Type Lipschitz selections in generalized 2-normed spaces

Analysis in Theory and Applications Volume 24 Number 3 (2008), 205–210 DOI10.1007/s10496-008-0205-2

1-TYPE LIPSCHITZ SELECTIONS IN GENERALIZED 2-NORMED SPACES Sh. Rezapour (Azarbaidjan University of Tarbiat Moallem, Iran) I. Kupka (Physics and Informatics of Comenius University, Slovakia) Received Sept. 12, 2007

Abstract. We shall introduce 1-type Lipschitz multifunctions from R into generalized 2-normed spaces, and give some results about their 1-type Lipschitz selections. Key words: multifunction, 2-normed space, Lipschitz selection AMS (2000) subject classification: 46B25, 41A65

1

Introduction

There are many papers about selection theory, Lipschitz multifunctions and applications (for example [2], [3], [4], [6], [7]). The concept of linear 2-normed spaces was introduced by Gahler ¨ [1] in 1965 . Later, Lewandowska extended the concept to generalized 2-normed spaces in 1999[5] . Let X be a linear space of dimension greater than 1 over K, where K is the real or complex numbers field. Suppose ., . is a non-negative real-valued function on X × X satisfying the following conditions: (i) x, y = 0 if and only if x and y are linearly dependent vectors. (ii) x, y = y, x for all x, y ∈ X . (iii) λ x, y = |λ | x, y for all λ ∈ K and all x, y ∈ X . (iv) x + y, z ≤ x, z + y, z for all x, y, z ∈ X . Then ., . is called a 2-norm on X and (X , ., .) is called a linear 2-normed space. Every 2-normed space is a locally convex topological vector space. In fact, for a fixed b ∈ X , pb (x) = x, b, x ∈ X , is a seminorm, and the family P = {pb : b ∈ X } of seminorms generates a locally convex topology on X .

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Defintion 1.1[5] . Let X and Y be linear spaces, D a non-empty subset of X ×Y such that for every x ∈ X , y ∈ Y the sets Dx = {y ∈ Y : (x, y) ∈ D},

Dy = {x ∈ X : (x, y) ∈ D}

are linear subspaces of the spaces Y and X , respectively. A function ., . : D −→ [0, ∞) is called a generalized 2-norm on D if it satisfies the following conditions: (N1 ) x, α y = |α |x, y = α x, y, for all (x, y) ∈ D and each scalar α . (N2 ) x, y + z ≤ x, y + x, z, for all (x, y), (x, z) ∈ D. (N3 ) x + y, z ≤ x, z + y, z, for all (x, z), (y, z) ∈ D. Then, (D, ., .) is called a 2-normed set. In particular, if D = X ×Y , (X ×Y, ., .) is called a generalized 2-normed space. Moreover, if X = Y , then the generalized 2-normed space is denoted by (X , ., .). For example, let A be a Banach algebra and a, b = ab for all a, b ∈ A. Then, (A, ., .) is a generalized 2-normed space. Also, for each two complex normed spaces (X , .1 ) and (Y, .2 ), (X ×Y, ., .) is a complex generalized 2-normed space, where x, y = x1 y2 .

Definition 1.2[5] . Let (X ×Y, ., .) be a generalized 2-normed space. Then  (a) The family β of all sets defined by ni=1 {x ∈ X : x, yi  < ε }, where n ∈ N, y1 , ..., yn ∈ Y and ε > 0, forms a complete system of neighborhoods of zero for a locally convex topology in X. 

(b) The family β of all sets defined by ni=1 {y ∈ Y : xi , y < ε }, where n ∈ N, x1 , ..., xn ∈ X and ε > 0, forms a complete system of neighborhoods of zero for a locally convex topology in Y. The above topologies are denoted by the symbols τ (X ,Y ) and τ (Y, X ), respectively.

2 (K, y)-Lipschitz Selections Definition 2.1. Let (X × Y, ., .) be a generalized 2-normed space, y ∈ Y , E a subset of R and F : E → 2X a multifunction. We say that F is (K, y)-Lipschitz selection whenever Hdy (F(t1 ), F(t2 )) ≤ K|t1 − t2 |, for all t1 ,t2 ∈ E, where Hdy (F(t1 ), F(t2 )) is defined by max{ sup

inf a − b, y, sup

a∈F(t1 ) b∈F(t2 )

inf c − d, y}.

c∈F (t2 ) d∈F(t1 )

Lemma 2.1. Let (X × Y, ., .) be a generalized 2-normed space, h > 0, y ∈ Y , a ∈ R and F : [a, a + h] → 2X a (K, y)-Lipschitz multifunction. Let 0 < r < K and b ∈ F(a). Then, there r exists a so called (K + , y)-Lipshitz function f : [a, a+h] → X such that f (a) = b, f ([a, a+h]) ⊆ 2 3 {x ∈ X : x − b, y ≤ Kh} and dy ( f (t), F (t)) = inf  f (t) − x, y < r, for all t ∈ [a, a + h]. 2 x∈F(t)

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r 1 h Proof. Let x0 = b and choose a natural number n such that K < min{ , }. Define n 6 3 h ti = a + i, for i = 0, 1, 2, · · · , n. There exists an element x1 ∈ F(t1 ) such that n x1 − x0 , y ≤ Hdy (F(t1 ), F(t0 )) +

r rh h ≤ (K + ). 2n n 2

Thus, we can find a set {x0 , x1 , · · · , xn } such that xi ∈ F(ti ) and h r xi+1 − xi , y ≤ (K + ), n 2

for i = 0, 1, 2, · · · , n − 1.

Now, we define the function f : [a, a + h] → X by n n f (t) = (t − ti )xi+1 + (ti+1 − t)xi , forall t ∈ [ti ,ti+1 ], i = 0, 1, 2, · · · , n − 1. h h r We show that f is a (k + , y)-Lipschitz function. Firstly, suppose that t,t  ∈ [ti ,ti+1 ] for some 2 i ∈ {0, 1, · · · , n − 1}. Then, n r  f (t) − f (t  ), y = |t − t  |xi+1 − xi , y ≤ (K + )|t − t  |. h 2

(1)

Now, suppose that ti ≤ t < ti+1 < · · · < tk−1 < t  ≤ tk for some i, k ∈ {0, 1, · · · , n}. Then, by (1)  f (t) − f (t  ), y ≤  f (t) − f (ti+1 ), y + · · · +  f (t  ) − f (tk−1 ), y r r r ≤ (K + )|t − ti+1 | + · · · + (K + )|t  − tk−1 | = (K + )|t  − t|. 2 2 2 Let t ∈ [a, a + h]. Then, t ∈ [ti ,ti+1 ] for some i ∈ {0, 1, 2, · · · , n − 1}. Choose x ∈ F(t) such that xi+1 − x , y < (K + 2r )(ti+1 − t). Thus,  f (t) − x , y ≤  f (t) − xi , y + xi − xi+1 , y + xi+1 − x , y h r r r < (K + )(t − ti ) + (K + ) + (K + )(ti+1 − t) 2 n 2 2 h r h r r h r = (K + )(ti+1 − ti ) + (K + ) ≤ 2 + (K + ) ≤ 2 + r < r. 2 n 2 n 2 6 n Hence, dy ( f (t), F(t)) < r, for all t ∈ [a, a + h]. Finally, note that 3 r  f (t) − b, y =  f (t) − f (a), y ≤ (K + )|t − a| ≤ Kh, 2 2 for all t ∈ [a, a + h].

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Proposition 2.2. Let (X × Y, ., .) be a generalized 2-normed space, h > 0, y ∈ Y , a ∈ R, F : [a, a + h] → 2X a (K, y)-Lipschitz multifunction with closed values, b ∈ F(a) and each ybounded sequence in X has a y-convergent subsequence. Then, F has a (K, y)-Lipshitz selection on [a, a + h]. 1 Proof. Choose a natural number m such that < K. By Lemma 2.1, we obtain a sequence m 1 , y)-Lipshitz function on [a, a + h], fi ([a, a + h]) ⊆ {x ∈ X : { fi }∞ i=m so that each fi is a (K + 2i 1 3 x − b, y ≤ Kh} and dy ( fi (t), F (t)) < , for all t ∈ [a, a + h]. Let ε > 0 be given and choose 2 2i 0<δ < Then

ε . 3(K + 1)

ε  fi (t) − fi (t  ), y < (K + 1)|t − t  | < , 3

for all t,t  ∈ [a, a + h] with |t − t  | < δ and all i ≥ m. We can take some points t1 , · · · ,ts ∈ E =   [a, a + h] Q such that [a, a + h] = si=1 {t ∈ [a, a + h] : |t − ti | < δ }. Since E is countable and { fi }∞ i=m is y-bounded sequence in X and every y-bounded sequence in X has a y-convergent ∞ subsequence, there is a subsequence { fi j }∞j=m of { fi }∞ i=m such that { fi j (t)} j=m is y-convergent ε for each t ∈ E. Choose a natural number N such that  fi j (tl )− fi j (tl ), y < , whenever j, j ≥ N 3 and 1 ≤ l ≤ s. If t ∈ [a, a + h], then |t − tl | < δ , for some 1 ≤ l ≤ s. Hence,  fi j (t) − fi j (t), y ≤  fi j (t) − fi j (tl ), y +  fi j (tl ) − fi j (tl ), y + fi j (tl ) − fi j (t), y < ε , for all j, j ≥ N. Thus, { fi j }∞j=m is y-uniformly convergent on [a, a + h]. Now, define f : [a, a + h] → X by f (t) = lim fi j (t). Then, f is (K + ε , y)-Lipshitz for each ε > 0, so f is a (K, y)j→∞

Lipshitz function. Finally, suppose that ε > 0 is given and choose a sufficiently large index j ε ε such that dy ( fi j (t), F(t)) < and supt∈[a,a+h]  fi j (t) − f (t), y < . Thus, dy ( f (t), F (t)) = 0 2 2 and so f is a (K, y)-Lipshitz selection of F.

Remark 2.1. By using similar proofs, we can show that Lemma 2.1 and Proposition 2.2 hold for [a − h, a]. Theorem 2.3. Let (X × Y, ., .) be a generalized 2-normed space, y ∈ Y , F : R → 2X a (K, y)-Lipschitz multifunction with closed values and each y-bounded sequence in X has a yconvergent subsequence. Then, F has a (K, y)-Lipshitz selection on R.

Proof. Let b ∈ F(0). By using Proposition 2.2, we can find a sequence { fn }∞ n=1 of (K, y)Lipshitz selections of F such that each f2n is defined on [2n, 2n + 2], each f2n+1 is defined on [−2n − 2, −2n], f2n (2n + 2) = f2n+2 (2n + 2), f2n+1 (−2n − 2) = f2(n+1)+1 (−2n − 2) and f1 (0) =

Analysis in Theory and Applications, Vol. 24, No.3 (2008) f2 (0) = b. Now, define f : R → 2X by ⎧ ⎨ f (t), 2n f (t) = ⎩ f 2n+1 (t),

209

t ∈ [2n, 2n + 2], t ∈ [−2n − 2, −2n].

Then, f is a (K, y)-Lipshitz selection of F on R.

3 1-type Lipschitz Selections Definition 3.1. (i) Let (X ×Y, ., .) be a generalized 2-normed space, E a subset of R and F : E → 2X a multifunction. We say that S ⊆ Y is a finite set for F whenever HdS (F(t1 ), F(t2 )) < ∞, for all t1 ,t2 ∈ E, where HdS (F(t1 ), F(t2 )) = sup Hdy (F(t1 ), F(t2 )). y∈S

(ii) Let (X × Y, ., .) be a generalized 2-normed space, E a subset of R and F : E → 2X a multifunction. We say that F is 1-type K-Lipschitz multifunction whenever HdS (F(t1 ), F(t2 )) ≤ K|t1 − t2 |, for all t1 ,t2 ∈ E and each finite set S ⊆ Y for F.

Example 3.2. Let (X , .1 ) and (Y, .2 ) be real normed spaces and x, y = x1 y2 , for all x ∈ X , y ∈ Y . Then, (X × Y, ., .) is a real generalized 2-normed space. Also, let E be a subset of R, F : E → 2X a multifunction and S a bounded subset of Y . Then, F is a 1-type K-Lipschitz multifunction on E. Lemma 3.1. Let (X ×Y, ., .) be a generalized 2-normed space, h > 0, a ∈ R and F : [a, a+ h] → 2X a 1-type K-Lipschitz multifunction. Let 0 < r < K and b ∈ F(a). Then, there exists a 3 1-type (K + 2r )-Lipshitz function f : [a, a+ h] → X such that f (a) = b, f ([a, a+ h]) ⊆ BS (b, Kh) 2 3 and dS ( f (t), F (t)) < r, for all t ∈ [a, a + h] and all finite set S for F, where BS (b, Kh) = {x ∈ 2 3 X : x − b, y ≤ Kh, for all y ∈ S} and dS ( f (t), F (t)) = sup dy ( f (t), F(t)). 2 y∈S Proof. Let S be a finite set for F and y ∈ S. A similar proof as Lemma 2.1 shows that there exists a (K + 4r , y)-Lipshitz function f : [a, a + h] → X such that f (a) = b, f ([a, a + h]) ⊆ r 5 {x ∈ X : x − b, y ≤ Kh} and dy ( f (t), F (t)) < , for all t ∈ [a, a + h]. Also, note that the 4 2 construction of the function f doesn’t depend to y. Thus, these relations hold for all y ∈ S. 3 r Hence, f is a 1-type (K + )-Lipshitz function on [a, a + h] such that f ([a, a + h]) ⊆ BS (b, Kh) 2 2 and dS ( f (t), F (t)) < r, for all t ∈ [a, a + h]. Finally by using Lemma 3.1, Proposition 3.2 and the similar proof as 2.2 and 2.3, we can provide the following results.

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Proposition 3.2. Let (X ×Y, ., .) be a generalized 2-normed space, h > 0, a ∈ R, F : [a, a+ h] → 2X a 1-type K-Lipschitz multifunction with closed values and each S-bounded sequence in X has a S-convergent subsequence for each finite set S for F. Then, F has a 1-type K-Lipshitz selection on [a, a + h]. A similar result holds on [a − h, a]. Theorem 3.3. Let (X × Y, ., .) be a generalized 2-normed space, F : [a, a + h] → 2X a 1-type K-Lipschitz multifunction with closed values and each S-bounded sequence in X has a S-convergent subsequence for each finite set S for F. Then, F has a 1-type K-Lipshitz selection on R.

References [1] Gahler, ¨ S., Linear 2-normierte Raume, ¨ Math. Nachr, 28 (1956), 1-45. [2] Gurican, ˇ J. and Kostyrko, P., On Lipshitz Selections of Lipshitz Multifunctions, Acta Mathematica Universitatis Comenianae, 66:67 (1985), 131-135. [3] Kupka, I., Continuous Selections for Lipshitz Multifunctions, Acta Mathematica Universitatis Comenianae, Vol. LXXIV, 1(2005), 133-141. [4] Kupka, I., Continuous Multifunctions from [-1,0] to rmark having no continuous selection, Publ. Math. Debrecen, 48: 3-4(1996), 367-370. [5] Lewandowska, Z., Linear Operators on Generalized 2-normed Spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 42-90:4(1999), 353-368. [6] Repovs, ˇ D. and Semenov, P., Selections of Convex Bodies and Splitting Problem for Selections, J. Math. Anal. Appl., 334 (2007), 646-655. [7] Wang, L., An Iteration Method for Nonexpansive Mappings in Hilbert Spaces, Fixed Point Theory and Applications, Vol. 2007, Article ID 28619, 8 pages, doi:10.1155/2007/28619.

Sh. Rezapour Department of Mathematics Azarbaidjan University of Tarbiat Moallem Azarshahr, Tabriz, Iran E-mail: [email protected] I. Kupka Faculty of Mathematics Physics and Informatics of Comenius University Mlynska´ dolina, 842 48 Bratislava Slovakia E-mail: [email protected]