Central European Journal of Mathematics

DOI: 10.2478/s11533-006-0023-8 Research article CEJM 4(4) 2006 547–561

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field M. Abad1∗ , J.P. D´ıaz Varela1 , B.F. L´opez Martinolich2 , M. del C. Vannicola2 , M. Zander1 1

Departamento de Matem´ atica, Universidad Nacional del Sur, 8000 Bah´ıa Blanca, Argentina

2

Departamento de Matem´ atica, Universidad Nacional del Comahue, 8300 Neuqu´en, Argentina

Received 18 April 2005; accepted 4 June 2006 Abstract: In this paper we give a term equivalence between the simple k−cyclic Post algebra of order p, Lp,k , and the finite field F (pk ) with constants F (p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ1 of the variety V(Lp,k ) generated by Lp,k into the variety V(F (pk )) generated by F (pk ) and an interpretation Φ2 of V(F (pk )) into V(Lp,k ) such that Φ2 Φ1 (B) = B for every B ∈ V(Lp,k ) and Φ1 Φ2 (R) = R for every R ∈ V(F (pk )). c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Varieties, Interpretations, equivalence, finite fields, post algebras MSC (2000): 06D25, 12E20, 03G25

1

Introduction

The original motivation behind the research presented in this paper came from the works of Moisil [12, 13] and Cendra [5]. In those papers, Moisil established relationships between some finite fields F (2k ) and some cyclic Boolean algebras, but it was Cendra in [5] who solved the problem of giving a constructive method to define a simple k−cyclic Boolean algebra A; T  on a given finite field F (2k ), k ≥ 1, and conversely. In this paper we solve a similar problem for any finite field F (pk ); +, ·, F (p) and k−cyclic Post algebras of order p, p prime, k ≥ 1. In fact, we describe a constructive method to transform ∗

E-mail: [email protected]

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a field F (pk ); +, ·, F (p) into a k−cyclic Post algebra of order p, p prime, k ≥ 1, we express the Post operations as terms in the language of fields, and conversely, the field operations as terms in the language of cyclic Post algebras. Notice that we have written F (pk ); +, ·, F (p) to denote a field with pk elements to emphasize that we are considering the elements of the prime field F (p) as constants (nullary operations), and consequently a term in the language of fields is a polynomial with coefficients in F (p). As it is standard practice, in this paper we will frequently use the same symbol to denote an algebra and its underlying universe. Let V(Lp,k ) denote the variety generated by the k−cyclic simple Post algebra of order p Lp,k , and let V(F (pk )) represent the variety generated by the field F (pk ); +, ·, F (p). An algebra A; F  is said to be functionally complete if any function f : An → A can be represented as a polynomial with coefficients in A. If two algebras A; F  and B; F   have the same universe and the same clone of n-ary term operations, that is, Clon (A) = Clon (B) for all positive integer n, we say that A and B are term equivalent. Examples of functionally complete algebras are the Galois fields F (pk ) (see [7]). Since the variety V(F (pk )) is arithmetical and the algebras Lp,k are simple, by [4, Th. 11.12, Ch. IV], they are functionally complete. Then the algebras F (pk ) and Lp,k are polynomially equivalent, as they are two functionally complete algebras with the same (finite) number of elements. In this paper we prove that the algebras F (pk ) and Lp,k are not only polynomially equivalent, but they are term equivalent. In fact, we prove that there exists an interpretation Φ1 of V(Lp,k ) in V(F (pk )) and an interpretation Φ2 of V(F (pk )) in V(Lp,k ) such that Φ2 Φ1 (B) = B for every B ∈ V(Lp,k ) and Φ1 Φ2 (R) = R for every R ∈ V(F (pk )). Section 2 collects the necessary facts about varieties, finite fields and cyclic Post algebras. Most of the material is folklore. In Section 3 we introduce the Lagrange polynomials on the field F (pk ) and also the Lagrange polynomials on the algebra Lp,k , the basic fact being that by means of such polynomials we can express any function on the field in terms of the Post operations, and any function on the algebra in terms of the field operations. Section 4 is devoted to prove the principal theorem of the paper, namely Theorem 4.1, which gives us the interpretations stated above. Finally, in Section 5, we illustrate the whole process with an example. It is a pleasure for us to thank W. Reartes for his assistance with software applications. The example given at the end of the paper would not have been possible without his invaluable help.

2

Definitions and Preliminaries

In order to keep this paper as self-contained as possible we give some definitions, properties and simple proofs about varieties, finite fields and cyclic Post algebras which can be found in the literature.

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For background material on varieties, the reader is referred to Burris and Sankappanavar [4]. The following definition of a variety is but one of several equivalent formulations. Given a class K of similar algebras, the class of all subalgebras, homomorphic images and direct products is denoted respectively by S(K), H(K) and P (K). If K ⊇ H(K) ∪ S(K) ∪ P (K), then K is a variety. The variety generated by K (the least variety containing K) is denoted V(K). In general, V(K) = HSP (K). If K is finite, say K = {A1 , . . . , An }, then V(K) will be denoted V(A1 , . . . , An ). We say that a variety V is interpretable [9, 10] in a variety W if for each V−operation Ft (x1 , . . . , xn ) there is a W−term ft (x1 , . . . , xn ) such that if A; Gt  is in W, then A; ft  is in V. Notice that the constants in the language of V must be interpreted as constants in the language of W. Intuitively, this means that each algebra in W can be turned into an algebra in V by defining the V−operations applying a uniform procedure. This notion can also be approached in the following way: there exists a functor Φ : W → V which commutes with the underlying set functors, that is, Φ W. ............................................... .V ...... .... ...... . . . . . . UW ........ UV .... ....... Sets is commutative. UV and UW are the forgetful functors which assign to each algebra its universe. Each functor Φ is called an interpretation of W in V. If A; Gt  is any algebra and for each V−operation Ft (x1 , . . . , xn ) there is a term ft (x1 , . . . , xn ) in the language of A; Gt  such that A; ft  is in V, the terms ft (x1 , . . . , xn ) define an interpretation of V in V(A; Gt ), the variety generated by the algebra A; Gt . One only has to observe that the evaluation of any term in an algebra B in V(A; Gt ), is determined by its evaluation in A and that both A; Gt  and B; Gt  satisfy the same Φ equations. We have V(A; Gt ) → V, and we say that Φ(A; Gt ) is an interpretation of V in V(A; Gt ). By an equivalence [10] of the varieties V and W is meant a pair of interpretations Φ1 of V in W and Φ2 of W in V such that Φ2 Φ1 = IdV and Φ1 Φ2 = IdW. Observe that in this sense, two varieties are equivalent iff they are term equivalent. Let F be an extension of a field K. An automorphism σ : F −→ F is said to be a K−automorphism if σ|`K = IdK . The group of all K−automorphisms of F is called the Galois group of F over K and it is denoted by AutK F . If H is a subgroup of AutK F , the set {x ∈ F : σ(x) = x, σ ∈ H} is called the fixed field of H. If F is an extension of a field K such that the fixed field of the Galois group AutK F is K, then F is called a Galois extension of K.

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Theorem 2.1. (The Normal Basis Theorem) Let F be a finite Galois extension of the field K, of degree n. Let G = {σ1 , σ2 , . . . , σn } be the Galois group of F . Then there exists an element w ∈ F such that {σ1 (w), σ2 (w), . . . , σn (w)} forms a basis of the K− vector space F . Every finite field F has pn elements, for some n ≥ 1, where p is the characteristic of F . Given a prime p and an integer n ≥ 1, there exists only one field F , up to isomorphism, with pn elements. The intersection F  of all subfields of F is called the prime field of F . If F has characteristic p, then F  ∼ = Zp . If F = F (pn ) is a finite field with pn elements, the multiplicative group F \ {0} = n {ε, ε2 , . . . , εp −1 = 1} is cyclic. The mapping σ : F (pn ) −→ F (pn ) defined by σ(x) = xp is a field automorphism, which is called the Frobenius automorphism. The group of automorphisms of F (pn ) is cyclic of order n and σ is a generator. As an application to the theory of switching circuits, G. Moisil introduced in 1954 [12] the symmetric Boolean algebras, that is, Boolean algebras with an automorphism of period 2, and later in 1965 [13], the cyclic Boolean algebras, in which the automorphism of period 2 is replaced by an automorphism of period k, for a given k. These algebras were also studied by A. Monteiro in 1978 [11]. As a natural generalization, M. Abad investigated in [1] the cyclic Post algebras of order n. Since key results concerning cyclic Post algebras are not widely known, we provide these for completeness. All the needed proofs for Post algebras can be found in Balbes and Dwinger [2] or in Boicescu et al. [3]. The following definition is equivalent to that given by Epstein [6]. Definition 2.2. A Post algebra of order n, n an integer ≥ 2, is a system A; ∧, ∨, 0, 1, ∼, n−2 {Ci }n−1 i=0 , {ei }i=1 , such that A; ∧, ∨, 0, 1 is a distributive lattice with 0 and 1, ∼, Ci are unary operations and ei are nullary operations fulfilling: (1) ∼∼ x = x, (2) ∼ (x ∧ y) = ∼ x∨ ∼ y, ∼ (x ∨ y) = ∼ x∧ ∼ y,  (3) Ci (x) ∧ Cj (x) = 0 for i = j, and n−1 i=0 Ci (x) = 1, (4) 0 = e0 ≤ e1 ≤ . . . ≤ en−1 = 1, (5) if x ∧ e1 = 0 then x = 0, (6) if x ∨ ei−1 = ei then x = ei ,  (7) for every x ∈ A, x = n−1 i=0 Ci (x) ∧ ei . It is known that if B(A) is the Boolean algebra of complemented elements of A, then x ∈ B if and only if x = Ci (y) for some i and some y ∈ A. The Ci (x) are unique, that is, for any given x ∈ A there is only one sequence of elements C0 (x), C1 (x), . . . , Cn−1 (x) satisfying (3) and (7) of Definition 2.2. The following example will play an important role. Let Ln be the set of all fractions

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j/(n − 1), with j = 0, 1, . . . , n − 1, considered as a sublattice of the real numbers, with ∼ (j/(n − 1)) = 1 − j/(n − 1) and Ci (j/(n − 1)) = 1 if j = i and Ci (j/(n − 1)) = 0 otherwise. Then Ln is a Post algebra of order n, where ej = j/(n − 1), 0 ≤ j ≤ n − 1. The variety of Post algebras of order n is a discriminator variety (see [4]), generated by the n-element chain. In particular, any finite Post algebra of order n is isomorphic to a direct product of copies of the n-element chain. Definition 2.3. A k−cyclic Post algebra of order n (n ≥ 2, k ≥ 1, n, k fixed) is a pair A; T , where A is a Post algebra of order n and T is a Postian automorphism of A such that T k (x) = x for every x ∈ A. Since the class of Post algebras of order n is equationally definable [3], it follows that the class of k−cyclic Post algebras of order n is also equationally definable. The set (Ln )k of all sequences x = (x1 , x2 , . . . , xk ), with xi ∈ Ln and with the pointwise defined operations, is also a Post algebra of order n. Now, if we put T (x1 , x2 , . . . , xk ) = (xk , x1 , x2 , . . . , xk−1 ), then T is an automorphism of (Ln )k such that T k (x) = x for every x ∈ (Ln )k , and then (Ln )k ; T  is a k−cyclic Post algebra of order n. We will denote by Ln,k the algebra (Ln )k ; T , where T is the automorphism described above. The simple algebras of the variety of all k−cyclic Post algebras of order n are the algebras Ln,d , with d a divisor of k [1].

3

Lagrange polynomials

The aim of this section is to obtain an effective representation of any function on a finite field F as a polynomial with coefficients in F , and an effective representation of any function on a finite cyclic Post algebra A as a (Post) polynomial with coefficients in A. Given a set with pk elements and a structure of field F = F (pk ) defined on it, let m F F ; +, · be the ring of functions f : F m → F where the operations + and · are defined by (f + g)(x1 , x2 , . . . , xm ) = f (x1 , x2 , . . . , xm ) + g(x1 , x2 , . . . , xm ), (f · g)(x1 , x2 , . . . , xm ) = f (x1 , x2 , . . . , xm ) · g(x1 , x2 , . . . , xm ). Let F [x1 , x2 , . . . , xm ]; +, · be the ring of all polynomials in m variables over the field F. m

F Lemma can be uniquely represented as a polynomial of the  3.1. Every function f ∈ F r r r form λi ·Mi , with i = (ri1 , ri2 , . . . , rim ) ∈ ([0, pk ))m , λi ∈ F and Mi = x1i1 ·x2i2 ·. . .·xmim . i

Proof. Every polynomial defines a function in a natural way [8]. So we have a mapping m ψ : F [x1 , x2 , . . . , xm ] → F F . This function ψ is one-to-one when restricted to the  r r r polynomials of the form p(x1 , x2 , . . . , xm ) = i λi ·Mi , λi ∈ F and Mi = x1i1 ·x2i2 ·. . .·xmim

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r

r

with rij < pk . Since there are pkm monomials Mi = x1i1 · x2i2 · . . . · xmim with 0 ≤ rij < pk , pkm   k pkm there are p polynomials of the form λi · Mi , and this is the number of functions i=1

m

f ∈ FF .

An effective representation for each function as a polynomial can be obtained by k considering the following Lagrange polynomials in F (pk ) = {0, ε, ε2 , . . . , εp −1 }. L0 (x) = (p − 1)xp

k −1

+ 1,

Lεi (x) = L0 (x + (p − 1)εi ). k

From the identity xp + (p − 1)x = 0, it follows that ⎧ ⎧ ⎪ ⎪ ⎨ 1 if x = 0 ⎨ 1 if x = εi . L0 (x) = and Lεi (x) = ⎪ ⎪ ⎩ 0 if x = 0 ⎩ 0 if x = εi m

So, for f ∈ F F ,



f (x1 , . . . , xm ) =

(α1 ,...,αm

f (α1 , · · · , αm ) · Lα1 (x1 ) · . . . · Lαm (xm ).

(I)

)∈F m

We now turn our attention to cyclic Post algebras. Consider again the simple k−cyclic Post algebra of order p, Lp,k = (Lp )k ; T . We are going to define the following operations ⊕ and ⊗ on (Lp )k . 

Ci (x ⊕ y) =

(Cs (x) ∧ Ct (y)),

s+t≡i mod (p)



Ci (x ⊗ y) =

(Cs (x) ∧ Ct (y)).

s·t≡−i mod (p)

Consequently, from (5) of Definition 2.2, 

p−1

x⊕y =

i=0



p−1

Ci (x ⊕ y) ∧ ei

and x ⊗ y =

Ci (x ⊗ y) ∧ ei .

i=0

The following results follow the Serfati’s paper [15] which studied some p−rings called Postian rings. Instead of s · t ≡ i(mod p) given in [15], the present paper proposes the change s · t ≡ −i(mod p). In the proof of the following lemma, recall that in (Lp )k , Cj (ei ) = 0 for i = j, and Cj (ei ) = 1 when i = j. Lemma 3.2. ei ⊕ ej = ek if and only if i + j ≡ k(mod p) and ei ⊗ ej = ek if and only if i · j ≡ −k(mod p).

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Proof. Suppose that ei ⊕ ej = ek .  Then Ck (ei ⊕ ej ) = Ck (ek ) = 1. Thus r+l≡k(mod p) (Cr (ei ) ∧ Cl (ej )) = 1, and then there exist r, l with r + l ≡ k(mod p) such that Cr (ei ) = 1 = Cl (ej ). We have that r = i and l = j, and so i + j ≡ k(mod p). Conversely, suppose that i + j ≡ k(mod p).  Then Ck (ei ⊕ ej ) = r+l≡k (mod p) (Cr (ei ) ∧ Cl (ej )) = Ci (ei ) ∧ Cj (ej ) = 1 = Ck (ek ).  For t = k, since 1 ≤ t ≤ p − 1, t ≡ i + j. So Ct (ei ⊕ ej ) = r+l≡t (mod p) (Cr (ei ) ∧ Cl (ej )) = 0 = Ct (ek ). The proof of the second part of the lemma is similar. 

Proposition 3.3. (Lp )k ; ⊕, ⊗ is a commutative ring with unit ep−1 = 1.

Proof. It is enough to prove that Lp ; ⊕, ⊗ is a commutative ring with unit ep−1 = 1. Let us see that the operation ⊕ is associative. For 0 ≤ i, j, k ≤ p − 1 we have ei ⊕ (ej ⊕ ek ) = ei ⊕ et = el if and only if j + k ≡ t(mod p) and i + t ≡ l(mod p). Then i + j + k ≡ l(mod p), that is, i + j ≡ l − k(mod p), so (ei ⊕ ej ) ⊕ ek = el−k ⊕ ek = el . Thus ei ⊕ (ej ⊕ ek ) = (ei ⊕ ej ) ⊕ ek . Similarly we can prove that ⊗ is associative. It is clear that ⊕ and ⊗ are commutative. For the distributive law, suppose that ei ⊗ ej = el , ei ⊗ ek = et and el ⊕ et = er . Then i · (j + k) ≡ i · j + i · k ≡ −(l + t) ≡ −r(mod p) and ei ⊗ (ej ⊕ ek ) = er . Then ei ⊗ (ej ⊕ ek ) = (ei ⊗ ej ) ⊕ (ei ⊗ ek ). Let us see that ep−1 is a unit. But ei ⊗ ep−1 = ej if and only if i · (p − 1) ≡ −j(mod p), and this is equivalent to i ≡ j(mod p). As 0 ≤ i, j ≤ p − 1, we have that j = i. In a similar way it can be proved that e0 is a zero element, and the negative of ei is ej with j ≡ −i(mod p).  The following properties can be easily seen, where px = x  . . . ⊕ x and  ⊕ x ⊕ xp = x  ⊗x⊗ . . . ⊗ x.

p times

p times

Lemma 3.4. For x ∈ (Lp )k , px = 0, xp = x, ∼ x = 1 ⊕ (p − 1)x and, over the chains of constants, ⎧ ⎪ ⎨0 if i = 0 p−1 . ei = ⎪ ⎩ ep−1 = 1 if i = 0 On the set Fm ((Lp )k ) of all functions f : [(Lp )k ]m → (Lp )k we can define a structure of ring by means of the following operations: (f ⊕ g)(x1 , x2 , . . . , xm ) = f (x1 , x2 , . . . , xm ) ⊕ g(x1 , x2 , . . . , xm ), (f ⊗ g)(x1 , x2 , . . . , xm ) = f (x1 , x2 , . . . , xm ) ⊗ g(x1 , x2 , . . . , xm ).

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Let (Lp )k [x1 , x2 , . . . , xm ]; ⊕ , ⊗ be the ring of all polynomials in m variables over the algebra (Lp )k , that is, expressions of the form  μi ⊗ Ni (x1 , x2 , . . . , xm ), i

where μi ∈ (Lp )k and Ni = 1 or Ni is a formal expression of the form (x1 )α1,1 ⊗ T (x1 )α1,2 ⊗ . . . ⊗ T k−1 (x1 )α1,k ⊗ (x2 )α2,1 ⊗ T (x2 )α2,2 ⊗ . . . ⊗ T k−1 (x2 )α2,k ⊗ . . . . . . ⊗ (xm )αm,1 ⊗ T (xm )αm,2 ⊗ . . . ⊗ T k−1 (xm )αm,k , where αi,j ∈ {0, 1, 2, p − 1}. Lemma 3.5. Every function f : [(Lp )k ]m → (Lp )k can be uniquely represented as a  k polynomial of the form i μi ⊗ Ni (x1 , x2 , . . . , xm ) where μi ∈ (Lp ) and Ni = 1 or Ni has the form (x1 )α1,1 ⊗ T (x1 )α1,2 ⊗ . . . ⊗ T k−1 (x1 )α1,k ⊗ (x2 )α2,1 ⊗ T (x2 )α2,2 ⊗ . . . ⊗ T k−1 (x2 )α2,k ⊗ . . . . . . ⊗ (xm )αm,1 ⊗ T (xm )αm,2 ⊗ . . . ⊗ T k−1 (xm )αm,k , where αi,j ∈ {0, 1, 2, p − 1}.  Proof. It is clear that the number of polynomials of the form i μi ⊗Ni (x1 , x2 , . . . , xm ), k k pkm with μi ∈ (Lp ) is (p ) , and this is the number of functions f : [(Lp )k ]m −→ (Lp )k . Consider now the following Lagrange polynomials La (x) ∈ (Lp )k [x]. L0 (x) = ∼ xp−1 ⊗ ∼ T (xp−1 ) ⊗ . . . ⊗ ∼ T k−1 (xp−1 ), La (x) = L0 (x ⊕ (p − 1)a). These polynomials satisfy ⎧ ⎪ ⎨ 1 if x = 0 L0 (x) = ⎪ ⎩ 0 if x = 0

and La (x) =

⎧ ⎪ ⎨ 1 if x = a ⎪ ⎩ 0 if x = a

.

This follows from the fact that xp−1 ∈ B((Lp )k ), and consequently, the operations ∼, T and ⊗ coincide with the ones given in [5] for the case of cyclic Boolean algebras. Then

f (x1 , . . . , xm ) =



f (α1 , . . . , αm ) ⊗ Lα1 (x1 ) ⊗ · · · ⊗ Lαm (xm ).

(α1 ,...,αm )∈Am

So the natural function τ : (Lp )k [x1 , . . . , xm ] → Fm ((Lp )k )

(II)

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is onto. Consequently, the representation of f ∈ Fm ((Lp )k ) given by  f (x1 , x2 , . . . , xm ) = μi ⊗ Ni (x1 , x2 , . . . , xm ) i



is unique.

4

Interpretations

The aim of this section is to give the announced interpretation Φ1 of the variety V(Lp,k ) in the variety V(F (pk )), and an interpretation Φ2 of the variety V(F (pk )) in V(Lp,k ). In the following theorem we obtain a structure of k-cyclic Post algebra of order p isomorphic to Lp,k from the field F (pk ), and conversely. Theorem 4.1. Given a finite field F (pk ) with pk elements, there exists a structure of simple k-cyclic Post algebra of order p defined on F (pk ) isomorphic to Lp,k = (Lp )k ; ∧, ∨, 0, 1, p−2 ∼, {Ci }p−1 i=0 , {ei }i=1 , T , such that 1. The constants ei are the elements of the prime field F (p). 2. The operations ∧ and ∨ are polynomials in F (p)[x1 , x2 ] of the form p2k ri1 ri2 i=1 λi · x1 · x2 , λi ∈ F (p), and the operations ∼, Ci and T are polynomials in k F (p)[x] of the form pi=1 λi ·xri , λi ∈ F (p), uniquely determined under the conditions rij < pk and ri < pk . 3. The operations + and · are uniquely determined polynomials in Lp [x1 , x2 ] of the form  i μi ⊗ Ni (x1 , x2 ) where μi ∈ {e0 , . . . , ep−1 } and Ni = 1 or Ni is a product (with ⊗ as product) of elements of the set {x1 , T (x1 ), . . . , T k−1 (x1 ), x2 , T (x2 ), . . . , T k−1 (x2 )}. Proof. Let us introduce an ordering on the prime field F (p) = {0, 1, . . . , p − 1} by 0 < p − 1 < p − 2 < . . . < 2 < 1. By defining e0 = 0 and ei = p − i, 1 ≤ i ≤ p − 1, ∼ ei = ep−i−1 , 0 ≤ i ≤ p − 1, and Ci (ej ) = 1 if j = i and Ci (ej ) = 0 otherwise , 0 ≤ i, j ≤ p − 1 the prime field becomes a Post algebra Lp . From now on, the proof follows closely the work of Cendra [5]. By Theorem 2.1, there exists w ∈ F (pk ) such that {w, σ(w), . . . , σ k−1 (w)} is a basis of the vector space F (pk ) over F (p). So given x ∈ F (pk ), we can write x=

k−1  i=0

λi (x)σ i (w),

λi (x) ∈ F (p).

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Thus x can be written x = λ0 (x), λ1 (x), . . . , λk−1 (x). As σ(λ0 (x), λ1 (x), . . . , λk−1 (x)) = λk−1 (x), λ0 (x), . . . , λk−2 (x), σ acts transitively on the coordinates λi (x), and consequently the only elements of F (pk ) fixed by the action of σ are 0, 0, . . . , 0, 1, 1, . . . , 1, . . . , p − 1, p − 1, . . . , p − 1. On the other hand, since F (p) is the fixed field of F (pk ), we have that for x ∈ F (p), x = λ(x), λ(x), . . . , λ(x) with λ(x) ∈ F (p). Consider the Post algebra A = (Lp )k . Observe that the Post algebra Lp and the field F (p) are equal as sets, as well as the Post algebra A and the field F (pk ). The constants of A are the k-tuples 0 = e0 = 0, 0, . . . , 0, 1 = ep−1 = 1, 1, . . . , 1 and ei = p − i, p − i, . . . , p − i with 1 ≤ i ≤ p − 2. 0, the first element of A, is the 0 of the field, 0 = 0, 0, . . . , 0. Define T : A → A by T (x) = σ(x) = xp . Then T (x ∧ y) = T (λ0 (x), λ1 (x), . . . , λk−1 (x) ∧ λ0 (y), λ1 (y), . . . , λk−1 (y)) = T (λ0 (x) ∧ λ0 (y), λ1 (x) ∧ λ1 (y), . . . , λk−1 (x) ∧ λk−1 (y)) = λk−1 (x) ∧ λk−1 (y), λ0 (x) ∧ λ0 (y), . . . , λk−2 (x) ∧ λk−2 (y) = λk−1 (x), λ0 (x), . . . , λk−2 (x) ∧ λk−1 (y), λ0 (y), . . . , λk−2 (y) = T (x) ∧ T (y). Arguing in a similar way with the operations ∨, ∼, {Ci }p−1 i=0 , and taking into account that T is bijective we have that T is an automorphism of Post algebras. Since it is clear that T k (x) = x, it follows that A; T  is a k−cyclic simple Post algebra. Now we prove that ∧, ∨, ∼, {Ci }p−1 i=0 and T are polynomials in F (p)[x1 , x2 , . . . , xm ]. We know that every function f : F m −→ F has a unique expression of the form km

f (x1 , x2 , . . . , xm ) =

p 

ηi Mi , ηi ∈ F (pk ),

i=1 r

r

where Mi = Mi (x1 , x2 , . . . , xm ) = x1i1 · . . . · xmim , 0 ≤ rij < pk , i = 1, 2, . . . , pkm . If x ∈ F (pk ), σ k (x) = xpk = x, then for each i ∈ {1, 2, . . . , pkm }, there exists j ∈ {1, 2, . . . , pkm } such that σ(Mi (x1 , x2 , . . . , xm )) = Mi (σ(x1 ), σ(x2 ), . . . , σ(xm )) = Mj (x1 , x2 , . . . , xm ), that is σ permutes the Mi . Since σ = T is a Post automorphism and also a field automorphism, it follows that f (σ(x ), σ(x2 ), . . . , σ(xm )) = σ(f (x1 , x2 , . . . , xm )) that is,  1  p ηi Mi (σ(x1 ), σ(x2 ), . . . , σ(xm )) = ηi Mi (σ(x1 ), σ(x2 ), . . . , σ(xm )), and consequently, p ηi = ηi . So ηi ∈ {0, 1, . . . , p − 1}, i = 1, 2, . . . , pkm . The conclusion follows from the fact that ∧, ∨, ∼, {Ci }p−1 i=0 and T are functions from m F into F , for some m, commuting with T . On the other hand, we know that every function f : Am −→ A has a unique representation of the form  f (x1 , x2 , . . . , xm ) = f (α1 , . . . , αm ) ⊗ Lα1 (x1 ) ⊗ . . . ⊗ Lαm (xm ). (α1 ,...,αm )∈Am

Arguing as above, we obtain that in particular, the functions f1 (x, y) = x + y and f2 (x, y) = x · y can be given in terms of the operations of A; T .

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557

As an immediate consequence of Theorem 4.1 we have the following (see [10], Th. 4.140). Corollary 4.2. The varieties V(Lp,k ) and V(F (pk )) are equivalent, that is, there exists an interpretation Φ1 of V(Lp,k ) in V(F (pk )) and an interpretation Φ2 of V(F (pk )) in V(Lp,k ) such that Φ2 Φ1 (B) = B for every B ∈ V(Lp,k ) and Φ1 Φ2 (R) = R for every R ∈ V(F (pk )). The following corollary is a consequence of the proof of Theorem 4.1. Corollary 4.3. Every function f : [Lp,k ]n → Lp,k commuting with T can be represented by a term in the language of V(Lp,k ). Similarly, every function f : [F (pk )]n → F (pk ) commuting with σ can be represented as a polynomial with coefficients in F (p). Observe that in the proof of the previous theorem we give a general procedure, by means of Lagrange polynomials, to obtain explicitly the cyclic Post operations as terms in the language of fields, and conversely. In particular, the algebras Lp,k and F (pk ) with constants F (p) are term equivalent. We illustrate this process in the following section.

5

Examples

Consider first the field F (3) ∼ = Z3 = {0, 1, 2}. If we consider the order 0 < 2 < 1, and the following operations C0 , C1 , C2 we have the Post structure L3 on the set {0, 1, 2}. r1 r2 r0

x

C0 (x) C1 (x) C2 (x)

0

1

0

0

2

0

1

0

1

0

0

1

Observe that the constants of the algebra are e0 = 0, e1 = 2 and e2 = 1. The Lagrange polynomials in F (3) are L0 (x) = 2x2 + 1, L1 (x) = 2x2 + 2x, and L2 (x) = 2x2 + x. Hence we can use the formula (I) in order to express the operations ∧, ∨, ∼, C0 , C1 , C2 as terms in the language of fields, that is, in terms of the operations + and · and the elements of the prime field F (3): x ∧ y = x2 y 2 + 2x2 y + 2xy 2 + 2xy, x ∨ y = 2x2 y 2 + x2 y + xy 2 + xy + x + y, ∼ x = 2x + 1, C0 (x) = 2x2 + 1, C1 (x) = 2x2 + x, C2 (x) = 2x2 + 2x, T (x) = x. Conversely, the Lagrange polynomials for the Post algebra 0 < 2 < 1 are

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L0 (x) = (2 ⊗ x2 ) ⊕ 1, L1 (x) = (2 ⊗ x2 ) ⊕ (2 ⊗ x), L2 (x) = (2 ⊗ x2 ) ⊕ x, and by means of formula (II), the field operations + and · can be expressed as x+y =x ⊕ y x · y = x ⊗ y. (Recall that the operations ⊕ and ⊗ are given in terms of ∧, ∨ and the Ci ). Now consider a field with 32 elements, F (32 ) = F (3)[x]/(1 + x2 ) = {0, 1 + x, 2x, 1 + 2x, 2, 2 + 2x, x, 2 + x, 1}. Since ε = 1 + x is a generator of the multiplicative group F (32 ) \ {0}, it follows that F (32 ) = {0, ε, ε2 , ε3 , ε4 , ε5 , ε6 , ε7 , ε8 }. Let σ be the Frobenius automorphism of F (32 ), σ(x) = x3 . By the Normal Basis Theorem, there exists w ∈ F (32 ) such that {w, σ(w)} is a basis of F (32 ) as an F (3)-vector space. So, every x ∈ F (32 ) can be written x = λ0 (x)w + λ1 (x)σ(w), with λ0 (x), λ1 (x) ∈ F (3). This element w can be any element in F (32 ) that satisfies that (see [8])      w ϕ(w)     0.  =    ϕ(w) w  Taking w = ε we have the following identification. x ∈ F (32 ) x = λ0 (x), λ1 (x)

0

ε

ε2

ε3

ε4

ε5

ε6

ε7

ε8

0, 0

1, 0

1, 2

0, 1

1, 1

2, 0

2, 1

0, 2

2, 2

We know that if we consider the ordering 0 < 2 < 1 on F (3) we obtain the Post algebra L3 . Now we consider on F (32 ) a structure of Post algebra A, where A = L23 and ∧, ∨, ∼, C0 , C1 , C2 are defined componentwise, and the constants of A are 0 = e0 = 0, 0 = 0, e1 = 2, 2 = 1 and 1 = e2 = 1, 1 = 2 (the constants are the elements of the prime field). In addition, as σ|`F (3) = id, if we define T = σ we obtain the following 2-cyclic Post algebra of order 3 A; T  associated to the field F (32 ).

M. Abad et al. / Central European Journal of Mathematics 4(4) 2006 547–561

1 =s ε4 = e2

A

@ @

ε2 s

@ @

ε s

@ @

s e1@@

@ s ε6 @ @

@s ε3

@

@s ε @ @

@s 7 ε

5

@s

0 = e0

559

x ∈ F (32 )

∼x

T (x)

C0 (x) C1 (x) C2 (x)

0 = 0, 0

1, 1

0, 0

1, 1

0, 0

0, 0

ε = 1, 0

0, 1

0, 1

0, 1

0, 0

1, 0

ε2 = 1, 2

0, 2

2, 1

0, 0

0, 1

1, 0

ε3 = 0, 1

1, 0

1, 0

1, 0

0, 0

0, 1

ε4 = 1, 1

0, 0

1, 1

0, 0

0, 0

1, 1

ε5 = 2, 0

2, 1

0, 2

0, 1

1, 0

0, 0

ε6 = 2, 1

2, 0

1, 2

0, 0

1, 0

0, 1

ε7 = 0, 2

1, 2

2, 0

1, 0

0, 1

0, 0

ε8 = 2, 2

2, 2

2, 2

0, 0

1, 1

0, 0

Now we will give the Post operations in terms of the operations of the field F (32 ) and the elements of the prime field F (3). The Lagrange polynomials for F (32 ) are L0 (x) = ε4 x8 + 1, Lε (x) = ε4 x8 + ε5 x7 + ε6 x6 + ε7 x5 + x4 + εx3 + ε2 x2 + ε3 x, Lε2 (x) = ε4 x8 + ε6 x7 + x6 + ε2 x5 + ε4 x4 + ε6 x3 + x2 + ε2 x, Lε3 (x) = ε4 x8 + ε7 x7 + ε2 x6 + ε5 x5 + x4 + ε3 x3 + ε6 x2 + εx, Lε4 (x) = ε4 x8 + x7 + ε4 x6 + x5 + ε4 x4 + x3 + ε4 x2 + x, Lε5 (x) = ε4 x8 + εx7 + ε6 x6 + ε3 x5 + x4 + ε5 x3 + ε2 x2 + ε7 x, Lε6 (x) = ε4 x8 + ε2 x7 + x6 + ε6 x5 + ε4 x4 + ε2 x3 + x2 + ε6 x, Lε7 (x) = ε4 x8 + ε3 x7 + ε2 x6 + εx5 + x4 + ε7 x3 + ε6 x2 + ε5 x, Lε8 (x) = ε4 x8 + ε4 x7 + ε4 x6 + ε4 x5 + ε4 x4 + ε4 x3 + ε4 x2 + ε4 x. Then, as ε4 = 2, applying formula (I), we obtain x ∧ y = x6 [y 6 + 2y 4 + 2y 3 + 2y 2 ] + x4 [2y 6 + 2y 4 + y 2 + 2y] + x3 [2y 6 + y 3 + y 2 + 2y] + x2 [2y 6 + y 4 + y 3 + y 2 ] + x[2y 4 + 2y 3 + 2y], x ∨ y = x6 [2y 6 + y 4 + y 3 + y 2 ] + x4 [y 6 + y 4 + 2y 2 + y] + x3 [y 6 + 2y 3 + 2y 2 + y] + x2 [y 6 + 2y 4 + 2y 3 + 2y 2 ] + x[y 4 + y 3 + y + 1] + y, ∼ x = 2x + 2, C0 (x) = x6 + x4 + 2x2 + 2, C1 (x) = x6 + x4 + 2x2 + x, C2 (x) = x6 + x4 + 2x2 + 2x, T (x) = x3 . Similarly, the Lagrange polynomials for the 2−cyclic Post algebra A of order 3 are (we adopt the convention that in absence of parenthesis, ⊗ is performed first, followed by ⊕)

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L0 (x) = x2 ⊗ (T (x))2 ⊕ e1 ⊗ x2 ⊕ e1 ⊗ (T (x))2 ⊕ 1, Lε (x) = x2 ⊗ (T (x))2 ⊕ x ⊗ (T (x))2 ⊕ e1 ⊗ x2 ⊕ e1 ⊗ x, Lε2 (x) = x2 ⊗ (T (x))2 ⊕ e1 ⊗ x2 ⊗ T (x) ⊕ x ⊗ (T (x))2 ⊕ e1 ⊗ x ⊗ T (x), Lε3 (x) = x2 ⊗ (T (x))2 ⊕ x2 ⊗ T (x) ⊕ e1 ⊗ (T (x))2 ⊕ e1 ⊗ T (x), Lε4 (x) = x2 ⊗ (T (x))2 ⊕ x2 ⊗ T (x) ⊕ x ⊗ (T (x))2 ⊕ x ⊗ T (x), Lε5 (x) = x2 ⊗ (T (x))2 ⊕ e1 ⊗ x ⊗ (T (x))2 ⊕ e1 ⊗ x2 ⊕ x, Lε6 (x) = x2 ⊗ (T (x))2 ⊕ x2 ⊗ T (x) ⊕ e1 ⊗ x ⊗ (T (x))2 ⊕ e1 ⊗ x ⊗ T (x), Lε7 (x) = x2 ⊗ (T (x))2 ⊕ e1 ⊗ x2 ⊗ T (x) ⊕ e1 ⊗ (T (x))2 ⊕ T (x), Lε8 (x) = x2 ⊗ (T (x))2 ⊕ e1 ⊗ x2 ⊗ T (x) ⊕ e1 ⊗ x ⊗ (T (x))2 ⊕ x ⊗ T (x). By formula (II), we have that x + y = x ⊕ y, and x · y = T (x) ⊗ y ⊕ x ⊗ y ⊕ x ⊗ T (y) ⊕ e1 ⊗ T (x) ⊗ T (y).

Acknowledgment We would like to thank the referees for their thoughtful comments and suggestions. They led to a much improved presentation of the paper.

References [1] M. Abad: “Cyclic Post algebras of order n”, An. Acad. Brasil. Ciˆenc., Vol. 53(2), (1981), pp. 243–246. [2] R. Balbes and P. Dwinger: Distributive Lattices, University of Missouri Press, Columbia, MO., 1974. [3] V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu: Lukasiewicz-Moisil Algebras, Annals of Discrete Mathematics, Vol. 49, North-Holland, Amsterdam, 1991. [4] S. Burris and H. Sankappanavar: A Course in Universal Algebra, Graduate Texts in Mathematics, Vol 78, Springer, Berlin, 1981. [5] H. Cendra: “Cyclic Boolean algebras and Galois fields F (2k )”, Portugal. Math., Vol. 39(1–4), (1980), pp. 435–440. [6] G. Epstein: “The lattice theory of Post algebras”, Trans. Amer. Math. Soc., Vol. 95, (1960), pp. 300–317. [7] K. Kaarly and A.F. Pixley: Polynomial Completeness in Algebraic Systems, Chapman and Hall, Boca Raton, 2001. [8] S. Lang: Algebra, Addison-Wesley Publishing Company, CA., 1984. [9] R. Lewin: “Interpretability into L  ukasiewicz algebras”, Rev. Un. Mat. Argentina, Vol. 41(3), (1999), pp. 81–98. [10] R. McKenzie, G. McNulty and W. Taylor: Algebras, Lattices, Varieties, , Vol. I, Wadsworth and Brooks, Monterey, CA, 1987.

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[11] A. Monteiro: “Alg`ebres de Boole cycliques”, Rev. Roumaine de Math´ematiques Pures Appl., Vol. 23(1), (1978), pp. 71–76. [12] G. Moisil: Algebra schemelor cu elemente ventil, Seria St. nat. 4-5, Revista Universitatii C.I. Parhon, Bucharest, 1954, pp. 9–15. [13] G. Moisil: “Alg`ebres universelles et automates”, In: Essais sur les Logiques non Chrysippiennes, Editions de L’Academie de la Republique Socialiste de Roumanie, Bucharest, 1972. [14] S. Rudeanu: Boolean functions and equations, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. [15] M. Serfati: “ Introduction aux Alg`ebres de Post et `a leurs applications (logiques a` r valeurs-´equations postiennes-grapho¨ıdes orient´es)”, Cahiers du Bureau Universitaire de Recherche Op´erationnelle Universit´e Paris VI. S´erie Recherche, Vol. 21, (1973), pp. 35–42.

DOI: 10.2478/s11533-006-0027-4 Research article CEJM 4(4) 2006 562–572

A refined Newton’s mesh independence principle for a class of optimal shape design problems Ioannis K. Argyros∗ Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

Received 11 April 2006; accepted 24 July 2006 Abstract: Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Mesh independence principle, Newton’s method, radius of convergence, optimal space design MSC (2000): 65K10, 49K20, 49M15, 49M05, 47H17

1

Introduction

In this study we are concerned with the problem min F (u) u∈U

(1)

where F (u) = J (u, S (u) , z (u)) + 2ε u − uT 2 , ε ∈ R, functions uT , S, z, and J are defined on a function space (Banach or Hilbert) U with values in another function space V. Many optimal shape design problems can be formulated as in (1) [6]. In the excellent paper by W. Laumen [6] the mesh independence principle [1] (see also [2]) was transferred to the minimization problem by the necessary first order condition F  (u) = 0 in U. ∗

E-mail: [email protected]

(2)

I.K. Argyros / Central European Journal of Mathematics 4(4) 2006 562–572

563

The most popular method for solving (2) is given for n ∈ N by Newton’s method F  (un−1 ) (w) (v) = −F  (un−1 ) (v) un = un−1 + w, where F, F  and F  also depend on functions defined on the infinite dimensional Hilbert space V. The discretization of this method is obtained by replacing the infinite dimensional space V and U with the finite dimensional subspaces V M , U M and the discretized Newton’s method    M  M    M w v = −FN uM v FN uM n−1 n−1 M M uM n = un−1 + w ,

Here we show that under the same hypotheses and computational cost a finer mesh independence principle can be given.

2

The mesh independence principle

Let u0 be chosen in the closed ball U∗ = U (u∗ , r∗ ) = {u ∈ U : u − u∗  ≤ r∗ , r∗ > 0} in order to guarantee convergence to the solution u∗ . The assumptions concerning the cost function FN , which are assumed to hold on a possible smaller ball Uˆ∗ = U (u∗ , rˆ∗ ) with rˆ∗ ≤ r∗ are stated below. Assumption C1. There exist positive constants L0 , L and δ such that for all u, v ∈ U∗ F  (u) − F  (u∗ ) ≤ L0 u − u∗  F  (u) − F  (v) ≤ L u − v    F (u∗ )−1  ≤ δ. (i)

Assumption C2. There exist uniformly bounded Lipschitz constants LN , i = 1, 2, such that (1) FN (u) − FN (v) ≤ LN u − v , for all u, v ∈ Uˆ∗ , N ∈ N, (2) FN (u) − FN (v) ≤ LN u − v , for all u, v ∈ Uˆ∗ , N ∈ N. (i)

Without loss of generality, we assume LN ≤ L, i = 1, 2, for all N. (1) (1) Assumption C3. There exist a sequence zN with zN → 0 as N → ∞, such that (1) FN (u) − F  (u) ≤ zN , for all u ∈ Uˆ∗ , N ∈ N, (2) FN (u) − F  (u) ≤ zN , for all u ∈ Uˆ∗ , N ∈ N. (2)

(2)

Assumption C4. There exists a sequence zN with zN → 0 as N → ∞ such that for all N ∈ N there exists a uˆN ∈ U N × Uˆ∗ such that  N  uˆ − u∗  ≤ z (2) . N

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I.K. Argyros / Central European Journal of Mathematics 4(4) 2006 562–572

Assumption C5. FN and FN correspond to the derivatives of FN . The cost function F is assumed to be twice continuously Fr´echet differentiable. Therefore, its first derivative is also Lipschitz continuous: ˆ u − v , F  (u) − F  (v) ≤ L

for all u, v ∈ U∗ .

ˆ ≤ L. Without loss of generality we assume L Remark 2.1. In general L0 ≤ L

(3)

holds and LL0 can be arbitrarily large. If L0 = L our Assumptions C1–C5 coincide with the ones in [6, p. 1074]. Otherwise our assumptions are finer and under the same computational cost since in practice the evaluation of L requires the evaluation of L0 . This modification of the assumptions in [6] will result in larger convergence balls U∗ and Uˆ∗ which in turn implies a wider choice of initial guesses for Newton’s method and finer bounds on the distances involved. This observation is important in computational mathematics. We now justify the claims made in the previous remark, as follows:            δ F  (u∗ ) − FN uˆM  ≤ δ F  (u∗ ) − F  uˆM  + F  uˆM − FN uˆM    (2) (1) ≤ δ L0 zM + zN ≤ δˆ z<1 hold for a constant zˆ if M and N are sufficiently large. It also follows by the Banach  −1 Lemma on invertible operators that F  uˆM exists and     δ   M −1  ˆ = δ. ≤ F uˆ 1 − δˆ z We showed in [3] that if r∗ ≤

2 1 , < (2L0 + L) δ δL0

(4)

then the estimates δ F  (ui ) − F  (u∗ ) ≤ δL0 ui − u∗  ≤ δL0 r∗ < 1 hold, which again also imply the existence of F  (ui ) with    F (ui )−1  ≤

δ ˆ = δ. 1 − δL0 r∗

Hence by redefining δ by δˆ if necessary we assume that    F (ui )−1  ≤ δ, for all i ∈ N       M −1  FN uˆ  ≤ δ, for all uˆM ∈ U M , N ∈ N,

(5)

(6) (7)

I.K. Argyros / Central European Journal of Mathematics 4(4) 2006 562–572

for M and N satisfying

 δ

(2) L0 zM

+

(1) zN



≤ δˆ z < 1.

565

(8)

The next result is a refinement of Theorem 2.1 in [6, p. 1075] which also presents sufficient conditions for the existence of a solution of the problem   min FN uM (9) u∈U M

and shows the convergence of Newton’s method for M, N → ∞. Theorem 2.2. Assume C1–C5 hold and parameters M, N satisfy



 1  (1) 1 (2) ZM N = 2δ max {1, L0 } + zN + zM ≤ min r∗ , . 2δ δL ˆ Then the discretized Newton’s method has a local solution uM ∗ ∈ U∗ satisfying  M  u∗ − u∗  < zM N .

(10)

(11)

Proof. We apply the Newton-Kantorovich theorem [5] to Newton’s method starting at uM ˆM to obtain the existence of a solution uM 0 = u ∗ of the infinite dimensional minimization problem. Using C2–C4, (7) and (10) we obtain in turn     −1   M   2h = 2δL FN uˆM FN uˆ         −1   ≤ 2δL FN uˆM  FN uˆM           ≤ 2δ 2 L FN uˆM − F  uˆM  + F  uˆM − F  (u∗ )    (1) ≤ 2δ 2 L zN + L0 uˆM − u∗    1  (1) (2) 2 ≤ 2δ L max {1, L0 } + zN + zM 2δ ≤ δLzM N ≤ 1 (12) which imply the required assumption 2h < 1 (for the quadratic convergence).   We also need to show U uˆM , r (h) ⊂ U (u∗ , rˆ∗ ) . By C4 it suffices to show √ 1  (2) r (h) = 1 − 1 − 2h ≤ rˆ∗ − zM . δL

(13)

But by (10) and the definition of r (h) we get  (1) (2) r (h) = 2δ max {1, L0 } zN + zM   1  (1) (2) (2) < 2δ max {1, L0 } + zN + zM − zM 2δ (2)

≤ zM N − zM (2)

≤ rˆ∗ − zM ,

(14)

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I.K. Argyros / Central European Journal of Mathematics 4(4) 2006 562–572

which gives (13).  M  Hence, there exists a solution uM ˆ , r (h) such that ∗ ∈ U u  M     M  M u∗ − u∗  ≤ uM   < zM N − z (2) + z (2) = zM N . + u ˆ − u ˆ − u ∗ ∗ M M which completes the proof of the theorem.

(15) 

Remark 2.3. If equalities hold in (7) then our Theorem 2.2 reduces to Theorem 2.1 in [6]. Otherwise it is an improvement (and under the same computational cost) since zˆ, M, N, zM N are smaller and δˆ (i.e., δ), r∗ , rˆ∗ are larger than the corresponding ones in [6, p. 1075] and our condition (12) is weaker than the corresponding (2.5) in [6] (i.e., set L0 = L in (12)). The claim made in Remark 2.1 is therefore justified and our theorem extends the applicability of the mesh independence principle.   1 to be as large as possible. It then immeRemark 2.4. In practice we want min r∗ , δL diately follows from (4), (12) and (14) that the conclusions of Theorem 2.2 hold if zM N given in (10) is replaced by



2δL 1  (1) 1 (2) 1 zM N = max {1, L0 } + . (16) zN + zM ≤ min r∗ , L0 2δ δL0 Another way is to rely on our Theorem 2 in [3] using the weaker (than (12)) NewtonKantorovich-type hypothesis       M −1   M  h0 = (L0 + L) δ FN uˆ FN uˆ  ≤ 1 (17) or as in (16) for (17) to hold we must have  

L 1  (1) (2) 2 δ max {1, L0 } + zN + zM h0 < L0 1 + L0 2δ 0 ≤ δL0 zM N ≤ 1, provided that 

L 1  (1) (2) δ max {1, L0 } + = 1+ zN + zM L 2δ 

0 1 . ≤ min r∗ , δL0 

0 zM N

(18)

  The other hypothesis for the application of our Theorem 2 in [3] is U uˆM , r1 (h) ⊆ U (u∗ , rˆ∗ ) , where  (1) (2) r1 (h) = 2δ max {1, L0 } zN + zM . Hence we arrive at:

I.K. Argyros / Central European Journal of Mathematics 4(4) 2006 562–572

567

0 Theorem 2.5. Under the hypotheses of Theorem 2.2 with zM N replaced by zM N (given in (15)) the conclusions of this theorem hold.

So far we showed that a solution  M    uM ˆ , r (h) (or uˆM , r1 (h) ) ⊂ U (u∗ , rˆ∗ ) ∗ ∈ U u of the discretized minimization problem exists. Next, in the main results of this study we show two different ways of improving the corresponding Theorem 2.2 in [6, p. 1076], where it was shown that the discretized M Newton’s method converges to the solution uM ∗ for any u0 ∈ U (u∗ , r1 ) for sufficiently small r1 . In order to further motivate the reader let us provide a simple numerical example. Example 2.6. Let U = R, U∗ = U (0, 1) and define the real function F on U∗ by F (x) = ex − 1.

(19) (1)

(2)

Then we obtain using (19) that L = e, L0 = e − 1 and δ = 1. We let zN = 0, zM = Set r∗ = rˆ∗ and N = 0. The radius of convergence given in [6, p. 1075] is r∗L = rˆ∗L =

1 . M

2 = .24525246, 3δL

where as by (4) ours is given by r∗ = rˆ∗ =

2 = .324947231. (2L0 + L) δ

(20)

That is, (3) holds as a strict inequality and r∗L = rˆ∗L < r∗ = rˆ∗ . The condition (2.4) used in [6, p. 1075] corresponding to ours (10) is given by



 1  (1) (2) L L 1 zM N = 2δ max {1, L0 } + zN + zM ≤ min rˆ∗ , . 2δ δL

(21)

We can tabulate the following results containing the minimum M for which conditions (8), (10), (16), and (15) are satisfied. The above table indicates the superiority of our results over the ones in [6, p. 1075]. We can now present a finer version than Theorem 2.2 in [6] of the mesh independence principle.

568

I.K. Argyros / Central European Journal of Mathematics 4(4) 2006 562–572

M

zM N (21)

r∗L (18)

zM N (10)

zM N (16)

zM N (15)

r∗ (20)

27

.238391246

.24525296

.164317172

.259945939

.212131555

.324947231

22 19

.319024562 .23350335

18

.318197333

Table 1 Comparison Table. Theorem 2.7. Suppose assumptions C1–C5 are satisfied and there exist discretization parameters M and N such that

 1 rˆ∗ 1 zM N ≤ min , . (22) 6 4 (2L0 + 3L) δ + 1 M Then the discretized Newton’s method converges to uM ∗ for all starting points u0 ∈ U (u∗ , r1 ) , where

 3 1 rˆ∗ r1 = min , . (23) 4 (2L0 + L) δ 2

Moreover, if

where

  M u0 − u0  ≤ τ, 1

 + u − u  zM N 0 ∗  2 τ= 1  b2 + b2 − 6Lδ 2 + u0 − u∗  zM N 2

(24)

and

  1  b = 1 + zM N − 2δL uM 0 − u∗ 2 the following estimates hold for ci ∈ R, i = 1, 2, 3, 4, n ∈ N:  M    M 2 un+1 − uM  ≤ c1 uM , ∗ n − u∗   M u − un  ≤ c2 zM N    M n   FN un − F (un ) ≤ c3 zM N and

(25)

 M  un − uM  ≤ un − u∗  + c4 zM N . ∗

Proof. We first show the convergence of the discretized Newton’s method for all uM 0 in a suitable ball around u∗ . Since the assumptions of Theorem 2.2 are satisfied, the existence ˆ of a solution uM ∗ ∈ U∗ is guaranteed. We shall show that the discretized Newton method M converges to  uM ∗ if u0 ∈ U  (u∗ , r2 ) , where 1 rˆ∗ r2 = min (2L0 +L)δ , 2 .

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The estimates       M   M M    u∗ − u∗  + uM ≤ 2  uM 0 − u∗ ∗ − u ∗ + u0 − u ∗ ≤ 2zM N + r2 ≤ rˆ∗    M  M   M M   ⊂ Uˆ∗ . Hence, Assumptions C1–C5 hold in U uM . imply U uM ∗ , u 0 − u∗ ∗ , u 0 − u∗ We can also have            F (u∗ )−1  FN uM − F  (u∗ ) ≤ δ FN uM − FN (u∗ ) + FN (u∗ ) − F  (u∗ ) ∗ ∗      ≤ δ L uM ∗ − u∗ + zM N ≤ (δL + 1) zM N δL + 12 < 1, ≤ (20 + 3L) δ + 1

(26)

where we used (1) zn ≤ 2δ1 zM N (by (10)).  −1 It follows by (26) and the Banach Lemma on invertible operators that FN uM ∗ exists and     δ   M −1    . FN u∗ ≤ 1 − δL + 12 zM N By the theorem on quadratic convergence of Newton’s method and since all assumptions hold, the convergence to uM ∗ has been established. Using a refined formulation of this theorem given by us in [3], the convergence is  M  guaranteed for all uM 0 ∈ U u∗ , r3 , where r3 = Therefore we should show

2   −1  . (2L0 + L) FN (uM ∗ )

  U (u∗ , r2 ) ⊂ U uM ∗ , r3

or equivalently  M   M    u0 − uM  ≤ u0 − u∗  + u∗ − uM  ∗ ∗ ≤ r2 + zM N 1 ≤ + zM N (2L0 + L) δ 1 + (2L0 + L) δ = (2L + L) δ   0 2 1 − LδzM N − 12 zM N ≤ (2L0 + L) δ 2   ≤ −1  = r3 . (2L0 + L) FN (uM ∗ ) M Hence, the discretized Newton’s method converges to uM ∗ for all u0 ∈ U (u∗ , r2 ) such that (25) holds for c1 = δL.

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Next, induction is used to show   M un − un  ≤ τ ≤ c2 zM N

(27)

3 for all uM 0 ∈ U (u∗ , r1 ) , r1 = 4 r2 , where τ is given by   2 12 + u0 − u∗  zM N  τ=   2 b + b2 − 6Lδ 12 + u0 − u∗  zM N  1 + u − u  zM N 0 ∗ ≤ 2 =: c2 zM N 2 b    with b = 1+ 12 zM N −2δL uM 0 − u∗ . The constant τ is well defined, since the inequalities   1 2Lδ + 1 1 1 6Lδ + u0 − u∗  zM N ≤ < and b ≥ 1 − 2δLρ1 ≥ 2 4 ((2L0 + 3L) δ + 1) 4 2   imply b2 ≥ 14 ≥ 6Lδ 12 + u0 − u∗  zM N . While the assertion (27) is fulfilled by the assumption for the case n = 0, the induction step is based on the simple decomposition  M −1    M   M     FN ui ui − ui − FN uM + FN (ui ) (28) uM i+1 − ui+1 = FN ui i    M    −1   + FN ui − FN (ui ) F (ui ) F (ui )

+ FN (ui ) F  (ui )−1 F  (ui ) − F  (ui ) +F  (ui ) − FN (ui )} .

Assumptions C1–C4, equation (27), and the definition of zM N imply        M        + δ FN (ui ) − F  (ui ) − F u − F δ FN uM (u ) ≤ δ F (u ) i i i  N i (1) ≤ δ Lτ + zN 1 ≤ δLτ + zM N 2 δLzM N + 2 u0 − u∗  δLzM N 1 ≤ + zM N 1 − 2δL u0 − u∗  2 1 1 δL + 4 3 ≤ <1 (2L0 + 3L) δ + 1     −1   . We obtain resulting in the inequality FN uM  ≤ 1− Lδτ +δ 1 z i ( 2 MN )    M   M 2 1 2  1  M     FN ui  ≤ Lτ ,   ui − ui − FN uM + F (u ) ≤ − u L u i i i N i 2 2 and the convergence assertion ui − u∗  ≤ u0 − u∗  yields     M       FN ui − FN (ui ) F  (ui )−1 F  (ui ) ≤ L uM i − ui ui − ui+1  ≤ 2Lτ u0 − u∗  .

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The assumptions of the theorem lead to    FN (ui ) F  (ui )−1 F  (ui ) − F  (ui ) ≤

≤ −FN (ui ) (ui+1 − ui ) + F  (ui ) (ui+1 − ui ) ≤ FN (ui ) − F  (ui ) ui+1 − ui 

≤ zn(1) 2 u0 − u∗  1 ≤ zM N u0 − u∗  δ (1)

and F  (ui ) − FN (ui ) ≤ zn ≤ ties complete the proof by

1 z . 2δ M N

Using the decomposition (28) the last inequali-

 M  ui+1 − ui+1  ≤

   δ 1 2 1 zM N   ≤ Lτ + 2L u0 − u∗  τ + + u0 − u∗  2 2 δ 1 − Lδτ + 12 zM N = τ. The last equality is based on the fact that τ is equal to the smallest solution of the   quadratic equation 3Lδτ 2 − 2bτ + 2zM N 12 + u0 − u∗  = 0. Finally, inequality (27) is shown by         M FN un − F  (un ) ≤ FN uM − FN (un ) + FN (un ) − F  (un ) n    ≤ L  uN n − un + zM N ≤ (Lc2 + 1) zM N =: c3 zM N and inequality (24) results from  M   M      un − u M    − (un − u∗ ) ≤ uM ∗ n − u n + u∗ − u ∗ ≤ c2 zM N + zM N ≤ (c2 + 1) zM N =: c4 zM N  Remark 2.8. The upper bounds on zM N and r1 were defined in [6] by

 1 rˆ∗ 1 min , 6 4 6Lδ + 1 and 3 min 4

1 rˆ∗ , 3Lδ 2

(29)

 (30)

respectively. By comparing (22), (23) with (29) and (30) respectively we conclude that our choices of zM N and r1 (or r2 or r3 ) are finer than the ones in [6]. However we leave the details to the motivated reader.

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References [1] E.L. Allgower, K. B¨ohmer, F.A. Potra and W.C. Rheinboldt: “A mesh-independence principle for operator equations and their discretizations”, SIAM J. Numer. Anal., Vol. 23, (1986). [2] I.K. Argyros: “A mesh independence principle for equations and their discretizations using Lipschitz and center Lipschitz conditions, Pan”, Amer. Math. J., Vol. 14(1), (2004), pp. 69–82. [3] I.K. Argyros: “A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space”, J. Math. Anal. Appl., Vol. 298(2), (2004), pp. 374-397. [4] I.K. Argyros: Newton Methods, Nova Science Publ. Corp., New York, 2005. [5] L.V. Kantorovich and G.P. Akilov: Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982. [6] M. Laumen: “Newton’s mesh independence principle for a class of optimal design problems”, SIAM J. Control Optim., Vol. 37(4), (1999), pp. 1070–1088. [7] W.C. Rheinboldt: “An adaptive continuation process for solving systems of nonlinear equations”, In: Mathematical models and Numerical Methods, Banach Center Publ., Vol. 3, PWN, Warsaw, 1978, 129–142.

DOI: 10.2478/s11533-006-0026-5 Research article CEJM 4(4) 2006 573–584

An existence result for an interior electromagnetic casting problem Mohammed Barkatou1∗ , Diaraf Seck2† , Idrissa Ly2‡ 1

Facult´e des Sciences, D´epartement de Math´ematiques et Informatique, Universit´e Chouaib Doukkali, B.P. 20, El Jadida, Maroc 2

Facult´e des Sciences Economiques et de Gestion, B.P. 5683, Dakar, S´en´egal

Received 23 February 2006; accepted 8 June 2006 Abstract: This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C 2 . Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Free boundary, maximum principle, shape optimization, shape derivative MSC (2000): 35B50, 35J05, 35R35

1

Introduction

  Let f be a positive function in Lp RN , p > 2 and N ≥ 2, with compact support K. Let σ and k be two strictly positive constants. Our aim here is to solve the following free boundary problem : ⎧ ⎪ ⎪ Find an open and bounded set Ω in RN containing strictly K ⎪ ⎪ ⎨ (ECP ) and a function uΩ ∈ H01 (Ω) such that : ⎪ ⎪ ⎪ ⎪ ⎩ −ΔuΩ = f in Ω, uΩ = 0 and |∇uΩ |2 = σH∂Ω + k 2 on ∂Ω, ∗ † ‡

E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

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where H∂Ω denotes the mean curvature of ∂Ω. This problem is known as an interior electromagnetic casting problem and has been studied from different points of view in [7, 12, 13, 18] and [10]. In this paper, using the implicit function theorem, the authors gave a sufficient condition of the existence of regular critical points of some energy functional assiociated with the problem (ECP ) in two dimensions. Note that in [10], the sufficient condition of existence of a solution for the problem (ECP ) given is not explicit. Using the shape derivative, the problem (ECP ) can be seen as the Euler equation of the following problem of minimization ([19] and [17]): (OP ) Find Ω ∈ Oad such that J(Ω) = min J(ω), ω∈Oad

where Oad is a set of admissible domains and  J(ω) = − |∇uω (x)|2 dx + σP (ω) + k 2 V (ω) ω

where P (ω) (resp. V (ω)) denotes the perimeter (resp. the volume) of ω and uω , the solution of the Dirichlet problem ⎧ ⎪ ⎨ −Δuω = f in ω, P (ω, f ) ⎪ ⎩ uω = 0 on ∂ω. In this paper, we begin by showing the existence of a minimum for the shape optimization problem (OP ). The minimum we obtain is of class C 2 (this regularity allows us to use the shape derivative). Then, using the optimality condition, the monotonicity of the mean curvature (for the open sets of class C 2 ) and the maximum principle, we will give a sufficient condition in order that the minimum obtained resolves the problem (ECP ). This condition is very simple to verify (in section 6, we will give two examples where this condition is explicit).

2

Notations and preliminary results

Let D be a given bounded open set of RN (N ≥ 2). D will contain all the sets we shall use. Now, since we must solve the shape optimization problem associated to the functional J, we need to work in some class of domains which must be compact for some topology and on which we can have : (1) the continuity result for the Dirichlet problem P (ω, f ); and, (2) the lower semi-continuity of the perimeter and the volume. As we will use the shape derivative, we need the equivalence between the Hausdorff topology and the compact convergence (see Definitions 2.2 and 2.3), in order to prove the C 2 regularity of the minimum of J.

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The family of domains which have the C-GNP (see Definition 3.1) seems to be the appropriate class. Before introducing this class of domains, let us recall some definitions and results that we will be using in this paper. In the definitions that will follow, Ωn and Ω are open subsets of D. Definition 2.1. Let F1 and F2 be two different compact subsets of D. We call a Hausdorff distance of F1 and F2 , denoted by dH (F1 , F2 ), the following positive number: dH (F1 , F2 ) = max {ρ(F1 , F2 ), ρ(F2 , F1 )} , where ρ(Fi , Fj ) = maxd(x, Fj ) i, j = 1, 2 and d(x, Fj ) = min |x − y| . x∈Fi

y∈Fj

Definition 2.2. Let Fn and F be respectively the complements, in D, of Ωn and Ω. H We say that the sequence Ωn converges, in the Hausdorff sense, to Ω (Ωn −→ Ω) if lim dH (Fn , F ) = 0.

n→+∞

The notion of convergence that follows is less than the Hausdorff convergence, but it is very useful for the stability of the solution of elliptic problems when the domain varies ([11] and [14]). Definition 2.3. We say that the sequence Ωn converges, in the compact sense, to Ω, K (Ωn −→ Ω) if • every compact subset of Ω is included in Ωn , for n large enough, and c ¯ is included in Ωcn (the exterior of • every compact subset of Ω (the exterior of Ω) ¯ n ), for n large enough. Ω Definition 2.4. We say that the sequence Ωn converges, in the sense of characteristic L1

functions, to Ω, (Ωn −→ Ω) if χΩn converges to χΩ in Lploc (RN ), 1 < p < ∞, (χΩ is the characteristic function of Ω). Lemma 2.5. (Maximum principle([9]. Let U ⊂ RN be open and bounded, and   v1 , v2 ∈ C 2 U , with Δv1 ≤ Δv2 and v1 ≥ v2 on ∂U. Then the following hold: (1) v1 ≥ v2 in U. (2) If v1 (x) = v2 (x) for some x ∈ ∂U and v1 = v2 and if U satisfies an interior ball 2 1 (x) > ∂v (x), where ν is the unit outward normal vector on ∂U condition, then ∂v ∂ν ∂ν at x. (3) If v1 = v2 in U, then v1 > v2 in U.

3

Main results

Definition 3.1. [3] Let C be a compact convex subset of RN , the bounded domain ω saisfies C-GNP if:

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(i) ω ⊃ int(C), (ii) ∂ωC is locally Lipschitz, (iii) for every x ∈ ∂ωC, the inward normal ray to ω (if exists) meets C, (vi) for any c ∈ ∂C, there is an outward normal ray Δc such that Δc ∩ ω is connected. Remark 3.2. In [3], we prove that the boundary of a domain ω which satisfies CGNP has a uniform cone property ([5]) outside C. Moreover, even though cusps can be formed at the points of ∂ω ∩ ∂C (One can consider, in two dimensions, the convex C = [−1, 1] × {0} and the domain ω = B(−1, 1) ∪ B(1, 1), observe that ω satisfies CGNP but it isn’t Lipschitz). It is then shown that these cusps are not sharper than (i.e. contain) a canonical cusp (which is obtained by revolving the cusp between two touching circles of large radius around its axis). In particular, this implies that every point of ∂ω is regular for the Dirichlet problem, as one can easily verify the Wiener citerion. Let C denotes the convex hull of K. If we put OC = { open subset ω of D, ω has the C-GNP} , we have the following results which we will use thereafter: Proposition 3.3. If ωn ∈ OC , then there exists an open subset ω ⊂ D and a subsequence H K (again labeled ωn ) such that (i ) ωn −→ ω, (ii) ωn −→ ω, (iii) χωn converges to χω in L1 (D) and (iv) ω ∈ OC . For the proof of this proposition, see Theorem 3.1 in [3]. H

Proposition 3.4. Let {ωn , ω} ⊂ OC such that ωn −→ ω. Let un and uω be respectively the solutions of P (ωn , f ) and P (ω, f ). Then un converges strongly in H01 (D) to uω (un and uω are extended by zero in D). The second proposition is proven for N = 2 or 3 (see Theorem 4.3 in [3]). Let us now state our main results, which we will prove in Section 5. Theorem 3.5. Let L be a compact subset of RN . Let fn be a positive sequence of functions with compact support in L. We assume that the fn are of class C 3 and 2 ∂ fn ∂ 3 fn ∂fn ∂xi ≤ M, ∂xi ∂xj ≤ M, ∂xi ∂xj ∂xk ≤ M, where M is a strictly positive constant and is independent of n. Define a sequence Ωn , by Ωn = {x ∈ L : fn (x) > 0} and suppose there exists α > 0 such that |fn (x)| + |∇fn (x)| ≥ α for all x in L. If the Ωn have the C-GNP, then there exists Ω of class C 2 and a subsequence (still denoted by Ωn ) such that Ωn converges in the compact sense, to Ω.

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The previous theorem allows us to get Theorem 3.6. There exists Ω ∈ OC which is of class C 2 and minimizes the functional J on OC . One would like to say that the minimum obtained in Theorem 3.6 is a solution of the problem (ECP ). It is not so simple. First of all, it should be noted that, without any assumptions on f , σ and k, the problem (ECP ) does not have, in general, a solution. Let Ω (a lipschitz open set) be a solution of the free boundary problem (ECP ) and uΩ ∈ H 2 (Ω). Now, if   Ω

−ΔuΩ =

f Ω

then, by Green formula, we obtain 

∂uΩ − = ∂ν ∂Ω

 f. Ω

But since,

∂uΩ = |∇uΩ | = σH∂Ω + k 2 , ∂ν one deduces the following necessary condition of existence :  

(N C) f dx = σH∂Ω + k 2 ds −

Ω

∂Ω

In addition, the boundary ∂Ω of the minimum Ω of the functional J can come to touch C. We will give, in the following theorem, a sufficient condition in order that Ω contains strictly C and that |∇uΩ (x)|2 = σH∂Ω + k 2 on ∂Ω. Theorem 3.7. Let σ and k be two strictly positive constants and let f be a positive function of Lp (D) (p > 2) with compact support K which has a nonempty interior. Let C be the convex hull of K. Let Ω be a minimum of the functional J on OC . Let uΩ and uC be, respectively, the solutions of Dirichlet problems P (Ω, f ) and P (int(C), f ) . Ω Suppose that uC ∈ C 2 (C), uΩ ∈ C 2 (C) and ∂u are continuous on ∂Ω. If C satisfies an ∂ν interior ball condition, and |∇uC |2 > σH∂C + k 2 on ∂C then

⎧ ⎪ ⎨ C is strictly contained in Ω; and, ⎪ ⎩ |∇uΩ |2 = σH∂Ω + k 2 on ∂Ω.

(1)

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Optimality condition

As it was mentioned in the introduction of this paper, we are going to use the standard tool of the domain derivative to write down the optimality condition. Let Ω be the minimum of the functional J obtained in Theorem 3.6 and put J(Ω) = J1 (Ω) + σP (Ω) + k 2 V (Ω) , where  J1 (Ω) = −

Ω

|∇uΩ (x)|2 dx.

  Suppose that uΩ ∈ H 2 (Ω) . Let us consider a deformation field θ ∈ C 2 RN ; RN , then the classical Hadamard formula yields for the derivative of J with respect to the displacement θ (or in the direction θ). As   2 dJ1 (Ω; θ) = − |∇uΩ (x)| θ.n dσ − 2 ∇uΩ .∇u (x)dx Ω

∂Ω

where n is the outward normal vector to ∂Ω and u the derivative of uΩ which can be defined as the solution of the following problem : ⎧ ⎪ ⎨ −Δu = 0 in Ω (2) ∂u ⎪ ⎩ u = − Ω θ.n on ∂Ω, ∂n   H∂Ω θ.n dσ and dV (Ω; θ) = θ.n dσ, dP (Ω; θ) = ∂Ω

then



dJ(Ω; θ) =

  σH∂Ω + k 2 θ.n dσ −

∂Ω



∂Ω

2



|∇uΩ (x)| θ.n dσ − 2 ∂Ω

Ω

∇uΩ .∇u (x)dx

(3)

Now, using the Green formula, we can evaluate dJ(Ω; θ) by ⎧   ⎪ 2 ⎨ dJ(Ω; θ) = (σH + k ) θ.n dσ − |∇uΩ (x)|2 θ.n dσ ∂Ω ∂Ω ∂Ω   ∂u ⎪  ⎩ +2 Ω uΩ .Δu (x)dx − 2 ∂Ω uΩ θ.n dσ. ∂n According to (2) and (3) ,  dJ(Ω; θ) = ∂Ω



σH∂Ω + k

2



 θ.n dσ −

|∇uΩ (x)|2 θ.n dσ.

∂Ω

Now, since Ω is the minimum for the functional J, dJ(Ω; θ) ≥ 0 for every admissible displacement θ. Therefore    σH∂Ω + k 2 − |∇uΩ (x)|2 θ.n dσ ≥ 0 for every admissible displacement θ. ∂Ω

(We mean by admissible displacement the one which allows us to keep C-GNP).

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Let us denote by Nx the inward normal at some point x of ∂Ω. One can see that ∂Ω is decomposed as follows: ∂Ω = Γ0 ∪ Γ1 ∪ Γ where ⎧ ⎪ ⎪ Γ0 = ∂Ω ∩ ∂C ⎪ ⎪ ⎨ Γ1 = {x ∈ ∂ΩC such that Nx is tangent to ∂C} ⎪ ⎪ ⎪ ⎪ ⎩ Γ = ∂Ω (Γ0 ∪ Γ1 ) . On Γ, all displacements θ are admissible, so by using θ (resp. −θ) and the density of the functions θ · n in L2 (Γ), one derives |∇uΩ (x)|2 = σH∂Ω + k 2 on Γ.

(4)

In the same way, the admissible displacements θ on Γ0 must satisfy θ · n ≥ 0. Hence, we get |∇uΩ (x)|2 ≤ σH∂Ω + k 2 on Γ0 . (5) Lastly, if γ is some connected component of Γ1 as admissible displacement, one can take θ such that for all x ∈ γ, θ(x) · n(x) = φ(x) (where φ is increasing or decreasing, according to the position of γ with respect to C). The optimality condition becomes    σH∂Ω + k 2 − |∇uΩ (x)|2 φ dσ ≥ 0. (6) γ

5

Proofs of the main theorems

Before proving Theorem 3.5, let us state and prove the following lemma. Lemma 5.1. Let L be a compact subset of RN . Let fn be a sequence of functions defined as in Theorem 3.5. Suppose that Ω is an open subset of L such that Ω = {x ∈ L : h(x) > 0} and ∂Ω = {x ∈ L : h(x) = 0} , where h is a continuous function defined in L. If the fn converge uniformly to h in L, then the Ωn converge in the compact sense, to Ω. Proof (of the Lemma 5.1). (1) Let K1 be a compact subset of Ω. If β1 = inf h, β1 > 0 and there exists n1 ∈ N K1

such that for all n ≥ n1 , |fn − h|L∞ (K1 ) < β1 . This implies that for all x ∈ K1 , fn (x) > h(x) − β1 ≥ 0 and then K1 is contained in Ωn , for n ≥ n1 . c (2) Let K2 be a compact subset of Ω . By hypothesis, Ω = Ω∪∂Ω = {x ∈ L : h(x) ≥ 0}. If β2 = maxh, β2 < 0 and there exists n2 ∈ N such that for all n ≥ n2 , |fn − h|L∞ (K2 ) < K2

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−β2 .This implies that for all x ∈ K2 , fn (x) < h(x)−β1 ≤ 0 and then K1 is contained c c in Ωn , for n ≥ n2 because {x ∈ L : fn (x) < 0} ⊂ Ωn .  Proof (of Theorem 3.5). Let C 2 (L) be the set of all functions of class C 2 defined on L. Consider the norm on C 2 (L) , defined by 2g ∂g ∞ | ∂x | + | ∂x∂i ∂x |L∞ . |g| := |g|L∞ + L i j We will apply the Ascoli theorem to the set B = {fn }(n∈N) . We can easily verify that {fn } is bounded and closed. By the Mean Value Theorem, we show that {fn } is equicontinuous



2  ∂ fn n (we can obtain the same results for ∂f and ) .B is compact, so there exist a ∂xi ∂xi ∂xj

function h ∈ C 2 (L) and a subsequence of fn (still labled fn ) such that fn converges to h. Now, let Ω = {x ∈ L : h(x) > 0} , and show that ∂Ω = {x ∈ L : h(x) = 0} . ∂Ω ⊂ {x ∈ L : h(x) = 0} is always true, it remains to show the other inclusion. Let x ∈ L such that h(x) = 0. As |fn (x)| + |∇fn (x)| ≥ α for all x in L, |∇h (x)| ≥ α > 0. Consider the function φ : R −→ R, φ(t) = h(x + t∇h(x)) with φ(0) = 0 and  φ (0) = |∇h (x)|2 . For xn = x + n1 ∇h(x), φ( n1 ) > 0 and there exists n0 ∈ N such that for all n ≥ n0 , xn ∈ Ωn and xn converges to x ∈ Ω. This implies that x ∈ ∂Ω, that is {x ∈ L : h(x) = 0} ⊂ ∂Ω. It then obvious that Ω is of class C 2 . Now, according to Lemma 5.1, the Ωn converge in the compact sense, to Ω.  Proof (of the Theorem 3.6). Using the variational formulation of the Dirichlet problem P (ω, f ), where f is chosen to be positive function with compact support, we have   2 |∇uω (x)| dx = f uω . ω

ω

If uD denotes the solution of the Dirichlet problem P (D, f ), by the maximum principle, 0 ≤ uω ≤ uD and so    2 dx ≥ − f uD J(ω) = − f uω + σP (ω) + k ω

ω

D

and inf J exists. Let Ωn be a minimizing sequence in OC (the Ωn can be defined as in Theorem 3.5). According to Theorem 3.5, there exist an open set Ω of class C 2 , and a K subsequence of Ωn (still denoted by Ωn ) such that Ωn −→ Ω. Proposition 3.3 implies H that Ωn −→ Ω, V (Ωn ) converges to V (Ω) and P (Ω) ≤ lim inf P (Ωn ). Proposition 3.4 n→+∞   implies that D f un χΩn converges to D f uΩ χΩ . Hence, J(Ω) ≤ lim inf J(Ωn ). Now as Ω ∈ OC , then J(Ω) = M inJ(ω).

n→+∞

ω∈OC



Proof (of the Theorem 3.7 ). ∂Ω = ∂C, otherwise Ω = int(C), H∂C = H∂Ω and uΩ = uC . But the optimality condition (5) gives |∇uC |2 = |∇uΩ |2 ≤ σH∂Ω + k 2 = σH∂C + k 2 on ∂C,

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which contradicts (1) . Now, suppose that ∂Ω ∩ ∂C = ∅. As uΩ and uC are in C 2 (C) , ΔuΩ = −f = ΔuC in int(C) and uΩ ≥ 0 = uC on ∂C, the maximum principle implies that uΩ ≥ uC in int(C). But uΩ = uC in int(C), then uΩ > uC in int(C). Now, since C satisfies the interior ball condition, uΩ = uC on ∂Ω ∩ ∂C, then Part 2 of Lemma 2.5 gives ∂uΩ ∂uC < on ∂Ω ∩ ∂C, ∂n ∂n ∂uΩ , or again, since |∇uΩ | = − ∂n |∇uC | < |∇uΩ |

on ∂Ω ∩ ∂C.

As |∇uΩ |2 ≤ σH∂Ω + k 2 on ∂Ω ∩ ∂C then |∇uC |2 < σH∂Ω + k 2 on ∂Ω ∩ ∂C. But H∂Ω ≤ H∂C (because Ω and C are of class C 2 and int(C) ⊂ Ω), so |∇uC |2 < σH∂C + k 2 on ∂C which is absurd. It then follows that C is strictly contained in Ω. Now, ∂Ω = Γ1 , otherwise ∂C is the evolute to ∂Ω and so ∂Ω ∩ ∂C = ∅, which Ω is continuous contradicts the fact that C is strictly contained in Ω. As we assume that ∂u ∂ν on ∂Ω, |∇uC |2 = σH∂Ω + k 2 on ∂Ω. 

6

Examples

Example 6.1. Consider the case where C is some closed ball B(0, R) and f is like in Theorem 3.7. Let EN be the fundamental solution of −Δ in RN . It is given by ⎧ ⎪ ⎪ EN (x) = − 1 log |x| for N = 2 ⎪ 2π ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ EN (x) =

1 1 kN |x|N −2

for N ≥ 3,

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(2(2−N )πN/2 ) where kN = . Γ(N/2) One knows that the solution of the Dirichlet problem P (B(0, R), f ) is given by (see for instance [6]),     |y| x Ry EN (x − y) − EN uC (x) = − f (y) dy R |y| B(0,R) and



R2 − |y|2

|∇uC (x)| = cN B(0,R)

R |x − y|N

f (y) dy for x ∈ ∂C,

where cN is a positive constant depending on N. As H∂C =

1 , R

the condition (1), becomes 

R2 − |y|2

cN B(0,R)

N

R |x − y|

 f (y) dy >

σ + k 2 for R

|x| = R.

(7)

Example 6.2. Once more again, C = B(0, R). Assume that f is radially symmetric. Then, the Dirichlet problem P (B (0, R) , f ) becomes N −1  uC = f (r) for r ∈ ]0, R[ , r uC (R) = 0.    By the first equation, rN −1 uC = −rN −1 f (r) . As uC (R) = 0, we get 

−uC −

r

N −1



uC (r) = R

N −1



uC (R) +



R

sN −1 f (s) ds.

r



As r → 0, we shall have rN −1 uC (r) → 0 (otherwise we get a distributional contribution to ΔuC at the origin). Thus  R 1  −uC (R) = N −1 sN −1 f (s) ds, R 0 R

and the condition (1), is given by RN1−1 0 sN −1 f (s) ds > Rσ + k 2 .

7

Final remarks

Remark 7.1. The hypothesis in Theorem 3.5 about the local regularity are not too restrictive because of, for instance, results due to E. Dibenedetto [8], J.L. Lewis [15] and G.M. Lieberman [16]. Remark 7.2. The case p = 2 and σ = 0 was studied in [2] where the authors gave an analogous sufficient condition of existence of a solution for the quadrature surfaces free boundary problem in two and three dimensions. They also showed that the sufficient condition given in Example 2. is a necessary one.

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Remark 7.3. Consider the following free boundary problem ⎧ ⎪ ⎪ −Δp uΩ = −div(|∇uΩ |p−2 ∇uΩ ) = f in Ω, ⎪ ⎪ ⎨ (I) uΩ = 0 on ∂Ω, ⎪ ⎪ ⎪ ⎪ p ⎩ |∇uΩ | = √ σH∂Ω + k p on ∂Ω, where H∂Ω denotes the mean curvature of ∂Ω, σ > 0, p > 1 and p = 2. As far as the author knows, this problem is still open. Using essentially the γconvergence [4] and the Hopf’s comparison principle [20], we can show [1] that the problem (I) has a solution if

|∇uC | > p σH∂C + k p on C.

References [1] M. Barkatou: “Sufficient condition of existence for a free boundary problem for the p-Laplacian”, Ann. Sci. Math. Qu´ebec, Vol. 26(2), (2002), pp. 123–132. [2] M. Barkatou: D. Seck and I. Ly: “An existence result for a quadrature surface free boundary problem”, Centr. Eur. J. Math., Vol. 3(1), (2005), pp. 39–57, to appear. [3] M. Barkatou: : Some geometric properties for a class of non Lipschitz-domains, New York J. of Math, Vol. 8, (2002), pp. 189–213. [4] D. Bucur and P. Trebeschi: “Shape Optimization Problems Governed by Nonlinear State Equations”, Proc. Roy. Sc. Edinburgh, Vol. 128A, (1998), (1998), pp. 945–963 [5] D. Chenais: “On the existence of a solution in a domain identification problem”, J. Math. Anal. Appl., Vol. 52, (1975), pp. 189–289. [6] R. Dautray and J.L. Lions: Analyse math´ematique et calcul num´erique pour les sciences et les techniques, Vol. I et II, Masson, Paris, 1984. [7] J. Decloux: On the two-Dimensional Magnetic Shaping Problem Without Surface tension, Ecole Polytechnique F´ed´erale de Lausanne, Suisse, 1990. [8] E. Dibenedetto: : “C 1+α local regularity of weak solutions of degenerate elliptic equations”, Nonlinear Analysis., Vol. 7, (1983), pp. 827–850. [9] D. Gilbarg and N.S. Trudinger: Elliptic partial equations of second order, SpringerVerlag, 1983. [10] M. Hayouni and A. Novruzi: “Sufficient condition for existence of solution of a free boundary problem”, Quart. Appl. Math., to appear. [11] A. Henrot: “Continuity with respect to the domain for the laplacian: a survey”, Control and Cybernetics, Vol. 23(3), (1994), pp. 427–443. [12] A. Henrot and M. Pierre: “About Critical Points of The Energy in an Electromagnetic Shaping Problem”, In: J.P. Zol´esio (Ed): Lecture Notes in Control and Information Sciences, Boundary Control and Boundary Variation, Vol. 178, Sophia Antipolis, 1991, pp. 238–252 [13] A. Henrot and M. Pierre: “About Existence of Equilibrium in Electromagnetic Casting”, Quater. Appl. Math., Vol. XLIX, (1991), pp. 563–575.

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[14] M.V. Keldyˇs: “On the solvability and the stability of the Dirichlet problem”, Amer. Math. Soc. Trans., Vol. 2–51, (1966), pp. 1–73. [15] J.L. Lewis: Regularity of the derivatives of solutions to certain degenerate elliptic equations., Indiana Univ. Math. J., Vol. 32, (1983), pp. 849–858. [16] G.M. Lieberman: “Boundary regularity for solutions of degenerate elliptic equations”, Nonlinear Anal., Vol. 12, (1988) pp. 1203–1219. [17] F. Murat and J. Simon: “Quelques r´esultats sur le contrˆole par un domaine g´eom´etrique”, Publ. du labo. d’Anal. Num., Paris VI, (1974) pp. 1–46. [18] A. Novruzi: Contribution en optimisation de formes et Applications, Thesis(PhD), Universit´e Henri Poincar´e Nancy, 1997. [19] J. Sokolowski and J.P. Zolesio: Introduction to shape optimization : shape sensitity analysis, Springer Series in Computational Mathematics, Vol. 10, Springer, Berlin, 1992. [20] P. Tolksdorf: “On the Dirichlet problem for quasilinear equations in domains with conical boundary points”, Comm. Partial Differential Equations, Vol. 8(7), (1983), pp. 773–817.

DOI: 10.2478/s11533-006-0028-3 Research article CEJM 4(4) 2006 585–593

Bayoumi Quasi-differential is different from Fr´ echet-differential Aboubakr Bayoumi∗ Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden Al-Azhar University, College of Science, Mathematics Dept., Naser City 11884, Cairo, Egypt

Received 29 September 2005; accepted 26 July 2006 Abstract: We prove that the Quasi Differential of Bayoumi of maps between locally bounded F spaces may not be Fr´echet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F -spaces here are not necessarily locally convex c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Infinite-dimensional spaces, nonlocally convex F -spaces, quasi-differential (Super or Bayoumidifferential), Fr´echet differential, optimization theory in finite-dimensional spaces MSC (2000): 32k1, 46A16

1

Introduction

In this paper we study the relationship between the concepts of Quasi-Differentiability (Q-differentiability) and Fr´echet-Differentiability (F -Differentiability) of maps between locally bounded F -spaces E and F . By an F -space we mean a space metrized by an F -norm. Their classes will be denoted by QD(E, F ) and F D(E, F ) respectively. Let E and F be p-normed and q-normed spaces respectively (0 < p, q ≤ 1), and U a nonempty open subset of E. For f, g : U → F , and a ∈ U , we say that f and g are quasi tangent (or pq-tangent) to each other at a if lim f (x) − g(x)1/q /x − a1/p = 0.

x→a ∗

E-mail: [email protected]

(1)

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In 1995 we gave the following definition depending on (1): Let E and F be p-normed and q-normed spaces, respectively (0 < p, q ≤ 1) and U a nonempty open set in E. A mapping f : U → F is said to be quasi-differentiable (Bayoumi differential or pqdifferentiable) at a ∈ U , if there exists a continuous linear map Ta ∈ L(E, F ) such that f and the continuous affine linear mapping E  x → f (a) + Ta (x − a) are pq-tangent at a, that is, lim f (x) − f (a) − Ta (x − a)p /x − aq = 0. (2) x→a

Ta is called the quasi-differential of f at a (or pq-differential) and is denoted by Df (a). It is to be noted that if E and F are both p-normed spaces or quasi-normed spaces with the same quasi-norm constants, then condition (2) turns out to be as the following classical one: lim f (x) − f (a) − Ta (x − a)/x − a = 0. x→a

We have seen in [2] that quasi-differentiability for a mapping between locally bounded F -spaces is a continuous linear operator. This is similar to that for the Fr´echet ones for normed spaces. However, our goal here is to prove that: The two concepts of differentiability are totally different. But the new one, that is the quasi-differentiability, is more suitable for all F -spaces which are locally convex or not. It may play roles in analysis and applied mathematics which cannot be done by the classical Fr´echet differentiability. It is worthwhile to point out that this new concept of differentiability has been discovered while the author was working on finding theorems for the mean value in real and complex locally bounded F -spaces; see [2, 3].

2

Finite-dimensional case

The following lemma shows the equivalence between the quasi-differentiability (Q-differentiability) and Fr´echet-differentiability (F -differentiability) in finite-dimensional spaces. Lemma 2.1. Let E and F be any two finite-dimensional p-normed spaces (1 > p > 0) and let f be a map between them. Then f is Q-differentiable if and only if f is F differentiable. Proof. Note that

f (x) − f (a) − Ta (x − a)1/p =0 x→a x − a1/p lim

if and only if

(3)

f (x) − f (a) − Ta (x − a) =0 x→a x − a lim

for a continuous linear mapping Ta ∈ L(E, F ), where a, x ∈ E. We know that finitedimensional topologies are equivalent. 

A. Bayoumi / Central European Journal of Mathematics 4(4) 2006 585–593

587

The following theorem shows that we still have the linear transformation, defined by the Jacobian matrix, to find quasi-differentials when we deal with finite dimensional spaces. Theorem 2.2. Let E and F be any two finite-dimensional p-normed and q-normed spaces, whose dimensions are n and m respectively (1 > p, q > 0). Let U ⊂ E be open and f : U → Rm a function defined by f (a) = (f1 (a), . . . , fm (a)), where fi : U → R (1 ≤ i ≤ m). Then if f is quasi-differentiable at a ∈ U , each of ∂fi the partial derivatives ∂x (a) exists, (1 ≤ i ≤ m, 1 ≤ j ≤ n). Furthermore, the quasij n differential Df (a) : R → Rm is the linear transformation defined by the Jacobian matrix of f at a. Proof. Let L = Df (a) : Rn → Rm . Let e1 , . . . , en and e1 , . . . , em be the standard basis for Rn and Rm . Assume (aij ) is the Jacobian matrix of f at a, so that Lej =

m 

aij ei ,

1≤j≤n

i=1

Since f is quasi-differentiable at a, it follows that f (a + hej ) − f (a) − L(hej )1/q = 0, x→a hej 1/p lim

1≤j≤n

Now 1/q

lim

x→a

f (a + hej ) − f (a) − hL(ej ) |h|



n  fi (a + hej ) − fi (a) = lim − aij |q | x→a h i=1

1/q = 0;

hence each term in the sum must tend to zero. Therefore ∂fi (a) fi (a + hej ) − fi (a) = lim = aij , x→a ∂xj h This proves that

∂fi (a) ∂xj

1 ≤ i ≤ m, 1 ≤ j ≤ n,

exists and equals aij , completing the proof of the theorem.



The following gives an example of a non quasi-differentiable function. Example 2.3. Let E = lp , 1 > p > 0. The function f (x) : lp → R defined by f (x) = |x1 | is not quasi-differentiable at 0. Notice that [|x − 0 − T0 (x)|q ]1/q ||x1 | − αx1 | = lim  . 1/p x→0 x→0 ( x − 0 |xi |p )1/p lim

(4)

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This shows that the limit does not exist for any constant α since it depends on the way that x1 approaches 0 That is, f is not quasi-differentiable at 0; i.e., f ∈ / QD(lp ; R) . Remark 2.4. For E = lp1 = lp ∩ R, we can also take f (x) = |x1 |p : lp1 → R to have

   |x1 |p  ||x1 |p − αx1 | lim − α = lim  x→0 x1 →0 |x1 | x1

which does not also exist if p equals, say, 1/2 .

2.1 Maxima and minima at interior points We shall be concerned here with maxima and minima at points of an open set in a finitedimensional p-normed space E (0 < p < 1). We show that we still have the quadratic forms defined by the Hessian matrix of the second order quasi-differentials of the given function f: U →R to find out local maximum and local minimum of f . As in the classical case, we say that a point v0 ∈ U is a local maximum for f if f (v) ≤ f (v0 ) for all v in some ball about v0 Similarly, v1 is a local minimum for f if f (v) ≥ f (v1 ) for all x in some ball about v1 . Just as in the classical case, we say that the interior point v0 is a critical point of f if the quasi-differential, f  (v0 ) = Dv0 f = 0, the zero transformation. One reason for this definition is the following theorem. Theorem 2.5. If v0 is an interior local maximum or local minimum point of f: U →R and f is quasi differentiable near v0 , then v0 is a critical point of f . Proof. This is similar to the classical proof for differentiability of functions.



Remark 2.6. According to Theorem 2.5, it suffices to find the maximum and minimum of some functions f . We know that for some purposes U may be a compact set, and f (v) = 0 if v is a boundary point for U and that f > 0 somewhere in U .

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589

2.2 Criteria for local maxima and minima Let U be an open subset of Rn , let f : U → R be a function with a continuous second order quasi-derivative, and let v0 be a point of U which is a critical point of f . We would like to have a method of determining whether v0 is a local maximum, or a local minimum or neither. We shall look at the second quasi-derivatives. We use the Hessian of f at v, H(f )(v), which is a matrix formed from the second partial derivatives. If the partial derivatives are continuous as we are assuming, then H(f )(v) is a symmetric matrix. By the arc segment Avv0 from v0 to v we mean the set   Avv0 = w ∈ U ; w = (1 − λ)1/p v0 + λ1/p v, 0 ≤ λ ≤ 1 ,

0 < p < 1.

If p = 1, Avv0 becomes the line segment from v0 to v. It is proved in [3] that the unit ball BE of a locally bounded space E contains Avv0 if v0 and v are in BE . Theorem 2.7. Let U be an open set in Rn and f: U →R a function having continuous second order partial derivatives. Let v0 be a critical point of f and let H(f )(v0 ) be the Hessian of f at v0 . (i) If H(f )(v0 ) is positive definite, then v0 is a local minimum. (ii) If H(f )(v0 ) is negative definite, then v0 is a local maximum. (iii) If H(f )(v0 ) is indefinite, then v0 is neither a local maximum nor a local minimum. Proof. The proof is like that for functions on R; see [8].

3



Infinite-dimensional case

The following example is for a non Fr´echet and non quasi-differentiable function. Example 3.1. Put E = lp , F = R, a = 0. We are going to establish a function f : E → F which is neither Fr´echet differentiable nor quasi-differentiable at a = 0, i.e., f∈ / QD ∪ F D. That is, f (x) − f (a) − Ta (x − a)1/q = 0 x→a x − a1/p lim

even when p = q = 1. Consider the following function f (x) =

∞ 1−1/p  1 1

n

· xn , x = (xn ) ∈ lp ,

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and the following set which approaches zero in l1   A1 = x(n) ; x(n) = (0, . . . , 0, n−1/p , 0, . . . ), n ∈ N . Now  1 (1−1/p) ∞  ∞ ( ) .x − f (x) − T0 (x) n 1 n 1 αn xn   lim = lim n x→0 x →0 |xi | x  1 (1−1/p) −1/p ( n ) ·n − αn xn  = lim n→∞ ( n1 )1/p   | 1 − αn ( n1 )1/p | = lim n 1 1/p = lim αn − n−1+1/p  = ∞ n→∞ n→∞ (n) which never goes to zero. That is, f ∈ / F D. Notice that x(n) → 0 in l1 . In the same way we can show that f ∈ / QD. In fact, let f be given above as a function on lp .  1 1−1/p   ∞ .xn − βn xn 1/q f (xn ) − T0 (xn )1/q 1 (n)  lim = lim xn →0 xn →0 xn 1/p ( |xi |p )1/p | 1 − βn ( n1 )1/p | = lim n 1 1/p = lim |n1/p−1 − βn | = ∞; n→∞ n→∞ (n)  βn xn , ( βn ) ∈ l∞ , and xn → 0 in lp . that is, f ∈ / QD, where T0 (x) = The following example shows that the class of F -differentiable maps may not be a subset of the class of quasi-differentiable maps. Example 3.2. There is a Fr´echet-differentiable map f : E → F from a p-normed space E into a q-normed space F (1 > p, q > 0) which is not quasi-differentiable. That is F D(E; F )  QD(E; F ).

(6)

More precisely, the function G : L1/2 (I) → R given by  G(f ) = f  =

1

0

|f |1/2 dx

has this property. Proof. Take first E = Lp (I), the space of integrable functions on I = [0, 1], and F = R, the real field with the q-norm, f = 0, the zero function in Lp (I). Consider the function G : Lp (I) → R given by  G(f ) = f  =

0

1

|f |p dx

(7)

A. Bayoumi / Central European Journal of Mathematics 4(4) 2006 585–593

Then

591

G(f ) − G(0) − 0)1/q f 1/q 1 = lim = lim , f →0 f →0 f 1/p f →0 f 1/p−1/q f − 01/p lim

So the limit does not exist when p1 > 1q , that is, for all values of (1 > q > p > 0), say for p = 1/2. Notice that the dual space of Lp (I) equals {0}. Therefore, the function G is not quasi-differentiable at f = 0. However, we notice that L1 (0, 1) ⊂ L1l2 (0, 1), For if f ∈ L1 (0, 1), then  1 0

|f

1/2

 .1|dx ≤

0

1

(8)

1/2  |f |dx ·

1/2

1

1dx

0

that is, f ∈ L1/2 (0, 1). Therefore we have considered our function f in L1 (0, 1)∩ L1/2 (0, 1). We claim that G(f ) = f  is a Fr´echet differentiable function at 0. Notice that  T0 (x) ∈ (L1 ) , the dual space which is isomorphic to L∞ . Now G(f ) − G(0) − T0 (f − 0) |G(f ) − T0 (f )| = lim f →o f →0 f − 0L1 f L1 |f  − T0 (f )| = lim =0 f →0 f L1 lim

1  as f → 0 in L1 , where T0 (f ) ∈ (L1 )  L∞ may be taken equal to f  = 0 |f |1/2 dx, since f is considered as an element in L1 . This completes the proof of the example.  In what follows we prove that the class of Q-differentiable maps may not be contained in the class of F -differentiable maps. Theorem 3.3. Not every quasi-differentiable map f from a p-normed space E into a q-normed space F is Fr´echet-differentiable, that is QD(E; F )  F D(E; F ).

(9)

Proof. As in the above example, we take E = Lp (I), the space of integrable functions on I = [0, 1], F = R, with the q-norm, and f = 0, the zero function in Lp (I). Consider the function G : Lp (I) → R given by

 G(f ) = f  =

0

1

|f |p dx.

We have seen that G(f ) − G(0) − 0)1lq f 1/q = lim x→a f →0 f 1/p f − 01/p 1 = lim . 1/p−1/q f →0 f  lim

(10)

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Now if 1q − p1 > 0, that is, when q < p, the limit will approach to zero and G(f ) will be a quasi-differentiable function. But if q = p (say, p = q = 1/2) we have non-zero limit since f 1/p = 1. (11) f →0 f 1/p So G(f ) will never be a Fr´echet-differentiable function. In addition, we note that for q > p, f is not quasi-differentiable.  lim

Remark 3.4. The relation between the classes of Q-differentials and F -differentials may be summarized so far by the following diagram. (We point out here that the term quasidifferentiable has been used in a quite a different way by some mathematicians, see Dem’yanov/Vasil’ev, Nondifferentiable Optimization, Optimization Software, Inc., 1985. Therefore the term “pq-differentiable” or “super-differentiable” or “Bayoumi differentiable” may also be used here instead of; see the Mathematical Reviews 2004). Gateaux differentiable maps

Quasi-differentiable maps

Frechet-differentiable maps

QD

FD

c_infinity F-differential intersects with c_infinity Q-differential

Fig. 1 Relation between the different classes of differentibility’s.

References [1] A. Bayoumi: Foundations of Complex Analysis in Non Locally Convex Spaces, Functions theory without convexity condition, North Holland, Mathematical studies, Vol. 193, 2003. [2] A. Bayoumi: “Mean-Value Theorem for complex locally bounded spaces”, Communication in Applied Non-Linear Analysis, Vol. 4(3), (1997).

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[3] A. Bayoumi: “Mean-Value Theorem for real locally bounded spaces”, Journal of Natural Geometry, London, Vol. 10, (1996), pp. 157–162. [4] A. Bayoumi: “Fundamental Theorem of Calculus for locally bounded spaces”, Journal of Natural Geometry, London, Vol. 15(1–2), (1999), pp. 101–106. [5] A. Bayoumi: “Mean-Value Theorem for Definite Integral of vector-valued functions of p-Banach spaces”, Algebra, Groups and Geometries, Vol. 22(4), (2005). [6] A. Bayoumi: “Bolzano’s Intermediate-Value Theorem for Quasi-Holomorphic Maps”, Central European Journal of Mathematics, Vol. 3(1), (2005), pp. 76–82. [7] B.S. Chae: Holomorphy and calculus in normed spaces, Marcel Dekker, 1985. [8] l.J. Corwin and R.H. Szczarba: Multivariable Calculus, Marcel Dekker, 1982. [9] S. Rolewicz: Metric linear spaces, Monografje Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk, 1972.

DOI: 10.2478/s11533-006-0032-7 Research article CEJM 4(4) 2006 594–599

A note on absolute weighted mean summability factors † ¨ Hu ¨seyin Bor∗ , Hikmet Seyhan Ozarslan Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey

Received 15 June 2005; accepted 23 June 2006 ¯ , pn ; δ | summability Abstract: In this paper we have proved a main theorem concerning the | N k ¨ methods, which generalizes a result of Bor and Ozarslan [3]. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Absolute summability, quasi power increasing sequence MSC (2000): 40D15, 40F05, 40G99

1

Introduction

A positive sequence (γn ) is said to be quasi β-power increasing sequence if there exists a constant K = K(β, γ) ≥ 1 such that Knβ γn ≥ mβ γm

(1)

holds for all n ≥ m ≥ 1. We denote by BVO the BV ∩ CO , where CO and BV of the  null sequences and sequences with bounded variation, respectively. Let an be a given α α infinite series with partial sums (sn ). We denote by σn and tn the n-th Ces`aro means of order α, with α > −1, of the sequence (sn ) and (nan ), respectively, i.e., σnα ∗ †

E-mail: [email protected] E-mail: [email protected]

n 1  α−1 = α A sv An v=0 n−v

(2)

¨ H. Bor, H.S. OZarslan / Central European Journal of Mathematics 4(4) 2006 594–599

tαn =

n 1  α−1 A vav , Aαn v=1 n−v

595

(3)

where Aαn = O(nα ), α > −1, Aα0 = 1 and Aα−n = 0 f or n > 0.  The series an is said to be summable | C, α |k , k ≥ 1 and α > −1, if (see [4]) ∞ ∞   1 α k k−1 α α k n | σn − σn−1 | = | t | < ∞, n n n=1 n=1

(4)

(5)

and it is said to be summable | C, α; δ |k , k ≥ 1, α > −1 and δ ≥ 0, if (see[5]) ∞ 

n

δk+k−1

|

σnα

−

α σn−1 |k =

n=1

∞ 

nδk−1 | tαn |k < ∞.

(6)

n=1

Let (pn ) be a sequence of positive numbers such that Pn =

n 

pv → ∞ as n → ∞,

(P−i = p−i = 0, i ≥ 1).

(7)

v=0

The sequence-to-sequence transformation n 1  un = pv sv Pn v=0

(8)

¯ , pn ) mean of the sequence (sn ), generated by the defines the sequence (un ) of the (N sequence of coefficients (pn ) (see [6]).  ¯ , pn | , k ≥ 1, if (see [1]) The series an is said to be summable | N k ∞ 

(Pn /pn )k−1 | Δun−1 |k < ∞

(9)

n=1

¯ , pn ; δ | , k ≥ 1 and δ ≥ 0, if (see [2]) and it is said to be summable | N k ∞ 

(Pn /pn )δk+k−1 | Δun−1 |k < ∞,

(10)

n pn  =− Pv−1 av , Pn Pn−1 v=1

(11)

n=1

where Δun−1

n ≥ 1.

¯ , pn ; δ | summability In the special case when δ = 0 (resp. pn = 1 for all values of n) | N k ¯ , pn | (resp. | C, 1; δ | ) summability. is the same as | N k k ¨ Quite recently Bor and Ozarslan [3] proved the following theorem.

¨ H. Bor, H.S. OZarslan / Central European Journal of Mathematics 4(4) 2006 594–599

596

Theorem 1.1. Let (Xn ) be a quasi β-power increasing sequence for some 0 < β < 1. Let there be sequences (βn ) and (λn ) such that | Δλn |≤ βn ,

(12)

βn → 0 as n → ∞,

(13)

∞ 

n | Δβn | Xn < ∞,

(14)

n=1

| λn | Xn = O(1) as n → ∞. If

n  | sv |k = O(Xn ), v v=1

then the series



m  pn | sn |k = O(Xm ), P n=1 n

as

n→∞

as m → ∞

(15)

(16)

(17)

¯ , pn | for k ≥ 1. an λn is summable | N k

Remark 1.2. For the work of [3], we have realized that the condition (λn ) ∈ BVO should added in the statement of Theorem 1.1.

2

The main result

¯ , pn ; δ | summaThe aim of this paper is to prove the following general theorem for | N k bility methods.

Theorem 2.1. Let (λn ) ∈ BVO and let (Xn ) be a quasi β-power increasing sequence for some 0 < β < 1. If the conditions (12)-(15) of Theorem 1.1 are satisfied and if (pn ) is a sequence such that m  Pn ( )δk−1 | sn |k = O(Xm ), m → ∞, (18) p n n=1

then the series



m  Pn | sn |k ( )δk = O(Xm ), m → ∞, pn n n=1    ∞ δk  Pn δk−1 1 Pv 1 ( ) =O , p P p P n n−1 v v n=v+1

¯ , pn ; δ | for k ≥ 1 and 0 ≤ δ < 1/k. an λn is summable | N k

(19)

(20)

¨ H. Bor, H.S. OZarslan / Central European Journal of Mathematics 4(4) 2006 594–599

597

Remark 2.2. It may be noted that, if we take δ = 0 in this theorem, then we get Theorem 1.1 In this case condition (20) reduces to m+1 

m+1  pn 1 1 1 = ( − ) = O( ) as m → ∞, P P Pn−1 Pn Pv n=v+1 n n−1 n=v+1

which always holds. We need the following lemma for the proof of our theorem.

Lemma 2.3 ([7]). Under the conditions on (Xn ), (βn ) and (λn ) as taken in the statement of the Theorem 2.1, the following conditions hold, when (14) is satisfied: nβn Xn = O(1) as n → ∞, ∞ 

βn Xn < ∞.

(21) (22)

n=1

3

Proof of the Theorem 2.1

¯ , pn ) mean of the series  an λn . Then, by definition and changing Let (Tn ) denotes the (N the order of summation, we have Tn =

v n n 1   1  pv ai λ i = (Pn − Pv−1 )av λv . Pn v=0 i=0 Pn v=0

Then, for n ≥ 1, we have Tn − Tn−1

n pn  = Pv−1 av λv Pn Pn−1 v=1

n−1 n−1 pn  pn  pn =− p v λv sv + Pv Δλv sv + λn sn Pn Pn−1 v=1 Pn Pn−1 v=1 Pn = Tn,1 + Tn,2 + Tn,3 .

Since | Tn,1 + Tn,2 + Tn,3 |k ≤ 3k (| Tn,1 |k + | Tn,2 |k + | Tn,3 |k ), to complete the proof of the Theorem 2.1, it is enough to show that ∞  n=1

(Pn /pn )δk+k−1 | Tn,r |k < ∞ f or

r = 1, 2, 3.

(23)

¨ H. Bor, H.S. OZarslan / Central European Journal of Mathematics 4(4) 2006 594–599

598

Now, applying H¨older’s inequality with indices k and k  , where get that m+1 

δk+k−1

(Pn /pn )

| Tn,1 | = O(1) k

n=2

= O(1)

m+1  n=2 m+1  n=2

×{

δk−1

(Pn /pn )

(Pn /pn )δk−1

1

n−1 

Pn−1

v=1

= O(1) = O(1) = O(1)

m  v=1 m 

n−1  (Pn−1 ) { pv | sv || λv |}k −k

1

n−1 

Pn−1

v=1

p v | sv | | λv | k

v=1

pv | sv |k | λv |k

m+1 

(Pn /pn )δk−1

n=v+1

1 Pn−1

(Pv /pv )δk−1 | sv |k | λv |

v=1 m−1 

v  Δ | λv | (Pi /pi )δk−1 | si |k

+ O(1) | λm |

= O(1)

+ k1 = 1 and k > 1, we

pv }k−1

v=1

= O(1)

1 k

m−1  v=1 m−1 

m 

i=1

(Pv /pv )δk−1 | sv |k

v=1

| Δλv | Xv + O(1) | λm | Xm βv Xv + O(1) | λm | Xm

v=1

= O(1) as m → ∞, by virtue of the hypotheses of the Theorem 2.1 and Lemma 2.3. Again applying H¨older’s inequality, as in Tn,1 , we have that m+1 

δk+k−1

(Pn /pn )

| Tn,2 | ≤ k

n=2

≤

m+1  n=2 m+1 

δk−1

(Pn /pn )

(Pn /pn )δk−1 (Pn−1 )−k

n=2

×(

n−1  (Pn−1 ) { Pv | Δλv || sv |}k −k

1

n−1 

Pn−1

v=1

= O(1)

m+1 

= O(1)

v=1

Pv | Δλv || sv |k

v=1

Pv | Δλv |)k−1

(Pn /pn )δk−1

n=2

m 

v=1 n−1 

Pv βv | sv |k

1

n−1 

Pn−1

v=1

m+1  n=v+1

Pv βv | sv |k

(Pn /pn )δk−1

1 Pn−1

¨ H. Bor, H.S. OZarslan / Central European Journal of Mathematics 4(4) 2006 594–599

= O(1) = O(1)

m 

(Pv /pv )δk vβv

v=1 m−1 

Δ(vβv )

v=1

v 

599

| sv |k v

(Pr /pr )δk

r=1

| sr |k r

m k  δk | sv | (Pv /pv ) + O(1)mβm v v=1

= O(1) = O(1) = O(1)

m−1  v=1 m−1  v=1 m−1 

| Δ(vβv ) | Xv + O(1)mβm Xm | (v + 1)Δβv − βv | Xv + O(1)mβm Xm v | Δβv | Xv + O(1)

v=1

m−1 

| βv | Xv

v=1

+ O(1)mβm Xm = O(1) as m → ∞, in view of the hypotheses of the Theorem 2.1 and Lemma 2.3. Finally, as in Tn,1 , we have that m  n=1

δk+k−1

(Pn /pn )

| Tn,3 | = O(1) k

m 

(Pn /pn )δk−1 | sn |k | λn |

n=1

= O(1) as m → ∞.

Therefore, we get that (23). This completes the proof of the Theorem 2.1. If we take pn = 1 in this Theorem 2.1, then we have a new result concerning the | C, 1; δ |k summability factors for infinite series.

References [1] H. Bor: “On two summability methods”, Math. Proc. Cambridge Philos. Soc., Vol. 97, (1985), pp. 147–149. ¯ , pn ; δ | summability of factored Fourier series”, J. [2] H. Bor: “On local property of | N k Math. Anal. Appl., Vol. 179, (1993), pp. 646–649. ¨ [3] H. Bor and H.S. Ozarslan: “On the quasi power increasing sequences”, J. Math. Anal. Appl., Vol. 276, (2002), pp. 924–929. [4] T.M. Flett: “On an extension of absolute summability and some theorems of Littlewood and Paley”, Proc. London Math. Soc., Vol. 7, (1957), pp. 113–141. [5] T.M. Flett: “Some more theorems concerning the absolute summability of Fourier series”, Proc. London Math. Soc., Vol. 8, (1958), pp. 357–387. [6] G.H. Hardy: Divergent Series, Oxford University Press, Oxford, 1949. [7] L. Leindler: “A new application of quasi power increasing sequences”, Publ. Math. Debrecen, Vol. 58, (2001), pp. 791–796.

DOI: 10.2478/s11533-006-0025-6 Research article CEJM 4(4) 2006 600–623

Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL and MV-algebras∗ Leonardo Manuel Cabrer† , Sergio Arturo Celani‡ , Departamento de Matem´ aticas, Facultad de Ciencias Exactas, Universidad Nacional del Centro, Pinto 399, 7000 Tandil, Argentina

Received 13 July 2005; accepted 13 June 2006 Abstract: In this work we give a duality for many classes of lattice ordered algebras, as Integral Commutative Distributive Residuated Lattices MTL-algebras, IMTL-algebras and MV-algebras (see page 604). These dualities are obtained by restricting the duality given by the second author for DLFIalgebras by means of Priestley spaces with ternary relations (see [2]). We translate the equations that define some known subvarieties of DLFI-algebras to relational conditions in the associated DLFI-space. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Bounded distributive lattices with fusion and implication, Priestley duality, residuated distributive lattices, MTL-algebras, IMTL-algebras, MV-algebras MSC (2000): 03G10, 03B50, 06D35,06D72

1

Introduction

In recent years many varieties of algebras associated to multi-valued logics have been introduced . The majority of these algebras are commutative integral distributive residuated lattices [8] with additional conditions, as for example, the variety of MV-algebras [4], the variety of BL-algebras, and the varieties of MTL-algebras and ITML-algebras [5, 6], recently introduced. All these classes of algebras are bounded distributive lattices with two additional binary operations ◦ and → satisfying special conditions. On the other hand, in [2], the varieties DLF of DLF-algebras, DLI of DLI-algebras and DLFI ∗

The authors are supported by the Agentinian Consejo de Investigaciones Cientificas y Tecnicas (CONICET). † E-mail: [email protected] ‡ E-mail: [email protected]

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601

of DLFI-algebras were introduced and studied. These algebras are bounded distributive lattices with one or two additional binary operations ◦ and → satisfying certain minimal conditions. The variety of DLFI-algebras is a common generalization of many varieties of algebras including residuated lattices [9], relevant algebras [18], MV-algebras, etc. The main objective of [2] was to develop a Priestley duality for these varieties, and in special, to give a topological characterization of the congruences in these varieties. The representation given in [2] is a generalization of the representation developed by A. Urquhart in [18] for Relevant algebras by means of relational Priestley spaces. In this paper we investigate a topological representation for some algebraic structures of fuzzy logic using the results given in [2]. In particular, we shall give a duality for integral commutative distributive residuated lattices, MTL-algebras, IMTL-algebras and MValgebras. In Section 2 we will recall the Priestley duality given in [2] for DLI-algebras, DLFalgebras and DLFI-algebras. In Section 3 we shall prove that a DLI-algebra or a DLFalgebra satisfies certain identities if and only if its dual space satisfies certain first order conditions. Some of these correspondences are acquaintances in relevance logic and relevant algebras [18], although our presentation is different. In Section 4 we shall give the mentioned Priestley dualities for integral commutative distributive residuated lattices, MTL-algebras, IMTL-algebras and MV-algebras. In section 5 we will discuss how the dualities given by the second author in [2] are related to other known representations and dualities for lattice with different operators.

2

Preliminaries

In this section we recall the Priestley´s duality for bounded distributive lattices (see [13, 14]), the definitions of DLI, DLF, and DLFI-algebras and its Priestley´s style duality given in [2]. We will use the following conventions. Given a set X and a subset Y ⊆ X we will note X\Y = {x ∈ X : x ∈ / Y }. When there is no risk of misunderstanding we will note c Y instead of X\Y . Given a bounded distributive lattice A = A, ∨, ∧, 0, 1, if H ⊆ A, we will note F (H) and I(H) to the filter and the ideal generated by H respectively. The collection of filters of A will be denoted by F i (A). We will note X (A) to the set of prime filters of A. Given a poset X, ≤, a set Y ⊆ X is called increasing if for every x ∈ Y and every y ∈ X, if x ≤ y then y ∈ Y . Dually, Y ⊆ X is said to be decreasing if for every x ∈ Y and every y ∈ X, if y ≤ x then y ∈ Y . We will note [Y ) ((Y ]) to the least increasing (decreasing) set that contains Y. If Y = {y} then we will note [Y ) by [y) and (Y ] by (y] , and we will say that a set Z ⊆ X is principal increasing (decreasing) if there exists x ∈ X such that Z = [x) (Z = (x]). The set of all increasing subsets of X will be denoted by Pi (X) and the power set of X by P(X). If X1 , ≤1  and X2 , ≤2  are posets, a map f : X1 −→ X2 is increasing when for every x, y ∈ X1 such that x ≤1 y it holds that

602

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f (x) ≤2 f (y). A Priestley space is a triple X, ≤, τ  where X, ≤ is a poset and X, τ  is a topological Stone space (compact, Hausdorff and 0-dimensional) that is totally separated in the order, i.e., for every x, y ∈ X such that x ≤ y, there is a clopen (closed and open in τ ) increasing set U ⊆ X such that x ∈ U and y ∈ / U . If X, ≤, τ  is a Priestley space, the set of all clopen increasing subsets of X will be denoted by D(X). Since D(X) is a ring of sets, D(X), ∪, ∩, ∅, X is a bounded distributive lattice. We will denote by PR the category whose objects are Priestley spaces and whose arrows are continuous and increasing functions. Given a bounded distributive lattice A, consider the topology τA on X (A) generated by the subbase whose elements are the sets of the form β(a) = {P ∈ X (A) : a ∈ P } and β(a)c = X (A) \ β(a), for each a ∈ A. Then X (A), ⊆, τA is a Priestley space. The map β : A → D(X (A)) is a bounded lattice isomorphism. For every Priestley space X, ≤, τ , the map ε : X → X (D(X)) given by ε (x) = {U ∈ D (X) : x ∈ U }, for x ∈ X, is an order-isomorphism and a homeomorphisms of topological spaces. Moreover, there is a duality between the algebraic category of bounded distributive lattices and the category PR. For details on Priestley duality see [13]. Now we recall the definitions of DLI, DLF, and DLFI-algebras and some results given in [2]. Definition 2.1. An algebra A = A, ∨, ∧, →, 0, 1 of type (2, 2, 2, 0, 0) is a Distributive Lattice with Implication (or DLI-algebra) if and only if A, ∨, ∧, 0, 1 is a bounded distributive lattice and the following conditions are satisfied: I1. (a → b) ∧ (a → c) ≈ a → (b ∧ c). I2. (a → c) ∧ (b → c) ≈ (a ∨ b) → c. I3. a → 1 ≈ 1. I4. 0 → a ≈ 1. An algebra A = A, ∨, ∧, ◦, 0, 1 of type (2, 2, 2, 0, 0) is a distributive lattice with Fusion (or DLF-algebra) if and only if A, ∨, ∧, 0, 1 is a bounded distributive lattice and the following conditions are satisfied: F1. a ◦ (b ∨ c) ≈ (a ◦ b) ∨ (a ◦ c). F2. (a ∨ b) ◦ c ≈ (a ◦ c) ∨ (b ◦ c). F3. a ◦ 0 ≈ 0 ◦ a ≈ 0. An algebra A = A, ∨, ∧, ◦, →, 0, 1 is a distributive lattice with Fusion and Implication (or DLFI-algebra) if and only if A, ∨, ∧, ◦, 0, 1 is a DLF-algebra and A, ∧, ∨, →, 0, 1 is a DLI-algebra. Let A be a DLFI-algebra. Let us recall that for any a, b, x ∈ A, if a ≤ b, then x → a ≤ x → b, b → x ≤ a → x, a ◦ x ≤ b ◦ x, and x ◦ a ≤ x ◦ b. We will note by DLF, DLI, DLFI the varieties of DLF, DLI and DLFI-algebras respectively.

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603

Example 2.2. In any distributive lattice A, ∨, ∧, 0, 1 we can define at least four binary operations on its domain such that the algebra A = A, ∨, ∧, →, 0, 1 be a DLI-algebra. Indeed, any of the following binary functions →i : A × A → A, for 1 ≤ i ≤ 4, define structures of DLI-algebra: a →1 b = 1.

a →2 b =

a →3 b =

a →4 b =

⎧ ⎪ ⎨ 1

if b = 1 or a = 0,

⎪ ⎩ 0

if b = 1 and a = 0.

⎧ ⎪ ⎨ 1

if a ≤ b,

⎪ ⎩ 0

if a  b.

⎧ ⎪ ⎨ 1

if a = 0,

⎪ ⎩ b

if a = 0.

Example 2.3. As in the case of implication in any distributive lattice A, ∨, ∧, 0, 1 we can define at least four binary operations on its domain such that the algebra A = A, ∨, ∧, ◦, 0, 1 is a DLF-algebra. The following binary functions ◦i : A × A → A, for 1 ≤ i ≤ 4, define structures of DLF-algebra: a ◦1 b = 0.

a ◦2 b =

a ◦3 b =

a ◦4 b =

⎧ ⎪ ⎨ 1

if a = 0 and b = 0,

⎪ ⎩ 0

if a = 0 or b = 0.

⎧ ⎪ ⎨ a

if b = 0,

⎪ ⎩ 0

if b = 0.

⎧ ⎪ ⎨ b

if a = 0,

⎪ ⎩ 0

if a = 0.

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Example 2.4. Let us consider a relational structure F = X, ≤, R, T , where X, ≤ is a poset, and R and T are ternary relations defined on X such that for each S ∈ {R, T }, if (x, y, z) ∈ S & x ≤ x & y  ≤ y & z ≤ z  , then (x , y  , z  ) ∈ S. It is easy to prove that A (F) = Pi (X) , ∪, ∩, ∗, → ∅, X is a DLFI-algebra, where the operations → and ∗ are defined by: U → V = {x ∈ X : For every y, z ∈ X, ( (x, y, z) ∈ T and y ∈ U ) implies z ∈ V } .

U ∗V

= {z ∈ X : There exists (x, y) ∈ U × V, (x, y, z) ∈ R} .

The algebra A (F) is called the DLFI-algebra associated with F. Now we shall see different examples of DLFI-algebras connected with ordered algebraic structures associated with known multi-valued logics. Definition 2.5. An integral commutative residuated lattice, or ICR-lattice, is a DLFIalgebra A = A, ∧, ∨, ◦, →, 0, 1 satisfying the following axioms: (R) a ◦ b ≤ c if and only if a ≤ b → c, for any a, b, c ∈ A, (A) a ◦ (b ◦ c) ≈ (a ◦ b) ◦ c. (I) 1 ◦ a ≈ a. (C) a ◦ b ≈ b ◦ a. The class of ICR-lattices is a variety and we will denote it by ICRL. For a survey on residuated lattices in general and further references see [9]. Now, we shall recall the definitions of some known subvarieties of ICRL. Let V be a subvariety of ICRL and let Γ be a finite set of equations in the algebraic language of the ICR-lattices. We shall denote by V +Γ the subvariety of V whose elements satisfies the equations Γ. Let us consider the following subvarieties of ICRL: MT L = ICRL + {(a → b) ∨ (b → a) ≈ 1} IMT L = MT L + {a ≈ (a → 0) → 0} . MV

= IMT L + {(a → b) → b ≈ (b → a) → a} .

The variety MT L is the variety of Monoidal T–norm based Logic algebras introduced in [5], the variety IMT L is the variety of Involutive MTL-algebras defined in [6], and MV is the variety of MV-algebras (see [4]). For the rest of the paper we are going to denote the algebraic category whose objects are the members of certain variety the same way that we denote the variety, e.g. we will denote by MT L the variety of MTL-algebras and the algebraic category whose objects are MTL-algebras.

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2.1 Representation and duality In this subsection we recall the Priestley duality for DLF, DLI, and DLFI-algebras. For the proof of the results of this subsection see [2]. Let A be a DLFI-algebra and let F, G ∈ F i (A). We define the following subsets of A: F →G

= {a ∈ A : there exist (f, g) ∈ F × G such that f ≤ g → a}.

F ◦G

= {a ∈ A : there exist (f, g) ∈ F × G such that f ◦ g ≤ a}

It is easy to see that for every F, G ∈ F i (A), F ◦ G and F → G ∈ F i (A). Definition 2.6. Let A be a DLFI-algebra. Let us define in X (A) the ternary relations RA and TA as follows: (P, Q, D) ∈ TA if and only if P → Q ⊆ D,

(P, Q, D) ∈ RA if and only if P ◦ Q ⊆ D. Example 2.7. Let A = A, ∧, ∨, 0, 1 be a bounded distributive lattice and let us consider the DLI-algebras Ai = A, ∧, ∨, →i , 0, 1 for 1 ≤ i ≤ 4 defined in Example 2.2. It holds TA1 = ∅, TA2 = X (A)3 , TA3 = {(P, Q, D) : Q ⊆ D} and TA4 = {(P, Q, D) : P ⊆ D}. Example 2.8. Let A = A, ∧, ∨, 0, 1 be a bounded distributive lattice and let us consider the DLF-algebras Ai = A, ∧, ∨, ◦i , 0, 1 for 1 ≤ i ≤ 4 defined in Example 2.3. It holds RA1 = ∅, RA2 = X (A)3 , RA3 = {(P, Q, D) : P ⊆ D} and RA4 = {(P, Q, D) : Q ⊆ D}. The following results are going to be useful in the proof of Theorems 3.1 and 3.5. Theorem 2.9. Let A be a DLFI-algebra. Let F, G ∈ F i (A) and P ∈ X (A). Then the following propositions hold: (1) If F → G ⊆ P , then there exist Q, D ∈ X (A) such that F ⊆ Q, G ⊆ D and Q → D ⊆ P. (2) If F ◦G ⊆ P, then there exist Q, D ∈ X (A) such that F ⊆ Q, G ⊆ D and Q◦D ⊆ P . Theorem 2.10. Let A be a DLFI-algebra, a, b ∈ A and P ∈ X (A). Then the following propositions hold: (1) a → b ∈ P if and only if for every Q, D ∈ X (A) such that a ∈ Q and P → Q ⊆ D then b ∈ D. (2) a ◦ b ∈ P if and only if there exist Q, D ∈ X (A) such that a ∈ Q, b ∈ D and Q ◦ D ⊆ P.

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Now we are going to recall the definition of DLI, DLF, DLFI-spaces and their morphisms. Definition 2.11. Let us consider the relational structure X, ≤, R, τ , where X, ≤, τ  is a Priestley space and R is a ternary relation defined on X . (1) We will say that X, ≤, R, τ  is a DLI-space if and only if for all U, V ∈ D (X), U → V ∈ D (X) , and (x, y, z) ∈ R whenever ε (x) → ε (y) ⊆ ε (z), for any x, y, z ∈ X, where → is defined in Example 2.4. (2) We will say that X, ≤, R, τ  is a DLF-space if and only if for all U, V ∈ D (X), U ∗ V ∈ D (X), and (x, y, z) ∈ R, whenever ε (x) ∗ ε (y) ⊆ ε (z), for any x, y, z ∈ X, where ∗ is defined in Example 2.4. If T is a ternary relation defined on X, we will say that X, ≤, R, T, τ  is a DLFIspace if and only if X, ≤, R, τ  is a DLF-space and X, ≤, T, τ  is a DLI-space. Remark 2.12. Note that if X, ≤, S, τ  is a DLF or a DLI-space, for all x, y, z ∈ X, if (x, y, z) ∈ S and x ≤ x, y  ≤ y and z ≤ z  , then (x , y  , z  ) ∈ S. Definition 2.13. An i-morphism, between DLI-spaces X1 , ≤1 , T1 , τ1  and X2 , ≤2 , T2 , τ2  , is a map f : X1 −→ X2 with the following properties: (1) f is increasing and continuous, (2) If (x, y, z) ∈ T1 , then (f (x) , f (y) , f (z)) ∈ T2 , (3) If (f (x) , y  , z  ) ∈ T2 , then there exist y, z ∈ X1 such that (x, y, z) ∈ T1 , y  ≤ f (y) and f (z) ≤ z  . An f -morphism, between DLF-spaces X1 , ≤1 , R1 , τ1  and X2 , ≤2 , R2 , τ2  , is a map f : X1 −→ X2 with the following properties: (1) f is increasing and continuous, (2) If (x, y, z) ∈ R1 , then (f (x) , f (y) , f (z)) ∈ R2 , (3) If (x , y  , f (z)) ∈ R2 , then there exist x, y ∈ X1 such that (x, y, z) ∈ T1 , x ≤ f (x) and y  ≤ f (y). An f i-morphism, between DLFI-spaces X1 , ≤1 , R1 , T1 , τ1  and X2 , ≤2 , R2 , T2 , τ2  , is a map f : X1 −→ X2 such that it is an f -morphism between the DLF-spaces X1 , ≤1 , R1 , τ1  and X2 , ≤2 , R2 , τ2  and that is an i-morphism between the DLI-spaces X1 , ≤1 , T1 , τ1  and X2 , ≤2 , T2 , τ2 . Definition 2.14. We will call FI to the category which have DLI-spaces as objects and i-morphims as arrows. In similar way we define FF to be the category of DLI-spaces, and FFI the category of DLFI-spaces. Let A be a DLFI-algebra and consider the structure: X (A) = X (A) , ⊆, RA, TA, τA , where RA and TA are the relations given in Definition 2.6. In [2] the second author proves that X (A) is a DLFI-space, called the DLFI-space associated with A.

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For every DLFI-space X, D (X) = D (X) , ∩, ∪, ∗, →, ∅, X is a DLFI-algebra. Moreover, there is a categorical duality between the categories DLFI and FFI . In [2] it is proved that DLI is dually equivalent to FI and DLF is dually equivalent to FF . Remark 2.15. There are others representations and dualities for lattices with different additional operators, given by A. Urquhart [18], V. Sofronie-Stokkermans [15, 17], N.G. Martinez [10, 11], and N.G. Martinez & H.A. Priestley [12] to mention some of them. In section 5 we will study the relation between these representations and dualities and the duality described in this section.

3

Some extensions of DLI-algebras and DLF-algebras

In this section we give a translation of some algebraic conditions defined on DLI-algebras or DLF-algebras into first-order conditions in its dual space. Notation: During the rest of this paper we will use the following conventions: If ψ and ϕ are formulas in the language of DLI-algebras (DLF) , we denote by ϕ  ψ the equation ϕ ∧ ψ ≈ ϕ. If A is a DLI-algebra (DLF), we will note A  ϕ  ψ if and only if ϕ  ψ is valid in A. We will note ⇔ instead of “if and only if”. Theorem 3.1. Let A be a DLI-algebra and let X = X (A) be the set of prime filters of A and let T = TA be the ternary relation given in Definition 2.6. Then: (1) A  a  (a → b) → b ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies (Q, P, D) ∈ T . (2) A  a → (b → c)  b → (a → c) ⇔ ∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (D, Z, W ) ∈ T ) implies (∃K ∈ X, (P, Z, K) ∈ T and (K, Q, W ) ∈ T ). (3) A  a → b  (b → c) → (a → c) ⇔ ∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (D, Z, W ) ∈ T ) implies ∃K ∈ X, ((P, Z, K) ∈ T and (Q, K, W ) ∈ T ). (4) A  (a ∧ b) → c  a → (b → c) ⇔ ∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (D, Z, W ) ∈ T ) implies ∃K ∈ X, (Q ⊆ K and Z ⊆ K and (P, K, W ) ∈ T ). (5) A  a → (b → c)  (a ∧ b) → c ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies ∃K ∈ X, ((P, Q, K) ∈ T and (K, Q, D) ∈ T ). (6) A  (a ∧ b) → c  (a → c) ∨ (b → c) ⇔ ∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (P, Z, W ) ∈ T ) implies (∃H, K ∈ X, (Q ⊆ H and Z ⊆ H and (K ⊆ D or K ⊆ W )) and (P, H, K) ∈ T ). (7) A  a  b → (a ∧ b) ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies (P ⊆ D and Q ⊆ D). (8) A  a  b → b ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies Q ⊆ D. (9) A  a  a → a ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies (P ⊆ D or Q ⊆ D). (10) A  a  b → a ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies P ⊆ D. (11) A  a ∧ (a → b)  b ⇔ ∀P ∈ X, (P, P, P ) ∈ T .

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(12) A  a → (b ∨ c)  (a → b) ∨ (a → c) ⇔ ∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (P, Z, W ) ∈ T ) implies (∃H, K ∈ X, (Q ⊆ H or Z ⊆ H) and K ⊆ D and K ⊆ W and (P, H, K) ∈ T ). (13) A  (a → b) ∧ (b → c)  a → c ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ T implies ∃K ∈ X, ((P, Q, K) ∈ T and (P, K, D) ∈ T ). (14) A  1 ≈ (a → b)∨(b → a) ⇔ ∀P, Q, D, Z, W ∈ X, (P, Q, D) ∈ T and (P, Z, W ) ∈ T implies Q ⊆ W or Z ⊆ D. (15) A  1 → a  a ⇔ ∀P, Q ∈ X, P ⊆ Q implies ∃D ∈ X, (P, D, Q) ∈ T . Proof. We will prove 2, 3, 6, 7, 12, 13 and 14. The other conditions are similar and left to the reader. 2. ⇒) Let P, Q, D, Z, W ∈ X such that (P, Q, D) , (D, Z, W ) ∈ T , i.e., P → Q ⊆ D and D → Z ⊆ W . We will prove that: (P → Z) → Q ⊆ W . Let a ∈ (P → Z) → Q, then there exist p ∈ P, q ∈ Q and z ∈ Z such that: p ≤ z → (q → a) . Thus, by hypothesis, p ≤ q → (z → a) ∈ P . Since P → Q ⊆ D, z → a ∈ D. As D → Z ⊆ W, we have that a ∈ W . Thus, by Theorem 2.9, there exists K ∈ X such that P → Z ⊆ K and K → Q ⊆ W , i.e., (P, Z, K),(K, Q, W ) ∈ T . ⇐) Let us suppose that there exist a, b, c ∈ A such that a → (b → c)  b → (a → c). Then, a → (b → c) ∈ P and b → (a → c) ∈ / P , for some P ∈ X. By Theorem 2.10 there exist Q, D ∈ X such that P → Q ⊆ D, b ∈ Q and a → c ∈ / D. Since a → c ∈ / D, there exist Z, W ∈ X such that D → Z ⊆ W , a ∈ Z, and c ∈ / W. Then we have that (P, Q, D) , (D, Z, W ) ∈ T , by hypothesis, there exists K ∈ X such that P → Z ⊆ K and K → Q ⊆ W . Since a → (b → c) ∈ P , we deduce that c ∈ W , which is a contradiction. 3. ⇒) Let P, Q, D, Z, W ∈ X such that P → Q ⊆ D and D → Z ⊆ W . Let us prove that: Q → (P → Z) ⊆ W . If a ∈ Q → (P → Z), there exists x ∈ P → Z such that x → a ∈ Q and there exists z ∈ Z such that z → x ∈ P . By hypothesis, z → x ≤ (x → a) → (z → a) ∈ P . Since P → Q ⊆ D, (z → a) ∈ P. As D → Z ⊆ W , a ∈ W . Therefore, by Theorem 2.9, there exists K ∈ X such that P → Z ⊆ K and Q → K ⊆ W. The proof of the only if part is easy and left to the reader. 6. ⇒) Let P, Q, D, W, Z ∈ X such that P → Q ⊆ D and P → W ⊆ Z. Consider the filter P → F (Q ∪ W ) and the ideal I = Dc ∩ Z c . We will prove that: P → F (Q ∪ W ) ∩ I = ∅.

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Suppose the contrary. Then there exist q ∈ Q, w ∈ W and x ∈ I such that (q ∧ w) → x ∈ P . By hypothesis, (q → x) ∨ (w → x) ∈ P. Since P is prime, q → x ∈ P or w → x ∈ P . Thus, x ∈ D or x ∈ Z, which is a contradiction. By Theorem 2.9 and Prime Filter Theorem there exist H, K ∈ X such that P → H ⊆ K, Q ∪ W ⊆ H and K ⊆ D or K ⊆ Z. The proof of the only if part is easy and it is left to the reader. 7. ⇒) Let P, Q, D ∈ X such that P → Q ⊆ D. Consider p ∈ P . By hypothesis p ≤ 1 → (p ∧ 1) = 1 → p, then p ∈ P → Q. Thus, P ⊆ D. Now consider q ∈ Q. By hypothesis 1 ≤ q → (1 ∧ q) = q → q ∈ P , then q ∈ D. Thus, Q ⊆ D. ⇐) Let us suppose that for some a, b ∈ A, a  b → (a ∧ b). Then there exists P ∈ X such that a ∈ P, b → (a ∧ b) ∈ / P. Since b → (a ∧ b) ∈ / P, there exist Q, D ∈ X such that P → Q ⊆ D, b ∈ Q and a ∧ b ∈ / D. Since P ⊆ D and Q ⊆ D, a, b ∈ D. Thus a ∧ b ∈ D, which is a contradiction. 12. ⇒) Let P, Q, D, Z, W ∈ X such that P → Q ⊆ D and P → Z ⊆ W . Let us consider the filter P → (Q ∩ Z) and the ideal I = I (Dc ∪ W c ). We will prove that: P → (Q ∩ Z) ∩ I = ∅. Suppose the contrary, then there exist q ∈ Q ∩ Z, d ∈ / D and w ∈ / W such that q → (d ∨ w) ∈ P . So (q → d) ∨ (q → w) ∈ P . Since P is prime, q → d ∈ P or q → w ∈ P . If q → d ∈ P and taking into account that P → Q ⊆ D, we have that d ∈ D, which is a contradiction. If q → w ∈ P we arrive at a similar contradiction. Thus, P → (Q ∩ Z) ∩ I = ∅. Then there exists K ∈ X such that P → (Q ∩ Z) ⊆ K and K ∩ (Dc ∪ W c ) = K ∩ (D ∩ W )c = ∅, i.e., K ⊆ D ∩ W . Since Q ∩ Z is a filter, by Theorem 2.9, there exists H ∈ X such that Q ∩ Z ⊆ H and P → H ⊆ K. ⇐) Suppose that there exist a, b, c ∈ A such that a → (b ∨ c)  (a → b) ∨ (a → c), then there exists P ∈ X such that a → (b ∨ c) ∈ P and (a → b) ∨ (a → c) ∈ / P. Thus there exist Q, D, Z, W ∈ X such that P → Q ⊆ D, P → Z ⊆ W , a ∈ Q ∩ Z, b ∈ / D and c∈ / W . By the hypothesis, there exist H, K ∈ X such that Q ∩ Z ⊆ H, K ⊆ D ∩ W and P → H ⊆ K. Since a ∈ H and a → (b ∨ c), b ∨ c ∈ K ⊆ D ∩ W , which is a contradiction. 13. ⇒) Let P, Q, D ∈ X such that P → Q ⊆ D. Let us prove that: P → (P → Q) ⊆ D. Let x ∈ P → (P → Q) , then there exists y ∈ P → Q such that y → x ∈ P and there exists q ∈ Q such that q → y ∈ P. Thus (q → y) ∧ (y → x) ≤ q → x ∈ P which implies that x ∈ P → Q ⊆ D.

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By Theorem 2.9, there exists K ∈ X such that P → Q ⊆ K and P → K ⊆ D. ⇐) Suppose that there exist a, b, c ∈ A such that (a → b) ∧ (b → c)  a → c, then there exists P ∈ X such that (a → b) ∧ (b → c) ∈ P and a → c ∈ / P . By Theorem 2.10 there exist Q, D ∈ X such that a ∈ Q, c ∈ / D and P → Q ⊆ D. By hypothesis there exists D1 ∈ X such that P → Q ⊆ D1 and P → D1 ⊆ D. Since a ∈ Q, b ∈ D1 . Then c ∈ D, because b → c ∈ P, which is a contradiction. 14. ⇒) Let P, Q, D, W, Z ∈ X such that P → Q ⊆ D and P → W ⊆ Z. Suppose that there exist a ∈ Q\Z and b ∈ W \D. Since 1 = (a → b) ∨ (b → a) ∈ P , if a → b ∈ P, then b ∈ D, and if b → a ∈ P, then a ∈ Z. In any case we arrive at a contradiction. The proof of the only if part is easy and it is left to the reader.  Using the previous Theorem we can prove the following results. Corollary 3.2. Let A be a DLI-algebra satisfying the equation: a  (a → b) → b. Then the following propositions hold: (1) A  a → (b → c)  b → (a → c) ⇔ A  a → b  (b → c) → (a → c). (2) A  a  b → b ⇔ A  a  b → a ⇔ A  a  b → (a ∧ b). (3) A  a → (b → c)  (a ∧ b) → c ⇔ A  (a → b) ∧ (b → c)  a → c. Corollary 3.3. Let A be a DLI-algebra satisfying the equation: a  b → b.

(a)

Consider the following propositions: (1) A  a → (b ∨ c)  (a → b) ∨ (a → c). (2) A  (a ∧ b) → c  (a → c) ∨ (b → c). (3) A  1 ≈ (a → b) ∨ (b → a). Then 1 implies 3 and 2 implies 3. Proof. First we will prove that 1 implies 3 . Let P, Q1 , Q2 , D1 , D2 ∈ X (A) such that P → Q1 ⊆ D1 and P → Q2 ⊆ D2 . By hypothesis 1 and item 12 of Theorem 3.1, there exist Q, D ∈ X (A) such that Q1 ⊆ Q or Q2 ⊆ Q and D ⊆ D1 ∩D2 and (P, Q, D) ∈ T . By condition (a) and item 8 of Theorem 3.1, we have that Q ⊆ D. Thus Q1 ⊆ Q ⊆ D ⊆ D2 or Q2 ⊆ Q ⊆ D ⊆ D1 . Finally, by item 14 of Theorem 3.1, (a → b) ∨ (b → a) ≈ 1 is valid in A. Now we will prove that 2 implies 3. Let P, Q1 , Q2 , D1 , D2 ∈ X (A) such that P → Q1 ⊆ D1 and P → Q2 ⊆ D2 . By hypothesis 2 and item 6 of Theorem 3.1, there exist Q, D ∈ X (A) such that Q1 ∪ Q2 ⊆ Q and D ⊆ D1 or D ⊆ D2 and P → Q ⊆ D. If D ⊆ D1 . By condition (a) and item 8 of Theorem 3.1, we have that Q2 ⊆ Q ⊆ D ⊆ D1 . If D ⊆ D2 by the same argument we have that Q1 ⊆ D2 . Thus by item 14 of Theorem 3.1, (a → b) ∨ (b → a) ≈ 1 is valid in A. 

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Theorem 3.4. Let A be a DLI-algebra. Suppose that the following equations are valid in A : (1) a  b → b. (2) 1 → a  a. (3) a → (b ∨ c)  (a → b) ∨ (a → c). Then, A  [(a → b) → b] ∧ [(b → a) → a]  a ∨ b. Proof. Suppose that there exist a, b ∈ A such that [(a → b) → b] ∧ [(b → a) → a]  a ∨ b. Then there exists P ∈ X (A), such that [(a → b) → b] ∧ [(b → a) → a] ∈ P , a ∈ / P and b ∈ / P . By condition 2, 1 → a ∈ / P and 1 → b ∈ / P . By Theorem 2.10, there exist / D1 and b ∈ / D2 . Q1 , Q2 , D1 , D2 ∈ X (A) such that P → Q1 ⊆ D1 , P → Q2 ⊆ D2 , a ∈ By condition 3 and by item 12 of Theorem 3.1, there exist Q, D ∈ X (A) such that P → Q ⊆ D, Q1 ∩ Q2 ⊆ Q, D ⊆ D1 and D ⊆ D2 . By conditions 1, 3 and Theorem 3.3, 1 = (a → b) ∨ (b → a) ∈ Q. If a → b ∈ Q, since (a → b) → b ∈ P and P → Q ⊆ D, we have that b ∈ D, which is a contradiction. If b → a ∈ Q we arrive at a similar contradiction. Thus, [(a → b) → b] ∧ [(b → a) → a] ≤ a ∨ b.  The following theorem gives a characterization of algebraic conditions in DLF-algebras. These results, and more general ones, has been obtained by Urquhart in [18] for relevant algebras, but here we will give a proof just using the results of [2]. Theorem 3.5. Let A be a DLF-algebra and let X = X (A) be the set of prime filters of A and R = RA the ternary relation given in Definition 2.6. Then: (1) A  (a ◦ b) ◦ c  a ◦ (b ◦ c) ⇔ ∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ R and (D, Z, W ) ∈ R) implies ∃K ∈ X, (P, K, W ) ∈ R and (Q, Z, K) ∈ R. (2) A  a ◦ (b ◦ c)  (a ◦ b) ◦ c ⇔ ∀P, Q, D, Z, W ∈ X. ((P, Q, D) ∈ R and (Z, D, W ) ∈ R) implies ∃K ∈ X, (Z, P, K) ∈ R and (K, Q, W ) ∈ R. (3) A  a ◦ b  b ◦ a ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ R implies (Q, P, D) ∈ R. (4) A  1 ◦ a  a ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ R implies Q ⊆ D. (5) A  a ◦ 1  a ⇔ ∀P, Q, D ∈ X, (P, Q, D) ∈ R implies P ⊆ D. (6) A  a  a ◦ 1 ⇔ ∀P ∈ X, ∃Q ∈ X, (P, Q, P ) ∈ R. (7) A  a  1 ◦ a ⇔ ∀P ∈ X, ∃Q ∈ X, (Q, P, P ) ∈ R. Proof. We will prove 1, 4 and 6, the other are similar and left to the reader. 1. ⇒) Let P, Q, D, Z, W ∈ X such that (P, Q, D) , (D, Z, W ) ∈ R, i.e., P ◦ Q ⊆ D and D ◦ Z ⊆ W . We will prove that: P ◦ (Q ◦ Z) ⊆ W . Let y ∈ P ◦ (Q ◦ Z), then there exist p ∈ P, q ∈ Q, z ∈ Z such that p ◦ (q ◦ z) ≤ y. By hypothesis, (p ◦ q) ◦ z ≤ y. Since p ◦ q ∈ D, (p ◦ q) ◦ z ∈ D ◦ Z ⊆ W . By Theorem 2.9, there exists K ∈ X such that Q ◦ Z ⊆ K and P ◦ K ⊆ W, i.e., .there exists K ∈ X such that (Q, Z, K),(P, K, W ) ∈ R.

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⇐) Suppose that there exist a, b, c ∈ A such that (a ◦ b) ◦ c  a ◦ (b ◦ c), then there exists W ∈ X such that (a ◦ b) ◦ c ∈ W and a ◦ (b ◦ c) ∈ / W . By Theorem 2.10, there exist D, Z ∈ X such that a ◦ b ∈ D, c ∈ Z and D ◦ Z ⊆ W and there exist P, Q ∈ X such that a ∈ P , b ∈ Q and P ◦ Q ⊆ D. By hypothesis, we have that there exists K ∈ X such that Q ◦ Z ⊆ K and P ◦ K ⊆ W . Thus b ◦ c ∈ K and a ◦ (b ◦ c) ∈ W, which is a contradiction. 4. ⇒) Let P, Q, D ∈ X such that P ◦ Q ⊆ D. By hypothesis for every q ∈ Q, 1 ◦ q ≤ q ∈ P ◦ Q ⊆ D. ⇐) Let us suppose that there exists a ∈ A such that 1 ◦ a  a, then there exists D ∈ X such that 1 ◦ a ∈ D and a ∈ / D. By Theorem 2.10, there exist P, Q ∈ X such that a ∈ Q and P ◦ Q ⊆ D. Then a ∈ Q ⊆ D, which is a contradiction. 6. ⇒) Let P ∈ X. By hypothesis, for every p ∈ P, p ≤ p ◦ 1. Then P ◦ {1} ⊆ P. By Theorem 2.9 there exists Q ∈ X such that P ◦ Q ⊆ P . ⇐) Let suppose that there exists a ∈ A such that a  a ◦ 1. There exists P ∈ X such that a ∈ P and a ◦ 1 ∈ / P . Thus P ◦ {1}  P. Therefore P ◦ Q  P for every Q ∈ X, which is a contradiction with the hypothesis. 

4

Duality for subvarieties of Commutative Residuated Lattices

Using the results of the previous sections we are going to develop topological dualities for some known algebraic subcategories of ICRL. We will first give a duality for ICRL, and subsequently we will extend this duality for MT L, IMT L and MV.

4.1 Integral Commutative Residuated Lattices Theorem 4.1 (Duality for ICR-lattices). The category ICRL is dually equivalent to the full subcategory of FFI whose spaces satisfy the following conditions: (1) T = R. (2) ∀x, y, z ∈ X, (x, y, z) ∈ T implies (y, x, z) ∈ T . (3) ∀x, y, z, v, w ∈ X, ∃s ∈ X, ((x, y, z) ∈ T and (z, v, w)) ∈ T implies ((x, v, s) ∈ T and (s, y, w) ∈ T ). (4) ∀x ∈ X, ∃y ∈ X, (x, y, x) ∈ T . (5) ∀x, y, z ∈ X, (x, y, z) ∈ T implies y ≤ z. Proof. Let us consider A a DLFI-algebra. By Theorem 3.5 we have that A satisfies (A), (I) and (C) if and only if its dual DLFI-space X (A) satisfies the conditions from 2 to 5. We are going to prove that A satisfies (R) if and only if TA = RA.

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Suppose first that A satisfies (R). Consider P, Q ∈ X (A), then: P → Q = {x ∈ A : there exist p ∈ P and q ∈ Q such that p ≤ q → x} = {x ∈ A : there exist p ∈ P and q ∈ Q such that p ◦ q ≤ x} = P ◦ Q. Thus, TA = RA. For the converse suppose that TA = RA and let a, b, c ∈ A. By Theorem 2.10 we have that: a ◦ b ≤ c ⇔ for every P, Q ∈ X (A) such that a ∈ P and b ∈ Q, c ∈ P ◦ Q ⇔ for every P, Q ∈ X (A) such that a ∈ P and b ∈ Q, c ∈ P → Q ⇔ a ≤ b → c. Thus, A satisfies (R).



The DLFI-spaces satisfiyng the conditions of Theorem 4.1 will be called ICR-spaces. The full subcategory of FFI whose objects are ICR-spaces will be noted by ICRL∗ .

4.2 MTL-algebras and IMTL-algebras Let us recall that the variety MT L of Monoidal T-norm based Logic algebras, or MTLalgebras, introduced in [5], can be defined as the subvariety of integral commutative residuated whose algebras satisfy the prelinearity equation: MT L = ICRL + {(a → b) ∨ (b → a) ≈ 1} . Theorem 4.2 (Duality for MTL-algebras). The category MT L is dually equivalent to the full subcategory of ICRL∗ whose spaces satisfy: ∀x, y, z, v, w ∈ X , ((x, y, z) ∈ T and (x, v, w) ∈ T ) implies (y ≤ w or v ≤ z) . Proof. Immediate from Theorem 4.1, and item 14 of Theorem 3.1.



Now, we shall study the representation for the variety IMT L of IMTL-algebras. The variety IMT L is defined by: IMT L = MT L + {(a → 0) → 0 ≈ a} . First we will need some previous results. By item 2 of Theorem 4.1, and item 1 of Theorem 3.1, every ICR-lattice satisfies the identity a  (a → b) → b, and in particular a  (a → 0) → 0. By items 4 and 5 of

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Theorem 4.1, and by items 10 and 15 of Theorem 3.1, we have that every ICR-lattice satisfies the identity: a ≈ 1 → a. (for an algebraic proof of these properties see [9]). Lemma 4.3 ([6, Proposition 3]). Each MTL-algebra is a subdirect product of linearly ordered MTL-algebras. Corollary 4.4. The following equation: (a ∧ b) → c ≈ (a → c) ∨ (b → c) is valid in every MTL-algebra A. Proof. Since the equation (a ∧ b) → c ≈ (a → c) ∨ (b → c) is valid in every lineraly ordered MTL-algebra, by the previous Lemma we deduce that it is valid in each MTLalgebra.  Remark 4.5. A proof of the previous Corollary can be deduced from the results of Ward and Dilworth in [19], or from the results of J.B. Hart, L. Rafter and C. Tsinakis in [8]. Notation. If T is a ternary relation over a set X, for every x, y ∈ X we are going to note T (x, y) = {z ∈ X : (x, y, z) ∈ T } . Lemma 4.6. Let A be a DLI-algebra and let X = X (A) and T = TA. Then A  1 → 0 ≈ 0 ⇔ ∀P ∈ X, ∃Q ∈ X, T (P, Q) = ∅. Proof. ⇒) Let P ∈ X. By hypothesis 0 ∈ / P → {1}, then there exist Q, D ∈ X such that P → Q ⊆ D = A, i.e., T (P, Q) = ∅. ⇐) Let suppose that 1 → 0 = 0, then there exist P ∈ X such that 1 → 0 ∈ P . Thus for every Q ∈ X, 0 ∈ P → Q which is a contradiction.  Theorem 4.7. Let A be a DLI-algebra, X = X (A) and T = TA. Suppose that A satisfies the following identities: (1) 1 → 0 ≈ 0. (2) (a ∧ b) → c ≈ (a → c) ∨ (b → c). Then the following propositions are equivalent: a. A  (a → 0) → 0  a. b. ∀P, Q ∈ X, If (∀D ∈ X, T (P, Q) = ∅ implies T (Q, D) = ∅), then Q ⊆ P .

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Proof. a ⇒ b). Let P, D ∈ X such that for every Q ∈ X, T (P, Q) = ∅ implies T (Q, D) = ∅, i.e., if P → Q = A, then Q → D = A. Let a ∈ D. We will prove that: P → [a → 0) = A. If we suppose that P → [a → 0) = A, there exists Q ∈ X such that a → 0 ∈ Q and P → Q = A. Thus Q → D = A. But a ∈ D and a → 0 ∈ Q, implies that 0 ∈ Q → D, which is a contradiction. Then 0 ∈ A = P → [a → 0), i.e., there exists p ∈ P such that p ≤ (a → 0) → 0 ≤ a. Thus a ∈ P . b ⇒ a). Let a ∈ A. If a = 0, then (0 → 0) → 0 = 1 → 0 = 0 ≤ 0 = a and the result follows. Suppose that there exists a = 0 such that (a → 0) → 0  a. Then there exists P ∈ X such that (a → 0) → 0 ∈ P and a ∈ / P . Let us consider the set: NP = {Q ∈ X : P → Q = A} . By condition 1 and Lemma 4.6, we have that NP = ∅. Now let us consider the set F = {H ∈ F i (A) : a ∈ H and for every Q ∈ NP , Q → H = A} , ordered by inclusion. Since (a → 0) → 0 ∈ P , if Q ∈ Np , then a → 0 ∈ / Q. Thus Q → [a) = A.  Consequently [a) ∈ F. Let us consider a chain {Hi : i ∈ I} , in F with I = ∅. Let G = {Hi : i ∈ I} . We prove that for each Q ∈ NP , Q → G = A. If we suppose the contrary there exist Q ∈ NP , q ∈ Q, and h ∈ G such that q ≤ h → 0. As h ∈ G, there exists i0 ∈ I such that hi0 ∈ Hi0 . So, Q → Hi0 = A, which contradicts the fact that Hi0 ∈ F. Thus, G ∈ F. By Zorn’s Lemma there exists an element D ∈ F which is maximal. We will prove that D ∈ X. Let suppose that there exist b, c ∈ A such that b ∨ c ∈ D but b, c ∈ / D Let us consider the filters Db = F (D ∪ {b}) and Dc = F (D ∪ {c}). Since Db , Dc ∈ / F, there exist Qb , Qc ∈ NP such that Qb → Db = A and Qc → Dc = A. Since P → Qb = A and P → Qc = A, we have by condition 2 and item 6 of Theorem 3.1, that there exist Q ∈ NP such that Qb ∪ Qc ⊆ Q. Then Q → Db = A = Q → Dc . So there exist q ∈ Q and pb , pc ∈ D, such that q ≤ (pb ∧ b) → 0 and q ≤ (pc ∧ c) → 0. Considering p = pb ∧ pc ∈ D, by condition 2 we have: q ≤ ((p ∧ b) → 0) ∧ ((p ∧ c) → 0) ≤ ((p ∧ b) ∨ (p ∧ c)) → 0 ≤ (p ∧ (b ∨ c)) → 0. Thus, Q → D = A, which is a contradiction with the fact that D ∈ F. Then D ∈ X and for every Q ∈ NP , Q → D = A. By hypothesis, a ∈ D ⊆ P, which is a contradiction.

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Therefore, (a → 0) → 0 ≤ a for every a ∈ A.



Using Theorems 4.7 and 4.2 we can conclude that: Theorem 4.8 (Duality for IMTL-algebras). The category IMT L is dually equivalent to the full subcategory of ICRL∗ satisfying the following conditions: (1) ∀x, y, z, v, w ∈ X, ((x, y, z) ∈ T and (x, v, w) ∈ T ) implies (y ≤ w or v ≤ z). (2) ∀x, y ∈ X, If (∀z ∈ X, T (x, z) = ∅ implies T (z, y) = ∅) then y ≤ x.

4.3 MV-algebras Let us recall that MV-algebras are definable as IMTL-algebras satisfying the identity: (a → b) → b ≈ (b → a) → a.§ Note that if an MTL-algebra A satisfies the identity (a → b) → b ≈ (b → a) → a, then it is an IMTL-algebra. Thus we have that the variety MV-algebras can be defined as: MV = MT L + {(a → b) → b ≈ (b → a) → a} . We shall use this characterization to give a duality for MV-algebras. Theorem 4.9. Let A be a DLI-algebra. If A satisfies: (1) (a ∧ b) → c ≈ (a → c) ∨ (b → c) . (2) 1 → a  a. Then the next propositions are equivalent: a. A  (a → b) → b  a ∨ b. b. For every P ∈ X (A) , for every b ∈ / P, if K ∈ X (A) is such that (for every Q ∈ X (A) (if b ∈ / P → Q implies that b ∈ / Q → K)) then K ⊆ P . Proof. a ⇒ b) Let us suppose that there are P, K ∈ X (A) and b ∈ / P such that for every Q ∈ X (A) (if b ∈ / P → Q then b ∈ / Q → K). Let a ∈ K, If (a → b) → b ∈ / P , then there exists Q ∈ X (A) such that a → b ∈ Q and b ∈ / P → Q. Thus b ∈ / Q → K, which is a contradiction with the fact the a ∈ K. Then (a → b) → b ≤ a ∨ b ∈ P . Since b ∈ /P and P is a prime filter, we have that a ∈ P . Thus K ⊆ P . b ⇐ a) Let us suppose that there exist a, b ∈ A such that (a → b) → b  a ∨ b. Then there exists P ∈ X (A) such that (a → b) → b ∈ P and a ∨ b ∈ / P . Consider NP,b = {Q ∈ X (A) : b ∈ / P → Q} , by condition 2 and item 15 of Theorem 3.1, we have that there exists Q ∈ X (A) such that P → Q ⊆ P. Therefore NP,b = ∅. Now consider the set F = {F ∈ F i (A) : a ∈ F and b ∈ / Q → F , for every Q ∈ NP,b } . §

see [6]

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Since (a → b) → b ∈ P and b ∈ / P → Q, if Q ∈ NP,b then a → b ∈ / Q. Thus we have that b ∈ / Q → [a) ,i.e. [a) ∈ F. By condition 1 and using the same argument of the proof of Theorem 4.7 we have that there exists K ∈ X (A) such that a ∈ K and for every Q ∈ NP,b , b ∈ / Q → K. Then a ∈ K ⊆ P, which is a contradiction. Therefore. (a → b) → b ≤ a ∨ b, for every a, b ∈ A.  It’s known (see [6, pg 275]) that every MTL-algebra A satisfies: a ∨ b ≈ ((a → b) → b) ∧ ((b → a) → a) , it also can be deduced from Theorem 3.4 and the fact that A  a  (a → b) → b. From that, we have that in every MTL-algebra A, the following statements are equivalent: (1) A  (a → b) → b  a ∨ b. (2) A  (a → b) → b ≈ (b → a) → a. By this observation and the previous Theorem we obtain the following result: Theorem 4.10 (Duality for MV-algebras). The category of MV-algebras is dually equivalent to the full subcategory of ICRL∗ whose spaces satisfy: (1) For every x, y, z, v, w ∈ X if (x, y, z) , (x, v, w) ∈ T , then y ≤ w or v ≤ z. (2) For every x ∈ X and every U ∈ D (X) such that x ∈ / U, if z ∈ X is satisfies that c (for every y ∈ X T (x, y) ∩ U = ∅ implies T (y, z) ∩ U c = ∅), then z ≤ x.

5

Relationship with others Representations

There are a large number of representations and dualities for distributive lattices developed by different motivations. In this section we will describe how some of these representations or dualities are related to the one studied in this paper. All the dualities considered in this section have a similarity, they are based on Priestley duality for distributive lattices (bounded or unbounded), i.e., they use Priestley spaces and some additional structure to represent the implication (and other operators in the case of Urquhart duality for relevant algebras and Sofronie-Stokkermans duality for lattices with operators).

5.1 Relevant Algebras The duality developed by the second author in [2] is an extension of the duality given by A. Urquhart in [18]. Urquhart gives a duality for relevant algebras which are DLFIalgebras with an antihomomorphism (Ockham negation) and a distinguished element. Since in relevant algebras the implication is the left residual of the fusion, Urquhart use the same ternary relation to represent the implication (see Theorem 4.1). This ternary relation is exactly the one used by the second author to represent the fusion. In the same paper Urquhart give a general result for translating the validity of some inequalities in

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the algebra to the validity of some first order formula over its dual space. This result is developed for inequalities between some special terms constructed just using the fusion connective. We think that it is possible to give more general results of this kind for terms with includes the implication connective, but this is out of the purpose of this paper.

5.2 Lattices with Operators In [15], Sofronie-Stokkermans develop a duality for a large classes of lattices with different kinds of operators, by means of Priestley spaces with relations and functions. SofronieStokkermans extend this duality for more general classes of operators in [17]. In her notation it is possible to describe a DLFI-algebra as a distributive lattice with two binary operations one of type 1, 1 → 1 (fusion) and other of type 1, −1 → −1 (implication). Sofronie-Stokkermans’s representation for the fusion (type 1, 1 → 1) is exactly the same ternary relation that we use. For the implication Sofronie-Stokkermans use a ternary relation R→ . We give here an equivalent definiton of R→ : (P, Q, D) ∈ R→ if and only if p → q ∈ / D for every p ∈ P and q ∈ / Q, It is easy to see that: (P, Q, D) ∈ R→ if and only if (D, P, Q) ∈ T . Then R→ and T are interdefinables. In [17], Sofronie-Stokkermans gives a way to check the validity of certain formulas in a class of lattice with operators. This technique requires that the class of lattices and its related frames satisfy certain properties. The first condition is that the class of frames is defined by first order conditions and the second is that the canonicity of the class of lattice with operators considered. In the case of DLI, DLF and DLFI-algebras we describe a large list of equations which are characterized by first order conditions in the related spaces (Theorems 3.1 and 3.5). So for the subvarieties determined by these equations we have that the dual spaces are described by first order conditions. This allow us to think that the results of Sofronie-Stokkermans hold in these varieties. We only need to prove the canonicity of these subvarieties of DLI, DLF and DLFI-algebras. This result was already proved for some of these varieties by the authors and it will appear in a future work, where we will study some logics related to DLI, DLF and DLFI-algebras.

5.3 Implicative Lattices In [10] N.G. Martinez gives a duality between Implicative lattices and Bounded Priestley spaces with a continuous binary function, and improve its results in [11]. We recall the definition of implicative lattices.

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Definition 5.1. An implicative lattice A, ∧, ∨, → is an algebra of type (2, 2, 2) such that A, ∧, ∨ is a distributive lattice and satisfies the following equations: (1) a → (b ∧ c) ≈ (a → b) ∧ (a → c). (2) (a ∨ b) → c ≈ (a → c) ∧ (b → c). (3) a → (b ∨ c) ≈ (a → b) ∨ (b → c). (4) (a ∧ b) → c ≈ (a → c) ∨ (b → c). We will say that A, ∧, ∨, →, 0, 1 is a Bounded implicative lattice if the reduct A, ∧, ∨, → is an implicative lattice and A, ∧, ∨, 0, 1 is a bounded lattice. For every distributive lattice A we will note S (A) = X (A) ∪ {∅, A} . In [11] Martinez define for every implicative lattice the following function Φ : S (A) × S (A) → S (A) :  Φ (P, Q) = {y : x → y ∈ Q} . x∈P

It follows straightforwardly that if P, Q ∈ X (A) , Φ (P, Q) = Q → P (see page 605). Then the ternary relation TA (Definition 2.6) can be recovered using Φ by (P, Q, D) ∈ TA if and only if Φ (Q, P ) ⊆ D. Items 6 and 12 of Theorem 3.1 give conditions over the ternary relation which determine when a DLI-algebra is a bounded implicative lattice. It is interesting to note that, in the other way, there are simple properties of Φ which determine when a bounded implicative lattice is a DLI-algebra. It follows from Definitions 2.1 and 5.1 that a bounded implicative lattice is a DLIalgebra if and only if the following conditions hold: a → 1 ≈ 1 and 0 → a ≈ 1. Given a bounded implicative lattice A = A, ∧, ∨, →, 0, 1 , it is easy to prove that: A  a → 1 = 1 ⇔ Φ (P, Q) = ∅ for every P, Q ∈ X (A) .

(B)

A  0 → a = 1 ⇔ Φ (A, Q) = A for every Q ∈ X (A) . If A is a DLI-algebra which satisfies items 3 and 4 of Definition 5.1, then it is easy to see that P → Q ∈ X (A) for every P, Q ∈ X (A) . Thus TA (P, Q) is a principal increasing set. Using the correspondences (B) and the previous observation we have that the function Φ can be described using the ternary relation TA by: ⎧ ⎪ ⎪ A if P = A and Q = ∅, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ A if Q = A and P = ∅, Φ (P, Q) = ⎪ ⎪ ∅ if P = ∅ or Q = ∅, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D if P, Q ∈ X (A) and TA (P, Q) = [D) . In [12] N.G. Martinez and H.A. Priestley give a duality between implicative lattices and bounded Priestley spaces with a special function.

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If A is an implicative lattice they define μ : D (S (A)) × S (A) −→ S (A) by: μ (β (a) , P ) = Pa = {x ∈ A : x → a ∈ / P}. As in (B) we can give some properties of μ which are equivalent to the fact that a bounded implicative lattice A is a DLI-algebra. A  a → 1 = 1 ⇔ μ (β (1) , P ) = ∅ for every P ∈ S (A) \ {∅} , A  0 → a = 1 ⇔ μ (β (0) , P ) = A for every P ∈ S (A) \ {∅} . It is easy to see that Pa is the maximal element of S (A) such that a ∈ / P → Pa . Thus given a bounded implicative lattice A which is a DLI-algebra we have that if P ∈ X (A) , then: {Q ∈ X (A) : T (P, Q) ∩ β c (a) = ∅} = {Q : a ∈ / P → Q} = {Q : for every x ∈ Q, x → a ∈ / P} = (Pa ] . It follows that μ can be defined by ⎧ ⎪ ⎪ ∅ if P = A, ⎪ ⎪ ⎨ μ (β (a) , P ) = A if P = ∅, ⎪ ⎪ ⎪ ⎪ ⎩ D if P ∈ X (A) and {Q ∈ X (A) : T (P, Q) ∩ β c (a) = ∅} = (D] . Thus clearly the function μ can be obtained from the ternary relation TA. In the other way, if we define Tµ ⊆ (X (A))3 by  (P, Q, D) ∈ Tµ if and only if D ∈ {βA (a) : a ∈ A and Q  Pa } then Tµ = TA. Since a ∈ P → Q if and only if there exist x ∈ Q such that x → a ∈ P, if and only if Q  Pa , we have that P  → Q ⊆ D if and only if a ∈ D (i.e. D ∈ βA (a)) for {βA (a) : a ∈ A and Q  Pa } every a such that Q  Pa , .i.e. D ∈ We can conclude that in the case of bounded implicative lattices which are DLIalgebras the dualities given in [10, 12] and in [2] are interdefinables.

5.4 WH-algebras In [3] Celani and Jansana introduce the variety of Weakly Heyting algebras, or WHalgebras, and give a Priestley style duality between this algebras and Priestley spaces endowed with a binary relation. Definition 5.2. A DLI-algebra A is a WH-algebra if the following conditions hold: (1) (a → b) ∧ (b → c)  a → c, (2) a → a ≈ 1.

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Using items 8 and 13 of Theorem 3.1 we can conclude that a DLI-algebra A is a WH-algebra if and only if TA satisfies the following properties: (1) ∀P, Q, D ∈ X (A), ∃Z ∈ X (A), (P, Q, D) ∈ TA implies that (P, Q, Z) ∈ TA and (P, Z, D) ∈ TA. (2) ∀P, Q, D ∈ X (A), (P, Q, D) ∈ TA implies that Q ⊆ D. Given WH-algebra A, in [3] Celani and Jansana define a binary relation over X (A) as follows (P, Q) ∈ SA ⇔ for every a, b ∈ A, (a → b ∈ P and a ∈ Q, then b ∈ Q).

(1)

Let A be a WH-algebra. Let TA be the ternary relation defined on X (A) as in Definition 2.6, and let us define a binary relation STA as follows: (P, Q) ∈ STA if and only if (P, Q, Q) ∈ TA. Then it is easy to see that SA = STA . So, the binary relation SA can be obtained from TA. Now, let us consider the binary relation SA defined in (1). Let us define a ternary relation TSA as follows:  TSA = (P, Q, D) ∈ X (A)3 : ∃F ∈ X (A) , Q ⊆ F ⊆ D and (P, F ) ∈ SA . We will see that TA = TSA . Let (P, Q, D) ∈ TA . Let F = P → Q. Since, a → a = 1 for all a ∈ A, Q ⊆ P → Q. Then Q ⊆ F ⊆ D. We prove that (P, F ) ∈ SA. Let a → b ∈ P and a ∈ F . Then there exists q ∈ Q such that q → a ∈ P . So, (q → a)∧(a → b) ≤ q → b ∈ P, i.e. b ∈ Q → P = F Thus, (P, Q, D) ∈ TSA . ” Suppose that (P, Q, D) ∈ TSA , then there exists F ∈ X (A) such that Q ⊆ F ⊆ D and (P, F ) ∈ SA. Let a, b ∈ A such that a → b ∈ P and a ∈ Q. Then b ∈ F, and since (P, F ) ∈ SA, b ∈ F . So, b ∈ D. Therefore, (P, Q, D) ∈ TA. We conclude that TA = TSA . Since the variety of WH-algebras is a subvariety of DLI and the binary relation defined in [3] can be obtained from the ternary relation used in this paper, we conclude that the duality given in [3] can be deduced from the duality given in [2] using the results of this paper.

6

Conclusions and future work

In this paper we gave a duality for many algebraic categories of algebras related to fuzzy logics. These dualities are an extension of the duality given by the second author in [2]. The method was to translate the equations which define the subvarieties in conditions over the dual spaces. In the case of ICRL, MT L and IMT L these conditions are of first order, but this was not the case for MV. Some open questions, also raised by one of the referees, are: Is there a generalized way to give the translations for some classes of equations? Is it possible to determine when an equation has a first order translation?

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The problem of canonicity of almost all the varieties definable by the equations given in this work is solved by the authors and will be showed in a future paper. It is interesting to note that M. Gehrke and H.A. Priestley in [7] had proved that the variety MV is not canonical. This variety is the only one, in this paper, for which we cannot obtain first order conditions to determine its dual spaces.

Acknowledgment We would like to thank the referees for their observations and suggestions which have contributed to improve this paper.

References [1] R. Dwinger and P.H. Balbes: Distributive Lattices, University of Missouri Press, Columbia, M, 1974. [2] S.A. Celani: “Distributive lattices with fusion and implication”, Southeast Asian Bull. Math., Vol. 28, (2004), pp. 999–1010. [3] S.A. Celani and R. Jansana: “Bounded Distributive lattices with Strict Implication”, Math. Log. Quart., Vol. 51(3), (2005), pp. 219–246. [4] R. Cignoli, I.M.L. D’Ottaviano and D. Mundici: Algebraic Foundations of ManyValued Reasoning, Trends in Logic, Vol. 7, Kluwer Academic Publishers, 2000. [5] R. Cignoli, F. Esteba, L. Godo and F. Montagna: “On a class of left continuous t-norms”, Fuzzy Set. Syst., Vol. 131, (2002), pp. 283–296. [6] F. Esteba and L. Godo: “Monoidal t-norm based logic: towards a logic for left continuous t-norms”, Fuzzy Set. Syst., Vol. 124, (2001), pp. 271–288. [7] M. Gehrke and H.A. Priestley: “Non-canonicity of MV-algebras”, Houston J. Math., Vol. 28(3), (2002), pp. 449–455. [8] J.B. Hart, L. Rafter and C. Tsinakis: “The Structure of Commutative Residuated Lattices”, Int. J. Algebr. Comput., Vol. 12(4), (2002), pp. 509–524. [9] P. Jipsen and C. Tsinakis: A Survey of Residuated Lattices, Ordered Algebraic Structures , Kluwer Academic Publishers, Dordrecht, 2002, pp. 19–56. [10] N.G. Martinez: “A topological duality for some lattice oredered algebraic structures including l-groups”, Algebra Univ., Vol. 31, (1996), pp. 516–541. [11] N.G. Martinez: “A simplified duality for implicative lattces and l-groups”, Studia Log., Vol. 56, (1994), pp. 185–204. [12] N.G. Martinez and H.A. Priestley: “On Priestley spaces of lattice-ordered algebraic structures”, Order, Vol. 15 (1998), pp. 297–323. [13] H.A. Priestley: “Ordered topological spaces and the representation of distributive lattices”, Proc. London Math. Soc., Vol. 24, (1972), pp. 507–530. [14] H.A. Priestley: “Stone Lattices: a topological approach”, Fund. Math., Vol. 84, (1974), pp. 127–143.

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[15] V. Sofronie-Stokkermans: “Duality and canonical extensions of bounded distributive lattices with operators, and applications to the semantics of non-classical logics I”, Studia Log., Vol. 64, (2000), pp. 93–132. [16] V. Sofronie-Stokkermans: “Duality and canonical extensions of bounded distributive lattices with operators, and applications to the semantics of non-classical logics II”, Studia Log., Vol. 64, (2000), pp. 151–192. [17] V. Sofronie-Stokkermans: “Resolution-based decision procedures for the universal theory of some classes of distributive lattices with operators”, J. Symb. Comput., Vol. 36, (2003), pp. 891–924. [18] A. Urquhart: “Duality for Algebras of Relevant Logics”, Studia Log., Vol. 56, (1996), pp. 263–276. [19] M. Ward and R.P. Dilworth: “Residuated lattices”, Trans. Amer. Math. Soc., Vol. 45, (1939), pp. 335–354.

DOI: 10.2478/s11533-006-0029-2 Research article CEJM 4(4) 2006 624–634

Limit theorems for number of diffusion processes, which did not absorb by boundaries∗ Aniello Fedullo1† , Vitalii A. Gasanenko2‡ 1

Universita Degli Studi Di Salerno, 84084 Fisciano (SA) Italia 2

Institute of Mathematics, National Academy of Science of Ukraine, 01601, Kiev, Ukraine

Received 10 October 2005; accepted 5 June 2006 Abstract: We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Stochastic differential equations, solution of parabolic equations, eigenvalues problem, Poisson random measure, generating function MSC (2000): 60J60

Let us consider the set of independent random diffusion processes ξk (t), k = 1, N , t ≥ 0, ξk (0) = γk , γk ∈ Q ⊂ Rd . Let the domain Q ⊂ Rd be open connected region and it is limited by the smooth surface ∂Q. All processes ξk (t) are diffusion processes with absorption on the boundary ∂Q. These processes are solutions of the following stochastic differential equations in Q dξ(t) = a(t, ξ(t))dt +

d 

(k)

bi (t, ξ(t))dwi (t)

(1)

i=1

ξ(t) ∈ Rd ; x ∈ Rd ∗

bi (t, x),

a(t, x) : R+ × Rd → Rd .

This research was partially supported by the Ministry of Education and Science of Ukraine, project 01.07/103 and University of Salerno, Italy. † E-mail: [email protected] ‡ E-mail: [email protected] or [email protected]

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with an initial condition: ξ(0) = γk ∈ D. (k) Here W (k) (t) = (wi (t), 1 ≤ i ≤ d), 1 ≤ k ≤ N are independent in totality of d-dimensional Wiener processes. Thus, these processes have the identical diffusion matrices and shift vectors, but they have different initial states. Let Q be bounded and the boundary ∂Q is the Lyapunov surface C (1,λ) . The initial number and positions of processes is defined by the random Poisson measure μ(·, τ ) in Q: mk (A, τ ) −m(A,τ ) P (μ(A, τ ) = k) = , e k! where m(·, τ ) is finitely additive positive measure on Q for fixed τ . We are going to investigate distribution of the number of the processes ξk (t), which is in Q for all moments of time t ≤ τ . This problem was offered in [1] as the mathematical model of practice problem. We will do the following suppositions with respect to functions a(t, x), bi (t, x), i = 1, d. They are sufficient for existence of unique solutions of equations (1) [2, p. 470]. There is such L that for functions bij (t, x), ai (t, x) the following inequalities take place d 

|a(s, x) − a(s, y)| +

|bk (s, x) − bk (s, y)| ≤ L|x − y|

k 2

|a(s, x)| +  for all x, y ∈ Rd . Here |y| =

d 

|bk (s, x)|2 ≤ L2 (1 + |x|2 )

k d  i=1

yi2

(2)

 12 .

We will define matrix σ = B T B, B = (bij (t, x)), σ = (σij (t, x)), 1 ≤ i, j ≤ d. We will consider the following parabolic boundary problem d d ∂u(t, x) ∂ 2 u(t, x)  ∂u(t, x) 1 σij (τ − t, x) + ai (τ − t, x) = ∂t 2 j,i=1 ∂xi ∂xj ∂xi i=1

u(0, x) = 1, x ∈ Q;

u(t, x) = 0, x ∈ ∂Q,

0≤t≤τ

(3)

t ∈ [0, τ ]

In addition to (2) we will assume that the coefficients of operator part of problem (3) satisfy to the H¨ older condition on variable t with index 0 < α < 12 . max i

|ai (t, x) − ai (t , x)| <∞ |t − t |α (x,t),(x,t ) sup

|σi,j (t, x) − σi,j (t , x)| < ∞, max sup i,j (x,t),(x,t ) |t − t |α ¯ here (t, x), (t, x ) ∈ [0, τ ] × Q.

(4)

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We will also assume that operator part of (3) is uniformly parabolic [3, p. 20]. It means that the following condition is executed ν| z|2 ≤



σij (t, x)zi zj ≤ μ| z|2

¯ (t, x) ∈ [0, τ ] × Q.

(5)

i,j

Here ν, μ there are the fixed positive numbers, and z = (z1 , · · · , zd ) there is an arbitrary real vector. In what follows, the conditions (2),(4),(5) are executed and region Q is limited by the Lyapunov surface ∂Q of class C (1,λ) . These suppositions guarantee existence of unique classical solution of problem (3) [3, p. 469]. It is known [2], that u(τ, x) is equal to probability of remaining in the region Q at time instant τ of a diffusion process from (1), which occurs at the point (0, x) at the initial moment ( ξ(0) = x, x ∈ Q). We denote by η(τ ) the number of remaining processes in the region Q at time instant τ . We introduce the following sets for y ∈ [0, 1] A(τ, y) = {x ∈ Q : u(τ, x) ∈ [0, y]} ,

mτ (y) := m(A(τ, y)).

As the function u(τ, x) : Q → [0, 1] is continuous at fixed τ , we have that the sets A(τ, y) are measurable and the definition of the measure m(·) on these sets is correct. Theorem 1. If the conditions (2), (4), (5) are fulfilled and the ∂Q is Lyapunov surface, then random value η(τ ) has Poisson distribution: P (η(τ ) = k) = with parameter a(τ ) =

1 0

ak (τ ) −a(τ ) , e k!

k = 0, 1, ...,

τ > 0,

ydmτ (y).

The analogy theorem was proved in [1] for case if d = 2 and if the region Q is circle: Q = {(x, y) : x2 + y 2 ≤ R2 }. The proof of Theorem 1 repeats the proof theorem from [1] almost word for word. Thus we have the exact formula of distribution function of η(τ ). However, the definition of the sets A(τ, y) with help analytical formula is difficult problem. We have difficulties in calculation of the function mτ (y) in consequence of this. Therefore it would be desirable to obtain an approximation for distribution function of η(τ ). Further we will prove such approximate formula for special case. Note that the authors of the article [4] investigated this special case, when the initial number and positions of diffusion processes are defined by the determinate limited measure N (B, τ ), where N (B, τ ) is equal to the number of points γk in a set B and N = N (Q, τ ) < ∞ for fixed τ > 0. We consider the following case a(t, x) = a = (0, . . . , 0), bi (t, x) = bi = (bi1 , . . . , bid ), 1 ≤ i ≤ d;  

d

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We define the matrix σ = B T B, B = (bij ), 1 ≤ i, j ≤ d  2 σ = (σij ), 1 ≤ i, j ≤ d and the differential operator A : σij ∂x∂i ∂xj . 1≤i,j≤d

Let σ be a matrix with the following property 

σij zi zj ≥ μ| z|2 .

1≤i,j≤d

Here μ, is a fixed positive number, and z = (z1 , · · · , zd ) is an arbitrary real vector. This operator acts in the following space HA = {u : u ∈ L2 (Q) ∩ Au ∈ L2 (Q) ∩ u(∂Q) = 0} with inner product (u, v)A = (Au, v). Here (, ) is inner product in L2 (Q). The operator A is positive operator. It is known [5] that the following eigenvalues problem Au = −λu,

u(∂Q) = 0

has infinite set of real eigenvalues λi → ∞ and 0 < λ1 < λ2 < · · · < λs < · · · . The corresponding eigenfunctions f11 , . . . , f1n1 , · · · , fs1 , . . . , fsns , · · · form the complete system of functions both in HA and L02 (Q) := {u : u ∈ L2 (Q)∩u(∂Q) = 0}. Here the number nk is equal to multiplicity of eigenvalues λk . We also assume that the σ-additive measure ν is given on the Σν - algebra sets of Q, ν(Q) < ∞. All eigenfunctions fij : Q → R1 and all measures m(·, τ ) are (Σν , ΣY ) measurable. Here ΣY is the system of Borel sets of R1 . Let ⇒ denotes weak convergence of random values or measures. The notation is fairly standard. However, for convenience of the reader the following is recalled. We consider probabilistic measures Pτ and P , which are generated by a distribution functions Fτ (y) = P (η(τ ) ≤ y) and F (y) = P (η ≤ y). These measures are given on the Σν . They are defined by the following relations uniquely Pτ (−∞, y] = Fτ (y), P (−∞, y] = F (y). As usual, we define weak convergence   of η(τ ) and of Pτ in the form f (y)Pτ (dy) → f (y)P (dy) under τ → ∞ for all bounded continuous functions f on R1 . As we investigate a random values with range in set of numbers {0, 1, 2, ...}, then weak convergence is equivalent to following convergences under τ → ∞: • convergence of generating functions   η(τ ) Es = P (η(τ ) = k)sk → Esη = P (η = k)sk , 0 ≤ s ≤ 1. The common k≥0

k≥0

convention 00 = 1 is used. • Pτ (y) → P (y) for any singleton {y}.

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We assign  τ g(τ ) = exp − λ1 . 2 Theorem 2. We suppose that m(·, τ ) holds the condition lim m(B, τ )g(τ ) = ν(B),

τ →∞

B ∈ Σν .

Then η(τ ) ⇒ η if τ → ∞ where η has the Poisson distribution function with the n1    f1i (x)c1i , c1i = f1i (x)dx. parameter a = F (x)dν(x) and F (x) = i=1

Q

Q

Proof. We consider the following initial-boundary problem 1  ∂ 2u ∂u σij = ∂t 2 1≤i,j≤d ∂xi ∂xj

x ∈ Q;

u(0, x) = 1 if x ∈ Q; u(t, x) = 0 if x ∈ ∂Q,

t≥0

(6)

We designate through γk = (xk1 , · · · , xkd ) the initial position of k-th process. We define the value of u(τ, γk ). We define a particular solution of (6) in the form u(t, x) = u1 (t)u2 (x). The ordinary argumentation leads to definition of joined constant λ: 1 ∂u1 Au2 = −λ. = u1 ∂t u2 We obtain the following system of problem owing to the latter one 2

Au2 = −λu2 ; u2 (∂Q) = 0. (7) ∂u1 λ (8) = − u1 ; u1 (0) = 1 ∂t 2 It is clear that u1 (t, λ) = exp(− 2t λ) is the solution of (8). The solution of (7) was described above. We assume that the system of functions {fij (x), i ≥ 1, 1 ≤ j ≤ ni } is orthonormalized with respect to the space L02 (Q). The general solution of the problem (6) has the following form ∞ 

nj  t u(t, x) = exp(− λj ) cjm fjm (x), 2 m=1 j=1

where the coefficients cjm are equal to coefficients of decomposition of the initial value  (unit) by the system of functions fjm : cjm = fjm (x)dx. The Parseval - Steklov equality Q

is true for these coefficients:

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nj ∞  

c2jm = |Q|.

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(9)

j=1 m=1 n1 

We assign F (x) =

c1i f1m (x). The function F (x) is continuous and bounded

m=1

¯ Since u(t, x) is probability, it is not difficult to show that F (x) ≥ 0 for all function on Q. x ∈ Q. Let M = sup F (x). We introduce the following sets x∈Q

Bk,n

 Mk M (k + 1) = x∈Q: < F (x) ≤ n n

Here 0 ≤ k ≤ n − 1 and n > 1. Let us denote by ζk,n (τ ), 1 ≤ k ≤ n the number of unabsorbed processes at time instant τ , which occur in the region Bk,n at initial time. These values are independent in totality by assumption. As we assume that the diffusion processes are independent, then according to the formula of total probability the distribution function of ζk,n (τ ) is defined by the following formula P (ζk,n (τ ) = l) =

∞ 

P (μ(Bk,n , τ ) = d)×

d=l l 



×

d 

u(τ, γik )

1≤i1 ,··· ,il ≤d,im =ij ,m=j k=1

(1 − u(τ, γis )),

l = 0, 1, . . . .

s=l+1,is ∈(i / 1 ,··· ,il )

Here γij ∈ Bk,n . The summation is taken over all collections of l different integer numbers from the set of integer numbers {1, 2, . . . , d}. We also use common convention 0 l   = = 1. k=1

s=l+1

We set ak,n (τ ) = min u(τ, x),

a ¯k,n (τ ) = 1 − ak,n (τ );

bk,n (τ ) = max u(τ, x),

¯bk,n (τ ) = 1 − bk,n (τ ).

¯k,n x∈B

¯k,n x∈B

Now Jk,n (l, τ ) :=

∞  md (Bk,n , τ ) d=l

d!

exp(−m(Bk,n , τ ))Cdl alk,n (τ )¯bd−l k,n (τ ) ≤

≤ P (ζk,n (τ ) = l) ≤ ∞  md (Bk,n , τ ) d=l

Further

d!

ad−l exp(−m(Bk,n , τ ))Cdl blk,n (τ )¯ k,n (τ ) =: Ik,n (l, τ ).

(10)

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d−l ∞ ¯  bk,n (τ )m(Bk,n , τ ) (m(Bk,n , τ )ak,n (τ ))l Jk,n (l, τ ) = exp(−m(Bk,n , τ )) = l! (d − l)! d=l =

(m(Bk,n , τ )ak,n (τ ))l exp(−bk,n (τ )m(Bk,n , τ )); l!

By analogy: (m(Bk,n , τ )bk,n (τ ))l exp(−ak,n (τ )m(Bk,n , τ )). l! We introduce the following generating functions Ik,n (l, τ ) =

ϕ(τ, s) = ϕk,n (τ, s) =





(11)

sl P (η(τ ) = l).

l≥0

sl P (ζk,n (τ ) = l),

k = 0, n − 1,

0 ≤ s ≤ 1.

l≥0

By construction, η(τ ) can be represented as η(τ ) = ζ1,n + · · · + ζn−1,n (τ ). Thus, ϕ(τ, s) =

n−1 

ϕk,n (τ, s).

(12)

k=0

Combining (10)-(12), we conclude that exp{(sak,n (τ ) − bk,n (τ ))m(Bk,n , τ )} ≤ ϕk.n (τ, s) ≤ ≤ exp{(sbk,n (τ ) − ak,n (τ ))m(Bk,n , τ )} and exp

 n−1 

 (sak,n (τ ) − bk,n (τ ))m(Bk,n , τ )

≤ ϕ(τ, s) ≤

k=0 n−1  ≤ exp{ (sbk,n (τ ) − ak,n (τ ))m(Bk,n , τ )}.

(13)

k=0

Since the function u(τ, x) is the continuous function in x ∈ Q, such points x∗ , x∗ ∈ ¯k,n exist that the following equalities take place B nk   τ τ ak,n (τ ) = exp(− λ1 )F (x∗ ) + exp(− λk ) ckm fkm (x∗ ), 2 2 m=1 k≥2 nk   τ τ ∗ exp(− λk ) ckm fkm (x∗ ), bk,n (τ ) = exp(− λ1 )F (x ) + 2 2 m=1 k≥2

here x∗ := x∗ (k, n, τ ), x∗ := x∗ (k, n, τ ).

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Now, we can rewrite the sums in exponent from (13) in the following forms n−1 

τ (sF (x∗ ) − F (x∗ )) exp(− λ1 )m(Bk,n , τ )+ 2 k=0

nj  τ   τ + exp(− λ1 )m(Bk,n , τ ) exp − (λj − λ1 ) cjm (sfjm (x∗ ) − fjm (x∗ )), (14) 2 2 m=1 j≥2 k=0 n−1 

n−1 

τ (sF (x∗ ) − F (x∗ )) exp(− λ1 )m(Bk,n , τ )+ 2 k=0

nj  τ   τ + exp(− λ1 )m(Bk,n , τ ) exp − (λj − λ1 ) cjm (sfjm (x∗ ) − fjm (x∗ )), (15) 2 2 m=1 j≥2 k=0 n−1 

We calculate limit of (14) if τ → ∞. The first sum of (14) converges to the following limit under the condition of theorem n−1 

sF (x∗ )ν(Bk.n ) −

k=0

n−1 

F (x∗ )ν(Bk.n ).

k=0

This is difference of two integral sums, which has the following limit under n → ∞ (see [6])  (s − 1) F (x)ν(dx). Q

We assign

nj  τ  sτ (x) = exp − (λj − λ1 ) ckm fkm (x). 2 m=1 j≥2



We consider sums of eigenfunctions in the form e(x, λ) =

nk 

2 fkm (x),

λk ≤λ m=1

The following result is proved in the monograph [7, Thm. 17.5.3] sup x∈Q

 d e(x, λ) ≤ Cλ 2 .

Asymptotic characteristic of eigenvalues λj under j → ∞ is defined by the following inequalities [5, sec. 18] 2

2

c1 j d ≤ λj ≤ c 2 j d ,

where c1 , c2 = const.

The latter one, (9) and the Caushy-Bunyakovskii inequality lead to the following convergence under τ → ∞

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   nj nj  τ    2  2  |sτ (x)| ≤ exp − (λj − λ1 ) cjm fjm (x) ≤ 2 m=1 m=1 j≥2   nj  d  τ ≤C λj2 exp(− (λj − λ)) c2jm ≤ 2 m=1 j≥2    nj   d  ≤C λj exp(−τ (λj − λ1 )) c2jm → 0. 

j≥2 m=1

j≥2

Thus, the second sum of (14) converges to zero. Similar considerations were applied to (15). Proof is complete.



Example. Now we apply the general approach to the particular case. We consider the case if Q is circle Q = {(x, y) : x2 + y 2 ≤ r02 }. We assume that the diffusion processes occurs at the point (xk , yk ) ∈ Q at the initial time. The processes are described in Q by the following stochastic differential equations dξ(t) =

2 

bi dwi (t)

(16)

i=1

ξ(0) = ξ0 = (xk , yk ), where b1 = (σ, 0), b2 = (0, σ) and W (t) = (wi (t), i = 1, 2) is a 2-dimensional Wiener process. We assume that the equation (16) defines a diffusion process with absorption on the boundary ∂Q = {(x, y, z) : x2 + y 2 = r02 }. In follows that J0 (x), J1 (x) are Bessel functions of zero and first order. They are defined as solutions of the following equations d2 y 1 dy n2 + ) = 0, + (1 − dx2 x dx x2 √ y(x0 ) = 0, (x0 = λr); |y(0)| < ∞; for n = 0 and n = 1. (0) The value of μm is equal to m-th root of the equation J0 (μ) = 0 [8, 9]. Let mes(·) denotes the Lebesgue measure. We set   (0) τ  σμ1 2 f (τ ) := exp − . 2 r0 We suppose that m(·, τ ) holds the condition m(·, τ )f (τ ) ⇒ mes(·) if τ → ∞.

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In this case the system of problems (7), (8) has the following form

u2 = −μu2 , (x, y) ∈ C;

u2 (x, y) = 0 if x2 + y 2 = r02 ,

(17)

2

∂u1 σ (18) = − μu1 , u1 (0) = 1. ∂t 2 According to the general approach for construction of solution u(t, x, y) (see, for example, [8, sec. IV]) we rewrite the problem of (17) in polar coordinates: u3 (r, ϕ) := u2 (r cos ϕ, r sin ϕ). The u3 is the solution the following problem ∂ 2 u3 1 ∂u3 1 ∂ 2 u3 + + μu3 = 0, + ∂r2 r ∂r r2 ∂ϕ2 u3 (r0 , ϕ) = 0. We obtain

u(t, x, y) = u(t, r) =

∞ 

 c m J0

(0) μm

r0

m=1



⎛

r exp ⎝−

t 2



(0) σμm

2 ⎞ ⎠,

r0

 −1 (0) (0) where cm = 2 μm J1 (μm ) .   (0) μ is a strictly decreasing function if 0 ≤ r ≤ r0 . Thus we The function J0 r10 r can construct the partitions Bk,n by the following partitions

 r0 k  2 r0 (k + 1) 2 ˜ Bk,n = (x, y) ∈ C : < x +y ≤ , n n

0 ≤ k ≤ n − 1.

˜k,n ) = g( k+1 ) − g( k ), where g(x) = πr02 x2 , 0 ≤ x ≤ 1. Now mes(B n n Finally, the parameter of Poisson distribution is equal to 1 −1 (0) (0) (0) 2 a = 2 μ1 J1 (μ1 ) 2πr0 J0 (μ1 x)xdx = π 

0



2r0 (0)

μ1

2 .

We used the following known relation αJ0 (α) = [αJ1 (α)] [8, p. 466] for calculation of the latter integral.

Acknowledgment The authors thank the referee for his helpful comments.

References [1] V.A. Gasanenko and A.B. Roitman: “Rarefaction of moving diffusion particles”, The Ukrainian Math. J., Vol. 56, (2004), pp. 691–694

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[2] I.I. Gikhman and A.V. Skorokhod: Introduction to the Theory of Random Processes, Nauka, Moscow, 1977, p. 568. [3] O.A. Ladigenskaya, V.A. Solonnikov and N.N. Uraltseva: Linear and Quasilinearity Equations of Parabolic Type, Nauka, Moscow, 1963, p. 736. [4] A. Fedullo and V.A. Gasanenko: “Limit theorems for rarefaction of set of diffusion processes by boundaries”, Theor. Stochastic Proc., Vol. 11(27), (2005), pp. 23–28. [5] S.G. Mihlin: Partial Differential Linear Equations, Vyshaij shkola, Moscow, 1977, p. 431. [6] A.N. Kolmogorov and S.V. Fomin: Elements of Theory of Functions and Functional Analysis, Nauka, Moscow, 1972, p. 496. [7] L. H¨ ormander: The Analysis of Linear Partial Differential Operators III, SpingerVerlag, 1985, p. 696. [8] A.N. Tikhonov and A.A. Samarsky: The Equations of Mathematical Physics, Nauka, Moskow, 1977, p. 736. [9] E. Janke, F. Emde and F. Losch: Special Functions, Nauka, Moskow, 1968, p. 344.

DOI: 10.2478/s11533-006-0024-7 Research article CEJM 4(4) 2006 635–647

On the dynamics of equations with infinite delay∗ Dalibor Praˇz´ak† Charles University Prague, 186 75 Prague 8, Czech Republic

Received 3 May 2006; accepted 6 June 2006 Abstract: We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Equations with infinite delay, exponential attractor, fractal dimension MSC (2000): 37L25, 37L30, 34K17

1

Introduction

There are many examples of models-arising in areas such as biology, economics, and materials science-in which the rate of change at time t depends not only on the present state of the system, but also on its history during some time interval [t − τ, t]. Mathematically, these models are characterized by evolutionary equations with delay. The more general case τ = ∞ yields so-called equations with infinite delay. There is extensive literature on the study of various aspects of equations with delay. However, it seems to us that, in most papers, a rather specific form of equation is studied. From the specific form, one can usually obtain a correspondingly specific description of the dynamics at hand. Yet, the general case-delayed equations in abstract form-remains to be subjected to more study. Herein lies the objective of the present paper. We consider systems of ordinary differential equations (ODEs) with infinite delay, where the underlying phase space X is characterized by an exponentially decaying weight. ∗ †

Research supported by the project LC06052 of the Czech Ministry of Education. E-mail: [email protected]ff.cuni.cz

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The main assumption is the Lipschitz continuity of the right-hand side with respect to the norm of X. Note that systems with bounded delay, or equations where the dependence on the past is given by convolution with an exponentially decaying kernel, are special cases of our general formulation. (More on the latter class of models follows.) Our main theorem is the existence of an exponential attractor. We also estimate its fractal dimension in terms of relevant constants. The possible applications of our results are twofold. On the one hand, the technique we use is based on the ”short trajectory” method (cf. Lemma 4.2), which, from what we know, occurs for the first time in [10]. This approach is very elementary and it is not difficult to see that the extension to partial differential equations (PDEs) is possible under suitable conditions. We believe that, in fact, the approach suggests a more general method of constructing exponential attractors, or finite-dimensional attractors-attractors that will cover a wide class of nonlinear dissipative evolutionary PDEs with infinite delay. The idea of constructing attractors in the space of trajectories is not new. Note that, in [1, 2], the existence of ”trajectory attractors” is proved for certain classes of PDEs with infinite delay. With regard to the finite-dimensionality of attractors, we also build on the approach of [11], where problems with bounded delay are studied. On the other hand, we find it no less important to consider the dynamics of ODEs with infinite delay. We observe that a general class of dissipative PDEs (without delay) can, using a suitable projection, be reduced to a system of ODEs with infinite delay, a system that preserves the large time dynamics of the original PDEs [5, 9, 13]. We discuss these problems in the last section. Let us now compare our results with other recent publications and also try to put them into a somewhat broader perspective. In [7], we find interesting results for a system in which dependence on the past is realized by convolution with kernels of the form k (s) =

1 s k0 ,  

(1)

where k0 (·) has exponential decay. The authors show that, for every  ∈ (0, 0 ], the problem has an exponential attractor E . In addition, the authors give rigorous proof of the expectation that, for  → 0+, E approach the attractor of the undelayed system. In our general formulation, we have not been able to recover such a specific result, but have identified an interesting analogy (at least a formal one). The estimate of the dimension of the attractor we obtain depends only on L/γ, where L is the Lipschitz constant for the dependence on the past, and γ measures the rate of the decay of the corresponding weight as s → −∞. From (1), one has L ∼ −1 , γ ∼ −1 , so our estimate remains bounded even if  → 0. Finally, we note that the presence of infinite delay essentially makes the problem hyperbolic in character. This is seen explicitly in the setting where the dependence on the past is given by convolution with a sufficiently well-behaved kernel. One can then rewrite the problem in terms of a new function-a so-called ”summed past history” (see [4])-that satisfies a particular hyperbolic equation.

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637

While, in our abstract formulation, we have not recovered a certain specificity, we have found a similarity between our approach and [12] or [3], where a nonlinear hyperbolic problem is treated using a similar kind of squeezing property. The paper is organized as follows: In section 2, we describe our system and fix the notation. Section 3 presents preliminary results concerning the theory of exponential attractors. It also gives two covering lemmas that are needed for the explicit dimension estimate. The main result-the existence of an exponential attractor for the system of delayed ODEs and an explicit dimension estimate-is proved in section 4 (Theorem 4.5). In the last section, we briefly discuss an application to systems that arise as suitable projections of dissipative PDEs.

2

Equation

We consider a system of ODEs with infinite delay. Express the system in the form dt p(t) = F (pt ) ,

(2)

where p(t) : R → RM is the unknown function. A natural phase-space for the problem is   X = χ(s) : (−∞, 0] → RM ; χ(s) is continuous , endowed with the norm (γ > 0) χX = sup |χ(s)|eγs .

(3)

s≤0

Throughout the paper, we adopt the convention that, if p(s) is a time-dependent function, then pt is an element of X defined by pt (s) = p(t + s) ,

s ≤ 0.

Our equation is equipped with the initial condition p0 = χ ∈ X .

(4)

We assume that F : X → RM is Lipschitz continuous. That is, |F (χ) − F (χ)| ˜ ≤ Lχ − χ ˜ X

χ, χ˜ ∈ X .

(5)

It is not difficult to adapt standard ODE theory to prove that (2), (4) is well posed and that one has the unique global solution p(t) : R → RM . Denoting St χ = p t , we define the solution operator St : X → X to problem (2). In our abstract formulation, we assume that there exists a more regular set   B = χ ∈ X; |χ(s)| ≤ K1 , Lip χ ≤ K2 ,

(6)

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It is positively invariant and uniformly absorbing, or at least uniformly exponentially attracting for (St , X). This is a natural assumption with regard to applications (cf. the last section). Hence, it will suffice to construct the exponential attractor for the dynamical system (St , B). The dimension estimate we derive will be in terms of M , γ and L, but independent of K1 , K2 . The convention is that c1 , c2 , . . . are generic constants that are independent of M , γ and L, and that change in meaning, depending on context. The reasonable assumption L ≥1 (7) γ simplifies certain expressions.

3

Preliminaries

Let X be a bounded, complete metric space. Typically, X is a closed subset of a Banach space from which it inherits a metric. The fractal dimension of A ⊂ X is defined as dimfX (A) = lim sup →0+

ln NX (A, ) − ln 

where NX (A, ) is the smallest number of closed sets of diameter 2 that cover A. We assume that St : X → X is a semigroup of operators, S0 = I and St+s = St Ss . The set E ⊂ X is called an exponential attractor to (St , X) provided (1) E is compact, (2) St (E) ⊂ E, t ≥ 0, (3) dimfX (E) < ∞, (4) distX (St X , E) ≤ c exp(−βt), t ≥ 0. As is well-known, the concept of an exponential attractor was proposed in [6] in response to specific defects associated with the usual notion of the global (universal) attractor. A primary feature of the exponential attractor is its robustness with respect to various approximations, including singular perturbations (see [8]). Extensive literature focuses on proving the existence of exponential attractors for various dissipative equations. It turns out that, as soon as one can prove that the global attractor has finite fractal dimension, one can almost always construct an exponential attractor. If S : X → X is a continuous mapping, the concept of exponential attractor extends in an obvious way to the (discrete) dynamical system (S n , X ). One usually first constructs the exponential attractor for a certain discrete subgroup of St , generated by S = St∗ with suitably-chosen t∗ . The following lemma gives a useful sufficient condition. Lemma 3.1. Let S : X → X be Lipschitz continuous, and let there exist θ ∈ (0, 1) and a constant K > 0 such that NX (S(F ), θρ) ≤ K (8)

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for any ρ > 0, F ⊂ X with diamX (F ) ≤ 2ρ. Then, the dynamical system (S n , X ) has an exponential attractor E ∗ , and ln K . (9) dimfX (E ∗ ) ≤ − ln θ Proof. Starting with F = X , R = diam(X )/2, one obtains NX (S n (X ), θn R) ≤ K n .

(10)

by induction. However, from this condition-which is, in fact, equivalent to the existence of the exponential attractor-one can employ a standard construction to obtain the conclusion of the lemma, including the explicit estimate (8). See e.g. [11, Theorem 1.1] for details.  Once the exponential attractor is constructed for the discrete subgroup, it is a matter of routine to extend it for the entire dynamics. Lemma 3.2. Let E ∗ be an exponential attractor for (S n , X ), where S = St∗ with some fixed t∗ > 0. Assume that St x is locally Lipschitz continuous w.r. to t and x. Then there exists an exponential attractor E to (St , X) such that dimfX (E) ≤ dimfX (E ∗ ) + 1 .

(11)

Proof. Set E = F(E ∗ × [0, t∗ ]), where F : (x, t) → St x. Given that F is Lipschitz, one has dimfX (E) ≤ dimfX (E ∗ ) + dimfR ([0, t∗ ]) = dimfX (E) + 1. It is straightforward to check the other properties of the exponential attractor. (See [6, Chapter 3].)  Since we want to estimate the dimension of E explicitly, we need to compute some coverings. Lemma 3.3. Let r ≤ R. It follows that BRM (0, R) can be covered by m balls of radius r, where  M 3R m≤ . (12) r Proof. Let Bj , j = 1, . . . m be a maximal system of disjoint balls of radius r/2, centered in BRM (0, R). Clearly, the same balls with radius r cover BRM (0, R). Otherwise, the system is not maximal. Moreover, Bj ⊂ BRM (0, R + r/2) and, by volume comparison,  r M

r M m ≤ R+ 2 2    M M 2R 3R m≤ ≤ , +1 r r using 1 ≤ R/r.





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Lemma 3.4. Let A ≥ 1/5, B > 0. The set   M = χ ∈ C([0, t∗ ]); |χ(t)| ≤ A , Lip χ ≤ B can be covered by K balls of radius 1 in the space C([0, t∗ ]), where ln K ≤ c M (t∗ B + 1) ln(A + 1) .

(13)

Proof. Choose points t0 = 0 < t1 < · · · < tn = t∗ such that ti+1 − ti ≤ δ := 1/(5B). This can be accomplished with n ≤ t∗ /δ + 1 ≤ 5Bt∗ + 1. Furthermore, there exist points xj , j = 1, . . . m in RM such that BRM (xj , 1/5) cover BRM (0, A). According to Lemma 3.3, one has m ≤ (15A)M . Now, consider the set N of functions χ(t) : [0, t∗ ] → RM such that χ(t) takes only the values xj for t = ti and is linear in [ti , ti+1 ]. Clearly, if K = #N , one has ln K = ln mn+1 ≤ M (5t∗ B + 2) ln 15A ≤ c M (t∗ B + 1)(ln A + 1) . The proof will be finished once we show that the balls B(χ, 1), χ ∈ N cover the set M. Let ψ ∈ M be arbitrary. From the preceding discussion, there exists χ ∈ N such that |χ(ti ) − ψ(ti )| ≤ 1/5, i = 0, . . . , n. We claim that |χ(t) − ψ(t)| ≤ 1 for all t ∈ [0, t∗ ]. It suffices to show this for fixed t ∈ [ti , ti+1 ], i. One has |χ(t) − ψ(t)| ≤ |χ(t) − χ(ti )| + |χ(ti ) − ψ(ti )| + |ψ(ti ) − ψ(t)| ≤ |χ(t) − χ(ti )| + 1/5 + δB ≤ |χ(t) − χ(ti )| + 2/5 . Yet, as a consequence of the piecewise linearity of χ, one has |χ(t) − χ(ti )| ≤ |χ(ti+1 ) − χ(ti )|. Also, |χ(ti+1 ) − χ(ti )| ≤ |χ(ti+1 ) − ψ(ti+1 )| + |ψ(ti+1 ) − ψ(ti )| + |ψ(ti ) − χ(ti )| ≤ 1/5 + δB + 1/5 ≤ 3/5 . 

The proof is finished.

4

Exponential attractor

In this section, we establish our main result: we construct an exponential attractor to (2) and give the estimate of its fractal dimension. The dissipation that is encoded in our equation is described in the following lemma. On the surface, it looks more like an assertion about the space X, and not about the equation itself. Lemma 4.1. Let η : (−∞, t] → RM be given. It follows that η t X ≤ e−γt η 0 X + ηC([0,t]) .

(14)

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Proof. By (3), η t X = sup |η(t + s)|eγs s≤0   = max sup |η(t + s)|eγs , sup |η(t + s)|eγs s≤−t

s∈[−t,0]

  ≤ max e−γt η 0 X , ηC([0,t]) ≤ e−γt η 0 X + ηC([0,t]) .

 Note that (14) is a kind of ”squeezing property”. To handle the second term on its right-hand side, we need the continuity of the evolution operator. Without concrete assumptions on its nonlinearity, the only available method is the long-established ”shorttrajectory trick” used in [10]. Lemma 4.2. Set

1 , (15) 2L where L is the Lipschitz constant from (5). Let p and q be solutions to (2), and z = p − q. Then (16) zC([0,]) ≤ 3z 0 X . =

Proof. Subtract the equations for p and q, and multiply by z/|z| to get dt |z(t)| ≤ |F (pt ) − F (q t )| ≤ Lz t X ≤ Lz 0 X + LzC([0,]) ,

∀t ∈ [0, ]

using (14). Pick t0 ∈ [0, ] such that |z(t0 )| = zC([0,]) . Integration over t ∈ [0, t0 ] yields |z(t0 )| ≤ |z(0)| + Lz 0 X + L|z(t0 )| (1 − L)|z(t0 )| ≤ (1 + L)z 0 X as |z(0)| ≤ z 0 X . Given that L = 1/2, the conclusion follows.



Using the equation once more, we estimate the difference of the time derivatives. Thus, as a result of the compact embedding of C 1 ([0, ]) into C([0, ]), we establish a kind of ”smoothing property”. Lemma 4.3. Using the previous notation, one finds dt zC([0,]) ≤ 3L z 0 X . Proof. Note that, in the proof of Lemma 4.1, we established   z t X ≤ max e−γt z 0 X , zC([0,t]) .

(17)

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Taken together with (5) and (16), we have dt zC([0,]) ≤ sup |F (pt ) − F (q t )| t∈[0,]

≤ L sup z t X t∈[0,]

  ≤ L sup max e−γt z 0 X , zC([0,t]) t∈[0,]

≤ 3Lz 0 X .  Now we can establish the ”discrete version” of the exponential attractor. Theorem 4.4. Set S = S . Then the semigroup (S n , B) has exponential attractor E ∗ , and dimfX (E ∗ ) ≤ c M (γ −2 L2 + 1) ln(γ −1 L + 1) . (18) Proof. We will use Lemma 3.1. Let F ⊂ B, diamX (F ) ≤ 2ρ. We cover S(F ) with sets Fj , j = 1, . . . , K that have diamX (Fj ) ≤ 2ρθ, where 1 θ = (1 + e−γ ) , 2

(19)

and K is independent of F , ρ. Clearly, F ⊂ BX (ψ0 , 2ρ), with ψ0 ∈ F arbitrary. By Lemmas 4.2 and 4.3, the set F |[0,] is contained ‡ in ψ0 |[0,] + M, where   M = χ ∈ C([0, ]); |χ(t)| ≤ 6ρ , Lip χ ≤ 6Lρ . We can find ηj ∈ F , j = 1, . . . , K such that the sets   Gj = ψ ∈ F ; ψ − ηj C([0,]) ≤ ρ cover F , where 1 (20)  = (1 − e−γ ) . 2 According to the scaling argument, this covering is equivalent to the problem of Lemma 3.4 with A = 6−1 , B = 6L−1 . Note that, since (7), γ ≤ 1/2 and, consequently,  ≤ 0.2. Thus, A ≥ 1/5, as required in Lemma 3.4. Hence, the number K is estimated by ln K ≤ c M ( L−1 + 1) ln(−1 + 1) .

(21)

Clearly, Fj = S(Gj ) cover S(F ). We claim that diamX (Fj ) ≤ 2ρθ. Let ψ, ψ˜ ∈ Gj be arbitrary. Using (14) with t = , from (19) it follows that (20) ˜ X + ψ − ψ ˜ C([0,]) ˜ X ≤ (θ − )ψ − ψ S(ψ) − S(ψ) ≤ (θ − )2ρ + 2ρ = 2θρ . ‡

To simplify the notation, we implicitly extend the elements of X to t ∈ [0, ] by solving (2).

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There remains the matter of evaluating the estimate for K. Using (7), (15) −1 =

2 γ ≤c . −γ/2L 1−e L

Similarly, − ln θ = − ln[(1 + e−γ/2L )/2] ≥ c

γ . L 

Hence, (9) and (21) yield (18). The main result now follows readily.

Theorem 4.5. Let the assumptions of section 2 hold. Then the dynamical system (St , X) associated with (2) has exponential attractor E. Its fractal dimension is estimated as dimfX (E) ≤ c M L2 γ −2 ln(Lγ −1 + 1) .

(22)

Proof. By a previous theorem, we have an exponential attractor E ∗ to (S n , B), where S ∗ = S . We want to apply Lemma 3.2, thus reducing the problem to the verification of the Lipschitz continuity of St χ w.r. to both χ and t. However, by (14) and (16), we have (t ∈ [0, ]) St χ − St χ ˜ X ≤ e−γt χ − χ ˜ X + χ − χ ˜ C([0,]) ≤ 4χ − χ ˜ X. For continuity in time, we observe St χ − StˆχX = sup |χ(t + s) − χ(tˆ + s)|eγs ≤ K2 |t − tˆ| , s≤0

where χ is an element of B. There remains the matter of evaluating the dimension estimate. Inasmuch as (7), we substitute (18) into (11), yielding (22). 

5

Application to projected PDEs

In the last section, we discuss an interesting application of the previous results. We look at a system of ODEs with infinite delay that arises as an equivalent description of the large time dynamics of an evolutionary PDE. Consider an abstract dissipative PDE in the form dt u + Au + R(u) = f .

(23)

Here, u = u(t) ∈ H is the unknown, where H is a suitable Hilbert space, A is a linear elliptic operator, and R is some lower-order nonlinearity. The dimension of H is infinite, and the relevant question is whether the large time behavior of the solutions is, in some sense, finite-dimensional. A partial answer-one that holds for numerous physically-interesting instances of (23)-is given by the existence of a global attractor A with finite (fractal) dimension (cf. [14]).

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The best kind of attractor is the so-called inertial manifold. Its existence implies that the large time dynamics of a given PDE can be described by a system of ODEs. Unfortunately, the known conditions under which this manifold can be constructed (e.g. large gaps in the spectrum of A) are too restrictive to cover a sufficiently wide class of problems. An important example where the existence of an inertial manifold remains an open problem is given by the 2d Navier-Stokes equations. It turns out, however, that, under general conditions, one can rewrite the large time dynamics of (23) as a system of ODEs with infinite delay. These so-called ”inertial manifolds with delay” were first treated in [5] (see also [9]). Here, following [13], we outline an alternate construction. We omit most technical details. Nonetheless, the key assumption that enables the construction is the squeezing property: there exist θ ∈ (0, 1), t∗ , c1 > 0 and finite-dimensional projector P such that u(t + t∗ ) − v(t + t∗ )H ≤ c1 P (u(t + t∗ ) − v(t + t∗ ))H + θu(t) − v(t)H

(24)

for any u and v solutions to (23) in the global attractor A. Setting p(t) = P u(t), from (23) we obtain dt p + Ap + P R(u) = g .

(25)

We denote g = P f and, henceforth, identify P H with RM . The key step in closing this projected equation in terms of just p is the following ”recovering lemma”. It states that the solutions on the attractor are uniquely determined by the P projections of its past values. Lemma 5.1. Set   A0 = u(s) : (−∞, 0] → H; u(0) ∈ A, u solves (23) ,   T = χ(s) : (−∞, 0] → RM ; ∃u ∈ A0 , χ(s) = P u(s) . Then there exists a mapping E : T → A such that, if u ∈ A0 and χ(s) = P u(s), E(χ) = u(0). Moreover, E is Lipschitz continuous from X (given in (3)) to H as long as γ > 0 is small enough. Proof. Let u(t), v(t) ∈ A0 . In virtue of (24), u(0) − v(0)H ≤ c1 P u(0) − P v(0)H + θu(−t∗ ) − v(−t∗ )H . Applying (24) repeatedly to the last term, one has u(0) − v(0)H ≤ c1

n−1  k=0

θk P u(−kt∗ ) − P v(−kt∗ )H + θn u(−nt∗ ) − v(−nt∗ )H .

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We let n → ∞. Since u and v are bounded, denoting χ = [P u]0 , ψ = [P v]0 enables us to obtain ∞  u(0) − v(0)H ≤ c1 θk |χ(−kt∗ ) − ψ(−kt∗ )| k=0

= c1

∞ 

∗

θk eγkt |χ(−kt∗ ) − ψ(−kt∗ )|e−γkt

k=0

≤ c1 χ − ψX

∞ 

∗

∗

θk−1 eγkt .

k=1

For γ sufficiently small, the last sum converges, thus establishing the desired result.



We see, then, that the elements p(t) of T are the solutions of ˜ t) = g , dt p + Ap + B(p ˜ : T ⊂ X → RM is Lipschitz, one can extend it to a ˜ = P R(E(pt )). Since B where B mapping B : X → RM that has the same Lipschitz constant. Moreover, based on the boundedness of T , we also have B(χ) = 0 if χX ≥ R0 . See [13, Lemma 2.2] for details. Thus, we obtain the projected (reduced) version of (23) dt p + Ap + B(pt ) = g .

(26)

The Lipschitz continuity of B ensures unique solvability-an important specification in proving the one-to-one correspondence of the solutions of (26) on T and the solutions of (23) on A (see [13, Theorem 3.3]). We have already shown that (26) has a global attractor Ar ⊂ X. It is easy to see that T (the P -projected negative trajectories on A ⊂ H) is an invariant subset of Ar . The question was whether the dynamics on Ar are really larger than the dynamics on A. We can now show that the dynamics are, in fact, not substantially larger insofar as (26) has an exponential attractor. In particular, Ar has finite dimension. We first show that there exists a more regular set B ⊂ X, one that attracts the solutions exponentially. Lemma 5.2. There exists K1 , K2 > 0 such that the set   B = χ ∈ X; |χ(s)| ≤ K1 , Lip χ ≤ K2 , is positively invariant and uniformly exponentially attracting for (26). Consequently, for any solution χ, there exist c, t0 > 0, depending only on χ0 X , such that distX (χt , B) ≤ ce−γt ,

∀t ≥ t0 .

Proof. Take K1 such that B(χ) = 0 Ap · p − g · p ≥ |p|2

if χX ≥ K1 ;

(27)

if |p| ≥ K1 .

(28)

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Further, set K2 = |g| + AK1 + sup |B(χ)| . χ∈X

(The supremum is finite because B is Lipschitz and has bounded support.) Note that any solution χ to (26) satisfies |dt χ(t)| ≤ K2 , t > 0. Moreover, the choice of K1 ensures that, if |χ(t)| ≤ K1 for t ≤ 0, then |χ(t)| ≤ K1 for all t. As a result, B is positively invariant. To prove that B is uniformly exponentially attracting, take an arbitrary solution χ. Let t0 ≥ 0 be the smallest number such that |χ(t0 )| ≤ K1 . By (27)-(28), such a t0 exists, depending only on |χ(0)| ≤ χ0 X , and, moreover, |χ(t)| ≤ K1 for ∀t ≥ t0 . Set χ(t), t ≥ t0 χ(t) ˜ = . χ(t0 ), t < t0 Clearly, χ˜t ∈ B for all t ∈ R. We claim that χt − χ˜t X → 0 exponentially, which is the desired conclusion. For t ≥ t0 , we have ˜ + s)|eγs χt − χ˜t X = sup |χ(t + s) − χ(t s≤0

= sup |χ(t + s) − χ(t ˜ + s)|eγs s≤t0 −t   ≤ e−γ(t−t0 ) sup |χ(t + s)| + K1 eγ(t+s−t0 ) s≤t −t  0 t0  −γ(t−t0 ) χ X + K1 . ≤e  Now, we can state the main theorem. Theorem 5.3. Equation (26) has an exponential attractor Er ⊂ X. Proof. This follows immediately from Theorem 4.5. For, writing F (χ) = g − Aχ(0) − B(χ), we see that (26) is a special case of (2). All the assumptions are verified above. 

References [1] V.V. Chepyzhov, S. Gatti, M. Grasselli, A. Miranville and V. Pata: “Trajectory and global attractors for evolution equations with memory”, Appl. Math. Lett., Vol. 19(1), (2006), pp. 87–96. [2] V.V. Chepyzhov and A. Miranville: “On trajectory and global attractors for semilinear heat equations with fading memory”, Indiana Univ. Math. J., Vol. 55(1), (2006), pp. 119–167. [3] I. Chueshov and I. Lasiecka: “Attractors for second-order evolution equations with a nonlinear damping”, J. Dynam. Differential Equations, Vol. 16(2), (2004), pp. 469– 512.

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[4] C.M. Dafermos: “Asymptotic stability in viscoelasticity”, Arch. Rational Mech. Anal., Vol. 37, (1970), pp. 297–308. [5] A. Debussche and R. Temam: “Some new generalizations of inertial manifolds”, Discrete Contin. Dynam. Systems, Vol. 2(4), (1996), pp. 543–558. [6] A. Eden, C. Foias, B. Nicolaenko and R. Temam: Exponential attractors for dissipative evolution equations, Vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris, 1994. [7] S. Gatti, M. Grasselli, A. Miranville and V. Pata: “Memory relaxation of first order evolution equations”, Nonlinearity, Vol. 18(4), (2005), pp. 1859–1883. [8] S. Gatti, M. Grasselli, A. Miranville and V. Pata: “A construction of a robust family of exponential attractors”, Proc. Amer. Math. Soc., Vol. 134(1), (2006), pp. 117–127 (electronic). [9] J.K. Hale and G. Raugel: “Regularity, determining modes and Galerkin methods”, J. Math. Pures Appl. (9), Vol. 82(9), (2003), pp. 1075–1136. [10] J. M´alek and J. Neˇcas: “A finite-dimensional attractor for three-dimensional flow of incompressible fluids”, J. Differ. Equations, Vol. 127(2), (1996), pp. 498–518. [11] D. Praˇz´ak: “A necessary and sufficient condition for the existence of an exponential attractor” Cent. Eur. J. Math., Vol. 1(3), (2003), pp. 411–417. [12] D. Praˇz´ak: “On the dimension of the attractor for the wave equation with nonlinear damping”, Commun. Pure Appl. Anal., Vol. 4(1), (2005), pp. 165–174. [13] D. Praˇz´ak: “On reducing the 2d Navier-Stokes equations to a system of delayed ODEs”, In: Nonlinear elliptic and parabolic problems, Vol. 64 of Progr. Nonlinear Differential Equations Appl., Birkh¨auser, Basel, 2005, pp. 403–411. [14] R. Temam: Infinite-dimensional dynamical systems in mechanics and physics, Vol. 68 of Applied Mathematical Sciences, 2nd ed., Springer-Verlag, New York, 1997.

DOI: 10.2478/s11533-006-0031-8 Research article CEJM 4(4) 2006 648–655

σ-asymptotically lacunary statistical equivalent sequences Ekrem Sava¸s1∗ , Richard F. Patterson2† 1 Y¨ uz¨ unc¨ u Yıl University Education Faculty, Department of Mathematics, 65080 Van, Turkey 2

Department of Mathematics and Statistics, University of North Florida, Jacksonville, Florida, 32224, USA

Received 15 June 2005; accepted 23 August 2006 Abstract: This paper presents the following definitons which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let θ be a lacunary sequence; Two nonnegative sequences [x] and [y] are Sσ,θ -asymptotically equivalent of multiple L provided that for every  > 0      xσk (m)  1 lim − L ≥  = 0 k ∈ Ir :  r hr yσk (m) Sσ,θ

uniformly in m = 1, 2, 3, . . . , (denoted by x ∼ y) simply Sσ,θ -asymptotically equivalent, if L = 1. Using this definition we shall prove Sσ,θ -asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and analogues results of Das and Patel in [1] shall also be presented. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: σ-asymptotically, Equivalent of multiple L MSC (2000): 40A99, 40A05

1

Introduction

Let l∞ and c be the Banach spaces of bounded and convergent sequences x = (xk ) normed by ||x|| = supk |xk |, respectively. ∗ †

E-mail: [email protected] E-mail: [email protected]

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Let σ be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functional φ on l∞ is said to be an invariant mean or a σ-mean if and only if i ) φ(x) ≥ 0 when the sequence x = [xk ] is such that xk ≥ 0 for all k, ii ) φ(e) = 1 where e = (1, 1, 1, . . .), and iii ) φ(x) = φ(xσ(k) ) for all x ∈ l∞ . Throughout this paper we shall consider the mapping σ has having on finite orbits, that is σ k (m) = m for all nonnegative integers with k ≥ 1, where σ m (k) is the m-th iterate of σ at k. Thus σ-mean extends the limit functional on c in the sense that φ(x) = lim x for all x ∈ c. Consequently, c ⊂ Vσ where Vσ is the set of bounded sequences all off whose σ-mean are equal. In the case, when σ(m) = m+1, the σ-mean is often called the Banach limit and Vσ is the set of almost convergent sequences, which was presented by Lorentz in [6]. In 1993 Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003 Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. In addition to the extensions above Patterson and Sava¸s incorporated lacunary sequences into these notions in [11]. This paper extends the definitions presented in [11] to a generalized notion of σ-convergence, namely Sσ -asymptotically equivalent of multiple L. In addition to these definition, natural inclusion theorems shall also be presented.

2

Definitions, and Notations

Definition 2.1 (Marouf, [9]). Two nonnegative sequences [x], and [y] are said to be asymptotically equivalent if xk lim =1 k yk (denoted by x∼y).

Definition 2.2 (Fast, Fridy; [2, 4]). The sequence [x] has statistic limit L, denoted by st − lim s = L provided that for every  > 0, 1 lim {the number of k ≤ n : |xk − L| ≥ } = 0. n n The next definition is natural combination of Definitions 2.1 and 2.2. Definition 2.3 (Patterson, [10]). Two nonnegative sequence [x] and [y] are said to be asymptotically statistical equivalent of multiple L provided that for every  > 0,     xk 1  lim {the number of k < n :  − L ≥ } = 0 n n yk

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(denoted by x ∼L y), and simply asymptotically statistical equivalent, if L = 1. In addition, these two definitions were also extended to incorporate lacunary sequence concept. Following these results Patterson and Sava¸s [11] introduce two new definitions, namely asymptotically lacunary statistical equivalent of multiple L and strong asymptotically lacunary equivalent of multiple L. By a lacunary θ = (kr ); r = 0, 1, 2, ... where k0 = 0, we shall mean an increasing sequence of non-negative integers with kr −kr−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr−1 , kr ] and kr hr = kr − kr−1 . The ratio kr−1 will be denoted by qr . Definition 2.4 (Patterson and Sava¸s [11]). Let θ be a lacunary sequence; the two nonnegative sequences [x] and [y] are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every  > 0      xk   1  lim  k ∈ Ir :  − L ≥   = 0 r hr yk SL

(denoted by x ∼θ y) and simply asymptotically lacunary statistical equivalent, if L = 1. Definition 2.5 (Patterson and Sava¸s [11]). Let θ be a lacunary sequence; the two nonnegative sequences [x] and [y] are strong asymptotically lacunary equivalent of multiple L provided that    1   xk − L = 0, lim  r hr yk k∈I r

NθL

(denoted by x ∼ y) and simply strong asymptotically lacunary equivalent, if L = 1. Following these results we shall now introduce three new notions, Sσ -asymptotically equivalent of multiple L, Sσ,θ -asymptotically equivalent of multiple L, and strong σasymptotically equivalent of multiple L. Then we use these definitions to prove Sσ asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and Das and Patel’s results in [1]. Definition 2.6. Two nonnegative sequences [x] and [y] are Sσ -asymptotically equivalent of multiple L provided that for every  > 0       xσk (m) 1 lim − L ≥  = 0 the number of k ≤ n :  n n yσk (m) S

uniformly in m = 1, 2, 3, . . . , (denoted by x ∼σ y) and simply σ-asymptotically equivalent, if L = 1.

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Definition 2.7. Let θ be a lacunary sequence; Two nonnegative sequences [x] and [y] are Sσ,θ -asymptotically equivalent of multiple L provided that for every  > 0      xσk (m)  1 k ∈ Ir :  lim − L ≥  = 0 r hr yσk (m) Sσ,θ

uniformly in m = 1, 2, 3, . . . , (denoted by x ∼ y) simply Sσ,θ -asymptotically equivalent, if L = 1. Definition 2.8. Let θ be a lacunary sequence; Two nonnegative sequences [x] and [y] are strong σ-asymptotically lacunary equivalent of multiple L provided that    1   xσk (m) − L = 0, lim  r hr y k k∈Ir

σ (m)

Nσ,θ

(denoted by x ∼ y)and simply strong σ-asymptotically lacunary equivalent, if L = 1.

3

Main Result

Theorem 3.1. Let θ = {kr } be a lacunary sequence then Nσ,θ

Sσ,θ

(1) if x ∼ y then x ∼ y, Sσ,θ

Nσ,θ

(2) if x ∈ l∞ and x ∼ y then x ∼ y, and Sσ,θ

Nσ,θ

(3) x ∼ y ∩ l∞ = x ∼ y ∩ l∞ . Nσ,θ

Proof. Part (1): If  > 0 and x ∼ y then    xσk (m)    ≥ − L y k   k∈Ir

σ (m)

  xσk (m)  k∈Ir & y −L≥

   xσk (m)    − L y k  σ (m)

k

  σ (m)    xσk (m)       − L ≥   . ≥   k ∈ Ir :  y k σ (m)

Sσ,θ

Sσ,θ

Therefore x ∼ y. Part (2): Suppose [x] and [y] are in l∞ and x ∼ y. Then we can assume that     xσk (m)  − L ≤ M for all k and m. y k σ (m) Given  > 0         xσk (m) 1 1   xσk (m)    − L = − L   y k hr k∈I  yσk (m) hr x  σ (m)   σ k (m) k∈Ir & y −L≥

r

σ k (m)

1 + hr



x   σk (m)  k∈Ir & y −L< k

   xσk (m)   − L y k σ (m)

σ (m)       xσk (m)   M    + . k ∈ I : − L ≥  ≤ r y k   hr  σ (m)

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Therefore x ∼ y. Part (3): follows from (1) and (2).



Lemma 3.2. Suppose for given 1 > 0 and every  > 0 there exist n0 and m0 such that        xσk (m) 1    − L ≥   < 1 , 0≤k ≤n−1:  n yσk (m) S

for all n ≥ n0 and m ≥ m0 then x ∼σ y. Proof. Let 1 be given. For every  > 0, choose n10 and m0 such that        xσk (m) 1    < 1 ,  − L ≥  0 ≤ k ≤ n − 1 :     n yσk (m) 2

(1)

for all n ≥ n10 and m ≥ m0 . It is sufficient to prove that there exists n01,1 such that for n ≥ n01,1 and 0 ≤ m ≤ m0        xσk (m) 1  − L ≥   < 1 , (2) 0 ≤ k ≤ n − 1 :   n yσk (m) If we let n0 = max{n10 , n01,1 } (2) will be true for n > n0 and for all m. Once m0 has been chosen, m0 is fixed, so       xσk (m)    0 ≤ k ≤ m0 − 1 :   ≥   = M. − L  y k   σ (m) Now taking 0 ≤ m ≤ m0 , and n > m0 , we have .            xσk (m)    xσk (m)  1  1 − L ≥   ≤  0 ≤ k ≤ m0 − 1 :  − L ≥   0 ≤ k ≤ n − 1 :   n yσk (m) n y k  σ (m)     xσk (m)    − L ≥   +  m0 ≤ k ≤ n − 1 :  yσk (m)      xσk (m)    1 M   − L ≥   +  m0 ≤ k ≤ n − 1 :  ≤ n n yσk (m) 1 M + , by(2). ≤ n 2 Thus for n sufficiently large      xσk (m)   M 1  1 0 ≤ k ≤ n − 1 :  + < 1 . − L ≥   ≤  n yσk (m) n 2 Thus (2) holds. This yields the result.



Theorem 3.3. Sσ,θ = Sσ for every θ. Proof. Let x ∈ Sσ,θ , then from Definition 2.7 assures us that, given 1 > 0 there exist  > 0 and L such that      xσk (m)  1   k ∈ Ir :  − L ≥  < 1 , hr yσk (m)

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for r ≥ r0 and m = kr−1 + 1 + u where u ≥ 0. Let n ≥ hr and write n = ihr + t with 0 ≤ t ≤ hr and i an integer. Since n ≥ hr and i ≥ 0. We obtain the following:          xσk (m)     xσk (m)   1  1        0 ≤ k ≤ n − 1 : 0 ≤ k ≤ (i + 1)h − L ≥  ≤ − 1 : − L ≥  r y k   y k   n n σ (m) σ (m)    i   xσk (m)   1      jh ≤ k ≤ (j + 1)h − 1 : − L ≥  = r r  y k   n σ (m)

j=0

(i + 1)hr 1 n 2ihr 1 for i ≥ 1. ≤ n ≤

For

hr n

≤ 1, since

ihr n

≤ 1. Therefore        xσk (m) 1   ≥   ≤ 21 .  − L 0 ≤ k ≤ n − 1 :   y k n σ (m)

Hence Lemma 3.2 implies Sσ,θ ⊆ Sσ . It is also clear that Sσ ⊆ Sσ,θ for every θ. This completes the proof.  Theorem 3.4. Let θ = {kr } be a lacunary sequence with lim inf qr > 1, then Sσ,θ

S

x ∼σ y implies x ∼ y. Proof. Suppose first that lim inf qr > 1, then there exists a δ > 0 such that qr ≥ 1 + δ for sufficiently large r, which implies hr δ . ≥ kr 1+δ S

If x ∼σ y, then for every ε > 0 and for sufficiently large r, we have  1  k ≤ kr kr 

      xσk (m)    k ∈ Ir :  ≥  − L  y k   σ (m)      xσk (m)   δ 1    . k ∈ I : − L ≥  ≥ r y k   1 + δ hr  σ (m)

    xσk (m)   1 :  − L ≥   ≥ yσk (m) kr

This completes the proof. Theorem 3.5. Let θ = {kr } be a lacunary sequence with supr qr < ∞, then Sσ,θ

S

x ∼ y implies x ∼σ y. Proof. If supr qr < ∞, then there exists B > 0 such that qr < B for all r ≥ 1. Let Sσ,θ

x ∼ y and ε1 > 0 . There exists R > 0 and ε > 0 such that for every j ≥ R

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 1  k ∈ Ij Aj = hj 

    xσk (m)   :  − L ≥   < 1 , for all m. yσk (m)

We can also find K > 0 such that Aj < K for all j = 1, 2, ..... Now let n be any integer with kr−1 < n < kr , where r > R. Then       xσk (m)  1    ≤ 1 − L ≥  k ≤ n :     n yσk (m) kr−1 1 = kr−1 1 + kr−1 + = + + +

         k ≤ kr :  xσk (m) − L ≥   y k     σ (m)         k ∈ I1 :  xσk (m) − L ≥    y k    σ (m)         k ∈ I2 :  xσk (m) − L ≥    y k   σ (m)       xσk (m)   1  k ∈ Ir :  ≥  − L ··· + y k   kr−1  σ (m)      xσk (m)   k1   ≥  k ∈ I : − L 1 y k   kr−1 k1    σ (m)      k2 − k1  k ∈ I2 :  xσk (m) − L ≥   y k   kr−1 (k2 − k1 )    σ (m)    xσk (m)    kR − kR−1    − L ≥   k ∈ IR :  ··· +  kr−1 (kR − kR−1 ) y k  σ (m)     xσk (m)    kr − kr−1  k ∈ Ir :  − L ≥   ··· +   kr−1 (kr − kr−1 ) y k σ (m)

= + ≤ ≤

k2 − k1 kR − kR−1 k1 A1 + A2 + · · · + AR kr−1 kr−1 kr−1 kR+1 − kR kr − kr−1 AR+1 + · · · + Ar k k  r−1  r−1   kR kr − kR sup Aj + sup Aj kr−1 kr−1 j≥1 j≥R kR K + B. kr−1



This completes the proof.

Theorem 3.6. Let θ = {kr } be a lacunary sequence with 1 < inf r qr ≤ supr qr < ∞, then Sσ,θ

S

x ∼ y ⇔ x ∼σ y. Proof. The result clearly follows from Theorem 3.4 and 3.5.



Remark 3.7. In the case when σ(n) = n + 1, the above results reduces to the results for almost convergence.

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References [1] G. Das and B.K. Patel: “Lacunary distribution of sequences”, Indian J. Pure Appl. Math., Vol. 26(1), (1989), pp. 54–74. [2] H. Fast: “Sur la convergence statistique”, Collog. Math., Vol. 2, (1951), pp. 241–244. [3] J.A. Fridy: “Minimal rates of summability”, Can. J. Math., Vol. 30(4), (1978), pp. 808–816. [4] J.A. Fridy: “On statistical sonvergence”, Analysis, Vol. 5, (1985), pp. 301–313. [5] J.A. Fridy and C. Orhan: “Lacunary statistical sonvergent”, Pacific J. Math., Vol. 160(1), (1993), pp. 43–51. [6] G.G. Lorentz: “A contribution to the theory of divergent sequences”, Acta. Math., Vol. 80, (1948), pp. 167–190. [7] Mursaleen: “Some new spaces of lacunary sequences and invariant means”, Ital. J. Pure Appl. Math., Vol. 11, (2002), pp. 175–181. [8] Mursaleen: “New invariant matrix methods of summability”, Quart. J. Math. Oxford, Vol. 34(2), (1983), pp. 133, 77–86. [9] M. Marouf: “Asymptotic equivalence and summability”, Int. J. Math. Math. Sci., Vol. 16(4), (1993), pp. 755–762. [10] R.F. Patterson: “On asymptotically statistically equivalent sequences”, Demonstratio Math., Vol. 36(1), (2003), pp. 149–153. [11] R.F. Patterson and E. Sava¸s: “On asymptotically lacunary statistically equivalent sequences”, (in press). [12] P. Schaefer: “Infinite matrices and invariant means”, Proc. Amer. Math. Soc., Vol. 36, (1972), pp. 104–110.

DOI: 10.2478/s11533-006-0030-9 Research article CEJM 4(4) 2006 656–668

Unsaturated solutions for partial difference equations with forcing terms Zhi-Qiang Zhu1 , Sui Sun Cheng2∗ 1

Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou 510665, P.R. China 2

Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, R.O. China

Received 7 June 2006; accepted 24 July 2006 Abstract: The concept of unsaturated infinite double sequence is introduced by making use of frequency measures. Unsaturated solutions are then studied for a partial difference equation. Conditions for all solutions to be unsaturated are obtained. Since unsaturated solutions are oscillatory, our results yield oscillation criteria. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Partial difference equation, frequency measure, unsaturated solution, oscillation criteria MSC (2000): 39A11

1

Introduction

Let N = {0, 1, 2, ...} and Z+ = {1, 2, 3, ...} . In this paper, we are concerned with partial difference equations of the form Δ1 u(i−1, j)+Δ2 u(i, j −1)+P1 (i, j)u(i−1, j)+P2 (i, j)u(i, j −1)+P3 (i, j)u(i, j) = f (i, j) (1) 2 for (i, j) ∈ Z+ , where f (i, j), P1 (i, j), P2 (i, j) and P3 (i, j) = −2 are all real functions defined on Z+2 , Δ1 u(i, j) = u(i + 1, j) − u(i, j) and Δ2 u(i, j) = u(i, j + 1) − u(i, j). Such an equation may arise from numerical simulations of solutions of partial difference ∗

E-mail: [email protected]

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equations [1, 2]. For example, consider an initial-boundary problem of the form aux + ut = 0, x > 0, t > 0, u(x, 0) = g(x), x > 0. By means of the finite difference method for calculating an approximate solution u(ih, jτ ) = ui,j of this problem, we are led to the following difference equation h a(ui,j − ui−1,j ) + (ui,j − ui,j−1 ) = 0, i, j = 1, 2, 3, ..., τ which can be regarded as a special case of (1). Equation (1) may also arise when stationary heat distribution is sought in the dynamic heat diffusion of a plane lattice molecules. More specifically, consider the temperature distribution of a large set of molecules in the plane. Assume that the molecules are bonded in a regular manner such that each molecule can be labeled by (i, j) where i, j ∈ Z. Let (t) the temperature of each molecule in the time period t ∈ N be denoted by uij . In the time (t) (t) period t, if the temperature uij of the (i, j) molecule is lower than the temperature ui+1,j of the neighboring molecule (i+1, j), then heat will flow from the (i+1, j) molecule to the (i, j) molecule. The amount of increase of temperature in molecule (i, j) from time t to (t+1) (t) time t + 1 is ui,j − uij , and it is reasonable to postulate that the increase is proportional (t) (t) to the difference ui−1,j − uij , that is,   (t+1) (t) (t) (t) uij − uij = α ui−1,j − uij , where α is a proportionality constant. Similarly, heat may flow from the other three neighbors. Thus, it is reasonable that the total effect under the superposition principle is         (t+1) (t) (t) (t) (t) (t) (t) (t) (t) (t) uij − uij = α ui−1,j − uij + β ui,j−1 − uij + γ ui+1,j − uij + δ ui,j+1 − uij . (2) If we assume further that each molecule can also allow a control mechanism (such as heat storing or releasing mechanism) or a perturbation, then a more general evolutionary equation may result:     (t+1) (t) (t) (t) (t) (t) uij − uij = α ui−1,j − uij + β ui,j−1 − uij     (t) (t) (t) (t) + γ ui+1,j − uij + δ ui,j+1 − uij + Q(i, j). A special solution may arise that is independent of time. Such a solution u(i, j) then satisfies 0 = α (u(i − 1, j) − u(i, j)) + β (u(i, j − 1) − u(i, j)) +γ (u(i + 1, j) − u(i, j)) + δ (u(i, j + 1, j) − u(i, j)) + Q(i, j). In particular, when γ = 0 = δ, we end up with the equation αΔ1 u(i − 1, j) + βΔu2 (i, j − 1) + Q(i, j) = 0,

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which is of the form (1). A solution of (1) is a double sequence {u(i, j)} defined for (i, j) ∈ N 2 \{(0, 0)} which renders (1) into an identity for each (i, j) ∈ Z+2 by substitution. In case P3 (i, j) = −2 for (i, j) ∈ Z+2 , the existence of solutions of (1) is easily established. Indeed, since (1) can be rewritten as (2 + P3 (i, j))u(i, j) + (P1 (i, j) − 1)u(i − 1, j) + (P2 (i, j) − 1)u(i, j − 1) = f (i, j), if boundary distributions of the form u(0, j) = ϕj , j ∈ Z+ ,

(3)

u(i, 0) = φi , i ∈ Z+

(4)

and are given, we can calculate from (1) u(1, 1); u(1, 2), u(2, 1), u(2, 2); u(1, 3), u(2, 3), u(3, 1), u(3, 2), u(3, 3); . . . successively in an unique manner. We remark that since we will be concerned with asymptotic properties of solutions of (1), we may, for the sake of convenience, extend the domain of definition of a solution of (1) by including the point (0, 0). Henceforth, a solution of (1) is a double sequence u = {u(i, j)} defined for (i, j) ∈ N 2 which renders (1) into an identity for each (i, j) ∈ Z+2 by substitution. As usual, a solution of (1) or, in general, a double sequence {u(i, j)} defined on N 2 is said to be eventually positive if there is a positive integer M such that u(i, j) > 0 for all i ≥ M and j ≥ M . Eventually negative, eventually nonpositive and eventually nonnegative solutions are similarly defined. {u(i, j)} is said to be oscillatory if for any subset of N 2 of the form {(i, j) ∈ N 2 : i, j ≥ T } , there are (α, β), (s, t) in it such that u(α, β)u(s, t) ≤ 0. In this note, we will be concerned with solutions of (1) which have “unsaturated upper positive parts”. A solution with “unsaturated upper positive part” cannot be eventually positive nor eventually nonpositive. Thus by finding unsaturated solutions, we are able to find oscillatory solutions as well.

2

Main results

To define unsaturated solutions, we need to recall some terminologies in [3]. Let S be a countable set. The notation |S| denotes the number of elements of S. The union and intersection of two sets S1 and S2 are denoted by S1 + S2 and S1 · S2 respectively. A lattice point z = (i, j) is a point in the plane with integer coordinates. A neighbor of lattice point (i, j) means the lattice point (i − 1, j), (i + 1, j), (i, j + 1) or (i, j − 1). Let Ω be a subset of N 2 . The lattice points z1 , z2 , ..., zn is said to form a path with terminals z1 and zn if z1 is a neighbor of z2 , z2 is a neighbor of z3 , etc. The subset Ω of N 2 is said to be connected if any two of its points are terminals of a path in Ω. A nonempty

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connected subset Ω of N 2 is called a domain. Let Ω be a domain of N 2 and z ∈ N 2 , z is said to be an exterior boundary point of Ω if it does not belong to Ω but has at least one neighbor in Ω. The set of all exterior boundary points of Ω will be denoted by ∂Ω. For any z = (i, j) ∈ N 2 , let zL = (i − 1, j), zR = (i + 1, j), zT = (i, j + 1) and zD = (i, j − 1). Further, we let ∂L Ω = {z ∈ ∂Ω : zR ∈ Ω}, ∂R Ω = {z ∈ ∂Ω : zL ∈ Ω}, ∂T Ω = {z ∈ ∂Ω : zD ∈ Ω} and ∂D Ω = {z ∈ ∂Ω : zT ∈ Ω}. For k ∈ N, let   Wk = (i, j) ∈ N 2 : max {i, j} = k . For any n ∈ N and any subset Ω of N 2 , let Ω(n) = {(i, j) ∈ Ω ∩ Wk : k ≤ n}, X n Ω = {(i + n, j) : (i, j) ∈ Ω} and Y n Ω = {(i, j + n) : (i, j) ∈ Ω}. Let α, β, γ, δ ∈ N with α ≤ β   and γ ≤ δ, the union of δn=γ βm=α X m Y n Ω will be denoted by Xαβ Yγδ Ω. We can easily see that for any lattice point (i, j) ∈ N 2 , it holds that (i, j) ∈ N 2 − Xαβ Yγδ Ω ⇔ (i − s, j − t) ∈ N 2 − Ω

(5)

for α ≤ s ≤ β and γ ≤ t ≤ δ, where (i − s, j − t) is null when i < α or j < γ. The frequency measure on lattice planes has been defined in [3]. A variant is defined below. Definition 2.1. Let Ω be a subset of N 2 . If lim sup n→∞

|Ω(n) | (n + 1)2

exists, then this limit, denoted by μ∗ (Ω), will be called the upper frequency measure of Ω. Similarly, if |Ω(n) | lim inf n→∞ (n + 1)2 exists, this limit, denoted by μ∗ (Ω), will be called the lower frequency measure of Ω. If μ∗ (Ω) = μ∗ (Ω), then the common limit denoted by μ(Ω) will be called the frequency measure of Ω. For any double sequence x = {x(i, j)}(i,j)∈S and any real numbers c1 and c2 , let (x > c1 ) = {(i, j) ∈ S : x(i, j) > c1 }. The set (x ≥ c1 ), (x < c1 ) etc. have similar meanings. Definition 2.2. Let u = {u(i, j)}i,j∈N 2 be any double sequence. If μ∗ (u > 0) = ω ∈ (0, 1), then u is said to have unsaturated upper positive part. If μ∗ (u > 0) = ω ∈ (0, 1), then u is said to have unsaturated lower positive part. The double sequence u is said to have unsaturated positive part if μ∗ (u > 0) = μ∗ (u > 0) = ω ∈ (0, 1). A double sequence u with unsaturated negative part can be defined accordingly. It is easy to see that if u = {u(i, j)} is eventually positive or eventually nonpositive, then μ∗ (u > 0) = 1 or μ∗ (u > 0) = 0 respectively. Hence, a sequence u = {u(i, j)} with

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unsaturated positive part cannot be eventually positive nor eventually nonpositive. In other words, u is oscillatory. The following properties of frequency measures hold. They are similar to those in [3]. Lemma 2.3. Let Ω and Γ be subsets of N 2 . The following hold: (i) if Ω ⊆ Γ, then μ∗ (Ω) ≤ μ∗ (Γ) and μ∗ (Ω) ≤ μ∗ (Γ); (ii) μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ) ≤ μ∗ (Ω + Γ) ≤ μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ); (iii) μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ) ≤ μ∗ (Ω + Γ) ≤ μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ); (iv) μ∗ (Ω) + μ∗ (N 2 − Ω) = 1; (v) if Ω · Γ is finite, then μ∗ (Ω) + μ∗ (Γ) ≤ 1. Lemma 2.4. Let Ω be a subset of N 2 and α, β, γ and δ be integers such that α ≤ β and γ ≤ δ. Then μ∗ (Xαβ Yγδ Ω) ≤ (β − α + 1)(δ − γ + 1)μ∗ (Ω) and μ∗ (Xαβ Yλτ Ω) ≤ (β − α + 1)(δ − γ + 1)μ∗ (Ω). From Lemma 2.3, we can easily obtain the following result. Lemma 2.5. If Ω1 , ..., Ωn be subsets of N 2 , then  n  n n  

Ωi ≤ μ∗ (Ωi ) − (n − 1)μ∗ Ωi μ∗ i=1

and μ∗

 n 

i=1

Ωi

≤ μ∗ (Ω1 ) +

i=1

(6)

i=1 n 

μ∗ (Ωi ) − (n − 1)μ∗

 n

i=2

Ωi

.

(7)

i=1

Next, we will make use of the frequency measures defined above to establish some criteria for solutions of (1) to have unsaturated positive parts. Before doing so, we quote a result in [3]. Lemma 2.6. Let Ω be a finite domain of N 2 and suppose {u(i, j)} is a bivariate sequence such that Δ1 u(i − 1, j) and Δ2 u(i, j − 1) are defined on Ω. Then    Δ1 u(i − 1, j) = u(i − 1, j) − u(i, j) as well as

(i,j)∈Ω

(i,j)∈∂R Ω

(i,j)∈∂L Ω







Δ2 u(i, j − 1) =

(i,j)∈Ω

(i,j)∈∂T Ω

u(i, j − 1) −

u(i, j).

(i,j)∈∂D Ω

2.1 Nonhomogenous case This section is concerned with (1) under the condition that the forcing term is not identically zero.We will denote the set {a, a + 1, · · · , b} of integers by Z[a, b] and the Cartesian

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product of Z[a, b] and Z[c, d] by Z[a, b] × Z[c, d]. Theorem 2.7. Suppose that   P3 P3 ∗ ∗ = ω1 , μ P2 > 1 + = ω2 , μ∗ (P3 > −2) = ω3 μ P1 > 1 + 2 2 + ∗ ∗ μ (f > 0) = ωf , μ (f < 0) = ωf− . Suppose further that there exists a constant ω0 ∈ (0, 1) such that 9(ω1 + ω2 + ω3 + ωf + ω0 ) < 1,

(8)

where ωf = max{ωf− , ωf+ }. Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part. Proof. We assert that μ∗ (u > 0) ∈ (ω0 , 1). Otherwise, there are two cases, that is μ∗ (u > 0) ≤ ω0 or μ∗ (u > 0) = 1. In case μ∗ (u > 0) ≤ ω0 , in view of Lemma 2.3(ii),(iv), Lemma 2.4 and (8), we have

  P3 P3 2 1 1 μ∗ N − X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f > 0) 2 2  2  ∗ 1 1 +μ N − X−1 Y−1 (u > 0)

  P3 P3 ∗ 1 1 = 2 − μ X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f > 0) 2 2  1 1  −μ∗ X−1 Y−1 (u > 0)   P3 P3 ∗ ∗ ∗ ∗ ≥ 2 − 9 μ (P1 > 1 + ) + μ (P2 > 1 + ) + μ (P3 > −2) + μ (f > 0) + μ∗ (u > 0) 2 2 > 1. Then, by Lemma 2.3(v), the intersection

  P3 P3 2 1 1 N − X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f > 0) ∩ 2 2  2  1 1 N − X−1 Y−1 (u > 0) is infinite. This, together with (5), implies that there exists a lattice point (ac , bc ) ∈ N 2 with (ac − 1, bc − 1) ∈ N 2 such that when (i, j) ∈ Z[ac − 1, ac + 1] × Z[bc − 1, bc + 1], P3 (i, j) ≤ −2, P3 (i, j) P3 (i, j) , P2 (i, j) ≤ 1 + , f (i, j) ≤ 0. 2 2 From (1) and (10), we obtain for (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1] that

(9)

u(i, j) ≤ 0, P1 (i, j) ≤ 1 +

(10)

Δ1 u(i − 1, j) + Δ2 u(i, j − 1) ≤ 0, (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1].

(11)

Let us set Ω = {(ac , bc )} and view P1 (ac , bc ) as P1 , P2 (ac , bc ) as P2 and P3 (ac , bc ) as P3 . According to (11), we need to deal with three cases.

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(i) Δ1 u(ac − 1, bc ) ≤ 0 and Δ2 u(ac , bc − 1) ≤ 0. Then, by Lemma 2.6 and (10), we have from (1) that     u(i − 1, j) − u(i, j) + u(i, j − 1) − u(i, j) 0≥ (i,j)∈∂R Ω

P1



(i,j)∈∂L Ω

u(i − 1, j) + P2

(i,j)∈Ω

u(i, j) −

(i,j)∈∂L Ω

+P1





(i,j)∈∂L Ω

u(i, j) + 

u(i, j) + (2 + P3 )

(i,j)∈∂D Ω



≥−

u(i, j)

(i,j)∈Ω

(i,j)∈∂D Ω

u(i, j) + P2

(i,j)∈∂D Ω



u(i, j − 1) + P3

(i,j)∈Ω



=−

(i,j)∈∂T Ω





u(i, j) + P1

(i,j)∈∂L Ω+∂D Ω

(i,j)∈∂L Ω

(i,j)∈∂L Ω+∂D Ω

(i,j)∈∂L Ω



u(i, j)

(i,j)∈Ω



u(i, j) + P2

u(i, j)

(i,j)∈∂D Ω

P3  P3  +(1 + ) u(i − 1, j) + (1 + ) u(i, j − 1) 2 2 (i,j)∈Ω (i,j)∈Ω   u(i, j) + P1 u(i, j) =− +P2



u(i, j) + (1 +

(i,j)∈∂D Ω

≥−



P3 ) 2

u(i, j) + 2(1 +

(i,j)∈∂L Ω+∂D Ω

= (1 + P3 )





u(i, j)

(i,j)∈∂L Ω+∂D Ω

P3 ) 2



u(i, j)

(i,j)∈∂L Ω+∂D Ω

u(i, j),

(i,j)∈∂L Ω+∂D Ω

this means that P3 ≥ −1, which is contrary to (9). (ii) Δ1 u(ac − 1, bc ) ≥ 0 and Δ2 u(ac , bc − 1) ≤ 0. Note from (9)-(10) that (1) implies 0 ≥ Δ2 u(i, j − 1) + P3 (i, j)u(i, j), (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1], so that by Lemma 2.6 , we have  u(i, j − 1) − 0≥ (i,j)∈∂T Ω



u(i, j) + P3

(i,j)∈∂D Ω



u(i, j)

(i,j)∈Ω

= −u(ac , bc − 1) + (1 + P3 )u(ac , bc ) ≥ −u(ac , bc − 1) + (1 + P3 )u(ac , bc − 1) = P3 u(ac , bc − 1). Now we see that P3 ≥ 0 which is contrary to (9). (iii) Δ1 u(ac − 1, bc ) ≤ 0 and Δ2 u(ac , bc − 1) ≥ 0. Since (1) implies 0 ≥ Δ1 u(i − 1, j) + P3 (i, j)u(i, j), (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1], by similar arguments as above, we are led to the same contradiction as P3 ≥ 0. In case μ∗ (u > 0) = 1, then in view of Lemma 2.3(iv), μ∗ (u ≤ 0) = 0. Note from (8) that 9(ω1 + ω2 + ω3 + ωf ) < 1.

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663

By the same reasons as above, we see that  

P3 P3 2 1 1 N − X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f < 0) ∩ 2 2  2  1 1 N − X−1 Y−1 (u ≤ 0) is infinite. Hence, there exists a lattice point (ac , bc ) ∈ N 2 with (ac − 1, bc − 1) ∈ N 2 such that u(i, j) > 0, P1 (i, j) ≤ 1 +

P3 (i, j) P3 (i, j) , P2 (i, j) ≤ 1 + , f (i, j) ≥ 0, P3 (i, j) ≤ −2 2 2

for (i, j) ∈ Z[ac − 1, ac + 1] × Z[bc − 1, bc + 1]. By similar arguments as above, we may  show that μ∗ (u > 0) = 1 does not hold. The proof is complete. We remark that the conditions P1 (i, j) ≤ 1 + P3 (i, j)/2 and P2 (i, j) ≤ 1 + P3 (i, j)/2 in (10) can be replaced by P1 (i, j) ≤ 0 and P2 (i, j) ≤ 0. We remark further that the proof above requires the inequality μ∗ (Ω + Γ) ≤ μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ) and the right side has been magnified. Theorem 2.8. Suppose that μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (P3 > −2) = ω3 , μ∗ (f > 0) = ωf+ , μ∗ (f < 0) = ωf− , μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] = ω + and μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f < 0)] = ω − . Suppose further that there exists a constant ω0 ∈ (0, 1) such that 9(ω1 + ω2 + ω3 + ωf + ω0 − 3ω) < 1,

(12)

where ωf = max{ωf− , ωf+ } and ω = min{ω − , ω + }. Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part. Proof. In case μ∗ (u > 0) ≤ ω0 . In view of Lemma 2.3(iv), Lemma 2.4, (6) in Lemma 2.5 and (12), we have   1 1 μ∗ N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0)]   1 1 Y−1 (u > 0) +μ∗ N 2 − X−1  1 1  Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0)] = 2 − μ∗ X−1  1 1  Y−1 (u > 0) −μ∗ X−1 ≥ 2 − 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (P3 > −2) + (f > 0)

− 3μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] + μ∗ (u > 0)} > 1.

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Then, by Lemma 2.3(v), the intersection 

   1 1 1 1 N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0)] ∩ N 2 − X−1 Y−1 (u > 0)

is infinite. Similar to those in Theorem 2.7, we will arrive at a contradiction. Consequently, μ∗ (u > 0) ≤ ω0 does not hold. Analogously, we can show that μ∗ (u > 0) = 1. That is, μ∗ (u > 0) ∈ (ω0 , 1). The proof is complete.  Theorem 2.9. Suppose that μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (f > 0) = ωf+ , μ∗ (f < 0) = ωf− and μ∗ [(P1 > 0) · (P2 > 0) · (f > 0)] = ω + , μ∗ [(P1 > 0) · (P2 > 0) · (f < 0)] = ω − . Suppose further that there exist a constant ω0 ∈ (0, 1) such that μ∗ (P3 ≤ −2) > 9(ω1 + ω2 + ωf + ω0 − 2ω),

(13)

where ωf = max{ωf− , ωf+ } and ω = min{ω − , ω + }. Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part. Proof. Similar to the proofs to Theorem 2.7-2.8, we need only to prove that μ∗ (u > 0) ≤ ω0 and μ∗ (u > 0) = 1 do not hold. Indeed, by Lemma 2.3(iv), we have   1 1 1 = μ∗ N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)]  1 1  Y−1 [(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] +μ∗ X−1   1 1 Y−1 [(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] + ≤ μ∗ N 2 − X−1 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (f > 0) + μ∗ (u > 0)

−2μ∗ [(P1 > 0) · (P2 > 0) · (f > 0)]}   1 1 < μ∗ N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] + μ∗ (P3 ≤ −2) when μ∗ (u > 0) ≤ ω0 , and   1 1 1 = μ∗ N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)]  1 1  Y−1 [(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)] +μ∗ X−1   1 1 Y−1 [(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)] + μ∗ (P3 ≤ −2) < μ∗ N 2 − X−1 when μ∗ (u > 0) = 1 (which implies that μ∗ (u ≤ 0) = 0). Hence, the intersections  and



 1 1 N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] ∩ (P3 ≤ −2)

 1 1 N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)] ∩ (P3 ≤ −2)

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665

are infinite. The rest of the proof is analogous to those as above and we will reach the fact that μ∗ (u > 0) ∈ (ω0 , 1). The proof is complete.  Note that   1 1 μ∗ N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)]  1 1  Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)] = 1 − μ∗ X−1

≥ 1 − 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (P3 > 0) + μ∗ (f > 0) + μ∗ (u > 0)} +27μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] . Thus, when 27μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] > 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (P3 > 0)} − 1, it follows that

  1 1 μ∗ N 2 − X−1 Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)] > 0. 1 1 By Definition 2.1, the set N 2 −X−1 Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)] is infinite and we have

P1 (i, j) ≤ 0, P2 (i, j) ≤ 0, P3 (i, j) ≤ −2, f (i, j) ≤ 0, u(i, j) ≤ 0 for some domain. Repeating the proofs as above, we may then obtain the following result. Theorem 2.10. Suppose that μ∗ (P1 > 0) = ω1 , μ∗ (P3 > −2) = ω3 ,

μ∗ (P2 > 0) = ω2 ,

μ∗ (f > 0) = ωf+ ,

μ∗ (f < 0) = ωf−

and μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] = ω + , μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f < 0)] = ω − . Suppose further that there exist a constant ω0 ∈ (0, 1) such that ω>

ω1 + ω2 + ω3 + ωf + ω0 1 − , 3 27

(14)

where ωf = max{ωf− , ωf+ } and ω = min{ω − , ω + }. Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part.

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2.2 Homogeneous case In this section,we are concerned with (1) under condition that the forcing term is identically zero.The proofs are the same as Theorems 2.8-2.10 and hence omitted. Theorem 2.11. Suppose that μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (P3 > −2) = ω3 and μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2)] = ω. Suppose further that there exists a constant ω0 ∈ (0, 1) such that 9(ω1 + ω2 + ω3 + ω0 − 2ω) < 1.

(15)

Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part. Theorem 2.12. Suppose that μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 and μ∗ [(P1 > 0) · (P2 > 0)] = ω. Suppose further that there exist a constant ω0 ∈ (0, 1) such that μ∗ (P3 ≤ −2) > 9(ω1 + ω2 + ω0 − ω).

(16)

Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part. Theorem 2.13. Suppose that μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (P3 > −2) = ω3 . Suppose further that there exist a constant ω0 ∈ (0, 1) such that μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2)] >

ω1 + ω2 + ω3 + ω0 1 − . 2 18

(17)

Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part.

3

Examples

We consider two examples. Example 3.1. Consider the partial difference equation with forcing term Δ1 u(i − 1, j) + Δ2 u(i, j − 1) − u(i − 1, j) − u(i, j − 1) − 2u(i, j) = f (i, j),

(18)

Z.-Q. Zhu, S.S. Cheng / Central European Journal of Mathematics 4(4) 2006 656–668

667

where P1 (i, j) = P2 (i, j) = −1, P3 (i, j) = −3 and ⎧ ⎪ ⎨ 1 i = 30m or j = 30n, m, n ∈ N f (i, j) = . ⎪ ⎩ 0 otherwise Then μ∗ (P1 > 0) = μ∗ (P2 > 0) = μ∗ (P3 > −2) = 0, μ∗ (P3 ≤ −2) = 1, μ∗ (f > 0) = and μ∗ (f < 0) = 0. Moreover, μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2)] = 0,

1 30

(19)

μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] = 0 as well as μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f < 0)] = 0. 1 , we see that (8) and (12)-(14) hold. By Theorems 2.7-2.10, every solution 18 u = {u(i, j)} of (18) has unsaturated upper positive part and hence it is oscillatory. Given ω0 =

Example 3.2. Consider the partial difference equation Δ1 u(i − 1, j) + Δ2 u(i, j − 1) − u(i − 1, j) − u(i, j − 1) − 6u(i, j) = 0,

(20)

where P1 (i, j) = P2 (i, j) = −1 and P3 (i, j) = −6. Similar to Example 3.1, we have 1 μ∗ [(P1 > 0) · (P2 > 0)] = 0 and (19) holds. On the other hand, when given ω0 = , 10 (15)-(17) are satisfied. By Theorems 2.11-2.13, every solution u = {u(i, j)} of (20) has unsaturated upper positive part and hence is oscillatory. In fact, u = {(−1)i+j+1 } is such a solution of (20).

4

Remarks

We obtained conditions for all solutions of our equations to be unsaturated. This is the first time an unsaturated solution is introduced, and hence there are no previous results for comparison. However, we have mentioned the fact that unsaturated solutions are also oscillatory. Hence our previous results are oscillation theorems. Usually, oscillation theorems are proved by showing neither eventually positive nor eventually negative solutions can exist. Hence these theorems do not provide any clues to how frequent solutions oscillate! Although frequently oscillatory solutions are also introduced in studying the oscillatory behavior of partial difference equations (see e.g. [4, 5]), these solutions can be different from unsaturated solutions. To see this, let u = {u(i, j)}(i,j)∈N 2 be defined by ⎧ ⎪ ⎪1 i < j ⎪ ⎪ ⎨ u(i, j) = 0 i = j , ⎪ ⎪ ⎪ ⎪ ⎩ −1 i > j

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Z.-Q. Zhu, S.S. Cheng / Central European Journal of Mathematics 4(4) 2006 656–668

then μ(u ≤ 0) = μ(u ≥ 0) = 1/2 and hence u is not frequently oscillatory of upper degree 1/2. Yet, u has unsaturated positive part because μ(u > 0) = 1/2! As our final remark, we mention that similar ideas can be used to establish criteria for solutions of (1) with unsaturated upper negative parts.

References [1] Y.Z. Lin and S.S. Cheng: “Stability criteria for two partial difference equations”, Comput. Math. App., Vol. 32(7), (1996), pp. 87–103. [2] Y.Z. Lin and S.S. Cheng: “Bounds for solutions of a three-point partial difference equation”, Acta Math. Sci., Vol. 18(1), (1998), pp. 107–112. [3] S.S. Cheng: Partial Difference Equations, Taylor and Francis, 2003. [4] C.J. Tian and B.Q. Zhang: “Frequent oscillation of a class of partial difference equations”, J. Anal. Appl., Vol. 18(1), (1999), pp. 111–130. [5] S.L. Xie and C.J. Tian: “Frequent oscillatory criteria for partial difference equations with several delays”, Comput. Math. Appl., Vol. 48, (2004), pp. 335–345.