + 1, fc + 2,. .. , k + p (mod n)

' (-iyFk+iti(Zi+i,Zi+2,...,Zp) = <

( — l)n~l

(i =

Fi+k+l,n-i(Zk+l,Zk+2,

Zi+k+i,Zi+k+2,..

.,Zp+k)

n-k,...,n-p),

• • • ,Zp+k+i, (i — n -p+

n i+1

, (-l) - Fi+k+i,n-i(Zk+1,Zk+2,..

.,Zp+k+i)

l,...,p-

1),

(i-p,p+l,...,n-

1).

Now, we will introduce the following notation „ _ f , / (-l) < + n -*flH-U+n-*-l 9i~ti+{ (_i)i+*-i F . > 1 + f c _.

(i = l,...,k + l-p), (i = k + 2-p,...,k).

{2 di})

-

If we substitute the function fk+x determined by Eq. (2.29) into Eq. (2.20), in view of Eq. (2.30) we obtain the functional equation (2.19). On the basis of Eq. (2.30) and the general solution of the functional equation (2.19), we obtain that the general solution of the functional equation (2.20) is determined by n—p

fr{Z\, Z2, . . . , Z p ) = 2_j(~iy j=l

-Pr«(Zj+l, Zj+2, . . . , Z p )

Derived Cyclic Functional Equation

17

p-l

+ J2 (-l) i_1 F ri (Z i+i,Zi_|_2)- •• , Z p , Z i , Z 2 , . . .

,Zp+i)

&—r + ^^(~"^-)n

l

^ + r , n - i ( Z i , Z 2 , . . . , Zp-j-j)

i=p

+ (— l ) n

r

-ffc,n-fc+l-r(Zi, Z2, • • • , Z p + & _ r )

n—p +

^

("I)*

-Pri(Zi+i,Zi+2, • •• , Zp)

i=n—r+1 p-i +

(— 1 ) "

2 ^

!

-Pi+r,n-i(Zl, Z2, . . . , Z p + j , Z j + i , Zj+2j • • • , Z p )

i=max(n—r+l,n—p+1) n-1

+

(-^n~iFi+r,n-i(Zi,Z2,...,Zp+i)(r

J2

=

l,2,...,n-k-p+l);

i=max(n—r+l,p)

(2.31) n—p / r ( Z l , Z 2 , . . . , Zp) = 2 ^ ( - l )

l _

Frt(Zi+l,Zj+2, •••, Zp)

i=l fc-r +

2^,

("•'•)'

•Pri(Zi+l,Zj+2, . . . , Z p , Z i , Z 2 , . . . , Z p + j )

i=n—p+1

r

+ ( — 1)

F,.,fc_,.+i(Zk_r_|_2, Zfc_r+3, . . . , Z p , Z i , Z 2 , . . . , Z p + ^ _

r +

i)

n—p +

^

(— I ) '

-Fri ( Z j + 1 , Z j + 2 , . . . , Z p )

i = n —r+1 p-l +

2-^i

(—-U

i=max(n—r+l,n—p+1)

-fi+r,n-j(Zl, Z2, ..., Zp+j, Z j

+ 1

, Zj+2, ..., Zp)

18

General Classes of Cyclic Functional

Equations

n-1

+

(-l)n~iFi+r,n-i(Z1,Z2,...,Zp+i)(r

Yl

=

n-k-p+2,...,p-l);

i=raax(n- r+l,p) k—r / r ( Z i , Z 2 , . . . , Z p ) = 2_^(~^~ i=l + ( — 1) ~TFrtk+l-r(Zk+2-r,

Fri(Zi+i,Zi+2,

Zk+3-r,

•••

,Zp)

• • • : Zp)

n—p

+

2_^ (—l) t_ FTi(Zi+i,Zi+2,...

,Z P )

i=n—r+1 P-l + ^L/ ( _ 1)™ i=max(n—r+l,n—p+1)

l

-Fi+r,n-t(Zl,Z2, . . . ,Zp+i,Zj_|_i,Zi+2,. . . ,Zp)

n-1

+

X)

(-l)n_ii?i+r,n-i(Zi,Z2,...,Zp+i)

(r = p , p + l , . . . , A ) .

i=max(n—r+1 ,p)

On the basis of Eqs. (2.29) and (2.31) we obtain that the general solution of the functional equation (2.20) in the case p < f c < n - l i s determined by Eq. (2.15), where k must be replaced by k + 1. • Therefore, the solution of the research problem given in [D. S. Mitrinovic and D. Z. Djokovic (1963)] is presented by this theorem. As particular cases see the results given in [M. Hosszii (1961); D. S. Mitrinovic (1963B)]. Example 2.1 For n = 8, p = 5 and k = 6 the complex vector functional equation Eq. (2.14) becomes / l ( Z l , Z 2 , Z 3 , Z 4 , Z 5 ) + /2(Z2,Z3,Z 4 ,Z5,Z 6 ) + /3(Z3,Z4,Z 5 ,Z 6 ,Z 7 ) +fi{Zi,

Z5, ZQ, Z7, Zg)+/5(Z5, Z6, Z7, Zg, Zi) + /e(Z6, Z7, Zg, Zi, Z2) = O,

whose general solution is given by /l(Zl,Z2,Z3,Z4,Z5) = Fn(Z2,Z3,Z4,Z5) - .^12^3, Z4, Z5) +Fi 3 (Z4,Z 5 ) — Fi4(Z5,Zi) - F 6 3(Zi,Z 2 ),

Derived Cyclic Functional Equation

19

/ 2 ( Z i , Z 2 , Z 3 , Z 4 , Z 5 ) = i*2i(Z2,Z3,Z4,Z 5 ) - F 2 2 ( Z 3 , Z 4 , Z 5 )

+F 2 3 (Z 4 , Z5) - F 2 4 (Z 5 , Zi) - F n ( Z i , Z 2 , Z 3 , Z 4 ), /3(Zl,Z 2 ,Z3,Z4,Z 5 ) = i 7 3 1 (Z 2 ,Z3,Z4,Z 5 ) — F 32 (Z3,Z4, Z 5 ) +F 3 3 (Z4,Z 5 ) + F 1 2 (Zi,Z2,Z 3 ) - i r 2i(Zi,Z2,Z3,Z 4 ),

/4(Z 1 ,Z2,Z3,Z4,Z 5 ) = F4i(Z 2 ,Z3,Z4,Z 5 ) - F4 2 (Z 3 , Z4, Z 5 ) - F i 3 ( Z 1 , Z 2 ) +F22(Zi,Z2,Z 3 ) - F3i(Zi,Z 2 ,Z3,Z4),

/5(Zl,Z2,Z 3 ,Z4,Z5) = F5i(Z2,Z3,Z4,Z 5 ) + Fi4(Zi,Zs) — F 2 3(Zi,Z 2 ) + F32(Zi,Z 2 ,Z3) — F4i(Z 1 ) Z2,Z3,Z 4 ), /6(Z 1 ,Z 2 ,Z3,Z4,Z 5 ) = F 6 3(Z 4 ,Z 5 ) + F24(Zi,Z 5 ) — F33(Zi,Z 2 ) + -F 4 2(Zi,Z2,Z3) — F5i(Zi,Z 2 ,Z3,Z4), where Fij are arbitrary complex vector functions from V. Now we will give two particular cases of the above theorem. Theorem 2.4 The general solution of the derived cyclic complex vector functional equation n

£)/i(ZilZj+1>...,Zi+p_1) = 0

( p < n < 2 p - l ; Z n + j = ZO

(2.32)

is given by n—p

/ r ( Z i , Z 2 > . . . , Z„) = ^ ( - l J ' - ^ Z i + i , Z i + 2 , . . . , Z p ) »=i

(2.33)

General Classes of Cyclic Functional

20

Equations

min (n—r,p—1)

+

2^i

(~^y

^ r i ( Z j + l , Z j + 2 , . . .,Z p ,Zi,Z 2 , . . .,Zp+i)

i=n—p+l p-1

+

(-l) n _ 1 -fi+r,n-i(Zl,Z 2 , . . .,Z p+ j,Zj + i,Zj_|_2, . . .,Z p )

2_j

i=max (n—r+l,n—p+1)

+ X>l) n ~ i i r «+r,n-*(Zi>Z2,...,Z p+i )

C1 ^ r ^ n )>

i=p

Wiere F^ are arbitrary complex vector functions from V. Proof. The proof of this theorem immediately follows from the previous theorem for k = n. • This theorem generalizes the results given in [D. Z. Djokovic (1964)]. Theorem 2.5 The general solution of the basic cyclic complex vector functional equation n

53/(Zi>Zj+i,...,Zj+p_i) = 0

(p < n< 2p - 1; Zn+i = Zt)

(2.34)

is given by /(ZllZ2,...,Zp)=Fo(Zl!Z2,...,Zp_1)-Fo(Z2jZ3j...,Zp)

(2.35)

p-[n/2]

+ 2_^

[-fi(Zi,z2,...,Zj,zn_p+j+i,...,zp)

— F!i(Z p _j + i,..., Z p , Zi, Z 2 , . . . ,Z 2 p _ n _j)J, where Fi (0 < i < p— [n/2]) are arbitrary complex vector functions from V. Proof. By summing up the functions fi (1 < i < n) determined by Eq. (2.33) and putting / i = f2 = ••• = fn = f, we obtain Eq. (2.35), where

Derived

Cyclic

Functional

Equation

21

we have introduced the following notations 1

F 0 ( Z i , Z 2 , . . . ,Z p _i)

=

G>(Zi,Z 2 , . . . , Z p _ r )

_

n—p n-p

rr

—y n

^ Gr(Zt,Zi+i,... ,Zp_r_i+i),

r=l

i=l

/

y(-l)

r r

i ir(Zi,Z2,. • . ,Zp_r),

i=l

•Fi(Zl,Z2,. . . , Z j , Z n _ p + i + i , . . . , Z p ) n—p+» ( ~ l ) i + 1 " """ / y •Pr,p-t(Zl,Z2,. . . , Z j , Z n _ p + j + i , . . . ,Zp) n p—i

— ^•FV,n-p+i(Z n -p+i+l,. . .,Zp,Zi,Z2,. . . ,Zj) r=l

(1<» < p - [ ( n +l)/2]). In particular, if n = 2m, we get -Pp-m(Zl, Z2, . . . , Z p _ m , Z m + i , . . . , Z p ) m

/ _ i-jp-m+l

=

/ n

y -Prm(Zl,Z2,

. . . , Z p _ m , Z m + i , . . . , Zp)

1

r=l

where • ^ r + m , T T I ^ I i • • • > "p—mi

— Frm(Zm+i,...

^ m + l i • • • j ^p)

, Z p , Z i , . . . , Z p _ m ) (1 < r < m). D

We have noticed that in [P. M. Vasic and R. Z. Djordjevic (1965)] special generalized cases of Eqs. (2.14), (2.32) and (2.34) are considered. They are solved in a complicated manner using a cyclic operator. At the end of the present section we give two more general theorems obtained in [I. B. Risteski (to appear A)]. Theorem 2.6 The general solution of the derived cyclic complex vector functional equation n

/

y/i(Zi,Zt+1,...,Zj+p_1)

= O

( Z n + i = Zj)

(2.36)

22

General Classes of Cyclic Functional

Equations

is given by / r ( Z r , Z r + i , . . . ,Z r + p _i)

(2.37)

r-l

— 2_/_1)r

^ J > ( { Z r , Z r + i , . . . , Z r + p _ i } n {Zj,Zj+i,.

..

,Zj+p-i})

i=i n

+

( - l ^ ^ ^ j Z r j Z r + i , . . . , Z r + p _ i } ft { Z j , Z j + i , . . . , Z j + p _ i } )

^ j=r+l

(1 < r < n), luftere F r j (1 < r < n - 1, r + 1 < j < n) are arbitrary complex vector functions from V such that

}n{z j( Z j + i » •

• • > Zj+p—i)) — A r j

{z r ,z r + i,... ,z r+p _i} n {Zj,Zj+i,... ,Zj+p_i} = 0, V

where Arj are constant vectors from V and ^ = O for a > v. a

Proof. We will prove the assertion of the theorem by mathematical induction. For n — 2, the equation (2.36) becomes /1(Z1,Z2,---,Zp) + /2(Z2)Z3,---)Zp+1) = 0 .

(2.38)

Putting Zi = Ci into the equation (2.38), we get /2(Z 2 ,Z3, • • • ,Z P + 1 ) = - F i 2 ( Z 2 , Z 3 , • • • ,Z P )

(2.39)

where the following notation is introduced Fi 2 (Z 2 ,Z3, • • • ,Z p ) = / i ( C i , Z 2 , • • • ,Z P ). If we put Eq. (2.39) into Eq. (2.38), we get / i ( Z i , Z 2 , • • • , Zp) = F 1 2 (Z 2 , Z 3 , • • • , Z p ).

(2.40)

Therefore, for i = 1,2 from Eq. (2.37) we obtain Eqs. (2.40) and (2.39), respectively, which means that the theorem holds for n = 2.

Derived Cyclic Functional

23

Equation

For some fixed n let us suppose that the general solution of the functional equation (2.36) is given by Eq. (2.37). Now, we will consider the functional equation n+l

^fli(Zi,Zi+1,---,Zi+p_1) = 0.

(2.41)

t=i

If we put Zj = Ci (1 • • • , Z n + p )

(2.42)

n

=

/ ,(-l)"-Fj,n+l({Z n +l|Z n + 2,- •• ,Zn+p}n{Zj,Zj+i,---

,Zj+p_i}).

3=1

Substituting Eq. (2.42) into Eq. (2.41) and introducing the new notations /i(Zj,Zj+i, • • • , Z; + p _i) = <7j(Z;,Zj + i,-- • , Z ; + p _ i )

(2-43)

n ?

+

(-l) i t,n+l({Z 7 l + i,Z„ + 2,-• • , Z n + p } n { Z j , Z i + i , • • • ,Zj + p _i}) (1 < i < n ) ,

we obtain the equation (2.36). According to the inductive hypothesis, the general solution of this equation is given by the formulae Eq. (2.37). Therefore, from Eqs. (2.37), (2.42) and (2.43) we get <7i({Zj,Zj+i, • • • ,Z; + p _i}) i-1

= 2^(-i)t+1-Fji({Zi,Zj+i,--- jZj+p-i}n{Zj,Zj+1)-••

,Zj+p-i})

3=1

+ 2_/ ( — 1 ) J - F ii({ z i' Z «+i'" *" > z t + P - i } n { Z j > z j + i > ' " > z i+P-i}) j=i+l — (-l)nFitTl+i({Zn+i,Zn+2,-

• • ,Zn+p} (1

n {Z;,Z; + i,- • • , Zj+p_i})

24

General Classes of Cyclic Functional

Equations

i.e., <7i(Zj, Z{+i, • • • , Z j + p _ i ) i-l = 2^(—i)'

^j'»({Zi,Zf + i,• • • , z i + p _ i } n {Zj,Zj+i,-•

n+l + 2 ^ (—i)3Fij{{Zi,Zi+i,---

• ,Zj+p_i})

,Zi+p-i}r\{Zj,Zj+i,-•

•,Zj-+P_i})

j=t+i

(1
•

This theorem generalizes Theorem 2.4 where it is assumed that p < n < 2p—l. Some other particular cases of the equation (2.36) were considered in [P. M. Vasic (1965); P. M. Vasic (1967)] under the assumption that the functions and the independent variables are real. E x a m p l e 2.2 equation

The general solution of the complex vector functional

/ i ( Z i , Z 2 ) + / 2 ( Z 2 , Z 3 ) + / 3 ( Z 3 , Z 4 ) + / 4 ( Z 4 , Zj) = O, which is a particular case for n = 4 and p = 2 of the equation (2.36), is given by /i(Zi,Z2)

=

F 1 2 ( Z 2 ) - y l 1 3 + i ; i4(Zi),

/ 2 (Z 2 ,Zs)

=

— i r 12 (Z 2 ) — i r 2 3(Z 3 ) + >124,

/3(Z3,Z4)

=

^13+f23(Z 3 )+i ; 34(Z4),

/4(Z 4 , Zi)

=

- i r 1 4 ( Z 1 ) - A24 - F 3 4(Z4),

where Fij (1 < i < 3; 2 < j < 4) are arbitrary complex vector functions from V and Aij (i = 1,2; j — 3,4) are arbitrary constant complex vector also from V. E x a m p l e 2.3

Now we will consider the functional equation /1(Z1)Z2)Z3) + /2(Z2,Z3,Z4) +

/ 3 ( Z 3 , Z 4 , Z 1 ) + / 4 (Z4,Z 1 ,Z 2 ) = 0 ,

(2.44)

Derived Cyclic Functional Equation

25

where fi : V3 i-> V (1 < i < 4). This equation is a particular case for p = 3 and n = 4 of the functional equation (2.36). According to the Theorem 2.6, the general solution of the functional equation (2.44) is /i(Zi,Z2)Z3)

=

F12(Z2,Z3)-F13(Z1,Z3)+F14(Z1,Z2),

(2.45)

/2(Z2,Z3,Z4)

=

-F12(Z2,Z3)-F23{Z3,Z4)

(2.46)

/3(Z3,Z4!Z1)

=

F 1 3(Z 1 ,Z 3 )+F 2 3(Z3,Z4)+F34(Z 1 ) Z4),

(2.47)

/4(Z4,Z1,Z2)

=

-F14(ZUZ2)-F24(Z2,Z4)-F34(ZUZ4),

(2.48)

+ F24(Z2,Zi),

where Fi 2 , F13, F 1 4 , F23 and F34 are arbitrary complex vector functions from V. If we put into Eq. (2.36) fi = Oj/ (1 < i < n) where at are complex constants, then we obtain a special case which will be also solved here in the next theorem. Theorem 2.7 The general solution of the basic cyclic complex vector functional equation with complex coefficients n

Y,aif(Zi,Zi+1,---

,Zi+p-X)=0

(Zn+i = Zi)

(2.49)

is given by f(ZuZ2,---,Zp) n

i—l

t=i

[j=i

(2.50)

= J2<*i E(- 1 ) i + l c " + 1 _ i G ^({ z ----' z ^-i} n { z ^---,z j + p _ 1 }) n

+

£ (-iycn+1-'Gy({z«, • • •, zi+p_x} n {zh • • •, zi+p_i» J=i+1

where C is a cyclic operator such that CG(Zi, • • •, Z n ) = G(Z 2 , • • •, Z„, Zj); dj (1 s). a

Proof. If we apply the operator Cn+1~i to both sides of Eq. (2.37) (with fi replaced by a,if) and then multiply by still indefinite complex constants

26

General Classes of Cyclic Functional

a

i (1 < i 5= n)i

we

Equations

have

o i a i C n + 1 - 7 ( Z i ) Z i + l l - . - ,Z i + p _!)

(2.51)

i-l

a; ^ ( _ 1 ) i + 1 ( J „ + 1 _ i F j i i { z ^

=

. ^z . + p _ i } n

{z^

; Z j + p _!})

J=I

+

53 (-l^c^-'^-aZi,. • -,zi+p^} n {z,-, • • •,zj+p_1}) (1 -.-,zH.p_1}n{z,-,-..,zi+p_1})

= J2 i=l

3=1

n

+

(-i)jC"+1-%-({zi,---,zi+p_1}n{zi,---,zi+p_1})

£ j=i+l

From the equation (2.49) and the equality (2.52) we find n

/ n

a

° = Z)M J2 ^ r=l

n

I n

r=l

»=1

n-1

+ £

\

Cr~lf{Zx,Z2, • • • ,ZP)

\i=l

i-l

^ - l j ^ c + ' - ' f y a z , , • • -, zi+p_x} n {Zj, • • -, zj+p_!}) 3=1

(-i)^" +r -%({z i ,--- ! z i+p _ 1 }n{z,v,z J+p _ 1 })

j=i+l

n—1

n

53 J ] (-l)i+1ai53arC"+r-iF:/i({Zi!...,Zi+p_1}n{ZJ-,---,Zj+p_1})

j=li=j+l

r=l

Derived Cyclic Functional Equation n—1

27

n

+ E E (-i)iai^arC"+'-%({zi,--.,zi+p_1}n{zj,.-- ,zj+p_!}) i=l.;'=i+l

r=l

(-i)iEK"+r-%({zi,-,zi+H}n{zj,..-,zj+H})

=E E i=l j=i+l

V"=l

- a j C " + r ^ F i i ( { Z i , • • -, Z i + P _!} n {Zj, • • ; Z j + P _!})]

n—1

n

= E E (-i)^"^-* x t=l j=i+l

x J^ar[air-1Fii({Zi,.---,Zi+p_1}n{Z,v--,Zj+p_1}) lr=l

— a^C

Fjj({Zi, • •-jZj+p-i} n {Z 2 j_j,Z2i-j+i,-• •,Z 2 j_j+ p _i})J >.

Now, we may determine the constants a; (1 < i < n) from the identities n

J2ar

[a i C r - 1 Pi,-({Z i , • • •, Z i + P _ i } n {Zj, • • •, Z j + P _ i } )

(2.53)

r=l

-ajC~

Fij({Zi,- • -.Zj+p-i} n {Z2i-j,Z2i-j+i, (1 a i j T = S,

(2.54)

and because it has a unique solution T, by introducing the new functions Fij n

i=l

=Gij

(1 < i < n - 1;

i + 1 < j < n)

(2.55)

General Classes of Cyclic Functional

28

Equations

there follows (2.50). E x a m p l e 2.4

•

We consider the functional equation

f(Z1,Z2,Z3)-f(Z2,Z3,Z4)

+ f(Z3,Z4,Z1)-f(Z4,Z1,Z2)

=

0.(2.56)

According to Eqs. (2.45), (2.46), (2.47) and (2.48) we obtain /(Zi,Z2,Z3)

=

F12(Z2,Z3)-Fi3(Z1,Z3)+F14(Z1,Z2),

-/(Zi.Za.Zs)

=

-F12(Z1,Z2)-F23(Z2,Z3)

/(Zi.Za.Zs)

=

F 1 3(Z3,Z 1 )+F 2 3(Z 1 ,Z 2 ) + JF34(Z3,Z2),

-/(Zi.Za.Zs)

=

+

F24(Z1,Z3),

-F14(Z2,Z3)-F24(Z3,Z1)-FM(Z2,Zi).

On the basis of the identities Eq. (2.53), we may determine the constants a, (i = 1,2,3,4) as follows 0iiFi2{Z2 +a1F12(Z4

Z3) Zi)

- a i F i 3 ( Z i Z3) - a i F i 3 ( Z 3 Zi) aiFu(Zi +aiF14(Z3

z2) z4)

— 02-^23 ( Z 2 Z s ) -«2-f23(Z4 Zi) 02-^24 ( Z l Z S )

-

ctiFi2(Z3,Z4) + a2Fi2(Zi,Z2; a 2 -Fi2(Z3,Z 4 ) - a 1 F i 2 ( Z i , Z 2 ) +

0:2^12 ( Z 2

= o,

z2) z4)

= 0,

z4) z2)

= 0,

+ a 3 Fi 3 (Z 3 ,Zi^ + a 1 F i 3 ( Z 2 , Z 4 ) + a 3 Fi3(Zi,Z 3 ^ + a 1 Fi 3 (Z 4 ,Z 2 )

-

a3-Fi3(Z4

-

a3-fl3(Z2

-

a 4 Fi4(Z 2 ,Z 3 ; -aiFi4(Z2,Z3)

+

-

a4Fu(Z4,Zi]

-

+ +

03^23 ( Z i , Z 2 ]

+ <x2F23(Z3,Z4

Q!3-P23(Z3,Z4]

+ a2F23(Zi,

— 0:4-^24 ( Z 3 , Z x ) -

a4Fu(Z3

<xiFu(Z4,Zi) + a4FuCZ-i — OJ3.F23 ( Z 2

a2F24(Z2,Z4) + a 4 F 2 4 (Z 4

-

"4-^24 ( Z i , Z 3 ;

- a 2 F 2 4(Z4,Z 2 ' + a 4 F 2 4 (Zi

z2) z4)

-

0:4-^34 ( Z 2 , Z i ]

- a 3 .F34(Z 4 ,Z 3 ) + a 4 F 3 4 ( Z 3

-

04^34 ( z 4 , z 3_,; -

+a3F34(Z1

z3)

Z2 - a 3 F2 3 (Z 4 Z i ) = 0,

+02-^24 ( Z 3 Z i ) «3^34(Z3

z3)

a2Fi2(Z3 Z i )

0 : 3 ^ 3 4 ( Z..2 , Z _, i)

+

z2) z4)

= 0,

z2) a 4.F 3... 4(Z!_. z 4.,) =

0,

i.e., we obtain ai = -<*2,

" i = 0:3,

c*i = -<x4,

a2 = -a3,

which means that a\ — —a2 = a3 = —a4 — 1.

a2 = a4,

a3 =

-a4,

Paracyclic Functional Equation

29

The general solution of the functional equation (2.56) is

/(Z 1 ,Z 2 > Z3) = F(Zi,Za) + n z 2 > Z 3 ) + G ( Z 1 > Z 3 ) - G ( Z 3 > Z 1 )

where

3

F(Zi,Z2)

=

-F 14 (Z 1 ,Z 2 ) + F 1 2 ( Z 1 , Z 2 ) + F 2 3 ( Z 1 , Z 2 ) + F34(Z 2 ,Zi),

G(Zi,Z 3 )

=

—Fi 3 (Zi,Z3) — i i 24(Zl,Z3).

Paracyclic Functional Equation

Let V be a complex vector space with complex dimension n, and let the complex vectors Xj, Yj € V (1 < i,j < n) be given as in the previous section. Throughout the section C, D, Ci and Di are constant vectors in V. Also, let there exist mappings / ; : Vp+q t-> V (1 < i < k). Now we will consider the following paracyclic complex vector functional equation of the first kind k /

y/t(Xj,Xj+i,...,

(k < n;

Xj+p-i, Yj, Y j + i , . . . , Yj.|_q,_i)

X n + j = Xj,

=

O

(3.1)

Y n +j = Yj).

In order to determine the general solution of the functional equation (3.1), we must distinguish the following six cases: 1° q < 2q - 1 < p = n,

2° q < p = n < 2q - 1,

3° q < p < n < 2q - 1 < 2p - 1, 4° q
+ q-l

T h e o r e m 3.1

+ < n < 2p-l,

q-K2p-l, 6° q < p <2q-l<

2p-l

< n.

If q < 2q — 1 < p = n, then the general solution of the

30

General Classes of Cyclic Functional

Equations

functional equation (3.1) is given by / r ( X i , X 2 ) . . . , X n , Yx, Y 2 , . . . , Y,)

(3.2)

min (k—r, q — 1) =

(—1)

2_^ i=\

- P H ( X J + X , X ; + 2 , . . . , X j , Yj+x, Yj+2, . . . , Y ? )

q-\ +

2_j ("l)* i=n—r+1

^ r i ( X j + x , X j + 2 , • • • >Xj, Yj+x, Yj+2, . . . ,

Yq)

min (k—r,n—q) +

( - 1 ) , _ ^r»(Xi+i,Xj+2,. . . ,Xj)

2^/ i=q n—q

+

2_^ (-l)n_t-Pi+r,n-t(Xx,X2, . . . ,Xn) i=max (n—r+1, q) k—r

+

+

2_^ (—1)" i=n—q+1 n-1

l

-fi+r,n-i(Xx,X2,.. . , X „ , Yx, Y2, . . . , Yg+j)

(— 1 ) "

2^i

J

-fi+r,n-i(Xx,X2,. . . , X n , Yx, Y2, •. . , Yg+j)

i=max (n—r+l,n—g-t-l)

(1 < r
(3.3)

min (fc —r,n—q) =

2^i

(~l)l~

^r«(Xj+l,Xj+2, •• •, X j , Y j + 1 , Yj+2, . . . ,

t=l n—q +

2_^ ( — 1)* i=n—r+1

-Fri ( X j + x , X j + 2 , . . . , X j , Y j + x , Y j + 2 ; • • • > Yq)

Yq)

Paracyclic Functional Equation

31

min (fc—r, q — 1)

+

( — 1)* • P r i ( X i + i , X i + 2 , . . . , X j ,

^ i=n—q+1

Y j + 1 , Yj+2, • • • , Yq, Y 1 ; Y 2 , . . . , Yq+i)

9-1

+

2_^i

\~^-l

-Fi+r,n-i(Xi,X2,. . . , X n ,

i=max (n—r+l,n—g+1)

Y i , Y 2 , . . • , Y q + i , Y j + i , Yj+2, • • • ,Yq) k—r

+

/ ,(~l)"~*-F]j+r,n-i(Xi,X2,. . . , X n , Y i , Y2, . . . ,Yq+i) i=q ra-1

+

( —l) n _ t -Fi+r,n-i(Xi,X2,. . . , X n , Y i , Y2, . . . , Y , + j )

2-j i=max (n—r+1, q)

(1 < r < fc), where Fij are arbitrary complex vector functions from V. Proof. The proof of this theorem is analogous to that of the previous Theorem 3.1. • Theorem 3.3 If q < p < n < 2q — \ < 2p — 1, then the general solution of the functional equation (3.1) is / r ( X i , X 2 , . . . , X p , Y i , Y 2 , . . . , Yq)

(3.4)

min(n—p,k—r) =

2_s i-\

(~1)'

- f r i ( X t + i , X j + 2 ; • • • ) X p , Y j + i , Y t + 2 ! • • • j Yq)

n—p

+

2^, (— I ) * " • ^ r i ( X i + i , X j + 2 , . . . , X p , Y j + i , Y j + 2 , . . . , Y g ) i=n—r+l min (fc—r,n—q)

+

2-i

( _ 1 ) I _ •Pri(Xi + i,Xt + 2 ,. . . ,Xp,

i=n—p+1

X l , X 2 , . . . , Xj+p, Y j + i , Yj+2) • • • , Yq)

32

General Classes of Cyclic Functional Equations

+

n—q 2__, ("I)' i—max (n—p+l,n—r+1)

^ri(Xj+i,Xj+2, • • • ,X p ,

X i , X 2 , . . . , Xj+p, Yj+i, Yi+2, • • • , Yg)

min (fc—r, 7—1) +

2^ i=n—g+1

(-I)'"

-fVt(Xj+i,Xj+2,...,Xp,

X i , X 2 , . . . , Xj+p, Yi+i, Y j + 2 , . . . , Yg, Y i , Y 2 , . . . , Yj_|_g)

+

2^1 (— 1 ) " i=max (n—g+l,n—r+1)

'•fi+r,n-t(Xi,X2,. .. , X j + p ,

X i + l , X j + 2 ) . . . , X p , Y i , Y 2 , . . . , Y ; + g , Y j + 1 , Y j + 2 , . . . , Yq)

min (fc—r,p—1) +

(—l)n_l-Fi+r,n-t(Xi,X2,. • . , X j + p ,

2 ^

Xj+i, Xj+2, • • •, X p ,Y i , Y 2 , . . . , Yj+?) p-1 '•ft+r.n-t(Xi,X2,. . . , Xj+p, i=max (g,n—r+1) Xi+i,Xj_).2, • • • > X p , Y i , Y 2 , . . . , Y j + 9 ) ft-r 1

Pi+r,n—i(Xi,X2,..

. , Xj+p, Y i , Y 2 , . . . , Yj+9)

n-1 +

2^i ("I)" *-f1»+r,n-t(Xi,X2,...,Xj+p, Yi, Y2,. . . , Yi+9) t=max (p,n—r+1)

(1 < r < jfe),

Paracyclic Functional Equation where F^ are arbitrary complex vector functions T h e o r e m 3.4 general solution

Ifq
33

from V.

+ q - l < 2 p - l , equation (3.1) is given by

then

/ r ( X i , X 2 , . . . , X P , Y i , Y 2 ) . . . , Yq)

the

(3.5)

min (n—p,k—r)

—

2^/

(~^y

-Frt(Xj+i,Xj + 2, • • • , X p , Y j + i , Y j + 2 , . . . , Y g )

1=1

n—p

+

2 _ , (~1Y~ i=n—r+1

• f r i ( X i + i , X j + 2 , • • • , X p , Y j + i , Y j + 2 , . . . , Yq)

min (k—r, q—1)

+

]r

(-lr^x^ -^•i+2) • • • > -*-p>

i=n—p+1 X l , X 2 , . . • , X j + p , Y j + i , Yj_)-2, . . • , Yq)

q-1

+

2_^

(—^)l_ ^rt(Xi+i,Xi+2,. • • ,Xp,

«=max (n—p+1,n—r+1) X i , X 2 , . . . , X j + p , Y i + i , Y j + 2 , . . . , Yg)

min (A—r,n—g)

+

( - 1 ) ' _ • P , r i ( X i + i , X i + 2 , . . . , X p , X i , X 2 , . . . ,Xj+p)

2J i=q n—q

+

(-l)n-1-Fi+r,n-i(Xi,X2,...,X;+p,Xj+i,Xj+2,...,Xp)

22

i=max (n—i+1, g) min (fc—r,p—1)

+

2_^

( _ l ) n _ t - f i + r , n - i ( X l , X 2 , . . . ,Xj+p,

i=n—g+1

Xi+l,Xj+2,- ••jXp, Y i , Y2, . . . ,

Yi+g)

34

General Classes of Cyclic Functional

+

P-l / J { — 1) i=max(n—g+l,n—r+1)

Equations

*-fi+r,7i-i(Xi, X 2 , . . . , X j + p ,

X j + i , X j + 2 , . . . , X p , Y i , Y 2 , . . . , Yj+g)

A—r + ^^(_I)™

8

-Fi+r,n-i(Xi,X2,. . . , Xi+p, Y i , Y2, . . . , Y j + ? )

i=p n-1 +

(— 1 ) "

^^

l

-Pf+r,n-i(Xi,X2, . . . ,Xi_)_p,Yi,Y2, . . . , Y j + g )

i=max (p,n—r+1)

(1

where Fij are arbitrary complex vector functions from V. We can prove the previous two theorems in the same way as the following theorem. Theorem 3.5 If q < p < p+q—1 < n < 2p— 1, then the general solution of the functional equation (3.1) is given by the formulae / r ( X i , X 2 , . . . ,X P , Y i , Y 2 , . . . , Y,)

(3.6)

min (q—l,k—r) (—I)*

=

2_^ t=l

+

2^i i-^1 i=n—r+1

^rt(Xj+i,Xj+2) . .. , X p , Y j + i , Yj+2, . . . ,

9-1 ^rt(Xj+i,Xi+2)- ••,Xp, Yj+i, Yj+2,. . . , Yg)

min (&—r,n—p) +

2^i

(— 1 ) * _ - P r i ( X i + i , X t + 2 , • • • , X p )

i=q n—p i+l,Xj+2,...,Xp) i=max (q,n—r+1) min (fe — r,p—1) +

2^i (~^T i=n—p+1

^rt(Xi+i,Xi+2,. . .,Xp,Xi,X2, . . . ,Xj+p)

Yq)

Paracyclic Functional Equation

35

p-1 2_j (— 1 ) " (n—p+l,n—r+1)

+ i=max

l

-Pi+r,n-i(Xi,X2, . . . ,Xj+p,

Xi+i,Xj+2) ••• )Xp)

min (A — r,n—q) +

2^i i=p

(—l)n_t-Fi+r,n-i(Xi,X2, . . . ,Xj+p)

n—g +

^

(-l)"_t-Fi+r,n-i(Xi,X2, ••. ,Xj+p)

i=max (p,n—r+1) *:—r +

(— l ) n _ l - P i + r , n - i ( X i , X 2 , . . . , X i + p , Y i , Y 2 , . . . , Y j + g )

2^

i=n—g+1 n-1 +

(— l ) " - t - F i + r , n - i ( X i , X 2 , . • • , X j + p , Y i , Y 2 , . . . , Y j + , )

2^

i=max(n—
(1 < r < fc), where F^ are arbitrary complex vector functions from V. Proof.

The proof of this theorem is based on mathematical induction.

For k — 2 the functional equation (3.1) has the form / i ( X i , X 2 ) . . . , X p , Y i , Y 2 , . . -,Y,) +

(3.7)

/ 2 ( X 2 , X 3 , . . . , X p + i , Y 2 , Y 3 , . . . , Y g + i ) = O.

Putting Xp+i = C, Y 9 + i = D into the equation (3.7), we obtain / i ( X i , X 2 , . . . , X p , Y i , Y 2 , . . . , Yq) =

•Fii(X 2 ,X 3 ,... ,X P , Y 2 , Y 3 , . . .

,Yq).

If we put Eq. (3.8) into Eq. (3.7), we have =

—

/ 2 ( X 2 , X 3 , . . . , X p + i , Y2, Y 3 ) . . . , Y g + 1 ) ^ l l ( X 2 , X 3 , . . .,Xp, Y2, Y 3 ) . . . , Yq),

(3.8)

36

General Classes of Cyclic Functional

Equations

i.e.,

f2(X1,X2,... =

-

,XP, Y l t Y a , . . . , Y,)

(3.9)

F i i ( X i , X 2 , . . . , X p _ i , Y i , Y 2 , . . . , Y9_i).

For k = 2 and r = 1 and r = 2, from Eq. (3.6) we deduce Eqs. (3.8) and (3.9), which means that the theorem holds for k = 2. Now we will suppose that the general solution of the functional equation k

/ „ff»(Xj,Xj + i,... , X j + p _ i , Yi, Y j + i , . . . , Y i + ? _x) = O

(3.10)

is given by (3.6) with fr replaced by gr. Let us consider the following functional equation /

y/i(Xj,Xt+i,...,Xj+P_i,

Yj, Y j + i , . . . , Y i + q ^ i ) = O.

(3.11)

i=l

We will distinguish the following five cases: 1° Let 1 < k < q. The substitutions Xj = d Yi—Di

for for

i^k + l,k + 2,...,k + p i ^ k + 1, k + 2 , . . . , k + q

(mod n), (mod n),

(3.12)

transform the equation (3.11) into fk+i (Xfc+i, X / . + 2 , . . . , X p + t , Yfc+1, Yfc + 2,..., Yg+fc)

(3.13)

n-l =

2^i ( " • ' • ) " *-' i i+*+l,n-i(Xfc+i,Xjfc- ) -2 I • • • ,Xj_|-p + fc, i=n—k

Yfc+l, Yft +2 , • • • > Yi+q+k ). Putting Eq. (3.13) into Eq. (3.11) and introducing new functions by 9i = fi + (-l)k+1-%k+i-i

(1 < i < A),

(3.14)

we obtain the equation (3.10). According to Eqs. (3.6) and (3.14), the

Paracyclic Functional Equation

37

general solution of the equation (3.11) is fr(Xi, X 2 , . . . , X p , Y i , Y 2 , . . . , Yq)

(3.15)

k—T

—

/ - ^ ( —1)*

-Fri(Xj+i,Xi+2,- . . , X p , Yj+i, Yj+2, . . . ,

Yq)

i=l 9-1 +

irrt(Xj+l,Xj+2,- •• ) X p , Y i + i , Yj+2, • . . ,

J_j ("I)' i=n—r+1

Yq)

n—p

+

( — I)'

2L>

^ri(Xj+i,Xj+2,- • • ,X p )

i=max (q,n—r+1) p-1 +

(-l)7l~*-Ft+r,n-i(Xi,X2,. . . ,Xi+p,Xi+i,Xi+2, ••• ,Xp)

2^

i=max (n—p+l,n—r+1) n—q

+

2_^i

(~1)" '-PVfr.n-i(Xi,X2,. .. ,Xj+ p )

i=max (p,n—r+1) n-1 + 2^ ( _ 1)" *-Fi+r,n-i(Xi,X2, . • . , X i + p , Y i , Y2, . . . , Yj+9) i=max (n—g+l,n—r+1)

+ ( — 1)

r r

i rfc+i_r(Xfc_r+2,Xfc_r+3) • • • >Xp, Yfc_r+2, Yft_r+3, . . . ,

Yq)

k+l-r —

(— I ) '

2L,

^rri(Xj+i,Xj+2,. . . , X p , Y j + 1 , Yj+2, . . . ,

Yq)

t=l

q-1 +

2^

(— 1 ) * ~ • f r t ( X j + i , X i + 2 , . . . , X p , Y i + i , Y j + 2 , - " )

Y9)

i=n—r+1 n—p +

2^ ( —1)*~ - F r t ( X j + i , X j + 2 , . . . , X p ) i=max (q,n—r+1)

p-1 + 2 ^ (—l)n_t-Pi+r,n-i(Xi,X2, . . . ,Xj+p,Xj+1,Xt+2) ••• ,Xp) i=max (n—p+l,n—r+1)

38

General Classes of Cyclic Functional Equations

• P i + r , n - i ( X i , X 2 , . . . , Xj+p) i=max (p,n—r+1) n-1

+

( _ 1)™ l -Fi+r,n-i(X 1 ,X 2 ) . . . , X i + p , Y i , Y 2 , . . . , Y i + 9 )

2^

i=max (n—
(1 < r < k). On the basis of the expressions Eq. (3.13) and (3.15), if 1 < k < q, the theorem holds for k + 1. Thus it holds for all such k, and also for k = q. 2° Let g < f c < n — p + 1 . equation (3.11) becomes

For the values Eq. (3.12) the functional

/fc+i(Xfc + i,Xfc + 2,..., X p+ fc, Yfc + i, Yfc + 2,..., Y9+fc)

(3.16)

n—q —

2^i (~*•)" '•^ 1 «+*+l.n-i(^*+liXfc+2) • • • jXj+p+fc) i=n—fc n-1 '•Pi+fc+l,n-i(X/, + 1 ,Xfc + 2! • • • ,Xj + p + fc, i=n—g+1 Yjfe+i, Y t + 2 , . . . , Yj+^+fc).

Now we will introduce the notations Eq. (3.14). On the basis of the expressions Eqs. (3.16) and (3.14), the equation (3.11) becomes Eq. (3.10). By using the inductive hypothesis and by virtue of the expression Eq. (3.14), we can conclude that the general solution of the equation (3.11) is given by the following equalities: / r ( X i , X 2 , . . . , X p , Y i , Y2, . . • , Yq) 9-1

=

2_^(~I)* i=l

^ r i ( X j + i , X j + 2 i . . . , X p , Y j + i , Yj+2, • • • ! Y ? )

k-r + ^ ( —1)'~ • F ' r i ( X j + i , X j + 2 , . . . , X , , ) i=q + ( — 1)

r

Fr]fc+i_r(Xj;_r+2,Xfc_r+3, . . . ,Xp)

Paracyclic Functional Equation

39

8-1

+

2^/ (~^Y i=n—r+1

^ri(Xi+i,Xj+2,. . . , X p , Yj+i, Yj+2,. . . , Yg) n—p

+

2__/ (—I)' i=max (g,n—r+1)

-Pri(Xj+i,Xj+2, •• • , X p )

p-1 +

^

(— 1)"

t

-Pi+r,n-t(Xi,X2, . . . , X i + p , X t + i , X j + 2 , . . . , X p )

i=max (n—p+l,n—r+1) n—9 2LK (_l)n_l-fi+r,n-i(Xl'^2, ••• ,Xj+p) i=max (p,n—r+1)

+

n-1 J

- P i + r , n — i ( X i , X 2 , . . . , X j + p , Y j , Y 2 , . . . , Yj_|_ ? )

i=max(n-q+l,n—r+1)

( l < r < * - g + l); / r ( X i , X 2 , . . . ,Xp, Yi, Y 2 , . . . ,

Yq)

k-r =

2 l , ( —1)*

^ r i ( X j + i , X t + 2 , . . . , X p , Y ; + 1 , Yj+2, . . . , Yg)

i=l 9-1 +

2^1 (~l)l~ i=n—r+1 n—p +

•fr»(Xi+i,Xi+2,... ,Xp, Y i + i , Y j + 2 , . . . , Yg)

/ ^ (— l ) t _ • F r i ( X i + l , X j + 2 , . . . , X p ) i=max (g,n—r+1)

p-1 • P i + r , n - i ( X i , X 2 , . . . , X j + p , X j + i , Xj_|-2; • • • j X p ) i=max (n—p+l,n—r+1) n—q

+

2~2

(-1)n" ' • F i + r , n - i ( X i , X 2 , . . . , Xj_|_p)

i=max (p,n—r+1) n~\ • f i + r , n - i ( X i , X 2 , . . . , X j + p , Y i , Y 2 , . . . , Yj_|_ g ) i=max(n—g+l,n—r+1) + ( — 1) r p 1 7 . i fc+i_ r (Xfc_ r +2) X f c _ r + 3 , . . . , X p , Y f c _ r + 2 , Y f c _ r + 3 , . . . , Y g )

40

General Classes of Cyclic Functional

(r = k-q

+

Equations

2,...,k),

or in a general form, / r ( X i , X 2 , . . . , X p , Y i , Y 2 , . . . , Yg)

(3.17)

min (q—l,fc+l —r)

=

2^i

(~1Y

^r»(Xi+i,Xj+2)"-)Xp, Yf+i, Yj+2,. . . , Y g )

i=l k+l-r

+

2^, (~ •*•)'

^V»(Xi+i,Xj+2,...,X p )

i=q

+

( _ ^)*

2^1

^r»(Xi+i,Xj+2, . . . , X p , Yj+1, Yi + 2, . . . , Yq)

z=n—r+1 +

n—p 2^1 (— I ) ' i=max (q,n—r+1)

-Pri(Xj+i,Xj + 2, . . . , X p )

p-1 + ^ ( _ 1 ) " *-Pi+r,n-i(Xi,X2,. . . , X j + p , X j + i , X j + 2 , - • • , X p ) i=max (n—p+l,n—r+1) n—9 + 2^ ( — 1 ) " ' - P i + r . n - i ( X i , X 2 , . . . ,Xj+p) i=max (p,n—r+1) +

n-1 2^ (_1)" i=max(n-q+l,n—r+1)

4

^i+r,n-i(Xi,X2, ... , X j + p , Y 1 , Y 2 , ... ,Yj+,)

(1 < r < fc). Therefore, on the basis of the expressions (3.16) and (3.17), we can conclude that the theorem holds in this case too, and also for A; = n — p + 1. 3° Let n -p+ find

1 < k < p. If we substitute Eq. (3.12) into Eq. (3.11), we

fk+i(Xfc+i,Xfc+2,... =

,Xp+fc, Yjfe+i, Yfc + 2 ) ... ,Yq+k)

(3.18)

P-i 2-i ( _ •*•)" '•^i+*+l,n-»(Xfc+l,Xfc+2,.. .,Xj+ p +fc,X*+i+i,Xfc+2+j,.. . , X t + p ) i=n—k

Paracyclic Functional Equation

41

'•fi+fc+l,n-t(Xfc + i,Xfc + 2 ,. . . , Xj+p+fc) t=p n-1 '•Fi+A+l,n-i(Xfc+i,Xfc + 2, . . . , X; + p + fc , i=n—g+1 Y * + l , Yji+2, . . . , Y i + g + f c ).

If we substitute Eq. (3.18) into Eq. (3.11) and if we take into account the transformation Eq. (3.14), then we obtain the equation (3.10). On the basis of the expression Eq. (3.14) and the inductive hypothesis, we find that the general solution of the functional equation (3.11) is given by the following formulae / r ( X i , X 2 , . . . , X p , Y i , Y 2 , . . . , Yg) =

8-1 2_^{~ 1)'

- F r i ( X j + i , X t + 2 , . . . , X p , Y j + i , Y j + 2 , . . . , Yq)

1=1

n—p i+l)Xj+2i • • • > i=q +

k—T 2_^i ( ~ * ) ' i=n—p+1

+

( — 1)

-^ri(Xj+i,Xj+2,.. . , X p , X i , X 2 , . . . ,Xj+p)

_r r

i r ^ + i _ r ( X f c + 2 _ r , X f c + 3 _ r , . . . , X p , X i , X 2 , . . . ,Xfc+i_ r + p )

,-1

+

2^i ( _ I ) ' " ^ r t ( X i + i , X i + 2 i • • • >X p , Y j + i , Yj+2, . . . , Yq) i=n—r+1 n—p

+

2__,

( - I ) * " •fri(Xi + i,Xj + 2, . . . ,X p )

i=max (q,n—r+1) p-1 + 2^, ( —l)n_t-Fi+r,n-i(Xi,X2,. . . ,X,+p,Xj+i,Xt+2, . . • ,Xp) i=max (n—p+1,n—r+1)

+

2^,

(~1)" l -Fi+r,n-i(Xi,X2, . . . ,Xj+p)

i=max (p,n—r+1)

42

General Classes of Cyclic Functional

Equations

n-1 +

2^i (— 1 ) " ' • P « + r , n - i ( X i , X 2 , . . . , X j + p , Y i , Y 2 , . . . , Y j + g ) i=m ax (n—g+1, n—r+1)

(1 < r < fc+p-n); /r(Xi,X2, . . . ,X p , Yi, Y2, . . . , Y,) g-i =

E^-^)'

^ri(Xj+i,Xj+2, . • . ,X p , Yj+i, Yj+2,. . . , Yq)

t=l

A:—r +

^^(

—

1)'

^ri(Xj+i,Xj+2,...,Xp)

9-1

+

(-l)

E

l _

-Fri(Xi+i,Xi+2,...,Xp,Yi+i,Yi+2,..., Y,)

i=n—r+1 n—p + 2__, { — 1)* i=max (q,n—r+1)

-Fr«(X{+i,X;+2, • • • , X p )

p-1 +

^_,

( — 1)™ ' • F i + r , n - i ( X i , X 2 , . . . , X i + p , X j + i , X i + 2 , . . . , X p )

i=max (n—p+l,n—r+1) n—g + E (*"•'•)" i=max (p,n—r+1)

l

-^i+r,n-i(Xi,X2,. .. ,Xj+p)

n-1 i

Fi+r,n-i(X\,

X 2 , . . . ,Xj_|_p, Y i , Y 2 , . . . , Y j + g )

i=max (n—g+l,?!—r+1)

+ ( — 1)

_r

-Fr,A+l-r(Xj;+2_r, Xfc+3_r, ..., X p )

(r = k + p - n + 1,... ,k - q + 1);

Paracyclic Functional Equation

43

/ r ( X 1 , X 2 , . . . ,Xp, Y i , Y2, . . . , Yg) k—r ^ ( — I )

=

1

•PVt(Xi+i,Xi+2,... , X P , Yj+i, Y j + 2 , . . . , Yg)

9-1

+

2^i (~^Y i=n—r+1

-Pri ( X j + i , X j + 2 , . . - , X p , Y j + i , Y j + 2 , . . . ,

Yq)

n—p +

(—1)'~ •fri(Xj+i,Xj+2, . . . , X p )

2_^ i=max (g,n—r+1) p-1

+

^ ,

(~1)

- P i + r . n - i ( X i , X 2 , ••• , X j + p ,

i=max (n—p+l,n—r+1) X i + i , X j + 2 , . . ., X p ) n—q

+

2~2

(-1)"" 1

Fi+r,n-i(X-i,

X2, . . . , Xj+p)

i^max (p,n—r+1) n-1 2_^i ( — 1)™ ' • f i + r , n - i ( X i , X 2 , . • . , X j + p , Y i , Y 2 , . . . , Y j + g ) i=max(n—q+l,n—r+1) + ( — 1) r F r ] f c + i _ r ( X f c + 2 - r , X f c + 3 _ r , . . . , X p , Y f c + 2 _ r , Y f c + 3 _ r , . . . , Yq) +

(r = k-q

+

2,...,k).

We can write the above equalities in a general form / r ( X i , X 2 , . . . , Xp, Y i , Y2,. . . , Yq)

(3.19)

min (q—l,fc+l—r) =

2_^

(— I ) *

•Pri(Xi+i,Xi_(-2, . . . , X p , Y j + i , Yj+2) • • • ; Y g )

i=l min (n—p,fc+l — r) +

^

+

fc+l-r 2 ^ (— ^ ) * i=n—p+1

(_1)J

^ri(Xi+i,Xi+2,...,Xp)

-fri(Xi+i,Xi+2,. . . , X p , X i , X 2 , • . . ,Xj+p)

44

General Classes of Cyclic Functional Equations

9-1 +

( _ I)'"

2^i

-^rt(Xj + i,Xj+2,- . . , X p , Y j + i , Y j + 2 , . . . , Yq)

i—n—r+1 n—p +

(_1)'

2^,

i=max

_

^r»(Xi+i,Xj+2, •.. ,Xp)

(q,n—r+1)

p-1 +

^ ^

(""I)"

' • P ' » + r , n - i ( X i , X 2 , . . . ,Xj_|_p,

t=max (n—p+l,n—r+1) Xj+i, Xj+2 > • • •j Xp)

n—q +

2 ^

(—1)"

*-Fi+r,n-i(Xi,X2,. . . ,Xj+p)

i = m a x (p,n—r+1) n-1 * - f i + r , n — t ( X i , X 2 , • • • , X i + p , Y i , Y 2 , . . . , Y^_j_g) i = m ax ( n—^4-1, n-^-l)

(1 < r < k). On the basis of the equalities (3.18) and (3.19), the theorem holds for this case as well, and also for k = p. 4° Let p < k < n - q + 1. Putting Eq. (3.12) into Eq. (3.11), we obtain /fc+i(Xjfc+i,XA;+2,... ,Xp + f c , Yfc+1, Yk+2,...

,Yq+k)

n—p =

/\_^ (~I)*

-pA+l,i(Xj + ft +1 ,Xj + fc + 2, . . . ,Xp+fc)

i=n—k p-1 " J -Fi+*+l,n-i(X<; + i,Xfc + 2,. . . , X j + f c + p , i=n—p+1 Xj+fc+i, Xj+^ + 2, . . . , Xfc+p)

n—g + /

y (~l)"~'-Pi+fc+l,n-i(X/. + i,Xfc- l -2,.

. . , Xj+p+fc)

(3.20)

Paracyclic Functional

+

Equation

n-1 2^i ( ~ l ) n ~ 1 F i + k + l , n - i { X k + l , X k + 2 , i=n—q+1

45

•• • ,Xj+p+/5,

Yfc+i, Yfc+2, . . . , Y j + ? + f c ).

If we substitute fk+i tutions

given by Eq. (3.20) into Eq. (3.11), by the substi-

_ / Si + {-l)n-k+iFk+i,n-k+i-i ~{ /i + f - l ^ - ^ H i - i

9i

(1<*<*-P+1), (i = k-p + 2,...,k),

,o9n ^Zi)

we obtain the equation (3.10). According to (3.21), the general solution of the functional equation (3.11) is given by / r ( X i , X 2 , . . . ,X P , Y i , Y 2 , . . . , Y,) ?-i =

/_^{~I)'"

-Fri(Xi+i,Xj+2,. . . , X p , Yj+i, Yj+2, . . . ,

Yq)

j=l

n—p +

/ , ( ~ 1 ) ' ~ Fri ( X j + i , X j + 2 , . • . , X p ) i=9 p-l

+

2^i

(— I ) * " • ^ r t ( X i + i , X i + 2 , . . . , X p , X i , X 2 , . . . , X j + p )

i=n—p+1 fc—r i(Xi,X2,... ,Xi+p) t=p

+ (_l)"-*+'-1Ffc+1,n_ft_1+t.(X1,X2,..., Xfc+1_r+p) 9-1

+

2^ (-l) i=n—r+1

l _

- F r i ( X j + i , X j + 2 , . . . , X p , Y j + 1 , Y j + 2 , . . ., Y g )

n—p

+

2^

( - l ) l _ -Fri(Xt+i,Xj + 2 , . . . ,X p )

i=max (g,n—r+1) p-l + 2^i (—I)" '•fi+r,n-i(Xi,X2,. . . ,Xj+p, i=max (n—p+1,n—r+1) Xj+i, Xj+2,. . . , Xp)

46

General Classes of Cyclic Functional

Equations

n—q Z

-fi+r,n-i(Xi, X2, . . . , Xj+p)

i=max (p,n—r+l) n-1 +

2^i ( — I)™ i=m ax (rt—q+-l, ?ir-r+l)

4

- f i + r , n - t ( X i , X 2 , . . • , X j + p , Y i , Y 2 , . . . , Yj-|-g)

(1 < r < fc-p+1); / r ( X i , X 2 , . . . , X p , Y j , Y 2 , . . . , Yq) 9-1

= 2^(—I)1 i=l

^r»(Xj+i,Xj+2, . . . ,Xp, Yi+i, Yi+2, . . . , Yg) n—p

+ ^ ( —l)l~ Fri(Xi+i,Xj+2, . . . ,Xp) i=9

+

k-r ^ , (— 1 ) ' i=n—p+1

-Pri(Xj+i,Xi+2, • •• , X p , X i , X 2 , . . . ,Xj+p)

i r 7 . ) < ; +i_ r .(XA+2_r,Xfc_)-3_ r , . . . , X p , X i , X 2 , . . . , X f c + i _ , . + p )

+ ( — 1)

g-1 +

2 ^ ( _ 1)' i=n—r+1

^ri(Xj+i,Xj+2,. . . ,Xp, Y j + i , Yj+2,- • •, Yg)

n—p + 2L< (~1)*~ i=max (g,n—r+1)

•Pri(Xi+i,Xj+2,...,Xp)

p-1 + 2-j (— 1 ) " i=max (n—p+l,n—r+1)

J

-Pi+r,n-t(Xi,X2,.. . , X j + p , X i + i , X j + 2 ; ••• ,Xp)

n—9 + ^ (— 1 ) " ' - P i + r . n - t ( X i , X 2 , . . . , X j + p ) i=max (p,71—r+1)

Paracyclic Functional Equation

47

n-l

+

( - 1 ) " t -Pi+r,n-i(Xi,X2,...,Xj +p ,Yi,Y2,. . . , Yi+q)

2-u

t=max(n-g+l,n-r+l)

(r = k-p

+ 2,

...,k+p-n);

/r(Xi,X2,. . • ,X p , Yi, Y2,. . . , Y,) 9-1

^(_1)1

=

^ri(Xi + i,Xi + 2,. . . ,X p , Y i + 1 , Yj + 2 ) • • • , Yg)

i=l fc — T

/ ,(~1)'~ -P1ri(Xj+1,Xt+2,...,Xp)

+

t=«

+ ( — 1)

_r

-Pr,*;+l-r(Xfe+2-r> X f c + 3 _ r , . . . , X p )

8-1

+

( —1)*_

2_/

^ri(Xj + i,Xj+2,. . . ,X p , Yi + 1 , Yj+2,. . . , Yq)

i=n—r+1 n—p

+

2_^

(~1)'~ •frt(Xi+i,Xj+2, . . . ,X p )

i=max (g,n—r+1) p-1

+ j=max

^

(—l) n ~ l i r i+ r] „_i(Xi,X2,... ,Xt + p,X{ + i,Xi+2, • • • ,Xp)

(n—p+l,n—r+1) n—g

i(Xi,X 2 ,. . . ,Xt +p ) i=max (p,n—r+1) n-l

+

2_,

(-l) n_! -Fi+r,n-i(Xi,X 2 , . . . ,Xi + p , Yi, Y2, . . • , Y i + g )

i=max(n—g+l,n—r+1)

(r =fc+ p + 1 - ra,..., A: - g + 1); / r ( X i , X 2 ) . . . , X p , Yj, Y 2 , . . . , Yq)

48

General Classes of Cyclic Functional k-r = / l~lj i=l

+ ( — 1) ~T

+

Equations

Fri ( X j + i , X j + 2 ) . . . , X p , Y j + 1 , Y j + 2 , . . . , Y g )

Frtk-r+l(X-k-r+2,Xk-r+3,

2^, (~^y~ i=n—r+l

• • •, X p , Y j f c _ r + 2 , Y f c _ r + 3 , . . . ,

Fri(X.i+i,Xi+2,...,X.p,Yi+i,Yi+2,-•

•

Yq)

,Yq)

n—p

+

(~1) 1 _ Fri(X-i+i,Xi+2,..

2^

.,Xp)

i=max (q,n—r+1) p-1 +

( _ 1)" '-fi+r.n-i(Xl,X2,. .. , X i + p , X j + i , X i + 2 , . . . , X p )

2^

t=max

(n—p+l,n—r+l) n—q

+

2_^

(—l) n _ '-Pi+r,n-i(Xi,X 2 ,. . . , X i + p )

i=max (p,n—r+1) 71-1

+ 2^ ( — 1)™ * - ' r i + r , n - i ( X i , X 2 , . . . , X j + p , Y i , Y 2 , . . . , Y j + ? ) i=max(n—g+l,n—r+1)

(r = k-q

+

2,...,k).

Prom the above equalities we obtain / r ( X i , X 2 , . . . , X p , Y i , Y 2 , . . . , Yq)

(3.22)

min (g—1,A;+1—r) —

(— I ) '

2^/ i=l

-fri ( X j + i , X j + 2 , . . . , X p , Y j + i , Y j + 2 , • • • , Y ? )

min (n—p,k+l —r) +

2^, i=q

( _ 1 ) , _ •fri(Xi+i,Xj+2,. ..,Xp)

min (p—l,fc+l —r) +

2^ i=n—p+1

(""I)1

•fri(Xj+i,Xi+2,.. . ,Xp,Xi,X2, . . . ,Xj+p)

Paracyclic Functional Equation

49

k+l-r l

Fi+r,n—i

( X i , X 2 , • . . , Xj_|_p)

i=p 9-1

+

2^/ (~^y i=n—r+l

- P r t ( X i + i , X j + 2 , - • • , X p , Y j + l , Y j _ | _ 2 , . • . , Yq)

n—p

+ 2^, (~1)' t=max (<;,n—r+1) P-l ^__,

+

JPri(Xi + i,Xi + 2,. • • ,Xp)

(—l)n-1-Fi+r,n-»(Xi,X2, . . . , X j + p , X i + i , X i + 2 , • •• ,Xp)

i=max (n—p+l,n—r+1) n—g '•Fi+r,n-t(Xi,X2,. .. , Xj+p) i=max (p,n—r+1) n-1 +

^ ^

(—I)"

l

-f(i+r,n-i(Xi,X2,. . . ,Xj+p, Y i , Y2, . . . , Y i + 9 )

i=max(nr-g+l,n^r+l) (1 < r < k). Therefore, the theorem is proved for p < k < n — q + 1. 5° Let n - q + 1 < k < n. If we put Eq. (3.12) into Eq. (3.11), we get fk+i (Xjt+i, Xfc+2 J • • •, Xp+fc, Yk+i, Y/j + 2,..., Yq+k)

(3.23)

«-i =

2—1 ( ~ *•)' i=n—k

^*+i,«(Xi+fc+i,Xj+jfe+2,...,Xp+fc, Yi+k+l,

Yj+fc+2, . . . ,

Yq+i-)

n—p

+

^ ( - l ) i - 1 * f c + i , i ( X i+k+l, X j + f c + 2 , . . . , X p + f c ) i=q p-1

+

2-*/ (~^)n

t

^t+A+l,n-j(Xfc+i,Xi;+2,. ..

,~X.i+k+p,

i=n—p+l

Xj+fc+i, H.i+k+2, • - • , Xfc + p)

General Classes of Cyclic Functional

50

l

Equations

-F(i+fc+l,n-t(Xfc+i,X/;+2, • • • , X j + p + f c )

i=p n-1 "t-fi+fc+l,n-i(Xfc+i,Xfc+2,- • • )Xj+j,+fc, i=n—q+1 Yfc+1, Yfc+2, . . . , Yi+q+k

)•

Substituting the function fk+i determined by Eq. (3.23) into Eq. (3.11) and using the substitutions Eq. (3.21), we obtain the equation (3.10). Therefore, the general solution of the functional equation (3.11) in the case considered will be / r ( X i , X 2 , . . . , X p , Y i , Y2, . . • , Yq) =

+

9-1 2-^( — 1 ) 1 _ ^ r » ( X j + i , X j + 2 ) • . • , X p , Y j + i , Y j + 2 , . . . , i=l n—p /_^i~

1)*

Yq)

- F H ( X J + I , X J + 2 , . . . ,X P )

i=q p-1 +

2^1 i-^)' i—n—p+1

^ r t ( X i + i , X j + 2 , . . . , X p , X i , X 2 , . .. , X j + p )

n—q + / „(~1)" z=p

+

t

-Pi+r,n-»(Xi,X2,. . . , X , + p )

A—r 2_< ( _ l ) n - * - P , t + r , n - « ( X i , X 2 , . . . , X , + p , Y i , Y 2 , . . . , Y j + 9 ) i=n—q+1

+

^ ( — 1) Z i=n—r+1

-Fri(Xt+i,Xi+2,. • - , X p , Y i + i , Yi+2? • • • > Y g )

n—p + Z^ (~1)*~ i=max (
^ri(Xi+i,Xi+2,.-.,Xp)

Paracyclic Functional Equation

51

p-1

i(Xi, X 2 , . . . , X i + P , X i + i , X i + 2 , • . . , Xp) i=max (n—p-\-l,n—r+l) n—q

'•f»+r,n-»(Xi,X2,. . • ,Xj_)-p) i=max (p,ra—r+1) n-1

+

(-l)" _ 1 -Fi+r,n-i(Xi,X 2 ,. . . ,Xj + p, Y i , Y 2 , . . • , Yj+g)

2^

t=max(n—g+l,n—r+1) + (— l ) n _

r

~ F f c + i i n _ f c + r . _ i ( X i , X 2 , . . . , X f c + i - r + p , Y i , Y 2 ) . . .,

Yk+i-r+q)

(1 < r
=

2 ^ ( — I ) * " -Pri (Xi + i, X j + 2 ) . .. , X p , Y j + 1 , Y j + 2 , . .. ,Y g ) i=l

+

•P,«(Xj4-i,Xi+2> • • • >Xp)

2^^~^' p-1

+

( _ 1 ) 1 _ •Pri(Xi+i,Xj+ 2) . . . , X p , X i , X 2 , . . . ,Xj + p )

^ i=n—p+1 fc-r

+

^ (

—

1)" l -Pi+r,n-i(Xi,X 2 , . . . ,Xj+p)

t=p

+ (-l)n

r

Ffc + l i „_ f c + r _i(Xi,X 2 ,... ,Xfc_ r+ i+p)

q-l

+

(—^)*

2^/

^ V t ( X ; + i , X j + 2 , . . . ,X P , Yj+i, Y j + 2 , . . . , Y g )

i=n—r+1 n—p + 2_j (— 1 ) * i=max (g,n—r+1)

^rt(Xj+i,Xj+2,...,Xp)

52

General Classes of Cyclic Functional

Equations

p-1 Fi+r,n-i(Xi,X2,.

. . ,Xj-)-p,Xj+i, Xj+2) • • • j X p )

t=max (n—p+l,n—r+1) n—7

+

(_1)n"

E

*-fi+r,n-i(Xi, X2, . . . , X j + p ) i=max (p,n—r+1) n-1 *-f»+r,n-t(Xi,X2, . . . ,Xj+p, Y i , Y2, . . . , Yj+g) i=max(n—g+l,n—r+1)

(r = g + k - n + 1,..., k - p + 1); / r ( X i , X 2 , . . . , X p , Y i , Y 2 , . . • , Yq) 9-1 2^(~ i=l n—p

—

+

E (

I)*

_

• f r i ( X j + i , X i + 2 , . . . , X p , Y j + i , Yj_|_2,-• • , Y g )

•*•)*

^ri(Xj+i,Xj+2,...,Xj,)

«=9 +

fc—r ^ ("I)' i=n—p+1

^r»(Xj+i,Xj+2,.. . , X p , X i , X 2 , .. . ,Xj+p)

9-1

+ E

(-l)i_1^ri(Xi+i,X i+2,

• • •, X p , Y j + i , Y i + 2 , . . . , Y , )

i—n— r + 1 n—p + 2^i (~tyl i=max (g,n—r+1)

^r»(Xi+i,Xi+2,. . . ,Xp)

p-1 • M i + r , n — t ( X i , X 2 , • • • ,X^_f_p, X j + i , X $ + 2 , • • • ) X p ) i=max (n—p+1,n—r+1)

+ 2_^ ( " I ) " l-fi+r,n-t(Xi,X2,. .. , X i + p ) i=max (p,n—r+1)

Paracyclic Functional Equation

53

n-1 +

(— 1 ) "

2^i

,

-^i+r,n-i(Xi,X2,. . . , X i + p , Y i , Y2, . . . , Yi+g)

i=raax(n-g+l,n-r+l) + ( — 1)

_r

Fr]fc+i_r(Xfc+2-r,Xfc+3_r, . . . , X p , X i , X 2 , ••.

,%k+l-r+p)

(r = k - p + 2,... ,p - n + k); /r(Xi,X2, . . •,Xp, Yi, Y2,. . . , Y,)

=

^__,( — 1 ) '

• F r i ( X j + 1 , X j + 2 , • • • ) X p , Yi-|_i, Y j + 2 , • • • i Y g )

1=1

k—r +2__,(—l) 1

F7.i(Xj+i,Xj+2,... ,Xp)

i=g + (—1)

_r

-Pr,*!+l-r(X*;+2-r) X f c + 3 _ r , . . . , X p )

9-1

+

2L/ ( — I ) ' " " • f i , i ( X i + i , X j + 2 , . . . , X p , Y j + i , Y,-)-2, . . • , Y 9 ) i=n—r+1 n—p +

(— 1 ) * ~

2_,

•fr»(Xi+i,Xj+2,...,Xp)

i=max (q,n—r+1) P-1 ^

+

(— l ) n ~ ' i i i + r , n - i ( X i , X 2 , . •• , X i + p ,

i=max (n—p+l,n—r+1) Xi+l,Xi+2i •• •> Xp) n—9 +

2^

(—1)" '•fi+r,n-t(Xi,X2, ••. ,Xj+p)

i=max (p,n—r+1) n-1 A-i+p, i=max(n—q+l,n—r+1) Yi, Y2,. .. , Yj+,)

54

General Classes of Cyclic Functional

Equations

(r = p - n + k + 1,..., k - q + 1); / r ( X i , X 2 , . . . ,X p , Yl, Y2,. . . , Yq) k—r ~

^(~1)

1

^ r t ( X i + l , X j + 2 , . . • , X p , Y j + i , Y i + 2 , . . . , Yq)

i=l r

+ ( — 1)

Fr^k-r+\ (Xfc+2-D Xft+3_r, . . . , X p ,

Yfc+2-n Yfc+3_r, . . . , Yq)

9-1

+

2^i (~tyl i=n—r+1

-Pri(Xi + i,X i+ 2,. . . ,X p , Y i + 1 , Yj + 2,. . . , Yq)

+ 2^, ( —I)* i=max (?,n—r+1)

•fri(Xi+i,Xj+2I-• • ,Xp)

p-1 + 2_^ \~~ 1)™ i=max (n—p+l,n—r+1)

^i+r,n-i(Xi,X2,. ..,Xj+p,Xi+i,Xj+2i ••• |Xp)

n—q '•Pi+r,n-i(Xi,X2, . . . , Xj+p) i=max (p,n—r+1) n-1 +

2^i

(~^)

l

-^i+r,n-t(Xi,X2, .. . ,Xj+p,

i=max(n—g+l,n—r+1) Y l , Y 2 , . . . , Y'j+g)

(r =fc-g+ 2,...,fc)We can write the previous equalities in the following general form /r(Xi,X 2 ,...,Xp,Yi,Y2,.. -,Y ? )

(3-24)

min(q>—l,fc+l—r) =

2-j i=l

(_1)*~ •fri(Xi+i,Xj+2,...,Xp, Yj + i, Yi+2,..., Y g )

Paracyclic Functional Equation

55

min (fc+1—r,n—p)

+

i-^)1

E

Fri(Xi+ i , X j + 2 ) . .. ,X p )

i=g min (fc+1—r,p—1) +

E i=n—p+1

(~^)*

^ri(Xj+i,Xi+2, . . . ,Xp,Xi,X2, . . . ,Xj+p)

min (fc+1—r,n—q)

+

( _ l ) n _ t - f i + r , n - i ( X i , X 2 , . . . ,X i + p)

E

fc+l-r +

E

( — 1 ) " ' • ^ 1 * + r , n - t ( X i , X 2 , . . . , X i + p , Y i , Y j , . . ., Y j + g )

i=ra—g+1

+

9-1 E ( — •'•)' i=n—r+1

^ ? r t ( X i + i , X i + 2 , . . . ,Xp, Y j + i , Yj+2, . . . ,

Yq)

n—p + E ( —•*•)* i=max (g,n—r+1)

•FVi(Xi+i,Xj+2, . . . , X p )

p-1 +

E ,

(— 1 )

-Pi+r,n-i(Xi,X2, . . . ,Xi+p,Xj+i,Xj-|-2,. . . ,Xp)

t=max (n—p+1,n—r+1) n—q *-fi+r,n-i(Xi, X2, . . . , Xj+p) i=max (p,n—r+1) n-1 •^i+r,n-i ( X i , X 2 , . . . , X j + p , Y j , Y 2 , . . . , Y j + g ) i=max(n^-g+l,n—r+1)

(1 < r < fc). Therefore, the theorem holds for n - q + 1 < k < n.

•

General Classes of Cyclic Functional Equations

56

Now we will solve two particular cases of the equation (3.1). a) By p u t t i n g k = n into equation (3.1), we obtain the functional equation n

5 ^ / j ( X i , X j + i , . • • ) X j + p _ i , Y j , Y j + i , . . . , Yi+q-i)

= O.

(3.25)

Therefore, if we put k = n into Eqs. (3.2), (3.3), (3.4), (3.5) a n d (3.6), t h e n we obtain the general solution of the functional equation (3.25) in the cases considered. For example, if we put k = n into Eq. (3.4), we obtain t h a t t h e general solution of the functional equation (3.25) for q < p < n < 2q — 1 < 2p — 1 is given by the formulae / r ( X i , X 2 , . . . , X P , Y i , Y 2 , . . . , Yq) =

n—p ^(~1)

1 -

Fri(Xi+i,Xi+2,

(3.26)

• • • ,XP, Yj+i, Y i + 2 , . . . , Y,)

t=i n—q

+ J2 (-l)i_1-Fri(X i+ljXi_|_2

3

• - • )Xp,

i=n—p+1

X i , X 2 , • • • ,Xp+», Y i + i , Yj+2, • • • , Yq) min (n—r, g—1)

» + i ) X j + 2 , . . . , X p , X i X 2 , . . . , Xp+j, i=n—q+l Y»+i, Yj+2j • . • , Yq, Y i , Y 2 , . . .

,Yg+i)

9-1

+ 2_j ( — 1) i=max (n—q-\-l,n—r+1)

t

- f i + r , n - i ( X i , X 2 , . . . ,Xp+j,Xi_|_i,Xj_|-2, . . . , X p ,

Y l , Y 2 , . . . , Y g + j , Y j + i , Y j + 2 , . . . , Yq)

P-l l

Fi+r,n-i ( X i , X 2 , . . . , Xp+j,

X j + l , X j + 2 , . . . , X p , Y i , Y2, . . .

,Yq+i)

Paracyclic Functional Equation

%

+ Z~/(~*)"

Fi+r,n-i(X-i,X.2,

57

• • • , X j + p , Y i , Y2, . . . , Yg+j)

i=p

(1 < r < n), where Fy are arbitrary complex vector functions from V. The functional equation (3.1) for q < p < 2q - 1 < 2p - 1 < n had not been previously investigated, but for this case we must additionally determine the general solution of the equation (3.25). Now we will give the following result which treats a more general case than the previous one. Theorem 3.6

The general solution of the functional equation (3.25) for

—-— > max (p, q) is given by the formulae Li

/ r ( X i , X 2 , . . . , X P , Y i , Y 2 , . . . , Yq) =

(3.27)

r

i r(Xi,X2,... ,Xp_i, Yi, Y 2 , . . . , Yg_i)

— ^r+l(X2,X3,. . . , X p , Y 2 , Y 3 , . . ., Yq) (l
Fn+1 = Fx),

where FT are arbitrary complex vector functions from V. Proof. Using the conventions fr = fr+m X r = X r + n and Y r = Y r + „ , the equation (3.25) in an expanded form can be written in the following way / r ( X r , X r - | - i , . . . jXr+p-i, Y r , Y r _j-i,..., Yr.|_g_i) (3.28) +

/r+l(X r .4-i,X r + 2) • • • >X r + p , Y P +i, Y r +2, • • • i Yr+q)

+•••

~r

Jr+n—p V-^-r+n—p j -"-r+n—p+1 > • • • j ^-r+n—1j

+

fr+n—p+1 ( X r + n _ p + i , X r + „ _ p + 2 j • • • > X r + n _ i , X r ,

+

/ r + n — l ( X r + n _ i , X r , . . . ,X r -(-p_2, Y r 4 . „ _ i , Y r , . . . , Y r + g _ 2 ) = O .

" r + n - p i I r+n—p+1 j • • • i * r+n—p+q—1/

* r+n—p+1) * r+n—p+2j • • • j * r+n—p+q) T ' ' '

Assuming that p > q (for p = q there are just slight modifications in the following formulae) and putting Y r + g = Y r + 9 + i = • • • = Y r + n _ i = C

General Classes of Cyclic Functional Equations

58

and X r + P = X r + p + i = • • • = X r + n _ i = C into Eq. (3.28), where C is a fixed vector from V, we obtain / r . ( X r , X r _ | _ i , . . . , X r + P _ i , Y r , Y r + i , . . . , Y r _). g _i) +

/ r + i ( X r + i , X r + 2 , . . . , X r + p _ i , C, Y r + i , Y r + 2 , . . . , Y r -t- 9 _i, C)

+

/r+2(X r +2,X r _(-3, . . . ,~X.r+p-\,C,C,

+

• • • + /r+p-l(X r -)-p_i, C, . . . , C) + fr+p{C, C,. . . ,C)

+

fr+p+l(C, C, . . . , C) + • • • + fr+n-p\C,

+

fr + n — p + i \ C , C , . . .

+

fr+n-p+2(C,C,

+

/ r + n - l ( C i X r , X r + i , . . . , X r + p _ 2 , C , Y r , . . . , Y r + g _ 2 ) = O.

(3.29)

Y r + 2 , Yr_|_3, . . . , Y r + a _ i , C, C)

C,C,. . . ,C)

,C,Xr,C,C,...,C)

. . . ,C, X r , X r + i , C , C, . . . ,C) + •••

If we further substitute Y r + 9 _ i — C and Xr^_p_i = C in Eq. (3.29), this yields / r ( X r , X r + i , . . . ,X r .+ p _2,C, Y r , Y r + i , . . . , Y r -|_ ? _2,C)

(3.30)

+

/ r + l ( X r+1 j X r + 2 , . . . , Xr_)-p_2, C, C, Y r + i , Y r + 2 > • • • j Y r + g _ 2 , C,C)

+

• • • + / r + p - 2 ( X r + p _ 2 , C, . . . , C) + / r +p_i(C, C,. . . ,C)

+

fr+p(C, C, . . . , C) +

+

fr +7i—p+i \C, C,...,

+

Jr+n-p+2\C, C, . . . , C, X r , X r + i , C, C,. . . , C) + •••

+

/r+n-l(C) X r , X r + i , . . . , Xr-)-p_2, C, Y r , . . . , Y r + g _ 2 ) = O.

1- fr+n-p(C, C, X r , C, C,...,

C,. . . ,C) G)

Subtracting Eq. (3.30) from Eq. (3.29), we get the formula -"-r+1 j • • • > -"-r+p—li * rj * r+l> • • • i ' r + g - 1 )

(3.31)

X r + i , . . . , X r + p _ 2 , C, Y r , Y r + i , . . . , Y r + g _ 2 , C7) —

/r+i(Xr+i,Xr+2,...,Xr+p_i,<_7, Y r + i , Yr+2j • • • > Yr+g_i,(7)

+

/ r + l ( X r+1 j X r + 2 , . . . , X r + p _ 2 , C, C, Y r + i , Y r + 2 , . . . , Y r + 9 _ 2 , C, C)

—

/r+2(X r -|-2, X r + 3 , . . . , X r + p _ i , C, C, Yr_|_2) Y r + 3 , . . . , Y r _|_ g _i, C, (7)

+

• • • + fr+p-2 ( X r + p _ 2 , C, . . . , C)

~

/ r + p - i ( X r + p _ i , C , . . . ,C) + / r + p _ i ( ( 7 , C , . . . ,C),

which holds for every r = 1,2,..., n.

Paracyclic Functional

Equation

59

Let us put now <7r(Xi,X 2 ,... , X p _ i , Y i , Y 2 , . . . , Y , _ i )

(3.32)

=

/r(Xi,X2,...,Xp_i,C, Yi, Y 2 , . . . ,

Yq-i,C)

+

/r+l(X2,X3, . . . , X p _ i , C , C, Y 2 , Y3, . . . , Y 9 _ i , C , C)

+

••• + / r +p_2(X p _i, C,..., C)

and AT = / r + p _ i ( C , C , . . . ,C).

(3.33)

Since / r = /r+m we conclude that gr = gr+n- Putting in Eq. (3.25) X r = Y r = C (1 < r < n), we find n

^ A

r

= 0.

(3.34)

r=l

According to Eqs. (3.32) and (3.33), the formula Eq. (3.31) can be written in the form / r ( X i , X 2 , . . . ,Xp, Y i , Y 2 , . . . , Yq) =

flv(Xi,X2,...

-

gr+1(X2,X3,...,Xp,Y2,Y3,...,Yq)

(3.35)

,Xp_i, Yi, Y2,..., Y,_i) + Ar

(1 < r < n).

Finally, with the notations *i

=

5i,

F2 F3

=

02-41,

=

9 3 - A i -

K

=

gn-Ai-A2

A2,

An-t,

the expression Eq. (3.35) can be written in the form Eq. (3.27). Thus we have proved that Eq. (3.27) is a consequence of Eq. (3.25) if n + 1 > 2 max(p, q). Conversely, a straightforward computation shows that the functions Eq. (3.27) satisfy the equation (3.35) for arbitrary functions Fr from V. D

General Classes of Cyclic Functional Equations

60

b) Now we will consider the functional equation n

^ ^ / ( X j , X j + i , . . . , X i + p _ i , Yj, Yj+i,..., Yj+,_i) = O

(3.36)

i=l

(X n +i = Xj, Y n +i = Yj) which is a particular case of the functional equation (3.25). In order to determine the general solution of the functional equation (3.36), we will distinguish the following cases: 1° q < 2q - 1 < p = n, 2° q < p = n < 2q - 1, 3° q < p < n < 2q - 1 < 2p - 1, 4° q
+ q-l

+
q-K2p-l, 6° q < p < 2q-l

< 2p-l

< n.

Theorem 3.7 If q < 2q — 1 < p = n, then the general solution of the functional equation (3.36) is given by /(X1,X2,...,X„,Y1,Y2,...,Yg) =

(3.37)

•Pb(Xi,X 2 ) ... , X n , Y i , Y 2 l . . . , Y ? _ i )

— F o ( X 2 , X 3 , . . . , X n , X i , Y2, Y 3 , . . . , Y q ) , where Fo is an arbitrary complex vector function from V. Proof. If we put k = n into Eq. (3.2), then by summing up the functions /r (1 < r < n) and putting /1 = / 2 = ••• = / „ = / , we obtain the formula Eq. (3.37). • In the same way the following theorems can be proved: Theorem 3.8 If q < p = n < 2q — 1, then the general solution of the functional equation Eq. (3.36) is given by the formula / ( X i , X 2 , . . . , X n , Y i , Y 2 , . . . , Yq) =

(3.38)

-Pb(Xi,X 2 ) ... , X n , Y i , Y 2 , . . . , Y 9 _i)

— F o ( X 2 , X 3 , . . . , X n , X i , Y 2 , Y 3 , . . . , Yq) +

g-ln/2] /.*/ [•* r »P^n-g+«+l)X n _ 9+ i + 2,...,Xj + i, i=l * l > * 2 > - - - » » t i » n - ) + i + l ] Y n _ 9 +t_)_2i • • • j Yq)

Paracyclic Functional Equation

61

— Fi[Ki,X.2, •.. , X n , Y ? _ j + i , Yg_i+2> • • •, Y g , Y i , Y 2 , . . . , Y 2g _j+ n )] > where Fi (i = 0,1,... V.

,q—[n/2]) are arbitrary complex vector functions from

Theorem 3.9 If q < p < n < 2q — 1 < 2p — 1, then the general solution of the equation (3.36) is f(X1,X2,...,Xp,Y1,Y2,...,Yq)

(3.39)

=

•Fb(Xi,X 2 ; ..., X p _ i , Y i , Y 2 , . . . , Y , _ i )

-

Fo(X2,X3,...,Xp,Y2,Y3,...,Yg)

p-q + / j [Fi(Xi,X2,...

, X j , Xn_p+i-|-i, Xn_p+j_)_2) • • • jXp,

»=i * n—p+i+l i * n—p+i+2 > • • • > * q)

— Fi(X.p-.i+i, X p _j+2) • • • j X p , X i , X 2 , . . . , X 2 p - n - i ,

Y i , Y 2 , . . . , Yp + ? _ n _j)j p-[n/2] +

/ j \Fi\A.\, i=p-q+l

X 2 , . . . , X i , X n _p+j+i,X n _p4.i_)-2) • • • >Xp, * 1 > * 2> • • • > * i+q—pi

* n — p + i + l > * n—p+i+21

• • • j I 9J

— Fj(Xp_i+i, Xp_j+2, . . . , X p , X i , X 2 , . . • , X 2 p _ n _ j , * p—i+1;

* p—i+2> • • • ) I g j ' l ) * 2 ) . . . j

1 p + « — n—ijj >

w/iere Fj (i = 0 , l , . . . , p — [n/2]) are arbitrary complex vector functions from V. Theorem 3.10 The general solution of the functional equation (3.36) for q < p <2q — 1
Fo(Xi,X2,

• • • ,Xp_i, Y i , Y 2 , . . . , Y g _ i )

(3.40)

62

General Classes of Cyclic Functional Equations

—F 0 (X2,X3,... , X p , Y2, Y 3 ) . . . , Yq) p+q—n—1

+

/ j

l-Fi(Xi, X2, . . . , X j , X n _ p - ) - t + i , X n _ p + i + 2 ! • • • > X p ,

j=l

' n - p + i + l i * n—p+i+2) • • • j * q)

— Fi(X.p-i+i,

Xp_j+2, . . ., X p , X i , X2, . . . , X 2 p - n - t ) Y i , 1 2 , . . ., Y p + g - n - j j J

p-[n/2] +

/ j [Fi[X.i, X 2 , . . . , X j , X n _ p + j . | _ i , X„_p_|_j_|_2J • • • j X p ) i=p+g—n — ,Tj(Xp_j-|-i, Xp_j+2j • • • j X p , X i , X2, . . . ,X2p_ n _j)J ,

where Fi (i = 0, l,...,p— from V.

[n/2]) are arbitrary complex vector functions

Next we will prove the following theorem. T h e o r e m 3.11 The general solution of the equation (3.36) for q < p < p + q — 1 < n < 2p — 1 is given by /(X1,X2,...,Xp,YllY2)...,Y,) =

Fo(Xi,X2,... ,Xp_i, Yi, Y2,..., Y,_i)

-

F o ( X 2 , X 3 , . . . ,XpY 2 , Y 3 , . . . , Yq)

(3.41)

p-[n/2] +

/

j

[ i ' t ( X i , X 2 , . . . ,Xt,X n _p_)_t+i, X n _p^-i+2) • • • )Xp)

t=l

— i*t(Aj,_i+i,Xp_i^2, • • • j X p , X i , X2, . . . , X2p—„—i)\ ,

where Fi (i = 0, l , . . . , p — [n/2]) are arbitrary complex vector functions from V. Proof. By summing up the functions fr (1 < r < n) determined by Eq. (3.6) and putting / x = / 2 = •••/„ = / , we obtain Eq. (3.41), where we

Paracyclic Functional

Equation

63

introduced the notations F0(Xi,X2,... ,Xp_i, Yi, Y2,..., Yg_i) q—l r 7 J / y <jr(Xj,Xi-)-i,..., X p _ r + i _ i , Y j , Yj_)_i,..., Y g _ r + j _ i ) _r=l i = l n—p r + 2_j ^ G r ( X i , X j + i , . . . , X p _ r + i _ i ) r=q i = l G r ( X i , X 2 , . . . , Xp_r, Y i , Y2, . . . ,

Xq-r)

n ^ ( - l ) r i ? i r ( X 1 , X 2 , . . . ,Xp_r, Yi, Y2,..., Yq_r) i=l

=

(l
(-1)

l,...,n-p);

Fk ( X i , X 2 , . . . , Xfc, X n _ p + f c + i , X n _ p + f c + 2 , . • •, X p ) 'n—p+k

k+l

/

n

,

•Pr,p-*;(Xi,X2,...)Xfc,

, r=l X n _ p + f c + i , X„_p+fc+2 j • • •, X p )

p—k —

2_^Fr
. . . , X p , X i , X 2 , . . . , Xfc)

( l < * X p ) ,

r=l

where •'V+m.ml-'*-!) • • • > J*-p—mi -"-m+1; • • • > -^-p)

64

General Classes of Cyclic Functional

Equations

= —^rm(X m +i,... , X p , X i , . . . , X p _ m )

(1 < r < m). D

We have not considered the case q max(p,q),

T h e o r e m 3.12

then the general solution of the

functional equation (3.36) is given by /(X1)X2,...,Xp,Y1,Y2,...,Y,) =

(3.42)

r

i ( X i , X 2 , . . . , X p _ ! , Y i , Y 2 , . . . , Yg_i)

— F ( X 2 , X 3 , . . . , X P , Y 2 , Y 3 , . . . , Y g ), where F is an arbitrary complex vector function from V. Proof. A straightforward calculation shows that every function / of the form Eq. (3.42) satisfies the functional equation (3.36). We have to prove the converse, i.e., that from Eq. (3.36) it follows that / has the form Eq. (3.42). Let C be a fixed vector from V. For X; = Yj = C (1 q (for p = q there are just slight modifications in the following formulae) and n > 2p — 1 and putting Y g + 1 = Y 9 + 2 = • • • = Y n = C and X p + 1 = X p + 2 = • • • = X n = C into Eq. (3.36), by virtue of (3.43) we obtain /(X1,X2,...,Xp,Y1,Y2,...,Yg) +

/ ( X 2 , X 3 , . . . , X P , C , Y 2 , Y 3 , . . . , Y q , C ) + •••

+

/(XP,C,...)C) + /(C,C,...,C,X1,C,C,...,C)

+

f(C,C,...,C,XuX2,C,C,...,C)

+

/(C ) Xi,X2,...,Xp_i,C,Yi,Y2,...,Y 9 _i) = O.

+ •••

(3.44)

Semicyclic Functional Equation

65

If we further substitute Yq = C and X p = C in the last equation, we get /(Xi,Xa>... ,Xp_i,C, Yi, Y 2 , . . . , Y,-!, O

(3.45)

+

/ ( X 2 , X 3 , . . . , X p _ i , C , C,Y 2 , Y 3 , . . . , Yq-i,C,C)

+ •••

+

/(X P _ 1 ,C7,...,C) + / ( C , C , . . . , C , X 1 , C , C , . . . , C )

+

/ ( C , C , . . . , C , X 1 , X 2 , C , C , . . . , C ) + •••

+

/ ( C > X i , X 2 , . . . , X p _ i , C , Y i , Y 2 , . • •, Y , _ i ) = O.

Subtracting Eq. (3.45) from Eq. (3.44), we find /(Xi,X2>...IXp>Y1>Y2>...,Y,)

(3.46)

=

/ ( X i , X 2 , . . . ,Xp_i,C, Yi, Y 2 , . . . , Yg_i,C)

—

/ ( X 2 , X 3 , . . . , X P , C , Y2, Y 3 , . . . , Y q , C )

+

/ ( X 2 , X 3 , . . . , X p _ i , C, C, Y 2 , Y 3 , . . . , Y g _ i , C,C)

— / ( X 3 , X 4 , . . . , X P , C , C, Y3, Y 4 , . . . , Yq,C, +

C)

••• + / ( X p _ l l C > . . . , C ) - / ( X p , C > . . . l C 7 ) .

Putting F ( X i , X 2 , . . . , X p _ i , Y i , Y 2 , . . . , Y g _i) =

/ ( X i , X 2 , . . . ,Xp_i,C, Yi, Y 2 , . . . ,

+

/ ( X 2 , X 3 , . . . ,Xp_i,C,C, Y2, Y 3 , . . .

+

••• + / ( X p - i . C - . - . C ) ,

the equality (3.46) takes on the form Eq. (3.42).

Yq^i,C) ,Yq-i,C,C)

•

For particular cases see the results obtained in [D. S. Mitrinovic (1963C); P. M. Vasic et al. (1965)].

4

Semicyclic Functional Equation

Let V be a complex vector space, and let Z; (1 < i < n) be complex vectors as in Sec. 1. Also, let S£ (0 < k < n) be the set of all strictly increasing mappings of the set {1,2,...,/:} into { 1 , 2 , . . . , n } . Let fr (r e S%) be mappings

66

General Classes of Cyclic Functional

Equations

We will solve the following semicyclic complex vector functional equation / , /r(Z r (i), Z r ( 2 ),. . ., Zr(fc)) = O.

T h e o r e m 4.1 given by

(4.1)

The general solution of the functional equation (4.1) is

/r(Z r (i),Z r (2),...,Z r (jt)) 2^i -*rn>(Zrp(i),Zrp(2),...,Zrp(;fe_i)) pes*.!

=

(4.2) (T £ S%),

where rp(i) = r(p(i)), and Frp are arbitrary complex vector functions from the vector space V such that 2J-Prp(Z t (i),Z ( ( 2 ),...,Z f ( f c _ 1 )) = O

(tEiSi-i),

(4.3)

rp=t

where the sum is extended over all r € S% and p € S^_1 such that rp — t. Proof. have

Let fT be denned by Eq. (4.2) and let Eq. (4.3) hold. Then we

/y

=

=

/r(Z r (i),Z r (2),...,Z r (j.))

2_^ 2^i ^n>(Z r p(i),Z rp (2),.. .,Z rp (j|._i)) ••esr pes*.! z2

22

• F rp(Z((i),Z t ( 2 ),...,Z t ( fc _i)) = O.

t e 5 J _ j rp=t

Hence, such functions satisfy the functional equation (4.1). Conversely, if fr (r € S%) is any solution of Eq. (4.1), we have to prove that the functions fr admit the representation Eq. (4.2) with the conditions Eq. (4.3). For fixed r let us put Zj = C into Eq. (4.1) for i ^ r(j) (1 < j < k). Then Eq. (4.1) yields the following (not unique)

Semicyclic Functional Equation

67

representation /r(Z r (i),Z r ( 2 ),. . .,Z r ( fc )) =

2^, G r p (Z r p( I ),Z r p(2) ) ...,Z r p (fc_i)). (4.4)

For an arbitrary £ € 5^_ x let -fft( z t(i), z t(2),---)Z 4 ( fc _i)) = 22 <J r rp (Z ( ( 1 ),Z t (2),...,Z f ( fc _i)).

(4.5)

rp=t

The equation (4.1) can be written in the form 53

fli(Zt(i)>Zt(2)>...,Zt(t_1))

= 0.

(4.6)

Let P ' be the set of all t £ S j j ^ such that Ht ^ O and P " = S£_x \ P ' . The equation (4.6) is reduced to

53 Ht(Zt{1),Zm,..., V _ x ) ) = O.

(4.7)

teP' We can suppose that among all representations of the form Eq. (4.4) of the functions fr we have taken that (or one of those) for which the number s of the elements of the set P' is minimal. If s = 0, we can take Frp = Grp and the theorem is proved. The case s = 1 is impossible in view of Eq. (4.7). Thus we can suppose that s > 1. Let t be some fixed element from P ' . Putting Z, = C into Eq. (4.7) for i ± t(j) (1 < j < k - 1), we obtain Ht("Z>t(i), Zt(2), • • •, Zt(fc_i)) = - 5 J JrpCZtih), Z t (j 2 ),..., Z t ( i m ) ) ,

(4.8)

where J r p ( Z t ( i l ) , Z t ( i 2 ) , . . . , Z t ( i m ) ) (m < k - 2) is obtained from G>p(Zrp(1), Z r p ( 2 ), • • •, Z rp ( fc _ : )) by putting Z{ - C for all i but t(l),t(2), ... ,t(k — 1). The sum on the right-hand side of Eq. (4.8) is extended over certain (not all) pairs of indices r and p. Consider a certain summand JroPo on the right-hand side of Eq. (4.8). Let wo 6 S% and VQ € Sj^_1 be such that UQVQ = *• We can construct a sequence of ordered pairs (U0,VQ),

( « 0 , W o ) , ( U l>^l)> ( " l . ^ l ) . •••>

which satisfy the following conditions

(uq,wq)

68

General Classes of Cyclic Functional Equations

2° («„«;,) = (r0,po); 3° Ui_iiOi_i = UjUj (1 < i < q); 4° the sequence UiiUj(l),... ,UiWi(k — 1) (0 Vi and GUi tWi, respectively. We have Hf = Ht + JroPO. On the other hand, if t £ P", i.e. Ht = O, then also Hf = O. If the same procedure is applied to all summands of the right-hand side of Eq. (4.8), we conclude that the new function Ht is identically the zero vector. This contradicts the minimum property of the number s. Hence, s = 0 which proves the theorem. • This theorem generalizes the results given in [D. Z. Djokovic (1965A)]. E x a m p l e 4.1

If n = 5 and k = 4, the equation (4.1) is

/(Zi,Z2,Z3,Z4) + +

0(Zi,Z 2 ,Z3,Z 5 ) + /i(Zi,Z 2 ,Z4,Z 5 ) i(Z1}Z3,Z4,Z5)+j(Z2,Z3,Z4,Z5)

= O.

Its general solution is given by /(Zi,Z2,Z3,Z4)

= +

/i(Zi,Z2,Z3) +/2(Zi,Z2,Z4) /3(Zl,Z 3 ,Z4) + / 4 ( Z 2 , Z 3 , Z 4 ) ,

g{ZuZ2,Z3,Z5)

=

5l(Z1,Z2,Z3)+52(Z1,Z2,Z5)

+

5 3 (Zi,Z 3 ,Z 5 ) + 5 4 ( Z 2 , Z 3 , Z 5 ) ,

=

/ix(Zi,Z 2 ,Z4) + /i 2 (Zi,Z 2 ,Z 5 )

+

/ i 3 ( Z i , Z 4 , Z 5 ) + /i 4 (Z 2 ,Z 4 ,Z 5 ),

=

i!(Zi,Z 3 ,Z4) + i 2 ( Z i , Z 3 , Z 5 )

+

i3(Zi,Z 4 ,Z 5 ) + i 4 ( Z 3 , Z 4 , Z 5 ) ,

=

j1(Z2,Z3,Z4)+j2(Z2,Z3,Z5)

+

j3(Z2,Zi,Z5)

/i(Zi,Z2,Z4,Z5) z

»( ijZ3,Z4,Z5)

j(Z2,Z3,Z4,Z5)

+

u(Z3,Z4,Z5),

Special Cyclic Functional Equation

69

where / i ( Z i , Z 2 , Z 3 ) + 9i (Zi

Z3)

/ 3 ( Z i , Z 3 , Z 4 ) + /n(Zi / 4 (Z 2 ,Z3,Z 4 ) + j i ( Z a

z4) = o, z4) = o, z4) = o,

g2(Z1,Z2,Z5)

ZB)

/3(Zi,Z3,Z4) + n(Zi + /u(Zi

=

=

O,

O,

93(Zi,Z3,Z 5 ) + i 2 (Zi

z5) = o,

94(Z2,Z 3 ) Z 5 ) + J2(Z 2

ZB)

=

O,

/i 3 (Zi,Z 4 ,Z 5 ) + i 3 (Zi

Z5)

=

O,

/i 4 (Z 2 ,Z 4 ,Z 5 ) + j 3 ( Z 2

Z5)

=

O,

j4(Z3,Z4,Z5) + j4(Z3

ZB)

=

O.

Hence we may take / i , / 2 , / 3 , / 4 ,92,93,94 /i 3 , hi, i 4 to be arbitrary complex vector functions from the complex vector space V and 91 = - / l ,

5

/*1 = -J*2, /i2 = - 9 2 ,

»1 = ~ / 3 i ii = -93,

jl = ~/4, h = ~94,

h = -h,3,

J3 = -hi, ji = -u-

Special Cyclic Functional Equation

The notations for the vectors in this section are the same as in Sec. 1. Let V be the vector space and let there exist mappings fi-.Vi+1^V

(l
and

gi

: V i + 2 •-» V

(l
Now we will prove the following result. Theorem 5.1 The general solution of the special cyclic complex vector functional equation n

2_^fi(Zi,Zi+1,--i=l

n—1

, Z2<) + ^ 9 i ( Z i , Z 3 , - - - , Z 2 i + i , Z 2 i + 2 ) = O i=l

15 given by /i(Zi,Z2) = Jff1(Z1)-Fi(Z3)>

(5.1)

70

General Classes of Cyclic Functional

Equations

/t(Zi,Zj + i, • • • ,Z2j) = ( - l ) t F i _ i ( Z i , Z j + i , - • • ,Z2i-2) +(—I) 1

Gj_i({Zj,Zi + i, • • • , z 2 ,} n { z 1 ; z 3 , • • • ,z 2 i_i},Z2i) +(-iyFi(Zi+l,

Z i + 2 , • • • , Z 2i )

(2 < i < n),

•fn(Z n +l,Z n +2,-• • ,Z2n) = O,

(5.2)

5»(Zi) Z 3 , • • • , Z2i+i, Z2J+2) = ( - l ) t i ? i ( Z 1 , Z 3 , • • • , Z 2 j-i) + ( — 1)* Gi({Zi + i,Zj + 2, ' ' " , Z2»+2} H {Z 1 ; Z 3 , • • • ,Z2i+l},Z2i+2) +(-l)itfi+1(Z1,Z3).-.,Z2i+1)

(l
i ? n ( Z i , Z 3 , - - • , Z 2 n-l) = O, where Fi, Gi, Hi (1 Z 4 ) + fli(Zi, Z 3 , Z 4 ) = O,

(5.3)

whose solution according to [D. S. Mitrinovic (1963A)] is /1(Z1,Z2)

=

H1(Z1)-F1(Z2),

/2(Z2)Z3,Z4)

=

F1(Z2)-Gi(Z3,Z4),

g^{Zx,Zz,Zi)

=

- / / 1 ( Z 1 ) + G 1 (Z 3 ! Z 4 ).

(5.4)

Thus, the theorem holds for n = 2. Suppose that theorem holds for any fixed n and let us consider the equation n+l

2_^fi(Zi,Zi+i,--t=l

n

, Z 2 i) + ^ 0 j ( Z i , Z 3 , - - - ,Z 2 i+l,Z2i+2) = O.

(5.5)

i=l

By putting Z* = d (1 < i < n), where C, = const into Eq. (5.5), we see that fn+i can be represented in the form / n + l ( Z n + l , Z n + 2 , • • • ,Z 2n +2) = (—1)™ F n ( Z n + i , Z n + 2 ) ' • • ,Z2 n )

(5.6)

Special Cyclic Functional

Equation

71

+ ( — l ) n G n ( { Z n + i , Z n + 2 , • • • , Z 2 n + 2 } f"l {Zi,Z3,- • • ,Z2 n +l},Z2 n +2), where F„ and Gn are arbitrary functions. By a substitution of Eq. (5.6) into Eq. (5.5), we obtain the equation n

n

2^, Ai(Zj,Zj + i, • • • ,Z 2 i) 4- 2_^Bi(Zi,Z3, i=l

• • • ,Z 2 t+i,Z 2 n +2) = O,

(5.7)

i=l

where we introduced the notations Ai = fh

Bi = gi

(l
n+1

An = fn + (-l)

Fn,

Bn=gn

(5.8)

+ (-1)"G„.

By putting Z; = d (i = 2,4, • • • ,2n) into Eq. (5.7) we obtain -B n (Zi,Z3,- • • ,Z2„+l,Z 2 n+2) = ( — l ) n # n ( Z l , Z 3 , • • • ,Z2„_l),

(5.9)

where ifn is an arbitrary function. On the basis of the expression Eq. (5.9), the equation (5.7) takes on the following form n

n—1

^ A i ( Z i , Z i + 1 , - - - , Z 2 i ) + ^ D i ( Z i , Z 3 , - - - , Z 2 i + i , Z 2 i + 2 ) = 0 , (5.10) i=i

t=i

where we introduced the notations Di = Bt

D„_i = 5 n _ 1 + ( - l ) n H „ .

(l
(5.11)

The functional equation (5.10) is an equation of the form Eq. (5.1). According to the inductive hypothesis, the general solution of the equation (5.10) is given by equalities of the form Eq. (5.2) with / ; replaced by Ai and & replaced by Di. By virtue of Eqs. (5.11), (5.9), (5.8) and (5.6) we deduce that the general solution of the equation (5.5) is given by fi(Zi,Zi+1,'--

,Z2i) = Ai(Zi,Zi+l,---

/ n ( Z n , Z„+i, • • • , Z 2 n )

= +

,Z2i)

An(Zn,

(l
Zn-)_i, • • • , Z2„) r

( — l ) " i „ ( Z „ + i , Z n + 2 , • • • ,Z2n),

/ n + l ( Z n + l , Z „ + 2 , • • • ,Z2n+2) — ( ~ 1 ) "

Fn ( Z n + i , Z n + 2 , • • • , Z 2 n )

72

General Classes of Cyclic Functional Equations

+(—i) n G n ({z n + i,z n + 2, • • •, Z271+2} n {Zi,Z3, • • • ,z 2n +i},Z2„ + 2), S i ( Z l , Z 3 , - - • ,Z2i+l,Z2i+2) = Z?j(Zi,Z3,--- ,Z2i+l,Z2i+2)

(1 n_i(Zi,Z3, • •-,Z2n-l,Z2n)

+

(-1)"

i7„(Zi,Z3, • •

,Z2n+i,Z2„+2) = ( - l ) n i ? n ( Z l , Z 3 , - - -

-,Z2n-\),

,Z2n-l)

G n ( { Z n + i , Z n + 2 , • • • , Z2n+2} H { Z i , Z 3 , - • • ,Z2n+l}>Z2n+2)-

This theorem generalizes the result given in [R. Z. Djordjevic (1965A)].

6

Condensed Cyclic Functional Equation

Consider the condensed cyclic complex vector functional equation h

E ^ Z i ( i ) > Z i ( 2 ) > - - - > Z i ( S i ) ) = °>

(6-1)

j=l

where 1 < st < n, fc : VSi •-> V (1 < i < k), ~i(v) 6 {1,2,...,n} 1
for

<Si.

For two sets of indices {i{l),i(2), . . . ,i(si)} we denote their intersection by {ij(l),ij(2),...

and {j(l),j(2),... ,ij(tij)}.

,j(sj)}

Theorem 6.1 The general solution of the condensed cyclic complex vector functional equation (6.1) is i-\ Z

Z

Z

fi( l(l)' i(2)>---' l(Si))

~

z2FJi('Z'ji(l)'--->ZJi(tji)) 3=1 k

~

z2 j=i+l

F

i3(ZTj(i)>--->zTj(tii))i

(6-2)

Condensed Cyclic Functional

where Fij(Zjj,1-),...

,Zjj,t..^)

73

Equation

are arbitrary functions from V for 1 v. a

Proof. We will prove the theorem by induction. For k = 2, equation (6.1) becomes /l(ZT(l)>ZT(2)'"->ZT(s1)) + /2(Z2(1)>Z2(2)'" -'Z2(s2))

=

^-

It is obvious that the functions / i and fa depend just on the variables z

l2(i)> z i2(2)>- • - . Z i ^ t i : , ) .

t h u s

w e

m a

w r i t e

y

/l(Zl(l)'Zr(2)'---»ZT(si))

=

-

"'?12(ZIi(l)>ZT2(2)'---'Z12(t12))'

/2(Z2(l)»Z2(2)»---'Z2(s2))

=

•Pl2(Z12(l)'Z12(2)'---'Zl2(ti2))

for an arbitrary function F\i : V*12 •-> V, i.e., for k = 2 the general solution of Eq. (6.1) is given by Eq. (6.2). For some fixed k suppose that the general solution of the functional equation (6.1) is given by the formula Eq. (6.2). Now, let us consider the equation fc+i

E/«( z ;(i)> z ;(2).---> z 7( Si )) =

a

(6-3)

i=l

If we put Zj = d for i & {k+T(l),k~+l{2),.. representation

.,k+l(sk+1)},

we obtain the

/*+i( z FfT(i)> ZFPT(2)> • • •' Zfc+T(sfc+i)) k

=

2 ^ F3,k+l ( Z pTT(l)' Z P+I(2) » • • • ' Z p+I(ty, fc+ i))

where ^\*+l( Z J>+I(l)> ZIfc+T(2)' • • • ' =

~

/i(zJ(i).Zj(2).---'Zi(,3))

z

Z

JJ+l(ti,h+1))

- = c ' for » (2 {* + !(!), * + l(2),...,fc + l ( » i + 1 ) }

(6-4)

74

General Classes of Cyclic Functional

Equations

If we substitute Eq. (6.4) into Eq. (6.3) and denote 5i(Zi(l)>Zi(2)>--->Zi(Si))

=

+

/*(Zi(l)>Zi(2)>---'Z7(Si)) -Fi,fc+i( Z i^+T(i)j Z Mfc+T(2)' • • • ' Z 7 ^ + T ( t i , f e + 1 ) ) '

then equation (6.3) becomes k

X] fl ' i ( Z i(l)' Z i(2)'---' Z i(s0)

=

°"

i=l

By assumption its general solution is i-l Z

Z

Z

5i( i(l) I i(2)'---' i( S i ))

~

-

2^^'*( Z J*(l)'" , ' Z ii(*J«))

22

Fi Z

i( ij(i)>--->Zij(tij))>

j=i+l

and hence i-l /i(z7(i)>Zi(2)>--->Zi(Si))

-

-

2^fi»(Zji(i)>---'Zj7(ij.o) i=i fc+i

J2

F

ij(Zij(l)'--->Zij(Ui))

for 1 < i < k. Together with Eq. (6.4) this yields Eq. (6.2) with instead of k.

k+1 •

The last theorem which is obtained in [I. B. Risteski and V. C. Covachev (2000)] generalized all previous results given in this chapter.

Chapter 2

Functional Equations with Operations between Arguments

In this chapter some classes of complex vector functional equations are solved, namely such equations in which operations between arguments appear. The results presented in this chapter are given in [I. B. Risteski et al. (1999); I. B. Risteski et al. (2000B); I. B. Risteski et al. (to appear A); K. G. Trencevski et al. (1999)].

7

Operator Functional Equation

Here a new linear operator \t is defined such that $ o $ = O. The general analytic solution of the vector functional equation * / = O is given. The results presented here are obtained in [I. B. Risteski et al. (1999)]. Definition 7.1. Let V and V be complex vector spaces. For an arbitrary mapping / : V n _ 1 i-» V (n > 1) we define a mapping \ t / : Vn •-*• V by (*/)(Z1>Z2>...,Zn_1,Z„) =

(7.1)

1

( - l ) " - / ^ ! , Z 2 , . . . , Z„_i) - / ( Z 2 , Z 3 , . . . , Z„) n-1

+

2 ^ ( - i ) 4 + 1 / ( Z i , z 2 , . . . , Z j + Zj+i,.. . , z n _ i , z n ) . i=l

If n = 1, we define * / = O. Remark 7.1. The definition of the operator * is a variation of the formula giving the differential of the bar construction [S. MacLane (1963)]. 75

76

Functional Equations with Operations between

Arguments

For an arbitrary mapping f : V n _ 1 i-> V it holds

Theorem 7.1

(tf o * ) / ( Z i , Z 2 , . . . , Zn> Z n + 1 ) = O. Proof. obtain

(7.2)

Applying the operator \P to the mapping \£/ : V" H V , we

(*o*)/(Zi,Z2,...,Zn,Zn+1) =

( - 1 ) " ( * / ) ( Z ! , Z 2 , . . . , Z„_i, Z„) - ( * / ) ( Z 2 ) Z 3 ) . . . , Z„, Z n + 1 )

+

^ ( - l ) i + 1 ( * / ) ( Z ! , Z 2 , ...,Zi

n

+ Zi+U...,

Z n> Z n + i ) .

i=l

Using the definition of the operator * inside the sum and replacing i by i + 1 in some of the terms, we can write down the previous equality as (*o*)/(Zi,Z2>...,

(7.3)

= ( - l ) " ( * / ) ( Z i , Z 2 , . . . , Z n _!, Z n ) - (*/)(Z 2 , Z 3 , . . . , Z„, Z„ + 1 )

+ < ^ ( _ l ) » + i [ ( _ i ) « - i / ( Z l , Z 2 , . . . , Zi + Zi+1,...,

Z B _!, Z n )

i=l

+ / ( Z 2 , Z 3 , . . . , Zj + i + Z j + 2 , . . . , Z n , Z „ + i ) j + / ( Z i , Z 2 ) . . . , Z„_i) — / ( Z 3 , . . . , Z„, Z n + i ) n—1 i—1

Z j + l , . . ., Zj+i + Z;+ 2 , . . . , Z n , Z„+i) i=2 J=l

n—2

+ /

y

n

/

y

(-1) 4

i=l j=i+2

,?

/ ( Z i , Z 2 , . . . , Z i + Z i + i , . . .,Zj + Z j - + i , . . . , Z n , Z n + 1 )

Operator Functional Equation

77

In the parentheses {• • • } the terms for j = i, i + 1 are omitted because they cancel each other. Further we will consider the double sum n—1 i—1

/] ^ ( - l ) t + J / ( Z i , Z 2 , . . . , Z j + Z j + i , . . . , Z i + 1 + Z j + 2 , . . . , Z n , Z n + i ) i=2 j=l

n

i-1

— 2 _ , Z ^ ( - ^ ) t + J + / ( Z i , Z 2 , . . . , Zj + Z j + i , . . . , Z t + Z { + i , . . . , z n , z n + i ) i=3 j = l ra—2

= /

n

/ ! • (—-0*

y

3

/ ( Z i , Z 2 , . . .,Zj + Z j + i , . . .,Zj + Z j + i , . . . , Z n , Z n + i )

j = l i=j+2 n—2

= -2_j

n

\~i ( _ i ) , + : ' / ( Z i , Z 2 , . . . , z , + Z j + i , . . . , Z j + Z j + i , . . . , z „ , z n + i ) .

t=l j=i+2

Since the double sums in Eq. (7.3) cancel each other, we obtain Z n - i ) Z n , Zn_|-i) = ( - l ) n ( * / ) ( Z ! , Z 2 , . . . , Z n _ ! , Z n ) - ( * / ) ( Z 2 , Z 3 , •.., Z„, Z n + 1 ) n-1

+ Z(-i) i + 1 [(-i)n_1/(Zi, z2,..., ii + zi+1,..., zn_!, zn) i=l

+ / ( Z 2 , Z 3 , . . . , Z i + 1 + Zj+2,..., Z„, Z„+i) I + / ( Z i , Z2,..., Z n _i) — / ( Z 3 , . . . , Z n , Z n + i )

= (-irw)(Zi, z2>..., zn_i, z„) - (*/)(z2, z3,..., z„, z n+ o + ( - l ) » - i [ ( - l ) » - 1 / ( Z 1 , Z 2 ) . . . , Z n _ 0 - / ( Z 2 , Z 3 , . . . , Zn)

78

Functional Equations with Operations between

Arguments

n-1

+ £ ( - i ) i + 7 ( Z i , z 2l ..., Zi + zi+1,..., zn_u z n )] + [(-i) B - 1 /(z 3 , z 3> ..., z„) - /(z 3 ,..., zn> z n+1 ) n-1

+ ^ (

—

I)*

/(Z2, Z3, . . ., Z; + i + Z j + 2 , . . ., Z„, Z n + i ) I

1=1

=

(-l)n(*/)(Z1>Z2,..,Zn_i,Zn)-(*/)(Z2>Z3j...)Zn,ZB+i)

+(-l)"-1(*/)(Z1,Z2,...,Zn_1,Zn) + (*/)(Z2,Z3,...,Z„,Zn+1) = 0 , which yields the validity of (7.2). • This formula shows that the kernel of the operator \P contains all mappings of the form \ t / . The next theorem provides a complete description of this kernel. T h e o r e m 7.2

The general solution of the operator equation */(Zi,Z2,...,

)= O

(7.4)

in the set of analytic functions f : Vn i-¥ V (n > 1) is given by f(Zu Z 2 ) . . . , Z„) = ( * F ) ( Z i , Z 2 , . . . , Z„_i, Z n ) + L(Zi, Z 2 , . . . , Z n ), (7.5) where F : V " - 1 i-> V' is an arbitrary analytic function andL is an arbitrary mapping: Vn i-+V'(n> 1) linear with respect to each argument. Proof. First note that if n = 1, the equation (*/)(Z1,Z2) = 0 is the Cauchy functional equation /(Z1+Z2)-/(Zi)-/(Za) = 0. The general analytic solution of this equation is / ( Z ) = AZ, where A is an (s x r)-matrix with arbitrary complex constant entries (r = dim V and s = dim V ) . About the solution of the Cauchy matrix functional equation see [0. E. Gheorghiu (1963)] and [A. Kuwagaki (1962)].

Operator Functional

Equation

79

Now let n > 2. The operator equation (7.4) is equivalent to (-l)"/(Z1,Z2,...,Zn)-/(Z2!Z3,...,Zn+1)

(7.6)

n

+

^ ( - l ) i + 1 / ( Z i , Z 2 , . . . , Zi + Z i + 1 , . . . , Z„, Z n + i ) = O.

Note that it is sufficient to prove the theorem if dim V = 1 and the general case is just a consequence. So let us assume that dim V = 1. Note also that / given by Eq. (7.5) is a solution of Eq. (7.4), but we want to prove that each solution is included in Eq. (7.5). Let dim V = r and let Zj = (z»i,--- ,Zir)T (1 < i < n + 1). By differentiating the equation (7.6) partially with respect to z„+i,„ (1 < v < r) at Z„ + i = O, we obtain the following system of r equations " 5 — / ( Z i , Z 2 , . . . , Z n ) = — p„(Z 2 l Z 3 , . . . , Z n ) oznv n-\

^ ( - l ) i + V ( Z i , Z 2 , . . . , Z i + Zi+1,...,Zn_1)Zn)

+

(l
where d p.. / ( Z i , Z 2 , . . . , Z n _ i , Z )

=

{-l)npv{Z1,Z2,...,Zn-i)

z=o for Z = (ti,...,

£ r ). After integration of this system we obtain

/ ( Z i , Z 2 , . . . , Z n ) = R(Zi, Z 2 , . . . , Z n _i) — P ( Z 2 , Z 3 , . . . , Z n )

(7.7)

n-l

+ ^(_1)J

- P ( Z i , Z 2 , . . . ,Zj + Z j + i , . . . , Z „ _ i , Z n ) ,

where d dZn-l,v

P ( Z i , Z 2 , . . . , Z„_i) = p v (Z!, Z 2 , . . . , Z n _ x )

(1 < v < r),

and R is an arbitrary analytic function with respect to Z» (1 < i < n — 1). We write -R(Zi,Z 2 ) ... , Z n _ i ) = (—l) n _ P ( Z i , Z 2 ) . . . , Z n _ i ) + < 5 ( Z i , Z 2 , . . . , Z n _ i ) ,

80

Functional Equations with Operations between

Arguments

so that Eq. (7.7) becomes /(Zi,Z2,...,Zn)

=

(*P)(Zi,Z2>...,ZB_i,Zn)

+

<2(Zi,z 2 ,.. • , z „ _ i ) ,

(7.8)

with <5 analytic function of Zi, Z 2 , . . . , Z n _ i . If / ( Z i , Z 2 , . . . , Z„_x, Z„) is a solution of Eq. (7.4), then (*Q)(Z1,Z2>...>Z„_i,Zn) = 0 , because ( * o $ ) P = O. Thus <2 satisfies an equation of the form Eq. (7.4) with n replaced by n — 1. If n = 2, then Q(Z) = AZ. Otherwise we may assume that Q is given by an equality of the form Eq. (7.8) (n replaced by n — 1) and complete the proof by induction. • In other words, the general analytic solution of the functional equation (7.6) is given by /(Z1,Z2,...,Zn_i,Zn)

(7.9)

= (-ly-^Zi, z2>..., z„_!) - F(z2, z 3 ,..., zn) n-l

+ ^(_l)

l +

-P 1 (Zi,Z 2 ,...,Zj + Z j + i , . . . , Z n _ i , Z „ ) + L ( Z i , Z 2 , . . . ,Z„),

i=l

where P1 is an arbitrary analytic function and L is a linear in each argument mapping. As particular cases of the operator equation (7.4), we consider the following functional equations given in [S. Kurepa (1956); D. S. Mitrinovic (with the collaboration of P. M. Vasic, 1986)] . E x a m p l e 7.1

If n = 2, then the functional equation (7.6) becomes

/ ( Z i , Z 2 ) - / ( Z 2 , Z 3 ) + / ( Z i + Z 2 , Z s ) - / ( Z i , Z 2 + Z 3 ) = O. According to Eq. (7.9), the general analytic solution of this functional equation is given by / ( Z i , Z 2 ) = F(Zj + Z 2 ) - F(Zi) - F(Z 2 ) + L(Zi, Z 2 ). E x a m p l e 7.2

If n = 3, the functional equation (7.6) is

- / ( Z i , Z 2 , Z 3 ) - / ( Z 2 , Z 3 ,Z 4 ) + /(Zx + Z 2 , Z 3 , Z 4 )

Generalized Functional Equation

81

- / ( Z i > Z 2 + Z3,Z4) + / ( Z i , Z 2 , Z 3 + Z 4 ) = 0 . The general analytic solution of this equation is given by /(Z1,Z2)Z3) = F ( Z 1 + Z 2 , Z 3 ) + F(Z1)Z2) -F(ZUZ2 E x a m p l e 7.3

+ Z3) - F ( Z 2 , Z 3 ) +

L(Zi,Z2,Z3).

If n = 4, the functional equation (7.6) takes on the form

/ ( Z i , Z 2 , Z 3 , Z 4 ) — /(Z 2 ,Z 3 ,Z4,Z5) + / ( Z i + Z 2 , Z 3 , Z 4 , Z 5 ) — / ( Z i , Z 2 + Z 3 , Z 4 , Z 5 ) + / ( Z i , Z 2 , Z 3 + Z 4 , Z 5 ) - / ( Z i , Z 2 , Z 3 , Z 4 + Z 5 ) = O. According to Eq. (7.9), the general analytic solution of this functional equation is given by / ( Z i , Z 2 , Z 3 , Z 4 ) = F(Zi + Z 2 , Z 3 , Z 4 ) + F ( Z i , Z 2 , Z 3 + Z 4 ) - F ( Z 2 , Z 3 , Z 4 ) - F ( Z i , Z 2 + Z 3 , Z 4 ) - F ( Z i , Z 2 ) Z 3 ) + L(ZU Z 2 , Z 3 , Z 4 ), where F is an arbitrary analytic function, and L is an arbitrary linear in each argument mapping. This method for solving functional equations does not appear in the other references [J. Aczel (1966); M. Ghermanescu (1960); M. Kuczma (1968); G. Valiron (1945)]. 8

Generalized Functional E q u a t i o n

In this section one generalized complex vector functional equation is solved. Further some particular cases are given. Let V, V be arbitrary complex finite dimensional vector spaces. Now we will prove the following result obtained in [I. B. Risteski et al. (2000B)]. T h e o r e m 8.1

The general analytic solution of the functional equation

( - l ) n / n + i ( Z i , Z 2 , • • • ,Zn) — /n+2(Z2,Z3,- • • ,Zn+1) n

+

Z ) ( _ 1 ) i + 1 ^ ( Z l ' Z 2 ' ' • ' ' Zi + Z<+1' • • ' ' Z"> Z "+l) = ° i=l

(8.1)

82

Functional Equations with Operations between

Arguments

is given by / i ( Z i , Z 2 , - • • ,Z n ) = L ( Z i , Z 2 , • • • ,Z„)

(8.2)

i 7 i n (Zi,Z 2 , • • • ,Z„_i) - i?i(Z 2 ,Z 3 ,- • • , Z n )

+(-1)" n-l

+ 2^(~^y+

Jpii(Zi,Z 2 , • • • , Zj + Zj+i, • • • , Z n _ i , Z n ) ,

/»(Zi,Z 2 ) - • • ,Z„) = L ( Z i , Z 2 ) - • • , Z n )

(8.3)

+ ( - l ) n + 1 F i n ( Z 1 , Z 2 , • • • , Z n _ 0 - Hi(Z2, Z 3 , • • • , Z n ) + 2_>( _ 1)

•Ffc,i-i(Zi,Z 2) - • • ,Z/(. + Zfc+i, • • • , Z n _ i , Z „ )

*=i *=i n - lI

+ ^ ( - l j ' + ^ t t C Z i , Z 2 , • • • , Zfc + Z f c + 1 , • • • , Z n _x, Z n )

(2 < i < n)

/n+i(Zi,Z2, • • • ,Zn) = L(Zi,Z2)- • • ,Zn) +(—l) n

(8.4)

F n n ( Z i , Z 2 , • • • ,Z n _!) - i ? n + 1 ( Z 2 , Z 3 ) - • • ,Z„)

n-l

+ ^ (

—

I)*

-Fira(Zl,Z2,- • • , Zj + Zj + i, • • • , Z n _ i , Z n ) ,

i-1

/n+2(Zi,Z 2 , • • • ,Z„) = L ( Z i , Z 2 , • • • , Z n ) +(—l) n

(8.5)

i J n + i ( Z i , Z 2 , • • • , Z n _ ! ) - i J i ( Z 2 , Z 3 , • • • ,Z„)

n-l

+ 2-^,(~ 1)*

# i + i ( Z i , Z 2 ) - • • , Zj + Zj + i,- • • , Z n _ i , Z n ) ,

t=l

where the unknown functions / i , • • • , / n +2 ore mappings from V" mio V wMe Fjj (1 • V linear with respect to each argument.

Generalized Functional

Equation

83

Proof. Note that it is sufficient to prove the theorem if dim V = 1, and then the general case is just a consequence. So let us assume that dim V' = 1, and let dim V — m and Zj = (zn, • • • , Zim)T (1 < i < n + 1). We also note that the sum of two solutions of Eq. (8.1) is also a solution of this equation, because of its linearity. Moreover, fi(ZuZ2,

• • • ,Zn) = L(Zi,Z2, • • • ,Zn)

(1 < * < n + 2)

is a solution of this equation. This solution vanishes when we put Zj = O for any j € {1, • • • , n} what we will often do henceforth, that is why we just add it to the solution obtained below. In order to prove that Eqs. (8.2), (8.3), (8.4) and (8.5) give the general analytic solution of Eq. (8.1), first we differentiate Eq. (8.1) partially with respect to each coordinate zu, z\2, ••• , z\m and then in the result obtained we will put Zi = O. By using the notations wrfiC2i,Z2,---

,Zn)

z=o

= / M _i,„(Z 2 > Z 3 , • • • , Z„)

(2 < i < n + 1)

for Z = (ti, • • • , tm), we obtain "5

/l(Z2,Z3, • • • , Z „ + i ) = /lli/(Z2 + Z3,Z4, • • • , Z „ + i )

(8.6)

OZ2v

— /l2v(Z2, Z3 + Z4, • • • , Z„ + i) + • • • + (— l)"/l,n-l,i/(Z2, Z 3 , • • • , Z„_i, Z n + Z n + i ) + ( - l ) n + 1 / i n V ( Z 2 , Z 3 , • • • , Z„)

(1 Z3,-" ,Zn) = /Hv(Z2,Z3,--- ,Zn) 5z 2 ^

(l
Then by an integration of Eq. (8.6) we obtain / i ( Z 2 , Z 3 ) - • • ,Z„_|_i) = .Fii(Z 2 + Z 3 ,Z4, • - • , Z „ + i ) - F i 2 ( Z 2 , Z 3 + Z4,--- , Zn+1)H

(8.7)

h ( - l ) " i ; i , n - l ( Z 2 , - - - ^ n - i j Z n + Zn+x)

84

Functional Equations with Operations between

+ (—1)"

Arguments

F i „ ( Z 2 , Z 3 , • • • , Z „ ) — Hi(Z3,Z^,-

• • ,Zn+i)

where Hi(Z3, Z4, • • • , Z n + i ) is an arbitrary differentiable vector function. By a substitution of Eq. (8.7) into Eq. (8.1), we obtain J p l l ( Z l + Z 2 + Z 3 , Z 4 , - •• , Z n + i ) — i r i 2 ( Z 1 + Z 2 , Z 3 + Z 4 , Z 5 , • • • , Z n + i ) (8.8) -I

1- ( — l ) n f l n _ 1 ( z 1 + Z 2 , Z 3 , • • • , Z n _ i , Z „ + Z n + i )

+(—l)n

i r i n ( Z i + Z 2 , Z 3 , • • • , Z n ) — Hi(Z3,Zi,

• • • ,Z„)

+ (—l)"/n+l(Zi,Z2, • • • ,Zn) — /n+2(Z2,Z3, • • • ,Zn+i) n

+ ^(-l)

t +

/i(Zi,Z2,--- ,Zj_i,Zj + Zj+i,--- , Z n + i ) = O.

If we p u t Z2 = O in Eq. (8.8) a n d replace Z ; + i by Z ; (2 < i < n), we obtain f2(Z1,Z2,---

,Zn)

= FniZt

+Z2,Z3,-

• • ,Zn)

(8.9)

—F 2 2(Zi, Z 2 + Z3, • • • , Z n ) + • • •

+ ( — l)ni<2In-l(Zl,Z2)-- • , Z n _ 2 , Z n _ 1 + Z„)

+(-l)n

F2n(Zi,Z2,- • • ,Zn_i) — #2(Z2,Z3,- • • ,Z„),

where •F2i(Zi,Z 2 ) - • • , Z n _ ! ) = F i j ( Z i , Z 2 , - • • , Z „ _ i ) — / j + 1 ( Z i , 0 , Z 2 , • • • , Z n _ i ) (2 < i < n) and if2(Z2,Z3,---,Z„)=F1(Z2,Z3,---,Zn)+/„+2(0,Z2,Z3,---,Zn).

Generalized Functional Equation

85

By a substitution of Eq. (8.9) into Eq. (8.8), we obtain -•fl2(Zl+Z2,Z3 + Z4,Z5, • • • ,Z„_)_i) + Fi3(Zi+Z2,Z3,Z4+Z5, • • • , Z n + i ) - • • • + ( - l ) " ^ - ! ^ ! + Z 2 , Z 3 , • • • , Z„_ 1 ; Z n + Z n + i ) (—1)"

Fin(Zi

(8.10)

+ Z2,Z3, • • • ,Zn) + F22(Zi,Z2 + Z3 + Z ^ Z s , • • • , Z n + i ) — F23(Zi,Z2 + Z3,Z4 + Z5,-• • , Z n + i )

-I

f. (-1)™

F 2 ) n _ i ( Z i , Z 2 + Z 3 ) - • • , Z n _ i , Z n 4- Z n + i )

+ ( - l ) n + 2 F 2 n ( Z 1 , Z2 + Z 3 l Z 4 , • • • , Z n ) —ifi(Z 3 ,Z4,- • • , Z n + i ) + if 2 (Z 2 + Z 3 ,Z4,- • • , Z n + i ) + ( — l ) n / n + 1 ( Z i , Z 2 , • • • ,Z„) - / n + 2 ( Z 2 , Z 3 ) - • • , Z „ + i ) + ^ ( - l ) i + 1 / i ( Z i , Z 2 , - - - , Z i _ 1 ) Z i + Z i + 1 , - - - , Z n + Zn+1) = 0 . Now by induction we will prove that the first k (2 < k < n) unknown functions fj (1 < j < k) which satisfy the functional equation (8.1) can be represented in the following form Eq. (8.7), fj(Zlt

Z 2 , • • • , Z„) = F u - i C Z i + Z a , Z 3 , • • • , Z n )

(8.11)

—F2j_i(Zi,Z2 + Z 3 , • • • , Z n ) + • • • + ( - i y F i _ 1 , i _ 1 ( Z 1 , Z 2 , • • • . Z , - ! + Z,-, • • • ,Z„) +(-l)'+1Fjj(Z1,Z2,--.,Zj+Zj+1,---,Zn) + ( - l ) J + Fj i j + i(Zi,Z2,--- , Z J + i + Z j + 2 , - - - , Z n ) H

h (-l) n Fj ) 7 V _i(Zi,Z 2 ,-- • , Z n _ 2 , Z n _ i + Z n )

+(_l)«+iir.n(Zl,z2,

• • • , Z n _ i ) - Hj(Z2,Z3,

• • • ,Zn)

(2 < j < A),

86

Functional Equations with Operations between

Arguments

and the remaining n — k + 2 unknown functions will satisfy the following equation ( _ l ) * + i F l f c ( Z l + z2, Z 3 , • • • , Zk+1 + Zk+2, • • • , Z n + 1 ) + (-1)

+

(8.12)

i jl 1]fc+ i(Z 1 + Z 2 , Z 3 , - - - ,Zk+2 + Z/fe+3,--- , Z n + i )

+ • • • + (-l)nF1,n_1(Z1

+ Z a , Z 3 , • • • , Z„ + Z n + i )

+ (-l)"+1Fln(Z1+Z2,Z3,---,Zn) -[(-1) +(-1)

+

F2*;(Zi,Z2 + Z 3 , Z 4 ) - • • ,Z fc+ i + Z fc+ 2,- • • , Z n + i )

F2,fc + i(Zi,Z 2 + Z 3 , Z 4 ) - • • ,Zfc+2 + Z f c + 3 , • • • , Z n + i ) + • • • + ( - l ) " F 2 i „ _ i ( Z 1 , Z 2 + Z 3 , Z4, • • • , Z n + Z n + i ) +(-l)"+1F2n(Z1,Z2 + Z3,Z4,---,Zn)]

+ • • • + ( - l ) * - ^ - ! ) * - ^ ^ ! , • • • , Zk + Zk+1 + Zk+2r +(—1) H

Fk>k+1(Zi,

• • , Zn+1)

• • • , Zfc + Zfc + i,Z fc+2 + Zjfc+3, • • • , Z n + i )

+ (—l)™Ffc,„_i(Zi,- • • , Zfc + Zfc + i,Z fc+2 , • • • , Z n + Z n + i ) + ( ~ l ) n + ^*n(Zl,-"" >Zfc + Zfc+i,--- ,Z n )] —Hi(Z3, Z 4 , • • • , Z„ + i) + i? 2 (Z 2 + Z 3 , Z 4 , • • • , Z n + i ) 1- (—1) Hk(Z2,-

•• ,Zk + Zk+ir • • , Z „ + i )

+ ( - l ) n / n + i ( Z i , Z 2 , - - • ,Z„) - / „ + 2 ( Z 2 , Z 3 , - •• , Z n + i ) n

+ E (-ir +1 / i (Zi,---,z i +z i+1 ,---,z n+1 ) = o. Indeed, from Eqs. (8.7), (8.9) and (8.10) it follows that the statement is true for A; = 2. We will suppose that it holds for k (2 < k < n — 1). If we

Generalized Functional Equation

87

put Z f c + 1 = O and replace Z i + i by Z, (A; + 1 < i < n) in Eq. (8.12), we obtain / f c + i ( Z i , Z 2 , - - - , Z n ) = Fik(Zi

+ Z2,Z3,--- ,Zn)

(8.13)

—Fik ( Z i , Z 2 + Z 3 , • • • , Z n ) + • • • +(-1)

+1

-Ffefc(Zi,--- ,Zfc_i,Zfc + Zfc+i,-- • , Z n )

+[/fc+2(Zi, • • • ,Zfc,0,Zfc + i + Zfc+2,- • • , Z n ) —-Fi,fc+i(Zi + Z 2 , Z 3 , • • • ,Zfc,0,Zfc+i + Zft+2, • • • , Z n ) +-p2,fc+i(Zi,Z 2 + Z 3 , Z 4 , • • • , Zfc,0, Zfc+i + Z A + 2 , • • • , Z n ) - • • • +(—1) ~~ Fk-\,k+i{Zi,--

-,Zk-2,Zk-i

+ Z f c , 0 , Zk+i + Z f c + 2 , Z j ; + 3 , • • - , Z n )

+ ( - 1 ) Ffc,fc + i(Zi, • • • , Z fc , Zfc+i + Zfc+2, • • • , Z n ) ]

-[A+3(Zi,- • •, Zfc,o, Zfc+i,Zfe+2 + Zfc +3) - • • , z „ ) -•f1i,fc+2(Zi + Z 2 , Z 3 , • • • ,Zjt,0,Zfc + i,Zfc + 2 + Z f c + 3 ,- • • , Z n ) + - F 2 , f e + 2 ( Z l , Z 2 + Z 3 , Z 4 , - • • , Z f c , 0 , Z f c + i , Z j f e + 2 + Zfc + 3, • • • , Z „ ) — • • •

+ ( - 1 ) ~~ i 7 fc-i,fc+2(Zi, • • •,Zfc_2,Zfc_ 1 + Z j ; , 0 , Z f c + i , Z f c + 2 + Zk+3,- • - , Z n ) + ( - 1 ) Fk,k+2(Zi,--H

,Zfc + i,Zjt + 2 + Zfc + 3 ,-- • , Z n ) ]

h (—1)"~ ~~ [ / n ( Z i , - • • , Z j t , 0 , Z f c + i , - • • , Z „ _ 2 , Z „ _ i + Z n ) —Fi,n-i{Zi

+ Z 2 , Z 3 , • • • , Z/t, O , Zfc + i, • • • , Z n _ 2 , Z n _ ! + Z n )

+ - f 2 , n - i ( Z i , Z 2 + Z 3 , Z 4 , - • • , Z f c , 0 , Z f c + i , - - • , Z „ _ 2 , Z „ _ i + Zn) — • • • + ( - 1 ) ~ Ffe_i i n-i(Zi,---,Zfc_2,Zfc_ 1 + Z f e , 0 , Z f c + 1 , - - - , Z „ _ 2 , Z n _ i + Z n ) +(—1) F j t , „ _ i ( Z i , • • • , Z n _ 2 , Z n _ i + Z n ) ]

88

Functional Equations with Operations between Arguments + (-l)"

[/n+l(Zi,--- ,Zfc,0,Zfc+i,--- , Z n _ i )

—Fin(Zi

+ 7i2,2i3,- • • ,Zjfe,0,Z&+i, • • • , Z n _ i )

+-p2n(Zl,Z2 + Z 3 , Z 4 , - • • , Z ^ , 0 , Z f c + i , • • • , Z „ _ i ) — • • • + ( — l ) ~ -F*:-i,7i(Zi, • • • , Zfc-2, Zfc_i + Zfc, O , Zfc +1 , • • • , Z n _ i ) +(-l)*Ffcn(ZllZ2,.-->ZB_i)] + (-1)

[/n+2(Z2,-" ) Z f t , 0 , Z * + i , - - - , Z n )

+ i ? i ( Z 3 , - - - , Zfc,0, Zjfc+i,- • • , Z „ ) - i f 2 ( Z 2 + Z 3 , Z 4 , • • • ,Zk,0,Zk+i, H

_

+ (—1)

Hk-i(Z1}-

• • • ,Zn)

• • ,Zfc_2,Zfc_i + Zk,0,Zk+i,

• • • ,Zn)

+(-l)fc-1iJA(Z2,Z3,---,Zn)]. Now we define (-l)*+2*i+i,*+<(Zi, Z2, • • • , Z „ _ 0 =

(8.14)

-Ffc+i+i(Zi,• • • )Zjfc,0,Zfc+i,"- , Z „ _ i )

-•F\,fc+i(Zi + Z 2 , Z 3 , • • • , Zfc, O , Zjfe+i, • • • , Z n _ i ) + ^ 2 , A + i ( Z i , Z 2 + Z 3 , Z 4 , - • • ,Zfc,0,Zfc+i,- • • , Z n _ i ) 1- (—1)

F f c - i ^ + ^ Z i , - • • ,Zfc_2,Z;t_i + Zfc,0,Zfc + i, • • • , Z „ _ i )

+ ( - l ) * J F M + i ( Z 1 , Z 2 , - - - ,Z„_i)

(l
^ + i ( Z 2 , Z 3 , - - - ,Zn) = (-1) ~ [/n+2(Z2,--- ,Zfc,0,Zfc+i,--- ,Z„)

+tf1(z3,---,zfc,o, Zfc+i, • • • , Z n ) - f f 2 ( Z 2 4- Z 3 , Z 4 , - - - , Z f c , 0 , Z f c + i , - - - , Z n )

(8.15)

Generalized Functional Equation

-I

1- (—1)

89

i?fc_i(Z2,- • • ,Zjfc_2,Zfc_i + Zfc,0,Zfc + i, • • • , Z „ ) +(-l)fc-1Ffc(Z2)Z3,---,Zn)].

Finally Eq. (8.13) together with Eqs. (8.14) and (8.15) yield A + i ( Z i , Z 2 , • • • , Z n ) = .Fifc(Zi + Z 2 , Z 3 , • • • , Z ra )

(8.16)

--p2fc(Zi,Z2 + Z 3 , Z 4 , • • • , Z n ) + • • • +1

+(-1) +(-1) +(—1)

+2

- F * * ( Z i , - - - ,Zjt_i,Zfc + Zk+i,---

,Zn)

Ffc+i ] fc + i(Z 1 , • • • , Zjt, Zk+i + Zk+2, ••• , Z „ )

i r fc + 1 | fc + 2 (Zi, • • • ,Zfc + i,Zfc + 2 + Zfc +3 , • • • , Z „ )

-I— . 4. ( — l ) " i ^ + l n _ 1 ( Z i , • • • , Z n _ 2 , Z n _ i + Z n ) + +(-l)n

irfc+i!n(Zi,Z2,- • • , Z n _ i ) — iffc+1(Z2,Z3, • • • ,Zra).

By a substitution of Eq. (8.16) into Eq. (8.12), after elimination of Fik,F2k,---

,Fkk

we obtain

( - l ) f c + 2 F i ^ + i ( Z i + Z 2 , Z 3 , - - • ,Zfc + i,Zfc + 2 + Z/t+3,--- , Z n + i ) + . . . + (-l)nFltn.1(Z1

(8.17)

+ Z 2 , Z 3 , • • • , Z„ + Z n + i )

+(-l)"+1Fln(Z1+Z2,Z3)---,Zn) -[(-1) -I

F2tk+l(Zi,Z2

+ Z 3 , Z 4 , - • • ,Zfc + 2 + Zfc + 3, • • • , Z n + i )

-I- ( —l) n i*2,„_i(Zi,Z2 + Z3,Z4, • • • , Z „ + Z n + i ) + ( - l ) n + 1 F 2 „ ( Z 1 , Z 2 + Z 3 , Z 4 , • • • , Z„)] + • • •

+ ( - l ) f c + 1 [ ( - l ) ' : + 2 ^ M + i ( z i . • • • > Z f c _i, Z * + Z f c + 1 ) Z f c + 2 + Z f c + 3 > • • • , Z n + 1 ) -I

1- ( — l ) " F / t i n _ i ( Z i , • • • , Z k - i , Z k + Zk+i,Zk+2,

••• ,Zn + Z n + i )

90

Functional Equations with Operations between Arguments + ( - l ) n + 1 F f e n ( Z i , - - • , Zjfe_i,Zjfe + Zk+i,---

,Z„)]

+(-l)*[(-l) fc + 2 J F fc+1 , fc+1 (Z i ) " " ' ,Zfc,Zfc + i+Zfc + 2 + Zfc + 3, • • • , Z n + i ) H + ( — l ) " F f t + 1 ] „ _ 1 ( Z 1 , - • • ,Zfc,Zfc + i 4- Zfc + 2 ,Zfc + 3 , • • • , Z n + Z n + i ) + ( - l ) n + 1 F i + i , n ( Z i , . - • • , Zfc, Z A + I + Zfc+2, • • • , Z n ) ]

— i f i ( Z 3 , Z4, • • • , Z n + i ) + i / 2 ( Z 2 + Z 3 , Z 4 , - • • , Z n + i ) h ( - 1 ) i7fc(Z 2 ,--- ,Zjfc_i,Zfc + Zfc+i,--- , Z n + i ) +(-1)

+

-ffjt+i ( Z 2 , • • • ,Zfc,Zfc +i 4- Z * + 2 , - - - , Z n + i )

+ (— l ) n / n + l ( Z i , Z 2 , - • • , Z n ) — /n+2(Z2,Z3, • • • , Z n + i ) n

+ 5 3 (-i) i + 1 / i (z 1 ,---,z i _ 1 ,z i + z i + 1 , . . - , z n + 1 ) = o . i—k+2

It follows from Eqs. (8.16) and (8.17) t h a t t h e statement also holds forfc+ 1. Hence, Eqs. (8.11) a n d (8.12) hold for 2 < fc < n. If we substitute fc = n in Eq. (8.12), we obtain {-l)n+lFln{Z1

+ Z2, Z3, • • • , Z n )

(8.18)

+ ( - i r + 2 F 2 n ( Z 1 , Z 2 + Z 3 , Z 4 , - - - , Z n ) + --+(—1) " Fn-itn(Zi, • • - , Z n _ 2 , Z n _ i + Z„) + (—1) n . F n n ( Z i , Z 2 , • • •, Z n _ i ) —i?l(Z3,Z4, • • • , Z n + i ) +i?2(Z2 + Z3,Z4, • • • , Z n + i ) — • • •

+(-l)"tf n (Z 2 ,---, Zn—1) Z n + Z n + i ) + ( — l ) n / n ^ _ i ( Z i , Z 2 , • • •, Z n ) —

/n+2(Z2,Z3, • • • , Z n + i ) = O .

Now if we substitute Z n + i = O in Eq. (8.18), we obtain / n + i ( Z i , Z 2 , - - - , Z n ) = Fi„(Zi + Z2,Z3,--- , Z n )

(8.19)

Generalized Functional Equation

91

—-p2n(Zl, Z 2 + Z 3 , Z4, • • • , Z „ ) + • • • +(—l)"Fn_i]7l(Zi, • • • , Zn_2, Zn_i + Zn) + ( — 1)"

i?nn(Zl,Z2,- • • ,Z„_i) — ifn+i(Z2,Z3, • • • , Zn)

where # n + i ( Z 2 , Z3, • • • , Z n ) = ( - l ) n

[ifi(Z3,Z4,-- • , Z „ , 0 )

—.ff2(Z2 + Z 3 , Z 4 , • • • , Z „ , 0 ) -I

1- ( — l ) n H n - i ( Z 2 , • • • , Z n _ 2 , Z n _ i + Z n , O )

+ ( _ 1 ) " + 1 F „ ( Z 2 ) Z3, • • • , Zn) + / n + 2 ( Z 2 , Z3, • • • , Z„, O)]. By a substitution of Eq. (8.19) into Eq. (8.18) we obtain — i ? l ( Z 3 , Z 4 , • • • , Z n + i ) + H2(Z2 + Z 3 , Z 4 , - • • , Z n + i ) + (—l)"if„(Z2, Z 3 , • • • , Z n _ 1 ; Z ra + Z n + 1 ) + ( ~ l ) n + -ffn+i(Z2,Z 3 ) - • • , Z n ) - / n + 2 ( Z 2 , Z 3 ) - • • , Z n + i ) = O , i.e., / n + 2 ( Z i , Z 2 , • •• , Z „ ) = i?2(Zi + Z 2 , Z 3 , - • • , Z „ )

(8.20)

-i/3(Zi,Z2 + Z3,Z4,--- ,Zn) H +(-l)nHn(Zi, +(—l)n

Hn+\(Zi,Z2,

• • • , Zn_2, Zn_! + Zn) • • • , Z n _ i ) - .Hi(Z2,Z3, • • •

,Zn).

By Eqs. (8.7) and (8.11) for k = n , Eq. (8.19) and (8.20) t h e theorem is proved. •

92

Functional Equations with Operations between

Arguments

This theorem is a generalization of Theorem 7.2. Indeed, in the special case when / ; = / (1 < i < n + 2) we obtain Theorem 7.2. As particular cases of the functional equations Eq. (8.1) we consider the following examples. E x a m p l e 8.1 Pexider form

If n = 1, then the functional equation (8.1) takes the

- / 2 ( Z i ) - / 3 (Z 2 ) + / ! ( Z ! + Z 2 ) = O. The general analytic solution of the equation is given by

/i(Z) / 2 (Z) h{Z)

= CZ+A + B, = CZ+A, = CZ+B,

where C is a constant matrix and A and B are constant complex vectors. E x a m p l e 8.2 h(ZuZ2)

If n = 2, then the equation (8.1) becomes - / 4 ( Z 2 , Z 3 ) + / i ( Z ! + Z 2 ,Z 3 ) - / 2 ( Z i , Z 2 + Z 3 ) = O.

According to the theorem, the general analytic solution of this functional equation is given by / i ( Z i , Z 2 ) = F n ( Z i + Z 2 ) - F 1 2 (Zi) - H1(Z2) + L(Z 1 ,Z 2 ), /2(Z1,Z2) = F 1 1 ( Z 1 + Z 2 ) - F 2 2 ( Z 1 ) - F 2 ( Z 2 ) + i(Z1,Z2), / 3 ( Z ! , Z 2 ) = F 12 (Z X + Z 2 ) - F22(Z1) - tf3(Z2) + L ( Z i , Z 2 ) , / 4 ( Z i , Z 2 ) = ff2(Zi + Z 2 ) - ff3(Z!) - i7 x (Z 2 ) + L(Zi, Z 2 ). E x a m p l e 8.3 If n = 3, the general analytic solution of the functional equation (8.1), which takes the form -h{Zu

Z 2 , Z3) - / 5 ( Z 2 , Z 3 , Z4) + A ( Z ! + Z 2 , Z 3 , Z 4 )

- / 2 ( Z l l Z 2 + Z 3 ,Z 4 ) + / 3 ( Z 1 , Z 2 , Z 3 + Z 4 ) = 0 ,

Generalized Functional

Equation

93

is given by /i(Zi,Za,Z3)

=

r

F11(Z1+Z2,Z3)~F12(Z1,Z2

+ Z3)

+i i 3 (Zi,Z2)

— H\(Z2,Z3)

/2(Zl,Z 2 ,Z3)

=

i r n ( Z i + Z2,Zs) — F22(Zi,Z2 + Z3)

+-F23(Zl,Z2)

-

H2(Z2,Z3)

/3(Zl,Z2,Z3)

=

-Fl2(Zi + Z2,Z 3 ) — F22(Z 1; Z2 + Z3)

+-p33(Zl,Z 2 )

-

i?3(Z2,Z 3 ) + L ( Z 1 , Z 2 , Z 3 ) ,

/4(Zl,Z2,Z3)

=

F\3(Zi +Z2,Z3)

+F33(Zi,Z2)

-

i?4(Z2,Z3) + £ ( Z 1 ; Z 2 , Z 3 ) ,

/5(Zi,Z2,Z3)

=

H2(Zi+

+ff 4 (Zi,Z 2 )

-

H1(Z2,Z3)

E x a m p l e 8.4

4- L(Z 1 ,Z2,Z3),

+

L(Zx,Z2,Z3),

- F23{Zi,Z2

+ Z3)

Z2,Zz) ~ Hs(Zi,Z2+ +

Z3)

L(Z1,Z2,Z3).

If n = 4, the giyen functional equation (8.1) becomes

/5(Zl,Z2,Z 3 ,Z4) — /6(Z2,Z3,Z4,Z 5 ) +

/ l ( Z l + Z2,Z3,Z4,Z 5 ) - / 2 ( Z i , Z 2 + Z3,Z4,Z 5 )

+

/ 3 ( Z 1 > Z 2 , Z 3 + Z 4 I Z B ) - / 4 ( Z I > Z 2 , Z 3 , Z 4 + ZB) =

The general analytic solution of this equation is /l(Zl,Z2,Z3,Z4) — Fu(Zi +

+ Z2,Z3,Z4) — Fi2(Zi,Z2 + Z3,Z4)

-Fl3(Zl,Z2,Z3 + Z4) — Fi4(Zi,Z2,Z3)

— iJi(Z2,Z 3 ) Z4) + L(Z 1 ,Z2,Z 3 ,Z4), /2(Zi,Z2,Z3,Z 4 ) r

=

i n ( Z i + Z2,Z 3 ,Z4) - F22(Zi,Z2

4- Z 3 ,Z4)

+

^23(Zl,Z2,Z3 + Z4) — F24(Zi,Z2,Z3)

— i?2(Z2,Z 3 ,Z 4 ) + L(Z 1 ,Z 2 ,Z 3 ,Z4),

0.

94

Functional Equations with Operations between Arguments

/3(Zi,Z2,Z3,Z 4 ) =

-Fl2(Zi + Z2,Z3,Z4) — F22(Zi,Z2 + Z3,Z.4)

+

^33(Zl,Z2,Z3 + Z4) — i r 34(Zi,Z2,Z3)

— H3(Z2,Z3,Z4)

+ L(Zi,Z2,Z3,Z 4 ),

/4(Zi,Z 2 ,Z3,Z4) =

-Fi3(Zi + Z 2 ,Z3,Z 4 ) - F 2 3(Zi,Z 2 + Z 3 ,Z 4 )

+

^33(Zi,Z 2 ,Z 3 + Z4) — -F44(Zi,Z 2 ,Z 3 )

— i?4(Z 2 ,Z 3 ,Z4) + L(Zi,Z 2 ,Z 3 ,Z4),

/5(Zl,Z2,Z3,Z 4 ) =

-Fl4(Zl + Z2,Z3,Z4) — F24(Zi,Z2 + Z3, Z4)

+

-p34(Zl,Z2,Z 3 + Z4) — F44(Zi,Z2,Z3)

— i?5(Z 2 , Z3, Z4) + L(Zi, Z2, Z3,Z4),

/6(Zi,Z 2 ,Z 3 ,Z4) =

H2(Zi + Z 2 ,Z 3 ,Z4) - ^ 3 ( Z 1 , Z 2 + Z 3 ,Z 4 )

+

-ff4(Zi,Z2,Z 3 + Z 4 ) — iJ 5 (Zi,Z2,Z 3 )

— i/i(Z2,Z 3 ,Z4) + L(Zi,Z2,Z3,Z 4 ). All the particular cases considered here generalize the results given in [S. Kurepa (1956); D. S. Mitrinovic and D. Z. Djokovic (1961); D. Z. Djokovic (1962); M. Hosszu (1963)].

9

Simple Functional Equations

Let V be a finite dimensional complex vector space and let us consider a mapping f : V2 >-> V. Throughout this section Zj (1 < i < 4p) are vectors in V. We assume that Z; = (zn(t), • • • ,Zin(t))T, where Zij(t) (1 V satisfying L(U!+U2,V)

=

L(Ui,V) + L(U2,V),

L(U,V!+V2)

=

L(U,Vi) + L(U,V2).

Now we will give the results obtained in [I. B. Risteski et al. (to appear A)]. T h e o r e m 9.1 The general continuous solution of the complex vector functional equation /(Z 1

)

-

/ ( Z 2 + --- + Z n + 1 , Z 1 )

+ / ( Z i + --- + Z„_i,Z„)

-

/ ( Z 2 + --- + Z n , Z „ + i ) + ---

+ / ( Z i + Z 2 ) Z 3 ) - / ( Z 2 + Z3,Z 4 )

+

/(Zi,Z2)-/(Z2>Z3) = 0

+

--- +

(9.1)

is #it;en 6j/ / ( U , V) = F ( U + V) - F(U) - F(V) + L(U, V) + L ( V , U ) ,

(9.2)

where F is an arbitrary complex vector function with values in V and L € B(V). Proof.

If we put Zi = O, then Eq. (9.1) reduces to / ( 0 , Z 2 ) = / ( Z 2 + -.- + Z n + i , 0 ) ,

from which we conclude that / ( O , U) = / ( V , O) = / ( O , O) = constant complex vector.

(9.3)

By putting Z 4 = Z 5 = • • • = Z n + i = O, the equation (9.1) according to Eq. (9.3) takes the following form /(Zx + Z2> Z 3 ) - / ( Z 2 + Z 3 , ZO + / ( Z 1 ; Z 2 ) - / ( Z 2 , Z 3 ) = O.

(9.4)

Rephrasing a result in [I. B. Risteski et al. (1999)], we see that Eq. (9.2) is the general continuous solution of the last equation. By a direct calculation we can verify that the function Eq. (9.2) satisfies the functional equation Eq. (9.1). •

96

Functional Equations with Operations between

Arguments

T h e o r e m 9.2 The general continuous solution of the complex vector functional equation / ( Z ! + Z 2 , Z 3 + Z4)

+

f(Z2 + Z3,Z4 + Z1)

(9.5)

+/(Z3 + Z4,Z1+Z2)

+

/ ( Z 4 + Z 1 , Z 2 + Z3)

- / ( Z 1 , Z 2 + Z 3 + Z4)

-

/ ( Z 2 , Z 3 + Z4 + Z1)

- / ( Z 3 , Z 4 + Z 1 + Z2)

-

/ ( Z 4 ) Z 1 + Z 2 + Z3)

- / ( Z 1 + Z 2 , Z 3 ) - / ( Z 2 + Z 3 ,Z 4 )

-

f(Z3 + Z4,Z1)-f(Z4

+/(Z1,Z4) + /(Z2,Z1)

+

/(Z3,Z2) + /(Z4,Z3) = 0

+

Z1,Z2)

is jiuen 61/ £/ie following formula / ( U , V) - F ( U + V) - F(V) + G(U) + L(U, V) + L(V, U),

(9.6)

where F and G are arbitrary complex vector functions with values in V, andL£B(V). Proof.

If we put Z 2 = U and Zi = Z 3 = Z 4 — O into Eq. (9.5), we have / ( O , U) = / ( O , O) = constant complex vector.

(9.7)

By putting Zi = U, Z 2 = V and Z 3 = Z 4 = O the equation (9.5) takes the following form [/(U, V) - / ( U , O)] - [/(V, U) - / ( V , O)] = O.

(9.8)

The function 5(U,V)

= /(U,V)-/(U,0)

(9.9)

satisfies the relations fl(U,V)=5(V,U)

and

g(UtO)

=g(0,U)

= 0.

(9.10)

Without difficulty one can verify that the function g(U, V) also satisfies

Simple Functional

Equations

97

the equation (9.5) g{Z1 + Z2,Z3 + Z4)

+ g(Z2 + Z3,Z4 + Z1)

+g(Z3+Z4,Z1+Z2)

+ g(Zi + Z1,Z2 + Z3)

-g(ZuZ2

+ Z3 + Z4)

-

g(Z2,Z3 + Z4 + Z1)

-g(Z3,Z4

+ Zx + Z2)

-

g(Z4,Zi

+ Z3,Z4)

-

g{Z3 + Z4,Z{) - g{Z4 +

+

g(Z3,Z2)

-g{Z1 + Z2,Z3)-g(Z2

+g(Z1,Z4)+ff(Z2,Z1)

+ Z2 + Z 3 ) + g(Z4,Z3)

ZuZ2)

= O.

If we put Z4 — O, we get g{Z2 + Z 3 , Zi) - g(Z2,Z3 + Zx) + g(Z3,Z2)

- g{Z3,Zx)

= O.

(9.11)

Consequently, the function g(V, V) in the form 5(U,

V) = F ( U + V) - F ( U ) - F ( V ) + L(U, V) + L(V, U)

(9.12)

satisfies the functional equation (9.11). Prom Eqs. (10.9) and (10.12) it follows that /(U,V)

=

F ( U + V) - F ( U ) - F ( V )

+

/ ( U , 0 ) + L(U,V) + L ( V , U ) ,

(9.13)

which means that / ( U , V) has the form Eq. (9.6). On the other hand, every function of the form Eq. (9.6) satisfies the complex vector functional equation (9.5). • Further let us assume that / : V3 *-¥ V. T h e o r e m 9.3

The general solution of the functional equation

/ ( Z x + Z2 + • • • + Z p , (9.14) Zp+l + Zp+2 + • • • + Z2p, Z 2p +i + Z2p_|_2 + • • • + Z 4 p )

+

/(Z 2 + Z3 + --- + Z p+1 , Zp+2 + Z p + 3 + • • • + Z 2 p +1 , Z 2p +2 + Z 2p+ 3 +

+

f- Zj ) + • • •

/ ( Z 4 p + Z 1 + --- + Zp_ 1 , Zp + Zp+i + • • • + Z 2 p _i, Z 2 p + Z 2p +i + • • • + Z4p_i) = O

is given by /(U,V,W)

= +

F 1 ( U , V + W ) - F 1 ( V , W + U) F 2 (U + V , W ) - F 2 ( W , U + V ) ,

(9.15)

98

Functional Equations with Operations between

Arguments

where Ft and F2 are arbitrary functions with values in V.

Proof. By a direct calculation we verify that Eq. (9.15) is a solution of the equation (9.14). Now we assume that / is some solution of the equation (9.14) and we will verify that / has the form Eq. (9.15). If we put Zi = U, Z p+ i = V, Z 2p+ i = W and Z, = O for i ^ l,p + 1,2p + 1 (1 < i < 4p) into Eq. (9.14), then we obtain p[/(U, V, W) + /(O, U, V + W) + /(W, O, U + V) + /(V, W, U)] = O or in a more convenient form, / ( U , V , W ) + / ( V , W , U ) = - / ( W , 0 , U + V ) - / ( 0 , U , V + W ) . (9.16) By permutations of the vectors U, V and W in the above relation, we get - / ( V , W , U ) - / ( W , U , V ) = / ( U , 0 , V + W ) + / ( 0 , V , W + U ) , (9.17) / ( W , U , V ) + / ( U , V , W ) = - / ( V , 0 , W + U ) - / ( 0 , W , U + V ) . (9.18) If we add together the last three Eqs. (9.16), (9.17) and (9.18), we obtain 2/(U,V,W)

= + -

/ ( U , 0 , V + W ) - / ( V , 0 , W + U) / ( 0 , V , W + U ) - / ( 0 , U , V + W) / ( W , 0 , U + V ) - / ( 0 , W , U + V).

(9.19)

Formula Eq. (9.19) implies 2 / ( W , 0 , U + V) = / ( W , 0 , U + V) + / ( 0 , 0 , U + V + W) - / ( U + V.O.W)

-

/ ( 0 , 0 , U + V + W) / ( O . W . U + V) / ( 0 , U + V,W)

or in a more convenient form =

/ ( W , 0 , U + V ) - / ( 0 , W , U + V) /(U + V , 0 , W ) + / ( 0 , U + V,W).

(9.20)

Functional Equations with Several Unknown Functions

99

If we use Eqs. (9.19) and (9.20), we find 4/(U,V,W) = 2[/(U,0,V + W)

-

/ ( V . O . W + U)]

+ 2 [ / ( 0 , V , W + U)

-

/ ( O . U . V + W)]

+/(U + V,0,W)

-

/ ( W . O . U + V) / ( 0 , W , U + V).

+ / ( 0 , U + V,W) If we put *i(U,V)

=

i[/(U,0,V)-/(0,U,V)],

F2(U,V)

=

i[/(U,0,V) + /(0,U,V)],

we obtain the form of the function / ( U , V, W).

10

•

Functional Equations with Several Unknown Functions

In this section some functional equations with several unknown functions and one operation of addition between arguments are solved. These results are obtained in [I. B. Risteski et al. (to appear A)]. Let V, V be finite dimensional complex vector spaces. Denote by V° the subspace of all real vectors in V (thus V = V° + iV°) and let £(V°, V ) be the space of all linear mappings V° H-> V . Lemma 10.1 Let f be a mapping V2 •-> V. Then the general continuous solution of the functional equation / ( Z i , Z2 + Z3 + Z 4 ) - f(Zi + Z 2 , Z 3 + Z 4 ) -

(10.1)

/(Zx + Z 4 , Z 2 + Z 3 ) + / ( Z i + Z 2 + Z 4 , Z 3 ) = O

is given by f(ZuZ2)

=

F 1 ( Z 1 + Z 2 ) R e Z 1 + F 2 ( Z 1 + Z 2 )ImZ 1

+

ff3(Zi+Z2)

(10.2)

+ L(Z1,Z2)-L(Z2,Z1),

where Hi : V (->• £(V°, V ) (i = 1,2), H3 : V H» V are arbitrary continuous complex functions, and L is a mapping V2 i->- V linear in each argument.

Functional Equations with Operations between

100

Arguments

Proof. First we note that / ( Z i , Z 2 ) = L(ZX, Z 2 ) - L ( Z 2 , Zi) is a solution of Eq. (10.1) which vanishes when we put any of its arguments equal to zero. Thus we can just add this solution to the one obtained below. If we introduce a new unknown function by f(Z1,Z2)=g(Z1,Z1+Z2),

(10.3)

then equation (10.1) becomes g(Zi, Z x + Z 2 + Z 3 + Z 4 ) - g(Zx + Z 2 , Z x + Z 2 + Z 3 + Z 4 ) -

g(Z1+Z4,Z1+Z2+Z3

+ Z4)

(10.4)

+

0 ( Z 1 + Z 2 + Z 4 , Z i + Z 2 + Z3 + Z 4 ) = O.

For Zi = O and Z 2 + Z 3 + Z 4 = T, Eq. (10.4) takes on the form g(Z2 + Z 4 , T) - fl(Z2, T) - 5 ( Z 4 ) T) + g(0, T) = O, i.e., h{Z2 + Z 4 , T) = h(Z2, T) + h(Z4, T),

(10.5)

where we have denoted /l(ZljT) = <7(Z1,T)-fl(0,T).

(io.6)

Therefore, the function h is additive in the first argument and the general continuous solution of the functional equation (10.5) is determined by h(ZuZ2)

= Hx(Z2)Re Zj. + H2(Z2)Im

Zu

(10.7)

where Hi (i = 1,2) are arbitrary continuous complex functions with values in£(V°,V). On the basis of the expressions Eqs. (10.7) and (10.6), for the function g we obtain P(Zi,Z2)

= ffi(Z2)Re Zi + tf2(Z2)Im Z1+H3(Z2),

(10.8)

where we put g(0, Z 2 ) = ^r 3 (Z 2 ). From Eqs. (10.8) and (10.3) there follows Eq. (10.2). In the subsequent Theorems 10.2-10.4 we assume that the mappings / i : V2

H»

V

and

ft : V3

H->

V'

(2 < i < n - 1).

•

Functional Equations with Several Unknown Functions

T h e o r e m 10.2

101

The functional equation

/ 1 ( Z 1 , Z 2 + Z3 + Z 4 ) + / 2 ( Z 2 , Z 3 , Z 4 + Z 1 ) + / 3 ( Z 3 , Z 4 ) Z 1 + Z 2 ) = O (10.9) has a general continuous solution given by /2(ZllZ2,Z3)

=' -f1(Z3,Z1

f3{Z1,Z2,Z3)

=

+ Z2) + F(Z2,Z3

+ Z1),

(10.10)

/ 1 ( Z 2 + Z 3 , Z 1 ) - / 1 ( Z 3 , Z i + Z 2 ) - F ( Z 1 , Z 2 + Z 3 ),

where F is an arbitrary continuous function V2 •-> V , and the continuous function f\ satisfies the functional equation (10.1). Proof.

If we put Z 4 = O into Eq. (10.9), we get / 2 ( Z 2 , Z 3 ) Z 1 ) = F(Z3,Z 1 + Z 2 ) - / 1 ( Z i , Z 2 + Z 3 ),

(10.11)

where F ( Z 1 , Z 2 ) = - / 3 ( Z 1 , 0 , Z 2 ) . For Z 2 = O the functional equation (10.9) in view of the relation Eq. (10.11) becomes / 3 ( Z 3 , Z 4 , Z 1 ) = / 1 ( Z 4 + Z 1 , Z 3 ) - / 1 ( Z 1 , Z 3 + Z 4 ) - J F ( Z 3 , Z 4 + Z 1 ) . (10.12) From Eqs. (10.11) and (10.12) there follows Eq. (10.10). By putting Eqs. (10.11) and (10.12) into Eq. (10.9), we deduce that the function / i must satisfy the equation (10.1). • T h e o r e m 10.3 tion

The general continuous solution of the functional equa-

/ 1 ( Z 1 , Z 2 + Z 3 + Z 4 + Z5)

+

/ 2 ( Z 2 , Z 3 ) Z 4 + Z 5 + Z1)

(10.13)

+/3(Z3,Z4,Z5 + Z1+Z2)

+

/ 4 ( Z 4 , Z 5 , Z 1 + Z 2 + Z3) = 0

is determined by f2(ZuZ2,Z3)

=

- / 1 ( Z 3 , Z 1 + Z 2 ) + F 1 ( Z 2 ) Z 3 + Z 1 ),

(10.14)

/3(Z1,Z2,Z3)

=

AtZa + Z ^ Z O - A ^ Z i + Za)

-

Fi(Zi,Za + Z 3 ) + Fa(Z 2 ,Z3 + Zi),

/4(Zi,Z2,Z 3 )

=

/ i ( Z 2 + Z 3 , Z 1 ) - / 1 ( Z 3 ) Z 1 + Z 2 ) - F 2 ( Z 1 ) Z 2 + Z 3 ),

where Fi (i = 1,2) are arbitrary continuous functions V2 i-> V , and / i is the general continuous solution of the equation (10.1).

102

Functional Equations with Operations between

Proof.

Arguments

If we put Z 4 = Z5 = O into Eq. (10.13), we obtain /2(Z2,Z3,Z1) = - / 1 ( Z 1 , Z 2 + Z 3 ) + , F 1 ( Z 3 , Z i + Z 2 ) ,

(10.15)

where F 1 (Z 1 ,Z 2 ) = - / 3 ( Z 1 , 0 , Z 2 ) - / 4 ( 0 , 0 , Z 1 + Z 2 ). For Z5 = Z 2 = O the functional equation (10.13) by virtue of the expression Eq. (10.15) becomes /3(Z3,Z4,Z1)

=

/ 1 ( Z 4 + Z 1 , Z 3 ) - / i ( Z 1 , Z 3 + Z4)

-

F 1 ( Z 3 , Z 4 + Z 1 ) + F 2 ( Z 4 , Z i + Z 3 ),

(10.16)

where we have denoted F 2 (Zi, Z 2 ) = —/ 4 (Zi, O, Z 2 ). By putting Z 2 = Z 3 = O into Eq. (10.13), on the basis of Eqs. (10.15) and (10.16) we obtain /4(Z4,Z5,Z1)

=

^(Zs + Z^Z,)

(10.17)

-

/ 1 ( Z 1 , Z 4 + Z 5 ) - F 2 ( Z 4 , Z 5 + Z 1 ).

From Eqs. (10.15), (10.16) and (10.17) there follows Eq. (10.14). By setting the functions fi (i = 2,3,4) determined by Eq. (10.14), i.e., Eqs. (10.15), (10.16), (10.17), into Eq. (10.13), we obtain the equation /1(Z1,Z2+Z3 + Z4+Z5)

-

/ 1 (Z 4 + Z 5 + Z 1 , Z 2 + Z 3 )

+ / 1 ( Z 4 + Z5 + Z1 + Z 2 ,Z 3 )

-

/ 1 (Z 5 + Z1 + Z 2 ,Z 3 + Z 4 )

- / 1 ( Z 1 + Z 2 + Z 3 ,Z 4 + Z 5 )

+

/ 1 (Z 5 + Z 1 + Z 2 + Z 3 ,Z 4 ) = 0 .

For Z5 = O this equation becomes just Eq. (10.1). Theorem 10.4 tion

(

z

•

The general continuous solution of the functional equa-

n

\

i>EZ; i=2

n-l

+

/

/

n+i-l

E ' ' ( Zi,Zi+i, £ i=2

\

j=i+2

\

Z,

=0, J

(10.18)

Functional Equations with Several Unknown

Functions

103

where n > 5 and Z n + ; = Zj for i = 1, • • • ,n — 2, is /2(Z1;Z2,Z3)

=

- / i ( Z 3 , Z 1 + Z 2 ) + F i ( Z 2 , Z 3 + Z 1 ), (10.19)

MZi.Za.Zs)

=

/ 1 ( Z 2 + Z 3 , Z 1 ) - / 1 ( Z 3 , Z 1 + Z2)

-

F i _ 2 ( Z 1 , Z 2 + Z 3 ) + J F i _ 1 ( Z 2 , Z 3 + Z1) (3 < i < n - 2),

/„-i(Zi,Z2,Z 3 )

=

fi(Z2 + Z3,Z1)-f1(Z3,Z1

+ Z2)

— ^ n - 3 ( Z l , Z 2 + Z 3 ), where Fi (1 < i < n — 3) are arbitrary continuous functions V2 •-»• V , and t/ie continuous function / i satisfies the functional equation (10.1). Proof. For n = 5 this theorem holds on the basis of Theorem 10.2. Now, we will assume that this theorem holds for some fixed n > 5. Then, Eq. (10.18) for n 4-1 has the form

(

n+l

z

n

\

f

n+i

\

i >j=2E z ; ) + i=2 E - M\ z*«z«+i> j =Ei + 2 z i / = °

(10-2°)

If we put Z„ + i = O into Eq. (10.20), then we will have

(

n

z

\

i »j =E2 z 0/

n-2

/

n+i-2

\

z z z + E ^ *> «+i' E 0/ i=2 \ j=i+2 n 2 I ~ \

(10-21)

+ 9n-l I Z n _i,Z„, 2 ^ Zj J = O,

V

J=I

/

where 5 n - l ( Z i , Z 2 , Z 3 ) = fn-i (Zi, Z 2 , Z 3 ) + / „ ( Z 2 , 0 , Z 3 + Zi).

(10.22)

On the basis on the inductive hypothesis, the general continuous solution of the functional equation (10.21) is given by Eq. (10.19), where fn-i is replaced by gn-iNow introduce the function Fn_2(Z1,Z2) = - / n ( Z i , 0 , Z 2 ) .

104

Functional Equations with Operations between

Arguments

By virtue of the expressions for the solution Eq. (10.19) and equality (10.22) we obtain /„_1(Z1,Z2,Z3)

=

/i(Z2+Z3,Z1)-/1(Z3,Z1+Z2)

(10.23)

— i*n- 3 (Zi,Z 2 + Z 3 ) + .F n _ 2 (Z 2 ,Z 3 + Zi). By setting Eq. (10.23) and the functions fc (1 < i < n — 2) determined by Eqs. (10.19) into Eq. (10.20), based on the hypothesis that the continuous function /1 satisfies the equation (10.1), we obtain /„(Zx, Z 2 ) Z 3 ) = A(Z 2 + Z 3 , Zi) - A ( Z 3 ) Z i + Z 2 ) - Fn-2(Z1,Z2

+ Z 3 ).

• 2

3

Theorem 10.5 Let f : V >-> V and g : V >->• V . continuous solution of the functional equation

(

n

\

Zi,£z;

n—1

+$>

I

n+i—1

Then the general

\

Zi,z i+1 , Yl Zi

=0

(10.24)

j=2 J i=1 \ j=i+2 J where n > 4 and Z n + ; = Zj for i = 1, • • • , n — 2, is determined by /(Z1,Z2)=JF(Z1+Z2), (10.25) g(Zu Z 2 , Z3) =

^ F ( Z ! + Z 2 + Z 3 ), n —1

where F is an arbitrary continuous complex vector function V \-¥ V. Proof.

If we put /1 = / , f2 = / 3 = • • • = / n - i = 5 into Eq. (10.18), we

obtain Eq. (10.24). Now by the substitution I

1

from Eq. (10.24) we get

(

z

n

\

i-EZi J=2

n—1

/

+ X> /

t=3

n+z—1

Zi.Zi+i, £ \

j=i+2

+ glZn^Zi+J^Zj]

\

Z,-

(10.26) j

=0.

Prom Eqs. (10.25) and (10.26) it follows that the function g satisfies the

Functional Equations with Several Unknown Functions

105

equation glZ2,Z3,Z1+^Zj j -5IZ„,Z2,Z1 + ^ Z

j

j =0

and, more generally,

(

t-l

n

Z i > Z i + 1 , £ z > + X) j=l

\

Z

/

i

j=i+2

=5

n-l

Z n ,Z 2 ) Z 1 + ^ Z

J

\

j=3

\

(10.27)

j

J

(2 < i < n - 1). On the basis of the relation Eq. (10.27), the functional equation Eq. (10.24) becomes flzu^Zj)

+ ( n - 2 ) J z n , Z 2 , Z 1 + ] T z i ) = O.

(10.28)

ra-1

If we put Z2 = Z n = O and 5Z Z_, = S, we obtain 3=3

F{Zi, S) + (n - 2)ff(O, O, Zj + S) = O. Thus /(Z1)S)=F(Z1+S),

(10.29)

where F(Zi) = - ( n - 2 ) f f ( 0 , 0 , Z i ) . On the basis of Eqs. (10.29) and (10.28) it follows that F

( ] C Z ' ) +(n-2)
=0,

i.e., fl(Zi,

z 2 , z3) =

-F(ZI + z2 + z s ). n—2

D

106

11

Functional Equations with Operations between

Arguments

Frechet's Functional Equations

First we will introduce the following notations. Let V be a finite dimensional complex vector space and let there exist a mapping f :V2 <->V. In this section Z* (0 < i < n + 1) will denote vectors in V. We assume that Zj = {zn{t),--- ,Zin(t))T, where the components Zij(t) (0 < i < n +1; 1 < j < n) are complex functions and O — (0, • • • , 0) T is the zero vector in V. We define multiplication of arbitrary two complex vectors U = {u\{t), • • • ,un(t))T and V = (vi(t), • • • ,vn(t))T in V as U V = («l(*)"l (*),•••

,Un{t)vn(t))T.

Now we will give the result obtained in [I. B. Risteski et al. (to appear A); K. G. Trencevski et al. (1999)]. Theorem 11.1 The general continuous solution of the complex vector functional equation / ( Z x + Z 2 + Z 3 + Z 4 , Z5) + / ( Z i + Z 2 + Z 3 , Z4 + Z5)

(11.1)

- / ( Z x + Z 2 , Z 3 + Z 4 + Z 5 ) - /(Zx, Z 2 + Z 3 + Z 4 + Z 5 ) + / ( Z 2 + Z 3 + Z 4 + Z 5 , Z 0 + / ( Z 2 + Z 3 + Z 4 , Z 5 + Zi) - / ( Z 2 + Z 3 , Z 4 + Z 5 + Zi) - / ( Z 2 , Z 3 + Z 4 + Z 5 + Zj) + / ( Z 3 + Z 4 + Z 5 + Z l 5 Z 2 ) + / ( Z 3 -I- Z 4 + Z 5 , Zi + Z 2 ) - / ( Z 3 + Z 4 , Z 5 + Zi -1- Z 2 ) - / ( Z 3 , Z 4 + Z 5 + Zi + Z 2 ) + / ( Z 4 + Z 5 + Zj + Z 2 , Z 3 ) + / ( Z 4 + Z 5 + Zi, Z 2 + Z 3 ) - / ( Z 4 + Z 5 , Z: + Z 2 + Z 3 ) - / ( Z 4 , Z 5 + Zi + Z 2 + Z 3 ) + / ( Z 5 + Zi + Z 2 + Z 3 , Z 4 ) + / ( Z 5 + Zi + Z 2 , Z 3 + Z 4 ) - / ( Z 5 + Zi, Z 2 + Z 3 + Z 4 ) - / ( Z 5 , Z x + Z 2 + Z 3 + Z 4 ) + / ( Z 1 ; Z 2 + Z 3 -I- Z 4 ) - /(Zx + Z 2 , Z 3 + Z 4 ) + / ( Z i + Z 2 , Z 3 ) - / ( Z i , Z 2 )

Frdchet's Functional

Equations

107

+ / ( Z 2 , Z3 + Z 4 + Z5) - / ( Z 2 + Z 3) Z4 + Z 5 ) + / ( Z 2 + Z 3 , Z 4 ) - / ( Z 2 , Z 3 ) + / ( Z 3 , Z 4 + Z5 + Zi) - / ( Z 3 + Z 4 , Z5 + Z J + / ( Z 3 + Z 4 , Z 5 ) - / ( Z 3 , Z 4 ) + / ( Z 4 , Z5 + Zi + Z2) + / ( Z 4 + Z 5 , Zx + Z2) + / ( Z 4 + ZB> Zi) - / ( Z 4 , Z 5 ) + / ( Z 5 , Z 1 + Z2 + Z 3 ) - / ( Z 5 + Z 1 ,Z 2 + Z 3 ) + / ( Z 5 + Z 1 , Z 2 ) - / ( Z 5 , Z 1 ) = O is given by /(U,V)

= F ( U + V) - F(U) - F(V) +

2

(11.2)

2

a(3U + 4V ) + £(3U + 4V) + L(U, V),

where F is an arbitrary complex vector function with values in V, and a, ft are complex constant matrices and L £ B(V). Proof. then

If we put Z : = U and Z 2 = Z3 = Z4 = Z5 = O in Eq. (11.1), 4/(U,0)-3/(0,U)=K = /(0,0).

(11.3)

If we replace Zx = U, Z 2 = V, Z 3 = W and Z 4 = Z 5 = O in (11.1), we find 3 / ( U + V + W , O) - 2 / ( 0 , U + V + W ) - / ( O , O) + / ( U + V, W ) - / ( W , U + V) + / ( V + W , U) - / ( U , V + W ) + / ( W + U , V ) - / ( V , W + U) = 0 , or, by virtue of Eq. (11.3), the above expression becomes /(U + V + W , 0 ) - / ( 0 , U + V + W) =

/ ( U + V , W ) - / ( W , U + V)

+

/(V + W , U ) - / ( U , V + W)

+

/ ( W + U , V ) - / ( V , W + U).

(11.4)

By putting Zi = U, Z 2 = V, Z 4 = W and Z 3 = Z 5 = O in Eq. (11.1) we obtain 2 / ( U + V + W , 0 ) - 2 / ( 0 , U + V + W ) + / ( U + V , W ) - 2 / ( W , U + V)

Functional Equations with Operations between Arguments

108

+2/(V + W, U) - /(U, V + W) + 2/(W + U, V) - 2/(V, W + U) +/(V, W) - /(U, V) + / ( U + V, O) + / ( O , U + V) + / ( 0 , W + U) - /(V, O) - /(O, W) - / ( O , U) = O. By application of Eq. (11.4), we find 4/(U + V + W , 0 ) - 4 / ( 0 , U + V + W) + / ( U , V + W ) - / ( U + V,W) +/(V, W) - /(U, V) + / ( U + V, O) + /(O, U + V) + / ( O , W + U) - / ( V , O) - /(O, W) - /(O, U) = O, or in view of Eq. (11.3) the above relation becomes = +

/ ( U + V, W) - /(U, V + W) + /(U, V) - /(V, W) (11.5) K - / ( 0 , U + V + W) + /(U + V , 0 ) + / ( 0 , U + V) /(0,U + W ) - / ( V , 0 ) - / ( 0 , U ) - / ( 0 , W ) .

By cyclic permutations of the vectors U, V and W in the above relation, we obtain = +

/ ( V + W, U) - /(V, W + U) + /(V, W) - / ( W , U) (11.6) K - / ( 0 , U + V + W) + /(V + W , 0 ) + / ( 0 , V + W) /(0,U + V ) - / ( W , 0 ) - / ( 0 , V ) - / ( 0 , U ) ,

= +

/ ( W + U, V) - /(W, V + U) + /(W, U) - /(U, V) (11.7) K - / ( 0 , U + V + W) + / ( W + U , 0 ) + / ( 0 , W + U) /(0,V + W ) - / ( U , 0 ) - / ( 0 , W ) - / ( 0 , V ) .

By adding together Eqs. (11.7), (11.6) and (11.5), and by using Eq. (11.4) we obtain =

/ ( U + V + W , 0 ) - / ( 0 , U + V + W) 2[/(0,U + V) + / ( 0 , V + W) + / ( 0 , W + U)]

+[/(U + V, O) + /(V + W, O) + / ( W + U, O)]

(11.8)

Prechet's Functional Equations

109

-2[/(0,U) + /(0,V) + /(0,W)] -[/(U,0) + /(V,0) + /(W,0)] + 3 K - 3 / ( 0 , U + V + W). It immediately follows that the function ¥>(U)

= /(U,0) + 2/(0,U)-3K

(11.9)

satisfies Prechet's functional equation [M. Frechet (1909)] ¥>(U + V + W ) - y>(U + V) - ¥>(V + W )

-(U) = AU 2 + BU, where A and B are constant complex matrices. For determination of the functions / ( U . O ) and / ( 0 , U ) we find the equations 4/(U,0)-3/(0,U)

=

K,

/(U,0) + 2/(0,U)

=

AU 2 + £ U + 3K.

By solving this system we obtain /(U,0)

=

3aU 2 + 3/3U + K,

/(O.U)

=

4 a U 2 + 4 / ? U + K,

where a = A/11 and j3 — B/ll.

From the equation (11.5) it follows that

/ ( U + V, W ) - / ( U , V + W ) + / ( U , V) - / ( V , W ) = 3aU 2 - 4 a W 2 + 6 a U V - 8 a V W + 3/?U - 4/JW. Using the notation ff(U, V) = a(3U 2 + 4V 2 ) + /J(3U + 4V), the above equations may be written in the form / ( U + V, W ) - / ( U , V + W ) + / ( U , V) - / ( V , W ) =

Functional Equations with Operations between

110

Arguments

Thus, the function / ( U , V ) — Z n + 1 ) - [ / ( Z o + Zi + --- + Z B _ i 1 Z n + Z n + i ) (11.10) +/(Zo + Zi +

h Z n _2 + Z„, Z n _i + Z n + i ) + • • •

+ / ( Z 0 + Z 2 + Z 3 + • • • + Z n , Zi + Z n + 1 ) + / ( Z i + Z 2 + • • • + Z„, Z 0 + Z n + 1 )] + [ / ( Z 0 + Z x + • • • + Z n _ 2 , Z n _i + Zn + Z„+i) + • • • ] + ( - l ) n [ / ( Z 0 , Zi + Z 2 + • • • + Z n + 1 ) + / ( Z ^ Z 2 + Z 3 + • • • + Z n + 1 + Z 0 ) + / ( Z „ , Z n + 1 + Z 0 + Z x + • • • + Z„_i)] = O is ^iuen by n

/ ( U , V ) = ^ U i F i ( U + V),

(11.11)

where Fi (1 

^ Z

i

=

T.

(11.12)

i=0

Using these notations and Eq. (11.10), we obtain Frechet's functional equation [M. Frechet (1909)] F(Z0 + Zi + • • • + Z„, T) - [F(Z 0 + Z x + • • • + Z n _ ! , T)

(11.13)

Functional Equations with Two Operations

111

+ F ( Z 0 + Zi + • • • + Z n _ 2 + z „ , T ) + • • • + F(Z0 + Z 2 + Z 3 + • • • + Z „ , T ) + F ( Z i + Z 2 + • • • + Z„,T)] + [F(Z 0 + Z x + • • • + Z n _ 2 ) T ) + • • •] - • • • + ( - l ) n [ F ( Z 0 , T ) + F ( Z 1 } T ) + F ( Z 2 , T ) + • • • + F ( Z n , T ) ] = O, whose general continuous solution according to [J. Aczel (1966)] is given by the formula n

F ( U , T) = J ^ U'Gi(T)

(11.14)

where d (1 < i < n) are arbitrary continuous functions with values in V. We put Eq. (11.14) into Eq. (11.12) and get Eq. (11.11). • E x a m p l e 11.1 A particular case of the above theorem for n = 2 is the following functional equation /(Zo + Z x + Z 2 , Z 3 ) - /(Zo + Zx> Z 2 + Z s ) - / ( Z 0 + Z 2 , Zi + Z 3 ) - / ( Z i + Z2 + Z 0 , Z3) + / ( Z 0 , Zx + Z2 + Z3) + / ( Z i , Z 2 + Z 3 + Z 0 ) + / ( Z 2 , Z 3 + Z 0 + Z 1 ) = O, whose general continuous solution is / ( U , V) = U 2 F 2 ( U + V) + UF X (U + V), where Fi and F2 are arbitrary continuous complex functions with values in V. 12

Functional E q u a t i o n s w i t h Two O p e r a t i o n s

In this section we will be able to solve some complex vector functional equations supplied by two operations, namely addition and multiplication, between arguments. For this purpose we use the same notations for the vectors given in the previous section and suppose that / : V2 i-+ V. Now we give the following results.

112

Functional Equations with Operations between

T h e o r e m 12.1

Arguments

The general solution of the functional equation

/ ( Z i + Z 2 ) Z 3 ) - / ( Z i • Z 2 , Z 3 ) - /(Zx + Z 2 , Z 3 • Z 4 )

(12.1)

+

/ ( Z 2 + Z 3 , Z 4 ) - / ( Z 2 • Z 3 , Z 4 ) - / ( Z 2 + Z 3 , Z 4 • ZO

+

/ ( Z 3 + Z 4 , ZO - / ( Z 3 • Z 4 , ZO - / ( Z 3 + Z4> Z x • Z 2 )

+

/ ( Z 4 + Zi, Z 2 ) - / ( Z 4 • Zx, Z 2 ) - / ( Z 4 + Zi, Z 2 • Z 3 ) = O

is (?it/en &«/ ifte following formula /(U,V)=F(U)-F(V),

(12.2)

where F is an arbitrary function with values in V. Proof.

If we put Z 3 = Z 4 = O into Eq. (12.1), we obtain /(Z1;Z2)

=

/(Z1-Z2,0) + /(0,Z1-Z2)

+

/(Z1,0) + /(0,Z2) + / ( 0 , 0 ) .

(12.3)

For Z 2 = Z 4 = O from Eq. (12.1) we get f(Z1,Z3)

+ f(Z3,Z1)

=

f(0,Z1)

+

f(Z1,0)

+

/(0,Z3) + /(Z3,0) + 2/(0,0).

The previous relation also holds if we put Z 3 = Z 2 , i.e., /(Z1,Z2) + /(Z2,Z1)

=

f(0,Z1)

+ f(Z1,0)

+

/(0,Z2) + /(Z2,0) + 2/(0,0).

(12.4)

After a permutation of Zi and Z 2 in Eq. (12.3), we obtain /(Za.ZO

= /(Z2-Z1,0) + /(0,Z2-Z1) (12.5) + /(Z2,0) + /(0,Zi) + / ( 0 , 0 ) . Since the multiplication between the vectors is commutative, we find / ( Z 1 ; Z 2 ) + / ( Z 2 , Zi) = 2/(Z x • Z 2 , 0 ) + 2 / ( 0 , Zj • Z 2 ) +

(12.6)

/ ( Z i , O) + / ( O , Zi) + / ( Z 2 , 0 ) + / ( O , Z 2 ) + 2 / ( 0 , 0 ) .

By comparison of Eqs. (12.4) and (12.6), we have /(Z1-Z2,0) + /(0,Z1-Z2) = 0.

(12.7)

Functional Equations with Two Operations

113

If we put U = Zi • Z 2 , the equality (12.7) takes on the form / ( U , 0 ) + / ( 0 , U ) = 0.

(12.8)

If we substitute Eq. (12.7) into Eq. (12.3), we get / ( Z i , Z 2 ) = / ( Z i , O) + / ( O , Z 2 ) + / ( O , O).

(12.9)

By virtue of Eq. (12.8), we may rewrite Eq. (12.9) as /(Z1,Z2) = / ( Z 1 , 0 ) - / ( Z 2 , 0 ) + / ( 0 , 0 ) .

(12.10)

If we put U = O into Eq. (12.8), we obtain 2/(0,0) = 0,

i.e.,

/(0,0) = 0,

and thus Eq. (12.10) becomes / ( Z 1 ; Z 2 ) = / ( Z 1 ; O) - / ( Z 2 , 0 ) .

(12.11)

Now, Eq. (12.2) follows immediately from Eq. (12.11), i.e., any solution of Eq. (12.1) has the form Eq. (12.2). Conversely, a direct calculation shows that the function Eq. (12.2) satisfies the equation (12.1) for an arbitrary F with values in V. • T h e o r e m 12.2

The general solution of the functional equation

n

^ [ / ( Z i + Zi+1, Zi+2) - / ( ^ • Zi+1, Zi+2) -

/ ( Z i + Zi+i, Zi+2 • Zi+s)] = O

(12.12)

( Z i + n = Z i ; n > 5)

is / ( U , V) = F(V) - F(V)

(12.13)

where F is an arbitrary complex vector function with values in V. Proof.

If we put Zi = O (1 < i < n) into Eq. (12.12), we find n/(0,0) = 0,

i.e.,

/(0,0) = 0.

By putting Zi = O (3 < i < n) into Eq. (12.12), we obtain / ( Z 1 ; Z 2 ) = / ( Z x • Z 2 , 0 ) + f(Q,Z1

-Z 2 ) + / ( Z i . O ) + / ( 0 , Z 2 ) . (12.14)

114

Functional Equations with Operations between

Arguments

Now, if we put Zj = O (i = 2, 4 0) + / ( 0 , Z 3 ) . For the sake of convenience we take Z 3 = Z 2 , then /(Z 1 ,Z 2 ) = /(Z 1 ,0) + /(0,Z 2 ).

(12.15)

From the equalities (12.14) and (12.15) it follows that /(Zi.Z2,0) + /(0,Z1-Z3) = 0, i.e., /(U,0) + /(0,U) = 0.

(12.16)

Therefore, the equality (12.15) becomes /(Z1,Z2)=/(Z1,0)-/(Z2,0)

(12.17)

which may be also rewritten in the form Eq. (12.13). Since any solution of the equation (12.12) has the form Eq. (12.13) and, conversely, the function Eq. (12.13) is a solution of the equation (12.12), it follows that the function Eq. (12.13) is the general solution of the equation (12.12). • Now assume that / : V3 >-¥ V. Theorem 12.3

The general solution of the functional equation

5

^__,[/(Zi + Zj+i, Zj+2 • Zj+3, Z;+4) + /(Zj • Z j + 1 , Z j + 2 , Zj+3 + Zj+4) (12.18) »=1

+ / ( Z j , Z; + i + Z j + 2 , Z; + 3 • Zj+4)] = O

(Zj + 5 = Zj)

is given by / ( U , V , W ) = F ( U , V, W ) - F ( V , W , U ) ,

(12.19)

where F is an arbitrary complex vector function with values in V. Proof. By a direct substitution of the function Eq. (12.19) into Eq. (12.18) we verify that it is a solution of the equation (12.18). Conversely, if we suppose that the function / ( U , V, W ) is a solution of the equation (12.18), we will prove that its form is given by Eq. (12.19).

Functional Equations with Two Operations

115

Prom Eq. (12.18) for Zx = Z 2 = Z 3 = Z 4 = Z 5 = O we obtain

15/(0,0,0) = 0, i.e., / ( 0 , 0 , 0 ) = 0.

(12.20)

If we put Zi = U, Z 2 = Z 3 = Z 4 = Z 5 = O into Eq. (12.18), in view of Eq. (12.20) we get 3[/(U, O, O) + / ( O , U, O) + / ( O , O, U)] = O, i.e., / ( U , O, O) + / ( O , U, O) + / ( O , O, U) = O.

(12.21)

If we substitute Z 5 = O in Eq. (12.18), making use of the equality (12.21), we obtain / ( Z i • Z 2 , Z 3 , Z 4 ) + /(Z 3 > Z4> Zi • Z 2 ) + / ( Z 4 , Zi • Z 2 , Z 3 ) +

/ ( Z 2 • Z 3 ) Z 4 , Z i ) + / ( Z 4 , Z 1 ; Z 2 • Z 3 ) + / ( Z i , Z 2 • Z 3 , Z 4 ) = O.

If we put Z 2 = I = (1,1, • • • , 1) T , then we obtain 2[/(Zi, Z 3 , Z 4 ) + / ( Z 3 ) Z 4 , Z 0 + / ( Z 4 , Zi, Z 3 )] = O. In other words, we have / ( Z l ; Z 2 , Z 3 ) + / ( Z 2 , Z 3 , Zx) + / ( Z 3 , Zi, Z 2 ) = O.

(12.22)

The general solution of the equation (12.22), according to [I. B. Risteski et al. (2000A)] is given by Eq. (12.19). • Theorem 12.4

The general solution of the functional equation

6

J^ifC^i

+ Zj+l, Zj+2 • Z; + 3 , Zj + 4 ) + / ( Z j • Zj + i, Z j + 2 , Z j + 3 + Z; + 4 ) (12.23)

t=i

+ / ( Z j , Zi+i + Zi + 2 , Zi + 3 • Z i + 4 )] = O

(Zi + 6 = Zj)

is given by /(U,V,W)=F(U,V)-F(V,W), where F is an arbitrary complex vector function with values in V.

(12.24)

116

Functional Equations with Operations between

Arguments

Proof. If we substitute the function Eq. (12.24) in Eq. (12.23), we obtain that Eq. (12.24) is a solution of the functional equation (12.23). Now we will prove that the converse is also true, i.e., that any solution of the equation (12.23) has the form Eq. (12.24). If we put Zj = O (1 < i < 6) into the equation (12.23), then we get 18/(0,0,0) = 0, i.e., /(0,0,0) = 0.

(12.25)

If we put Zi = U and Zi = O (2 < i < 6), we get 3[/(U, O, O) + / ( O , U, O) + / ( O , O, U)] = O, i.e., / ( U , O, O) + / ( O , U, O) + / ( O , O, U) = O.

(12.26)

Now, by putting Z 3 = Z 4 - Z 5 = Z 6 = O into Eq. (12.23) and taking into account Eqs. (12.25) and (12.26), we find /(Zx, Z 2 , 0 ) + / ( O , Zx, Z 2 ) + / ( O , O, ZO + / ( Z 2 , 0 , 0 ) = O.

(12.27)

If we substitute Z 2 = Z4 = Z 5 = Z& = O into Eq. (12.23), according to Eqs. (12.25) and (12.26) we obtain /(Z1,Z3,0) + /(Z3,0,Z1) + /(0,Z1,Z3) = 0. If we replace Z 3 by Z2 in the above expression, we get /(Z1,Z2,0) + /(Z2,0,Z1) + / ( 0 , Z 1 , Z 2 ) - 0 .

(12.28)

From the equalities (12.27) and (12.28) it follows that / ( Z 2 , 0 , ZO = / ( Z 2 , 0 , 0 ) + / ( O , O, Zi).

(12.29)

Functional Equations with Two Operations

117

If Z 5 = Z 6 = O in Eq. (12.23), we obtain f(Z1 + Z2,Z3-Zi,0)

+ f(Z1-Z2,Z3,Z4)

+ f(Z1,Z2

+

Z3,0)

+

/ ( Z 2 + Z 3 , 0 , 0 ) + / ( Z a • Z3, Z 4 , 0 ) + / ( Z 2 , Z 3 + Z 4 , 0 )

+

/ ( Z 3 + Z 4 , 0 , Zj) + / ( Z 3 • Z 4 , 0 , Z x ) + /(Z 3 > Z 4 , 0 )

+

/ ( Z 4 , 0 , Z 2 ) + / ( O , O , Zx + Z 2 ) + / ( Z 4 , 0 , Zi • ZB)

+

/ ( 0 , Z 1 - Z 2 , Z 3 ) + / ( 0 , Z 1 , Z 2 + Z3) + / ( 0 , Z 1 , Z 2 - Z 3 )

+

/ ( Z 1 ; Z 2 • Z s , Z 4 ) + / ( O , Z a , Z 3 + Z 4 ) 4- / ( O , Zx + Z 2 , Z 3 • Z 4 ) = O.

By using the equalities (12.28) and (12.29), the above equation reduces to the following simpler form f(Z1-Z2,Z3,Zi)+f(Z1,Z2-Z3,Z4)

+ f(Z2-Z3,Z4,0)

+

f(Z4,0,Z1-Z2)

+ / ( 0 , Z 1 ; Z 2 • Z s ) + / ( O , Zi • Z 2 ,Z 3 ) + / ( Z 3 , Z 4 , 0 ) + / ( Z 4 , 0 , Z x ) = O. In the above equation we put Z 2 = I = (1,1, • • • , 1) T , and then we obtain 2[/(Zj, Z 3 , Z 4 ) + / ( Z 3 , Z 4 , 0 ) + / ( O , Z x , Z 3 ) + / ( Z 4 , 0 , Z1)} = O, i.e., we get / ( Z i , Z 2 , Z 3 ) = - / ( Z 2 , Z 3 , 0 ) - / ( O , Z x ,Z 2 ) - / ( Z 3 , 0 , Zi).

(12.30)

We may show that the equation (12.23) is a consequence of Eqs. (12.28), (12.29) and (12.30). On the basis of the last three equalities, we obtain / ( Z i , Z 2 , Z s ) + / ( Z 2 , Z 3 , Zi) + / ( Z 3 , Z j , Z 2 ) =

[ - / ( Z 2 , Z 3 , 0 ) - / ( O , Z l f Z 2 ) - / ( Z 3 , 0 , Z1)]

+

[-f(Z3,Z1,0)-f(0,Z2,Z3)-f(Z1,0,Z2J]

+

[ - / ( Z i , Z 2 , 0 ) - / ( O , Z 3 , Z J - / ( Z 2 , 0 , Z 3 )]

=

- [ / ( Z 2 , Z s , O) + / ( O , Z 2 , Z 3 ) + / ( Z 3 , 0 , Z 2 )] +/(Z3,0,Za)-/(Za,0,Zs)

-

[/(Z3,Z1,0) + /(0,Z3,Z1) + /(Z1,0,Z3)] +/(Z1,0,Zs)-/(Zs,0,Z1)

-

[/(Z1,Z2,0) + /(0,Z1,Z2) + /(Z2,0,Z1)] +f(Z2,0,Z1) f(ZuO,Z2)

118

Functional Equations with Operations between

Arguments

=

/ ( Z 3 , 0 , 0 ) + / ( O , O, Z 2 ) - / ( Z 2 , 0 , 0 ) + / ( O , O, Z 3 )

+

/(Z1,0,0) + /(0,0,Z3)-/(Z3,0,0) + /(0,0,Z1)

+

/(Z2,0,0) + / ( 0 , 0 , Z 1 ) - / ( Z 1 , 0 , 0 ) + /(0,0,Z2) = 0.

From the last equality it follows that there exists a function G(U, V , W ) such that / ( Z i , Z 2 , Z 3 ) = G ( Z i , Z 2 , Z 3 ) - G(Z 2 , Z 3 , Z x ).

(12.31)

On the basis of the equalities (12.30) and (12.31), we obtain G(Z1,Z2,Z3)-G(Z2,Z3,Z1) =

-

G(Z2,Z3,0) + G(Z3)0,Z2)

-

G(0,Z1,Z2) + G(Z1,Z2,0)

-

G(Z3)0,Z1) + G(0,Z1,Z2).

(12.32)

If we substitute Z3 = O in the above equation, we obtain G(Z!,Z2,0) -G(0,Z!,Z2)

-

G(Z2,0,Z1) = - G ( Z 2 , 0 , 0 ) + G(0,0,Z2)

+ G(Z1,Z2,0)-G(0,0,Z1) + G(0,Z1,0)

or, written in a more convenient form, G(Z2,0,Z1) -G(0,0,Z2)

-

G(0,Z1;Z2)-G(Z2,0,0)

+ G(0,0,Z1)-G(0,Z1,0).

By using the last equality, the function Eq. (12.31) takes the following form /(Z1,Z2,Z3)

=

G(Z1,Z2,0)-G(Z2,Z3,0)

+G(Z3,0,Z2)-G(0,Z1,Z2)

+

G(0,Z1,Z3)-G(Z3,0,Z1)

= [G(Z1,Z2)0)-G(Z2,Z3,0)]

-

[G(0,Z1,Z2)-G(0,Z2,Z3)]

+[G(Z3)0,Z2)-G(0,Z2,Z3)]

-

[G(Z3,0,Z1)-G(0,Z1,Z3)}

= [G(Z1,Z2,0)-G(Z2,Z3,0)]

-

[G(0,Z1,Z2)-G(0,Z2,Z3)]

+[G(Z3,0,0)-G(0,0,Z3)

+

G(0,0,Z2)-G(0,Z2,0)]

-[G(Z3,0,0)-G(0,0,Z3)

+ 0(0,0,2^-0(0^,0)]

= [G(Z1,Z2,0)-G(0,Z1,Z2)

+

G(0,Z1,0)-G(0,0,Z1)]

-[G(Z2,Z3,0)-G(0,Z2,Z3)

+ G(0,Z2,0)-G(0,0,Z2)].

Functional Equations with Two Operations

119

Finally, if we denote F(ZUZ2)

=

G(Z1,Z2,0)-G(0,Z1)Z2)

+

G(0,Z1,0)-G(0,0,Z1),

then the function / ( Z i , Z 2 , Z3) can be rewritten in the following form f(ZuZ2,Z3)

= F(ZUZ2)

-

F(Z2,Z3),

which means that we have obtained the form given by Eq. (12.24). We conclude that the function Eq. (12.24) is the general solution of the equation (12.23). • The above results are obtained in [I, B. Risteski et al. (to appear A)].

Chapter 3

Functional Equations with Constant Parameters

Functional equations which contain constant parameters are studied in this chapter. Such equations with constant parameters solved here are the general parametric functional equation, the special parametric functional equation, expanded parametric functional equation and the general expanded parametric functional equation. The results presented here are obtained in [I. B. Risteski et al. (to appear B); I. B. Risteski et al. (2001A)]. First we introduce the following notations. Let N be the set of all positive integers. Let V, V be finite dimensional complex vector spaces and Zj, i G N , be vectors in V. We may assume that Zj = (zn(t), ••• , Zin(t))T, where Zy(t) (1 < j < n) are complex functions and O = (0, • • • , 0 ) T is the zero-vector in V or V . For a vector U € V denote by Re U (respectively Im U) the real (respectively imaginary) part of U. Moreover, we denote by V° the subspace of all real vectors in V (thus V = V° + iV°), and by £(V°, V ) the space of linear mappings V° •-> V . Let (m, n) be the greatest common divisor of m and n. 13

General Parametric Functional Equation

In the present section our object of investigation will be the following functional equation m-\-n

/TO—1

n—1

\

E -M £ am-x^Zi+j, £ an-^Zi+m+j i=l

\j=0

j=0

121

=O J

(13.1)

122

Functional Equations

with Constant

Parameters

where a is a complex number and fi : V2 i-+ V (1 < i < m+n) are unknown complex vector functions. The above equation for a = 1 was solved in [S. B. Presic and D. Z. Djokovic (1961)] under the assumption that the functions and variables are real. But the argument given there is valid only if the greatest common divisor of m and n is 1. Also, one special general case is solved in [D. Z. Djokovic (1965B)]. The theorems of [D. Z. Djokovic (1965B)] concerning the cases m ^ n should be modified to give the general continuous solutions. In the more general formulation as given in [D. Z. Djokovic (1965B)] they are invalid. Now we will give the following result which is obtained in [I. B. Risteski et al. (to appear B)]. Theorem 13.1 If a = 1, (m,ri) = 1 and m + n > 2, then the general continuous solution of the functional equation (13.1) is /i(U, V) = F1(XJ + V)Re U + F 2 (U + V)Im U + G 4 (U + V)

(13.2)

(1 £(V°, V ) (i = 1,2) and Gi : V ^ V (1 < i < m + n - 1) are arbitrary continuous complex vector functions. Proof. we set

We accept the convention to reduce the indices (mod m + n). If m+n

S = Ti

=

Y,ZU Zi + Z i + 1 + • • • + Z i + m _ i

(13.3) ;nS m +n

(l
+

n-l),

the vectors Tj (1 < i < m + n - 1) and S are independent since (m, n) = 1.

General Parametric

Functional

Equation

123

The equation (13.1) becomes m+n—1

/

/i

o

a

\

Ti + — — , — — - Ti V m +n m +n I x

i—i

13.4)

'

( mS Jm+n I — Ti — T2 — • • • — T m + n _ i + ; , \ m +n nS \ — + Ti + T2 + • • • + Tm+„_i m +n J

+

= O.

We introduce the new notations f i ( \ J + ^ - , \ m+n

^--v)=9i(\J,S) m+n J

( l < i < m + n),

i.e., " , U + V m +n ) The equation (13.4) is transformed into

(l
+ n).

(13.5)

m+n—l

J2

9i(Tt, S) + 3m+n(-Ti - T 2

T T O + n _i, S) = O.

(13.6)

By the substitution T i = T 2 = • • • = T r _ i = T r + i = • • • = T m + n _ i = O, we obtain gr(Tr,S)

=-gm+n(-Tr,S)

- Hr(S)

(1 < r < m + n - 1 ) .

(13.7)

Putting the equalities (13.7) into Eq. (13.6), we get Sm+n(-Tl — T 2 — • • • — Tm+n-i, m+n—1

=

£

S)

(13.8)

m+n—1

5m+n(-Ti,S)+

We conclude that the function K(V,S)=gm+n(\J,S)

^

^(S).

m+n—1 + —~^ J] Hi(S) m +• + rt. n — — }2. *—* rn.

(13.9)

i=l

satisfies the functional equation m+n—1

/r(Zi + z2 + "- + z m+n _i, s)= Y, i=l

K z

( i>s)-

(13-10)

124

Functional Equations

with Constant

Parameters

Using the continuity of K, from Eq. (13.10) we deduce that for fixed S we have tf(U,S)=ciReU

+ c 2 ImU.

The mappings ci,C2 £ £(V°, V) may depend upon S. Hence, K{V, V) = Fi (V)Re U + F2 (V)Im U,

(13.11)

where Fi : V i-> £(V°, V) are continuous functions. Prom Eqs. (13.9), (13.11) and (13.7) we obtain m+n—l

e?m+n(U,V)

=

0 P (U,V)

=

PKVJReU + ZaCVJImU-

m +n — 2 Jl(V)ReU + F 2 (V)ImU-fl'r(V)

i=l

(13.12)

m+n—l

+

1 E m + n —2 i = l

^i(V)

(l
From Eqs. (13.5) and (13.12) we deduce that / P (U,V)

= +

Fl(U

V)Ref^^) \ m +n J F2(V + V)Im ('"Y, , ^ V ) - # r ( U + V) m +n +

(13.13)

m+n—l

/ m + „(U + V)

E ff*(U + V)

+

m +n — 2

=

F 1 (U + V)Re

+

„ ,TT , , w F 2 (U v + V)Im

(l
i=l

'

riU — mV m +n /nU-mV\

V m+n

J

m+n—l

m + n —2

£

Hi(V + V).

»=i

By denoting - F i ( U + V)Re

m(U + V) m +n

F 2 (U + V)Im

m(V + V) m +n

m+n—l

1 + m + n — 2 J2 i=l

Hi(U + V)-Hr(U

+ V)=Gr(U

+ V)

General Parametric

Functional

(1 < r < m + n-F 1 (U + V)Re

m(U + V) m+ n

_.

125

1),

F 2 (U + V)Im

ro(U + V) m +n

m+n—1

—— Y,

m + n —2

Equation

Hi(V + V) = Gm+n(U + V),

from Eq. (13.13) we get Eq. (13.2). The converse can be established by a straightforward verification. E x a m p l e 13.1 tion

•

The general continuous solution of the functional equa-

/ i ( Z i + Z 2 , Z 3 ) + / 2 ( Z 2 + Z 3 , Zi) + /s(Z 3 + Zi, Z 2 ) = O is given by /!(U,V)

=

F 1 (U + V ) R e U + JF2(U + V ) I m U + Gi(U + V),

/2(U,V)

=

F 1 (U + V ) R e U + i^2(U + V ) I m U + G 2 (U + V ) ,

f3 (U, V)

=

-Fi (U + V)Re (U + 2V) - F2 (U + V)Im (U + 2V) - G 1 ( U + V ) - G 2 ( U + V)!

where Fi,F2 : V H > £(V°, V ) and G i , G 2 : V >-»• V are arbitrary continuous complex vector functions. Corollary 13.2 equation

The general continuous solution of the vector functional

m+n

2_, fft(Z« H

1- Zj +TO _i, Zi + Z 2 H

1- Z m + n ) = O

1=1

if (m, n) = 1 and m + n > 2 w given by 2 i (U,V) = F 1 ( V ) R e U + F 2 ( V ) I m U + G i (V)

^

( l < t < m + n),

Gi(V) = -m[F!(V)Re V + F 2 (V)Im V],

tofcere F i , F 2 : V i-+ £(V°, V ) , Gj : V H> V (1 < i < m + n - 1 ) are arbitrary continuous complex vector functions.

126

Functional Equations with Constant

Proof.

Put fi(U, V) = gi(U,U

Parameters

+ V) in Theorem 13.1.

•

T h e o r e m 13.3 The general continuous solution of the complex vector functional equation (13.1) if a = 1, (m,n) = d > 1, m/d = p, n/d = q and p + q > 2 is given by fid+j (U, V) = Fu (U + V)Re U + F2j (U + V)Im U + Gtj (U + V) (0
l<j
p+q-l

J2 G«(U) = Hj(V) -p[F y (U)ReU + F2i(U)ImU]

(13.14)

(l<J
£ # , - ( U ) = Ot

i^- : V K> £(V°, V ) Hr.V^V Gy : V i-* V

(» = 1,2; 1 < j < d), (l
(0 < i < p + q - 2; 1 < j < d)

are arbitrary continuous complex vector functions. Proof.

We set / i ( U , V ) = ffi(U,U + V)

( l < i < m + n)

(13.15)

and we obtain m+n

J2gi(Zi + Zi+1 + --- +

Zi + Z 2 + • • • +

) = O.

(13.16)

i=i

Let us introduce the new vectors Vi = Zi + Z i + i + • • • + Z i + d _ !

(1 < t < m + n)

(13.17)

so that V i + m + „ = Vf, and W = Z 1 + Z 2 + --- + Z m + n .

(13.18)

General Parametric

Functional

Equation

127

(1 < j < d).

(13.19)

They are not independent because p+q-l

Y,

Vid+j = W

i=0

The vectors Yi (1 < i < m + n — d) and W are independent because the rank of the matrix of linear forms determining them ism + n — d+1, which is easy to verify. In the sequel we will use all vectors given by Eqs. (13.17) and (13.18) but we must have always in mind that Eq. (13.19) holds. The equation (13.16) becomes m+n

Yl 9i(Yi + V i + d + • • • + V i + ( p _ 1 ) d ) W) = O. i=l

It can be written in the following form d

p+q-l

X ) z2 Sid+j(yid+j + V(i+1)d+j j=l

H

1- V( i+ p_i) d+J -, W) = O.

i=0

If we set here V

i d + i

-0

{0
+ q-2;

V( p + g _i)d+j = W

j = 1,2, • • • . r - l . r + l,--- ,d),

{j = 1,2,-•• , r - l , r + l,--- ,d),

we get p+q l

~ 2j

gid+r(Vid+r + V ( i + i ) d + r H

h V(i+p_1)d+r, W)

i=0

H (W) ?1 — = O P

(1 < r < d) and

YlHr(W)

= Q.

r=l

By using the corollary of Theorem 13.1 we get gid+r(\J, V) = Flr(V)Re\J (0
+ q-l;

+ F2r(V)ImU + l
Gir(V)

V

128

Functional Equations with Constant Parameters p+q-1

£

Gi P (V) = Hr(V) -p[Flr(V)ReV

+ F 2 r (V)ImV]

(1 < r < d), where F j r : V h > £(V°, V ) Gir.:V^V'

(t = 1,2; 1 < r < d),

( 0 V

(1 < r < d - 1)

are arbitrary continuous complex vector functions. By application of the equalities (13.15) these formulae give Eq. (13.14). It is easy to prove that the functions fi : V2 i-> V (1 < i < m+n) defined by Eq. (13.15) satisfy the complex vector functional equations (13.1). • E x a m p l e 13.2 tion

The general continuous solution of the functional equa-

/ i ( Z ! + Z 2 + Z 3 + Z 4 l Z 5 + Z 6 ) + / 2 ( Z 2 + Z 3 + Z 4 + Z B | Z 6 + Zi) + / 3 ( Z 3 + Z 4 + Z 5 + Z6> Zi + Z 2 ) + / 4 ( Z 4 + ZB + Z 6 + Zi, Z 2 + Z 3 ) + / 5 ( Z 5 + Z 6 + Zi + Z 2 , Z 3 + Z 4 ) + / 6 ( Z 6 + Zi + Z 2 + Z 3 , Z 4 + Z 5 ) = O is given by / i ( U , V) = F n ( U + V ) R e U + F 2 1 (U + V ) I m U + G 0 i(U + V), / 2 ( U , V) = F12(U + V ) R e U + F 2 2 (U + V ) I m U + G 0 2 (U + V), / 3 ( U , V) = Fn(V

+ V ) R e U + F 2 1 (U + V ) I m U + G n ( U + V),

/ 4 ( U , V) = F 1 2 (U + V ) R e U + F 2 2 (U + V)Im U + G 1 2 (U + V), / 5 ( U , V) = - F n ( U + V)Re (U + 2V) - F21 (U + V)Im (U + 2V) + # i ( U + V) - G 0 i(U + V) - G n ( U + V),

General Parametric Functional Equation

129

/ e ( U , V) = - F 1 2 ( U + V)Re (U + 2V) - F22(U + V)Im (U + 2V) - f l i ( U + V) - G 0 i(U + V) - G 1 2 (U + V), where Fir.V^C{V\V) Gir.V^V

(t = l,2), (i = 0,1; j = 1,2),

are arbitrary continuous complex vector functions. Theorem 13.4 is

The most general solution of (13.1) if a = 1 and m = n

/i(U,V) /m+i(U,V)

(1 V (1 Z3 + Z4) + + / 3 ( Z 3 + Z 4 , Zx + Z 2 )

+

/ 2 ( Z 2 + Z 3 ) Z4 + Z1) / 4 ( Z 4 + Zi, Z 2 + Z 3 ) = 0

is / i ( U , V)

,

/ 2 ( U , V)

ore arbitrary,

/3(u,v) =

^ ( U + VJ-ACV.U),

/4(U,V)

- ^ ( U + VJ-^V.U),

=

where i?i : V i-> V is an arbitrary function. T h e o r e m 13.5 J/ a m + n 7^ 1 and m ^ n, the general solution of the functional equation (13.1) is given by /i(U, V) = Fi(V + amV)-Fi+n(anV+V)

+ Ai

(1 < * < m + n), (13.21)

130

Functional Equations with Constant

Parameters

where Fi : V H-> V' (1 < i < m + n) are arbitrary complex vector functions, m+n

and Ai are arbitrary constant complex vectors such that ^2 A\ = O. i=l

Proof.

If we introduce new functions gi by the equation

/ i ( U , V ) = 5 i ( U + a m V , a n U + V)

( l < t < m + n),

(13.22)

then Eq. (13.1) becomes m+n

£> *=i

/ m—1

n—1

£ am-^Zi+j + J2"m+n-1-JZm+i+j,

\j=o

j=o m—1

n—1

\

}=0

j=0

J

i.e., m+n

Y, 9i

lm+n—1

S

m+n—1

\

o'Zn.+i-i-,-, J ] a''Zi_i_,-

= O.

(13.23)

Since a m + n ^ 1, this transformation is possible. Also we may introduce new vectors Vj by m+n—1

Vj =

5 3 aJZm+i-i-j i=o

(1 < i < m + n)

and then Eq. (13.23) takes the form m+n

53^(V i ; V i + n ) = 0.

(13.24)

i=0

By putting V,- = O (j = 1,2, • • • , i — 1, i + 1, • • • , i + n — 1, i + n + 1, • • • , m + n) we obtain ft(Vi, V i + n ) = fi(Vj) + G , ( V i + n )

(1 < t < m + n).

(13.25)

On the basis of the expression Eq. (13.25), the equation (13.24) becomes m+n

53t i i i( v 0 + G!i(Vi+„)] = O, j=i

General Parametric

Functional

Equation

131

or m+n

J2 [Fi(Vi) + Gm+i(Vi)] = O.

(13.26)

i=l

Prom Eq. (13.26) it follows that Gi+m(Vi)

= -Fi(Vi)

+ Ai

(1 < i < m + n),

(13.27)

where Ai are arbitrary constant complex vectors with the property m+n i=l

On the basis of the expression Eq. (13.27), the equality (13.25) has the form gi(XJ,V)=Fi(U)+Fi+n(y)+Ai

{l
+ n),

(13.28)

m+n

where ^

Ai = O.

i=l

On the basis of Eqs. (13.28) and (13.22), we obtain Eq. (13.21). Example 13.4

•

If a 3 ^ 1, the general solution of the functional equation

/ i ( a 2 Z i + a Z 2 + Z 3 , Z4)

+

/ 2 ( a 2 Z 2 + aZ 3 + Z 4 , Z J

+ / 3 ( a 2 Z 3 + aZ 4 + Zi, Z 2 )

+

/ 4 ( a 2 Z 4 + oZi + Z 2 , Z 3 ) = O

is given by /i(U,V)

=

F!(U + a 3 V ) - F 2 ( a U + V) + Ai,

/2(U,V)

=

F 2 (U + a 3 V ) - F 3 ( a U + V) + A 2 ,

/3(U,V)

=

F 3 (U + a 3 V ) - F 4 ( a U + V ) + A 3 ,

/4(U,V)

=

Fi(XJ + a3V)-F1(aU

+

V)-A1-A2-A3,

where Fj : V •-»• V (i = 1,2,3,4) are arbitrary complex vector functions, and Ai (i = 1,2,3) are arbitrary constant complex vectors. Theorem 13.6 If am+n ^ 1 and m = n, the most general solution of the functional equation (13.1) is / i + m ( U , V ) = - / i ( V , U ) + Ai

(l
(13.29)

132

Functional Equations with Constant

Parameters

where fi : V2 *-¥ V" (1 < i < m) and Ai (1 < i < m) are arbitrary complex m

constant vectors such that ^2 Ai = O. 8=1

Proof. By the transformations used in the proof of the previous theorem we may bring the equation (13.1) to the form Eq. (13.24). For Vj = O (j = 1,2, ••• ,i-l,i + l,--- ,i + m- l,i + m + 1, • • • ,2m) the equation (13.24) becomes 9i(Vi,

Vi+m) + gi+m(Vi+m, Vi) = Ai

(1 < * < m),

(13.30)

where Ai (1 < i < m) are arbitrary complex constant vectors. By substituting Eq. (13.30) into Eq. (13.1), we obtain that it must hold m i=l

On the basis of this equality and Eq. (13.30), we obtain Eq. (13.29). • E x a m p l e 13.5 If a 4 ^ 1, the most general solution of the functional equation / i ( a Z ! + Z 2 , aZ 3 + Z 4 ) + / 3 ( a Z 3 + Z 4 , aZ1+Z2)

+

/ 2 ( a Z 3 + Z 3 , aZ 4 + Zj)

+

/ 4 ( a Z 4 + Zlt aZ2 + Z 3 ) = O

is given by /i(U, V)

(i = l,2)

are

arbitrary,

/3(U,V)

=

-/1(U,V)+A,

/4(U,V)

=

-^(U.VJ-A,

where A is an arbitrary complex constant vector. If am+n = 1, then the functional equation (13.1) may be transformed in the following way. We introduce new vectors by the equality Vi = a1~iZi,

i.e.,

Zi = ai~1Vi

(l
+ n).

Then the equation (13.1) becomes m+n

£ j=l

j

?n—l

«m-2+i £

fi \

j'=0

n—1

Vi+j,

am+n-*+i

£ j=0

\

Vm+i+j

= O. J

(13.31)

General Parametric

Functional

Equation

133

Now, if we put 9i(V,

V) = fi(am-2+iU,

am+n-2+iV)

(1 E -+^ = ° The equation (13.33) is just Eq. (13.1) for a = 1.

T h e o r e m 13.7 Ifam+n = 1, (m,n) = 1 andm + n > 2, tten £/ie general continuous solution of the functional equation (13.1) is given by /j(U, V) = F i ( o n + 2 _ i U + a m + n + 2 - i V ) R e (a"* 2 - *!!) + F 2 ( a n + 2 - i U + a m + n + 2 - i V ) I m ( a ^ ^ U ) + Gi(an+2-*U +

(13.34) m+n+2 a

-iV)

(1 £(V°, V ) (» = 1,2) and G< : V ^ V (1 < t < m + n - 1) are arbitrary continuous complex vector functions. Proof. The proof immediately follows from Eqs. (13.33), (13.32) and Theorem 13.1. • T h e o r e m 13.8 Ifam+n - 1, (m,n) = d > 1, m/d = p, n / d = g a n d p + q > 2, then the general continuous solution of the functional equation (13.1) is fid+j (U, V) = FXi {an+2-l\5

+ om+n+2-iV)Re ( a ^ - ' U )

(13.36)

+F2j ( a ^ ^ U + a m + n + 2 - i V)Im (a""*" 2 -^) + Gy ( a ^ ^ U + am+n+2-i (0
l<j
V)

134

Functional Equations

with Constant

Parameters

SO that p+q-1

]T) Gij (U) = Hj(U) - p[F!j(U)Re U + F2j (U)Im U]

(13.37)

(l<J
£^•(11) = 0,

(13.38)

3=1

where Fir.V^C{V°,V) GyiV^V

(t = l,2; (0
Hj-.V-tV

l<j
+ q-2;

l<j
(1 < j < d - 1)

are arbitrary continuous complex vector functions. Proof. On the basis of the expressions Eqs. (13.33), (13.32) and Theorem 13.3 we derive the proof of the theorem. • T h e o r e m 13.9 If am+n = 1 and m = n, then the most general solution of the functional equation (13.1) is given by fi(U, V) /m+i(U,V)

(1 < i < m) = -

n+2 i

Hi(a

-V

/i(a

n+2 i

are arbitrary, + a ™+"+ 2 -*V)

- U , a ™+"+2-*v)

(13.39) (1 < i < m),

where H; : V 4 V' are arbitrary complex vector functions such that m

£ Hi(U) = O. i=l

Proof. The proof immediately follows from Eqs. (13.33), (13.32) and Theorem 13.4. •

Special Parametric

14

Functional

Equation

135

Special Parametric Functional Equation

All notations for the vectors are the same as in the previous section. Now, we will solve the following functional equation [I. B. Risteski et al. (to appear B)] m+n

I m—1

E / i=l

n—1

m

E a ~

1_Jz

\

^. E «

\j=0

n_1 J

~ Z i+m+j

j=0

= O,

(14.1)

J

which is obtained as a special case of the equation (13.1) for / ; = / (1 V, G : V2 t-> V are arbitrary complex vector functions. Proof.

We set / ( U , V) = g(U + amV,

anU + V)

(14.3)

into Eq. (14.1) and deduce that m+n

/m+1

n—1

E A E ^-^Zi+j

+ Y/am+n-1Zi+m+j,

m—1

n—1

\

3=0

j=0

J

E am+n-^Zi+j + ^ a " - 1 ^ Z i + r a + j

= O,

i.e., m+n

fm+n—I

E> i=i

E y

j=o

m+n—1

aJZ

»+—i-i. E j=o

\

iz

° *-i-; = °

(14-4)

/

This transformation of the equation (14.1) is possible since am+n ^ 1.

136

Functional Equations with Constant

Parameters

Now we introduce new vectors m+n+l

Vi=

Yl

^Zi-i-j

( l < t < m + n).

(14.5)

The linear forms Eq. (14.5) are independent since their determinant is (am+n

1 \m+n—1

Making use of these notations, the equation (14.4) becomes m+n

^2 g(Vi,Vi+n)

= O.

(14.6)

i=l

If m 7^ n, we set Vi = V 2 = • • • = V m _ i = V m + i = V m + 2 = • • • = V r a + B - i = O and we get 0 ( U , V ) = F ( U ) + *i(V).

(14.7)

We substitute g from Eq. (14.7) into Eq. (14.6) and obtain m+n

'£[F(Vi) + F1(Vi)} = 0, »=i

which implies that Fi(Vj) = —F(V<). Hence, ff(U,V)

= F(U)-F(V).

(14.8)

IfTO= n, the equation (14.4) yields

g(U,V)+g(V,U)

= 0,

i.e., g(V, V) = G(U, V) - G(V, U).

(14.9)

From Eqs. (14.3), (14.8) and (14.9) we conclude that Eq. (14.2) holds. It is easy to verify that Eq. (14.2) satisfies Eq. (14.1). • E x a m p l e 14.1 equation

If a3 ^ 1, then the most general solution of the functional

/ ( o Z i + Z 2 , Z 3 ) + / ( a Z 2 + Z 3 , Zi) + / ( a Z 3 + Z j , Z 2 ) = O is given by / ( U , V) = F ( U + a 2 V) - F ( a U + V),

Special Parametric

Functional Equation

137

where F : V t-¥ V is an arbitrary complex vector function. Example 14.2 equation

If a 4 ^ 1, the most general solution of the functional

/ ( a Z i + Z 2 , aZ 3 + Z 4 )

+

/ ( a Z 2 + Z 3 , aZ 4 + Zi)

+ / ( a Z 3 + Z4, a Z i + Z 2 )

+

/ ( a Z 4 + Z 1 ; aZ 2 + Z 3 ) = O

is given by / ( U , V ) = G(U + a 2 V, a 2 U + V) - G(a 2 U + V, U + a 2 V), where G : V2 i->- V is an arbitrary complex vector function. Theorem 14.2 Ifam+n — 1, (m,n) = 1 andm + n > 2, then the general continuous solution of the functional equation (14.1) is given by m+n

/(U,V)

J][JF1(aiU + ai+mV)Re(aiU)

=

(14.10)

i=l

+F2(aiV + a i + m V ) I m (o*U)] m+n—1

+

[G i (a i U + a i + m V ) - G i ( a i U + a m V)]

^ i=i

-

m[Fi (U + am V)Re (U + a m V ) + F 2 ( U + a m V)Im (U + a m V)],

wfcere F* : V H- £(V°, V ) (i = 1,2) and Gt : V H> V (1 < i < m + n - 1) are arbitrary complex vector functions. Proof. Let us put Z; = a i _ 1 T j (1 < i < m + n). The equation (14.1) becomes m+n

£ «=i

I

m—1

a™ + i - 2 2

/ \

n—1

T i + i , am+n~2+i

j=o

£

\

Tm+W

j=o

= O.

(14.11)

/

Now we make the substitutions /(am+i-2U,

m+n 2+i a

- V)

= fi(U, V)

(1 £ Tm+i+j = O. *=i

\i=o

i=o

(14.13)

/

The equation (14.13) is just Eq. (13.1) for a — 1, and its solution is determined by Theorem 13.1. By an application of Theorem 13.1, and by the equality (14.12) we get /i(U, V) = P i ^ + ^ U + a m + n + 2 _ < V ) R e ( a ^ ^ U ) +P2(an+2-iXJ

+ a ^ + ^ V ^ m (an+2^lJ)

(14.14)

+ Q ^ a ^ ^ U + a ™+"+ 2 -«V)

(1 £(V°, V ) (i = 1,2) and Q; : V >->• V (1 < i < m + n) are continuous complex vector functions. By addition of all equations (14.14) and putting Pi(U)

=

(m + n)P!(U),

P 2 (U) = (m + n)P 2 (U),

Qi(U)

=

(m + n)G„+2-i(U)

(i = 1,2, • • • ,m + n)

we obtain Eq. (14.10). E x a m p l e 14.3 tional equation

D

If a 3 = 1, the general continuous solution of the func-

/ ( a Z i + Z 2 , Z 3 ) + / ( a Z 2 + Z 3 , Zi) + / ( a Z 3 + Z l f Z 2 ) = O is given by /(U,V)

=

F i ( a U + V ) R e ( a U ) + P 2 ( a U + V)Im(aU)

+

P!(a 2 U + V)Re (a 2 U) + P 2 (a 2 U + V)Im (a 2 U)

-

Pi (U + a 2 V)Re (U + 2a 2 V) - P 2 (U + a2V)Im (U + 2a 2 V)

+

G1(aU + V)-G1{U

+

G 2 (a 2 U + a V ) - G 2 ( U + a 2 V),

+ a2V)

Special Parametric Functional Equation

139

where Ft : V •-> £(V°, V ) (i = 1,2) and G* : V (->• V (i = 1,2) are arbitrary complex vector functions. Theorem 14.3 Ifam+n = 1, (m,n) = d>l, m/d = p, n/d = q andp+ q > 2, then the general continuous solution of the functional equation (14.1) is given by /(U,V)

=

d-1

p+q-l

Yl

£

j=-t

[Fhj+2(an-id-jTJ

i=0

F2,j+2(an-id-j\J

+

+ a-id-:>V)Im{an-id-jV)

n id j

+

a-id-jV)Re(an-id-jU)

+

Gi,j+2(a - ~ V

(14.15)

id j

+

a- - V)},

so that p+q-l

J2 Gij (U) = Hj (U) - p[Fy (U)Re (U) + F2j (U)Im (U)] (1 < j < d) i=0

and d

.7=1

where F{j : V H> £(V°, V') Gij:Vi->V'

(t = 1,2;

1 < j < d),

(0
fi,- : V H> V

l<j
(1 < j < d - 1)

are arbitrary continuous complex vector functions. Proof. We can start from equation (14.11). (14.11) on the basis of Theorem 13.3 we get / ( U , V)

Prom Eqs. (14.10) and

=

P y ( a n - i < , - * + 2 U + am+n+2'id-jV)Re

(an-id-j+2U)

+

P2j(an-id-j+2XJ

+ am+n+2-id-jV)lm

(an-id-j+2U)

+

Qij(an-id-j+2\J

+ an+m+2-id-jV)

{0
p + q-l;

l<j
(14.16)

140

Functional Equations with Constant

Parameters

p+q-1

J2 QaCU) = K,-(U)-p[Py(U)Re(U) + P2i(U)Im(U)] (14.17) i=0

(1 < 3 < d), d

£^•(17) = 0,

(14.18)

where Pir.V^C{V°,V) Qij-.V^V

(t = l,2; (0
Kj-.V^V

l<j
+ q-2;

l<j
(1 < j < d - 1)

are continuous functions. We take into account Eqs. (14.17) and (14.18) and we add together all equations (14.16). In this way we obtain Eq. (14.16) with P y (U) = (m + n ) P y (U), Qij (U) = (m + n)Gy (U), (0
+ q-2;

P2j (U) = (m + n ) F 2 j (U), Kj (U) = (m + n)fl>(U) \<j
E x a m p l e 14.4 If a6 = 1, then the general continuous solution of the functional equation / ( a 3 Z i + a 2 Z 2 + a Z 3 + Z 4 , aZ5 + Z6) + f(a3Z2+a2Z3+aZ4

+ Z5, aZ 6 + Zi)

+ / ( a 3 Z 3 + a 2 Z 4 + a Z 5 + Z 6 , aZ1+Z2)+f(a3Zi+a2Z5+aZ6+Z1,

aZ2+Z3)

+/(o 3 Z5+a 2 Z6+aZi+Z 2 ,aZ3+Z4) + /(a 3 Z 6 +o 2 Zi+aZ2+Z3,aZ4+Z 5 ) = O is given by / ( U , V) = F n ( a U + a 5 V)Re (aU) + F 2 1 (aU + a 5 V)Im (aU) + F n ( a 3 U + aV)Re (o 3 U) + F 2 i ( a 3 U + oV)Im (a 3 U)

Special Parametric

Functional Equation

141

- F u ( a 5 U + a 3 V)Re (a 5 U + 2a 3 V) - F 2 1 (a 5 U + a 3 V)Im (a 5 U + 2a 3 V) + F 1 2 ( U + a 4 V)Re (U) + F 2 2 (U + a 4 V)Im (U) + F 1 2 ( a 2 U + V)Re (a 2 U) + F 2 2 (a 2 U + V)Im (a 2 U) - f \ 2 ( a 4 U + a 2 V)Re (a 4 U + 2a 2 V) - F 2 2 (a 4 U + a 2 V)Im (a 4 U + 2a 2 V) +G 0 i(oU + a 5 V) - G 0 i(o 5 U + a 3 V) + G 0 2 (aU + a 4 V) - G 0 2 (a 4 U + a 2 V) + G n ( a 3 U + aV) - G u ( o 5 U + a 3 V) + Gi 2 (o 2 U + V) - Gi 2 (o 4 U + a 2 V) + # i ( a 5 U + a 3 V) - # i ( a 4 U + a 2 V), where ^ - : V»->£(V°,V') (i,j = 1,2); G « : V 4 V (t = 0 , l ; j = 1,2) and .Hi : V i-> V are arbitrary continuous complex vector functions. T h e o r e m 14.4 If am+n = 1 and m = n, the most general solution of the functional equation (14.1) is given by m

/(U,V)

=

^ [ ^ ( a ' U , an+iV)-Fi{<JV,

an+iV)

i=l

+ f l i ( a n + i V + a i U)] >

(14.19)

m

£ff 4 (U) = 0, i=l 2

where F{ : V <-+ V (1 < i < m) and Ht : V ^ V (1 < i < m - 1) are arbitrary complex vector functions. Proof. We start again from the equation (14.11). According to Theorem 13.4 and Eq. (14.10) we have / ( U , V) = Pi(am-i+2V, /(U,V)

= -

a ™+»+2-* V )

(1 a

m+2 i

m

- V)

(14.20) (1 < i < m),

142

Functional Equations with Constant Parameters

By addition we get Eq. (14.19) with Pi(U, V) = 2 m F m _ i + 2 ( U , V),

Qi(U) = 2mff m _ i + 2 (U).

• E x a m p l e 14.5 equation

If a

4

= 1, the most general solution of the functional

/ ( a Z i + Z 2 , aZ 3 + Z 4 ) +f(aZ3

+ Z 4 , oZ x + Z 2 )

+

f(aZ2 + Z3,aZ4

+ Z1)

+

/(0Z4 + Z1, aZ 2 + Z 3 ) = 0

is given by / ( U , V)

=

Fi(aU, a 3 V) - Fi(aV, a 3 U) + F2{a2V, V)

-

F 2 (a 2 V, U) + ffi(a3U + oV) - # i ( U + a 2 V),

where Fj : V2 •-> V (i = 1,2) and F i : V 4 V are arbitrary complex vector functions. Now, as special cases we obtain the results given in [D. Z. Djokovic et al. (1966); R. Z. Djordjevic and P. M. Vasic (1967); D. S. Mitrinovic and J. E. Pecaric (1991)].

15

E x p a n d e d P a r a m e t r i c Functional E q u a t i o n

The notations for the vectors in this section are the same as in Sec. 13. In this section we will solve the following simple complex vector functional equation [I. B. Risteski et al. (2001A)] m+n+k

E «=i

fm—1

/

n—1 m-W

E a

Z<+i> E°n"1_iZ*+™+i'

y j=o

(15.1)

j=o k-l

E "-k-l-jry n

^i+m+n+j

=o

3=0 \Zm+n+k+i

= Zj),

where a is a complex number and / : V3 i-> V is an unknown complex vector function.

Expanded Parametric Functional Equation

143

The above equation for k — 0 was solved in the previous section. Also, the functional equation (15.1) for a = 1 was solved in [D. Z. Djokovic (1961)] under the hypothesis that the function and variables are real. Now we will solve the functional equation (15.1) in the following cases. I. Let o = 1. If we introduce the notations S=

m+n+k ] T Z,-

(15.2)

and / ( U > V , S ) = f l ( U , V , S + U + V),

(15.3)

then the equation (15.1) becomes m+n+k

I m—l

£

g

n—1

£

Zi+j, £

\

Zm+i+j, S U O .

If we introduce new variables T; (1 

then the equation (15.4) becomes k

/

c,

m—l

ma 2=1

^ (

o

E

f

ma

9

/ , i=n+k+l

9\ — ; \

3=0

m—l

—

^-\

m +n +k

z=k+l \ •m+n+k 1

+

n—1

na

J=0

n+k

u

a

v-^ „

«

+

^

na Ti+i

'

r~,—

/ , T^m+i+j, — ; j=0

\

^-\

m +n +k ~

]=0 n+k—1

)

m+A:—1

^

I T

-+"+^>

}=0 „

n—1

S

) \

~ r + > _,T m + j+j, S j=0

I

= O.

144

Functional Equations

with Constant

Parameters

By putting n S ^ +U> , fc+V, ^m + n + k m +n + k

S)=g(U,V,S)>

(15.6)

the equation (15.5) becomes k

I m—1

»=1

\ 3=0

n—1

E # E ^> E T ™+^ s ] k+n

+

E

m+k — 1

T

H I E

«+i' ~~ E

\j=0

m+n+A

/

E

"^"i+n+t+j, S

j=0 n+k—1

n—1

\

\

'^ m + i +i' / ,, T m + i + j , S I = O.

^ I ~~ E

i=n+k+l

( 15 - 7 )

3=0

Jm — 1

«=A+1

+

T

j=0

j=0

J

Now, we may suppose that the variable S is fixed, and we may put # ( U , V , S ) = /i(U,V).

(15.8)

In this case, the functional equation (15.7) takes the following form A:

I m —1

h

E ^+i+j

j=l

j=0

\j=0

E

I m—\

^ I E

i=k+l m+n+k

+

\

T

Y, \Y, «+;'' m+k +

n—1

T

E i=n+k+l

Ti+J>

_

E

\

^"i+n+i+j I

j=0 n+k+1

M ~" E \

J m+k — 1

\j=0 I

( l5 - 9 )

J n—1

^m+i+j, /

j=0

y

\

Tm+i+j I = O.

j=0

J

The complex vector function h has the following properties: 1°. If m
(15.10)

h(U, - U ) + /i(U, O) + h(0, - U ) = O,

(15.11)

/i(0,U) + / i ( 0 , - U ) = 0 ,

(15.12)

Expanded Parametric Functional Equation

h(U, V) = h(U + V, O) - h(Y, O) + h(0, V),

h(\J, O) + h(-U, O) = O,

h(U, V) = h(0, U + V) - h(0, U) + h(U, O).

145

(15.13)

(15.14)

(15.15)

2°. If m < n < k and m + n = k, then Eqs. (15.10), (15.11), (15.12), (15.13), (15.14) and (15.15) hold. 3°. If m < n < k and m + n > A;, then Eqs. (15.10), (15.11), (15.12), (15.13), (15.14) hold and /i(U, V) + h(V, - U - V) + h(-V

- V, U) = O.

(15.16)

4°. If m < n = k, then Eqs. (15.10), (15.11), (15.12), (15.13) and (15.16) hold. 5°. If m = n and 2m < &, then Eqs. (15.10), (15.11) hold and h(U, V) + h(V, O) + h(p, - U - V) + h(-V

- V, U) = O.

(15.17)

6°. If m = n and 2m = fc, then Eqs. (15.10), (15.11) and (15.17) hold. 7°. If TO = n < k and 2m > k, then Eqs. (15.10), (15.11) and (15.17) hold. 8°. If n < m < k and m + n < k, then Eqs. (15.10), (15.11), (15.14), (15.15), (15.12) and (15.13) hold. 9°. If n < m < k andTO+ n = k, then Eqs. (15.10), (15.11), (15.14), (15.15), (15.12) and (15.13) hold. 10°. If n < m < k and m + n > k, then Eqs. (15.10), (15.11), (15.14), (15.15), (15.12) and (15.16) hold. 11°. If n < m = k, then Eqs. (15.10), (15.11), (15.14), (15.15) and (15.16) hold. 12°. If m = n = k, then Eqs. (15.10), (15.11) and (15.16) hold. Now, we will prove the theorems which treat the functional equation (15.1). T h e o r e m 15.1

If a = 1, m,n < k, m ^ n and m + n ^ k, then the

146

Functional Equations with Constant Parameters

general continuous solution of the functional equation (15.1) is /(U,V,W)

u +v+w

=

(?i(U + V + W)He

+

G 2 (U + V + W)Im f u - m U + V + ^ V> ) \ m+n+k) ( u+V+W\ + + G 3 (U + V + W)Re V - n , V m+n+kJ

+

g4(U +

+

V+ W ) I

U - mm + n+k

m

(v-n

U

+ m

V +

n

^

where Gi : V *-> £(V°,V) (1 < i < 4) are arbitrary continuous complex vector functions. Proof. Let m < n < k and m + n < k. On the basis of the expressions Eqs. (15.13) and (15.15), we obtain that the complex vector function h satisfies the equation /i(U + V , 0 ) - / i ( 0 , U + V) = / i ( U , 0 ) - / i ( 0 , U ) + / i ( V , 0 ) - / i ( 0 , V ) , and hence we get h(V, O) - h(0, U) = G'Re U + G"Im U, where G',G" 6 £(V°, V ) are arbitrary continuous functions of S. Thus, the equation (15.15) has the form h(U, V) = h(0, U + V) + G'Re U + G"Im U.

(15.18)

If we substitute the function h determined by Eq. (15.18) into Eq. (15.9), on the basis of the expression Eq. (15.12) we obtain k

/

m+n-l

Ti

\

Jfe—1

2 > o, Y, +j - E

h

/

T

fc-1

\

°' E -+"+^ = °- (15-19)

i=l \ j=l J i=k+l \ 3=0 J If we put T i = U , Tk = V and TV, = O (j ^ l.jfc) into Eq. (15.19), then we obtain / i ( 0 , U + V) = / i ( 0 , U ) + h(0, V), and hence we deduce h(0, U) = G 3 Re U + G 4 Im U,

(15.20)

Expanded Parametric Functional Equation

147

where Gs,G^ £ £(V°, V ) are arbitrary complex vector functions of S. On the basis of the expressions Eqs. (15.20) and (15.18), we obtain h(V, V) = G i R e U + G 2 I m U + G 3 Re V + G 4 Im V,

(15.21)

where G1=G' + G 3 and G2 = G" + G 4 . Let m < n < k and m + n > k. On the basis of the expressions Eqs. (15.16), (15.13) and (15.12), we get h(U + V, O) - / i ( - U - V, O) - h(0, U + V) = h{V, O) - ft(-U, O) - h(0,U)

+ h(V, O) - h(-V,

O) - h(0, V),

and hence we can derive that h(V, O) - / i ( - U , O) - /i(0, U) = G 3 Re U + G 4 Im U,

(15.22)

where Gz,G± € £(V°, V ) are arbitrary complex vector functions of S. On the basis of the identity Eq. (15.22), the equation (15.13) becomes h(V, V) = h{V + V, O) - / i ( - U , O) + G 3 Re V + G 4 Im V.

(15.23)

If we substitute the function h determined by Eq. (15.23) into Eq. (15.9), we obtain k

/m+n—1

$> »=i

E i=fc+l

m+n+k

Y, <+i. o + Y, V i=i

m+n+k

-

\

T

/

i=fc+i

I m+k — 1

h

I

k~\

\

j=o

y

- £ T m + n + i + i , ° (15-24)

y

m+fc

/

n—1

T

E \

\

h

Tm+n+i+i, O - E ^ - E ^ >

j=0

/

i=l

\

\

0=0.

j=0

/

For T i = U, Tm+k = V and Tj = O (j ^ l , m + k) from Eq. (15.24) we have h(U + U, O) = /i(U, O) + /i(V, O), where /i(U, O) = G x Re U + G 2 Im U. On the basis of this, the equation (15.23) becomes Eq. (15.24).

148

Functional Equations with Constant

Parameters

Let n < m < k and m + n < k. On the basis of the equations (15.15) and (15.13) we obtain h(U + V, O) - h{0, U + V) = h{\3,0)

- h{0, U) + h(V, O) - h(0, V),

from where it follows that / i ( U , 0 ) - / i ( 0 , U ) = G i R e U + G£lmU.

(15.25)

On the basis of the equality (15.25), the equation (15.15) becomes /i(U, V) = h(0, U + V) + G'Re U + G"Im U.

(15.26)

If we substitute the function h determined by Eq. (15.26) into Eq. (15.9), we obtain the following equation k

I

m+n+l

\

m+n+k

I

k—\

\

*=1

\

3=0

J

i=k+l

\

j=0

J

By putting T i = U, T m + n = V and Tj = O (j ^ l , m + n) in Eq. (15.27) we have /i(0,U,V) = /i(0,U) + /i(0,V), from which we conclude that h(0, U) = G 3 Re U + G 4 Im U. On the basis of this, the equation (15.26) becomes Eq. (15.21). Let n <m < k and m+n > k. On the basis of Eqs. (15.15) and (15.16), we obtain that the function h satisfies the functional equation h(V + V, O) + h(Ot - U - V) - h(Ot U + V) = /i(U, O) + h(0, - U ) - h(0,U)

+ h(V, O) + h(0, - V ) - h(0, V).

Therefore, the function / i ( U , 0 ) + h(0, - U ) - h(0,U)

is determined

by h(U, O) + h(0, - U ) - h(0,V)

= GiReU + G2ImU.

(15.28)

The equation (15.15), on the basis of the equality (15.28), becomes h(U, V) = h(Q, U + V) - h(Q, - U ) + G'Re U + G"Im U.

(15.29)

Expanded Parametric

Functional

Equation

149

The function h needs to satisfy the equation (15.9). If we substitute Eq. (15.29) into Eq. (15.9), we get k

I

m+n—1

i=l

\

j=0

fc+1

/

T

\

m+n+k

J

i=fc+l

\

m+n+k

I

k—1

\

j=0

J

m—1

\

5 > o, J2 *+i + E \°> -E -+"+^ (15-3°) m—1

T

h

T

\ I

- E M ° > - E ^ - E M o, £T ro+n+i+j = o. i=l

\

j=0

y

i=n+A;+l

\

j=0

/

If we put T i = U, Tn+k = V and T,- = O (j ^ l,n + k) into Eq. (15.30), we obtain MO, U + V) = h(0, U) + h(0, V), from which it follows that h(0, U) = G 3 Re U + G 4 Im U. On the basis of the last equality, the equality (15.29) has just the form Eq. (15.21). Consequently, in all four cases the function h is determined by Eq. (15.21). We denote the arbitrary complex functions G, (1 < i < n) of S by Gi(S) (1 < i < 4). On the basis of that, the equality (15.21) and the transformations Eqs. (15.8), (15.6), (15.3) and (15.2) there follows the proof of the theorem. • Theorem 15.2 If a — 1, m ^ n and m + n = k, then the general continuous solution of the functional equation (15.1) is determined by /(U,V,W)

=

( u+V+W \ W u +V-A—I—:S-, U + V + W \ m+ n+ k J

_ F f_ u _v + *3I±v±w V + +

m+n+k (

Gi(U + V + W)Re

+WN J u+V+W\

U-m—-—^—r-\ \ m+n+k J ( u + V Z+ W G 2 (U + V + W)Im U - m—^ -7\ m+n+k

where F : V2 i-+ V" and G, : V t+ £(V°,V) (i = 1,2) are arbitrary continuous complex vector functions.

150

Functional Equations with Constant Parameters

Proof. Let m < n. As in the proof of the previous theorem, on the basis of the expressions Eqs. (15.13) and (15.15) we may show that the function h has the form determined by Eq. (15.18). If we substitute the function h determined as previously into Eq. (15.1), on the basis of the expression Eq. (15.12) we conclude that the equality is valid and, consequently, the function / i ( 0 , U ) may be arbitrary. According to Eq. (15.12), for / i ( 0 , U ) we can put h(0,U)

=

F(\J)-F(-V),

where F is an arbitrary continuous function. On the basis of this, the equality (15.18) becomes h{U, V) = F(U + V) - F(-\J

- V) + G i R e U + G 2 ImU.

(15.31)

Let n < m. As in this case the equalities (15.13) and (15.15) hold and the function h has the form determined by Eq. (15.18). If we substitute Eq. (15.18) into Eq. (15.9), we obtain that the equation (15.9) holds, which means that / i ( 0 , U ) is an arbitrary function for which Eq. (15.12) holds. On the basis of this, we obtain that the function h in this case has the form determined by Eq. (15.31). If we take into account that the arbitrary continuous complex vector functions Gi and G2 are Gi(S) and G2(S) respectively, on the basis of the expressions Eqs. (15.8), (15.6), (15.3) and (15.2), there follows the proof of the theorem. D Theorem 15.3 If a = 1 and m < n = k, the functional equation (15.1) has a general continuous solution determined by /(U,V,W)

=

U 4-V + W F (1 U + V - (m + n)—-—?—-, U + V + W v ' m+n+k U +V +W F [-V + n , , U + V + W

m+n+k

+ Gi(U + V + W)Re (V +

G 2 (U + V + W)Im

V

U + V + W 1

—

•m + n + k U + V + W 1 — m +n+k

where F : V2 <-+ V and G{ : V >-» £(V°,V) (i = 1,2) are arbitrary continuous complex vector functions.

Expanded Parametric

Functional

Equation

151

Proof. On the basis of Eqs. (15.13) and (15.16), as in the proof of Theorem 15.1, we conclude that Eq. (15.22) holds, i.e., Eq. (15.23) is valid. If we substitute Eq. (15.23) into Eq. (15.9), since n = k, we obtain an identity which means that for /i(U, O) we may take an arbitrary continuous function. On the basis of this, for the function /(U, V) we obtain ft(U,V) =P , (U + V ) - J F ( - V ) + G i R e V + G 2 IinV.

(15.32)

From Eq. (15.32), on the basis of the transformations Eqs. (15.8), (15.6), (15.3) and (15.2) and taking into account that Gi and Gi are arbitrary continuous functions of S, we obtain that the function / has the form given in the Theorem 15.3. • Theorem 15.4 If a = 1 and n < m = k, then the general continuous solution of the functional equation (15.1) is /(U,V,W)

( u + V-t-W = F U+V-(m +n)—^—^-, U+V + W \ m+n+ k _ p.f_u + roU±V±W + W N \ m +n +k ) ( U+V+W\ + Gl(u+ V+ W ) R e ( u - m ^ ± n i r ) / u+V+W + G 2 (U + V + W)Im ( U - m — — ? - f v ' \ m+n+k

where F : V2 H> V and Gt : V H- £(V°,V) {i = 1,2) are arbitrary continuous complex vector functions. Proof. On the basis of Eqs. (15.15), (15.16) and (15.28), we obtain that the function h satisfies the equation (15.29). If the function h determined by Eq. (15.29) is substituted into Eq. (15.9), then the equation (15.9) becomes an identity, which means that h(0,XJ) can be substituted by an arbitrary continuous function -F(U). On the basis of this, the equality (15.29) has the form /i(U, V) = F(U + V) - .F(-U) + GxRe U + G2Im U.

(15.33)

According to the transformations Eqs. (15.8), (15.6), (15.3) and (15.2), from the equation (15.33), by putting Gi = Gi(S) and G2 = G2(S), it

152

Functional Equations with Constant

Parameters

follows that the function / has the form which is given in the statement of this theorem. • Theorem 15.5 If a — 1, m = n < k and 2m ^ k, then the general continuous solution of the functional equation (15.1) is determined by ( U+V+W = F U-m—^ ±—-, U + V + W \ m+n+k ( u+V+W - WV-n—^ 2--, U +V + W \ m+n+k ( U+V+W + Gi(U + V + W)Re [U-m—^ I L T v

/(U,V,W)

'

+

\

m+n+k

U + V + W G 2 (U + V + W)Im U - m m +n +k

where F : V2 i-+ V and d : V (-> C(V°,V) continuous complex vector functions. Proof.

(i = 1,2) are arbitrary

Let 2m < k. On the basis of the expression Eq. (15.17), we obtain h{V, V) + h(V, O) + h(0, h(-U

- U - V) + h(-V - V, U) = O,

- V, U) + h(U, O) + h(0, V) + h(V, - U - V) = O,

h(V, - U - V) + h(-U - V, O) + h(0, U) + h(XJ, V) = O. If from these three equalities we eliminate / i ( - U — V, U) and h(V, - U - V), we obtain 2/i(U, V)

=

- / i ( - U - V, O) - h(Ot - U - V) + h(U, O)

-

h(0,U) +

h(0,V)-h(V,0).

If we substitute this expression for h into Eq. (15.9), we get k

I

m+n—1

\

*

/

m+n—1

5 > - £ Ti+j, o + j > »=i m+n+k

y

j=o

+ E ME »=*+!

y

lk — 1

Tm

\i'=0

°> - £

»=i y \

+

+"+^' ° /

j=o

m+n+A

£ »=*:+!

h

/

T

\

(15-34)

'+; J

k—1

\

T

\°> E m+n+i+i = O. \

j=0

/

Expanded Parametric

Functional

Equation

153

If we put T i = U, Tm+n = V and TV,- = O (j ^ l , m + n) into Eq. (15.34), we have h(U + V, O) + h(Ot U + V) + (k - l)[/i(V, O) + h(0, V)] +

/»(-U, O) + h(0, - U ) + fc[/i(-V, O) + MO, - V ) ] = O. (15.35)

For V = O, from Eq. (15.35) we obtain h(-U, O) + h(0, - U ) + h(U, O) + h(0, U) = O,

(15.36)

i.e., the equality (15.35) has the form h(XJ + V, O) + /i(0, U + V) = h(V, O) + h(0,U)

+ /i(V, O) + h(0, V).

Hence, it follows that h(U, O) + h(0, U) = GiRe U + G 2 Im U. Thus we obtain that the function /i(U, V) is determined by h(U, V) = /i(U, O) + h(0, V), or /i(U, V) = F(U) - F(V) + G i R e U + G 2 Im V,

(15.37)

where we have put ^ ( U ) = - / i ( 0 , U ) . Let 2m > k. If we put T i = U, Tk = V and Tj = O (j ^ 1, A;) into Eq. (15.34), we get /i(U, O) + h(0, U) + (m + n)[ft(V, O) + h(0, V)] +

h(-V

+

(m + n - l ) [ / i ( - V , 0 ) + / i ( 0 , - V ) ] = 0 ,

(15.38)

- V, O) + h(0, U - V)

and hence for V = O we obtain Eq. (15.36), i.e., the equality (15.38) has the following form h(XJ + V, 0) + h(0, U + V) = fc(U,0) + / i ( 0 , U ) + / i ( V , 0 ) + / i ( 0 , V ) . Hence it follows that the function h has the form Eq. (15.37). Thus, on the basis of the expressions Eqs. (15.37), (15.8), (15.6), (15.3) and (15.2), we conclude that the function / has the form given in the theorem. •

154

Functional Equations with Constant Parameters

Theorem 15.6 If a = 1 and 2m — 2n = k, then the general continuous solution of the functional equation (15.1) is given by / ( U , V, W )

=

F ( U , V + W ) - F(V, W + U)

+

G(U + V, W ) - G ( W , U + V ) ,

where F, G : V2 >-> V are arbitrary complex vector functions. Proof. From the equation (15.34), which holds in this case, there immediately follows the equality /i(U, O) + h(0, U) + h(-V, O) + h(0, - U ) = O, i.e., we may put /i(U,0) + /i(0,U) = 2 G ( U ) - 2 G ( - U ) ,

(15.39)

where G is an arbitrary complex vector function. Since by virtue of Eq. (15.14) it holds 2/»(U, V)

=

-h(-U

-

h(0,V) +

- V, O) - h(0, - U - V) + /i(U, O)

h(0,V)-h(V,0),

on the basis of the expression Eq. (16.39) we have h(U, V) = F ( U ) - F(V) + G(U + V) - G ( - U - V),

(15.40)

where we introduced the notation 2F(U) = / i ( U , 0 ) - / i ( 0 , U ) . From the equalities (15.40), (15.8), (15.6), (15.3) and (15.2) there follows the proof of the theorem. • Theorem 15.7 If a — 1 and m = n = k, then the general continuous solution of the functional equation (15.1) is given by / ( U , V, W ) = F ( U , V, W ) - F ( V , W , U), where F : V3 i-> V is an arbitrary complex vector function. Proof. If we put Ti = U, T m + 1 = V and X,- = 0 ( j ' / l , m + l ) into Eq. (15.9), then we obtain h(XJ, V) + h(V, - U - V) + h(-V - V, U) = O.

Expanded Parametric

Functional

Equation

155

According to [M. Ghermanescu (1940)] the general solution of this equation is given by h(V, V) = F ( U , V) - F ( V , - U - V),

(15.41)

where F : V2 i->- V is an arbitrary complex vector function. On the basis of the expressions Eqs. (15.41), (15.8), (15.6), (15.3) and (15.2), we conclude that the function / is determined by the form given in the theorem. • II. Let

m+n+k

^ 1. If we put

a

/(U,V,W)= k+m

g{V + a

m

V

+ a W,

(15.42) m+n

V + a

n

W

n+k

XJ + akV),

+ a XJ, W + a

then the functional equation (15.1) becomes m+n+k

I m —1

i=l

\j=0 ra-l fc-1

+ 2^, _n-1—jrj a

Am+i+j + 2_^i a

j=0 k—1 , \ "* m+n+k-1-jrj

n—1

E

a

Am+i+j

J'=0

a

+ 2_j j=0

Am+n+i+j,

j=0 m—1 , \ "* &m+n+i+]

+ 2__, j=0

m+n-l-jy a

^i+3'

fc-1

E

„fc-l-jy

+

n—1 n—1 ^am+n+fc-l-iz.+j. + ^a"+*-l-i j'=0

\ Z m +

.

j=0

+ j

.

=

0

,

J

i.e., m+n+k

/

I m+n+k — 1

y

5 I

2-(

i=0

\

j'=0

a3

m+n+fc—1

2m-l+i-j)

2^

0"'Z m + n _i + j_j,

j=0

m+n+k—1 / „

\ O^^m+n+k-l+i-j

I = O.

156

Functional Equations with Constant

Parameters

This transformation of the equation (15.1) is possible since am+n+k If we introduce new variables Tj by the relations

^ 1.

m+n+k — 1

Tj =

ajZm-i+i^j

^^ j=o

(1 < i < m + n + k),

then the previous equation becomes m+n+k

J2 g(Ti,Ti+n,Ti+n+k)

= O.

(15.43)

Now we will give the following results. Lemma 15.8 determined by 1°.

The general solution of the functional equation (15.43) is

g(\J, V, W ) = F(U) - F ( W ) - H(V) {m^n^k^m,

2°.

m ^ n + k,

n ^ m + k,

m = n + k),

g(U, V, W ) = F ( W ) - F(U) - G(U, V) - G(V, U) (m ^ n ^ k y£ m,

4°.

6°.

k = m + n),

g(U, V , W ) = G(U, V) - G ( V , W )

(m ^ n = k,

m ^

g(U, V, W ) = K(U, V) - tf(V, W ) + G(W, U) - G(U, W ) (m ^ n = k,

7°. m + n), 8°.

n — m + k),

g(U, V, W ) = F(U) - F(V) + G(V, W ) - G(W, V) (m ^ n ^ k ^ m,

5°. n + k),

k ^ m + n),

g(U, V, W ) = F(U) - F(V) - G(U, W ) - G(W, U) (m^n^k^m,

3°.

H(W)

g(XJ,V,W)

m = n + k),

= G(U, V) - G(W,U)

(m = n ? k,

k ?

g(V, V, W ) = K(V, V) - K(W, U) + G(V, W ) - G(W, V)

(m = n^k,

k = m + n),

Expanded Parametric

9°. k + m), 10°.

Functional

Equation

g(U, V , W ) = G(V,W) - G(W,U)

157

(m = A; ^ n,

n ^

g(V, V, W ) = K ( U , V) - iT(V,U) + G(V, W ) - G ( W , U ) (m = k^n,

n = k + m),

11°. 5 ( U , V, W ) = L(U, V, W ) - L(V, W , U ) where F,H : V >-* V, G,K : V2 ^> V and L : V3 ^ complex vector functions.

(m = n = k), V are arbitrary

Proof. Since the proofs of the particular cases are similar or completely the same, we will prove the lemma only in the cases 3° and 8°. 3°. If all variables in Eq. (15.43), except for Z;, Zi+n and Zj + n + jt, are equal to some constant, we obtain g(Zi, Zi+n, Zi+n+jfc) = F(Zi+n+k)

+ K(Zi, Zj+ n ),

i.e., g(\J,V,W)

= F(W) + K(TJ,V).

(15.44)

Hence, the equation (15.43) becomes m+n+k

m+n+k

Y, F(Zi)+ Yl K(ZuZi+n) = O. i=l

i=l

If we put Zr = O (r ^ i, i + n) in the above equation, we have F(Zi) + F(Zi+n)

+ K(Zh Zi+n) + K(Zi+n,

Zi) = O,

and thus it follows that K(V, V) = G(U, V) - G(V, U) - F ( U ) . Now, the equality (15.44) becomes g(U, V, W ) = F ( W ) - F(U) + G(U, V) - G(V, U).

8°. If we put Z r = O (r ^ i, i + n, i + n + k) into Eq. (15.43), then the equation (15.43) becomes g(Zi,Zi+n,Zi+n+k)

— K(Zi,Zi+n)

+ Ki(Zi+n,Zi+n+k)

+

K2(Zi+n+k,Zi),

158

Functional Equations

with Constant

Parameters

i.e.,

g(V,V,W)

= K(U,V)

+

K1(V,W)+K2(W,V),

where K, K\, K2 : V2 i-> V are arbitrary complex vector functions. By a substitution of the expression obtained for g into Eq. (15.43), we get m-\-n-{-k

m~\-n-\-k

m-\-n+k

J2 K(Zi,Zi+n)+

J2 K1(Zi,Zi+n)+

]T K2(Zi,Zi+n) = 0.

j=l

i=l

i=l

(15.45) Hence, for Zr (r ^ i,i + k) we obtain K2(Zi, Zi+k) + K2(Zi+k,Zi)

+ H(Zi) + H(Zi+k)

= O,

where H : V i-> V' is an arbitrary complex vector function. Prom the above equation it follows that the function K2 has the form K2(V, V) = G(U, V) - G(V, U) - H(V),

(15.46)

where G : V2 i-> V is an arbitrary complex vector function. If we put Eq. (15.46) into Eq. (15.45), we obtain m+n+k

Y,

[K{Zi,Zi+n)+Kl{ZuZi+n)

+ H{Zi)} = 0.

(15.47)

i=l

For Zr = O (r ^ i,i + n) from the above equation we obtain Ki(U,V)

= -K(U,V)

- 2H(\J) -

2H(V).

By putting the above obtained expression for K\ into Eq. (15.47), we have m+n+k

£ ff(z*) = o, i.e., ff (U) = O. On the basis of the previous equalities, the function g is determined by g(U, V, W ) = K(V, V) - K(W, U) + G(V, W ) - G(W, V).

Expanded Parametric Functional Equation

159

According to the previous lemma and the transformation Eq. (15.42), we obtain the following theorems. Theorem 15.9 If am+n+k ^ 1, m^n^k^m, m^n + k, n^m + k and k ^ m + n, then the general solution of the functional equation (15.1) is /(U,V,W)

=

F ( U + ak+mV

+

G{V + am+nW

+ a m W ) - F ( W + an+kU + anV)-G(W

+ an+kV

+ akV) + akV),

where F, G : V i-» V are arbitrary complex vector functions. Theorem 15.10 / / am+n+k ^ 1, m ^ n ^ k ^ m and m = n + k, then the general solution of the functional equation (15.1) is /(U,V,W)

=

F ( U + ak+mV

+ a m W ) - F ( V + a m + n W + a"U)

+

G(U + a * + m V + o m W , W + a n + f c U + a fc V)

-

G ( W + a n + * U + a*V, U + a f c + n V + a m W ) ,

where F : V i-> V' and G : V2 i-> V are arbitrary complex vector functions. Theorem 15.11 If am+n+k ^ 1, m ^ n ^ k ^ m and n = m + k, then the general solution of the functional equation (15.1) is /(U.V.W)

=

F ( W + a n + f c U + a*V) - F ( U + a * + m V + a m W )

+

G(U + a f c + m V + a m W , V + a m + n W + a n U )

-

G(V + a m + n W + o n U, U + ak+mV

+ amW),

iu/iere F : V n V and G : V2 (-»• V are arbitrary complex vector functions. Theorem 15.12 J/ a m + n + f c ^ 1, m ^ n ^ k ^ m and k = m + n, then the functional equation (15.1) nas a general solution /(U,V,W)

=

F ( U + a f c + m V + a ro W) - F ( V + a m + n W + a"U)

+

G(V + a m + n W + a n U, W + an+kU

-

G(W + an+kV

+ akV)

+ afcV, V + a m + " W + a n U),

where F : V H-> V and G : V2 H* V are arbitrary complex vector functions.

160

Functional Equations with Constant

Parameters

T h e o r e m 15.13 Ifam+n+k ^ 1 andm ^ n + k, then the general solution of the functional equation (15.1) is / (U, V, W )

=

F(V + ak+mV

-

F ( V + am+nW

+ amW,

V + am+nW

+ a n U , U + ak+mV

+ anV) +

amW),

where F : V2 i-> V w an arbitrary complex vector function. T h e o r e m 15.14 7/ a m + n + f c ^ 1, m ^ n = k and m = 2n, then the functional equation (15.1) ftas a general solution determined by /(U,V,W)

=

F ( U + a * + m V + a m W , V + a m + n W + a"U)

-

F ( V + a m + n W + a n U , W + a n + f c U + akV)

+

G ( W + a n + * U + a*V, U + ak+mV

-

G(U + ak+mV

+ amW,

+ amW)

W + a n + f c U + a*V),

where F, G :V2 *-> V are arbitrary complex vector functions. T h e o r e m 15.15 If am+n+k ^ 1, m = n •£ k and k ^ m + n, then the functional equation (15.1) has a general solution given by / ( U , V, W )

=

F ( U + ak+mV

+ a m W , V + a m + n W + anU)

-

F ( W + a n + f c U + afcV, U + ak+mV

+

amW),

where F : V2 •-> V is an arbitrary complex vector function. T h e o r e m 15.16 If am+n+k ^ 1, m = n ^ k and k - m + n, then the functional equation (15.1) Aas a general solution / ( U , V, W )

=

F ( U + a* + m V + a m W , V + am+nW

-

F ( W + an+kV

+

G(V + o m + n W + a n U , W + a n + f c U + a*V)

-

G ( W + a n + f c U + a fc V, V + a m + n W + a n U ) ,

+ akV, U + ak+mV

+ anV) +

amW)

where F, G : V2 t-> V are arbitrary complex vector functions. T h e o r e m 15.17 If am+n+k ^ 1, m = k ^ n and n ^ m + k, then the functional equation (15.1) /ias a general solution / ( U , V, W )

=

F ( V + a m + n W + a n U , W + a"+*U + akV)

-

F ( W + a n + f c U + a*V, U + ak+mV

+ amW),

General Expanded Parametric Functional Equation

161

where F : V2 >-¥ V is an arbitrary complex vector function. T h e o r e m 15.18 / / am+n+k ^ 1, m = k ^ n and n — m + k, then the functional equation (15.1) has a solution given by / ( U , V, W )

=

F ( U + o* + r a V + a m W , V + a r o + n W + a n U )

-

F ( V + a m + " W + anU, U + afc+mV + a m W )

+

G(V + a m + n W + a n U , W + a n + f c U + a*V)

-

G ( W + an+kXJ + a*V, U + a m + n V + a " W ) ,

where F , G : V 2 ^ V are arbitrary complex vector functions. T h e o r e m 15.19 If am+n+k ^ 1, m = n = k, then the general solution of the functional equation (15.1) is

/(U.V.W) k+m

+ amW,

=

F(U + a

V

-

F(V + am+nW

V + am+nW

+ a n U , W + an+kV

+ o"U, W + an+kXJ + akV) + akV, U + ak+mV

+

amW),

where F : V3 t-> V is an arbitrary complex vector function. III. If am+n+k = 1, this case is very difficult and up to now we are not able to solve the functional equation (15.1). 16

General Expanded Parametric Functional Equation

All notations for the vectors are the same as in Sec. 13. The general expanded parametric functional equation [I. B. Risteski et al. (2001A)] m+n+k

I m—1

n—1

*=i

\i=o

j=o k-1

/ Ja 3=0 \Jm+n+k+i

=

\ 3

Zi+m+n+j J = O J

Jit ^m+n+k+i = " « /

where a is complex number, will be solved here.

162

Functional Equations with Constant

Parameters

Further, we will consider the solving of the functional equation (16.1). For the equation (16.1) we will determine the general solution only if am+n+k

.J. j

Now, we transform Eq. (16.1). If we put /i(U,V,W)= k+m

Si(U + a

V

m

(16.2)

m+n

+ a W, V + a

W

+ o"U, W + a

n+fc

U + a*V),

then the equation (16.1) becomes m+n+k

Im—1

n—1

»=1

\ J=0

3=0 fc-1 _i_ \ ' n r a + * - l - i 7 . . + 2_^ a Am+n+i+3 > 3=0

n—1

t—1 ^ m + i + j + 2_j j=0

a J=0

m—1 + 2_^/ ° j=0

^m+n+1+3

k—1

«+J'

m—1 O

Am+n+i+j

j=0

+ 2_, j=0

a

^*+J n-1

\

+ Y/an+k-1-iZrn+i+j\ i=o

=0, /

i.e., m+n+k ^ »=1

Im+n+k—1 ft \

XI 3=0

m+n+fc—1 J

a Zm-l+i-j,

£ j=0

^Zm+n-l+i-j, m+n+k—1

/ , 3=0

(16.3) \

rfZm+n+k-l+i-j

J J

=

Since the linear forms m+n+k — 1

Tj =

£

o 3 Z m _i + j_j

(1 < i < m + n + k)

3=0

are linearly independent, it is possible to introduce new variables T , which are determined by the above relations.

O.

General Expanded Parametric

Functional Equation

163

Therefore, the functional equation (16.3) takes the following form m+n+k

Y,

9i(Ti,Ti+n,Ti+n+k)

= O.

(16.4)

For the last equation the following lemma holds. L e m m a 16.1 The general solution of the functional equation (16.4) is given by the equalities 1°.

9i(V,

V, W ) = Fi(U) - Fi+n+k(W)

- Hi(V) - Hi+k(W)

- Ait

m+n+k

E *= °

(m^n^k^m, 2°.

m ^ n + k,

gi(\J,V,W)

n^

= Fi(V)-Fi+k(W)

m + k,

k^m

+ Gi(\J,W)

Si(U, V, W ) = Fi(V)-Fi+k(W)-Gi+m(W,U)-Ai+m

+ n), (l
(m+1 < i < 2m),

m

E^ = ° i=l

(m ^ n ^ k ^ m, 3°. ft(U,

5i(U,V)W)

m^n

+ k),

= F i ( W ) - F i + m ( U ) + Gi(U,V)

V,W) = Fi(W)-.Fi+m(U)-Gi+m(V)U)-Ai+m

[rn^n^k^m, 4°. ffi(U,V,W)

fli(U,V,W)

(n + 1 < t < 2n),

n = m + k),

= F<(U)-Fi+n(V) + Gi(V)W)

= F«(U) - F i + n ( V ) -Gi+k(W,V)

i=l

(1 < » < n),

-Ai+k

(1 < t < A), (k +

l
164

Functional Equations with Constant

(m ^ n 7^ k ^ m, 5°.

Parameters

A; = m + n),

#(U, V, W) = Gi(U, V ) - G i + n ( V , W)+Ai

(1 < i < m+n+fc),

m+n+fc

E ^=° i=l

{m^n 6°.

— k,

m^n

+ k),

0,(11, V, W) = i^(U, V) - Ki+n(V, W) + Gi(W, U)

(1 < i <

TO),

W ) - G i + m ( U , W)

(m 7^ n = k, 7°. n + fc),

9 i (U,V,W)

(m+1 < » < 2m),

m — n + k),

= Gi(U,V)-Gi+n+*(W,U)+,4i

(1 < i
E ^=° i=l

(m = n^k, 8°. i < Jfe),

k ^m + n),

5i(U,V,W)=^(U,V)-JFfi+n+fc(W,U)+Gi(V,W)+^

(1 <

E^ = °' i=l S i (U,V,W)

= /ri(U,V)-^n+fc(W,U)-Gi(V,W) (m = n^k,

9°. n + k),

5 i (U,V,W)

k = m + n),

= Gi(U,V)-Gi+m+n(W,V) + ^i m+n+k

E ^=° i=l

(* + l < i < 2k),

(l
General Expanded Parametric Functional Equation

(m = k ^ n, 10°. i < n),

gi(U,V,W)

n^m

165

+ k),

= Ki(V,W)-Ki+k(W,U)+Gi(U,V)+Ai

(1 <

n

9i(U,V,W)

= ^(V.Wj-i^i+^W.Uj-Gi+nCV.U) (m = k^n,

11°.

fli(U,

(n + 1 < » < 2n),

n = m + k),

V, W ) = Li(U, V, W )

(1 < * < 2m),

ft(U, V, W ) = -L<(V, W , U ) - L i + 2 m ( W , U , V)

(2m + 1 < * < 3m),

(m = n = k). In all cases the functions Ft, Hi : V i-> V , Gt, Ki : V2 <-> V and are arbitrary complex vector functions such that i^-|_n_j_rn_^^ — Fi, Gi+n+m+k = Gi,- ••. Also, for the arbitrary constant complex vectors Ai the equalities Ai+n+m+k = Ai hold. Proof. Since the statements of the lemma for particular cases may be proven in a similar way, therefore it is sufficient to prove the lemma only for some cases, for example 4° and 5°. 4°. If in Eq. (16.4) all variables, except T j , T j + n and Ti+n+k, to some constant, we obtain

are equal

9i(Ti,Tj+„,Tj+ n +fc) = Fi(Ti) + ifj(Ti4- n ,Tj + „+fc), i.e., gi(U,V,W)

= Fi(V)+Ki(V,W),

(16.5)

where Fi and Ki (1
£ i=l

m+n+k

Fi(Ti)+

£ t=l

Ki(Ti,Ti+n+k)

= 0,

166

Functional Equations with Constant

Parameters

i.e., m+n+k

m+n+k

/ , Fi+n(Ti+n)

+

i=l

2_^ Ki(Ti,Ti+n+k)

= O.

i=l

If we put Tr = O (r = i + n, i + n + k) into the previous equality, we obtain Fi+n(Ti+n)

+

Fi+n+k(Ti+n+k)

+ Ki(Ti+n,Ti+n+k)

(16.6)

+

Ki+k(Ti+n+k: T^i+n) + A{+k = O,

where A{ are arbitrary constant complex vectors. From Eq. (16.6) we obtain Ki(V, W ) = -Ki+m(W,

V) - Fi+n(V)

- Fi+n+k(W)

-

Ai+k,

i.e., the equation (16.5) takes the form ft(U,V,W)

-

5i(U,V,W)

=

F{(U)+^(V,W)

(l
Fi(V)-Ki+k(W,V)

-

Fi+n(V)

- Fi+n+k(W)

- Ai+k

(k +

l
such that

If we introduce new functions Gi by Gi(V,W)

=

Ki(V,W)+Fi+n(V),

the previous equations take the following form 0i(U,V,W)

=

Fi(V)-Fi+n(V)

9i(V,V,W)

=

Fi(\J) -

-

Gi+k(W,V)-Ai+k

+ Gi(V,W)

(1 < * < *),

Fi+n(V)

where k i=l

(k +

l
General Expanded Parametric

Functional Equation

167

5°. If T j = O (j ^ i,i + n,i + n + k), from the equation (16.4) we obtain
+ Mi(Tj+ n ,T;+ n +fc),

i.e., 9i(V,

V, W ) = ^ ( U , V) + Mi(V, W ) ,

(16.7)

where Ki and M* are arbitrary complex vector functions. On the basis of the relation Eq. (16.7), the functional equation Eq. (16.4) becomes m+n+k

E

m+n+k

Ki(Ti,Ti+n)+

]T

M ; ( T i + n , T i + r i + j t ) = O,

i.e., m+n+k

]T

m+n+k

Ki(Ti,Ti+n)+

i=l

Yl

Mi+m+n(Ti,Ti+n)

= O.

(16.8)

i=l

If we put TV = O (r ^ i, i + n) into Eq. (16.8), then Ti+n) + Mi+m+nCFi, Tj_|_n) + Pj(Tj) — Qj+ n (Tj+ n ) — O, (16.9) where Pi and Qi are arbitrary complex vector functions. On the basis of the above equation (16.9), Eq. (16.8) becomes m+n+k

m+n+k

i=l

i=l

from which we o b t a i n m+n+k

Qi(Ti) = Pi(Tt) + Bt,

£

B

i = °-

i=l

Therefore, the equality (16.9) takes the form Ki(Ti,Ti+n)

+ Mt_|_m+fc(Tj,Ti+n) + Pi(Ti) — Pj+ n (Ti+ n ) — Bi+n = O.

On the basis of this equality and Eq. (16.7), we have 9i(V,

V, W ) = Ki(V, V) - Ki+n(V,

W ) + Pi+„(U) - P i + 2 n(V) +

Bi+2n,

168

Functional Equations with Constant

Parameters

I.e., m+n+k

gi(U,V,W)

= Gi(XJ,V)-Gi+n(V,W)+Ai,

£

A> = O,

where we put Gi(lJ,V)

= Ki(\J,V)

+ Pi+n(U),

Ai = Bi+2n.

D

On the basis of Lemma 16.1 and the transformations Eq. (16.2), there follow the theorems which treat the functional equation (16.1). Theorem 16.2 / / a m + n + k ^ 1, m^n^k^m, mj^n + k, n^m + k and k ^ m + n, then the general solution of the functional equation (16.1) is /<(U,V,W)

=

Fi(V + am+nW

+ anV) - Fi+k(W

+

G*(U + a f c + m V + a m W , W + a n + f e U + akV) (l
/i(U,V,W)

m+n

+ an+kTJ + akV)

m), n

W + o U ) - F i + i k ( W + o n + * U + a*V)

=

F i (V + a

-

Gi+m(W

+ an+kU + a*V, U + ak+mV

-

Ai+m

(m + 1 = 0, t=i

where Fi : V i-> V and Gj : V2 t-> V are arbitrary complex vector functions, and Aj are arbitrary constant complex vectors such that

m+n+k

^

Ai = O.

i=i

Theorem 16.3 / / a m + n + k ^ 1 , m^n^k^mandm-n the general solution of the functional equation (16.1) is /i(U,V,W)

=

F^V + am+nW

+ anXJ) - Fi+k(W

+

G<(U + a * + m V + amW,

+ k, then + an+kV

+ akV)

W + a n + f c U + akV)

(1 < i < m ) , /i(U,V,W)

=

Fi(V + am+nW

+ anV) - Fi+k(W

+ an+kXJ + akV)

-

Gi+m(W

+ an+kXJ + akV, U + a f c + m V + a m W )

-

Aj+m

(m + 1 V and d : V2 i-* V ore arbitrary complex vector functions, m

and Ai are arbitrary constant complex vectors such that ^2 Ai = O. t=i m+n+k

T h e o r e m 16.4 If a ^ 1, m ^ n ^ k ^ m and n = m + k, then the general solution of the functional equation (16.1) is fi(V, V, W )

=

Fi(W + an+k\J

+

d(U

+ ak+mV

+ akV) - Fi+m(U + amW,

+ ak+mV

+

amW)

V + a m + n W + o n U)

(1 V' and Gi : V2 i-> V' are arbitrary complex vector functions, n

Ai are arbitrary constant complex vectors such that ^2 Ai = O. T h e o r e m 16.5 If am+n+k ^ 1, m ^ n ^ k ^ m and k = m + n, then the general solution of the functional equation (16.1) is given by fi (U, V, W )

=

Fi (U + ak+mV

+ a r o W ) - F i + n (V + a m + " W + a n U )

+

Gi(V + a m + n W + a n U , W + o n + f c U + a*V) (1 < i < k),

fi(V, V, W )

k+m

V

+ amW)

+ a m + " W + a"U)

=

Fi(U + a

- Fi+n(V

-

Gi+k (W + a n + * U + a* V, V + a m + n W + a n U )

-

Ai+A

(* + 1 < i < 2k),

where F{ : V <-¥ V and G, : V2 »-• V are arbitrary complex vector functions, k

Ai are arbitrary constant complex vectors such that ^2 Ai = O. »=i m+n+k

T h e o r e m 16.6 / / a ^ 1, m ^ n = k and m j£ n + k, then the general solution of the functional equation (16.1) is fi(U, V, W )

=

Fi(XJ + ak+mV

-

Fi+n(V

+ a m W , V + a m + n W + anU)

+ am+nW

+ a n U , W + an+kV

(1 -> V are arbitrary complex vector functions, Ai are arbitrary m+n+k

constant complex vectors such that

Yl

Ai — O.

i=l

Theorem 16.7 If am+n+k / 1, m ^ n = i; and m = n + k, then the general solution of the functional equation (16.1) is determined by fi(XJ, V, W )

=

Fi(V + ak+mV

+ amW,

-

Fi+n(V

+

Gi(W + a n + * U + a fc V, U + ak+mV

+ am+nW

V + am+nW

+ anV)

+ o n U, W + an+kU

+ akV)

+ amW)

(1 < i < TO), fi(U, V, W )

=

Fi(V + ak+mV

+ amW,

-

Fi+n(V

-

Gi+m(U + a f c + m V + a m W , W + an+k\J

+ am+nW

V + a m + " W + anU)

+ a n U , W + an+kV

(m + l
+ akV) + akV)

2m),

where Fi, G, : V2 i-> V are arbitrary complex vector functions. Theorem 16.8 If am+n+k ^ 1, m = n ^ k and k ± TO + n, then the general solution of the functional equation (16.1) is /i(U, V, W ) = Fi(U + ak+mV - F i + n + f c ( W + an+kU

+ a m W , V + am+nW

+ akV, U + ak+mV

+ anU)

+ amW)

+ Ai

(1 V are arbitrary complex vector functions, Ai are arbitrary m+n+k constant complex vectors such that Yl Ai = O. i=l

Theorem 16.9 If am+n+k ^ 1 and k = m + n, then the general solution of the functional equation (16.1) is fi(V, V, W )

=

Fi(U + ak+mV

+ omW, V + a m + n W + anU)

-

Fi+n+k (W + an+kU + a*V, U + ak+mV

+

d(V

+ a m + n W + a n U , W + an+k\J (1 < * < *),

+ amW)

+ akV) + ^

General Expanded Parametric

/i(U, V, W )

Functional Equation

+ amW,

171

=

Fi(V + ak+mV

V + a m + n W + a"U)

-

Fi+n+k (W + an+k\J + a* V, U + ak+mV

-

Gi+Jfc ( W + an+kV

+ amW)

+ a fc V, V + a m + n W + a n U )

(fc + 1 V' are arbitrary complex vector functions, and Ai are k

arbitrary constant complex vectors such that ^2 Ai = O. i=l m+n+k

T h e o r e m 16.10 / / a ^ 1, m - k ^ n and n ^ m + k, then the general solution of the functional equation (16.1) is = Fi(U + a * + m V + a m W , V + am+nW

fi(V,V,W)

+ a"U)

- F i + m + n ( W + a" +fc U + afcV, V + a m + n W + a"U) + At (1 -+ V' are arbitrary complex vector functions,

and Ai are

m+n+k

arbitrary constant complex vectors such that

Yl

Ai = O.

i=l

T h e o r e m 16.11 / / am+n+k ^ 1, m = k ^ n and n = m + k, then the general solution of the functional equation (16.1) is determined by /i(U,V,W)

=

Fi(V + am+nW

-

Fi+k(W

+

d(U

+ a n U , W + an+kV

n+k

+a

+ ak+mV

\J

k

k+m

+ a V, U + a

+ amW,

V

V + am+nW

+ a* V) + amW) + anU) + ^

(1 < i < n ) , /i(U, V, W )

=

Fi(V + a m + n W + a"U, W + an+kU

+ akV)

-

F i + f c (W + an+kXJ + akV, U + ak+mV

-

G i + n ( V + o m + n W + anU, U + afc+mV + a m W )

+ amW)

(n + 1 V are arbitrary complex vector functions, and Ai are n

arbitrary constant complex vectors such that ^2 Ai = O. i=l

172

Functional Equations with Constant

Parameters

T h e o r e m 16.12 If am+n+k ^ 1, m = n = k, then the general solution of the functional equation (16.1) is /i(U,V,W) =

k+m

Fi(V + a

V

+ a m W , V + am+nW

+ o n U, W + an+kU

+ a*V)

(1 < i < 2m), ^(U.V.W) = -

-Fi(V+aTO+nW+anU, W+an+*U+a*V, U+a*+roV+omW) Fi+2m(W+an+kV+akV,V+ak+mV+amW,V+am+nW+an\J) ( 2 m + 1 < i < 3m),

where Fi : V3 i-» V are arbitrary complex vector functions. If a = 1 or am+n+k = 1 (a ^ 1), then the solution of the functional equation (16.1) is very complicated and up to now we have not been able to obtain it.

Chapter 4

Functional Equations with Constant Coefficients

In [I. B. Risteski (to appear A)] the solution of one class of homogeneous complex vector functional equations is obtained by the method of intersections, while in [I. B. Risteski and V. C. Covachev (2000)] several classes of homogeneous complex vector functional equations are solved on the basis of the well-known method of elimination of variables. For the purpose of expanding of our investigations, here we will propose a matrix method suitable for both homogeneous and nonhomogeneous complex vector functional equations with constant complex coefficients. The results presented here are obtained in [I. B. Risteski (to appear B)]. Now we will introduce the following notations. Let V be a finite dimensional complex vector space and let a mapping / : V" i-> V exist. Throughout this chapter Z; (1 < i < n) are vectors in V. We assume that Zj = (zn{t), • • • ,z,n(t))T, where Zij{t) (1 < i < n) are complex functions T and O = (0,0, • • • , 0) is the zero vector in V.

17

Matrix Equations

Let A be an n x n matrix. Suppose that by elementary transformations the matrix A is transformed into A = P1DP2, where Pi and P2 are regular matrices and D is a diagonal matrix with diagonal entries 0 and 1 such that the number of the units is equal to the rank of the matrix A. The matrix B = P2~1DP^1 satisfies the equality ABA = A. This means that the matrix equation AX A = A has at least one solution for X. 173

174

Functional Equations with Constant

Coefficients

If A satisfies the identity AT + kiA1"1

+ • • • + fcr_iA = O

(fcr-i/0),

then the matrix (Ar~2 + kxAr~3 + ••• + kr-2I)

X = —

{I is the unit n x n matrix)

/c,—l

is also a solution of the equation AX A = A. Now we will prove the following theorem. Theorem 17.1 If B 1° AX = O <£> 2° XA = O <£> 3° AX A = A o matrices); 4° AX = A <£> 5° XA = A «•

satisfies the condition ABA — A, then X = (I-BA)Q, (X andQ arenxm matrices); X — Q(I-AB), (X andQ aremxn matrices); X = B + Q- BAQAB (X and Q are n x n X = I + (I - BA)Q; X =I + Q(I-AB).

Proof. The theorem will be proved only for the case 3°, because the other cases can be proved analogously. 3°. Let X = B + Q- BAQAB. Further it holds AX A = ABA + AQA - ABAQABA,

=>

AX A = A + AQA - AQA

ABA = A

=>

AXA = A.

Conversely, assume that AX A — A, then it holds B + (X - B) - BA(X -

= X-

BAXAB

+ BABAB

= X-

B)AB

BAB + BAB = X.

Thus AXA = A

=>

X = B + Q-BAQAB,

for

Q = X - B.

n

Homogeneous

Functional Equations with Constant

Coefficients

175

If X is an n x fc-matrix, according to 1° all the solutions of the homogeneous system of equations Xi

x2

=0

have the following form Xi

x2

«2

=

(I-BA)

(«!,••• , u n are arbitrary). L «n J

18

Homogeneous Functional Equations with Constant Coefficients

Now we will prove the following results. Theorem 18.1 The general solution of the basic cyclic complex vector functional equation with complex constant coefficients n

B(/) = ^ o i / ( Z i , Z i + i > - - - , Z j + n _ 1 ) = 0

(Zn+i = Zi),

(18.1)

t=i

where a; formula

(1 < i < n) are complex constants, is given by the following

/(Zi,Z2,--- ,Zn) /(Z2,Z3,---,Z1) /(ZnjZi, ••• ,Zn_i) _

h{7i\,7i2,

= B

h(Z2,Z3,---

••• ,Zn)

,Zi)

/i(Zn,Zi, •• • ,Zn_i)

(18.2)

176

Functional Equations

with Constant

Coefficients

where

A =

' ax an

a2

•

an

' h

b2

•

ax

•

an-i

bn

h

•• •

. 0,2

a3

. b2

h

••

are nonzero nxn

ai

and

B =

.

•• •

bn bn-l

(18.3)

h

cyclic matrices with complex constant elements, such that AB = O,

(18.4)

O is the nxn zero matrix and h is an arbitrary complex vector function with values in V. Proof.

If we permute successively the vectors in Eq. (18.1), we get

Ol/(Zi, Z2, • • • , Tin)

+ 02f{2i2, Z3, • • • , Z„, Zi) + \-anf (Zn,Zi,-•• ,Zn-i) - O, On/(Zi, Z 2 , • • • , Z„) + a i / ( Z 2 , Z 3 ) • • • , Z n , Zi) + • • • + o,n-if(Zn,Zi, • • • ,Z„_i) = O, a2f(Zi,Z2,

••• , Zn)

+ +

a 3 / ( Z 2 , Z 3 , • • • , Z n , Z\) h a i / ( Z n , Z i , - •• ,Z„_i) = O,

i.e., in a matrix form AF = 0,

(18.5)

where A is given by Eq. (18.3), O

/ ( Z i , Z 2 ) - •• , Z n )

/ ( Z 2 , Z 3 , - - - ,Zi) F =

and

0 =

o

(18.6)

/(Z„,Zi,- • • ,Zn_i) _

o

We shall write / G F to express the first equality of Eq. (18.6). The necessary and sufficient condition for the system (18.5) to have a nontrivial solution is det A = 0.

(18.7)

Homogeneous Functional Equations with Constant Coefficients

177

Let / ( Z i , Z 2 , • • • ,Zn)

—y

bih(Zi,Zi+i,

• • • ,Zi+n^{)

(Zn+i

= Zj),

(18.8)

i=l

where bi (1 < i < n) are complex constants such t h a t the m a t r i x B defined by Eq. (18.3) satisfies Eq. (18.4) and h is an arbitrary complex vector function with values in V. By a cyclic permutation of the vectors in Eq. (18.8), we obtain / ( Z i , Z2, • • • , Z„) =

6ifc(Zi, Z 2 , • • • , Z „ ) + b2h(Z2,Z3r H h 6„/i(Zn, Zi, • • • , Z n _ i ) ,

• • , Zi)

/ ( Z 2 , Z 3 , • • • , Zx) =

&nMZi, Z 2 , • • • , Z „ ) + 6i/i(Z 2 , Z 3 , • • • , Z i ) H + &rj-iMZn,Zi, • • • , Z n _ i ) ,

/ ( Z n , Z i , • • • ,Z„_i) =

62/i(Zi,Z2, • • • ,Z„) + 63/i(Z2,Z3, • • • , Z i ) H \-bih(Zn,Zi,•• , Z „ _ i ) ,

i.e., in a m a t r i x form F = BH

(18.9)

where

H

MZi,z 2 , • • • , z n ) ft(Z2,Z3,--- ,Zi)

(18.10)

. Mz„,Zi,• • • ,z n _i) _ and F is defined in Eq. (18.6). After a multiplication of Eq. (18.9) by A, we have AF = ABE

= OH = 0,

which means t h a t the function / € F satisfies t h e functional equation (18.1) for any h G H. On the other hand, according to 1° of Theorem 17.1 each solution of equation (18.5) has the form F = (I-

BA)H,

Functional Equations with Constant

178

Coefficients

where B satisfies ABA = A. If we put B — I — BA, we have AB = A-ABA = O and B ^ O since det A = 0. Thus Eq. (18.9) where B satisfies Eq. (18.4) is the general solution of the system (18.5), i.e., Eq. (18.8) gives the general solution of the functional equation (18.1). • E x a m p l e 18.1 equation

By a cyclic permutation of the variables in the functional

/(Zl,Z2,Z3,Z4) - /(Z2,Z3,Z4,Zi) +

/(Z3,Z4,Z1,Z2)-/(Z4,Z1,Z2,Z3) = 0

we obtain the following system / ( Z 1 , Z 2 ) Z 3 , Z 4 ) - / ( Z 2 , Z 3 Z4,Zi)

z 2 ,z 3 ) = o,

+

/(Z3,Z4>Z1,Z2)-/(Z4,Z1

-

/ ( Z i , Z 2 , Z 3 , Z 4 ) + / ( Z 2 , Z 3 Z 4 ,Z!)

-

/(Z3,Z4,Z1,Z2) + /(Z4,Z1

z 2 ,z 3 ) = o,

/ ( Z 1 , Z 2 , Z 3 , Z 4 ) - / ( Z 2 , Z 3 Z 4 ,Z!)

z 2 ,z 3 ) = o,

+

/(Z3,Z4,Zi,Z2)-/(Z4,Z1

-

/ ( Z 1 , Z 2 , Z 3 ! Z 4 ) + / ( Z 2 , Z 3 Z 4 ,Zi)

-

/(Z3,Z4)Zl!Z2) + /(Z4,Z1

z 2 ,z 3 ) = o.

The matrix of coefficients of this system is

A =

1-1 1-1 -1 1-1 1 1-1 1-1 -1 1-1 1

Since det A = 0, the above system has a nontrivial solution. For the matrix A there exists a nonzero 4 x 4 cyclic matrix B, such that AB = O, i.e.,

B =

h

b2

63

b\ — b2 + b3 b3

bi b± - b2 + b3

&2

b3

h

62

b2

63

&2 + b3

&i

61 - 6 2 + 63

Homogeneous

Functional Equations with Constant

Coefficients

179

Therefore the general solution of the given functional equation is / ( Z i , Z 2 , Z 3 , Z 4 ) = b1h{Z1,Z2,Z3, +b3h{Z3,Zi,

Z 4 ) + b2h(Z2, Z 3 , Z 4 , ZO

Zu Z 2 ) + (&! - 62 + W Z 4 , Z i , Z a , Z 3 ),

where /i is an arbitrary complex vector function with values in V. Theorem 18.2

2/ the matrix A satisfies the condition

Am + XU™-1 + • • • + \m-iA

=0

(Am_!^0),

(18.11)

then the general solution of the functional equation (18.1) is given by F = — ^ - (A"1'1

+ AxA m - 2 + • • • + Xm-iI)H,

(18.12)

Am-l

where I is the n x n unit matrix and Aj (1 < i < m — 1) are complex numbers. Proof. The proof of this theorem is very easy. By multiplication of the formula Eq. (18.12) with A, we obtain AF = —!— (Am + AiA" 1 - 1 + • • • + Xm-iA)H

= OH = 0,

Xm-l

or in other words the function / 6 F satisfies the functional equation (18.1) for any h € H. Conversely, if F is a solution of AF = O, then obviously F satisfies the identity F = —!— (A™-1 + XxAm-2

+ ••• + Am_x I)F,

Am-l

i.e F can be represented by the formula Eq. (18.12) with H = F. Hence we proved that the function given by Eq. (18.12) is the general solution of Eq. (18.1). D Example 18.2

For the functional equation

2 / ( Z i , Z 2 , Z 3 ) - 3/(Z 2 , Z 2 , Z 3 ) + / ( Z 3 , Z 3 , Z 3 ) = O the matrix of coefficients is T 2

A =

-3

11

Functional Equations with Constant

180

Coefficients

The matrix A satisfies the following equation A3 -A2-2A

= O.

The required general solution of the above functional equation is • /(Zi,z2,zs) • /(Z2,Z2,Z3) = . /(Z3,Z3,Z3) _

1 L

'

h(Zi,Z2,Z3) --(A -A-2I) M 2 , Z 2 , Z 3 ) _ /i(Z 3 ,Z 3 ,Z 3 ) _ 2

Z

i.e.,

/ ( Z i , Z a , Z s ) = /i(Z 3 , Z 3 , Z 3 ) = p(Z 3 ), where p is an arbitrary complex vector function with values in V. This example shows that Theorem 18.2 can be also applied to equations not of the form Eq. (18.1). Theorem 18.3

The general solution of form / ( Z i , Z 2 , • • • ,Z n ) = i?(/l(Zi,Z 2 ) - • • ,Z„))

(18.13)

n

=

^ & t M Z i , Z ; + i , - • • ,Zj-)_„_i)

(z„ + j = Zj)

i=l

o/ f/ie functional equation (18.1) is reproductive (R(R(h)) only if the following condition is satisfied E{f) = O

/ =

R{f).

= R(h)) if and (18.14)

Proof. Assume that R(h) is the general solution of the equation E(f) = O. Let R(R(h)) = R(h) hold for every h. Then, from E(f) = O it follows that / = R(h) for some h, so that for the same h we have / = R(R(h)) = R(f). Conversely, let the condition E{f) = O => / = R(f) be satisfied. Since for every h it holds E(R(h)) = O, then according to the assumption we obtain R(h) = R{R(h)) for every h. D Example 18.3 We will determine the general reproductive solution for the functional equation given in Example 18.1. On the basis of the general solution, we obtain i2(/(Zi,Z 2 > Z3,Z 4 ))

Homogeneous

Functional Equations with Constant

Coefficients

181

= 6i/(Zi,Z2,Z3,Z4) 4-62/(Z2,Z3,Z4,Zi) + 6 3 / ( Z 3 , Z 4 , Zlt Z 2 ) + (61 - 62 + & 3 )/(Z 4 , Zi, Z 2 , Z 3 ) = &i/(Zi, Z 2 , Z 3 , Z 4 ) + 6 2 /(Z 2 , Z 3 , Z 4 , Zi) +&3/(Z3,Z4,Z1,Z2) + ( 6 i - 6 2 + 63)[/(Z1)Z2,Z3,Z4) _

/(Z2,Z3,Z4,Zi) + /(Z3,Z4,Zi,Z2)] = (26 1 -& 2 + 6 3 ) / ( Z i , Z 2 l Z 3 ) Z 4 ) +(-61 + 262 - 6 3 )/(Z 2 , Z 3 , Z 4 , Zx) + ( 6 i - 6 2 + 263)/(Z3,Z4,Z1,Z2).

The condition i ? ( / ( Z i , Z 2 , Z 3 , Z 4 ) ) = / ( Z i , Z 2 , Z 3 , Z 4 ) holds if 2b1-b2+b3

= l,

-61 4- 262 - b3 = 0,

61 - 62 + 263 = 0,

i.e., 61 = 3/4,

62 = 1/4

and

63 = - 1 / 4 .

Therefore, the general reproductive solution of the given functional equation is /(Zi,Z2,Z3,Z4)

=

3 1 - / i ( Z i , Z 2 , Z 3 , Z 4 ) 4- - / i ( Z 2 , Z 3 , Z 4 , Z i )

-

- / i ( Z 3 , Z 4 , Z i , Z 2 ) 4- - / i ( Z 4 , Z i , Z 2 , Z 3 ) .

Next we will give a procedure by which for every functional equation (18.1) in the case n = 3 we may determine the canonical equation which is equivalent to it. For every canonical equation we will determine the general and reproductive solution. Now we will consider the equation o i / ( Z i , Z 2 , Z 3 ) + a 2 / ( Z 2 , Z 3 , ZO + a 3 / ( Z 3 , Z 1 ; Z 2 ) = O.

(18.15)

Functional Equations with Constant

182

Coefficients

From Eq. (18.15) by a cyclic permutation of the variables we obtain the following system o i / ( Z i , Z 2 , Z 3 ) + a 2 / ( Z 2 ) Z 3 , Zi) + a 3 / ( Z 3 , Zi, Z 2 ) = O, a3/(Z1,Z2,Z3) + a1/(Z2,Z3,Z1) + a 2 / ( Z 3 , Z 1 ) Z 2 ) - 0 , a 2 / ( Z 1 ; Z 2 , Z 3 ) + a 3 / ( Z 2 , Z 3 , Z x ) + a1f{Z3,Z1,Z2)

(18.16)

= O.

The determinant of the system (18.16) is A = - ( a i + a2 + a 3 )[(a x - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - aj) 2 ]. There are four possible cases: a) ai + a2 + a 3 7^ 0 and (ai - o 2 ) 2 + (a2 b) ai 4- a 2 + a 3 ^ 0 and (ai - a 2 ) 2 + (a 2 c) ai + a 2 + o 3 = 0 and (ai - a 2 ) 2 + (a 2 d) ai + a 2 + a 3 = 0 and (01 - a 2 ) 2 + (a 2 In the case a) the system (18.16) is obviously

a 3 ) 2 + (a 3 - ax)2 ^ 0, a 3 ) 2 + (a 3 - a i ) 2 = 0, a 3 ) 2 + (a 3 - a^) 2 ^ 0, o 3 ) 2 + (a 3 - a i ) 2 = 0. equivalent to the equation

/ ( z l l z 2 ) z 3 ) = o. In the case b) we have a\ - (ai + 0,2)0.3 + a 2 - a\a2 + a\ — 0 or 03 = Ui + o 2 ± i ( a i If a 2 = ai, then also a 3 = a\ and ai + a 2 + a 3 7^ 0 implies ai 7^ 0. Thus the system (18.16) is equivalent to the equation f(Zu

Z 2 , Z 3 ) + f(Z2,Z3,Z1)

+ / ( Z 3 , Zi, Z 2 ) = O.

(18.17)

In the general case we can write a-3 =

l±i>/3 5

«i H

iTiVS ^

a,2

or, if CJ6 and LJ^1 are the primitive 6th roots of 1, then a 3 = aiUe + Now Eq. (18.15) takes the form aif{Zu

+

Z 2 , Z 3 ) + a2f (Z2,Z3,Z1) 1

(aia;a + 02a;^ )/(Z3,Z 1> Z2) = 0 .

CL2UQ1.

(18.18)

Homogeneous Functional Equations with Constant Coefficients

183

If a 3 = 0, we have ai + a2 7^ 0, ajWg + a 2 = 0 and the equation takes the form /(Zi,Z2,Z3)-W3/(Z2,Z3,Zi) = 0>

(18.19)

where w3 is a primitive third root of 1 (we assume LJ3 = w|). If a 2 = 0 or ax = 0, we obtain respectively the equations

f(Z1,Z2,Z3)+cj6f(Z3,ZuZ2)

= 0

and /(Z2,Z3,Zi) +w6-1/(Z3,Z1,Z2) = O which can be reduced to Eq. (18.19) by a cyclic permutation of the vectors. We will see that equation (18.18) can be always reduced to Eq. (18.17) or Eq. (18.19). To this end we shall use the following lemma. Lemma 18.4

Suppose that equation (18.15) can be written in the form O i l d / J Z i , Z 2 , Z 3 ) + c 2 /(Z 2 , Z 3 , Zi)] +

a 2 [c 1 /(Z 2 > Z 3 , Zi) + c 2 / ( Z 3 , Z 1 ; Z 2 )]

+

a 3 [c 1 /(Z 3 , Zi, Z 2 ) + c 2 / ( Z l l Z 2 , Z s )] = O,

(18.20)

where 011

o-i

OL$

a3

ax

a2

012

0:3

ai

^0.

(18.21)

Then equation (18.15) is equivalent to c i / ( Z 1 ; Z 2 , Z 3 ) + c 2 /(Z 2 , Z3> Zi) = O. Proof. system

(18.22)

By a cyclic permutation of Eq. (18.20) we derive the following

a i [ c x / ( Z i , Z 2 , Z 3 ) + c 2 / ( Z 2 , Z 3 , Zi)] +

a 2 [c 1 /(Z 2 , Z3> Z x ) + c 2 /(Z 3 , Zi, Z 2 )]

+

a 3 [ c i / ( Z 3 ) Zi, Z 2 ) + c2f (Z1,Z2, Z 3 )] = O,

Functional Equations with Constant

184

Coefficients

a 3 [ c 1 / ( Z l ! Z 2 , Z 3 ) + c 2 /(Z 2 , Z 3 , Zi)] + +

a1[c1/(Z2,Z3,Z1)+c2/(Z3,Z1,Z2)] a 2 [ C l / ( Z 3 , Zi, Z 2 ) + c 2 / ( Z 1 ; Z 2 , Z 3 )] = O, a 2 [ c i / ( Z i , Z 2 , Z 3 ) + c 2 / ( Z 2 ) Z 3 , Z : )]

+

a 3 [ C l / ( Z 2 , Z 3 , ZO + c 2 /(Z 3 , Z 1 ( Z 2 )]

+

a1[ci/(Z3,Z1,Z2) + c2/(Zi,Z2,Z3)] = 0 .

The determinant of this system is cci a3 a2

a2 ai a3

a3 a2 ai

^0,

thus we deduce Eq. (18.22).

•

We see that equation (18.18) can be written in the form Eq. (18.20): 0-[/(Z1,Z2,Z3)-w3/(Z2,Z3,Z1)] +

a2[f(Z2,Z3,Z1)-uJ3f(Z3,Z1,Z2)}

+

a1o;6[/(Z3,Z1)Z2)-a;3/(Z1,Z2,Z3)] = 0 .

The determinant in condition Eq. (18.21) is

0

a.2

aiu6

aiLJe

0

a2

a-2

aiLJ6

0

„3 = o2 -

-a?

If a 2 ^ af, then by Lemma 18.4 equation (18.18) is equivalent to Eq. (18.19). If o 2 = a\, then it is equivalent to Eq. (18.17). If a 2 = af, but a 2 ^ a i , then a 2 = aiw| or a 2 = ai^g. In the latter case we have ai + a 2 + a 3 = ai + aiWg 4- aiu>6 + ai^g = ai — ai^g + aiWg — ai = 0 which is a contradiction. Suppose that a 2 = aiw|. Then a 3 = 2aiW6 and equation (18.18) becomes / ( Z j , Z 2 , Z 3 ) + Wg7(Z2, Z 3 , Z J + 2w 6 /(Z 3 , Zi, Z 2 ) = O

Homogeneous Functional Equations with Constant Coefficients

185

/(Zi,Z2,Z3)-W3/(Z2,Z3,Zi) +

2w3[/(Z2,Z3,Z1)-W3/(Z3)Z1,Z2)]

+

0- [/(Zs.Zx.Za) - w 3 /(Z 1 ,Z 2 ,Z 3 )] - O.

The determinant 1

2w3

0

0 2u3

1 0

2u>3 1

= 1 + 8wf = 9 ^ 0,

thus by Lemma 18.4 equation (18.18) is equivalent to Eq. Eq, (18.19). In the case c) a\ + a2 + a3 — 0 and at least one o: of the inequalities ai — a 2 7^ 0, a 2 — — a03 7^ 00 and and 03 03 — — aj ai 7^ 7^ 00 is is valid. valid. Suppose, Suppos for the sake of 3 7^ a\ — a 2 \we have definiteness, that ai — a 2 7^ 0. In view of 03 = — ax o i / ( Z i , ZZ22,,ZZ33)) + oa 2 / ( Z 2a , ZZ33, ZZi) Z33,Z , 1,Z2) i ) + (-01 - aa22)f) / ((Z

= O,

or the form Eq. (18.20) 0-[/(Z1,Z2!Z3)-/(Z2,Z3,Z1)] + -

a2[f(Z2,Z3,Z1)-f(Z3,ZuZ2)] a 1 [ / ( Z 3 , Z 1 , Z 2 ) - / ( Z 1 , Z 2 , Z 3 ) ] = <0 .

The determinant in Eq. (18.21) is 0 -Oi

a2

a2 0 -ax

--ax ai

a2 0

-„3

ax — 7^0,0,the the equality equality a\a\ — —a\a\ == 00isispossible possible oonly for a2 = axu3 Since ai - aa22 7^ (u>3aaprimitive primitivethird thirdroot rootofof1), 1),then then03a3==—ai(l —ai(l+u> +u>3)3)arand we are led to a (u>3 a2)22 + + (a2 — a3)2 + (a3 — ai) ax)22 7^ contradiction with (ai — a2) ^ 0. Thus, in this case the equation (18.15) is equivalent to the equation f(Z1,Z2,Z3)-f(Z2,Z3,Z1)

=

0.

(18.23)

If ai — a 2 = 0, then at least one of the differences a2— 03 a n d 03 — a,\ is not 0 and we come to the same conclusion.

186

Functional Equations with Constant

Coefficients

In the case d) we have 03 = — a,\ — 0,2, a\ + a\a2 + a\ = 0, i.e., either ai = a 2 = 03 = 0 and the equation (18.15) reduces to the identity

0 = 0, or a2 = aitJ3, 03 = aiw|, where W3 is as above. Now equation (18.15) reduces to / ( Z ! , Z 2 , Z 3 ) + W 3 / ( Z 2 , Z 3 , Z 1 ) + a, 2 /(Z 3 ,Z 1 ,Z 2 ) = O.

(18.24)

On the basis of the exposition we conclude that the following lemma holds. Lemma 18.5 ing equations I.

The functional equation (18.15) is equivalent to the follow-

/(Z1>Z2,Z3) = O i /

01 + a2 + 03 7^ 0 II.

and

Eq. (18.17) or £g. (18.19) if

a\ + a 2 + 0-3 ^ 0 Ji7.

anii

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a\)2 — 0;

£g. (18.23) if

o-i + «2 + 0,3 = 0 JV.

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 7^ 0;

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (03 - ai) 2 7^ 0;

and

0 = 0 or Eq. (18.24) 1/

aj + a 2 + 03 = 0

and

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - aj) 2 = 0.

For any of the above equations we give formulae for the general solutions and formulae for the general reproductive solutions of these equations. Proposition 18.6

The equation (18.17)

/ ( Z l f Z 2 , Z 3 ) + / ( Z 2 , Z 3 , Z 0 + / ( Z 3 | Zi, Z 2 ) = O has a general solution given by /(Zi,Z2>Zs)

= -

b1h(Z1,Z2,Z3)

+ b2h(Z2,Z3,Z1)

(18.25)

(b1+b2)h(Z3,Z1,Z2),

where h is an arbitrary complex vector function with values in V and bi, b2 are arbitrary complex constants.

Homogeneous Functional Equations with Constant Coefficients Proof.

187

From the given equation we obtain the system f(ZuZ2,Z3)

+ f(Z2,Z3,Z1)

+ f(Z3,Z1,Z2)

=

O,

/(Z1>Z2,Z3) + /(Z2,Z3>Zi) + /(Z3,Zi>Z2)

=

O,

/(Z1,Z2,Z3) + /(Z2,Z3,Z1) + /(Z3,Z1,Z2)

=

O,

with matrix of coefficients A =

1 1 1

1 1 1 1 1 1

The general solution of this system, according to Eq. (18.2) is given by /(Zi,Z2>Z3) /(Za.Zs.ZO = B /(Z3,Z1,Z2) J

J»(Zi,Z 2 ,Z 3 ) /i(Z 2 ,Z 3 ,Zi)

h(Z3,Z1,Z2) J

where B is a general cyclic 3 x 3 matrix which satisfies the condition AB = O. On the basis of the formula Eq. (18.8), we may write / ( Z i , Z 2 , Z s ) = 6ift(Zi,Z 2 | Z 3 ) + 6 2 /i(Z 2 , Z 3 , Zi) - (6i +

b2)h(Z3,Z1,Z2),

which proves the proposition. Proposition 18.7

D

The reproductive solution of the equation (18.17)

/(Z1,Z2,Z3) + /(Z2,Z3,Z1) + /(Z3,Z1,Z2) = 0 is 2 1 1 /(Zi,Z2,Z3) = -/i(Zi,Z2,Z3) - -/i(Z2,Z3,Zi) - -/i(Z3,Zi,Z2), where h is an arbitrary complex vector function with values in V. Proof. This statement will be proved in the following way. From the general solution Eq. (18.25) we obtain fl(/(ZltZa,Z3))

=

61/(Z1,Z2,Z3) + 62/(Z2,Z3,Z1)

-

(&i+&2)/(Z 3 ,Z 1 > Z 2 )

=

6i/(Z1,Z2)Z3)+62/(Z2,Z3,Z1)

+

( & i + 6 2 ) [ / ( Z i , Z 2 , Z 3 ) + /(Z 2 ,Z3,Z 1 )]

=

(26i + b3)f(Z1,Z2,

Z 3 ) + (h + 26 2 )/(Z 2 , Z 3 , Zj).

188

Functional Equations with Constant

The condition R(f(Z1,Z2,Z3))

Coefficients

= / ( Z i , Z 2 , Z 3 ) is satisfied if

bi = 2/3

and

b2 = - 1 / 3 .

Thus / ( Z i , Z 2 , Z 3 ) = -h(ZltZ2,Z3)

- ^h(Z2,Z3,Z1)

-

^h(Z3,ZuZ2)

is a reproductive solution of the equation /(Z1,Z2,Z3) + /(Z2,Z3,Z1)+/(Z3,Z1,Z2) = 0 . In a similar way we can prove that the equation (18.19) has a general solution given by f(Z1,Z2,Z3)=b[h(Z1,Z2,Z3)+UJ3h{Z2,Z3,Z1)+ulh(Z3,Z1,Z2)], where b is a complex constant and h is an arbitrary complex vector function with values in V. Of course, in the last equality we can put 6 = 1 , including the arbitrary constant in the function h. Also, if we put b\ = b, b2 = btj3, then by virtue of the equality 1 + u>3 + ui3 = 0 we see that the general solution of Eq. (18.19) is of the form Eq. (18.25). On the other hand, it is sometimes convenient to keep this factor. For instance, we see that the reproductive solution of Eq. (18.19) is obtained for b — 1/3, i.e., / ( Z i , Z 3 > Z 3 ) = | [h(ZuZ2,Z3)

+u3h{Z2,Z3,Z1)+<4h{Z3,Zl,Z2)]

Similarly, the equation (18.23) has a general solution given by / ( Z i , Z 2 , Z 3 ) = h{Z1}Z2,Z3)

+ h(Z2,Z3,Z1)

+

h(Z3,ZuZ2)

and a reproductive solution given by / ( Z 1 , Z 2 , Z 3 ) = i [ M Z 1 , Z 2 , Z 3 ) + / l (Z 2 ,Z3,Z 1 ) + / l (Z3,Z 1 ,Z 2 )], and the equation (18.24) has a general solution given by /(Zi,Z2lZ3)

= -

b1h{Z1,Z2,Z3)-\-b2h{Z2,Z3,Z1) (wl&i+ws&ajMZs.Zi.Za)

.

Homogeneous Functional Equations with Constant

Coefficients

189

and a reproductive solution given by f(Z1,Z2,Z3)

= | / l ( Z 1 , Z 2 ! Z 3 ) - fh(Z2,Z3,Z1)

-

<

^h(Z3,Z1,Z2).

On the basis of the previous results the following two theorems hold. Theorem 18.8

The general solution of the equation

a i / ( Z i , Z 2 ) Z 3 ) + a 2 / ( Z 2 , Z 3 , ZO + a 3 / ( Z 3 , Zu Z 2 ) = O is given by the following formulae 1°. J / a i + a 2 + a 3 7^0 tften/(Z1,Z2,Z3) = 0 ;

and

(ai - a2)2 + (a2 - a3)2 + (az - ai)2 ^ 0,

2°. then

and

(ai - a 2 ) 2 + ( a 2 - a 3 ) 2 + (a 3 -a{)2

Ifai+a2

f(Z1,Z2,Z3)

+ a3^Q

= b1h(Z1,Z2,Z3)

+

= 0,

b2h(Z2,Z3,Z1)-(b1+b2)h(Z3,Z1,Z2)

(in particular, + cj3h(Z2, Z 3 , Z x ) + w 2 /i(Z 3 , Zi, Z 2 )

/ ( Z i , Z 2 , Z 3 ) = h(ZuZ2,Z3)

«/ a i , a 2 , a 3 are distinct nonzero numbers); 3°. t/ien

J / a i + a 2 + a3 = 0

and

f(Z1,Z2,Z3)=h(Z1,Z2,Z3) 4°. i/ien:

( a i - a 2 ) 2 + ( a 2 - a 3 ) 2 + ( a 3 - a i ) 2 ^ 0, +

7/ai+a2 + o 3 = 0

and

h(Z2,Z3,Z1)+h(Z3,Z1,Z2);

( a i - a 2 ) 2 + ( a 2 - a 3 ) 2 + ( a 3 - a i ) 2 = 0,

a) f{Z1,Z2,Z3)

=

&; f(zuz2,z3)

= b1h{z1,z2,z3) + b2h(z2,z3,z1) -

h(ZuZ2,Z3)

if ai = a 2 = a 3 = 0;

(w|6i+W3&2)MZ3,Z 1 ; Z 2 )

j / a 2 = aiw 3 , a 3 = aiwf, ai 7^ 0, where h is an arbitrary complex vector function with values in V. Theorem 18.9 Under the assumptions of the previous theorem denoted by 1° — 4° the general reproductive solutions are given by the following formulae

190

Functional Equations

1".

f(Z1,Z2,Z3)

with Constant

= Oi

2°. a) f(ZuZ2,Z3) if ai = a2 = a3 ^ 0;

=

lh(Z1,Z2,Z3)-±h(Z2,Z3,Z1)-±h(Z3,Z1,Z2)

b) f{ZuZ2,Z3) = f A(Zi,Z 2 ,Z 3 ) - fh(Z2,Z3,Zx) if ai, a2,a3 are distinct; 3°. 4°.

-

fh(Z3,Z1,Z2)

/ ( Z 1 , Z 2 , Z 3 ) = l[/i(Zi,Z 2 ,Z3) + MZ2,Z3,Z 1 ) + / l (Z3,Z 1 ,Z 2 )]; 0;

f(Z1,Z2,Z3)=h(Z1,Z2,Z3);

b)f (ZUZ2,Z3)

19

Coefficients

= |ft(Zi,Z 2 ,Z 3 ) - fh(Z2,Z3,Zx)

-

fh(Z3,ZuZ2).

Nonhomogeneous Functional Equations with Constant Coefficients

Next we will give the following results. Theorem 19.1 The basic cyclic complex vector nonhomogeneous functional equation with complex constant coefficients E(f)

=

/ , Qi/(Zj, Z , + i , • • • , Zj + n _i)

(19.1)

i=l

#(Zi,Z 2 ,-• • ,Z„)

(Z n + j = Zj)

iu/jere Cj (1 < i < n) are complex constants, has a solution if the right-hand side g satisfies g{Zi,Z2,--g(Z2,Z3,---

,Zn) ,Zi)

(AC +1)

(19.2) . 3(Zn,Zi,--- ,Z„_i)

where A is given by Eq. (18.3), C is any nonzero nxn cyclic matrix with complex constant entries satisfying AC A + A — O, O is the nxn zero matrix, I is the nxn unit matrix and O is defined as in Eq. (18.6). If Eq. (19.2) holds for some C, then the general solution of Eq. (19.1)

Nonhomogeneous

Functional Equations with Constant

is given by the following

Coefficients

191

formula / ( Z i , Z 2 ) - •• , Z „ ) /(Z2,Z3,-- • ,Zi) (19.3)

/(Zn,Zi, • • • ,Zn_i) _ 9(Zi,Z2,---,Zn) g{Z2,Z3,--,ZX)

/i(Zi,Z2,- • • ,Zn) h(Z2,Z3,--,Zi)

=

B

-c 5(Zn,Zi,- • • ,Zn_i)

. Mz„,Zi, • • • , z „ _ i ) .

where the nonzero n x n cyclic matrix B given by Eq. (18.3) satisfies condition AB = 0 and /i is an arbitrary Proof.

the

(19.4)

complex vector function

with values in V.

By a cyclic permutation of the vectors in Eq. (19.1) we get o i / ( Z i , Z 2 ) - •• , Z „ ) + o 2 / ( Z 2 , Z 3 , - - • , Z i ) +

• • • + o,nf(Zn,

Z i , • • • , Z n _ i ) = g(Z\, Z 2 , • • • , Z n ) ,

o n / ( Z i , Z 2 , • • • , Z n ) + axf(Z2, +

Z3, • • • , Zx)

• • • + a n _ i / ( Z „ , Z i , • • • , Z n _ i ) = g ( Z 2 , Z 3 , • • • , Z{),

(19.5)

a 2 / ( Z i , Z2, • • • , Zn) + a 3 / ( Z 2 , Z3, • • • , Zi) +

• • • + a i / ( Z n , Z i , • • • , Z„_x) =
i.e. in a m a t r i x form (19.6)

AF = G, where /(Zi,Z2,- • • ,Zn)

/(Za^s.-^-.Z!) F =

.(19.7)

and G = /(ZnjZj, • • • ,Zn_i) .

£/(Z n ,Zi,- • • , Z n _ i )

192

Functional Equations with Constant

Coefficients

Suppose that equation (19.6) has a solution F and that C satisfies AC A + A = 0. Then {AC + I)G = (AC + I)AF = {AC A + A)F = O, i.e., Eq. (19.2) must be satisfied. Conversely, let Eq. (19.2) hold for some cyclic matrix C. Then —CG is easily seen to be a solution of Eq. (19.6): A(-CG)

= -{AC + I)G + IG = IG = G.

Now let us prove that Eq. (19.3) is the general solution of the equation (19.1). Let / be a solution of the equation (19.1), which we will write in the form E{f) = g.

(19.8)

We denote by fh the general solution of the equation E(f) = O, and by fp we denote a particular solution of the equation (19.8). Then / = fh + fp is the general solution of the equation (19.8). Indeed,

E(h + fP) = E{fh) + E{fp) = g. On the other hand, let / be an arbitrary solution of the equation (19.8). Then E(f - fp) = E(f) - E(fp) = g-g

= 0,

i.e., f — fp is a solution of the associated homogeneous equation. So there exists a specialization fh of the expression fh such that

f-fP

= h,

i-e., f = fh+ fP-

Thus fh + fp includes all solutions of the equation (19.8). The general solution of the homogeneous equation E(f) — O given in a matrix form according to Theorem 18.1 is BH, where B and H are defined by Eqs. (18.3) and (18.10) respectively, and a particular solution of the equation E{f) = g in a matrix form is — CG, then F = BH — CG includes all solutions of the nonhomogeneous equation. On the other hand, every function of the form Eq. (19.3) satisfies the functional equation (19.1). • Next we will consider the functional equation a1f(Z1,Z2,Z3)+a2f(Z2,Z3,Z1)+a3f(Z3,Z1,Z2)=g(Z1,Z2,Z3).

(19.9)

Nonhomogeneous

Functional Equations with Constant

Coefficients

193

By a similar procedure as in the previous section one can prove the following lemma. Lemma 19.2 The functional equation (19.9) is equivalent to I. / ( Z i , Z 2 , Z 3 ) = ^[(a21-a2a3)g(ZuZ2,Z3)+(a23-a1a2)g(Z2,Z3,Z1) +

(al-a1a3)g(Z3,Z1,Z2)]

if oi + a2 + a3 ^ 0 II.

and

(ai - a2)2 + (02 - a 3 ) 2 + (03 - a i ) 2 ^ 0;

/ ( Z 1 , Z 2 , Z 3 ) + / ( Z 2 , Z 3 , Z 1 ) + / ( Z 3 , Z i ) Z 2 ) = ^ ( Z i ) Z 2 > Z 3 ) if

oi = a 2 = 03 ^ 0

or /(Zi>Z2>Z3)-w3/(Z2>Z3,Z1) a22g{Z3,Zl,Z2)

' -oiQ3^6g(Zi,Z 2 ,Z 3 ) + ofa>gg(Z 2> Z 3 ,Zi) + = <

if<4 ^af, ff(Zi,Z2,Z3)-2a;3g(Z2,Z3,Z1)+4a;|g(Z3,Zi,Z2) 9ai i / a 2 = aiW3

ai + o 2 + a 3 ^ 0 III.

and

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 = 0;

/(Z1,Z2,Z3)-/(Z2,Z3,Z1)

= -5

3 [aia 2 g(Zi, Z2> Z 3 ) + o 2 g(Z 2 , Z 3 , Z x ) + a2g(Z3, Zu Z 2 )]

0 2 — Oj

if ai + a 2 + 03 = 0 IV.

and

O = g(ZuZ2,Z3)

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 7^ 0; or

f(Z1,Z2,Z3)+oj3f(Z2,Z3,Zi)+uJ2f(Z3,Z1,Z2)

=

—g(Z1,Z2,Z3) Oi

if oi + 0 2 + 0 3 = 0

and

(ai — a 2 ) 2 + (02 - a 3 ) 2 + (03 — a i ) 2 = 0.

194

Functional Equations with Constant Coefficients

For each of the above equations we will determine the conditions which must be satisfied by the function g so that the equation should have a solution. Proposition 19.3 a1f(Z1,Z2,Z3)

The equation + a2f(Z2, Z 3 , Zi) + a3f(Z3,Z1,Z2)

= g(ZuZ2,

Z3)

whose coefficients satisfy the conditions ai = a2 = a3 ^ 0 has a solution if and only if the function g satisfies the condition g(ZuZ2,Z3) Proof.

- g(Z2,Z3,Z1)

= O.

In this case the equation considered is equivalent to the equation

a1[/(Z1,Z2,Z3) + /(Z2,Z3,Z1) + /(Z3,Z1,Z2)] =

g(Z1,Z2,Z3).

Since ax

1 1 1

1 1 1 1 1 1

then A2 = 3 a i A i.e. A\ -r1- }A + A = O, and hence we obtain C = 1

'

y3ai J

'

^-. 3ai

The condition ACG + G — O reduces to |5(Z1,Z2jZ3) - i5(Z2,Z3,Z1) - ^(Z3,Z1,Z2) = O and (see Lemma 18.4) eventually to g{ZuZ2,Z3)-g{Z2,Z3,Zl)

=

0.

In a similar way, we obtain the necessary and sufficient conditions for solvability of the other equations. Thus, we obtain the following result. Theorem 19.4

The functional equation

a i / ( Z i , Z 2 , Z 3 ) + a 2 / ( Z 2 , Z 3 , Zi) + o 3 / ( Z 3 , Zu Z 2 ) = g(Zu Z 2 , Z 3 ) has a solution if and only if the function g satisfies the following conditions

Nonhomogeneous Functional Equations with Constant Coefficients

1°.

p(Zi,Z2,Z3) is arbitrary if

ai + a2 + a3 ^ 0 2°.

195

and

{a\ - a2)2 + (02 - 03) 2 + (03 - a\)2 7^ 0;

g(Z1,Z2,Z3)-g(Z2,Z3,Z1)

= 0 or

2 5 (Z 1 ,Z 2 ,Z 3 )+W33(Z2,Z3,Z 1 )+a; f f (Z3,Z 1 ,Z 2 )

= 0

if a\ + 0,2 + a3 ^ 0 3°.

g(ZuZ2,Z3)

ai + 0,2 + a3 = 0 4°.

and

(a\ - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 = 0;

+ g(Z2,Z3,Z1) and

+ g(Z3,Z1,Z2)

= O if

(ai - a 2 ) 2 + (a 2 - c^) 2 + (03 - ai) 2 ^ 0;

3 = 0 or 0(Zi,Z2,Z3)-W3S(Z2,Zs,Zi)=O

if ai 4- a 2 + 03 = 0

and

(ai - a 2 ) 2 + (02 - a3)2 + (a3 — 01) 2 = 0.

Now we will find the general solution for any equation of Lemma 19.2. We will illustrate this only for the second equation. Proposition 19.5

The general solution of the equation

/ ( Z i , Z 2 , Z 3 ) + / ( Z 2 , Z3> Zi) + / ( Z 3 , Zi, Z 2 ) =

—g(Zl!Z2,Z3)

is given by / ( Z i , Z 2 , Z 3 ) = 6i/i(Zi, Z 2 , Z s ) +

b2h(Z2,Z3,Z1)

-(61 + 6a)MZs, Zi, Z 2 ) + — ff(Zi, Z 2 , Z 3 ), where h is an arbitrary complex vector function with values in V. Proof. is

The general solution of the corresponding homogeneous equation

/ ( Z i , Z 2 > Z 3 ) = M ( Z i , Z 2 , Z 3 ) + b2h(Z2,Z3,Z1)

- (&! +

b2)h(Z3,ZuZ2).

196

Functional Equations with Constant

Coefficients

The particular solution of the considered nonhomogeneous equation is /(Zi,Z2,Z3) = ^-»(Z1>Z2>Z3). Thus, the general solution of the given nonhomogeneous equation is / ( Z I , Z 2 , Z 3 ) = 6I/I(ZI,Z2,Z3) + &2/I(Z2,Z3,ZI)

-(&i+62)MZ3,Zi,Z2) + — 6ai Theorem 19.6

g{ZuZ2,Z3). g

The general solution of the equation

Oi/(Zi, Z 2 , Z 3 ) + a 2 / ( Z 2 , Z 3 , ZO + a 3 / ( Z 3 , Zlt Z 2 ) =

g(Z1,Z2,Z3)

in the cases given in the Theorem 19.4 is 1°. / ( Z l f Z 2 , Z 3 ) = £[(a? - a 2 a 3 )ff(Z 1 , Z 2 , Z 3 ) +(a?. - a i a 2 )ff(Z 2 , Z 3 , Zi) + (a^ - aia 3 )0(Z 3 , ZX, Z 2 )]

ai + a 2 + a 3 ^ 0

and

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 ^ 0;

2°. / ( Z 1 > Z 2 , Z 3 ) = 6iMZi,Z a ,Z3) + 62ft(Z2>Z3,Z1) -(bi+b2)h{Z3,Z1,Z2)

+ —g(Z1,Z2,Z3)

if

ai

= a2 = a3 ^ 0

or f(Z1,Z2,Z3)

= fc(Zi, Z 2 , Z 3 ) + w 3 /i(Z 2 , Z 3 , Zi) + w 2 /i(Z 3 , Z 1 ( Z 2 ) +

r

(af-2aia2cj6-Q|^)g(Zi,Z2,Z3)+a;3(Q^-aia2a;6-2a^)ff(Z2,Z3,Zi) 3(a| - a?) < ifa\^a\, U}3

v

-T—g{Z2,Z3,Z1) oai

ai + a 2 + a 3 ^ 0

ifa2 = a1u3

and

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 = 0;

Paracyclic Functional Equations with Constant

3°.

f(Z1,Z2,Z3) |

= b1[h(Z1,Z2,Z3)

Coefficients

+ h(Z2,Z3,Z1)

+

197

h(Z3,Z1,Z2))

(a2 - a^gjZuZ^Z^ + (Ql + 2a2)g(Z2,Z3,Z1) 3(al + a\a2 + a2,)

if ai +a2 + a3 — 0 4°.

f(Z1,Z2,Z3)

f{ZuZ2,Z3)

and

(ai - a2)2 + (02 - a 3 ) 2 + (a3 - a^)2 7^ 0;

= h(Z1,Z2,Z3)

or

=

blh(Z1,Z2,Z3)

+

b2h(Z2,Z3,Z!)

-

(u23b1+L>3b2)h(Z3,Z1,Z2)

+

—g{Z1,Z2,Z3) 00,1

if ai + a2 + a3 = 0 20

and (01 - a 2 ) 2 + (02 - 03)2 + (a3 - ai) 2 = 0.

Paracyclic Functional Equations with Constant Coefficients

Let V be a complex vector space with complex dimension n, and let the complex vectors Xj, Yj 6 V (1 < i, j < n) be given as above. Throughout this section Ci are constant complex vectors in V and let / : Vn+k i-> V. Now we will consider the following paracyclic complex vector functional equation of the first kind n ^ ^ a j / ( X j , X j + i , • • • , Xj+ n _i, Yj, Yj+i,- • • , Yi+k-i) (X n +i = Xj,

=O

(20.1)

Yn_|_j = Yj)

where aj (1 1) and k = n. Let us determine the general solution of the equation n ^ a j / ( X j , X j + 1 , - - - ,Xi+n_1,Yj) = 0 . i=l

(20.2)

198

Functional Equations with Constant

Coefficients

By a cyclic permutation of the vectors in Eq. (20.2), we obtain the matrix system (20.3)

AF = 0, where

A =

F

=

a-i an

a2 ai

. a2

a3

a-n O-n-1

(20.4)

d

O O

/ ( X 1 ; - - - , X n , Yi) /(X2,---, Xi,Y2) and

o=

o

. /(Xn,--- ,Xn_i,Yj,) _

For the system (20.3) the following theorem holds. T h e o r e m 20.1 The general solution of the functional equation (20.2) is given by the formula F = BH,

(20.5)

AB = 0,

(20.6)

if

where A and B are nonzero nxn the nxn zero matrix and

cyclic matrices given by Eq. (18.3), O is

/i(Xi,X2, • • • , X n ) /i(X2,X3,--- ,Xi)

H .

ft(Xn,Xi,---

(20.7)

,Xn_i) _

where h is an arbitrary complex vector function with values in V. Proof. If not all coefficients ai (1 < i < n) are 0, we can suppose without loss of generality that ai ^ 0. Then equation (20.2) is equivalent to the

Paracyclic Functional Equations with Constant

Coefficients

199

equation / ( X i , • • • , X „ , Yx)

=

— / ( X 2 , • • • , X n , X i , Y2) — • • •

—

/(Xn,Xi,--- ,Xn_i,Yn).

By putting Yj = C» (2 < i < n) where C* are arbitrary complex constant vectors from V, we obtain /(Xi,-- • ,Xn,Yi) —

02

/(X2,---,Xn,X1,C2)----

or en

/(X

(20.8)

Xi,--- Xn_i,Cn).

The right-hand side of the last equation depends on X j , • • • , X„ only. Denote this expression by /i(Xi, • • • ,X„). Therefore, Eq. (20.8) obtains the following form /(X 1)'

• '

, X „ , Yi) = /i(Xi, • • • , X n ) .

(20.9)

The formula Eq. (20.9) is the general solution of the equation (20.2) if and only if it holds 2_^dih(X-i,'X-i+i, • • • ,Xj+ n _i) — O.

(20.10)

t=i

The above equation is equivalent to the functional equation (18.1), and therefore Theorem 18.1 holds. Thus Eq. (20.5) is true. D Now, we will solve the functional equation (20.1) if k = n. By denoting the pairs (Xj,Yj) = Z; (1 < i < n), the functional equation (20.1) takes the form Eq. (18.1), then Theorem 18.1 holds, i.e., the general solution is given by F = BH, where

F =

/(Zi,Z2,- • • ,Zn) / ( Z 2 , Z 3 , - - - ,Zx) / ( Z n , Zi, • • • , Z n _i) _

200

Functional Equations

with Constant

Coefficients

/(Xi,X2,--- ,Xn, Yi, Y2)-• • ,Y„) / ( X 2 , X 3 , - - - ,Xi, Y2, Y3, • • • ,Yi)

. / ( X n , X i , ' " ,Xn_i, Yn, Yi, • • • ,Yn_i) /t(Xi,X2, • • • , X„, Y i , Y2)- • • , Y„) /i(X2,X3,--- , X i , Y 2 , Y 3 , - - - ,Yi) H = /i(Xn,Xi,--- ,Xn_i, Yn, Yi, • • • ,Yn_i) and B is given by Eq. (18.3). Next, we will consider the case 1 < k < n. To this end, instead of the equation (20.1) we will consider the equation o i / ( X i , - - - , X n , Y i , - - - , Yjt, Yfc + i, • • • , Y n ) +

(20.11)

a 2 / ( X 2 , • • • , X n , X i , Y 2 , • • • , Yfc, Yfc+i, • • • , Y n , Y i ) + • • •

+ a n / ( X n , X i , • • • , X r a _ i , Y „ , Y i , • • • , Y„+fc_i, Y n + ^ , • • • , Y n _ i ) = O . By a cyclic permutation of the vectors in the last equation, we obtain the m a t r i x system (20.3), where F =

(20.12)

/ ( X i , • • • , X n , Y i , • • • , Yfc, Yfc + i,- • • , Y n ) / ( X 2 , - - - , X n , X i , Y 2 ) - • • , Yfc, Yfc+i, • • • , Y n , Y i )

/ ( X r a , X j , - - - , X n _ j , Y n , Y i , • • • , Yn_|_fc_i, Y n + fc, • • • , Y n _ i ) A and O are as in Eq. (20.4). T h e necessary and sufficient condition for the system (20.3) with Eq. (20.12) t o have nontrivial solution is det A = 0. Since det A is cyclic, then its value is n-l

detA= ]]_E(ei), i=0

where Si (0 < i < n - 1) are distinct roots of t h e binomial equation b(x) = 1 - xn = 0.

Paracyclic Functional Equations with Constant

Coefficients

201

Therefore, the equation (20.11) has nontrivial solutions if and only if the characteristic equation E(x) = a,i + a^x + • • • + anxn~1 = 0 has common roots with the binomial equation b(x) = 1 — xn = 0. If this is so, we can write E(x) = P(x)D(x), b(x) = D(x)F(x), where D{x) is the greatest common divisor of the polynomials E(x) and b(x). The general solution of the equation (20.11) is given by the formula / ( X i , - - - , X n , Y i , - - - ,Yfc,Yfc+i,--- ,Y„) =

&i/i(Xi,- • • , X „ , Y i , - • • , Y m , Ym_|_i,- • • , Y n )

+

&2MX2, • •' , X n , X i , Y2, • • • , Y m , Y m + i , • • • , Y n , Y i ) + • • •

+

&s+i MXg-H, • • • , X„, X i , • • • , X s , * s+1 j ' • ' ) * m+si I m + s + l i " ' I ' D I M i ' " 1 ' 1 ) 1

where the complex numbers bi (1 < i < s + 1) are coefficients of the polynomial 61 + b2x + • • • + bs+ixs = F(x). The functional equation (20.1) will be called reduced equation of the equation (20.11). Now we will prove the following result. T h e o r e m 20.2

Every function f given by /(X1,---,Xn,Y1,---,Yn)

=

6i/i(Xi, • • • , X n , Y i , • • • , Y m )

+

&2MX2, • • • , X „ , X i , Y2, • • • , Y m + i ) + • • •

+

bs+ih(Xs+i,-

(20.13)

• • ,Xn,Xi,--- ,Xs,Yg+i,--- ,Ym+s),

satisfies the equation (20.1), where m — k — s for k > s, /i(Xi,--- , X „ , Y S , - - - , Y m ) is an arbitrary complex vector function with values in V and if k — s < 0, then h is an arbitrary complex vector function only of X j , • • • , X n with values in the same space V. Proof. We should prove that / = F(h) is solution of the equation (20.1), where h is an arbitrary complex vector function with values in V. Indeed, we have D(f) = D(F(h)) = b(h) = 0,

202

Functional Equations

with Constant

from where it follows that E(f) = P(D(f)) to prove.

Coefficients

= O, which we were required •

Next, we will solve the functional equation ai/(Xi,X2,X3,Yi,Y2)+a2/(X2,X3,Xi,Y2,Y3) +

(20.14)

a3/(X3,Xi,X2,Y3,Yi)=0.

By a procedure similar to that in the first section, we may prove the following lemma. Lemma 20.3 The functional equation (20.14) is equivalent to the equation I. f(X1,X2,X3,Y1,Y2) = Oif oi + fl2 + 03 7^ 0

and

(ai - O2)2 + (a 2 - a 3 ) 2 + (03 - a i ) 2 7^ 0;

//. / ( X i , X 2 , X 3 , Y i , Y 2 ) + / ( X 2 , X 3 ) X i , Y 2 , Y 3 ) + / ( X 3 , X i , X 2 ) Y 3 , Y 1 ) = 0 or f(X1,X2X3,Y1,Y2)-u3f(X2,X3,X1,Y2,Y3)

= 0

if ai + 02 + 03 7^ 0

and

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - a i ) 2 = 0;

///. / ( X 1 , X 2 , X 3 , Y i ) Y 2 ) - / ( X 2 , X 3 , X 1 , Y 2 , Y 3 ) = 0 if ai + a 2 + 03 = 0

and

(ai - a2)2 + (02 - A3)2 + (03 - Oi) 2 7^ 0;

IV. 0 = 0 or f(XuX2,X3,Y1,Y2)+tJ3f(X2,X3,X1,Y2,Y3)W3f(X3,X1,X2,Y3,Y1)

= 0 if ai + a 2 + a 3 = 0 Proposition 20.4

and

(ai - o 2 ) 2 + (02 - a 3 ) 2 + (a 3 - a i ) 2 = 0.

The functional equation

/(Xi,X2,X3,Yi,Y2)-/(X2,X3,Xi)Y2,Y3) = 0, has a general solution f(X1,X2,X3,Y1,Y2)^h(X1,X2,X3)

+ h(X2,X3,X1)

+

h(X3,X1,X2),

Paracyclic Functional Equations with Constant Coefficients

203

where h is an arbitrary complex vector function of the variables X i , X2, X3 with values inV. Proof.

The given equation may be written in the following form / ( X i , X 2 , X 3 , Y i , Y 2 ) = / ( X 2 , X 3 , X i , Y 2 , Y 3 ).

The left-hand side of the equation is independent of Y 3 and the right-hand side is independent of Y i , so we have / ( X 1 , X 2 , X 3 , Y i , Y2) = F ( X 1 , X 2 , X 3 , Y 2 )

(20.15)

/ ( X 2 , X 3 , X i , Y 2 j Y3) = F ( X 1 , X 2 , X 3 , Y 2 ) .

(20.16)

and

On the other hand, from Eq. (20.15) we find / ( X 2 , X 3 , X i , Y 2 , Y 3 ) = F ( X 2 , X 3 , X i , Y3), thus we have F ( X i , X 2 , X s , Y 2 ) = F ( X 2 , X 3 , X 1 , Y 3 ).

(20.17)

Since the left-hand side of the above equation is independent of Y 3 and the right-hand side is independent of Y 2 , we obtain F(X1,X2,X3,Y2) = G(X1,X2,X3).

(20.18)

On the basis of the equality (20.18), the formula Eq. (20.15) becomes / ( X 1 , X 2 , X 3 , Y 1 , Y2) = G ( X i , X 2 > X 3 ) .

(20.19)

The formula Eq. (20.19) gives a solution of the equation if and only if G ( X 1 , X 2 , X 3 ) - G ( X 2 , X 3 , X i ) = Ol whose general solution is given by G ( X i , X 2 > X 3 ) = /i(Xi > X 2 > X 3 ) + M X 2 , X 3 , X 1 ) + M X 3 , X 1 > X 2 ) . Therefore, the general solution of the equation is /(X1,X2,X3lY1)Y2)

=

/l(X1,X2,X3) + /i(X2,X3,X1)

+

h(Xs,Xi,X2).

204

Functional Equations

with Constant

Coefficients

On the basis of the previous results, the following theorem holds. Theorem 20.5

The general solution of the equation

a1f(X1,X2,X3,Y1,Y2)+a2f(X2,X3,XuY2,Y3) +

a3f(X3,X1,X2,Y3,Y1)=0

is given by the formulae 1°. / ( X 1 , X 2 ! X 3 , Y 1 , Y 2 ) = O i / ai + a2 + a3 ^ 0

and

(oj - a 2 ) 2 + (a2 - a 3 ) 2 + (03 - a i ) 2 ^ 0;

2°. / ( X l ! X 2 , X 3 ) Y 1 , Y 2 ) = M X i , X 2 , X 3 ) Y 1 ) - / 1 ( X 2 , X 3 , X 1 , Y 2 ) or / ( X i ^ X a . Y ! , Y2) = M X i , X 2 , X 3 )

+cj3h(X2,X3,X1)

+ w | / i ( X 3 , X i , X 2 ) if 01 + a2 + a3 ^ 0

and

3°. f(X1,X2,X3,Y1,Y2) +h(X3,XuX2) if 01 + a2 + a 3 = 0

(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (03 - a i ) 2 = 0; = h(X1,X2,X3)

and

+

h(X2,X3,X1)

(ai - a 2 ) 2 + (o 2 - a 3 ) 2 + (03 - 01) 2 ^ 0;

4°. / ( X 1 ) X 2 , X 3 , Y l l Y 2 ) = /i(X 1 ,X2,X3 1 Y 1 ,Y 2 ) or /(X1,X2,X3,Y1,Y2) = /i(X1,X2,X3,Y1)-W3MX2,X3)X1,Y2)

ai + 02 + 03 = 0

and

(ai - 02)2 + (02 - a 3 ) 2 + (03 - a i ) 2 = 0,

where h is an arbitrary complex vector function with values in V.

Chapter 5

Systems of Linear Functional Equations

In this chapter two types of systems of linear functional equations are solved, namely systems in which each equation contains all the unknown functions and systems in which not all equations contain all the unknown functions. The results presented here are obtained in [I. B. Risteski et al. (submitted)] (see also [I. B. Risteski et al. (2001B)]). 21

S y s t e m s in Which Each Equation Contains All Unknown Functions

Now we prove the following results. Theorem 21.1

The general solution of the system of functional equations

/o(Zi,Z 2 )

+ +

/ 1 (Z 2 ,Z3,Z 4 ) + 5i(Z 1 ,Z 3 ,Z4) (21.1) /2(Z3,Z 4 ,Z 5 ,Z 6 ) +02(Zl,Z3,Z5,Z 6 ) = O,

/o(Z 1 ,Z 2 )

+ +

/ 1 ( Z 2 l Z 3 , Z 4 ) + 5i(Zi > Z3,Z4) (21.2) <72(Z3, Z 4 , Z 5 j Z 6 ) + / 2 ( Z i , Z 3 ) Z 5 ) Z 6 ) = O,

is determined by /o(Z 1 ,Z 2 ) = F 1 ( Z 1 ) - F 2 ( Z 2 ) , / i ( Z i , Z 2 , Z 3 ) = F2(Z1) + Gi(Z 2 , Z 3 ), <7 1 (Z 1) Z 2) Z 3 )

=

- J F 1 (Z 1 ) + G 2 (Z 1 ,Z 3 ) - G 1 ( Z 2 ) Z 3 ) + G 2 (Z 2 ) Z 3 ) + A, 205

(21.3)

206

Systems

of Linear Functional

Equations

/ 2 ( Z i , Z 2 , Z 3 , Z 4 ) = - G 2 ( Z i , Z2) + G3(Z3, Z 4 ) , 0 2 ( Z i , Z 2 , Z 3 , Z 4 ) = - G 2 ( Z i , Z 2 ) - G 3 ( Z 3 , Z 4 ) - A, where Fi, F 2 , G, (i = 1,2,3) are arbitrary functions A is an arbitrary constant vector from V. Proof.

with values in V, and

By p u t t i n g Z; = C, (z = 3,4, 5,6) into Eq. (21.1), we obtain /o(Zi,Z2)=F1(Z1)-F2(Z2),

(21.4)

where we introduced the notations ^(Zi)

=

i^ 2 (Z 2 )

=

-firi(Zi,C3,C,4)-52(Z1,C3,C5)C6), /1(Z2,C3,G4) + /2(C3,G4,G5,G6).

By virtue of the expression Eq. (21.4), by putting Z,- = C,- (i = 1,5,6) into Eq. (21.1) we get / 1 ( Z 2 , Z 3 , Z 4 ) = J F 2 (Z 2 ) + G 1 ( Z 3 , Z 4 ) I

(21.5)

where Gi(Z3,Z4)

=

-Fi(C!)

-

g^Cu^.Z*)

—

/ 2 ( Z 3 , Z 4 , C 5 , C 6 ) -
If we p u t Z,- = Ct {i = 5, 6) into Eq. (21.1), in view of Eqs. (21.4), (21.5) we have 5 i ( Z i , Z 3 , Z 4 ) = -F1(Z1)

- G i ( Z 3 , Z 4 ) + G' 2 (Z 3 , Z 4 ) - ff(Zi, Z 3 ) ,

(21.6)

where we introduced the notations G 2 ( Z 3 , Z 4 ) = - / 2 ( Z 3 , Z 4 , G5, Ce),

H(Z\, Z3) = 0 2 ( Z i , Z 3 , G5, Ce).

By substituting Eqs. (21.4), (21.5) and (21.6) into Eqs. (21.1) a n d (21.2), the system becomes G2(Z73,Z4)-^(Z1,Z3) +

/ 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) + 0 2 ( Z i , Z 3 , Z 5 , Z 6 ) = O, G2(Z3,Z4)-Jff(Z1,Z3)

+

(21.7)

02(Z 3 , Z 4 , Z 5 , Z 6 ) + / 2 ( Z i , Z 3 , Z 5 , Z 6 ) = O.

(21.8)

Systems in Which. Each Equation

Contains All Unknown Functions

207

By putting Z; = C\ (i = 1,3) into Eq. (21.8), after replacing Z 4 by Z 3 and putting Zi = d into Eq. (21.7), the equations (21.7) and (21.8) become G'2(Z3)Z4)-/f(C1,Z3) +

(21.9)

/2(Z3,Z 4 ,Z 5 ,Z 6 ) + 5 2 ( C i , Z 3 , Z 5 , Z 6 ) = O, G ' 2 ( C i , Z 3 ) - # ( < ? ! , Ci)

+

(21.10)

52(Gi,Z 3 ,Z 5 ,Z 6 ) + / 2 ( C i , C i , Z 5 , Z 6 ) = O.

By subtracting Eq. (21.10) from Eq. (21.9), we obtain / 2 ( Z 3 ) Z 4 , Z 5 , Z 6 ) = F 3 (Z 3 ) - G' 2 (Z 3 , Z 4 ) + G 3 (Z 5 , Z 6 ),

(21.11)

where F 3 (Z 3 )

=

G'2(Gi,Z3)-i/(G1,Gi) +

G 3 (Z 5 ,Z6)

=

/ 2 ( G i , G i , Z 5 ,Z6).

ff(G1,Z3),

From Eqs. (21.7) and (21.11) we obtain g2(Z1, Z 3 , Z S) Z 6 ) = H(ZUZ3)

- F 3 (Z 3 ) - G 3 (Z 5 , Z 6 ).

(21.12)

We substitute the functions / 2 and #2 determined by Eqs. (21.11) and (21.12) (with Z3,Z 4 replaced by Zi,Z3 and vice versa) into the equation (21.8) and obtain G' 2 (Z 3 , Z 4 ) + H{Z3, Z 4 ) - F 3 (Z 4 ) =

(21.13)

,

G 2 (Z 1 ,Z 3 ) + F ( Z 1 ) Z 3 ) - F 3 ( Z 1 ) .

It is clear that both sides of this equality are a function just of Z 3 , say, F 4 (Z 3 ). Then we have G 2 (Z 3 , Z 4 ) + H(Z3, Z 4 ) = F3(Z4) + F 4 (Z 3 ).

(21.14)

If in Eq. (21.14) we replace Z 3 , Z 4 by Zi, Z 3 , we obtain G'2(Z1} Z3) + H(ZUZ3)

= F3(Z3) + F 4 (Zi).

The last two equalities, together with Eq. (21.13), yield F 3 (Zi) - F 4 (Zi) = F 3 (Z 3 ) - F 4 (Z 3 ).

(21.15)

208

Systems

of Linear Functional

Equations

It is clear that both sides of this equality are equal to a complex constant vector A, and F 4 (Z 3 ) = F 3 (Z 3 ) - A. On the basis of this, the equation (21.14) takes on the form H(Z3, Z 4 ) = - G 2 ( Z 3 , Z 4 ) + F 3 (Z 4 ) - A,

(21.16)

where we introduced a new function G 2 (Z 3 , Z 4 ) = G' 2 (Z 3 , Z 4 ) - F 3 (Z 3 ).

(21.17)

From Eqs. (21.16), (21.15), (21.12), (21.11), (21.6), (21.5) and (21.4) there follows the result Eq. (21.3). D This theorem generalizes the result given in [R. Z. Djordjevic (1965B)]. Theorem 21.2

The general solution of the system of functional equations

/o^Z,)

/o(Z 1 > Z 2 )

+

/1(Z2,Z3,Z4) +

i7l(Z1,Z3,Z4)

(21.18)

+

/ 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) + <72(Zi, Z 3 , Z 5 , Z 6 ) = O,

+

^(Z2,Z3,Z4)+/i(Z1,Z3,Z4)

+

/ 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) + g2(Zu Z 3 , Z 5 , Z 6 ) = O

(21.19)

is determined by fo(Z1,Z2) fi{ZuZ2, gi(Z1,Z2,

=

F1(Z1)-F2(Z2),

Z 3 ) = F2(Z1) + F 3 (Z 2 ) + G 2 (Z 2 , Z 3 ),

Z 3 ) = F 2 (Zi) - Gi(Z 2 , Z 3 ) - G 2 (Z 2 , Z 3 ),

f2{ZltZ2,

Z 3 , Z 4 ) = Gi(Zi, Z 2 ) - H(ZUZ3,

(21.20)

Z 4 ),

<72(Zi, Z 2 , Z 3 , Z 4 ) = - F i ( Z i ) - F 2 (Zi) - i? 3 (Z 2 ) + H(Z2, Z 3 , Z 4 ), tu/iere F, (i = 1,2,3), Gj (j = 1 , 2 ) and H are arbitrary functions values in V .

with

Systems in Which Each Equation

Proof.

Contains All Unknown Functions

209

If we put Z, = C,- (i = 3,4, 5,6) into Eq. (21.18), we obtain /0(ZljZ2)=F1(Z1)-F2(Z2),

(21.21)

where -Fi(Zi)

=

- # i ( Z i , C 3 , G 4 ) -52(Zi,C3,Cs,Ce),

i*2(Z2)

=

/l(Z2,C3,C4) +

f2{C3,Ci,C§,Cs).

By putting Z, = C,- (i = 2,4) into Eq. (21.18), by virtue of the expression Eq. (21.21) we get Js(Zi, Z 3 , Z 5 , Z 6 ) = - J ? i ( Z 0 - G(Z X , Z 3 ) + H(Z3, Z 5 , Z 6 ),

(21.22)

where H(Z3, Z 5 , Z 6 ) = -/i(C* 2 , Z 3 , C 4 ) - / 2 ( Z 3 , G4, Z 5 ) Z 6 ) + F 2 (C 2 ), G(Z!,Z 3 ) = - 5 i ( Z i , Z 3 , C 4 ) . By virtue of the expressions Eqs. (21.21) and (21.22), if we put Z,- = d (t = 1 , 2 ) into Eq. (21.18), we get / 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) = - f f (Z 3 , Z 5 , Z 6 ) + Gi(Z 3 l Z 4 ),

(21.23)

where we have put Gi(Z 3 ) Z 4 ) = - / i ( C 2 , Z 3 , Z 4 ) - 0i(Ci, Z 3 , Z 4 ) + G ( d , Z 3 ) + F 2 (G 2 ). In view of the expressions Eqs. (21.21), (21.22) and (21.23), the system of functional equations (21.18) and (21.19) becomes A ( Z 2 , Z 3 , Z 4 ) +
F 2 ( Z 2 ) - G ( Z 1 , Z 3 ) + G 1 (Z 3 ,Z 4 ) = 0 ,
-

(21.24)

(21.25)

F 2 ( Z 2 ) - G ( Z 1 , Z 3 ) + Gi(Z3,Z 4 ) = 0 .

By putting Z 2 = C 2 into Eq. (21.24), after exchanging the roles of Zi and Z 2 in Eq. (21.25), subtracting Eq. (21.24) from Eq. (21.25), we get / i ( Z 2 , Z 3 , Z 4 ) = F 2 (Z 1 ) + G(Z2, Z 3 ) - G ( Z ! , Z 3 ) + /i(G 2 , Z 3 , Z 4 ) - F 2 ( C 2 ) ,

210

Systems

of Linear Functional

Equations

or if we p u t Zi = C 2 , we obtain / i ( Z 2 ) Z 3 , Z4) = G ( Z 2 l Z3) + G 2 ( Z 3 , Z 4 ) ,

(21.26)

where G 2 ( Z 3 , Z4) = / l ( C 2 , Z3, Z4) — G(C2, Z3). From Eqs. (21.25) and (21.26) it follows t h a t gi(Z2,

Z 3 , Z4) = F 2 ( Z 2 ) - G i ( Z 3 , Z4) - G 2 ( Z 3 , Z 4 ) .

(21.27)

We substitute the functions /1 and gi determined by Eqs. (21.26) and (21.27) into the equation (21.24) and find G ( Z 2 , Z 3 ) - F2(Z2) = G ( Z 1 ; Z3) - F 2 ( Z i ) . It is clear t h a t both sides of this equality are equal to a function F 3 ( Z 3 ) (independent b o t h of Zi and Z 2 ) , thus G(Z2,Z3) = F2(Z2) + F3(Z3).

(21.28)

T h u s , on the basis of Eqs. (21.28), (21.27), (21.26), (21.23), (21.22) and (21.21), we obtain Eq. (21.20). • T h e o r e m 21.3 /o(Zi,Z2)

/o(Z!,Z2)

is determined

The general solution of the system of functional

equations

+

f1(Z2,Z3,Z4)

+ g1(Z1,Z3,Z4)

(21.29)

+

/ 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) + g2(Zlt

+

5l(Z2,Z3,Z4)

+

<72(Z3, Z 4 , Z 5 , Z 6 ) + / 2 ( Z i , Z 3 , Z 5 , Z 6 ) = O

Z3, Z5, Z6) = O,

+ /1(Z1,Z3,Z4)

(21.30)

by /o(Zi,Z2) = F 1 ( Z i ) - F 2 ( Z 2 ) ) / 1 ( Z 1 , Z 2 , Z 3 ) = F2(Z1) + G1(Z2,Z3),

gl(Z1,Z2,

Z3) = F 2 ( Z i ) + F(Z2)

+ F3(Z3)

- Gx(Z2,Z3)

-

A, (21.31)

/ 2 ( Z 1 ; Z 2 ) Z 3 , Z 4 ) = F 3 ( Z i ) - F ( Z i ) - F 3 ( Z 2 ) + G 2 ( Z 3 , Z 4 ) + A, g2(ZuZ2,

Z3, Z4) = F 3 ( Z i ) - F ( Z i ) - F 3 ( Z 2 ) - G 2 ( Z 3 , Z 4 ) ,

Systems in Which Each Equation

Contains All Unknown Functions

211

where F,- (i = 1,2,3), Gj (j = 1,2) and H are arbitrary functions with values inV, F = Fi + F 2 + F 3 and A is an arbitrary constant vector from V. Proof.

By putting Z, = C,- (i = 3,4,5,6) into Eq. (21.29), we obtain /o(Z 1 ,Z 2 ) = F 1 ( Z 1 ) - F 2 ( Z 2 ) ,

(21.32)

where Fi(Zi)

-

-5i(Zi,C3,C4)-52(Zi,C3,C5,C6),

^ 2 (Z 2 )

=

/l(Z 2 ,C 3 ,C4) + / 2 (C3,C4,C5,C 6 ).

If we put Z,- = Ci (i = 1,5,6) into Eq. (21.29), in view of the expression Eq. (21.32) we get / 1 (Z 2 ,Z 3 ,Z4) = JF2(Z2) + G 1 (Z3,Z 4 ) I

(21.33)

where Gi(Z 3 , Z 4 ) = -F^dyg^d,

Z 3 , Z 4 ) - / 2 ( Z 3 , Z 4 l C 5 , C 6 )-ff 2 (C 1 , Z 3 , C 5 , C 6 ).

By putting Z,- = d (i = 1,5,6) into Eq. (21.30), by virtue of the expression Eq. (21.32) we obtain <7i(Z2, Z 3 , Z 4 ) = F 2 (Z 2 ) + G(Z 3 , Z 4 ),

(21.34)

where we have put G(Z 3 ,Z 4 ) = - F 1 ( C 1 ) - / i ( C 1 , Z 3 , Z 4 ) - 5 2 ( Z 3 , Z 4 , C 5 , C 6 ) - / 2 ( C 1 , Z 3 , C 5 , C 6 ) . On the basis of the expressions Eqs. (21.32), (21.33) and (21.34), the system of functional equations (21.29) and (21.30) becomes Fi(Zi)

+

F 2 (Z!) + Gi(Z 3 ,Z 4 )

(21.35)

+G(Z3,Z4)

+

/2(Z3,Z4,Z5,Z6)+52(Z1,Z3,Z5,Z6) = 0,

Pi(Zi) +G(Z3lZ4)

+ +

F 2 (Zi) + Gi(Z 3 l Z 4 ) (21.36) 5 2 (Z3,Z 4 ,Z5,Z 6 ) + / 2 ( Z 1 , Z 3 , Z 5 , Z 6 ) = 0 .

We put Z; = C\ (i = 1,3) into Eq. (21.36), and further we replace Z 4 by Z 3 . Also, we put Zi = C\ into Eq. (21.35). By subtracting the equations

212

Systems

oj Linear Functional

Equations

obtained, we get / 2 ( Z 3 l Z 4 , Z 5 , Z 6 ) = F3(Z3) - G1(Z3, Z 4 ) + G 2 (Z 5 , Z 6 ) - G(Z 3 , Z 4 ), (21.37) where we introduced the notations F3(Z3) = Gi(Ci, Z 3 ) + G ( d , Z 3 ),

G 2 (Z 5 , Z 6 ) = / 2 ( C i , d , Z 5 , Z 6 ).

From the equation (21.35), on the basis of the expression Eq. (21.37), we get g2(Zx, Zs, Z 5 , Z 6 ) = -FiiZi)

- F2(Z1) - F3(Z3) - G 2 (Z 5 , Z 6 ).

(21.38)

We substitute the functions / 2 and g2 determined by Eqs. (21.37) and (21.38) into the equation (21.36) and find Fi(Zi) + F2{Zl) + F3(Zi) - Fi(Z 3 ) - F2(Z3) - F3(Z4) +

(21.39)

Gi(Z 3 ) Z 4 ) + G(Z 3 , Z 4 ) - Gi(Z!, Z s ) - G(Z Xl Z 3 ) = O.

If we put Zi = C\, the equation (21.39) becomes Gi(Z 3 ) Z 4 ) + G(Z 3 , Z 4 ) = Fi(Z 3 ) + F2(Z3) + F3(Z3) + F 3 (Z 4 ) - A, (21.40) since G 1 (G 1 ,Z 3 ) + G(G 1 ,Z 3 ) = F 3 (Z 3 ), 3

where we have put A = J2 F,(Ci). On the basis of the expression Eq. (21.40), the equalities (21.34) and (21.37) obtain respectively the forms 5l(Z2,Z3)Z4)

/2(Z3,Z4,Z5)Z6)

=

F2(Z2) + F(Z3)

+

JF3(Z4)-Gi(Z3,Z4)-A,

=

F 3 (Z 3 ) - F(Z3)

-

F 3 (Z 4 ) + G 2 (Z 5 ,Z 6 ) + A,

(21.41)

(21.42)

where we introduced the new function 3

F(Z 3 ) = ^ F , ( Z 3 ) . «'=i

(21.43)

Systems in Which Each Equation

Contains All Unknown Functions

213

On the basis of Eqs. (21.43), (21.42), (21.41), (21.38), (21.33) and (21.32), we obtain the equalities (21.31). • Theorem 21.4 /o(Z 1 ,Z 2 )

fo(ZuZ2) /o(Zi,Z 2 )

The general solution of the system of functional equations +

/ 1 (Z 2 ,Z 3 ,Z4) + <7i(Zi,Z 3 ,Z4)

(21.44)

+

/2(Z 3 , Z 4 , Z 5 , Z 6 ) + #2(Zi, Z 3 , Z 5 , Z 6 ) = O,

+

/1(Z2,Z3,Z4)+5i(Zi>Z3,Z4)

+

g2(Z3, Z 4 , Z 5 , Z 6 ) + / 2 ( Z i , Z 3 , Z 5 ) Z 6 ) = O,

+

5 -i(Z 2 ,Z 3 ,Z 4 )

+

/ 2 ( Z 3 , Z 4 l Z 5 , Z 6 ) + g2{ZltZ3,Z5,

(21.45)

+ /i(Z1,Z3,Z4)

(21.46) Z6) = O

is determined by the equalities /o(Z 1 ,Z 2 ) = F 1 ( Z 1 ) - F 2 ( Z 2 ) , / i ( Z ! , Z 2) Z 3 ) = F 2 (Zi) + Gi(Z 2 , Z 3 ), 5i(Zi, Z 2 , Z 3 ) = F 2 (Z X ) + Fx(Z 2 ) + F 2 (Z 2 ) - Gi(Z 2 , Z 3 ) + K, (21.47) / 2 ( Z j , Z 2 , Z 3 , Z 4 ) = - F i ( Z i ) - F 2 (Zi) + G(Z 3 , Z 4 ), flf2(Zi, Z2> Z 3) Z 4 ) = ^ ( Z i ) - F 2 ( Z 0 - G(Z 3 , Z 4 ) - K, where F\, F2, G, G\ are arbitrary functions with values in V', and K is an arbitrary constant vector from V. Proof. The equations (21.44) and (21.45) form the system of functional equations given by Eqs. (21.1) and (21.2). Consequently, the functions determined by the equalities (21.3) satisfy Eqs. (21.44) and (21.45). In order for the functions Eq. (21.3) to satisfy the equation (21.46) we must have F 1 (Z 1 ) + J F 2 ( Z 1 ) - G 2 ( Z 1 , Z 3 ) =

(21.48)

F 1 (Z 2 ) + F 2 ( Z 2 ) - G 2 ( Z 2 , Z 4 ) = 0 .

It is clear that both sides of Eq. (21.48) are equal to a complex constant vector M, thus G 2 (Zi,Z 3 ) = Fi(Zx) + F 2 (Z!) - M.

(21.49)

Systems

214

of Linear Functional

Equations

If we introduce a new function G by G(Z 1 ,Z 2 ) = G 3 (Zi,Z 2 ) + M and denote K = A - 2M, from Eqs. (21.49) and (21.3) there follows Eq. (21.47). •

22

S y s t e m s in Which Not All Equations Contain All Unknown Functions

T h e o r e m 22.1

The general solution of the system, of functional equations /o(Zi,Z 2 )

+

/i(Z2,Z3,Z4)

+

g1(Z1,Z3,Z4)

(22.1) = 0,

/o(Zi,Z 2 )

+

<7i(Z 2 ,Z 3l Z 4 )

+/i(Zi,Z3)Z4)

+

/2(Z3lZ4,Z5>Z6) = 0 ,

(22.2)

is given by the equalities /o(Zi,Z 2 )

-

/1(Z1,Z2,Z3)

=

Si(Zi,Z 2 ,Z 3 )

=

-F(Z1)-F{Z2), F(Z1)-G(Z2,Z3),

(22.3)

F(Z1)+G(Z2,Z3),

/2(Z1,Z2,Z3)Z4) = 0 , where F and G are arbitrary functions with values in V'. Proof. According to [D. S. Mitrinovic (1963A)] the general solution of the functional equation (22.1) is /o(Zi,Z 2 )

=

H1(Z1)-F1{Z2),

f1(Z1,Z2,Z3)

=

F1(Z1)-G1(Z2,Z3),

5i(Zi,Z2)Z3)

=

- ^ i ( Z 1 ) + G 1 (Z 2 ,Z 3 ),

(22.4)

where F\, H\ and Gi are arbitrary functions with values in V . We substitute the functions /o, f± and g\ determined by the equalities (22.4) into the equation (22.2). We must have F i ( Z 0 + ffi(Zi) - F!(Z 2 ) - H1(Z2) = - / 2 ( Z 3 , Z 4 , Z 5 , Z 6 ).

(22.5)

Systems in Which Not All Equations

Contain All Unknown Functions

215

It is obvious that both sides of this equality are constant. By putting Zi = Z 2 into Eq. (22.5), we get /2(Z3,Z4,Z5,Z6) = 0,

(22.6)

F i ( Z 0 + ffi(Z0 = Fi(Z 2 ) + # i ( Z 2 ) = 2A,

(22.7)

and further

where A is a constant vector from V . By introducing new functions F and G by the following equalities F(Z0

=

G(Zi,Z 2 )

=

FM-A, G1(Z1,Z2)-A,

on the basis of the expressions Eqs. (22.7), (22.6) and (22.4), we obtain Eq. (22.3). • T h e o r e m 22.2

The general solution of the system of functional equations

/o(Zi, Z 2 ) + / i ( Z 2 , Z 3 , Z 4 ) + <7i(Zi, Z 3 , Z 4 ) = O, /o(Z!, Z 2 ) + <7!(Z2, Z 3) Z 4 ) + / i ( Z ! , Z 3 , Z 4 ) + / 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) = O, /o(Z!,Z 2 )

+

/1(Z2,Z3,Z4)+5l(Z1,Z3,Z4)

+

/ 2 ( Z 3 , Z 4 , Z 5 , Z 6 ) + 5 2 ( Z i , Z 3 , Z 5 , Z 6 ) = O,

fc-1

/o(Zi, Z 2 ) + ^ 5 , ( Z j + i , Z, + 2 , • • • , Z 2 l + 2 ) 1=1

fc-1

+ 2^/«(Zi, z 3 , • • • , z 2 i + 1 , z 2 l + 2 ) + /fc(Zfc+i, Zfc+2, • • •, z2fc+2) = o , 1=1

fc /o(Zi, Z 2 ) + ^ / , ( Z i + i , Z, + 2 , • • • , Z 2 , + 2 ) 1=1

k

+

^ ^ ( Z i , Z 3 , • • • , Z 2 l + i , Z 2 i + 2 ) = O, 1=1

Systems

216

of Linear Functional

Equations

m— 1

/o(Z1,Z2)+^fir1(Z

i + l ) Z , - + 2, • • • , Z 2 i + 2)

«= 1 m— 1

+

A^ /i(Zl, Z 3 , • • • , Z2j + 1, Z2> + 2 ) + /m(Z m +l, Z m + 2 , • • • , Z 2m +2) = O, 1=1

m

/ 0 ( Z i , Z 2 ) -f- 2j/,-(Z,- + i, Z, + 2, • • • , Z 2 ,+ 2 ) >' = 1 m

+

2_^9i(Z>i, Z 3 , • • • , Z 2 l + i , Z 2 i+ 2 ) = O, 1=1

is given by /o(Z 1 ,Z 2 )

=

-F(Zi)-F(Z2),

/1(ZljZ2,Z3)

=

F(Z1)-G(Z2)Z3),

Si(Zi,Z2,Z3)

=

JP(Z1)+G(Z2)Z3),

fi=9i

= 0

{2
where F and G are arbitrary functions with values in V'. Proof.

The proof of this theorem follows from Theorem 22.1.

•

We have noticed that this approach to the solution of systems of linear complex vector functional equations is not considered in the references [W. Eichhorn (1963); M. Ghermanescu (1953); M. Ghermanescu (1955)].

PART 2

Nonlinear Complex Vector Functional Equations

Chapter 6

Quadratic Functional Equations

In this chapter two vector functional equations are considered. First one simple quadratic complex vector functional equation is solved, and after that investigations are made when the special real quadratic functional equation has nontrivial solutions. The results presented here were obtained in [I. B. Risteski and V. C. Covachev (submitted A)]. Now we will introduce the following notations. Let V be a finite dimensional complex vector space and let there exist a mapping / : V2 i-> V. In the next section Zj (0 < i < n) will denote vectors in V. We assume that Zj = (ZJI (£),••• ,Zin(t))T, where the components Zij(t) (0 < i < n; 1 < j < n) are complex functions and O = (0, • • • , 0) T is the zero vector in V. We define multiplication of arbitrary two vectors U = (ui(t),--- ,un(t))T and V = (vi(t),--- ,vn(t))T in V as U V = (ui(t)wi(*),-•• ,un(t)vn{t))T. 23

Simple Quadratic Functional Equation

Now we will prove the following results. Theorem 23.1

The general solution of the functional equation

F(Zo)Z1>Z2)---,Zn)

+ +

F ( Z o , Z 2 ) Z 3 > " - , Z n > Z 1 ) + --.

(23.1)

,

-f (Zo,Z n ,Z 1 ,Z2, • • • , Z n _ 2 , Z n _ i ) = O,

where F ( Z 0 | Zi, Z a , • • • , Z n ) = / ( Z 0 , Z P ) / ( Z „ Z r ) 219

(/ : V2 .-> V)

(23.2)

Quadratic Functional

220

Equations

is given by / ( U , V) = g(V)h(V) (g,h : p, q, r, n V. 2'. 3'. 4'. 5'. 6'.

- g(V)h(U)

(23.3)

V •-> V are arbitrary complex vector functions) in the case when satisfy one of the following six conditions: 2p — q + r — n, 2n = 3r - 3p (p
(g : V following I". 2". 3". 4". 5". 6".

(23.4)

H-> V is an arbitrary complex vector function) six conditions is satisfied: 2p = q + r-n, 2n ^ 3r - 3p (p
when one of the

/(U,V)=0

(23.5)

in all other cases. Proof. Between the parameters p, q, r there may exist one of the following relations: l°.p
4°.p
5°.q
F(Zo,Zi,Z2,- • • , Z n _ i , Z „ ) = G ( Z o , Z i + r , • • • , Z n , Z i , - • • , Z r ) , Z0 = U 0 , so that Zj+n — Zj

Zj = U „ _ r + i

(23.6)

(1 U i > . - - , U n _ 2 , U „ _ i ) = O>

(23.7)

Simple Quadratic Functional Equation

221

but from Eq. (23.2) it follows that GfUo.UL-.-.Un)

(23.8)

/(Uo,U„+f-r)/(U„+,-r,U„) /(Uo,Up_r)/(Un+,_r,Un) /(U0,Up_r)/(U,_r,U„)

(p < q < r), (q
In the cases 4°, 5°, 6° we put F(Z0,Zi,Z2,- • • ,Z„_i,Zn)

(23.9)

== G(Zo, Z r _ i , Z r _ 2 , • • • , Z i , Z „ , - - - , Z r ), Z0 = U 0 ,

Zj = U n _ r + i

(1 < i < n)

so that Zj+n — Zj (1 < j < n). Then the equation (23.1) reduces to the equation (23.7), but the function Eq. (23.2) becomes GCUo.Ui.-.-.U,,) /(U0,Ur_p)/(U„+r_„Un) /(U0,Ur_p)/(Ur_„Un) /(U0,Un+r_p)/(Un+r._g,Un)

(23.10) (p
In the functions given by Eqs. (23.8) and (23.10) the vector variables Uj (0 < i < n) are arranged by increasing indices. Consequently, it suffices to consider the functional equation G(Uo,Ui,U2,---,U„)

+

G ( U 0 , U 2 , U 3 , - - - , U „ , U 1 ) + --- (23.11)

+

G(Uo,Un)Ui,.-.,Un_2lUn_i) = 0,

where GfUo.ULUa.-.-.U,,)

=

/(U0,U*)/(UmiU„) (0 < k < m < n).

(23.12)

We will call the equation (23.11) with the condition Eq. (23.12) transformed equation. By putting Uj = U (0 < j < n) the transformed equation reduces to /(U,U) = 0.

(23.13)

222

Quadratic Functional

Equations

If we put Uj = X for all indices j distinct from k, m, n and U* = Y,

U m = U,

U n = V,

then by virtue of Eq. (23.13) the transformed equation becomes / ( X , Y ) / ( U , V) + / ( X , U)/(U 2 m _ f c , U m _*) /(X,V)/(Um_,,Un_*) = 0,

+

(23.14)

where Uj = U j _ n for j > n. In the equation (23.14) the vectors U 2m _fc, U m _fc, Un-k can be all equal to X, or they can take values among X , Y , U, V according to the following table: U2m_fc Um_fc U„_fc

Y Xor V U

Y X X

V Y U

V X Xor Y

X Y X

X X YorU

For instance, it cannot be U2m_fc = U, because in such a case the equation (23.14) would contain a term / ( X , U ) / ( U , U m _*) which is impossible in view of the substitution made. If U2m_fc = Um_fc = U n _ fc = X, by virtue of Eq. (23.13) the equation (23.14) reduces to /(X,Y)/(U,V) = 0. From this equation for X = U, Y = V we obtain / 2 ( U , V) = O, so in this case there exists only the trivial solution /(U,V) = 0.

(23.15)

Now we will consider all cases when in equation (23.14) besides / ( X , U ) / ( U , V) at least one more addend is not O. a) We will have \J2m-k = Y if and only if n = 2m — 2k. The equation (23.14) then becomes / ( X , Y ) / ( U , V) + / ( X , U ) / ( Y , U m _ fc ) +

(23.16)

/(X,V)/(U m _ f c ,U 2 m _ 3 fc) = 0 .

On the basis of the second addend of this equation, taking into consideration the condition 0 < k < m < 2m -2k, we can conclude that Um_fc = X. Consequently, the equation (23.16) takes on the form / ( X , Y ) / ( U , V ) + / ( X , U ) / ( Y , X ) + / ( X , V ) / ( X , U 2 m _ 3 * ) = O. (23.17)

Simple Quadratic Functional

223

Equation

The only possible values of U2m_3fc are (i) U2m_3fc = Y =*• m = n = 2k; (ii) U2m_3fc = U =>• m = 3k, n = 4k. However, the case (i) cannot occur because of the assumption 0 < k < m < n. In the case m — 3k, n = 4k the equation (23.17) takes the form / ( X , Y ) / ( U , V) + / ( X , U ) / ( Y , X) + / ( X , V ) / ( X , U) = O.

(23.18)

If n = 2m — 2k and m ^ 3fc, then the equation (23.17) becomes / ( X , Y ) / ( U , V) + / ( X , U ) / ( Y , X) = O.

(23.19)

b) U2m-fc = V can appear if and only if n = 2m —fc,and in this case the equation (23.14) takes the form /(X,Y)/(U,V) + /(X,U)/(VlUm_fc) +

(23.20)

/(X>V)/(UTO_t>U2m_2ik) = 0 .

Then the only possible values of U2m-2k are (iii) U2m-2fc = U =>• m — 2k,n = 3k; 3 (iv) U 2 m - 2 * = Y =>• m = - k, n = 2k. In the case (iii) we have \3m-k = Y and equation (23.20) reduces to the following equation / ( X , Y ) / ( U , V) + / ( X , U ) / ( V , Y) + / ( X , V ) / ( Y , U) = O.

(23.21)

In the case (iv), since k,m,n are natural numbers, there must exist a natural number v such that k = 2v, m = 3u, n = 4v. Then U„ = X because otherwise one of the equalities v — 2v, v = 3v, v = 4v should hold, which is impossible. Consequently, in this case the equation (23.20) takes the form / ( X , Y ) / ( U , V) + / ( X , U ) / ( V , X) + / ( X , V ) / ( X , Y ) = O.

(23.22)

If n = 2m — k and n ^ 2k, n ^ 3fc, then the equation (23.20) becomes / ( X , Y ) / ( U , V) + / ( X , U ) / ( V , X ) = O.

(23.23)

c) Um_fc = Y implies m = 2k, and the equation (23.14) gets the form / ( X , Y ) / ( U , V ) + / ( X , U ) / ( U 3 * , Y ) + / ( X , V ) / ( Y , U n _ f c ) = O. (23.24)

224

Quadratic Functional

Equations

From this equation we see that neither of U3fc and Un-k can be equal to Y. We can have U3fc = V and Un-k = U if m = 2k, n — 3k. But this is just case (iii) which leads to equation (23.21). If m = 2k, n / 3k, the equation (23.24) has the form f(X, Y ) / ( U , V) + / ( X , U ) / ( X , Y) + / ( X , V ) / ( Y , X) = O.

(23.25)

d) Un-k — Y appears only if n = 2k and equation (23.14) becomes / ( X , Y ) / ( U , V) + / ( X , U ) / ( U 2 m _ , , U m _*) +

(23.26)

/(X,V)/(Um_fc,Y)=0.

The equality U2m-k = Y would imply m = n — 2k which is impossible. If 3 JJ2m-k — V, then m = -k, n = 2k. But this is just the case (iv) which leads to equation (23.22). 3 If n = 2k and m ^ -k, then the equation (23.26) gets the form / ( X , Y ) / ( U , V) + / ( X , V ) / ( X , Y) = O.

(23.27)

e) U„_fc = U appears only if n = k + m. Then from Eq. (23.14) we obtain / ( X , Y ) / ( U , V) + / ( X , U ) / ( U 2 m _ f c , U r o _*) +

(23.28)

/(X,V)/(Um_fc>U) = 0 .

If U2m-/fc = Y, then m = 3k, n = 4k and this is case (ii) which leads to equation (23.18). If U 2m -ft = V, then m = 2k, n = 3k and Um-k — Y . This is case (iii) which leads to equation (23.21). When n = k + m and m ^ 3k, m ^ 2k, then the equation (23.28) becomes / ( X , Y ) / ( U , V) + / ( X , V ) / ( X , U) = O.

(23.29)

Therefore, to solve the transformed equation for distinct values of k, m, n we must solve the following Eqs. (23.18), (23.19), (23.21), (23.22), (23.23), (23.25), (23.27) and (23.29). The equations (23.18), (23.19), (23.22), (23.23), (23.27) and (23.29) for X = U and Y = V in accordance with Eq. (23.13) get the form /2(U,V) = 0

Simple Quadratic Functional

Equation

225

which means that the general solution in these cases is /(U,V) = 0.

(23.30)

The equation (23.21) has the general solution / ( U , V) = s(U)MV) - g(V)h(V),

(23.31)

where g, h : VH> V are arbitrary functions. In order to find the general solution of the vector equation (23.25), we write the scalar equation for the j - t h component, j = 1, • • • , n: / j ( X , Y ) / j ( U , V ) + / J ( X , U ) / j ( X , Y ) + / i ( X , V ) / i ( Y , X ) = 0 . (23.25,-) For each nontrivial solution /,• of the scalar equation (23.25,) there exists at least one pair of complex vectors (A,-, B,) such that /,(A,-, B,) ^ 0. For X = V = A , a n d Y = B , from Eq. (23.25,) it follows that /,(A,-,B,)[/,(U, A,) + / , ( A , , U ) ] = 0 or /,(A,-,U) = - / , ( U , A , ) .

(23.32)

By putting X = A,- and Y = Bj from Eq. (23.25,), on the basis of Eq. (23.32) we obtain / , ( U , V) = /,(A,-, V) - /,(A,-,U).

(23.33)

Introducing the notation <7j(U) = /,(A,-,U), the equality (23.33) becomes /,(U,V)=fl,(V)-5,(U).

(23.34,)

On the other hand, if /,• = 0 is a trivial solution of Eq. (23.25,), we can still write it in the form Eq. (23.34,), where gj is an arbitrary constant. So if we define g(U) = (51 (U), • • • ,gn(V))T, we have / ( U , V ) = <7(V)-«7(U).

(23.34)

Since the function Eq. (23.34) is really a solution of the equation (23.25), the general solution of this equation has the form Eq. (23.34), where g : V *->• V is an arbitrary function. On the basis of the above there follows the result:

226

Quadratic Functional

Equations

The general solution of the transformed equation is r s(U)/»(V) - 0(V)/i(U) / ( U , V) = I g(V) - g(U) I, 0

(m = 2fc, n = 3*), (m = 2k, n ? 3k), in all other cases,

(23.35)

where g, h : V f-> V are arbitrary complex vector functions. Theorem 23.1 immediately follows from the result obtained above and the changes Eqs. (23.6) and (23.9). • For illustration of the above theorem we will consider some examples. Example 23.1 The functional equation (23.1) with the condition Eq. (23.2) for p — 4, q = 6, r = 2 and n = 6 becomes /(Zo, Z 4 ) / ( Z 6 , Z a ) + / ( Z 0 , Z 8 ) / ( Z i , Z 3 ) + / ( Z 0 , Z 6 ) / ( Z 2 ) Z 4 ) +

/ ( Z 0 , ZO/fZs, Z 8 ) + / ( Z 0 , Z 2 ) / ( Z 4 , Z 6 ) + /(Zo, Z 3 ) / ( Z 5 l Zi) = O.

By putting Z 4 = B , Z6 = U and Z 2 = V and by replacing all the other variables by A, from the last equation we derive / ( A , B ) / ( U , V) + / ( A , U ) / ( V , B) + / ( A , V ) / ( B , U) = O whose general solution is given by the formula f(V,V)

= g(\J)h(V)

-

g(V)h(V).

E x a m p l e 23.2 The equation (23.1) with Eq. (23.2) for p = 5, q = 3, r = 2 and n = 5 takes the form /(Zo, Z 5 ) / ( Z 3 , Z 2 ) + /(Zo, Z i ) / ( Z 4 , Z 3 ) + /(Zo, Z 2 ) / ( Z 5 , Z 4 ) + / ( Z 0 , Z 3 ) / ( Z i , Z 5 ) + / ( Z 0 , Z 4 ) / ( Z 2 , Zi) = O. If we put Z5 = B , Z 3 = U and Z 2 = V, after we replace all the other variables by A, we obtain the following equation / ( A , B ) / ( U , V) + / ( A , V ) / ( B , A) + / ( A , U ) / ( A , B) = O. For U = A and V = B from the above equation we obtain / ( A , B) = —/(B, A), so that the general solution is the function / ( U , V ) = <7(V)- 5 (U), where/(A,U)=5(U).

Special Quadratic Functional Equation

227

Example 23.3 The equation (23.1) with Eq. (23.2) for p = 1, q = 2, r = 4 and n = 5 becomes / ( Z 0 , Z1)f(Z2,Zi)

+ / ( Z 0 , Z 2 ) / ( Z 3 , Z 5 ) + /(Zo, Z 3 ) / ( Z 4 , Zj)

+ / ( Z 0 , Z 4 ) / ( Z 5 , Z 2 ) + /(Zo, Z 6 ) / ( Z i , Z s ) = O. If we put Zo = Zi = Z 3 = Z 4 = U and Z 2 = Z 5 = V, we obtain / 2 ( U , V) = O =$> / ( U , V ) — O, which means that the general solution of the above functional equation is the trivial solution.

24

Special Quadratic Functional Equation

We will call special equation the functional equation (23.1) in the special case when the function F is given by the following formula F(Z0,Z1,-")Zn) = /(Z1,Zp)/(Z„ZP)>

(24.1)

where / : R 2 H 4 R . We do not solve the special equation in the general case, but we will investigate when this equation has nontrivial solutions. At the same time, we will give some necessary conditions for this equation to have other than trivial solutions. After that, we will give one hypothesis when this equation has nontrivial solutions. By putting Zp = Z r = V and replacing all the other variables by U, in view of / ( U , U) = 0 the special equation becomes /2(U,U) + /(V,Z2p_1)/(Zp+,_1,Zp+r_1) +

(24.2)

/(V > Z J H.p-i)/(Z, + P _ 1 ,Z 2 ) .-i) + / ( Z a i , V ) / ( Z a a , Z a s ) = 0>

where

011

r-p+l =_~ {* ^ ~+ r-p+l

(r-p+l>0), (r-p + l < 0 ) ,

_

r + q-p ", + q + r-p

(r + q-p> 0), (r + qp<0),

2r-p \ n + 2r-p

(2r-p>0), (2r-p<0).

•{_( a3

~

Quadratic Functional

228

Equations

We can write equation (24.2) in the following form a / 2 ( U , V) + 6/(U, V ) / ( V , U) + c/ 2 (V, U) = 0, where a, b,c(a ^ 0) are nonnegative integers such that \
(24.3) + b + c
(24.4)

If b = 0, by virtue of a > 0, c > 0 and / : R 2 •-»• R from Eq. (24.3) we get /(U,V)=0.

(24.5)

If c = 0, b ^ 0 (b ^ a), assuming the existence of a pair of real numbers (A, B) such that f(A, B) ^ 0, from Eqs. (24.3) and (24.4) we obtain the following system

af(A,B) + bf(A,B)f(B,A) af2(B,A) + bf(A,B)f(B,A)

= 0, = 0.

From this system we obtain f(A, B) — f(B, A) = 0 which contradicts the assumption f(A,B) ^ 0. Consequently, in this case we have Eq. (24.5). Now, we will investigate the case a,b,c^ 0. If a = b = c = 1, assuming that f(A,B) ^ 0, from Eqs. (24.3) and (24.4) we get

f2(A,B)

+ f(A,B)f(B,A)

+ f2(B,A)

= 0,

but this equation has no nontrivial real solutions, so that f(A, B) — f(B, A) = 0 and again we have Eq. (24.5). We obtain the same result in the cases when a = 2, b = c = 1 and a = b = 1, c = 2. Now, we will analyze the case when o = i / 0 and c = 0. From Eqs. (24.3) and (24.4) we have [/(U,V)+/(V,U)]2=0, i.e., / ( U , V ) + / ( V , U ) = 0.

(24.6)

The remaining possibility is a = c = 1 and b — 2. Then from Eq. (24.3) we again obtain Eq. (24.6).

Special Quadratic Functional

229

Equation

Consequently, a necessary condition for the existence of nontrivial solutions of the special equation is c = 0,

a = b (= 1, 2)

(24.7)

a = c = 1.

(24.8)

or 6 = 2,

The relations Eq. (24.7) for the case o = b — 1 will hold if and only if one of the following conditions is satisfied: 2r — 1 = p,

p + q — 1 ^ r,

r + q — p = p,

p + q-\

r + q — pj^p

^ r,

2r-l^p

(mod n), (mod n).

(24.9) (24.10)

The case a = 6 = 2, c = 0 i s impossible because then p + r — 1 = r (mod n) must hold, but this relation is also impossible. The relations Eq. (24.8) will hold if and only if p + q — 1 = r,

2r — l=p,

r + q — p=p

(mod n).

(24.11)

On the basis of this there follows the result: T h e o r e m 24.1 For the special equation to have nontrivial solutions one of the conditions Eqs. (24.9), (24.10), (24.11) must be satisfied. If there exist nontrivial solutions of the special equation, they have the property Eq. (24.6) so that all solutions must be of the following form / ( U , V) = G(U, V) - G ( V , U ) ,

(24.12)

2

where G : R i-> R is an appropriately chosen function. The simplest function of the form Eq. (24.12), except for the function / ( U , V) = 0, is the following function /(U,V)=5(U)-0(V).

(24.13)

Let us investigate, when the function given by Eq. (24.13) is a solution of the special equation for any function g : R i - > R . Then we must have g(Zi)g(Zq) -g(Zp)g(Zq) +g(Z2)g(Zq+1) -g(Zp+1)g(Zq+i)

+ +

g(Zi)g(Zr) g(Zp)g(Zr) g(Z2)g(Zr+1) g(Zp+i)g(Zr+1)

230

Quadratic Functional

+

Equations

•••

+g(Zn)g(Zq-1) -giZp^giZg-i)

+

g(Zn)g(Zr-i) 5(Zp_1)fl(Zr_i)=0.

The last equation will be an identity for all g in two cases: 1°. The terms of the first and third column of the above equation mutually cancel, and so do the terms of the second and fourth column. 2°. The terms of the first and second column mutually cancel, and so do the terms of the third and fourth column. This will happen if and only if 1°. 2r - 1 = p, 2q-l=p (mod n) or 2°. q + r — p = p, r + q — 1 = 1 (mod n). The conditions 1° and 2° indeed present sufficient conditions for the existence of solutions other than the trivial one of the special equation. Now, we will give the following hypothesis: Hypothesis 24.1 The special equation has nontrivial solutions if and only if one of the following two conditions holds: 2r-l=p, q + r—p = p,

2q-l=p

(mod n),

r +q— 1= 0

(mod n).

(24.14) (24.15)

If these conditions are satisfied, simultaneously a necessary condition for existence of nontrivial solutions holds. Example 24.1

For the equation

/ ( Z i , Z 3 ) / ( Z 5 , Z 2 ) + / ( Z 2 , Z 4 ) / ( Z 6 , Z 3 ) + / ( Z 3 , Z 5 )/(Z 7 > Z 4 ) +

/ ( Z 4 , Z 6 ) / ( Z 1 ) Z6) + / ( Z 5 , Z 7 ) / ( Z 2 , Z a ) + / ( Z 6 , Z 1 ) / ( Z 3 ) Z7)

+

/(Z7,Z2)/(Z4,Z1)=0

the necessary conditions are satisfied, but the conditions of the hypothesis are not. The above equation has only the trivial solution.

Chapter 7

Modified Quadratic Functional Equations

In this chapter the basic quadratic complex vector functional equation, permuted quadratic functional equation, and three modified quadratic complex vector functional equations are solved. The results presented here were obtained in [I. B. Risteski and V. C. Covachev (submitted B)]. The notation used is as in Chapter 6.

25

Basic Quadratic Functional Equation

Now the following result will be proved. Theorem 25.1

The general solution of the functional equation 2n-l

/

J

-F(Zl, Z»+i, Z; + 2 • • • , Z 2 „+i-l) — O

(25.1)

(Z 2 „+i-i = Zu 2 

V)

231

(25.2)

232

Modified Quadratic Functional

Equations

is given by the following formula f(V V) - { 9
n = 2,

,

where g,h : V *-> V are arbitrary functions. Proof. The function given by Eq. (25.3) is a solution of the equation (25.1). Conversely, we will prove that every solution of the equation (25.1) has the form Eq. (25.3). If we substitute Z» = U (1 < i < 2n) into Eq. (25.1), we obtain /(U,U) = 0.

(25.4)

The trivial solution / ( U , V) = O of the equation (25.1) is included in Eq. (25.3). Moreover, a zero component of a nontrivial solution can be represented in this form. Further we will seek only nontrivial solutions. If we put Z2fc+2 = Z 2 = V and Z 3 = Z 4 = • • • = Z 2 f c + i = Z 2 * + 3 = • • • = Z 2 n = Zi = U into the equation (25.1), on the basis of Eq. (25.4) we have pf(V,

V) + qf(V, V ) / ( V , U) + r / 2 ( V , U) = O,

(25.5)

where p>0, q>0, r>0 are integers. Since the function / ( U , V) = h(V) — h(U) must satisfy the equation (25.5) for every function h, we get the condition p + r = q. By a permutation of the variables U and V in Eq. (25.5) we obtain pf2(V, U) + gf(V, U ) / ( U , V) + rf(V,

V) = O.

(25.6)

By adding together the equalities (25.5) and (25.6), and by virtue of the condition p + r = q>0we have /(U,V) + /(V,U) = 0 .

(25.7)

If n = 2, then the equation (25.1) according to Eq. (25.2) becomes / ( Z 1 , Z 2 ) / ( Z 3 , Z 4 ) + / ( Z 1 , Z 3 ) / ( Z 4 ) Z 2 ) + / ( Z 1 , Z 4 ) / ( Z 2 , Z 3 ) = O. (25.8) We denote any nonzero component of a nontrivial solution again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors

Basic Quadratic Functional Equation

233

(A, B) (A, B € V) such that / ( A , B) ^ 0. By putting Zl = A, Z 2 = B into the scalar equation (25.8), we obtain

If we denote g(Z) = / ( A , Z ) / / ( A , B ) and h(Z) = / ( B , Z ) , by virtue of the property Eq. (25.7) the equality (25.9) becomes / ( Z 3 , Z 4 ) = g(Z3)h(Z4)

- 5 (Z 4 )/ l (Z 3 ).

Thus for n = 2 the theorem is proved. Now, we will consider the case when k = 1 and n > 2. If we put Z3 = Z 4 = • •• = Z 2 n _ 2 = Zi,

Z 2 n _i = U,

Z 2 n = V,

(25.10)

then the equation (25.1) takes the following form / ( Z i , Z 2 ) / ( U , V) + f(Zu +

U ) / ( V , Z2)

/(Z1,V)/(Z2,Z1) + /(Z1,V)/(Z1,U) = 0.

Again we denote by / any nonzero component of the solution / . By putting Zi = A, Z 2 = B ( / ( A , B ) ^ 0) and applying the property Eq. (25.7), from the previous equation we find / ( U , V) = &

^

[/(B, V

) - / ( A , V)] + / ( A , V).

(25.11)

For V = B , from the above equation there follows the equality / ( B , U) - / ( A , U) = - / ( A , B)

(25.12)

which holds for every U. If we take into account Eq. (25.12), from Eq. (25.11) it follows that / ( U , V) = / ( A , V) - / ( A , U) = h(V) -

h(V),

which means that the theorem for k = 1 and n > 2 is proved. For k > 1, the equation Eq. (25.1) with the substitutions given by Eq. (25.10) takes the form /(Z1,U)/(V,Z2) + /(Z1,V)/(Z1,U)+/(Z2,Z1)/(Z1,U) = 0.

234

Modified Quadratic Functional

Equations

If we put Zi = A and U — B into any nontrivial scalar equation in the last vector equation, we obtain /(V,Z2) = / ( A , Z 2 ) - / ( A , V ) , i.e.,

f(XJ,V) =

h(V)-h(U).

This completes the proof of Theorem 25.1. 26

•

Permuted Quadratic Functional Equation

The functional equations considered in the next sections will be solved only for n > 2. Now we will solve the permuted quadratic complex vector functional equation. Theorem 26.1

The general solution of the functional equation 2n-l / , F(Zi,

Z;+i, Z;+2 ••• , Z2n+i_i) =

O

(26.1)

t=l

( Z 2 n + i _ ! =Zi,2
1),

where F(Zi,Z2, k =

^/(ZW-LUO

.i=l

(26.2)

•• • , Z2„) "I

Vn—k •

J

£/(Z

2fc+2t-l, V j j

Li=l

(/ : V2 H. V; n > 2) Ui

€

{Z 2 ) Z 4 ,---,Z 2 J f e }

(l
Vj

€

{Z 2 f c + 2 ,Z 2 f c + 4,--- , Z 2 n } U s ^ U t and Vs^Vt

(l
is given by the following formula f(V,V)=g(V)-g(V), where g : V i-> V is an arbitrary function.

(26.3)

Permuted Quadratic Functional Equation

235

Proof. The function given by Eq. (26.3) is a solution of the equation (26.1) Really, by a substitution of Eq. (26.3) into Eq. (26.2), after a rearrangement of the terms we obtain -FXZl,Z2,- • • ,Z2n) -

[ff(Zi) - g(Z2) + g{Z3) -••• + g(Z 2 *_i) - g(Z2k)}

x

[S(Z 2 JH-I)

- g(Z2k+2) +

ff(z2fc+3)

x

+ g(Z2n-i) -

g(z2n)},

i.e., the same result as by the substitution of Eq. (26.3) into Eq. (25.2). Since the function Eq. (26.3) is a solution of the equation (25.1), it follows that Eq. (26.3) is a solution of the equation (26.1). Conversely, we will prove that every solution of the equation (26.1) has the form Eq. (26.3). If we substitute Zj = U (1 < i < 2n), the equation (25.1) becomes /(U,U) = 0.

(26.4)

The trivial solution / ( U , V) = O of the equation (26.1) is included in Eq. (26.3). Moreover, a zero component of a nontrivial solution can be represented in this form. Further we will seek only nontrivial solutions. If we put Z2 = Z2n — V and Z 3 = Z4 = • • • = Z 2 n _i = Z\ = U into the equation (26.1), on the basis of the above expression Eq. (26.4) we obtain p / 2 ( U , V) + qf(V, V ) / ( V , U) + rf (V, U) = O,

(26.5)

where p> 0, q>0, r > 0 are integers. Since the function Eq. (26.3) must satisfy the equation (26.5) for every function g, by a substitution of Eq. (26.3) into Eq. (26.5) we get the condition p + r = q. By a permutation of the variables U and V, from Eq. (26.5) we obtain pf2(V, U) + 9 / ( V , U ) / ( U , V) + r / 2 ( U , V) = O.

(26.6)

By adding together the equalities (26.5) and (26.6), and by virtue of the condition p + r — q>0we get /(U,V) + /(V,U) = 0.

(26.7)

Now we will distinguish the following two cases: 1°. The case when k = 1. In this case the function Eq. (26.2) becomes F(Z1,Z2,---,Z2n) = /(Z1,Z2)[/(Z3)V1)

236

Modified Quadratic Functional

Equations

+ / ( Z 4 , V2) + • • • + / ( Z 2 n _ ! , Vn_i)], Vi€{Z4,Z6,---,Z2n} Vi ? V j

if

(l
It is necessary to separately investigate the cases V„_i ^ Z 2 n and V„_i = Z 2 n . We denote any nonzero component of a nontrivial solution again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors (A,B) ( A , B 6 V) such that / ( A , B ) ^ 0. In the case V n _ i ^ Z 2 n the substitutions Zi = A,

Z2 = B ,

Z 3 = Z 6 = • • • = Z 2 n _ 3 - A,

Z 2 n _ ! = U,

V n _ i = V,

(26.8)

Vj = V 2 = • • • = V n _ 2 = A

transform any nontrivial scalar equation in Eq. (26.1) to the following form /(A,B)/(U,V)

+

/(A,U)[/(A,B) + /(A,V)]

+

/ ( A , V ) [ / ( B , A ) + / ( U , A ) ] = 0.

From the above equation, on the basis of the expression Eq. (26.7) we obtain /(U,V) = / ( A , V ) - / ( A , U ) . By introducing the notation / ( A , U ) = g(V), we obtain that the function / really is of the form Eq. (26.3). In the case when V„_i = Z 2 „ the substitutions Eq. (26.8) transform any nontrivial scalar equation in Eq. (26.1) to the following equation /(A,B)/(U,V)

+

/(A,U)[/(V,A) + /(A,B)]

+

/(A,V)[/(B,A) + /(A,U)] = 0

(26.9)

when Vi ^ Z4, and to the equation / ( A , B ) / ( U , V) + / ( A , U ) / ( V , B) + when Vi = Z4.

/(A,V)/(B,A) + /(A,V)/(A,U)=0

(26.10)

Permuted

Quadratic Functional

Equation

237

The equation (26.9) on the basis of the expression Eq. (26.7) yields f(U,V)=g{V)-g(V),

(26.11)

where the notation / ( A , U) = g(V) is used. From Eq. (26.10), for V = B , we obtain /(U,B) = / ( A , B ) - / ( A , U ) . By using this equality and by introducing the notation / ( A , U) = g(V), equation (26.10) becomes Eq. (26.11). 2°. The case when k > 1. Let Ui = Z 2 p (1 < p < k). Now we will distinguish two cases: 1 < 2p < k and k <2p < 2k. For the case 1 < 2p < k, we put Zi = U, Z 2 p = V , Z 2 n = Y and Z 2 = Z 3 = ••• = Z 2 p _i = Z 2p +i = ••• = Z 2 n _i = Z, so that the equation (26.1), on the basis of the expressions Eqs. (26.4) and (26.7), takes the following form /(Z,Y)/(U,V) + c/(Z,Y)/(V,Z)

+ a/(U,Z)/(V,Z) + 6/(U,Z)/(Y,Z)

(26.12)

+ # ( U , Z ) / ( Y , V ) + £ / ( U , Y ) / ( V , Z ) = O,

where a, b, c, d, e are integers. We notice that in this equation e = 0. Indeed, the term / ( U , Y ) / ( V , Z) in the equation (26.1) can only appear in the item F(Zi, Z 2 n , Z 2 , Z 3 , • • •, Z 2 n - i ) - The variable Z 2 p appears among the arguments of this function in (2p + 2)-nd place, so that by the above substitution we obtain / ( Z i , Z 2 „ , Z 2 , Z 3 , • • • ,Z 2 n _x) = [/(U, Y) + / ( V , Z ) ] ( n - k)f(Z, Z) = O. The function / ( U , V) = g(V) - g(U) must satisfy the equation (26.12) for every function g. By a substitution of Eq. (26.3) into Eq. (26.12) we come to the following system of equations a + b=l,

a — c — d=— 1,

b — c + d = Q, a — d = 0,

c = 1, a + b — c = 0.

From this system we obtain b = 1 - a,

c = 1,

d = a.

(26.13)

238

Modified Quadratic Functional

Equations

Then the equation (26.1) has the form /(Z,Y)/(U,V)

+

o/(U,Z)/(V,Z) +

(l-a)/(U,Z)/(Y,Z)

+ /(Z,Y)/(V,Z)

+

o / ( U , Z ) / ( Y , V ) = O.

(26.14)

For Z = A, Y = B from any nontrivial scalar equation in Eq. (26.14) we obtain /(A,B)/(U,V)

+

a / ( U , A ) / ( V , A ) + (1 - a ) / ( U , A ) / ( B , A)

+ /(A,B)/(V,A)

+

a / ( U , A ) / ( B , V ) = 0.

(26.15)

For U = B from this equality we obtain / ( B , V) + / ( V , A) + / ( A , B) = 0.

(26.16)

By virtue of (26.16) from (26.15) we get /(U,V) = / ( A , V ) - / ( A , U ) . If we put
+

/(Z,Y)/(U,V)

+

ai/(U,Z)/(V,Z)

+ & 1 / ( U , Z ) / ( Y , Z ) (26.17)

Cl/(Z,Y)/(V,Z)

+

d!/(U,Z)/(Y,V) +

ei/(U,Y)/(V,Z)

= O.

Similar to the case 1 < 2p < k, we come to the conclusions that £i = 1 and b\ = 1 — d\,

ci = 0,

ai = — &i.

Thus the equation obtained implies the equation (26.15). Consequently, in this case every solution has the form Eq. (26.3). •

27

First Modified Quadratic Functional Equation

In this section the first modified equation will be solved.

First Modified Quadratic Functional Equation

Theorem 27.1 tional equation

239

The general solution of the first modified quadratic func2n-l

£F(Zi,Zi+i>Zi+2,-",Z2n+i-i) = 0

(27.1)

i=l

(Z2n+i_1=Zi, 2 V 3 ) + --- + /(V f c _ 2 > U f c _i)-/(U f c > Vfc_ 1 )] x

x [ / ( Z 2 H I , W i ) + / ( Z 2 H 3 , W 2 ) + • • • + /(Z2„_i,Wn_fc)] =

fork

even,

[/(Z1,U1)-/(U2,V1) + /(V2,U3) /(U f c _i,V f c _ 2 ) + / ( V t - i . U * ) ] x

x [/(Z 2 f c + 1 ,W 1 ) + / ( Z 2 f c + 3 , W 2 ) + --- + / ( Z 2 n i W n _ f c ) ] (f : V2 H- V;

n > 2),

Uie{Z2,Z4,---,Z2fc} V > e { Z 3 > Z 5 , " - ,Z 2fe _i}

(l
W r G {Z 2fc+2 , Z 2 f c + 4 , • • • , Z 2 n } Us^Ut,

Vs^Vt,

fork odd

Ws^Wt

(1 < r < n - k), fors^t,

is given by f(U,V)=g(\)-g(V)

(27.3)

for k odd, where g : V ^ V is an arbitrary function, while for k even the components of the general solution of Eq. (27.1) are given by Eq. (27.3) or by / ( U , V ) = MV),

(27.4)

240

Modified Quadratic Functional

Equations

where h is a suitably chosen function. Proof. First we will find the general solution of the functional equation (27.1) for k odd. By putting Z» = U (1 < i < 2n) into Eq. (27.1) we obtain /(U,U) = 0.

(27.5)

For Zi = Z 2 „ = U and Z; = V (2 < i < 2n - 1), on the basis of the expression Eq. (27.5), from the equation (27.1) we get f (U, V) + / ( U , V ) / ( V , U) = O.

(27.6)

By a permutation of U and V, the last equality becomes / 2 ( V , U) + / ( V , U ) / ( U , V) = O.

(27.7)

If we add together Eqs. (27.6) and (27.7), we obtain [/(U,V) + / ( V , U ) ] 2 = 0 , i.e., /(U,V) = -/(V,U).

(27.8)

By using the relation Eq. (27.8), the equation (27.1) can be transformed into the equation (26.1), whose solution according to Theorem 26.1 is given by /(U,V)=5(V)-S(U).

(27.9)

Since the last function in Eq. (27.9) satisfies the equation (27.1), Theorem 27.1 is proved for the case k odd. Now, we will prove Theorem 27.1 for the case k even. For the components of the function / : V2 H-> V (denoted again by / ) there exist two possibilities. 1°. / ( U , U) = 0. By putting Zi = Z 2 n = U and Z{ = V (2 < i < In — 1), on the basis of the assumption / ( U , U) = 0, from Eq. (27.1) we obtain the relation Eq. (27.6). As above, we deduce Eq. (27.8). Thus in this case the equation (27.1) becomes the equation (26.1). Consequently, for k even the components of the general solution of the equation (27.1) satisfying / ( U , U) = 0 are given by the formula Eq. (27.9). 2°. / ( U , U) ^ 0. In this case there exists at least one vector C g V such that / ( C , C) ^ 0.

Second Modified Quadratic Functional

Equation

241

By putting Zi = U, Z 2 = V and Z* = C (3 < i < 2n), from the equation (27.1) we obtain (n - fc){[/(U, V) - K + 2(fc - 1)/(U, C)

(27.10)

-2(k-2)K-f{C,V)-f(V,C)]K +

[/(U, C) - K][f(C, V) + / ( V , C) + 2(n-k-

where we have put f(C,C) obtain

1)K}} = 0,

= K. From the last equation, for V = C, we f{V,C)

= K.

(27.11)

By using Eq. (27.11), from Eq. (27.10) we get / ( U , V) - h(V)

(h(XJ) = f(C,U)).

(27.12)

However, in the general case the function Eq. (27.12) is not a solution of the equation (27.1) for an arbitrary function h. O In the next section we will solve the second modified equation. 28

Second Modified Quadratic Functional Equation

Now, we will give the following result. Theorem 28.1

The functional equation 2n-l

£F(Zi>Zi+1>Zi+2,"-,Z2n+i-i) = 0

(28.1)

i=l

(Z2n+i.1=Zi,

2
where F(Z1,Z2,---,Z2„)

(28.2)

=

[/(Z 1 ,U 1 ) + / ( Z 3 , U 2 ) + --- + /(Z 2 f c _ 1 ,U f c )] x

x

[/(V1,W1)-/(W2,V2) + /(V3,W3) +/(V n _ f c ,W n _fc)]

= x

forn-k

/(Wn_*_i,Vn_fc_i) odd,

[/(Z 1 ,U 1 ) + / ( Z 3 , U 2 ) + --- + /(Z 2 *_ 1 > U*)] x [/(Vi, W i ) - / ( W 2 , V 2 ) + / ( V 3 , W 3 ) - • • • + / ( W -/(Vn-i,W„-(i)] forn-k even

B

.

M )

V

B

.

M

)

242

Modified Quadratic Functional

(f : V2 •-> V;

Equations

n > 2),

Uie{Z2,Z4,---,Z2fc}

(1 < » < * ) ,

V,- € {Z24+1, Z2fc+3, • • • , Z 2 n _!}

(1 < j < n - k),

W r 6 {Z2k+2, Z2fc+4, • • • , Z 2 n }

(1 < r < n - k),

Us^Ut,

Vs^Vt,

Ws^W

t

fors^t,

1°. ftas a general solution given by /(U,V)=0(U)-fl(V),

(28.3)

where g : V 4 V w an arbitrary function, if n — k is odd; 2°. If n- k is even and Vj = Z 2 t+2i+i, W j = Z2fc+2j+2 (1 < i, j < n — k), then the function / ( U , V) = tflS(U) - K2g(V)

(28.4)

is a solution of the equation (28.1). For k > 1 and ifi = K2 the function Eq. (28.4) is a general solution of the equation (28.1) i / / ( U , U ) = O holds; 3°. / / in Eq. (28.2) the variable Z 2 n appears as an argument of a function f before which the + sign stands, then the general solution of the equation (28.1) is the function (28.3) i / / ( U , U ) = O holds. Proof. 1°. By putting Zj = U (1 < i < 2n), from the equation (28.1) we obtain /(U,U) = 0.

(28.5)

According to this relation, for Zi = Z 2 „ = V and Z; = U (2 < i < 2ra — 1), from the equation (28.1) it follows that [/(U, V) + / ( V , U)]/(V, U) = O.

(28.6)

By a permutation of the variables U and V in Eq. (28.6) and adding together the relation obtained and Eq. (28.6), we deduce /(U,V) + /(V,U) = 0 .

Second Modified Quadratic Functional

Equation

243

On the basis of this equality, we can transform the equation (28.1) into the equation (26.1), whose general solution is given by /(U,V) = 0(V)-S(U). Since this function satisfies the equation (28.1) if n —fcis odd, this means that Theorem 28.1 is proved. 2°. It is not difficult to check that Eq. (28.4) is indeed a solution of the equation (28.1) if n —fcis even and when the variables Vj and W r (1 < j , r 1), by putting Z 2 = Z 3 = Z4 = U and Z 5 = Ze = • • • = Z 2 „ = Zi = V, from Eq. (28.1) we obtain [/(U,V) + / ( V , U ) - / ( U , U ) - / ( V , V ) ] [ / ( U , V ) - / ( U , U ) ] = O. (28.7) By a permutation of the variables, we have [/(V,U) + / ( U , V ) - / ( V , V ) - / ( U , U ) ] [ / ( V , U ) - / ( V , V ) ] = O. (28.8) If we add together Eqs. (28.7) and (28.8), we obtain / ( U , V) + / ( V , U) - / ( U , U) - / ( V , V) = O. Since / ( U , U ) = O, the last equation gives /(U,V) + /(V,U) = 0, which means that equation (28.1) in the case considered becomes Eq. (26.1). 3°. Under the assumptions of the theorem the substitutions Z2 = Z 2 n = U, Zj = V (3 < i < 2n - 1) and Z x = V transform the equation (28.1) into the following form / ( V , U ) [ / ( U , V) + / ( V , U)] = O.

(28.9)

By a permutation of the variables in this relation and adding together the relation obtained and Eq. (28.9), we get /(U,V) + /(V,U) = 0. Consequently, in this case the equation (28.1) becomes Eq. (26.1), whose general solution is the function Eq. (28.3). • We have not been able to obtain the general solution of the equation (28.1) if n —fcis even. In the last section we will solve the third modified equation.

244

29

Modified Quadratic Functional

Equations

Third Modified Quadratic Functional Equation

We will finish this chapter with the following result. Theorem 29.1

The general solution of the functional equation 4m—1

/ J ^ ? ( Z i , Z j + i , Z j + 2 ) - • • ,Z 4 m + j_ 1 ) = O i=i

( Z 4 m + i _ i = Zi;

(29.1)

2 Za)-/(Z4>Z3) + /(Z6>Z6) h / ( Z 4 r _ 3 , Z 4r _2) — / ( Z 4 r , Z 4 r _i)] X

X

[/(Z 4 r + i, Z 4 r + 2) — / ( Z 4 r + 4 , Z 4 r + 3) H

1- / ( Z 4 m _ 3 , Z 4m _2) — / ( Z 4 m , Z 4 m _i)] ( / : V 2 H + V;

m,r>

1)

has components which are either constants or functions of the form f{V,V)=g(Y)-g{V),

(29.3)

where g is an arbitrary function defined in V. Proof. For the components of the function / : V2 H> V (denoted again by / ) we have two possibilities: 1°. If / ( U , U ) = 0, then for every nontrivial solution / there exists at least one pair of constant complex vectors (A, B) (A, B € V) such that / ( A , B ) ^ 0. By putting Zi = U, Z 2 = V, Z 4 m _i = B and Zt = A (3 < i < Am — 2), we obtain / ( A , B ) / ( U , V) - / ( A , B ) / ( U , A) + / ( A , B ) / ( V , A) = 0, i.e., by introduction of the notation / ( U , A) = — g(U) we have Eq. (29.3). 2°. If / ( U , U) ^ 0, then there exists at least one constant vector C € V such that f(C, C) ^ 0. By the substitution Z 2 = U , Z 4 r + 2 = U and Z3 = Z 4 = • • • = Z 4 r + i = Z 4 r + 3 = • • • = Z 4 m = Zi = C, from the equation (29.1) we obtain [f{C, U) - f(C, C)}2 + 2[f(C, C) - / ( U , C)][f(C, C) - f(C, U)]

Third Modified Quadratic Functional Equation

+[f(C,C)-f(U,C)}2

245

= 0,

i.e., f(C,V)

+ f(V,C)

= 2K,

(29.4)

where we have put f(C, C) — K. If we substitute Z4 r _i = Z 4 r = Z 4 r +i = Z 4 r + 2 — U and the remaining variables by C, then from Eq. (29.1) it follows that - [ / ( U , U) - Kf + [f(C, U) - / ( U , U)][/(C, U) + / ( U , C) - 2K] = 0, and according to Eq. (29.4) we find /(U,U) =

tf.

(29.5)

By substituting Z 2 = Z 4 = Z 4 r + 2 = Z 4 r + 4 = C and the rest of the variables by U, by virtue of Eq. (29.5) from the equation (29.1) we obtain /(U,C) = /(C,U), so that from Eq. (29.4) it follows that / ( U , C ) = / ( C , U ) = tf. By putting Z 4 = Z 4 r + 4 = U, Z 3 = Z 4 r + 3 = V and substituting all the other variables by C, according to the equality / ( U , C) = / ( V , C) = K from the equation considered we obtain [/(U,V)-/(C,C)]2=0, i.e., f(V,V)=K

(a constant).

(29.6)

Since the functions given by Eqs. (29.3) and (29.6) are indeed solutions of the equation (29.1), the proof of the theorem is complete. D

Chapter 8

Expanded Quadratic Functional Equations

In this chapter expanded quadratic functional equations with functional arguments, with the same signs or alternating signs between the functions, and a generalized expanded quadratic functional equation are solved. The results presented here were obtained in [i. B. Risteski and V. C. Covachev (submitted C)]. The notation used is as in Chapters 6 and 7. We denote by I = (1,1, • • • , 1) T the unit vector in V.

30

Expanded Quadratic Functional Equations with Functional Arguments

Now the following results will be proved. Theorem 30.1

The general solution of the functional equation 3k

5 3 F ( Z 1 , Z I - + i , Z j + 2 - - - ,Z3fe+i) = 0

(30.1)

:' = 1

(Z3fc+i = Zi, 2 < » < 3 A ) , where F(U1,U2,...,U3fc+i) = /(U1,5(U2,U3,---,Ufc+1))x

(30.2)

x/(#(U f c + 2 , U f c + 3 , • • • , U2fc+i), g(U2k+2, U 2 f c + 3 , • • • , U 3 f c + 1 )), 247

248

Expanded

Quadratic

Functional

Equations

and the function g : Vk f-> V satisfies the condition g{U,I,I,---,I)

= g(I,U,l,...,I)

= --- = g(I,I,---,I,U)

= V,

(30.3)

is given by the following formula / ( U , V) = G(U)H(V)

- G(V)H(U),

(30.4)

where G,H : V i-> V are arbitrary functions. Proof. Indeed, a straightforward calculation shows that the function given by Eq. (30.4) satisfies the functional equation (30.1) for arbitrary functions G and H. Conversely, we will prove that every solution of the functional equation (30.1) has the form Eq. (30.4). The trivial solution / ( U , V) = O of the equation (30.1) is included in Eq. (30.4). Moreover, a zero component of a nontrivial solution can be represented in this form. Further we will seek only nontrivial solutions. If we put Z3 = Z4 =

• • • = Zfc = Zfc + 2 =

• • • = Tj2k + l = Z 2 A + 3 =

• • ' = 7i3k + l =

I

into the equation (30.1), according to the condition Eq (30.3) we obtain /(Zi,Z2)/(Zfc+1,Z2fc+2) +

(30.5)

/(Z1,Zfc+1)/(Z2fc+2,Z2) + /(Z1,Z2fc+2)/(Z2,Zfc+1)=:0.

From the last equation it follows that /(U,U) = 0. We denote any nonzero component of a nontrivial solution again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors (A,B) ( A , B e V) such that / ( A , B ) ^ 0. By putting Z x = A, Z 2 = B, Zfc+i = U and Z2fc+2 = V, the scalar equation (30.5) takes the form

If we substitute U = B and if we take into consideration the condition / ( U , U) = 0, from the above equation (30.6) it follows that /(B,V) = -/(V,B).

(30.7)

Expanded Quadratic Functional Equations with Functional Arguments

249

According to Eq. (30.7), the equation (30.6) takes the form

* u ' v ) = 7(^W /(B ' V) ~ 7 ^ / ( B ' u ) -

<30-8>

If we denote /(A,U) /(A,B)

G(U),

/(B,U) =

ff(U),

then we get / ( U , V) = G(U)ff(V) - G(V)ff(U), which means that Eq. (30.4) is the general solution of Eq. (30.1). Theorem 30.2

•

The general solution of the equation n-l

J2 F{ZltZi+u

Zi+2 • • • , Zn+,-_i) = O

(30.9)

i=l

(Zn+i^

= Z{, 2 < t < n - l ) ,

w/iere F(Ui,U2)---,Un) =

(30.10)

/(Ui,5i(U2)U3,---Ufc+1))x

x

[/(52(Ufc+2, Uk+3, • • • Uk+t+i),g3(Uk+i+2,

+

/(#3(Ufc + 2 ,Ufc+3, • • • , U f c + m + l),£f2(Ufc + m + 2,Ufc + m + 3, • • • , U „ ) ) ]

Uk+i+3, • • • , U„))

+ /(U1)52(U2,U3,---U£+1))x x

+

[f{9i(Ui+2,'Ui+3,---'Uk+t+i),g3{Uk+i+2,'Uk+t+3,--,

/ ( 5 3 ( U ^ + 2 , U ^ + 3 , • • • ,Ul+m+l),gi(Ul+m+2,Ui+m+3, +

,U„)) • • • , Un))]

/(U1,53(U2,U3,---Um+1))x

X

[/(5l(Um+2,Um+3,Ufc+m+i),52(Ufc+m+2,Ufc+m+3,

+

f{92 ( U m + 2 , U m + 3 , • • • , Ut+m+i),

• • • , U„))

gi ( U ^ + m + 2 , U f + m + 3 , • • • , U „ ) ) ]

(k + £+m+ 1 = n) and the functions gt (i = 1, 2, 3) satisfy the conditions gi(U,I,l,---,I)=gi(l,V,I,---,I)

= --- = gi{I,I,---,I,U) (. = 1,2,3),

=U

(30.11)

Expanded Quadratic Functional

250

Equations

is given by / ( U , V) = G(U)//(V) - G ( V ) # ( U ) ,

(30.12)

where G, H : V >-> V are arbitrary functions. Proof. It is easy to check that the function given by Eq. (30.12) is a solution of the equation (30.9). To this end it suffices to put Eq. (30.12) into Eq. (30.9). Conversely, we will prove that every solution of the equation (30.9), if the conditions Eq. (30.11) are satisfied, is given by Eq. (30.12). Without loss of generality of the proof, we may assume that k < I < m. If we substitute Z, — I (1 < i < 2n) and if we apply the properties Eq. (30.11), then the equation (30.9) takes the form / ( I , I) = O. The trivial solution / ( U , V) = O of the equation (30.9) is included in Eq. (30.12). Moreover, a zero component of a nontrivial solution can be represented in this form. Further we will seek only nontrivial solutions. First, we will find all solutions of the equation (30.9) for which the following condition holds: /(Z,I)=0.

(30.13)

We denote any nonzero component of a nontrivial solution again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors (A, B) (A, B € V) such that / ( A , B) ^ 0. If we substitute Zi = A, Z^+2 = U, Z/j+3 = B and Z2 = Z3 = • • • = Zk+i = Z/e+4 = • • • = Z n = I, according to Eqs. (30.11) and (30.13) from the equation (30.9) it follows that / ( I , U ) = 0.

(30.14)

Now, if we substitute Z 3 = Z 4 = • • • = Z/-+i = Zfc+4 = • • • = Zn = I, according to the equalities (30.13) and (30.14) the equation (30.9) becomes /( Z l> Z2)/(Zfc + 2 , Z fc+3 ) +

/ ( Z i , Z f c + 2 )/(Z f c + 3 , Z 2 ) + / ( Z i , Z f c + 3 )/(Z 2 ) Z fc+2 ) = 0.

The general solution of the above equation is given by Eq. (30.12), where G and H are arbitrary functions which according to Eq. (30.13) satisfy the

Expanded Quadratic Functional Equations with the Same Signs

251

condition G{V)H(I)

- G(I)H(U)

= O.

Now, we find all solutions of the equation (30.9) for which / ( Z , I) ^ O. In this case, for each component of such a solution (denoted again by / ) for which / ( Z , I ) ^ 0 there exists at least one vector C ^ I such that /(C,I)^0. For Zi = C, Z2 = U and Z3 = Z 4 = • • • = Z„ = I, according to Eq. (30.11) the equation (30.9) takes the form / ( C , I ) [ / ( U , I ) + / ( I , U ) ] = 0, which implies that /(U,I) = - / ( I , U ) .

(30.15)

If we substitute Zi — C, Z2 = U, Z3 = V, Z4 = Z5 = • • • = Z n = I, and if we take into consideration the property Eq. (30.15), then the equation (30.9) reduces to the following form f(C, I ) / ( U , V) - f{C, U ) / ( I , V) + f(C, V ) / ( I , U) = 0.

(30.16)

If we denote ^ H U G ( U )

and

/(I,U)=if(U),

we obtain / ( U , V) = G(U)H(V)

-

G(V)H(U),

which represents the general solution Eq. (30.12) according to the hypothesis / ( Z , I ) £ O. •

31

Expanded Quadratic Functional Equations with the S a m e Signs b e t w e e n the Functions

In this section we will prove the following results.

Expanded Quadratic Functional

252

Theorem 31.1

Equations

The general solution of the functional equation 2n-l

] T F(Z1)Zi+1)Zj+2)---,Z2n+1-_i) = 0

(31.1)

t=i

(Z 2 „ + ,--i = Z,-, 2
1),

w/iere F(Z1,Z2)-.-,Z2n_1,Z2„)

(31.2)

n-3

=

/ ( Z i , Z 2 ) 2 ^ ^ ' [/( Z J'+3. Z i+4) + / ( Z 2 n _ j _ i , Z 2 n _ j ) ] + an_2/(Z!, Z2)/(Zn+1, Zn+3)

( / : V2 -> V)

anii aj (0 < j < n — 2) are complex constants such that n-2

( 31 - 3 )

X>*o, is given by /(U,V)

G(U)-G(V), G(U)tf(V)-G(V)//(U),

n>2, n = 2,

l

"'

where G,H : V H V are arbitrary functions. Proof. Next, we will prove the theorem for n. > 2. If we put Zi = Z 2 = • • • = Z 2 n = U, then from Eq. (31.1) we obtain / ( U , U) = O for each U . The trivial solution / ( U , V) = O of the equation (31.1) is included in Eq. (31.4). Moreover, a zero component of a nontrivial solution can be represented in this form. Further we will seek only nontrivial solutions. We denote any nonzero component of a nontrivial solution again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors (A, B) (A, B G V) such that / ( A , B) ^ 0. According to the condition Eq. (31.3) there exists an index k 6 {0, 1, • • •, n — 2} such that a^ ^ 0. We will distinguish the following cases: 1°. Let k = 0. If we substitute Z 5 = Z 6 = ••• = Z 2 n = Zj = A, Z 2 = B, Z 3 = U

and

Z4 = V

Expanded Quadratic Functional Equations

with the Same Signs

253

into Eq. (31.1), we get a 0 [/(A, B ) / ( U , V) + / ( A , U ) / ( V , A)

(31.5)

+ /(A)B)/(A,U) + /(A,V)/(B,U)] +

ai

[/(A, B ) / ( V , A) + / ( A , B ) / ( A , V)] = 0.

For V = A, from the above relation it follows that / ( U , A) = —/(A, U). If we put U = B into the equation (31.5), then we obtain /(B,V) = - / ( V , A ) - / ( A , B ) . If we substitute G(U) = / ( U , A), from Eq. (31.5) it follows that /(U,V)=G(U)-G(V). 2°. Let 0 < * < n - 2. If we put Zl = Z 3 = • • • = Zfc+2 = Zfe+5 = • • • = 7i2n = A, Z2 = B, Z/j+3 = U, Zfc+4 = V, then the equation (31.1) takes the form ak+1 {/(A, B) [/(A, V) + / ( V , A)]} +

(31.6)

ak [/(A, B ) / ( U , V) + / ( A , B ) / ( A , U) + / ( B , A ) / ( A , V)] = 0.

For V = A from (31.6) it follows that / ( A , U) = - / ( U , A), i.e., /(U,V) = G(U)-G(V), where G(U) = / ( U , A ) . 3°. Let k = n — 2. If we substitute Zl = Z3 = • • • = Z n = Z„+3 = • • • = Z2n = A, Z2 = B ,

Z n +i = U,

Z„ + 2 = V

into (31.1), we get a„_ 2 [/(A, B ) / ( U , V) + / ( A , B ) / ( A , U) + / ( B , A ) / ( A , V)] = 0. For V = A this equation yields the equality /(A,U) = - / ( U , A ) .

(31.7)

Expanded Quadratic Functional Equations

254

According to this equality, the equation (31.7) becomes /(U,V)=G(U)-G(V), where G(U) = / ( U , A ) . We have shown that for n > 2 / ( U , V) has the form /(U,V) = G(U)-G(V),

(31.8)

where G is an arbitrary function from V. The proof of the theorem for n = 2 is very easy. In fact, for n = 2 from Eq. (31.1) there follows the equation /(Zl!Z2)/(Z3,Z4) +

(31.9)

/ ( Z i , Z 3 ) / ( Z 4 , Z2) + / ( Z j , Z 4 ) / ( Z 2 , Z 3 ) = O.

The above equation is the same as equation (30.5) (see also Eq. (25.8)), and according to Theorem 30.1 (or Theorem 25.1) it follows that / ( U , V) = G(U)ff(V) - G(V)ff(U), where G,H : V i-* V are arbitrary functions. Theorem 31.2

D

The general solution of the functional equation 2n-l

/ „ F(Zi\, Zt + 1 , Z,- + 2. ' ' • > Z 2 n + ! _l) — O

(31.10)

i=i

(Z 2 n +i'-

Z; : 2 < t < 2 n - 1)

where -F(Zi, Z 2 , • • • , Z 2 „) fc-i / y/(Z2j + l,Z2j+2) J =0

(31.11)

n-fc-1 / y aj/(Z2fc+j+i,Z2n_j)

/ : V2 — i >• V, a^ (0 < j < n — k — 1) are complex constants such that n — k~ 1

(31.12) /or n > 2 is given by the formulae /(U,V) = G(U)-G(V)

(31.13)

Expanded Quadratic Functional Equations

with the Same Signs

255

or / ( U , V) = G(U)tf (V) - G(V)H{\3),

(31.14)

where G,H : V >-)• V are arbitrary functions. Proof. In the previous theorem we proved that the function given by Eq. (31.13) is one solution of the equation (31.10). Now, we will prove the converse, i.e., that every solution of the equation (31.10) has the form Eq. (31.13) or Eq. (31.14). If we put Z, = U (1 < i < 2n) into equation (31.10), by virtue of the relation Eq. (31.12) we obtain /(U,U) = 0.

(31.15)

The trivial solution / ( U , V) = O of the equation Eq. (31.10) is included in both Eqs. (31.13) and (31.14). Moreover, a zero component of a nontrivial solution can be represented in each of these forms. Further we will seek only nontrivial solutions. We denote any nonzero component of a nontrivial solution of equation (31.10) again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors (A, B) (A, B 6 V) such that /(A,B)^0.

(31.16)

Since n > 2, we will distinguish the following three cases: 1°. Let k — 1. On the basis of the condition Eq. (31.12) there exists at least one coefficient cij, j £ {0,1, • • • , n — 2} which is distinct from 0. Let ar be the nonzero coefficient with the smallest index. In the case when 0 < 1r < n — 3, if we substitute Z2 = B, Z r +3 =

u,

z2n_r = v, Zi = Z3 = • • • = Z r +2 = Z r + 4 = • • • = Z 2 n - r - l = Z2„_ r +1 = • • • = 'Lin — A

into Eq. (31.10), then we obtain ar [/(A, B ) / ( U , V) + / ( A , B ) / ( A , U) + / ( B , A ) / ( A , V)] +

a2r+1[/(A,U)/(A,V) + /(U,A)/(A,V)]=0.

(31.17)

Expanded Quadratic Functional

256

Equations

If we put into Eq. (31.17) V = A, then we get / ( A , U ) = - / ( U , A), and on the basis of this Eq. (31.17) becomes /(U,V)=G(U)-G(V), where we put G(U) = / ( U , A). In the case when 2r > n — 3 and 3r ^ 2 n - 4 , if we put Z 2 = B, U, Z 2 „_ r = V,

Zr+3 =

Zi = Z3 = • • • = Z r + 2 — Z r +4 = • • • = Zi2n-r-l

— Z 2 n - r + l = • • • = Z2„ = A

into Eq. (31.10), we get ar [/(A, B ) / ( U , V) + / ( A , B ) / ( A , U) + / ( B , A ) / ( A , V)] +

a 2 „ _ 2 r _ 4 [ / ( A , U ) / ( A , V ) + / ( A , U ) / ( V , A ) ] = 0.

(31.18)

From the above equation for V = A we obtain / ( A , U) = —/(U, A), and on the basis of this (31.18) becomes /(U,V)=G(U)-G(V), whereG(U) = / ( U , A ) . Now, let 3r = 2n — 4 and let besides the coefficient ar which is distinct from zero, there exist a coefficient as (s > r) which is also different from zero. Then, by putting Z 2 = B , Z,+3 = U, Z2„- s = V and by substituting the other variables by A, the equation (31.10) becomes as [/(A, B ) / ( U , V) + / ( A , B ) / ( A , U) + / ( B , A ) / ( A , V)] +

a 2 n _ 2 , _ 4 [ / ( A , U ) / ( A , V ) + / ( A ) U ) / ( V , A ) ] = 0.

(31.19)

If we put V = A, from the above equation we obtain / ( A , U) = —/(U,A), and on the basis of this (31.19) becomes /(U,V) = G(U)-G(V), where G(U) = / ( U , A ) . However, if all the coefficients ao, 0,1, • • • ,an-2 except ar are equal to zero, by substituting Z 2 = B , Z r + 3 = U, Z 2 n - r = V and the other variables by A, from Eqs. (31.10) and (31.11) we obtain ar [/(A, B ) / ( U , V) + / ( A , U ) / ( V , B) + / ( A , V ) / ( B , U)] = 0.

(31.20)

Expanded Quadratic Functional Equations with the Same Signs

257

For V = B , from Eq. (31.20) it follows that /(B,U) = -/(U,B).

(31.21)

On the basis of the equalities (31.20) and (31.21), we get / ( U , V) = G(V)H(V)

- G(V)ff(U),

where we put G(U) = / ( A , U ) and H(U) = / ( B , U ) / / ( A , B ) . 2°. Let 1 < Ar < n — 1. Also in this case, from the condition Eq. (31.12) we conclude that at least one of the coefficients a, (0 < j < n — k — 1) must be different from zero. Let again ar be the nonzero coefficient with the smallest index. We will distinguish three possibilities: a) For r = 0, i.e., a^ ^ 0, if we put Z4 = Z 5 = • • • = Z2,, = Zi = A, Z2 = B , Z3 = U, from the equation (31.10) we obtain / ( A , U ) = - / ( U , A). By putting Z 5 = Z 6 = • • • = Z 2 „ = Zi = A, Z 2 = U, Z 3 = V, Z4 = B , the equation (31.10) becomes a0 [/(B, A ) / ( U , V) + / ( B , A ) / ( A , U) + / ( A , B ) / ( A , V)] +

a0[/(A,U)/(A,V) + /(A,U)/(V,A)]

+

0l

(31.22)

[/(B, A ) / ( A , U) + / ( A , B ) / ( A , U)] = 0.

On the basis of the equality / ( A , U ) = —/(U,A), the above equality (31.22) becomes /(U,V) = G(U)-G(V), whereG(U) = / ( U , A ) . b) For 0 < r < n - k - l , by putting Z 2 = B, Z r + 3 = U and by substituting the other variables by A, we deduce from Eq. (31.10) the following equation o r / ( A , B ) [ / ( A , U ) + / ( U , A ) ] = 0, from which we obtain /(A,U) = - / ( U , A ) .

(31.23)

Now, if we put Z 2 = U, Z 3 = V, Z r + 4 = B and if we substitute the other variables by A, then from Eq. (31.10) it follows that a r + 1 [/(A, B ) / ( A , U) + / ( B , A ) / ( A , U)] +

(31.24)

a r [ / ( B , A ) / ( U , V ) + / ( B , A ) / ( A , U ) + / ( A , B ) / ( A , V ) ] = 0.

Expanded Quadratic Functional

258

Equations

On the basis of the equalities (31.23) and (31.24) we obtain /(U,V)=G(U)-G(V), where we introduced the notation G(U) = / ( U , A). c) For r = n - k - 1, by putting Z 2 = B, Z n _/; + 3 = V, Z n _ A + 2 = U and by substituting the other variables by A, from Eq. (31.10) we obtain a n _ f c _!/(A, B) [/(U, V) + / ( A , U) + / ( V , A)] +

an_k_J(B,

(31.25)

A) [/(A, V) + / ( V , A)] = 0.

From Eq. (31.25), for V = A we get / ( A , U) = - / ( U , A ) , and on the basis of this Eq. (31.25) becomes /(U,V)=G(U)-G(V), where G(U) = / ( U , A ) . 3°. Let k = n — 1. In this case we have -F(Zi, Z 2 , • • • , Z 2n ) =

[/(Zi, Z 2 ) + / ( Z 3 , Z 4 ) + • • • + / ( Z 2 n _ 3 , Z 2 n _ 2 )] a 0 / ( Z 2 „ - 1 , Z 2 n ).

From the condition Eq. (31.12) it follows that ao ^ 0. Then we may divide the equation (31.10) by ao, but in this case we obtain a functional equation whose general solution according to [I. B. Risteski and V. C. Covachev (submitted B)] is given by /(U,V)=G(U)-G(V), where G : V >-> V is an arbitrary function.

•

n-k-l

Yl aj — 0, the function Eq. (31.13) is a solution of i=o the equation (31.10), but the question of generality of this solution remains open. Remark 31.1.

If

In the next section we will consider an expanded quadratic functional equation with alternating signs between the functions.

Expanded Quadratic Functional Equation with Alternating

32

Signs

259

Expanded Quadratic Functional Equation with Alternating Signs between the Functions

Here the following result will be proved. Theorem 32.1

The general solution of the functional equation 2n-l

/ , -F(Zl,Zi + i,Zj + 2, • • • ,Z2„-l,Z2n) — O

(32.1)

i=l

(Z 2 n + ,_i = Z i ;

2
whe •f (Zi, Z 2j • • • , Z 2 n _i, Z2„)

(32.2)

[f(Zi, Z2) - / ( Z 4 , Z3) + • • • - /(Z 2 f c _ 2 , Z 2fc - 3 ) + /(Z 2 fc-i, Z2fc)] n-fc-1

x =

<

£

aj/(Z2fc+j+i,Z 2 n -j)

(A euen),

3=0

[/(Zi, Z2) - / ( Z 4 , Z3) + • • • + /(Z 2 f c _ 3 , Z2fc_2) - /(Z 2 f c ) Z 2fc _i)] n-fc-l

x

2J

a

j/(Z2fe+j-(-i, Z 2 n _j ) (/:

(jfe odd)

V2^V),

and a.,- (0 < j < n — k — 1) are complex constants such that n-k-l

£

a,-#0,

(32.3)

j=o

/or A; odd is given by the formulae /(U,V) = G(U)-G(V)

(32.4)

/ ( U , V) = G(U)ff(V) - G(V)ff(U),

(32.5)

or

where G, H : V i-> V are arbitrary functions, while for k even it has components which are either constants or functions of the form Eq. (32.4). Proof. k odd.

Now, we will find the general solution of the equation (32.1) for

Expanded Quadratic Functional

260

Equations

By putting Z, = U (1 < i < 2n) from the equation (32.1) we obtain /(U,U)=0.

(32.6)

For Zi = Z 2 „ = U, Zj = V (2 < j < 2n - 1), on the basis of the equality (32.6), the equation (32.1) becomes

J2 « , [ / 2 ( U , V ) | / ( U , V ) / ( V , U ) ] = 0 ,

(32.7)

3=0

or, on the basis of the condition Eq. (32.3), we get /2(U,V) + /(U,V)/(V,U) = 0.

(32.8)

By a permutation of the variables U and V, the last equation becomes /2(V,U) + /(V,U)/(U,V) = 0.

(32.9)

By addition of Eq. (32.8) and (32.9), we obtain [/(U,V) + / ( V , U ) ] 2 = 0 , i.e.,

/(V,U) = -/(U,V).

(32.10)

Making use of the relation Eq. (32.10), we can reduce the equation (32.1) to the equation (31.10), whose general solution according to Theorem 31.2 is given by /(U,V) = G(U)-G(V)

(32.11)

/ ( U , V) = G(U)ff(V) - G(V)ff(U).

(32.12)

or

Since the functions given by Eqs. (32.11) and (32.12) satisfy the equation (32.1), Theorem 32.1 is proved for the case of odd k. Now, we will pass to the proof of Theorem 32.1 for the case of even k. First, we will assume that a component of / (denoted again by / ) satisfies / ( U , U) = 0. By putting Zx = Z 2 n = U and Z?- = V (2 < j < 2n - 1), on the basis of the assumption / ( U , U) = 0, from the equalities (32.1) and (32.2) we obtain the relation Eq. (32.7) from which, by virtue of the condition Eq. (32.3) we get Eq. (32.8). By a permutation of the variables U and

Expanded Quadratic Functional Equation with Alternating

Signs

261

V, from Eq. (32.8) we deduce Eq. (32.9). Adding together Eq. (32.8) and (32.9) we obtain Eq. (32.10), so that in this case the equation (32.1) with Eq. (32.2) reduces to the equation (31.10) with Eq. (31.11). Therefore, the general solution of the functional equation (32.1) with Eq. (32.2) for A; even is given by Eq. (32.11) i f / ( U , U ) = 0. Now, we assume that / ( U , U) ^ 0. According to this assumption, there exists at least one complex vector C such that f(C, C) ^ 0. For Zi = U, Z, = C (2 < i < 2n) the equation (32.1) reduces to f[V,C)=A,

(32.13)

where we put f(C, C) = A. By putting Zj. = U, Z 2 = V , Z, = C (3 i=o from which on the basis of the condition Eq. (32.3) it follows that /(U,V) = /(C,V).

(32.14)

If we put U = V into the above equality, we obtain / ( V , V) = f(C, V), i.e., we have /(U,V) = /(V,V) = /(C,V).

(32.15)

By virtue of the condition Eq. (32.3) at least one of the coefficients oo, ai, • • • , a n _/;_i is different from zero. Let a r be the nonzero coefficient with the smallest index. If we substitute Zi = Z3 = • • • = Z r +2 = Z r +4 = • • • = Ti^n = C and Z2 = Z r +3 = U, from Eqs. (32.1) and (32.2) on the basis of the equalities (32.13) and (32.15) we get ar[f(C,V)-f(C,C)]2 = 0, from which it follows that /(C,U)=A

(32.16)

By virtue of the equalities (32.14) and (32.16) we have /(U,V)=A

(32.17)

Expanded Quadratic Functional

262

Equations

This completes the proof of Theorem 32.1. Remark 32.1.

•

The functions given by Eqs. (32.4) and (32.17) are solutions n-k-l

of the equation (32.1) if

Yl

ij = 0, but the question of generality of these

3 =0

solutions remains open.

33

Generalized Quadratic Functional Equation

In this section we will solve the functional equation / ( Z x , Z 2 ) 5 (Z 3 , Z 4 ) + /(Zx, Z3)
/(Z 1 ,Z 4 )<7(Z 2 ) Z 3 ) = 0

(/,
(33.1)

2

V ^V)

which is more general than the equation (31.9). It is easy to see that if a component of / is identically 0, then the corresponding component of g may be arbitrary. Similarly, if a component of g is identically 0, then the corresponding component of / may be arbitrary. So in the next theorem we consider only solutions (/, g) of Eq. (33.1) for which no component of / or g is identically 0. Theorem 33.1 The general solution of the functional equation (33.1) is given by the formulae /(U.V)

=

Kl{V)H2(V)-K2{U)H1{\),

(33.2)

g(U,V)

=

Hi(\J)H2(V)-H2(\J)Hi(V),

(33.3)

where Hi, H2, A'i, K2 : V H V are arbitrary functions. Proof. Let (/, g) be a solution of Eq. (33.1) such that no component of / or g is identically zero. Then for any pair of components of / and g (denoted again by (f,g)) there exist at least two pairs of complex constant vectors (A,B) and {C,V) (which may coincide) such that the components f(A,B) and g(C,V) are both not zero. If we put Zi = A, Z 2 = Z3 = Z4 = B into equation (33.1), we obtain f(A,B)g(B,B)=0,i.e., g(B,B)=0.

(33.4)

Generalized Quadratic Functional

Equation

263

By putting Zx = A, Z 2 = Z 3 = B, Z 4 = U, the equation (33.1) on the basis of the equality (33.4) becomes g(B,V) + g{U,B) = 0, from which it follows that g(B,V)

= -g(V,B).

(33.5)

For Zi = A, Z 2 = B, Z3 = U, Z4 = V, the equation (33.1) becomes f(A, B)g(V, V) + / ( A U)g(V, B) + f{A, V)g(B, U) = 0. (33.6) If we introduce the notation # i ( U ) = f(A,U)/f{A,B) and tf2(U) = g(B, U), by virtue of the equalities (33.5) and (33.6) we obtain 5(U, V) = ffx(U)tf2(V) - H 1 (V)// 2 (U).

(33.7)

Now, if we put Zi = U, Z 2 = V, Z 3 = C, Z 4 = V, from the equation (33.1), on the basis of the equality (33.7), we have / ( U , V) = IU{V)H2{\)

- Hi(V)K 2 (U),

(33.8)

where we introduced the notations tfi(U)

=

7i 2 (U)

=

/^(cwu.zo-tfi^/cu.c)
g(c,v)

This completes the proof of Theorem 33.1.

•

Remark 33.1. If (/,g) is a solution of Eq. (33.1), for the corresponding components of / and g one of the following three possibilities may occur: a) The component of / is identically 0, the component of g is arbitrary; a) The component of g is identically 0, the component of / is arbitrary; c) The components of/ and g are given by formulae of the form Eqs. (33.2), (33.3).

Chapter 9

Higher Order Functional Equations

In this chapter some complex vector functional equations of higher order without parameters and with complex parameters are solved. The results presented here were obtained in [I. B. Risteski (submitted)]. The notation used is as in the previous chapters. 34

Higher Order Functional Equation without Parameters

In this section the following result will be proved. Theorem 34.1

The general solution of the functional equation 2n-l

/ „

F(7J\,

Z,- + I,

Z i + 2 , • • • , Z 2 n +i_i) = O

(34.1)

;=i

(Z 2 n + i _i = Zi, 2 1), (34.3)

is given by /(U,V)

=

g(U)h(V)-g(V)h(U),

/(U,V)

=

O,

n>2, 265

n = 2,

(34.4) (34.5)

Higher Order Functional

266

Equations

where g,h : V i-> V are arbitrary functions. Proof. If we put Z^ = U (1 < k < 2n), the equation (34.1) takes the form/(U,U) = 0. Now, we will distinguish two possibilities: 1°. Let n = 2. In this case the equation (34.1) becomes /(Z1(Z2)/(Z3)Z4) +

(34.6)

/ ( Z i , Z 3 ) / ( Z 4 , Z 2 ) + / ( Z L Z 4 ) / ( Z 2 , Z 3 ) = O.

This equation is the same as equation (25.8) (see also Eq. (31.9)) and according to Theorem 25.8 its general solution is given by the formula Eq. (34.4). 2°. Let n > 2. If we put Zk = U (k odd) and Zfc = V (k even), and if we take into consideration the property / ( U , U) = O, then from Eq. (34.1) it follows that n-l

^/"-,'(U,V)/i(V,U) = 0.

(34.7)

j'=0

By the substitutions Zx = Z 4 = U,

Z 2fc _i = U,

z2 = z 3 = v ,

z2fc = v

(3 < k < n) the functional equation (34.1) reduces to /"-1(U,V)/(V,U) = 0.

(34.8)

Also, the following equality holds /"-1(V,U)/(U,V) = 0.

(34.9)

Now, if we put Zi = Z 4 = Ze = • • • = Z 2 r = Z 2 r + 2 = U, Z 2 = Z 3 = Z 5 = • • • = Z 2 r _i = Z 2 r + 1 = V, (r + 2 < k < n;

Z2/j_i = U, Z2fc = V

1 < r < n)

the equation (34.1) becomes /n-r(U,V)/r(V,U) = 0

(l
(34.10)

Higher Order Functional

Equation with Complex

Parameters

267

According to equality (34.10), the equation (34.7) yields /(U,V) = 0

(n>2).

This completes the proof of Theorem 34.1.

35

a

Higher Order Functional Equation with Complex Parameters

In this section we will generalize the results given in the previous one. T h e o r e m 35.1

The general solution of the functional equation 2n-l

5 ^ aiF(ZiIZi+1,ZI+2,---,Z2n+i-i) = 0

(35.1)

i=i

(Z2n+1_1 = Zi; 2 1),

fc=i

is given by 2n-l

/(U,V)

=

g(U)g(V)

(n > 2)

if

£

a,-= 0,

(35.3)

t=i

/(U,V)

=

g(U)h(V) - g(V)h(U) (n = 2) if ai = a 2 = a 3 (^ 0),

(35.4)

/(U,V)

=

O

(35.5)

in all other cases,

where g,h : V t-+ V are arbitrary functions. Proof. First, we will prove the theorem for the case n = 2. Then the functional equation (35.1) becomes a1/(Z1,Z2)/(Z3,Z4) +

a 2 /(Zi, Z 3 ) / ( Z 4 , Z 2 ) + a 3 / ( Z ! , Z 4 ) / ( Z 2 , Z 3 ) = O.

(35.6)

Higher Order Functional

268

Equations

By a cyclic permutation of the vectors Z2,Z3,Z4 in equation (35.6) we obtain ai/(Zi,Z3)/(Z4lZ2) +

a2f(Zl,

(35.7)

Z 4 ) / ( Z 2 , Z 3 ) + a 3 / ( Z 1 ) Z 2 ) / ( Z 3 , Z 4 ) = O, a1/(Z1,Z4)/(Z2,Z3)

+

(35.8)

a 2 / ( Z i , Z 2 ) / ( Z 3 , Z 4 ) + a 3 / ( Z ! , Z 3 ) / ( Z 4 ) Z 2 ) = O.

The system of equations (35.6), (35.7) and (35.8) has a nontrivial solution if and only if the following condition is satisfied a\ a3 a2

a,2 a 3 oi a 2 a 3 a'i

0.

(35.9)

In all other cases the general solution of the functional equation (35.6) is

/(U,V) = 0. From Eq. (35.9) it follows that (ai + a2 + a 3 )[(ai - a 2 ) 2 + (a 2 - a 3 ) 2 + (a 3 - ax)2] = 0.

(35.10)

We will investigate the following cases: 1°. Let a\ + a 2 + a 3 = 0 and a\ = a2 (^ 0). Then the condition Eq. (35.9) is satisfied. The equation (35.6) has the form / ( Z i , Z 2 ) / ( Z 3 , Z 4 ) + / ( Z j , Z 3 ) / ( Z 4 , Z 2 ) = 2/(Z!, Z 4 ) / ( Z 2 , Z 3 ).

(35.11)

By a cyclic permutation of the vectors Z 2 , Z 3 , Z 4 from this equation we find / ( Z i , Z 3 ) / ( Z 4 , Z 2 ) + / ( Z i , Z 4 ) / ( Z 2 , Z 3 ) = 2/(Zx, Z 2 ) / ( Z 3 , Z 4 ).

(35.12)

If we eliminate the term / ( Z j , Z 2 ) / ( Z 3 , Z 4 ) from Eqs. (35.11) and (35.12), we obtain the equation /(Zx, Z 3 )/(Z 4 , Z2) = / ( Z x , Z 4 ) / ( Z 2 l Z 3 ).

(35.13)

We denote any nonzero component of a nontrivial solution of the equation (35.11) again by / , but now / will be a scalar function of two vector arguments. For such a component there exists at least one pair of constant complex vectors (A, B) (A, B G V) such that / ( A , B) ^ 0.

Higher Order Functional

Equation with Complex Parameters

269

By putting ly = A, Z 2 = U, Z 3 = V and Z 4 = B , the equation (35.13) becomes /(A,B)/(U,V) = /(A,V)/(B,U).

(35.14)

If we put U = B into the last equation, we have

On the basis of this equality, the equation (35.14) becomes /(U,V)=J(U)J(V),

(35.15)

where we put

Hf/(A,U,=S(U,. Really, the function given by Eq. (35.15) is a solution of the equation (35.11). 2°. Let ai + a2 + 03 = 0 and a\ ^ a 2 . Then the condition Eq. (35.9) is satisfied. For ai -f a2 + 03 = 0, the equation (35.6) becomes a i / ( Z i , Z 2 ) / ( Z 3 ) Z 4 ) + a 2 /(Zx, Z 3 ) / ( Z 4 , Z 2 )

(35.16)

= (ai + a 2 ) / ( Z 1 , Z 4 ) / ( Z 2 , Z 3 ) . If we suppose that a\ — 0, this allows us to divide by
(35.17)

For Zi = U, Z 2 = B , Z3 = Z 4 = A, by virtue of the above relation Eq. (35.17), the equation (35.16) reduces to the equation (a1-a2)/(A,B)/(U,A)=0,

270

Higher Order Functional

Equations

from which it follows that / ( U , A ) = 0.

(35.18)

By the substitutions Zi = U, Z 2 = V, Z 3 = A, Z 4 = B , the equation (35.16), on the basis of the equality (35.18), yields / ( U , V ) = 0.

(35.19)

Let / ( U , U) ^ O. In this case, for any component of the solution of the equation (35.16) (denoted again by / ) for which / ( U , U ) ^ 0 there exists at least one constant complex vector C such that f(C, C) ^ 0. If we put Zi = Z 2 = Z 4 = C, Z 3 = U, from Eq. (35.16) we obtain /(C,U) = /(U,C).

(35.20)

For Zi = Z 2 = C, Z 3 = U, Z 4 = V, in view of Eq. (35.20) we can write the equation (35.16) in the following form / ( u V)

= /(^/(C.v)

AU,Vj

f(C,C)

'

or /(U,V)=J(U)J(V),

(35.21)

with the notation f(C, U) = y/f(C,C)g(U). It is not hard to check that the function Eq. (35.21) is really a solution of the equation (35.16). Since Eq. (35.21) includes the trivial solution Eq. (35.19) as well as solutions with zero components, the general solution of the equation (35.16) is given by the formula Eq. (35.21). 3°. Let ai + a 2 + a 3 ^ 0. Adding together the equations Eqs. (35.6), (35.7) and (35.8), according to the condition ai + a 2 + a 3 ^ 0 we obtain the equation (34.6). According to Eq. (35.4) the solution of this equation has the following properties /(U,V) = -/(V,U),

/(U,U) = 0.

(35.22)

If we put Zi = Z 4 = U, Z 2 = Z 3 = V, from Eqs. (35.6) we find (oi-a2)/2(U,V) = 0. We will distinguish two cases: a\ ^ a2 and a\ = a 2 .

(35.23)

Higher

Order

Functional

Equation

with

Complex

Parameters

271

In the case ai ^ a 2 , starting from Eq. (35.23), we obtain Eq. (35.5). In the case a\ = a 2 , from the condition Eq. (35.10) we find o-i = 0,2 = ^ 3 ,

and the equation (35.6) is transformed into equation (34.6). As stated above, the general solution of this equation is the function Eq. (35.4). Thus the theorem is proved for n = 2. Now we pass to the proof of the theorem for n > 2. First we will investigate the case 2n-l

Ea^°-

By putting Z,- = U (1 < i < 2n), from the equation (35.1) we obtain the identity /(U,U)=0.

(35.24)

Next, we assume ai ^ 0. We may assume this without loss of generality since if ai — 0, then there must be at least one a,- ^ 0 (2 < i < 2n — 2), and by a cyclic permutation of the vectors Z2, Z3, • • • , Z2n we may achieve that the coefficient at the term / ( Z i , Z 2 )/(Z3, Z 4 ) • • • / ( Z 2 n - i , Z 2 n ) be different from zero. By introducing the substitutions Zi = Z3 = • • • = Z 2 n _i = U

and

Z 2 = Z4 = • • • = Z 2 „ = V

and taking into consideration the identity Eq. (35.24), the equation (35.1) takes the form n-l

^a2i+1/n-'(U,V)/i(V,U) = 0.

(35.25)

1=0

For Z3 = Z5 = • • • = Z 2 r + 1 = V, Zi = Z 2 l _i = U, (r + 2 < i < n;

Z 4 = Zg = • • • = Z 2 r + 2 = U, Z 2 = Z2* = V 1 < r < n)

Higher Order Functional

272

Equations

the equation (35.1) yields a1/"-r(U,V)/'-(V,U) = 0

(l
so that from Eq. (35.25) we deduce / ( U , V) = O. Now, we pass to the investigation of the case 2n-l

£ a, = 0. If we assume / ( U , U ) = O, then the general solution of the equa2n-l

tion (35.1) is Eq. (35.5), which may be proved as in the case ^Z ai ¥^ 0»=i

If we assume that / ( U , U ) •£ O, then for each component of / (denoted again by / ) such that / ( U , U) ^ 0 there exists at least one complex constant vector C such that f(C, C) ^ 0. There is at least one index r £ {1, 2, • • • , 2n — 1} such that n-2

^a

2 l + r

^0

where

aj - aj_ 2 n+i

(j > 2n - 1).

i=0

In fact, if such an index did not exist, then we would have the following system of 2rz — 1 linear homogeneous equations n-2

5^a2,-+r = 0

(l
i=0

which has only the trivial solution a i = a 2 = • • • = a 2 „ _ i = 0,

but this contradicts the assumption that at least one of these parameters is distinct from zero. Now we assume that n-2

^a

2 i + 1

^0.

(35.26)

1=0

We may assume this because the case when n-2

^a2i+1 = 0 i=0

n-2

and

^ i-o

a2i+r ? 0

(r £ {2, 3, • • • , 2n - 1})

Nonlinear

Operator Functional

273

Equation

can be reduced to the case Eq. (35.26) by a cyclic permutation of the variables Z 2 , Z3, • • • , Z 2 n and a simple renumeration of the variables. By putting Z 2 n _i = U, Z 2 n = V, Z,- =C

(1 2,-+i/n-1(C,C)/(U,V)

(35.27)

i=0 n-2

+

Y. «2.+2/ n - 2 (C', C)f(C, U ) / ( V , C) »'=0

+

a2n^fn'2(C,

C)f(C, U)/(G\ V) = 0.

Since n-2

n-2

0 2 n - l = — 2_^ Coi + 1 — 2_j a 2i' + 2, i=0 i=0

according to the assumption Eq. (35.26), for V = C\ from the equation (35.27) there follows f(C, U) = / ( U , C). By introducing the notation f(C, U) = g(\J) y/f(C,C), from Eq. (35.27) we get f{V,V)=g(V)g(V). 2n-l

Since in the case "^2 Hi — 0 this function is really a solution of the »=i

equation (35.1), this means that Theorem 35.1 is completely proved. 36

•

Nonlinear Operator Functional Equation

In this section a nonlinear operator functional equation of k-th order will be solved. Definition 36.1. Let ^ij be the operator which transposes (changes the places of) the i-th and j-th argument of the function F, i.e., ^ijF(Zi,

• • • , Zj_i, Zj, Zj+i, • • • , Zj_i, Zj, Zj+i, • • • , Z n )

— ^ ( Z i , • • • , Z;_i, Zj, Z,-+i, • • • , Zj_i, Zj, Zj+i, • • • , Z n ).

(36.1)

274

Higher Order Functional

Theorem 36.1

Equations

The general solution of the functional aF(Zi,Z2)---

equation (36.2)

,Zkn)

kn

—

2_^ ^nrF(Zi,

Z 2 , • • • , Z „ _ i , Z n , Z n + i, • • • , Zfc„),

r=n + l

where a is a complex parameter, fc-i

F(7,i, Z2, • • • , Zfcn) = YY fC^ni + l i Z n j + 2 , ' ' ' > Z m + n )

(36.3)

i=0

(/ : Vn !->• V;

n >'2, k >'2), has components given by (36.4)

f{Ui,U2,---,Vn) Hi(Ui) #2(U!)

#i(U

2

)

H2{XJ2)

•••

•• •

Hi(Un) H2(Un) if

= ^n(Ui)

/(Ui,U2, ••,un) =

^n(Uo)

•••

f .n^(u,-) or

/(Ui.L^-.^UnJ^O

a =

fi»(u») a = n(k — 1

if

(36.5)

0 i/

a#r(*-l)

where H, (1 < i' < n), K are arbitrary functions

(1 < r < n ) ,

(36.6)

in V.

Proof. For the proof of this theorem in the case a = k — 1, n > 2 we shall need the following result. Lemma 36.2

/ / A , (1 < i: < n) are constant

complex vectors such

/(Ai,A2,---,An)#0, then i/ie following

equality

(36.7)

holds

/(A„,U1]U2,...,Un.2,An)E0. Proof of Lemma

that

(36.8)

36.2. We will suppose that this is not true, i.e., / ( A „ , U L U 2 l • • • , U n _ 2 , A „ ) ^ 0.

(36.9)

Nonlinear

Operator Functional

Equation

275

By putting Z; = A,(1 < i < n), Z,. n + 1 = Z ( r + 1 )„ = A n (1 < r- < *? — 1), Zrn+j+1=Uj (l
(36.10)

x

[/(A1)A2

) / ( A „ , U 1 ; U 2 , • • • , U n _ 2 , A„)

+

/ ( A , , A 2 ) • • • , A„_!, U ! ) / ( A n , A n i U 2 , • • • , U n _ 2 , A„)

+

/ ( A i , A 2 , - - - , A n _ 1 , U 2 ) / ( A n , U 1 , A n , U 3 , - - - 1 U „ _ 2 , A n ) + ---

+

/ ( A i , A 2 , • • • , A „ _ i , U „ _ 2 ) / ( A „ , U 1 , • • • , U n _ 3 , A n , A„)] = 0.

According to the hypothesis Eq. (36.9), from Eq. (3.10) it follows that / ( A i , A 2 , • • • , A n _ a , A n ) / ( A „ , U 1 ( U 2 ) • • • , U„_ 2 , A„)

(36.11)

+

/(Ai,A2)--- ,An_1,U1)/(An,An,U2,--- ,Un_2,An)

+

/(Ax, A 2 , • • • , A„_i, U 2 ) / ( A n , U i , A n , U 3 , • • • , U n _ 2 , A„) + • • •

+

/ ( A i , A 2 , • • • , A „ _ 1 , U n _ 2 ) / ( A n i U i , • • • , U „ _ 3 , A n , A n ) = 0.

Let En-2 = {1,2,3, •• • ,n — 2}, and let Sr (0 < r < n - 2) be a subset of the set £' n _ 2 which contains r elements. For r = n — 2 we have 5„_2 = En-2- Putting into Eq. (36.2) Z, = A n (1 < i < kn), we obtain 7l-A-n, A n , • • • , A

n

)

=

0.

(36.12)

Now, we suppose that / ( A n , V 1 , V 2 ) - • • , V „ _ 2 , A„) :

(36.13)

ds, where

v

* € Sr,

' = {t

ie

En-2\sr.

(36.14)

Under this assumption we will show that / ( A n , W 1 , W 2 , - . . , W n _ 2 , A „ ) = 0,

(36.15)

Higher Order Functional

276

Equations

where |A„,

tesP_i,

P u t t i n g U ; = W,- (1 • • • , A „ _ i , A „ ) / ( A „ , W i , W 2 , • • • , W n _ 2 , A n ) = 0. From this relation (r > 1) we obtain / ( A n ) W 1 ) W 2 , - - - , W u _ 2 > A n ) = 0. Consequently, by induction we proved t h a t /(An)U1)U2).--,Un_2)A„) = 0 if exactly r (0 < r < n — 2) elements among U i , U 2 , • • • , U n _ 2 are equal to A n . For r = 0 we obtain / ( A „ I U 1 > U 2 l - - . , U „ _ 2 , A n ) = 0) which contradicts the hypothesis Eq. (36.9). Note t h a t if k = 2, we do not use the hypothesis Eq. (36.9). In this case Eqs. (36.10) and (36.11) are identical. T h e above proof yields directly Eq. (36.8). This completes the proof of the lemma. • We will prove Theorem 36.1 by induction. For n = 2, the functional equation (36.2) takes the form (A - l ) / ( Z i , Z 2 ) / ( Z 3 , Z 4 ) • • • / ( Z 2 f c _ 1 , Z2k) +

/(Zi,Z3)/(Z2,Z4)---/(Z2fc_1)Z2fc) /(Z1)Z4)/(Z3)Z2)--./(Z2fc-i,Z2fc)

+ ••• +

/(Z1)Z2fc_i)/(Z3)Z4)---/(Z2,Z2fc)

+

/(Z1,Z2,.)/(Z3,Z4).../(Z2fc_1)Z2).

(36.17)

Nonlinear

Operator Functional

277

Equation

If we substitute Z,- = U (1 < i < 2k), the above equation becomes /(U,U) = 0.

(36.18)

For any nonzero component of a nontrivial solution of Eq. (36.17) (denoted again by / ) there exists at least one pair of constant complex vectors (A, B) such that / ( A , B) ^ 0. Putting into the functional equation (36.17) Z2,-_i = A

(1 < * < * ) ,

Z2j=B

(2<j
Z 2 = U,

it takes the form / f c - 1 ( A , B ) [ / ( U , B ) + / ( B , U ) ] = 0, from which it follows that /(U,B) = -/(B,U).

(36.19)

For Z 2 l _i = A, Z 2 i = B (1 V)/(B,U)]. If we introduce the notations /(A,U) = /(A,B)

ffi(U),

/ ( B , U ) = /f 2 (U)

we obtain that the function /(U,V) =

Hl(\J) H2(V)

H^V) H2(V)

is the general solution of the functional equation (36.2) for n = 2 because it includes the trivial solution / ( X , Y) = 0. Now, we suppose that Theorem 36.1 holds for n — 1, i.e., the general solution of the functional equation (A-l)F(Z1,Z2,---,Zfc(n_1))

(36.20)

fc(n-l)

=

2_^ * n - I , r f ( Z l , Z 2 , • • • , Z„_i, Z„, • • • , Zfc(n_1)),

Higher

278

Order

Functional

Equations

where (36.21)

- F ( Z i , Z 2 , • • • , Zfc(„_i)) fc-1

=

_[ j_ / ( Z ( „ _ i ) i + i , Z ( n _ 1 ) , + 2 , • • • , Z(„_i),- + „_i), i =0

is given by

(36.22)

/(U1,U2,---,Un_1) HiiUi) H2(Ui)

ffn-ltUO

H1(V2) #2(U2)

ff„_i(U2)

••• •••

tfi(U„_i) /f2(U„_i)

•••

tfn-xfUn-i)

Let / ( A i , A 2 , • • • , A „ ) ^ 0 (here, as usual, / denotes a nonzero component of a nontrivial solution). If we put Zni+i=An

(0 < » < * - ! ) ,

/ ( A n , Z 2 , Z 3 , • • • , Z„) — # ( Z 2 , Z 3 , • • • , Z„) ( / : Vn 4 C ,

^ : V " - 1 M- C ) ,

according to L e m m a 36.2 from Eq. (36.2) we obtain (k — 1)#(Z 2 , • • • , Z „ _ i , Z n ) 5 ( Z n + 2 , Z n + 3 , • • •, Z 2 n ) • • •5(Z(fc_1)„ + 2, ' • •, Zfcn)

= # ( Z 2 , • • • , Z n _ i , Z n + 2 ) 5 ' ( Z „ , Z n + 3 , • • • , Z 2 „) • • •
+

••

+ # ( Z 2 ) - • • ,Z n _i > Zfcn)5'(Z„ + 2 , - • • , Z 2 „ ) • • • 5 , ( Z ( f c _ i ) „

(36.23)

+

2,' " • ,Zfcn-l,Z„)

Nonlinear

Operator Functional

Equation

279

According to the inductive hypothesis, we obtain t h a t the general solution of the equation (36.23) is given by 5

(U1,U2,-..,Un_1) = /(An,U1,U2l.--,Un_1) Hi(Ui)

#i(U2)

•••

ffi(Un_!)

H2{Vi)

H2(U2)

•••

tf2(U„-i)

Hn_1(Vl)

ifn_i(U2)

•••

ffn.^Un.x)

(36.24)

where #,• : V t - » C ( l < i < n — 1 ) are arbitrary functions. If we put into equation (36.2) Zi=Ai

(1 < » ' < « - 1 ) ,

J

ni?'-f'm — -"-m

Z

Z„ = U i ,

;i < j < k - 1; 1 < m < n ) ,

(fc-l)n + l = A n ,

Z(fc_i) n + r = U r

(2 < ? • < « ) ,

we obtain / ( A i , A2, • • • , An_!, A „ ) / ( U i , U2, • • • , Un) =

(36.25)

/(All---,An_1,Ui)/(An)U2,...,Un)

-

f(A1,..-,

A n _ i , U 2 ) / ( A „ , U i , U3, • • • , U„)

-

/(A!,--- , A n _ 1 , U n ) / ( A n , U 2 > - - , U n _ 1 , U 1 ) .

On the basis of the equality (36.24) we obtain / ( A n , • • • , U,-_!, U i , U I + 1 , • • • , U j - i , U i ( U i + 1 , • • • , U n _ i ) (36.26) =

- / ( A n , • • • , U,-_1,Uj,U,-+i, • • • , U j _ i , U i , U j + 1 , • • • , U „ _ i ) (1 • • • , U n ) has the form Eq. (36.4). It remains still to show that every function of the form Eq. (36.4) is really a solution of the equation (36.2). For this purpose, we will consider

Higher Order Functional Equations

280

the following identity tt(Z)

DU) = # ( Z )

0„_i,„ 7i(Z)

(36.27)

o,

where /

%(z) =

ffi(Zi) Hi(Z2)

V Hi(Z n _i)

#2(Zi)

^n(Zi)

H2(Z2)

Hn(Z2)

•f^2(Z n -l)

Hn{2in-i)

ff2(z„)

Hn(Zn)

#i(Zn.)+i)

# 2 ( Z „ j + i)

Hn(7inj+i)

\ J^l(Z n j + n )

//^(Znj + n)

Hn[7jnj+n)

/ -H(Z)

ffi(Z„)

I \

/

and O n _ i n is the zero (n - l ) x n matrix. According to Eq. (36.27), we conclude that the following identity holds A.--1

fe-i

£ ^a) n

H\{TLni + \) #2(Z„,- + i)

Hi(7jni + 2) H 2 ( Z n ! ' + 2)

••• •••

#l(Z„,-+n) i/2(Z n «+n)

# n ( Z n i + l)

ff„(Z„,-+2)

'•'

#n(Zm-+n)

! = 1

= 0. (36.28)

By evaluating the determinant D(j) according to the Laplace rule, we show that the function given by Eq. (36.4) is really a solution of the equation (36.2). This completes the proof of Theorem 36.1 for a = k - 1. Now we pass to the proof of Theorem 36.2 for a = 7i(k — 1). First, we suppose that / ( U , U, • • • , U) ^ O. Henceforth we denote by / any component of the function / : Vn t-> V for which / ( U , U, • • • , U) ^ 0. For such a component there exist at least one constant complex vector C for which f(C, C, •• • , C ) # 0 . If we put Z„+l- = U and substitute the rest of the variables by C, from Eq. (36.2) we obtain

f(C,

,C,V,C---,C)

=

f(C,C,---,C,U).

(36.29)

By putting Z n+! - = U, Z, l+ j = V (j > i) and substituting the rest of the variables by C, from the equation (36.2) by virtue of Eq. (36.29) we

Nonlinear

Operator Functional

Equation

281

obtain /(C,--,C,U,.--,V,.-.,C)

(36.30)

/(c,c,---,c,u);(c,c,---,c,v) f(C,C,--,C) We assume t h a t

/(c,---,u 1 ,...,c,u 2 ,--.,c,---,u„,c,---,c)

n/tc.c.-.-.cuo /"^(CC.-.-.CT)

(36.31)

•

If we put Z n + i l = U i , Z n + I - 2 = U 2 , • • • , Z n + i „ = U „ , Z n + i „ + 1 = U „ + 1 and substitute the rest of the variables by C, then the equation (36.2) becomes i//(C,C,-..,C)/(C,---,U1,--.1C,U2,---,U1„C)---,U„+1,-..)C) =

/ ( C , C, • • • , C, U ! ) / ( C , • • • , C, • • • , U 2 , • • • , U „ , • • • , U „ + l l • • • , C)

+

f(C, C, • • • , C, U 2 ) / ( C , . . . , U ! , • • • , U „ , • • • , U „ + 1 , • • • , C)

+

•••

(36.32)

+ /(c,c,---,c,u, + i )/(c,---,u 1 ,...,u 2 ,-..,u„--.,c,---,c). On the basis of the inductive hypothesis Eq. (36.31), from Eq. (36.32) it follows t h a t

/(
=

2 )

.••

rf/Cc.c.-.-.cuo V(c,c,-,c) •

,=s

,u„.---,!;.,+!,•••,C)

(36 33)

-

Therefore, we proved by m a t h e m a t i c a l induction t h a t Eq. (36.33) holds for every v < n.

Higher Order Functional

282

Equations

By putting Z,- = C (1 < i < n), Tin+j — Uj (1 < j < n), Z 2 n + S = C (1 < s < n(fc — 2)), from the equation (36.2) we obtain nf(C, C, • • • , C)/(U X , U 2 , • • • , U„)

(36.34)

=

f(C, C,---,C,

U i ) / ( C , U 2 , U 3 , • • • , U„)

+

f(C, C, • • • , C, U 2 ) / ( U ! , C, U 3 , U 4 l • • • , U n )

+ ••• +

f(C, C,---,C,

U n ) / ( U i , U 2 , • • • , U n _ ! , C).

On the basis of the equality (36.33), the last equality (36.34) becomes

Uf(c,c,---,c,vi) /(

U

l.

U

2 . - - - . U n ) = '= f „ - l

( C [ C )

...

i C )

•

(36-35)

By introducing the notation A'(U) = f(C, • • • , C, U ) / ^ / " - U C ' . C , - - - , ^ ) , we obtain that the function / in the case a = n(k — 1) really has the form Eq. (36.5). Now, we suppose that / ( U , U , - - - , U) = 0. Next, we will need the following result. L e m m a 36.3 to \Jn, then

If at. least one of the variables U i , U 2 , • • • , U n _ i is equal / ( U i , U 2 , - , U n ) = fl.

Proof of Lemma 36.3. Let En^x = { l , 2 , 3 , - - - , n - 1}, and let Sm (1 < m < n — 1) be a subset of the set En-\ which contains m elements. We h a v e / ( U n ) U „ , - - - ,U„) = 0 . Now, we suppose that /(Vi,V2)---)Vn_1,Un) = 0 holds, where v

. _fu„, \Y,-,

tG5m, iGE„.i\5m.

(36.36)

Nonlinear

Operator Functional

Equation

283

We will prove t h a t /(Wi,W2)---,W„_i)U„)=0,

(36.37)

where w

_ [U"> [Yi,

* e S'm-i, z'G£n_i\5m_i.

By substituting Z n m + 1 - = Z,- (0 < m < k — 1) into the equation (36.2) and by putting Z,- = W ; , on the basis of the assumption (36.36) we obtain ( f c _ l ) ( „ _ m ) / * ( W i , W 2 , - - - , W n _ i , U n ) = 0, from which we deduce Eq. (36.37) because k > 2 and in < n. Therefore, L e m m a 36.3 is proved by induction.

•

By putting Z n m + l - = U,- (0 < m < k — 1) and according to L e m m a 36.3, from the equation (36.2) we obtain ( i - l ) ( n - l ) / * ( U i , U 2 , . . . , U „ ) = 0. Since k > 1 and n > 1, from this we obtain t h a t the function / has the form Eq. (36.5). We may easily check t h a t the functions given by Eq. (36.5) satisfy the functional equation (36.2). We can do this by a direct substitution of Eq. (36.5) into Eq. (36.2). This completes the proof of Theorem 36.1 for the case a — n(k — 1). Now, we will pass to the proof of Theorem 36.1 for the case a ^ r(k — 1) (1 < r < n). In this case, L e m m a 36.3 also holds. By putting Znm+i = Z,- (0 < m < k — 1) and according to L e m m a 36.3, from the equation (36.2) we obtain [a-(*-l)]/fc(Zi,Z2).-.,Zn) = 0. Since a jL k — 1, from the above equality we immediately deduce the statement of Theorem 36.1 for the case a ^ r(k — 1) (1 < 7* < ?;.). Therefore, Theorem 36.1 has been completely proved. • We have not been able to solve equation (36.2) for a = r(k — 1) (2 < r < n — 1).

Higher Order Functional

284

Remark 36.1.

Equations

The function (36.38)

/(Ui,U2,-,U„) H1(U1)

H 1 (U 2 )

•••

H2(Ui)

H2{XJ2)

•••

Hi{\J.) H2(Vs)

ff.(Ui)

HS(U2)

•••

Hs(Us)

=

n ^i(u.). l=S + l

where s — n — r + 1, Hi (1 < i < n — r + 1) are arbitrary functions, is a solution of the equation (36.2) for o = r(k — 1) but the question of generality of this solution remains open. Also, for r = n the function given by Eq. (36.38) becomes Eq. (36.5). If we assume that

n ff1(z,-)=i, i=n+l

then the function Eq. (36.38) for r = 1 becomes Eq. (36.4). All this suggests us to put forth the following hypothesis. Hypothesis 36.1 The general solution of the functional equation (36.2) in the case a = r(k — 1) (1 < r < n) is given by the formula Eq. (36.38).

Chapter 10

Systems of Nonlinear Functional Equations

In this chapter complex vector systems of nonlinear partial functional equations are considered. They may be sorted in two types. The first three systems solved are considered as complex vector systems of partial quadratic functional equations with real parameters. The last three systems solved represent complex vector systems of partial functional equations of higher order without parameters. In this chapter the unknown functions depend on arguments in an arbitrary finite dimensional complex vector space V and take values in the field of complex numbers C so that their products make sense. As in Sec. 10, denote by V° the space of all real vectors in V (thus V = V° + iV°) and let C{V°, C) be the space of all linear mappings V° >->• C. The results presented in this chapter were obtained in [I. B. Risteski et al. (submitted)].

37

Systems of Quadratic Functional Equations

Consider the system of functional equations / ( Z i , Z 2) Z 3 , • • • , Z 2 „_i, Z 2 n ) + / ( Z i , Z 3 , Z 4 , • • • , Z 2 n , Z 2 ) + • • •

(37.1)

+ / ( Z i , Z 2 „ , Z 2 , • • • , Z 2 n _ 2 , Z 2 n - i ) +sgn a [ # ( Z i , Z 2 , Z 3 , • • • , Z 2 n _ i , Z 2 n ) + 5(Zi, Z 3 ) Z 4 ) • • • , Z 2 n , Z 2 ) + • • • + p(Zi, Z 2 n , Z 2 , • • • , Z 2 n _ 2 , Z 2 n _!)] = 0, MZi, Z 2 , Z 3 , • • • , Z 2 n _ i , Z 2 n ) + ft(Zi, Z 3 , Z 4 , • • • , Z 2 n , Z 2 ) + • • • 285

286

Systems of Nonlinear Functional Equations

+ h(Zi, Z 2 n , Z 2 , ' • ' , Z 2 n _ 2 , Z 2 n - i ) + fc(Zi, Z2, Z3, • • • , Z 2 n - 1 , Z2n) + fc(Zi, Z3, Z 4 , • • • , Z2n> Z 2 ) + • • • + fc(Zi, Z 2 n , Z 2 , • • • , Z 2 „ _ 2 , Z 2 n - l ) = 0, where

/(Z1)Z2)Z3,---,Z2n_i)Z2„) = x

[F(Zi, Z 2 ) + F{Z3, Z4) + • • • + F{Z2i-i, Z 2i )] x [F(Z 2 i + 1 , Z 2 „) + F ( Z 2 l + 2 , Z 2 n _ 0 + • • • + F(Zi+n,

(37.2) Zi+n+1)],

#(Zi, Z 2 , Z 3 , • • • , Z 2 n _i, Z 2 n ) =

[G(Zi, Z 2 ) + G(Z 3) Z 4 ) + • • • + G(Z 2l -_i, Z 2l )] x

X

[G(Z 2 t + i, Z 2 n ) + G(Z 2 i + 2 , Z 2 n - l ) + ' ' ' + G( Z 1 + n , Z i + n + i ) ] ,

^i(Zi, Z 2 , Z3, • • • , Z 2 n _ i , Z 2n ) =

[F{ZU Z 2 ) + F ( Z 3 , Z 4 ) + • • • + F(Z 2 ,-_i, Z2,-)] x

x

[G(Z 2 ;+i, Z 2 „) + G(Z 2 l + 2 , Z 2 n - i ) + • • • + G ( Z , + n , Z j + n + i ) ] , &(Zi, Z 2) Z3, • • • , Z 2 n _ i , Z 2u )

=

[G(Z 1( Z 2 ) + G(Z 3 ) Z 4 ) + • • • + G(Z 2 ,-_i, Z2<)] x

X

[F(Z2i + l, Z 2 n ) + F(Z2i+2, Z 2 n-l) + • • • + F(Zi+n, Z , + n + i)], 2

F, G : V 1—>• C are arbitrary complex functions, a is a real constant and n > 2, I < i < n — 1. Definition 37.1. A solution (F(U, V), G(U, V)) of system (37.1) is called isotropic if (F(U, V), G(U, V)) ^ (0, 0) and F 2 ( U , V ) + G 2 ( U , V ) = 0.

(37.3)

The general solution of system (37.1) includes all solutions of this system with the possible exception of the isotropic ones. Theorem 37.1 formulae

The general solution of the system (37.1) is given by the

F(U,V)

=

P(U)-P(V),

(37.4)

G(U,V)

=

Q(U)-Q(V),

(37.5)

where P and Q are arbitrary complex functions defined in V.

Systems

of Quadratic Functional

Equations

287

Each isotropic continuous solution of Eq. (37.1) has the form P ( U , V) = P(U) - P ( V ) ,

G(U, V) = ±i(P{U)

- P(V))

(i2 — —1; we take the same sign for all vectors U, V 6 V). Proof. Let F ( U , V), G(U, V) represent an isotropic continuous solution of system (37.1). This means that for any pair of vectors (U, V) G V2 we have G(U,V) = ±iP(U,V)

(37.6)

(both signs are possible). It is clear that the real dimension of V2 is 4 dim V, the real codimension of the set {(U, V) G V2 : P ( U , V) = 0} is 2. Hence its complement is a connected set in which the function G(U, V ) / P ( U , V) is continuous, thus it can take just one of the values i or — i. Henceforth, when we write ±i (as in Eq. (37.6)), we mean only one of these two signs. By virtue of Eq. (37.6) system (37.1) implies / ( Z i , Z 2 , • • • , Z 2 n - i , Z 2 n ) + / ( Z i , Z3, • • • , Z 2 n , Z2)

(37.7)

+ .-- + / ( Z 1 , Z 2 n , Z 2 , - - - , Z 2 n _ 1 ) = 0 . The general solution of this equation for n > 2 according to [D. S. Mitrinovic and S. B. Presic (1962A)] is Eq. (37.4): P(U,V) = P ( U ) - P ( V ) , where P(U) = F ( U , A) for some fixed vector A. Then G(U,V) = ± i ( P ( U ) - P ( V ) ) . Next we are looking for solutions of system (37.1) not satisfying Eq. (37.3) (anisotropic solutions). We will distinguish the following cases: 1°. Let a < 0. In this case the system of complex vector functional equations (37.1) will be / ( Z 1 ( Z 2 ) • • • , Z 2 n ) + / ( Z i , Z 3 , • • • , Z 2 n , Z2) H —

h / ( Z i , Z 2f! , Z 2 , • • • , Z 2 n _i)

#(Zi, Z 2 , • • • , Z 2 n ) — g(Zi, Z 3 , • • • , Z 2 n , Z 2 ) - • • • - g(Zi, Z 2 n , Z 2 , • • • , Z 2 n _i) = 0,

(37.8)

288

Systems

of Nonlinear Functional

Equations

h{7ii, Z 2 , • • • , Z 2 n ) + h(Zi, Z3, • • • , Z 2 n , Z 2 ) + • • • + /l(Zi, Z 2 n , Z 2 , • • • , Z 2 n _i) +

^(Zi, Z 2 , • • • , Z 2 n ) + k(Zlt Z 3 , • • • , Z 2 n , Z 2 ) + • • • + fc(Zi, Z 2 „, Z 2 , • • • + Z 2 n _ x ) = 0.

If we substitute all the variables Z,- (1 < i' < 2n) in the system (37.8) by U, then we obtain F 2 ( U , U) - G 2 (U, U) = 0,

2F(U, U)G(U, U) = 0,

so that F ( U , U ) = 0,

G ( U , U ) = 0.

(37.9)

For any nontrivial anisotropic solution of the system (37.8) there exists at least one pair of vectors (A, B) £ V2 such that F2(A,B) + G 2 ( A , B ) ^ 0 .

(37.10)

a) Let i = 1. If we execute the substitutions Z5 = Zg — • • • = Z 2 n = Zi = Z 2 = A, Z3 = B, Z4 = U, on the basis of the equalities (37.9) the system (37.8) becomes F ( A , B ) [ F ( U , A ) + F ( A , U ) ] - G ( A , B ) [ G ( U , A ) + G(A,U)] G ( A , B ) [ F ( U J A ) + F ( A , U ) ] + F ( A , B ) [ G ( U , A ) + G(A,U)]

=

0, (37.11) = 0.

In view of Eq. (37.10), from Eq. (37.11) it follows that F(A,U) = - F ( U , A ) ,

G(A,U) = - G ( U , A ) .

(37.12)

By putting Z 5 = Z 6 = • • • = Z 2 n = Zi = A, Z 2 = V, Z 3 = B, Z 4 = U into Eq. (37.8), on the basis of the expressions Eqs. (37.9) and (37.12) we get F(A, B)[F(U, V) - F ( U , A) + F ( V , A)] -

G(A, B)[G(U, V) - G(U, A) + G(V, A)] - 0, G(A, B)[F(U, V) - F ( U , A) + F ( V , A)]

+

F ( A , B)[G(U, V) - G(U, A) + G(V, A)] = 0,

(37.13)

Systems

of Quadratic Functional

Equations

289

whence P ( U , V ) - P ( U , A ) + P ( V , A ) = 0,

G(U, V) - G ( U , A) + G(V, A) = 0,

which implies Eqs. (37.4), (37.5): F ( U , V) = P(U) - P ( V ) ,

G(U, V) = Q(U) - Q(V),

where we have introduced the notations P(U) = P(U,A),

Q(U) = G ( U , A ) .

b) Let 1 < i < n - 1. For Z 4 = Z 5 = • • • = Z 2 n = 1X = A, Z 2 = B, Z 3 = U, the system (37.8) becomes Eq. (37.11) and again we have Eq. (37.12). By putting Z 5 = Z 6 = • • • = Z 2 n = Zj = A, Z 2 = U, Z 3 = V, Z 4 = B, on the basis of the equalities (37.9) and (37.12) the system (37.8) takes on the form Eq. (37.13). In view of Eq. (37.10) this again leads to Eqs. (37.4), (37.5), where P(U) = P ( U , A) and Q(U) = G(U, A). c) Let i = n — 1. If we substitute all variables Zi, Z3, • • • , Z 2 n _i (except for Z 2 and Z 2 n ) by A, and if we put Z 2 = U, Z 2 n = B , then the system (37.8) becomes Eq. (37.11), which implies Eq. (37.12). If we substitute in the system (37.8) Z3 = Z4 = • • • = Z 2n _2 = Zi = A,

Z 2 = V,

Z 2 n - i = B,

Z 2 n = U,

on the basis of equality (37.12) we obtain Eq. (37.13), and we find Eqs. (37.4), (37.5). Consequently, if a < 0, the theorem is proved. 2". Let a = 0. In this case the system (37.1) becomes /(Z1,Z2,Z3,---,Z2n_1,Z2n) +

/ ( Z i , Z3, Z 4 , • • • , Z 2 n , Z 2 ) + • • •

+

/ ( Z i , Z 2 n , Z 2 , • • • , Z 2 n _i) = 0, M z i ) Z2,' • • , Z 2 n ) + h(Zii, Z 3 , • • • , Z 2 „, Z 2 ) + • • • + /i(Zi, Z 2 n , Z 2 , • • • , Z 2ra _i)

+ fc(Zi, Z 2 , • • • , Z 2 n ) + fc(Zi, Z 3 , • • • , Z 2 n , Z 2 ) + --- +

fc(Z1,Z2„,Z2,---

,Z 2 „_i) = 0.

(37.14)

Systems

290

of Nonlinear Functional

Equations

The first equation of the system (37.14) is Eq. (37.7), hence its general solution for n > 2 according to [D. S. Mitrinovic and S. B. Presic (1962A)] is Eq. (37.4): F(V,V) where P{U) =

=

P(U)-P(V),

F{U,A).

In the second equation of the system (37.14), if we put Zi = B and substitute all the other variables Zj (2 < i < 2n) by A so that F(A, B) ^ 0, we obtain F ( A , B ) G ( A , A ) = 0, whence G(A,A) = 0.

(37.15)

Now, we will distinguish three cases. a) If i = 1, by putting Z 5 = Z 6 = • • • = Z 2 „ = Zx = A, Z 2 = V, Z 3 = B, Z4 = U, the second equation of the system (37.14) in view of the equalities (37.4) and (37.15) becomes F(A, B)G(U, V) + F(A, B)G(A, U) +

J F(B,A)G(A,V)

+

G(A,B)F(A,U) + G(B,A)F(A,V)

+

F ( A , V ) G ( A , U ) + F ( A , V ) G ( U , A ) = 0.

(37.16)

+ G(A,B)F(U,V)

If we put V = A into (37.16), on the basis of the expression Eq. (37.15) we obtain G(A,U) = —G(U, A), and then Eq. (37.16) becomes G(U,V) = Q ( U ) - Q ( V ) , whereQ(U) = G ( U , A ) . b) In the case 1 < i < n — 1, if we substitute all variables Zi, Z4, Z5, • - •, Z 2 n - i , Z 2 „ (except for Z2 and Z 3 ) by A, and if we put Z 2 = B , Z 3 = U, the second equation of the system (37.14) becomes F ( A , B ) [ G ( A , U ) + G(U,A)] = 0, whence G(A,U) = - G ( U , A ) .

(37.17)

Systems

of Quadratic Functional

Equations

291

For Z 5 = Z 6 = • • • = Z 2 „ = Zi = A, Z 2 = U, Z 3 = V, Z 4 = B, on the basis of the equality (37.17) and Eq. (37.4), the second equation of the system (37.14) takes on the following form

G(U,V) = Q ( U ) - Q ( V ) ,

where Q(U) = G ( U , A ) . c) If i = n - 1, for Z 3 = Z 4 = • • • = Z 2 „_i = Zi = A, Z 2 = U, Z 2 n = B, by virtue of the equality (37.4), from the second equation of the system (37.14) we obtain Eq. (37.17). By putting Z 3 = Z 4 = • • • = Z 2 „ - 2 = Zi = A, Z 2 = V, Z 2 n - i = B, Z 2 n = U, on the basis of the equality (37.4) and Eq. (37.17) from the second equation of the system (37.14) we find

G(U,V) = Q ( U ) - Q ( V ) , where Q(U) = G(U, A), i.e., Theorem 37.1 is proved if a = 0. 3°. If a > 0, then the system (37.1) becomes

/ ( Z i , Z 2 , Z 3 , • • Z 2 n ) + / ( Z i , Z 3 , Z 4 , • • • , Z 2 n , Z2) + • • • +

/(Zi,Z2n,Z2,

, Z 2 n _i) + g{Zu Z 2 , Z 3 l • • • , Z 2 n )

(37.18)

+

5(Z1,Z3,Z4l--

Z 2 n , Z 2 ) + • • • 4- 5(Zi, Z 2 n , Z 2 , • • • , Z 2 n _i) = 0,

h{Zl,Z2,Z3,--

Z 2n ) + h{Z\, Z 3 , Z 4 , • • • , Z 2 n , Z 2 ) + • • •

+

h(Zi, Z 2 n , Z 2 ,

, Z 2 n _i) + fc(Zi, Z 2 , Z 3 , • • • , Z 2 n )

+

fc(Zi,Z3)Z4)--

Z 2 n , Z 2 ) + • • • + fc(Zi, Z 2 n , Z 2 , • • • , Z 2 n _i) = 0.

If we introduce new functions S and T by the formulae

5(U,V)

=

F(U,V) + G(U,V),

T(U,V)

=

F(U,V)-G(U,V),

(37.19)

Systems

292

of Nonlinear Functional

Equations

the system (37.18) becomes s ( Z i , Z 2 , Z 3 , • • • , Z 2 n ) + « ( Z i , Z 3 , Z 4 , • • • , Z 2 n , Z2) + • • • +

*(Zi,Z2n>Z2,---,Z2„-i)+<(Zi,Z2lZ3,---,Z2n)

+

t(Zi,

+

s

—

i ( Z i , Z 3 , Z 4 , • • • , Z 2 n , Z 2 ) — • • • — t(Zi,

(37.20)

Z 3 , Z 4 , • • • , Z2n, Z 2 ) + • • • + i ( Z i , Z 2 n , Z 2 , • • • , Z 2 n - l ) = 0,

s ( Z i , Z 2 , Z 3 , • • • , Z 2 n ) + « ( Z i , Z 3 , Z4> • • • , Z 2 „ , Z 2 ) + • • • ( Z i , Z 2 n , Z 2 , • • • , Z 2 n - l ) — t(2ii,

Z2, Z 3 , • • • , Z2n) Z2n, Z 2 , ' • • , Zin-l)

— 0.

where s{Z\,

Z2, • • • , Z 2 n - 1 , Z2n)

=

[S{Z-i, Z 2 ) + 5 ( Z 3 ) Z 4 ) + • • • + 5 ( Z « _ i , Z2,-)] x

X

[5(Z2» + i , Z 2 n ) + 5 ( Z 2 i + 2, Z 2 „ _ i ) + • • • + 5 ( Z ! + n , Z j + „ + i ) ] , < ( Z i , Z 2 , • • ' , Z 2 n - 1 , Z2n)

=

[T(Zi, Z 2 ) + T ( Z 3 , Z 4 ) + • • • + T ( Z 2 , _ i , Z 2 i )] x

X

[T(Z2)' + l, Z 2 n ) + T(Zoi + 2, Z 2 n - 1 ) + --- + T(Z i + n , Z j + „ + i ) ] .

Comparing the equations of the system (37.20), we find s(Zi,Z2)Z3,--- ,Z2n-i,Z2„) +

s(Zi, Z3,Z4, • • • , Z2n,Z2) + • • •

+

« ( Z i , Z2n, Z2, • • • , Z 2 n - 2 , Z 2 n _ i ) = 0,

(37.21)

t{Zi, Z2, Z3, • • • , Z 2 n - i , Z 2 „) +

^(Zl, Z3, Z4, • • • , Z2n, Z2) + • • •

+

<(Zj, Z 2 n , Z2, • • • , Z2n-2, Z 2 n _ i ) = 0.

T h e general solution of the system (37.21) is 5(U,V) = M ( U ) - M ( V ) ,

T{U,V)

= H{U)-H(V)

where M and H are arbitrary complex functions. On the basis of the equalities (37.19) and (37.22), we obtain F ( U , V) = P ( U ) - P ( V ) ,

G(U, V) = Q(U) - Q(V),

(37.22)

Systems

of Quadratic Functional

Equations

293

where P(U) = i [ M ( U ) +

Q(U) = 1-{M(V) - H{TJ)].

ff(U)],

^

Next we consider the system of functional equations / ( Z a , Z 2 , Z 3 )/(Z 4 ) Z5, Z6) + / ( Z 2 , Zi, Z 4 )/(Z 3 , Z 5 , Z 6 ) +

/ ( Z i , Z 2) Z 5 )/(Z 3 ) Z 4l Z6) + / ( Z 2 ) ZX) Z 6 )/(Z 3 ) Z 4) Z 5 )

+

a[<7(Z!, Z 2 , Z 3 )5(Z 4 , Z 5 , Z 6 ) + <7(Z2) Zj, Z 4 ) 5 (Z 3 , Z5> Z6)

+

g(Z1,Z2, Z5)g(Z3, Z 4 , Z 6 ) + <;(Z2, Z1,Z6)g(Z3,

(37.23)

Z 4 , Z 5 )] = 0,

/ ( Z i , Z j , Z3)flf(Z4, Z 5 ) Z 6 ) + / ( Z 2 ) Zi, Z4)<;(Z3, Z 5 ) Z 6 ) +

/ ( Z 1 ; Z 2 , Z5)<;(Z3) Z 4 , Z 6 ) + / ( Z 2 ; Zi, Z 6 )^(Z 3 , Z 4 l Z 5 )

+

5(Zi,

+

<7(Zi, Z 2 , Z 5 ) / ( Z 3 ) Z 4 , Z 6 ) + j ( Z 2 , Zi, Z 6 ) / ( Z 3 ) Z 4) Z 5 )

+

% ( Z ! , Z 2 , Z 3 ) 5 (Z 4 , Z 5 l Z6)+g(Z2,Z1,Z4)g(Z3,

+

5(Zi,

Z 2 l Z 3 ) / ( Z 4 , Z 5 , Z 6 ) + <7(Z2) Zi, Z 4 ) / ( Z 3 , Z 5 , Z 6 ) Z 5 , Z6)

Z 2 , Z 5 ) 5 (Z 3 ) Z 4) Z 6 ) + jr(Za, Z1,Z6)g(Z3,

Z 4 , Z 5 )] = 0,

3

where /, (/ : V >-» C and a and 6 are real constants. Definition 37.2. Let 6 2 /4+a < 0. A solution (/(U, V, W ) , # ( U , V, W)) of system (37.23) is called isotropic if ( / ( U , V , W ) , j ( U , V , W)) £ (0,0),

(37.24)

2

2

/ ( U ) V , W ) + & / ( U , V , W ) 5 ( U , V , W ) - a < ? ( U , V , W ) = 0. The general solution of system (37.1) includes all solutions of this system with the possible exception of the isotropic ones. Theorem 37.2 The general solution of the system (37.23) is given by the following formulae 1° if b2/4 + a = -p2 {p > 0) : 2p/(U,V,W)

(37.25)

=

A[H1(V),H2(V),(2p

-

A[ffi(U), K2(V),

2PH3(W)

+ b)H3(W)]

+

ALffi(U), K2(V),

2pK3{W) -

+

A[K1(V),H2(V),2pK3(W)

+ bK3(W)} -

bH3(W)] bH3(W)},

Systems

294

of Nonlinear Functional

Equations

pg(V, V , W ) =

A[A- 1 (U),A' 2 (V),A' 3 (W)]

-

A[H1(V),H2(V),H3(W)}

+

A[A1(U),i/2(V),^3(W)]

+

A[Hi(U),if 2 (V),if3(W)]

(37.26)

(for the continuous isotropic solutions we have 2p/(U,V,W)

=

pg(\J,V,W)

(2pTbi)A[H1(V),H2(V),H3(W)},

=

(37.27)

±iA[H1(V),H2(V),H3(W)])]

2° if 6 2 /4 + a = 0 :
A[ff 1 (U), J ff 2 (V),/f 3 (W)] A[K1(\J),H2(V),H3(W)},

/(U.V.W) =

A[H1(U),H2(V),H3(W)-^A3(W)]

-

^A[A 1 (U),^ 2 (V), J tf 3 (W)];

3° if 6 2 /4 + a = p2

=

(2p-6)A[i/1(U)>F2(V),F3(W)]

+

(2p + 6)A[A 1 (U),A 2 (V),A 3 (W)] ) 2pg(\J,V,W)

~

(37.29)

(p > 0) : 4p/(U,V,W)

=

(37.28)

A[H1(JJ),H2(V),H3(W)} A[A1(U)IA2(V),A3(W)])

(37.30)

(37.31)

Systems

oj Quadratic Functional

Equations

295

where Hi, Ki (i,j = 1,2,3) are arbitrary complex functions defined in V, and A[H1(V),H2{V),H3(W)] Proof.

=

ffi(U) ffi(V) ffi(W)

H2{\3) H2(V) ff2(W)

H3(U) H3{V) #3(W)

If we introduce the function h by / i ( U , V , W ) = / ( U , V, W ) + - j ( U , V , W ) ,

(37.32)

the system of functional equations (37.23) becomes fc(Zi, Z 2 , Z 3 )/»(Z 4 , Z 5 , Z 6 ) + A(Z2) Zi, Z 4 )/i(Z 3 , Z 5 , Z 6 ) +

/i(Zi, Z 2 , Z 5 )/i(Z 3 , Z 4 , Z 6 ) + /i(Z 2) Zi, Z 6 )/i(Z 3 , Z 4 , Z 5 )

+

(6 2 /4 +

+

sr(Zi, Z 2 ) Z 5 ). g (Z 3) Z 4 , Z 6 ) + <7(Z2, Zi, Z 6 )g(Z 3 , Z 4 , Z 5 )] = 0,

(37.33)

a)[ f l f(Zi,Z2,Z3)p(Z4 I Z5,Z 6 )+5(Z2 ) Zi,Z 4 )j(Z3 ) Z5 ) Z6)

A(Zi, Z 2 , Z3)
A(Zi, Z 2 , Z 5 ) 5 (Z 3 , Z 4 , Z 6 ) + /i(Z 2 , Zi, Z6)flf(Z3, Z 4 ) Z 5 )

+

fl,(Zi,Z2,Z3)/i(Z4,Z5,Z6)

+

+ #(Z 2 ,Zi,Z 4 )/i(Z 3 ,Z 5 , Z 6 )

5 ( Z i , Z 2 , Z 5 )A(Z 3 , Z 4 , Z 6 ) +
According to [J. Aczel (1958)], we will distinguish three cases: 1°. Let b2/4 + a = -p2 (p > 0). In this case the system 37.33) will be /i(Z L , Z 2 , Z 3 )/i(Z 4 , Z 5 , Z 6 ) + A(Z2, Zi, Z 4 )/i(Z 3 , Z 5 , Z 6 ) +

/i(Z x , Z 2 , Z 5 )/i(Z 3) Z 4 , Z 6 ) + /i(Z 2 , Zi, Z 6 )ft(Z 3) Z 4 ) Z 6 )

-

[fc(Zi, Z 2 , Z3)fc(Z4, Z 5 , Z 6 ) + *(Z 2 , Z L Z4)fc(Z3, Z 5 , Z 6 )

(37.34)

+ fc(Zi, Z 2 , Z5)fc(Z3, Z 4 , Z 6 ) + fc(Z2) Zi, Z6)fc(Z3) Z 4 , Z 5 )] = 0, /i(Zi, Z 2 ) Z3)fc(Z4, Z 5 , Z 6 ) + /i(Z 2 , Zi, Z4)fc(Z3, Z 5 , Z 6 ) +

A(Zi, Z 2 , Z 5 )t(Z 3 , Z 4 , Z 6 ) + /i(Z 2) Zi, Z6)fc(Z3, Z 4 , Z 5 )

+ fc(Z!, Z 2 l Z 3 )/»(Z 4) Z 5 ) Z 6 ) + fc(Z2) Zi, Z 4 )A(Z 3) Z 5 , Z 6 ) +

k{ZuZ2,

Z 5 )/i(Z 3 , Z 4 , Z 6 ) + Ar(Z2) Zi, Z 6 )/»(Z 3 , Z 4 , Z 5 ) = 0,

where *(U,V)W)=pflr(U,V,W)

(37.35)

Systems

296

of Nonlinear Functional

Equations

Suppose that ( / ( U , V , W ) , j ( U , V , W ) ) is a continuous isotropic solution of system (37.23). The relation Eq. (37.24) by virtue of the changes Eqs. (37.32) and (37.35) becomes h2(U, V, W ) + A;2(U, V , W ) E O

(which justifies Definition 37.2), i.e., k(V, V, W ) = ±ih(V, V, W ) .

(37.36)

As in Theorem 37.1 we must take the same sign for all triples of vectors 3

(u,v,w)ev .

In view of Eq. (37.36) system (37.34) becomes h{Z1,Z2,Z3)h{Z4,Z5,Z6) +/i(Z 1 ,Z 2 ,Z 5 )MZ3,Z4,Z 6 )

+

h{Z2,Z1,Z4)h{Z3,Z5,Z6)

+

h{Z2,Z1,Z6)h{Z3,Zi,Zs)

(37.37) = 0.

The general solution of the equation (37.37) according to [L. Carlitz (1963)] is given by ft(U, V , W ) = A[H1(U),H2(V),H3(W)] {Hi : V H > C , i=

(37.38)

1,2,3),

where

are arbitrary complex functions and A, B, C are vectors from V such that h{A, B, C) # 0. From Eqs. (37.32), (37.35), (37.36) we find 2p/(U,V,W) = (2pTM)MU,V)W))

pg{\3, V, W ) = ±ih{V, V, W ) ,

and Eq. (37.38) leads to Eq. (37.27). Next we are looking for solutions of Eq. (37.34) not satisfying Eq. (37.36) (anisotropic solutions). By putting Zi = Z 2 = • • • = Z 6 = U, the system (37.34) becomes /i 2 (U, U, U) - fc2(U, U, U) = 0,

h(U, U, U)A(U, U, U) = 0,

whence / i ( U , U , U ) = 0,

* ( U , U , U ) = 0.

(37.39)

Systems

of Quadratic Functional Equations

297

For Zi = Z 2 - Z 3 = Z 4 = U, Z 5 = Z 6 = V, the system (37.34) yields / i 2 ( U , U , V ) - A ; 2 ( U , U , V ) = 0,

/i(U,U, V)fc(U, U, V) = 0,

so that we obtain / i ( U , U , V ) = 0,

Jfc(U,U,V) = 0.

If we put Z 2 = Z 4 = Z 5 = Z 6 = U, Z2-Z3obtain

V, from Eq. (37.34) we

2/i 2 (V, U, U) + h(U, V, U)/i(V, U, U) -

(37.40)

(37.41)

2

2Ar (V,U,U)-fc(U,V,U)fc(V,U,U) = 0, 4/i(V ) U,U)&(V,U,U) + /i(U ) V,U)fc(V,U,U)

+

/i(V,U,U)fc(U,V,U) = 0.

By putting Zi = Z 3 = V, Z 2 = Z 4 = Z 5 = Z 6 = U into Eq. (37.34), we obtain /i 2 (V, U, U) + 2h(V, U, U)/i(U, V, U) -

(37.42)

2

/ c ( V , U , U ) - 2 f c ( V , U , U ) f c ( U , V , U ) = 0, ft(U, V, U)fc(V, U, U) + ft(V, U, U)Jfc(V, U, U)

+

/i(V,U ) U)Jfe(U ) V,U) = 0.

Comparing Eqs. (37.41) and (37.42), we find h2(V, U, U) - k2{V, U, U) = 0,

/i(V, U, U)fc(V, U, U) = 0,

i.e., we obtain /»(V,U,U) = 0,

A;(V,U,U) = 0.

(37.43)

By the substitutions Zi = Z 4 = V, Z 2 = Z3 = Z5 = Z6 = U, the system (37.34) becomes /i 2 (U, V, U) - Ar2(U, V, U) = 0,

h(V, V, \J)k{U, V, U) = 0,

i.e., A(U,V,U) = 0,

A(U,V,U) = 0.

(37.44)

For any anisotropic solution of the system (37.34) there exist at least three vectors A, B, C £ V such that h2(A, B, C) + k2{A, B, C) ^ 0.

Systems

298

of Nonlinear Functional

Equations

If we put Zi = U, Z 2 = V, Z 3 - A, Z 4 = B, Z 5 = Z 6 = C into Eq. (37.34), on the basis of the equalities (37.39), (37.40), (37.43) and (37.44) we obtain h(U,V,C)

= -h(V,U,C),

*(U,V,C) = - & ( V , U ) C ) .

(37.45)

Using Eq. (37.45), the system (37.34) for Zl = A, Z 2 = B, Z 3 = Z 6 = C, Z 4 = U, Z 5 = V yields /i(U,V,G) = /i(G,U,V),

Jfc(U)V,C) =

fc(C,U,V).

(37.46)

If we put Zi = A, Z 2 = B, Z 3 = Z 5 = C, Z 4 = U, Z 6 = V, the system (37.34) becomes /i(C,U,V) = - / i ( U , C , V ) ,

fc(C,U,V)

= -fe(U)C,V).

(37.47)

On the basis of the equalities (37.44), (37.45) and (37.46), if we put Zi = A, Z 2 = B, Z 3 = C, Z 4 = U, Z 5 = V, Z 6 = W , the system (37.34) yields /i(U,V,W)

(37.48)

=

?[ffi(U)F(V, W ) - ff!(V)F(U, W ) + # i ( W ) F ( U , V)

-

A' 1 (U)G(V,W) + A ' i ( V ) G ( U , W ) - A ' i ( W ) G ( U , V ) ]

+

r [ # i ( U ) G ( V , W ) - # i ( V ) G ( U , W ) + Hl(W)G{V,

+

7 i i ( U ) J F ( V , W ) - A ' 1 ( V ) J F ( U , W ) + Ai(W) J F(U,V)],

V)

Jfe(U,V,W)

(37.49)

=

q[H1{U)G{V,W)-H1(V)G{U,W)

+

H1{W)G{U,V)

+

Ki(U)F{V,Yf)

+

r[#i(U)F(V,W)-

-

A' 1 (U)G(V,W) + A ' i ( V ) G ( U , W ) - A ' i ( W ) G ( U , V ) ] ,

- A j ( V ) F ( U , W ) + A'i(W)F(U, V)] fl"i(V)F(U,W)

+ ifi(W)F(U,V)

where we have introduced the notations ffi(U)

=

h(A,B,U),

K1{U) =

k(A,B,U),

F(U,V)

=

9

~

/i(U,V,G), G ( U , V ) = ik(U,V,C), (37.50) h{A,B,C) _ k{A,B,C) r h2{A,B,C) + k2{A,B,C)' ~ h2(A,B,C) + k*(A,B,C)'

Systems

of Quadratic Functional

299

Equations

For Zi = Z 4 = C, Z 2 = S, Z 3 = T, Z 5 = U, Z 6 = V in view of the notations Eq. (37.50), the system (37.34) becomes F(S, T ) F ( U , V) + F ( S , U ) F ( V , T) + F{S,V)F(T, -

U)

(37.51)

G(S,T)G(U, V) - G ( S , U ) G ( V , T ) - G(S, V ) G ( T , U ) = 0, F(S, T)G(U, V) + F(S, U)G(V, T) + F ( S , V)G(T, U)

+

G(S, T ) F ( U , V) + G(S, U) F ( V , T) + G(S, V ) F ( T , U) = 0.

The general solution of the system (37.51) according to [P. M. Vasic (1963)] is given by the formulae F(U,V)

=

A[jyi(U),g^(V)+r/^(V)]

+ G(U,V)

=

(37.52)

A{K2(\J),rH3(\)-qK3(V)], A[K2(U),qH3(V)

+

+ rK3(V)]

A[H2(\J),qK'3(\)-rH3(V)},

On the basis of the equalities (37.32), (37.35), (37.48), (37.49) and (37.52) we obtain Eqs. (37.25) and (37.26), with the following notations H3(U) = (q2-r2)H3(U)+2qrK3(\J),

tf3(U)

=

(r2-q2)K3(U)+2qrH3(U).

The functions / and g given by the formulae Eqs. (37.25) and (37.26) are really solutions of the system (37.23). Consequently, in the case fe2/4 + a = —p2 (p > 0) all functions / and g which are solutions of the system (37.23) and do not satisfy Eq. (37.24) are determined by the formulae Eqs. (37.25) and (37.26). The functions Hi, Ki (i,j — 1,2,3) which appear in these solutions are arbitrary complex functions defined in V. 2°. Let b2/4 + a = 0. In this case the first equation of the system (37.33) takes on the form Eq. (37.37): /i(Z 1 ,Z 2 ,Z 3 )/i(Z 4 ,Z5,Z 6 )

+

/i(Z 2 ,Z 1 ,Z 4 )/i(Z 3 ,Z 5 ,Z 6 )

+/i(Z 1 ,Z 2 ,Z 5 )/i(Z 3 ,Z4,Z 6 )

+

/i(Z 2 ,Z 1 ,Z 6 )/ 1 (Z 3 ,Z 4 ,Z 5 ) = 0,

and its general solution according to [L. Carlitz (1963)] is given by the equation (37.38): / l ( U , V , W ) = A[i/ 1 (U), J tf 2 (V),^ 3 (W)]

(Hi-.V^C,

t = 1,2,3),

Systems

300

of Nonlinear Functional

Equations

where HlilJ)

-

h(A,B,\J) h(A,B,C)>

^

2 ( U )

h(A,U,C) - h(A,B,Cy

ff(m_h(RTir, *(V)-h(B,U,C)

H

are arbitrary complex functions and A, B, C are vectors from V such that h(A,B,C)^0. If we put Zi = Z 2 = Z 4 = Z 5 = C, Z 3 = U, Z 6 = V, the second equation of the system (37.33) yields h{U,C,V)g{C,C,C)

= 0.

(37.53)

Using the equality ft(U,C, V) = —/i(U, V , C ) , which follows from the equation (37.38), and by putting U = A, V — B, from Eq. (37.53) we obtain g(C, C, C) = 0. For Zi = A, Z 2 = B, Z 3 = Z 4 = Z 5 = C, Zg = U the second equation of the system (37.33) becomes g(C, C, U) = 0. If we put Zi = A, Z2 = 5 , Z3 = Z5 = Z6 = C, Z 4 = U, the same equation yields 5>(U, C, C) = 0. Finally, the substitution Zi = Z 4 = Z 5 = C, Z 2 = U, Z 3 = A, Z 6 = B reduces the second equation of the system (37.33) to g(C, U, C) = 0. By putting 1X = U, Z 2 = V, Z 3 = A, Z 4 = 5 , Z 5 = Z 6 = C, on the basis of the above result we obtain <7(U,V,C*) = -<7(V,U,C).

(37.54)

For Zi = yl, Z 2 = B, Z 3 = Z 6 = C, Z 4 = U, Z 5 = V, the same equation yields <7(U,V,C) =
(37.55)

Also, based on the previous results, for Zi = A, Z 2 = B, Z 3 = Z5 = C, Z 4 = U, Z6 = V, the second equation of the system (37.33) is reduced to g{C,U,V)

= -g(\J,C,V).

(37.56)

By putting Zi = A, Z 2 = B, Z 3 = C, Z 4 = U, Z 5 = V, Z 6 = W , on the basis of Eqs. (37.54), (37.55) and (37.56) the second equation of the

Systems

of Quadratic Functional

Equations

301

system (37.33) becomes h(A, B, C)[g(XJ, V, W ) - ffi(U)5(V, W , C)

(37.57)

+

H1(V)g(V,

W , C) - Hx{yf)g{V, V, C)] + g(A, B, C)h(V, V, W )

-

g(A, B, U)/i(V, W , C) + g(A, B, V)h(U, W , C)

-

5(A,5,W)/l(U,V,C)

= 0.

For Zi = Z 4 = C, Z 2 = S, Z 3 = T, Z 5 = U, Z 6 = V and with the notations Eq. (37.50) the system (37.33) is reduced to F ( S , T ) F ( U , V ) + F(S,U)JF(V,T) + JF(S,V)F(T,U) F{S,T)G(XJ, +G{S,T)F(\J,V)

=

0,

=

0.

V) + F(S, U)G(V, T) + F ( S , V)G(T, U) + G{S,U)F{V>T)

+ G(S,V)F(T,U)

The solution of this system according to [P. M. Vasic (1963)] is given by the formulae F(U,V)

=

A[H2(U),H3(V)},

G(U,V)

=

A[K2(\J),H3(V)]

(37.58) +

A[H2(U),K's(V)-mH^(V)},

where K(m-9(A,\J,C) h2{l3)

-h(A,B,Cy

, , f l *
K

u n

g(A,g,C) r n - ^ ^ .

On the basis of the equalities (37.57) and (37.58) we obtain Eq. (37.28), where K3(U) = ^ ( U ) - 2mH3(U),

/^(U) -

g[A

B ' U) h(A,B,C)

On the basis of the expressions Eqs. (37.32), (37.52) and (37.28) we get Eq. (37.29). The functions given by Eqs. (37.28) and (37.29) are really solutions of the system (37.23). Consequently, the general solution of the system (37.23) in the case b2 + 4a = 0 is given by the formulae Eqs. (37.28) and (37.29), where Hi, Kj (i,j = 1,2,3) are arbitrary complex functions defined in V and m is an arbitrary constant.

302

Systems

of Nonlinear Functional

Equations

3°. Let 6 2 /4 + a = p2 (p > 0). Introducing the new functions M and N by the formulae

M(U,V,W)

=

h(U,V,W)+pg(\J,V,W),

W(U,V,W)

=

/>.(U,V,W)-ra(U,V,W),

(37.59) the system (37.33) becomes M(Z1,Z2,Z3)M(Z4,Z5,Z6)

+

M(Z2,Z1,Z4)M(Z3,Z5,Z6)

+ M ( Z 2 , Z 1 , Z 5 ) M ( Z 3 , Z 4 Z6)

+

M(Z2,Z1,Z6)M(Z3lZ4,Z5)

+7V(Z 1 ,Z 2 ,Z 3 )iV(Z4,Z5 Z 6 )

+

7V(Z 2 ,Z 1 ,Z 4 )7V-(Z 3 ,Z 5 ,Z 6 )

+7v-(Z 1 ,Z 2! Z 5 )7V(Z3 J Z 4 Z 6 )

+

M ( Z 1 , Z 2 l Z 3 ) M ( Z 4 , Z 5 Z6)

+

7V(Z 2 ,Z 1 ,Z 6 )7V(Z 3 ,Z 4 ,Z 5 ) = 0, (37.60) M(Z2,Z1,Z4)M(Z3,Z5,Z6)

+ M ( Z 1 , Z 2 , Z 5 ) M ( Z 3 ) Z 4 Z6) -Ar(Z 1 ,Z 2 ,Z 3 )Ar(Z 4 ,Z5 Z 6 )

+ -

M(Z2,Z1,Z6)M(Z3>Z4,Z5) iV(Z 2 ,Z 1 ) Z 4 )iV(Z3,Z 5 ,Z 6 )

-AT(Z 1 ,Z 2 ,Z 5 )Ar(Z 3 ,Z 4 Z 6 )

-

7V(Z 2 ,Z 1 ,Z 6 )Af(Z 3 ,Z 4 ,Z 5 ) = 0,

Comparing the equations of the system (37.60), we find M(Z1,Z2,Z3)M(Z4,Z5,Z6)

+

M(Z2,Z1,Z4)M(Z3,Z5,Z6)

+M(Z1,Z2,Z5)M(Z3,Z4,Z6)

+

AT(Z 1 ,Z 2 ,Z 3 )AT(Z 4 ,Z 5 ,Z 6 )

+

M ( Z 2 , Z 1 , Z 6 ) M ( Z 3 , Z 4 , Z 5 ) = 0, (37.61) 7V(Z 2 l Z 1 ,Z 4 )Ar(Z 3 ,Z 5 ,Z 6 )

+AT(Z 1 ,Z 2 ,Z 5 )7V(Z 3 ,Z 4 ,Z 6 )

+

AT(Z 2 ,Z 1 ,Z 6 )7V(Z 3 ,Z 4 ,Z 5 ) = 0.

The general solution of this system according to [L. Carlitz (1963)] will M(U,V,W)

=

A^ifUj.ffaCVj.ffafW)],

A'(U,V,W)

=

A[A' 1 (U),K 2 (V),A' 3 (W)],

(37.62)

where if^, A'j (i,j = 1,2,3) are arbitrary complex functions defined in V. On the basis of the equalities (37.32), (37.59) and (37.62), we find Eqs. (37.30) and (37.31). The functions given by Eqs. (37.30) and (37.31) are solutions of the system (37.23).

Systems

of Quadratic Functional

Equations

303

Consequently, the general solution of the system (37.23) in the case fc2/4 + a = p2 (p > 0) is given by Eqs. (37.30) and (37.31) where Hi, Kj (i,j = 1,2,3) are arbitrary complex functions denned in V. • Now we will prove the following general result. T h e o r e m 37.3

The general solution of the system of functional equations (37.63)

/(Zi,Z2,-..

Z n - i , Zn)/(Z„ + i, Z „+ 2, ' • • , Z 2 n )

-

/(Zi,Z2,-..

Z n - i , Z n +i)/(Z„, Z n + 2 , Z n + 3 , • • • , Z 2 n )

-

/(Zi,Z2,.-.

Z n - i , Z n + 2 ) / ( Z n+ l , Z „ , Z n + 3 , • • • , Z 2 „) — • • •

-

+

/ ( Z i , z 2 > . . . Z n - 1 , Z 2 n ) / ( Z n + l, Z n +2, • • • , Z 2 n - 1 , Z n ) s g n a [ # ( Z i , 2 J 2 , • • • Z n - 1 , Z n ) 5 ( Z n + l, Z n + 2, • • • , Z 2 n )

-

<7(Zi,Z2,---

Z n - 1 , Z n + l)^(Z n , Z n + 2 , Z „ + 3 , • • • , Z 2 n )

-

<7(Zi,Z2,.-.

Z n - 1 , Z n + 2 )<7(Z n + l, Z„, Z n + 3 , • • • , Z 2 n ) — • • •

-

Z n - 1 , Z 2 n)5(Zn + l, Z n + 2 , ' • ' , Z 2 n _ i , Z n )] = 0,

/ ( Z i , z 2 l . . . Z n - 1 , Zn)ff(Zn + l, Z „ +2) • • • , Z 2 n ) -

/(Zi.Za,...

Z n - 1 , Z n + l)
-

/(Zi.Za,-..

Z n - 1 Z n + 2)5,(Zn + l, Z n , Z n + 3 , • • • , Z 2 n ) — • • •

-

/ ( Z i , z 2 > . . . Z n - 1 ^2n Mz n + 1, Z n +2, • • • , Z 2 „_i, Z n )

+

Z n - 1 , Z n ) / ( Z n + l, Z „+ 2, • • • , Z 2 „)

-

j(Zi,Za,---

Z n - 1 , Zn + l)/(Z n , Z n+ 2, Z n + 3 , • • • , Z 2 n )

-

<7(Zi,Z 2 ,---

Z n - 1 , Z n + 2)/(Z n+ i, Z„, Z n + 3 , • • • , Z2n) — • • •

— ^(Zi,Z2,.--

Z n - 1 , Z 2 n ) / ( Z n + l, Z n+ 2) ' • • , Z 2 n-1, Z„) = 0,

where f,g : V" <-* C and a is a real constant, is given by the following formulae: 1°. if a < 0, /(Ui,U2,-,Un) 5(Ui,U2:---,Un)

=

A[ff1(Ui),ff2(U2))---,tfn(Un)]

+

AtA'^Ui), K2(U2),

=

i{A[Kl(V1),K2(V2),---,Kn(Vn)}

-

••• ,

(37.64)

Kn(\Jn)},

A[H1(\J1),H2(\J2),---,Hn(\Jn)}};

(37.65)

Systems

304

2°. if

of Nonlinear Functional

Equations

a>0, ,U„)

,U„)

=

A[H1(U1),H2(\J2),-

,tf»(U„)] (37.66)

+

A[K1(U1),K2(U2),-

,Kn(Un)},

=

A[ffi(Ui),ff 2 (U 2 ),-

,Hn{Vn)]

-

AfA^U^/i^U,),.

,An(Un)];

(37.67)

3° if a = 0,

/(u1;u2,.--

,U„)

=

A^fUO.ft^),.

,tf n (U„)]

^(Ui)

#i(U2)

•

#i(U„)

tf2(U0

# 2 (U 2 )

•

/fj(U„)

i/n(Ui)

ff n (U 2 )

•••

ff„(U„)

^Ux.^.-.-.Un)

(37.68)

(37.69)

=

A ^ U O , H2(\J2),

H3(V3),

• • • , #„(Un)]

+

Afif^Ui), A' 2 (U 2 ), tf3(U3), • • • , J7„(U„)] + • • •

+

A[Hl(V1),H2{U2),

• • • , ffn-iOJ,,..!), A n ( U n ) ] ,

where Hi, Ki (1 < i, j < n) are arbitrary complex functions defined in V. Proof. 1°. Let a < 0. Introducing new functions M and N by the formulae M(Ui;U2)---,Un) =

(37.70)

/(U1,U2,...,UB) + f , ( U 1 , U v , U B ) , N(\J1,V2,---

=

,Vn)

/(Ui,U2,---)Un)-z<,(U1,U2)---,Un),

(f2 = — 1) the system (37.63) becomes M{Z1:

Z 2 , • • • , Z„_i, Z„)Af (Z n + i, Z „ + 2 ) Z n + 3 , • • • , Z 2 n ) (37.71)

=

M ( Z i , Z 2 , • • • , Z„_i, Z n + i ) M ( Z n , Z n + 2 , Z„+3, • • • , Z 2 n )

+

M ( Z i , Z 2 , • • • , Z„_i, Z n + 2 ) M ( Z n + i , Z„, Z n + 3 , • • • , Z 2 n ) + • • •

+

Af(Zi,Z2,---, Zn-i, Z 2 n ) M ( Z n + i , Z n + 2 , • • • , Z 2 n -i, Z n ) ,

Systems oj Quadratic Functional Equations W(Z 1, Z 2 , • • • , Z n _ i , Zin)N(2in

305

+ i, Z n + 2, Z n + 3 , • • • , Z2n)

=

iV(Zi, Z2, • • • , Z „ _ i , Z„ + i)7V(Z n , Z n

+

7V(Z X ,Z 2 ,.-- , Z n _ i , Z n + 2 ) ^ ( Z n + 1 , Z n , Z „ + 3 l - - - ,Z 2 n ) + ---

+

JV(Zi,Z 2 ,

+

2, Z „ + 3 , • • • , Z2n)

• • •, z n _i, z2„)./v(zn+i, z n+2 , • • •, z 2 „_i, z„).

On the basis of the equalities (37.71) and (37.70), we find Eqs. (37.64) and (37.65), where Hi, Kj : V i-> C are arbitrary complex functions. 2°. Let a > 0. Introducing the functions M and N by M(U1,U2,---,Un)

=

/ ( U 1 , U 2 , - - - , U n ) + <7(U1,U2,.--,Un), (37.72)

iv(u 1 ,u 2 l ---,u n ) = / ( u 1 , u 2 , . . . , u n ) - 5 ( u 1 , u 2 , . . . , u n ) , the system (37.63) is reduced to Eq. (37.71). On the basis of the equalities (37.71) and (37.72) there follow Eqs. (37.66) and (37.67). 3°. Let a = 0. In this case, the first equation of the system (37.63) is reduced to / ( Z i , Z 2 , • • • , Z n _ i , Z n ) / ( Z n + i , Z n +2, • • • , Z2„) + 2,

(37.73)

=

/ ( Z i , Z 2 , • • • , Z„_i, Z„ + i ) / ( Z n , Z n

Z„ + 3, • • • , Z2n)

+

/ ( Z i , Z2, • • • , Z n _ i , Z„_|-2)/(Zn + l, Z „ , Z n + 3 , • • • , Z2n) + • • •

+

/ ( Z i , Z 2 , • • • , Z n _ i , Z 2 „ ) / ( Z n + i , Z n + 2 , • • • , Z 2 n _ i , Z„).

According to [P. M. Vasic (1963)], the general solution of the equation (37.73) is given by Eq. (37.68). Further in the proof of this theorem, we will use the following Lemma 37.4

/ / A , E V (1 < i < n) are such that /(A!,A2,---,An)^0,

then the following equality holds / ( A n , U 1 , U 2 , . . . , U n _ 2 , A n ) = 0.

(37.74)

Proof of Lemma 37.4- By putting Z; = A,Z„+i = Z 2 n = A n ,

Zj=\Jj-„-1

(1 < i < n), (i = n + 2,n + 3 , - - - , 2 n - l ) ,

Systems

306

of Nonlinear

Functional

Equations

the equality (37.73) becomes / ( A i , A 2 , - - . A ^ i . A ^ A n . U i . U a , - - - ,Un_2,An)

(37.75)

+

/(A1,A2,---,An_1,U1)/(An,AniU2>--- ,Un_2,An)

+

/ ( A i , A 2 l - • • , A „ _ 1 , U 2 ) / ( A „ , U 1 , A „ , U 3 , - • • ,U„_2, A n ) H

+

/ ( A i , A 2 , - - - , A „ _ i , U „ _ 2 ) / ( A n , U 1 , - - - , U „ _ 3 , A n , A„) = 0 .

Since the formulation of the lemma is exactly the same as of Lemma 36.2, and equation (37.75) is identical with Eq. (36.11), the proof of the lemma is completed in the same way as of Lemma 36.2. • We will prove that Eq. (37.69) holds by induction. If n — 2, then / ( U ! , U 2 ) = A[H1(U1),H2(U2)},

(37.76)

vhere ffl(Ul)

=

AA.Br

ff2(U2)

= /(B U2)

'

and / ( Z i , Z2)g(Z3, Z 4 ) - f(Z1,Z3)g{Z2,

Z 4 ) - / ( Z i , Z4)«/(Z3, Z 2 )

(37.77)

+g(Z1, Z 2 ) / ( Z 3 , Z 4 ) -
» = OT9(B'Us)" %^)S{B'Ul) +

7(ATB)-/(B'

U2)

- 7{A^rf{B'Ul)

- J(A^)

/ ( U l 1 U2)

(37J8)

-

Using the notations g ( A , U i ) _ K /TT x 7(ATB)--AI(UI)'

/ „ TT \ rs'tn \ »(B.UI) = M U I ) ,

ff(A~B)

/(ATB) = - m '

Systems

of Quadratic Functional

Equations

307

on the basis of the relations Eqs. (37.78) and (37.76) we find j(Ui,U2)

=

A[i7 1 (U 1 ),/^(U 2 )] + A[A' 1 (U 1 ), J ff 2 (U 2 )]

+

7nA[ J f/ 1 (U 1 ),// 2 (U 2 )],

i.e.,

g(Uu U 2 ) = AtA'^Ui), ff2(U2)] + A[H1(\J1), A 2 (U 2 )], where A' 2 (U) = A 2 (U) + m.i/ 2 (U) Assume that this theorem holds for n — 1, so that the general solution of the functional equation /(Z1:

i Z n _ 2 , Z n _ 1 )5i(Z n , Z n + i , • • • , Z 2 n _ 2 )

(37.79)

-

/(Zi,

, Z n _ 2 , Z n )#(Z n _i, Z„+i, • • • , Z 2 n _ 2 )

-

/(Zi;

, Zn_2, Z n + i ) ^ ( Z n , Z n _ i , Z n + 2 , • • • , Z2n_2) — • • •

-

/(ZL

, Z n _ 2 , Z 2 n _ 2 )g(Z n , Z„+i, • • • , Z 2 n _3, Z„_i)

+

, Z„_ 2 , Z n _ i ) / ( Z n , Z„+i, • • • , Z 2 n _ 2 )

-

g(Zu

, Zn_2, Z n ) / ( Z n _ i , Z n + i , • • • , Z2n_2)

-

s(Zi.

, Z n _ 2 , Z n + i ) / ( Z n , Z„_i, Z n + 2 , • • • , Z 2 n _ 2 ) — • • •

-

5(Zi,

i Z n _2, Z 2 n _ 2 ) / ( Z n , Z n + i , • • • , Z 2 n _3, Z n _i) = 0

is introduced by 0(U1,U2>...,Un_i) =

(37.80)

A[/fi(U1)lJf2(U2))---1ffI,.i(Ull.i)]

+

A ^ U j ) , A 2 (U 2 ), # 3 ( U 3 ) , • • • , ff„-i(U„_i)] + • • •

+

A[if!(Ui), # 2 ( U 2 ) , • • • , ffn_2(U„_2), K n - i f U , , . ! ) ] .

Let / ( A j , A 2 , • • • , A„) ^ 0. If we put Zi = A„, Z„+i = A„, /(An,U2,U3,---,Un)

=

F ( U 2 , U 3 , - - - ,U„),

flf(A„,U2)U3)---,Un)

=

G(U2,U3,---,Un),

the second equation of the system (37.63), according to the lemma, becomes .F(Z 2 , Z 3 , • • • , Z n _ i , Z n )G(Z n + 2 , Z n + 3 , • • • , Z 2 n )

(37.81)

Systems

308

of Nonlinear Functional

Equations

—

ir(Z2,Z3,'

, Z n _i, Z„+ 2)G(Z„, Z n + 3 , • • • , Z2n)

—

F(Z2,2>3r

, Z n _i, Z n + 3 ) G ( Z n + 2, Z„, Z n + 4 , • • • , Z2n)

,

—

-F (Z2,Z3,'

, Z n _i, Z2n)G(Z„ + 2, • • • , Z2n_l, Z2n)

+

G(Z2,Z3)-

, Z n _i, Z n ) F ( Z n + 2, Z n +3, • • • , Z2n)

—

G(Z2,Z3,-

, Z„_i, Z n + 2)i r (Z n , Z n + 3 , • • • , Z 2 „)

—

G(Z2,Z3)-

i Z„_i, Z n+ 3).F(Z n + 2, Z n , Z n + 4 , • • • , Z 2 n )

—

G(Z2,Z3)'

, Z„_i, Z2n)F(Zn

+

2, • • • , Z2n-1, Z„) = 0.

On the basis of the equalities (37.79) and (37.80), we find the general solution of the equation (37.81) which is G(U1,U2,---,Un_1) =

(37.82)

ff(AniUi,U2l."1Un.1)

=

A[(#i(U1)).ff2(U3),--.

,ff„_i(U„_i)]

+

A ^ i t U i ) , A'2(U2), /f3(U3), • • • , i / n - ^ U ^ x ) ] + • • •

+

A[H1(U1),H2(U2),--

,/f„_2(Un_2),A'n_1(Un_1)],

where # , ( U ) , A ^ ( U ) (1 < i , j < n — 1) are arbitrary complex functions defined in V. If we p u t Z;=Ai Z„=Ui,

Zn+i=An,

(l
(2<*
into the second equation of the system (37.63), we obtain (37.83)

/(Ai,A2)

,An)flf(Ui(U2)"-)Un)

=

/(Ai,A2,

,A„_1,Ui)(/(A„,U2,--- ,Un)

-

/(Ai,A2,

, A„_i,U2)(/(A„)Ui,U3, • • • ,U„) - •

-

/(Ai,A2,

,An_1,Un)5(An,U2,---

.Un-^Ui)

+

,AB.,,Ui)/(An,U2)-

,U„)

-

5(Ai,A2,'

,A„.1,U2)/(An,U1,U3,--,Un)-.

-

5,(Ai,A2,'

,An_1)Un)/(An,U2)---)UB_i,Ui)

-

g(A1,A2,-

,An)/(U1)U2,...,UB).

Systems

of Quadratic Functional

Equations

309

On the basis of the properties Eqs. (37.68) and (37.82), we obtain / ( A „ , - - - , U i . i j U . - . U i + i , - - - , U j _ i , U j , U f + i , - •• , U „ _ i ) =

— / ( A n , • • • , U j _ i , U j , U , + 1 , • • • , U j _ i , U,-, U j + i , • • • , U „ _ i ) ,

(37.84) # ( A „ , • • • , U , - _ i , U j , U ; + 1 , • • • , Uj_i,Uj,Uj + i, • • • , U „ _ i ) =

— # ( A n , • • • ,Ui-i,\Jj,Ui+i,

• • • , U J _ I , U J , Uj+i, • • • ,U„_i)

(1 < i< j < n - 1). By introducing the notations /(Ai,--- ,An_i,U) _ f(Au---,An) -^

ff(Ai,--l ( U )

'

g{M,---

,A„)

/(Ax.-.-.A,,)

,An_i,U) f(M,---,An) -

A l ( U )

'

= — m,

on the basis of the equalities (37.84), (37.83), (37.82) and (37.68) we find ff(Ui,U2,---,Un)

=

A[K1(V1),H2(V2),---,Hn(Vn)}

+

A[H1(V1),K2(IJ2),

+

A[H1(V1),

+

mAiH^V,),^^^,---

H3(U3), • • • , Hn(Un)] + • • •

• • • , Hn-iiVn-t),

Kn(Un)] ,Hn(Vn)},

i.e., g{Ui,\J2, • • • , U n ) is given by Eq. (37.69), where Kn(\Jn)+mHn(Un) is replaced by A ' n ( U n ) . Now we will prove t h a t the functions given by Eq. (37.64) and (37.65), (37.66) and (37.67), (37.68) and (37.69) are solutions of the system (37.63). We introduce the following notation similar to t h a t of Sec. 36: D{FUF2,---

,Fn,GuG2)

••• ,Gn) =

^(U) On-l,n f(U) G{V)

where /

F1(V1)

f2(Ui)

fn(Ui)

\

Fn(U2) T(XJ)

*i(U2)

F2(U2)

Fn(Un~l) J V F!(U„.i)

F2(U„_i;

Systems

310

of Nonlinear

Functional

Equations

I F1(Vn) Fi(Un+i)

F 2 (U n ) F2(Un+1)

Fn(Un) \ F„(U„ + 1 )

V Fl(V2n)

F 2 (U 2 n )

Fn(U2n)

T{\5) = /

G{XJ) is given by a formula of the same form with Fj replaced by Gj (1 < j < n) and On-i,n is the zero (n — 1) x n matrix. Let D(Hi, H2, • • • , Hn, H\, H2, • , Hn

0,

then it will be D{H1 H2,-D(Hl H2,--

• , H„ Hi,

H2,

• •

,Hn)

Hn K\,

K2,

• •

K2,--

• . Kn Hi,

H2,

• •

,Kn) ,Hn)

D(I
• , Kn Ki,

K2,

• •

J
D(Hl

H2,--

• ,

Hn Hi,

H2,

- -

, Hn)

- D{Hl H2,-- D{Kl K2,--

• ,

Hn Ki,K2,--

• ,

Kn Hi,

+

D{KX K2,--

• ,

D(Hi H2,--

+

+ + + +

D(Ki

• ,

,Kn) • •

,Hn)

Kn Ki,K2,

• •

,A'n) = 0,

•

,Hn Hi,H2,

• •

,Hn)

D{Ki K2,-

•

,Kn Hi,

• •

,Hn)

-

D(H1 H2,--

• ,

Hn

D(Ki

• , K„ A'i, K2,

+

D{HX H2,--

- D(Ki

+

K2r

I<2,-

• • ,

H2,

H2,

A'i, A' 2 , • •,Kn)

,Hn Hi,

H2,

Kn Hi,H2,

D{HX H2,--

•

D{KX K2,-

• , Kn

,Hn Ki,K2,--

• •

,Kn)

• •

,Hn)

• •

,Hn) ,Kn)

A'i, A' 2 , • •,A'n) = 0.

ua In a f the determinants which appear in nrf rule, if we evaluate By the Laplace the previous identities, we obtain that the functions given by Eqs. (37.66) and (37.67) in the case a > 0 are a solution of the system (37.63). In the case a < 0, we can analogously show that the functions Eq. (37.64) and (37.65) represent a solution of the system (37.63).

Systems

of Higher Order Functional

Equations

311

If a — 0, we will show that the functions Eqs. (37.68) and (37.69) are a solution of the system (37.63). The function Eq. (37.68) satisfies the first equation of the system (37.63). Now we will prove that the functions Eqs. (37.68) and (37.69) satisfy the second equation of the system (37.63). Let us consider in this case the following identity D(Hx + KUH2) +

D{HUH2

+

D(HUH2,

•••iHn,H1

+ KuH2,---1

Hn)

(37.85)

+ K2, H3, • • • , Hn, HX,H2 + K2,H3, • • - , # „ ) + ••• • • • , # „ + K„,HUH2,

• • • , # „ + Kn) = 0.

By the evaluation of the determinants which are found in the identity Eq. (37.85), we obtain that Eqs. (37.68) and (37.69) satisfy the second equation of the system (37.63). Consequently, the functions Eqs. (37.68) and (37.69) represent a solution of the system (37.63) if a — 0. D

38

S y s t e m s of Higher O r d e r F u n c t i o n a l E q u a t i o n s

T h e o r e m 38.1

The general solution of the system of functional equations

/ ( Z 1 ; Z 2 ) / ( Z 3 , Z 4 ) + / ( Z i , Z 3 ) / ( Z 4 , Z 2 ) + / ( Z i , Z 4 ) / ( Z 2 l Z 3 ) = 0, Z 4 ) + g{Zi, Z3)<7(Z4, Z 2 ) + g(Z1, Z 4 ) 5 (Z 2 ) Z 3 ) +

j ( Z i , Z 2 ) / 3 + 2 i ( Z 3 , Z 4 ) + g{Z1, Z 3 ) / 3 + 2 i ( Z 4 , Z 2 )

+

g(Zu Z 4 ) / 3 + 2 i ( Z 2 , Z 3 ) + g{Z3, Z 4 ) / 3 + 2 i ( Z i , Z 2 )

+

g(Z4, Z2)f+2i(Z1,

+

(3 + 2t)/(Zi, Z 2 ) / ( Z ! , Z 3 ) / ( Z i , Z 4 ) / ( Z 3 , Z 4 ) / ( Z 4 , Z 2 ) / ( Z 2 , Z 3 ) x

x

[f{Zu

+

/2(Z1,Z3)/2(Z4,Z2)],=0,

Z 3 ) + g(Z2, Z3)f3+2i(Z1,

Z 2 ) / 2 ( Z 3 ) Z4) + / ( Z i , Z 2 ) / ( Z i , Z 3 ) / ( Z 3 , Z 4 ) / ( Z 4 , Z2) (38.1)

/.(Zi, Z 2 )/i(Z 3 , Z 4 ) + h(Z1:Z3)h(ZA, +

Z4)

h(ZltZ2)[g(Zs,

Z4) + /

3+2l

+2i

Z2) 3 2

' ( Z 3 , Z4)] + ^

+ f (Z4,

Z 2 )] 3 + 2 ^

+

h{Zx, Z3)[g(Z4,Z2)

+

/i(Z 1 ,Z 4 )[ 5 (Z 2 ,Z 3 ) + / 3 + 2 i ( Z 2 , Z 3 ) ] 3 + ^

+

h{ZuZA)h{Z2,Z3)

Systems

312

+ x

of Nonlinear Functional

(3 + 2j)\g(Zu Z2) + f3+2i(Zu [sf(Zi, Z 4 ) + /

x fo(Z4) Z 2 ) + /

3+2,

Z2)][g(Z1, Z 3 ) + fs+2i(Zi,

"(Z 1 ) Z4)][ff(Z3, Z 4 ) + /

3+2i

( Z 4 , Z2)][<7(Z2, Z 3 ) + /

3+2,

Equations

3+2, 3+2

2

' ( Z 1 ( Z 4 )] [ 5 (Z 2 , Z 3 ) + /

Z 3 )] x

' ( Z 3 , Z 4 )] x

' ( Z 2 , Z 3 )] x

3+2i

( Z 2 , Z 3 )] 2

x

{[g{Zx, Z 4 ) + /

-

[ff(Zj, Z 2 ) + / 3 + 2 i ( Z i , Z 2 )]( 5 (Z 1 , Z 3 ) + / 3 + 2 *'(Z 1 , Z 3 )] x

x

[«7(Z3, Z 4 ) + / 3 + 2 , ' ( Z 3 , Z 4 )]fo(Z 4) Z2) + / 3 + 2 i ( Z 4 , Z2))Y = 0,

where f,g,h: V2 >-> C are arbitrary complex functions and i,j G {0, 1, 2}, is given by the formulae /(U,V) = Gi(U) Gi(V)

^(U) F 2 (V)

F1(U) Fx(V) G 2 (U) G 2 (V)

-

fi(U) Fi(V)

(38.2)

1

f 2 (U) F 2 (V)

3+2i

, (38.3)

«e {0,1,2} /»(U,V)

=

Hi(U) Hi(V)

1 1¥ 2 (V)

-

Gi(U) Gi(V)

G 2 (U) G 2 (V)

3+2j

, (38.4)

i e {0,1,2} where Fk, Gk, Hk (k = 1,2) are arbitrary complex functions defined in V. Proof. The first functional equation of the system (38.1) according to [D. S. Mitrinovic and S. B. Presic (1962B); D. S. Mitrinovic et al. (1963)] has a general solution given by Eq. (38.2). The complex function / ( U , V) satisfied the conditions / ( U , U ) = 0,

/ ( U , V ) + / ( V , U ) = 0.

(38.5)

The functions g(U, V) and /i(U, V) from the second, respectively third equation of the system (38.1) also satisfy the same conditions for Zi = Z 2 = Z 3 = Z 4 = U ^ O or Zi = Z 4 = U, Z 2 = Z 3 = V. Since / 3 + 2 ! ( Z r , Zfc) have distinct powers in the second equation of the system (38.1) for i G {0,1,2}, we obtain that the first equation of the system (38.1) implies / 3 ( Z ! , Z 2 ) / 3 ( Z 3 , Z4) + / 3 ( Z ! , Z 3 ) / 3 ( Z 4 , Z2) + / 3 ( Z 2 , Z 4 ) / 3 ( Z 2 , Z3) =

3/(Z1,Z2)/(Z1,Z3)/(Z1,Z4)/(Z3,Z4)/(Z4,Z2)/(Z2,Z3),

(38.6)

Systems

of Higher Order Functional

Equations

313

/ 5 ( Z 1 ; Z 2 ) / 5 ( Z 3 , Z4) + / 5 ( Z ! , Z 3 ) / 5 ( Z 4 , Z2) + / 5 ( Z ! , Z 4 ) / 5 ( Z 2 j Z3) -

| / ( Z i , Z 2 ) / ( Z j , Z 3 )/(Zx, Z 4 ) / ( Z 3 , Z 4 ) / ( Z 4 , Z 2 ) / ( Z 2 , Z 3 )

x

[/2(Z1,Z2)/2(Z3,Z4) + /2(Z1,Z3)/2(Z4,Z2) + /2(Z1,Z4)/2(Z2,Z3)]) / 7 ( Z X , Z 2 ) / 7 ( Z 3 , Z4) + / 7 ( Z i , Z 3 ) / 7 ( Z 4 , Z2) + / 7 ( Z i , Z 4 ) / 7 ( Z 2 ) Z3)

=

^/(Z1,Z2)/(Z1,Z3)/(Z1,Z4)/(Z3,Z4)/(Z4)Z2)/(Z2>Z3) x

x

[/ 2 (Z!, Z 2 ) / 2 ( Z 3 , Z 4 ) + / 2 ( Z ! , Z 3 ) / 2 ( Z 4 , Z 2 ) + / 2 ( Z ! , Z 4 ) / 2 ( Z 2 , Z 3 )] 2 .

If we apply the formulae Eq. (38.6) to the second functional equation of the system (38.1), then we obtain the following equation [g(Zu Z 2 ) + f+2'(Z1, + +

[ 5 (Z 1 ,Z 3 ) + /

3+2!

b(Z1,Z4) + /

3+2!

Z 2 )][ 5 (Z 3 , Z 4 ) + / 3 + 2 i ( Z 3 , Z4)]

(Z1,Z3)][5(Z4,Z2) + / (Z1,Z4)][5(Z2,Z3) + /

3+2

(38.7)

«(Z 4 ,Z 2 )]

3+2i

( Z 2 , Z 3 ) ] = 0.

By repeating the previous procedure to the last equation of the system (38.1), we obtain {h(Z1, Z 2 ) + [g(Zl, Z2) + f+2i(Zl, x

{/*(Z3, Z 4 ) + [«,(Z3, Z 4 ) + /

3+2 3

Z 2 )] 3 + 2 ^} x

(38.8)

3

' ( Z 3 , Z 4 )] +«} 2i

+

{/ l (Z 1 ,Z 3 ) + b ( Z 1 , Z 3 ) + / + ( Z 1 , Z 3 ) ] 3 + 2 ^ x

x

{/j(Z4, Z 2 ) + [ Z 2 )] 3 + 2 '}

+

{h(Z1, Z 4 ) + [g(Zx, Z4) + f3+2i(Z1,

x

{h(Z2, Z 3 ) + [sf(Z2l Z3) + / 3 + 2 ! ( Z 2 , Z 3 ) ] 3 + « } = 0.

Z 4 )] 3 + 2 i} x

The system of vector functional equations (38.1) with the unknown functions f,g,h: V2 i->- C and i,j G {0,1,2} is equivalent to the system formed by the first equation of the system (38.1) and the equations (38.7) and (38.8), whose general solution is given by the equations (38.2), (38.3) and (38.4). D

Systems

314

Theorem 38.2 equations

of Nonlinear Functional

Equations

The general solution of the following system of functional

f(Z1, Z 2 , Z 3 ) / ( Z 4 , Z 5 , Z 6 ) + / ( Z 2 ) Zj, Z 4 ) / ( Z 3 , Z Bl Z 6 ) +

/(Zx, Z 2 , Z 5 ) / ( Z 3 , Z 4 ) Z 6 ) + / ( Z 2 , Zj, Z 6 ) / ( Z 3 l Z 4 , Z 5 ) = 0, ^(Z!, Z 2 , Z 3 ) 5 (Z 4 ) Z 5 , Z 6 ) + ff(Z2) Zi, Z4)(?(Z3, Z 5 , Z 6 )

+ + +

g(Z1,Z2,

Z 5 ) 5 (Z 3 ) Z 4 , Z 6 ) +
Z 4 , Z5)

3

g(ZuZ2,Z3)f(Z4, flf(Z2,

(38.9)

Z 5 , Z 6 ) + <;(Z4, Z 5 , Z 6 ) / ( Z l l Z 2 , Z 3 ) 3

Z 1 ; Z 4 ) / ( Z 3 l Z 5 | Z 6 ) + <7(Z3, Z 5 , Z 6 ) / 3 ( Z 2 , Zi, Z 4 )

+

j ( Z i , Z 2 , Z 5 ) / 3 ( Z 3 , Z 4 , Z 6 ) +
+

3 ( Z 2 , Z i , Z 6 ) / 3 ( Z 3 , Z 4 , Z 5 ) + 5 ( Z 3 , Z 4 l Z 5 ) / 3 ( Z 2 ) Zi, Z 6 )

+

3/(Z1)Z2,Z3)/(Z2,Z1,Z4)/(Z4,Z5,Z6)/(Z3)Z5lZ6) x

x

[/(Z x , Z 2 , Z B )/(Z 3 ) Z 4 , Z 6 ) + / ( Z 2 ) Zi, Z 6 ) / ( Z 3 ) Z 4 ! Z 5 )]

+

3/(Z!, Z 2) Z 5 ) / ( Z 2 , Zi, Z 6 ) / ( Z 3 , Z 4 , Z 6 ) / ( Z 3 , Z 4 ) Z 5 ) x

x

[/(Zi, Z 2 , Z 3 ) / ( Z 4 , Z 5 , Z 6 ) + / ( Z 2 ) Zi, Z 4 ) / ( Z 3 , Z 5 ) Z 6 )] = 0,

where f, g : V3 t-> C are arbitrary complex functions, following formulae

/(U,V,W) =

Fx(U) Fi(V) Fi(W)

F 2 (U) F 2 (V) F 2 (W)

is given by the

F 3 (U) F 3 (V) F 3 (W)

(38.10)

j(U,V,W) Gi(U) Gi(V) d(W)

G 2 (U) G 2 (V) G 2 (W)

G 3 (U) G 3 (V) G 3 (W)

(38.11) F X (U) Fi(V) Fi(W)

F 2 (U) F 2 (V) F 2 (W)

F 3 (U) F 3 (V) F 3 (W)

3

where Ft, G, (?' = 1,2,3) are arbitrary complex functions defined in V. Proof. For the first equation of the system (38.9), the general solution is given by Eq. (38.10). We can rearrange the terms in the second equation of the system (38.9) as follows: [g{Zlt Z 2 , Z 3 ) + f{ZuZ2,

Z3)][ff(Z4) Z 5 , Z 6 ) + / 3 ( Z 4 , Z 5 , Z 6 )]

(38.12)

Systems

of Higher Order Functional

Equations

315

+

[ff(Z2) Zi, Z 4 ) + / 3 ( Z 2 , Zi, Z4)][sf(Z3) Z 5 , Z 6 ) + / 3 ( Z 3 , Z 5 , Z 6 )]

+

[ff(Zi, Z 2 l Z 5 ) + / 3 ( Z ! , Z 2 , Z5)][sf(Z3) Z 4 , Z 6 ) + / 3 ( Z 3 ) Z 4 ) Z 6 )]

+ fo(Z2, Zi, Z 6 ) + / 3 ( Z 2 , Zx, Z6)][fl(Z3, Z 4 ) Z 5 ) + / 3 ( Z 3 , Z 4 ) ZB)] = 0 by using the identity A3 + B3 + C3 + D3 = 3AB(C + D) + 3CD{A + B)

if

A + B + C+D = 0.

Therefore, the system (38.9) with two unknown functions f,g: V3 >->• C is equivalent to the system formed by the first equation of the system (38.9) and the equation (38.12), whose general solution is given by Eqs. (38.10) and (38.11). • T h e o r e m 38.3 The general continuous solution of the following system of functional equations / n ( Z i , Z 2 + Z 3 ) + / 2 i ( Z 2 , Z 3 -I- Zi) = / 3 i ( Z 3 , Zi + Z 2 ), / 1 2 ( Z i , Z 2 + Z 3 ) + 2/ 1 1 (Z 1 , Z 2 + Z 3 )/ 2 1 (Z 2 l Z 3 + Z x ) +

/22(Z 2 , Z 3 + Zi) = / 3 2 ( Z 3 , Zi + Z 2 ), /i3(Zi, Z 2 + Z 3 ) + 3/ 1 2 (Zi, Z 2 + Z 3 )/ 2 i(Z 2 , Z 3 + Zi)

+

3 / n ( Z i , Z 2 + Z 3 )/ 2 2 (Z 2 , Z 3 + Zi)

+

/23(Z 2 ,Z 3 + Zx) = / 3 3 ( Z 3 , Z i + Z 2 ), / i 4 ( Z i , Z 2 + Z 3 ) + 4/ 1 3 (Zi, Z 2 + Z 3 ) / 2 i ( Z 2 , Z 3 + ZO

+

6 / i 2 ( Z i , Z 2 + Z 3 ) / 2 2 ( Z 2 ) Z 3 + Zi)

+

4 / 1 1 ( Z 1 ) Z 2 + Z 3 )/23(Z 2 ,Z 3 + Z 1 )

+

/a4(Z 2 , Z 3 -f- Zi) = / 3 4 (Z 3 , Zi + Z 2 ), / i 5 ( Z i , Z 2 + Z 3 ) + 5/ X4 (Zi, Z 2 + Z 3 )/ 2 i (Z 2 , Z 3 + Zx)

+

10/ 1 3 (Z 1 ,Z 2 + Z 3 ) / 2 2 ( Z 2 , Z 3 + Z 1 )

+

10/ 1 2 (Z 1 ,Z 2 + Z 3 )/ 2 3(Z 2 ,Z 3 + Z 1 )

+

5 / u ( Z 1 ( Z 2 + Z 3 ) / 2 4 ( Z 2 , Z 3 + Zi)

+

/25(Z 2 , Z 3 + Zi) = / 3 5 ( Z 3 , Zi + Z 2 ),

(38.13)

Systems

316

of Nonlinear Functional

Equations

where fu, /2!-, fsi • V2 i-» C (1 < i < 5) are arbitrary complex functions, is given by the formulae /n(U,V)

=

F 1 (U + V ) R e ( 2 U - V ) + F 2 (U + V ) I m ( 2 U - V )

+

G;(U + V)

=

JFi(U

+

G 1 (U + V) + G 2 (U + V),

=

i i ( U + V ) R e ( 2 U - V ) + L 2 (U + V ) I m ( 2 U - V )

+

Jffi(U

=

Xi(U + V ) R e ( V - 2 U ) + L 2 ( U + V ) I m ( V - 2 U )

/3i(U,V)

/is(U,V) /32(U,V)

(»= 1,2),

(38.14)

+ V ) R e ( V - 2 U ) + JF2(U + V ) I m ( V - 2 U )

+ V) + / A ( U , V )

(i=l,2),

(38.15)

+ ff1(U + V) + ff2(U + V) + / 3 2 1 (U,V), /i3(U,V)

/3s(U,V)

/,- 4 (U,V)

/34(U,V)

/<8(U,V)

/35(U,V)

=

Mi(U + V ) R e ( 2 U - V ) + M 2 (U + V ) I m ( 2 U - V )

+

/ i i ( U + V) + 3 / ^ ( U , V ) / ! l ( U , V )

-

2/? 1 (U,V)

=

Mi(U + V ) R e ( V - 2 U ) + M 2 (U + V) Irn (V - 2U)

+

7i'1(U + V) + A'2(U + V)

+

3/32(U,V)/31(U,V)-2/331(U,V),

(i=l,2),

(38.16)

=

Pi(U + V) R e ( 2 U - V ) + P 2 (U + V ) I m ( 2 U - V )

+

Q i (U + V) + 4 / i 3 ( U , V ) / l l ( U , V ) - 1 2 / 1 - 2 ( U , V ) / 2 1 ( U , V )

+

3/? 2 (U,V) + 6/A(U,V)

=

Pi(U + V ) R e ( V - 2 U ) + F 2 (U + V ) I m ( V - 2 U )

+

Qi(U + V) + Q 2 (U + V) + 4/ 3 3 (U, V ) / 3 i ( U , V)

-

12/ 3 2 (U, V J / I J U , V) + 3/| 2 (U, V) + 6 & ( U , V),

(t=l,2),

(38.17)

=

i?i(U + V ) R e ( 2 U - V ) + JR2(U + V ) I m ( 2 U - V )

+

5,-(U + V ) + 5 / , - 4 ( U , V ) / j l ( U , V )

-

20/ l 3 (U, V)/A(U, V) + 60/ i 2 (U, V)/?! (U, V)

-

30/? 2 (U,V)/,- 1 (U,V) + 10/,- 3 (U,V)/ ! 2 (U,V)

-

24/? 1 (U,V)

=

i? 1 (U + V ) R e ( V - 2 U ) + i? 2 (U + V ) I m ( V - 2 U )

+

5!(U + V ) + 5 2 ( U + V)

+

(i=l,2),

5/34(U,V)/3i(U,V)-20/33(U,V)/|1(U,V)

(38.18)

Systems

of Higher Order Functional

Equations

+

60/32(U)V)/|1(U,V)-30/322(U,V)/31(U,V)

+

10/33(U,V)/32(U,V)-24/351(U,V),

317

Fit Li, Mi, Pi, Ri : V K> £(V°,C), G{, Hi, Kit Qif St : V ^ C (i = 1,2) are arbitrary complex functions. Proof. The general continuous solution of the first equation of the system (38.13) according to [S. B. Presic and D. Z. Djokovic (1961)] is given by Eqs. (38.14). Now, if we use the first equation of the system (38.13), then the second equation takes on the following form [/ 1 2 (Z 1 ,Z 2 + Z 3 ) - / 1 2 1 ( Z 1 ) Z 2 + Z3)] +

[ ^ ( Z a . Z a + Z i J - Z l ^ Z a . Z a + Zi)]

=

[/3 2 (Z 3 , Z! + Z 2 ) - / I ^ Z a , Zi + Z 2 )].

(38.19)

This equation has a form like the first equation of the system (38.13). If we take into account the previous two equations, then the third equation of the system (38.13) becomes [/i 3 (Zi, Z 2 + Z 3 ) - 3/ 1 2 (Z 1 , Z 2 + Z 3 ) / n ( Z i , Z 2 + Z 3 ) +

2/f 1 (Z 1 , Z 2 + Z 3 )] + [/ 23 (Z 2 , Z 3 + Zi)

-

3/ 2 2 (Z 2 , Z 3 + Z i ) / 2 i ( Z 2 ) Z 3 + Zi) + 2/ 2 3 1 (Z 2 , Z 3 + Zi)]

=

[/ 33 (Z 3 , Zi + Z 2 ) - 3/ 3 2 (Z 3 j Zi + Z 2 ) / 3 i ( Z 3 j Zi + Z 2 )

+

2 / | 1 ( Z 3 , Z 1 + Z 2 )].

(38.20)

The equation (38.20) has also a form like the first equation of the system (38.13). If we apply the above two equations, then the fourth equation of the system (38.13) takes on the form [/i 4 (Z!, Z 2 + Z 3 ) - 4/i 3 (Zi, Z 2 + Z 3 )/ 1 1 (Z 1 , Z 2 + Z 3 ) 3

+

12/ 1 2 (Z 1 ,Z 2 + Z 3 )/ 1 1 (Z 1 ) Z 2 + Z 3 )

-

3f^2(ZuZ2

+

[/ 24 (Z 2l Z 3 + ZO - 4/ 2 3 (Z 2 ) Z 3 + Z i ) / 2 i ( Z 2 , Z 3 + ZO

+

^ ^ ( Z a . Z a + Z i J / l J Z a . Z s + Z!)

-

3/222(Z2> Z 3 + Zi) - 6 / ^ 2 , Z 3 + Zi)]

+ Z 3 ) - 6f?l(Zi, Z 2 + Z 3 )]

(38.21)

Systems

318

of Nonlinear Functional

Equations

=

[/ 34 (Z 3 , Zi + Z 2 ) - 4/ 3 3 (Z 3 , Zi + Z 2 )/ 3 i(Z 3 , Zj + Z 2 )

+

12f32(Z3,Z1

-

3/| 2 (Z 3 , Zj + Z 2 ) - 6 ^ ( Z 3 ) Zi + Z 2 )].

+ Z 2 ) / 2 1 ( Z 3 , Z 1 + Z2)

Thus also the equation (38.21) has a form like the first equation of the system (38.13). Similarly, the last equation of the system (38.13) is reduced to the following equation [/is(Zi, Z 2 + Z 3 ) - 5/ 1 4 (Z 1 , Z 2 + Z 3 ) / n ( Z i , Z 2 + Z 3 ) (38.22) +

20/ 1 3 (Z 1 ,Z 2 + Z 3 ) / 2 1 ( Z 1 , Z 2 + Z 3 )

-

6 0 / ] 2 ( Z 1 , Z 2 + Z 3 )/ 1 3 1 (Z 1 ,Z 2 + Z 3 )

+

30/ ] 2 2 (Z 1 ,Z 2 + Z 3 ) / 1 1 ( Z 1 , Z 2 + Z 3 )

-

10/ 1 3 (Z!, Z 2 + Z 3 ) / i 2 ( Z i , Z 2 + Z 3 ) + 24/ 1 5 1 (Z 1 , Z 2 + Z 3 )]

+

[/2 5 (Z 2 , Z 3 + Zi) - 5/ 2 4 (Z 2 , Z 3 + Z!)/ 2 1 (Z 2 , Z 3 -f Zi)

+

20/ 2 3 (Z 2 ,Z 3 + Z 1 ) / 2 1 ( Z 2 , Z 3 + Z 1 )

-

60/ 2 2 (Z 2 ,Z 3 + Z 1 ) / 2 1 ( Z 2 , Z 3 + Z 1 )

+

3 0 / | 2 ( Z 2 , Z 3 + Z 1 ) / 2 i ( Z 2 , Z 3 + Z1)

-

10/ 2 3 (Z 2 , Z 3 + Z 1 )/ 2 2 (Z 2 , Z 3 + Zi) + 24/ 2 5 1 (Z 2 , Z 3 + Zi)

=

[/ 35 (Z 3 , Zj + Z 2 ) - 5/ 3 4 (Z 3 , Zi + Z 2 )/ 3 1 (Z 3 , Zi + Z 2 )

+

2 0 / 3 3 ( Z 3 , Z i + Z 2 ) / | 1 ( Z 3 , Z 1 + Z2)

-

6 0 / 3 2 ( Z 3 , Z 1 + Z 2 ) / 3 3 1 ( Z 3 , Z i + Z2)

+

3 0 / | 2 ( Z 3 ! Z 1 H - Z 2 ) / 3 1 ( Z 3 , Z 1 + Z2)

-

10/ 3 3 (Z 3 , Zi + Z 2 )/ 3 2 (Z 3 , Zi + Z 2 ) + / ^ ( Z a , Zi + Z 2 )],

which has a form like the first equation of the system (38.13). Thus system (38.13) with five functional equations and fifteen unknown functions /,j : V2 i-> C (i = 1,2,3; j = 1,2,3,4,5) is equivalent to the system formed by the first equation of the system (38.13) and the equations (38.19), (38.20), (38.21) and (38.22), whose general continuous solution is given by Eqs. (38.14), (38.15), (38.16), (38.17) and (38.18). • Such systems have not been taken into account in the references [j. Aczel (1966); Ya. Brodskiy and A. K. Slipenko (1987); L. I. Davidov (1977); M. Ghermanescu (1960); D. S. Mitrinovic and J. E. Pecaric (1991)].

Bibliography

Aczel, J. (1958) "Vorlesungen iiber Funktionalgleichungen I", Math. Nachr. 19, 87. Aczel, J. (1966) "Lectures on Functional Equations and Their Applications", Academic Press, New York-London. Aczel, J., Ghermanescu, M. and Hosszu, M. (1960) "On cyclic equations", Publ. Math. Inst. Hung. Acad. Sci. 5A, 215. Brodskiy, Ya. and Slipenko, A. K. (1987) "Functional Equations", Kiev (in Russian). Carlitz, L. (1963) "A special functional equation", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 97-99, 1. Davidov, L. I. (1977) "Functional Equations", Sofia (in Bulgarian). Djokovic, D. Z. (1961) "Solution d'une equation fonctionnelle cyclique", Bull. Soc. Math. Phys. R. P. Serbie 13, 185. Djokovic, D. Z. (1962) "Sur I'equation fonctionnelle /(i'3,£i + £2) — f(xi,x? + X3) + f(x2,xi) — f(x2,xs) = 0", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 70-76, 7. Djokovic, D. Z. (1964) "Generalization of a result of Aczel, Ghermanescu and Hosszu", Publ. Math. Inst. Hung. Acad. Sci. 9A, 51. Djokovic, D. Z. (1965) "General solution of a functional equation", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 132-142, 55. Djokovic, D. Z. (1965) "A special cyclic functional equation", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 143-155, 45. Djokovic, D. Z., Djordjevic, R. Z. and Vasic, P. M. (1966) "On a class of functional equations", Publ. Inst. Math. Beograd 6, (20), 65. Djordjevic, R. Z. (1965) "Solution d'une equation fonctionnelle a plusieurs fonctions inconnues", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 132-142, 51. Djordjevic, R. Z. (1965) "Solution d'une systeme d'equations fonctionnelles lineaires", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 1 4 3 319

320

Bibliography

155, 61. Djordjevic, R. 2. and Vasic, P. M. (1967) "O jednoj klasi funkcionalnih jednacina", Mat. Vesnik 4, (19), 33. Eichhorn, W. (1963) "Losung einer Klasse von Funktionalgleichungssystemen", Arch. Math. 14, 266. Frechet, M. (1909) "Une definition fonctionnelles des polynomes", Nouv. Ann. Math. 9, (4), 145. Gheorghiu, O. E. (1963) "Sur une equation fonctionelle matricielle", C. R. Acad. Sci. Paris 256, 3562. Ghermanescu, M. (1940) "Sur quelques equations fonctionnelles lineaires", Bull. Soc. Math. France 68, 109. Ghermanescu, M. (1953) "Un sistem de equa^ii funct;ionale", Acad. R. P. Romdne. Bui. Sti. Sec(. Sti. Mat. Fiz. 5, 575. Ghermanescu, M. (1955) "Sisteme de Ecua^ii Funct;ionale Liniare de Primul Ordin", Bucures.ti. Ghermanescu, M. (1960) "Ecuatii Funct;ionale", Editura Academiei Republicii Populare Romane, Bucureijti. Hosszii, M. (1961) "A linearis fiiggvenyegyenletek egy osztalyarol", Magyar Tud. Akad. Mat. Fiz. Oszt. Kozl. 11, 249. Hosszii, M. (1963) "On a class of functional equations", Publ. Inst. Math. Beograd 3, (17), 53. Kuczma, M. (1968) "Functional Equations in a Single Variable", Monografie Matematyczne 46, Paristwowe Wydawnictwo Naukowe, Warsaw. Kurepa, S. (1956) "On some functional equations", Glasnik Mat. Fiz. Astr. 11, 3. Kuwagaki, A. (1962) "Sur l'equation fonctionelle de Cauchy pour les matrices", J. Math. Soc. Japan 14, 359. MacLane, S. (1963) "Homology", Die Grundlehren der mathematischen Wissenschaften 114, Academic Press, Inc., Publishers, New York, SpringerVerlag, Berlin-Gottingen-Heidelberg. Mitrinovic, D. S. (1963) "Equation fonctionelle a fonctions inconnues dont toutes ne dependent pas du meme nombre d'arguments", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 115-121, 29. Mitrinovic, D. S. (1963) "Equations fonctionelles cycliques generalisees", C. R. Acad. Set. Paris 257, 2951. Mitrinovic, D. S. (1963) "Equations fonctionelles lineaires paracycliques de premiere espece", Publ. Inst. Math. Beograd 3, (17), 115. Mitrinovic, D. S. (with the collaboration of Vasic, P. M.) (1986) "Diferencijalne Jednacine. Zbornik Zadataka i Problema", Naucna Knjiga, Beograd. Mitrinovic, D. S. and Djokovic, D. 2. (1961) "Sur quelques equations fonctionnelles", Publ. Inst. Math. Beograd 1, (15), 67. Mitrinovic, D. S. and Djokovic, D. 2. (1963) "Neki nereseni problemi u teoriji funkcionalnih jednacina", Mat. Bibliot. 25, 153. Mitrinovic, D. S. and Pecaric, J. E. (1991) "Ciklicne Nejednakosti i Ciklicne Funkcionalne Jednacine", Naucna Knjiga, Beograd.

Bibliography

321

Mitrinovic, D. S. and Presic, S. B. (1962) "Sur Une Equation Fonctionnelle Cyclique Non Lineaire", C. R. Acad. Sci. Paris 254, 611. Mitrinovic, D. S. and Presic, S. B. (1962) "Sur Une Equation Fonctionnelle Cyclique d'Ordre Superieur", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 70-76, 1. Mitrinovic, D. S., Presic, S. B. and Vasic, P. M. (1963) "Sur Deux Equations Fonctionnelles Cycliques Non Lineaires", Bull. Soc. Math. Phys. R. P. Serbie 15, 3. Presic, S. B. and Djokovic, D. Z. (1961) "Sur une equation fonctionnelle", Bull. Soc. Math. Phys. R. P. Serbie 13, 149. Risteski, I. B. (to appear) "Solution of a class of complex vector linear functional equations", Missouri J. Math. Sci. Risteski, I. B. (to appear) "Matrix method for solving of linear complex vector functional equations", Internat. J. of Math, and Math. Sci. Risteski, I. B. (submitted) "Some higher order complex vector functional equations" . Risteski, I. B. and Covachev, V. C. (2000) "On some general classes of partial linear complex vector functional equations", Sci. Univ. Tokyo J. Math. 36, 105. Risteski, I. B. and Covachev, V. C. (submitted) "On some quadratic vector functional equations". Risteski, I. B. and Covachev, V. C. (submitted) "Some modified quadratic complex vector functional equations". Risteski, I. B. and Covachev, V. C. (submitted) "Some expanded quadratic complex vector functional equations". Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (1999) "A simple functional operator", New York J. Math. 5, 139. Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (2000) "Cyclic complex vector functional equations", Appl. Sci. 2, 13. Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (2000) "On a linear vector functional equation", in: Some Problems of Applied Mathematics, Eds. A. Ashyralyev, H. A. Yurtsever, Fatih University Publications, Istanbul, 174. Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (2001) "An expanded class of parametric linear complex vector functional equations", Balkan J. Geom. Appl. 6. Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (2001) "Systems of Functional Equations", in: Applications of Mathematics in Engineering and Economics'26, Eds. B. I. Cheshankov, M. D. Todorov, Heron Press, Sofia, 113. Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (to appear) "On some simple complex vector functional equations", / . of Univ. of Bra$ov. Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (to appear) "On a class of parametric partial linear complex vector functional equations", An. Univ. Craiova. Ser. Mat. Informat.

322

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Risteski, I. B., Trencevski, K. G. and Covachev, V. C. (submitted) "Complex vector systems of partial functional equations". Trencevski, K. G., Risteski, I. B. and Covachev, V. C. (1999) "On two complex functional equations of Frechet's type", in: Biomathematics, Bioinformatics & Applications of Functional, Differential, Difference Equations, Eds. Haydar Akca, Valery Covachev &; Elena Litsyn, Akdeniz Univ., 180. Valiron, G. (1945) "Equations Fonctionelles—Applications", Paris. Vasic, P. M. (1963) "Sur un systeme d'equations fonctionnelles", Glasnik Mat. Fiz. Astr. 18, 229. Vasic, P. M. (1963) "Equation fonctionnelle d'un certain type de determinants", C. R. Acad. Sci. Paris 256, 1898. Vasic, P. M. (1965) "Sur une classe d'equations fonctionelles lineaires", C. R. Acad. Sci. Paris 260, 5666. Vasic, P. M. (1967) "Resolution d'une classe d'equations fonctionelles lineaires", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 181-196, 53. Vasic, P. M. and Djordjevic, R. Z. (1965) "Sur I'equation fonctionelle cyclique generalisee", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 1 3 2 142, 33. Vasic, P. M., Janic, R. R. and Djordjevic, R. Z. (1965) "Sur les equations fonctionelles lineaires paracycliques de premiere espece", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 132-142, 39.

Index

special cyclic FEs, vii, ix, 3, 69 expanded parametric FE, viii, ix, 121, 142 FEs with constant parameters, viii, ix, 121 FEs with distinct functions, viii FEs with operations between arguments, vii, ix, 75 FEs with several unknown functions, ix, 99 FEs with two operations, viii, ix, 111 Frechet's FEs, viii, ix, 106, 109, 110 general expanded parametric FE, viii, ix, 121, 161 general parametric FE, viii, ix, 121 generalized equation of operator type, vii generalized FE, ix, 81 higher order FEs, viii, x, 265 with complex parameters, x, 265, 267 without parameters, x, 265 linear FEs, vii, ix, 1 (linear) FEs with constant coefficients, viii, x, 173 homogeneous, viii, x, 173, 175, 192, 195

cyclic matrix (matrices), 176, 178, 187, 190, 191, 192, 198 cyclic operator, 21, 25 cyclic permutation, 108, 177, 178, 182, 183, 191, 198, 200, 268, 271, 273 equation binomial, 200, 201 characteristic, 201 matrix, x, 173 reduced, 201 scalar, 225, 233, 234, 236, 238, 248 transformed, 221, 224, 226 FE = (complex vector) functional equation Cauchy (matrix) FE, 78 cyclic FEs, vii, ix, 3 basic cyclic FEs, vii, ix, 3, 5, 20, 25, 175, 190 condensed cyclic FEs, vii, ix, 3, 72 derived cyclic FEs, vii, ix, 3, 6, 8, 10, 19, 21 paracyclic FEs, viii, ix, x, 3, 29, 197 semicyclic FEs, vii, ix, 3, 65, 66

323

324

nonhomogeneous, viii, x, 173, 190, 192, 196 nonlinear FE(s), vii, x, 217 nonlinear operator FE, x, 273 operator FE, vii, ix, 75, 78, 80 quadratic FEs, viii, x, 219 basic quadratic FEs, x, 231 expanded quadratic FEs, viii, x, 247 with alternating sign between the functions, x, 247, 258, 259 with functional arguments, x, 247 with the same sign between the functions, x, 247, 251 generalized (expanded) quadratic FE, x, 247, 262 modified quadratic FEs, viii, x, 231, 238, 239, 241, 244 permuted quadratic FE, x, 231, 234 special (real) quadratic FE, x, 219, 227, 229, 230 special parametric FE, viii, ix, 121, 135 systems of (linear) FEs, viii, x, 205, 208, 210, 211, 213, 214, 215, 216

Index systems of (nonlinear) FEs, viii, xi, 285, 293, 303, 311, 313, 314, 315, 318 quadratic, viii, xi, 285 higher order, viii, xi, 285, 311 homogeneous system of equations, 175 Laplace rule, 280, 310 linear homogeneous equations, 272 matrix approach (method), viii, 173 Pexider form, 92 solution anisotropic, 287, 288, 296, 297 isotropic, 286, 287, 293, 294, 296 nontrivial, 176, 178, 225, 227, 228, 229, 230, 232, 235, 236, 244, 248, 250, 252, 255, 268, 269, 277, 278, 288, 306 reproductive, 180, 181, 186, 187, 188, 189 trivial, 225, 227, 230, 232, 235, 248, 250, 252, 255, 270, 272, 277

COMPLEX VECTOR FUNCTIONAL EQUATIONS by Ice Risteski & Valery Covachev (Bulgarian Academy of Sciences) The subject of complex vector functional equations is a new area in the theory of functional equations. This monograph provides a systematic overview of the authors' recently obtained results concerning both linear and nonlinear complex vector functional equations, in all aspects of their utilization. It is intended for mathematicians, physicists and engineers who use functional equations in their investigations.

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