KKM-Type Theorems for Best Proximal Points in Normed Linear Space

Transcription

1 International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, HIKARI Ltd, KKM-Type Theorems for Best Proximal Points in Normed Linear Space Hoonjoo Kim Department of Mathematics Education Sehan University 1113, Noksaek-ro, Samho-eup, Yeongam-gun Jeonnam, 58447, Korea Copyright c 2018 Hoonjoo Kim. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We generalize the R-KKM theorem in [4] to that for the intersectionally closed-valued maps and using this, we prove best proximity pair theorems for multimaps with (unionly) open fibers in normed linear spaces which generalize the results of [4]. Mathematics Subject Classification: 41A65; 46B20; 47H10; 47H04 Keywords: best proximal point, best proximity pair, R-KKM, unionly open, transfer open, intersectionally closed, transfer closed 1 Introduction and Preliminaries A multimap (or map) F : X Y is a function from a set X into the power set 2 Y of Y ; that is, a function with the values F (x) Y for x X. For A X, let F (A) = {F (x) : x A}. Let denote the closure of F. Let (M, d) be a metric space and A and B be nonempty subsets of M. Consider a multimap F : A M. A point x 0 A is called a best proximal point of F if d(x 0, F (x 0 )) = d(a, B). In this case, (x 0, F (x 0 )) is called a best proximity pair for F. Note that if d(a, B) = 0 and F is a single valued map, then the best proximal point is a fixed point of F.

2 604 Hoonjoo Kim The following notations are used in the sequel. A 0 = {x A : d(x, y) = d(a, B) for some y B}, B 0 = {y B : d(x, y) = d(a, B) for some x A}. The pair (A, B) is said to be a proximal pair if, for each (x, y) A B, there exists ( x, ỹ) A B such that d(x, ỹ) = d( x, y) = d(a, B). A pair (A, B) is a proximal pair if and only if A = A 0 and B = B 0. Sankar Raj and Somasundaram [4] introduced R-KKM maps as follows: Let (A, B) be a proximal pair of a normed linear space M and A be convex. The map F : B A is said to be an R-KKM map if for any {y 1,, y m } B, there exists {x 1,, x m } A with x j y j = d(a, B) for each j = 1,, m such that co{x 1,, x m } j=1,,m F (y j). They proved an extended version of the Fan-Browder multivalued fixed point theorem for the best proximal point setting; Theorem 1.1. Let (A, B) be a nonempty compact convex proximal pair in a normed linear space M. Let F : A B be a multimap such that (1) F (x) is convex for each x A; and (2) F 1 (y) is open for each y B. Then there is a w A such that d(w, F (w)) = d(a, B). Recently the author [1] showed that R-KKM maps are generalized KKM maps, applied R-KKM property to the map defined on product spaces, and proved new best proximal point theorems in normed linear spaces using the generalized KKM theorem. In this paper, we just generalize the R-KKM theorem of [4] to that for the intersectionally closed-valued maps and prove fixed points of best proximity pair theorem for maps with unionly open fibers in normed linear spaces in [1] with different method. We also prove fixed points of best proximity pair theorems for a family of maps with open fibers in normed linear spaces. These theorems generalize the results of [4]. 2 Main Results When A is a nonempty subset of a normed linear space M, let P A (y) = {x A : x y = d(y, A)}. The author [1] generalized R-KKM maps in [4] as follows: For I = {1,, n}, let (A, B i ) be a proximal pair of a normed linear space M, A be convex, and B := I B i. The map F : B A is said to be an R KKM map if for any {y 1 = (y 1 1,, y 1 n),, y m = (y m 1,, y m n )} B,

3 KKM-type theorems for best proximal points in normed linear space 605 there exists {x 1,, x m } A with x j i I P A(y j i ) for each j = 1,, m such that co{x 1,, x m } j=1,,m F (yj ). When Z is a set and X is a topological space, consider the following related three conditions for F : Z X; (a) z Z F (z) = z Z F (z) (F is intersectionally closed-valued). (b) z Z F (z) = z Z F (z) (F is transfer closed-valued). (c) F is closed-valued. Luc et al. [3] noted that (c) = (b) = (a). A multimap F : X Y is said to be unionly open-valued (resp., transfer open-valued) on X if and only if the multimap G : X Y, defined by G(x) = Y \F (x) for every x X, is intersectionally closed-valued (resp., transfer closed-valued) on X. See [3] and Tian [5]. Lemma 2.1. Let Z be a set, X be a topological space, Y be a closed subset of X and F : Z X be a transfer closed-valued map. And let S : Z Y be a map defined by S(z) = F (z) Y for z Z. Then S is transfer closed-valued. Proof. Since F (z) = F (z), S(z) = {F (z) Y } = { F (z)} Y = { F (z)} Y = {F (z) Y } S(z). The following KKM type theorem is Theorem 3.2 in [4]; Theorem 2.2. Let (A, B) be a nonempty proximal pair in a normed linear space M and F : B A be an R-KKM map. If, for each x B, F (z) is closed in M and there exists at least one z 0 B such that F (z 0 ) is compact in M, then {F (z) : z B} is nonempty. Theorem 2.3. Let (A, B) be a nonempty proximal pair of a normed linear space M, and F : B A be a multimap satisfying (1) F : B M is an R-KKM map; (2) F is intersectionally closed-valued; and (3) there exists at least one z 0 B such that F (z 0 ) is compact in M. Then z B F (z). Proof. Since F is an R-KKM map with closed values, by Theorem 2.2, we have F (z). Since F is intersectionally closed-valued, we have z B z B F (z) = z B F (z). The following is an existence theorem for the best proximity pairs;

4 606 Hoonjoo Kim Theorem 2.4. Let I = {1,, n} and for each i I, (A, B i ) be a nonempty convex proximal pair in a normed linear space M. Let A be compact. For each i I, let F i : A B i be a multimap such that (1) F i (x) is convex for each x A; (2) F 1 i (y i ) is open for each y i B i ; and (3) i I P A(y i ) for each (y 1,, y n ) B = B i. Then there is a w A such that d(w, F i (w)) = d(a, B i ) for each i I. Proof. Define F : A B by F (x) = F i (x) and G : B A by G(y 1,, y n ) = A\F 1 (y 1,, y n ). Then G(y 1,, y n ) = {x A : y i / F i (x) for some i I} = A\ I F 1 i (y i ). (a) Suppose that for some (y 1,, y n ) B, G(y 1,, y n ) =. Then F 1 (y 1,, y n ) = A, that is, (y 1,, y n ) F (x) for all x A. Since i I P A(y i ) and y B i = B i0, there exists a w A such that y i w = d(y i, A) = d(a, B i ) for all i I. Note that (y 1,, y n ) F (w) and for each i I, d(a, B i ) d(w, F i (w)) w y i = d(a, B i ). Therefore d(w, F i (w)) = d(a, B i ) for each i I. (b) Assume that G(y 1,, y n ) is nonempty for each (y 1,, y n ) B. By the hypothesis, G is nonempty closed-valued and G(y 1,, y n ) = (A\F 1 i (y i )) (y 1,,y n) B = I = I (y 1,,y n) B I (y 1,,y n) B (A\ F 1 i y i B i (A\F 1 i (y i )) (y i )) = I =. Therefore G is not an R-KKM map. Hence there exist {y 1 = (y1, 1, yn), 1, y m = (y1 m,, yn m )} and x j i I P A(y j i ) for j = 1,, m such that co{x 1,, x m } j=1,,m G(yj ). Choose w = j λ jx j co{x 1,, x m }\ j=1,,m G(yj ), i.e., w m j=1 F 1 (y j ). Therefore y j i F i(w) for all j = 1,, m and i I. Put z i := j λ jy j i. Since F i (w) is convex and y j i F i (w), z i F i (w) for each i I. For each i I, d(a, B i ) d(w, F i (w)) w z i = j λ jx j j λ jy j i j λ j x j y j i = j λ jd(a, y j i ) = d(a, B i). Therefore d(w, F i (w)) = d(a, B i ) for all i I. If I is a singleton, then P A (y) for each y B, since B = B 0. Therefore the condition (3) in Theorem 2.4 is automatically satisfied. Using Theorem 2.3 and the argument of the proof in Theorem 2.4, we obtain the following theorem which extends Theorem 1.1;

5 KKM-type theorems for best proximal points in normed linear space 607 Theorem 2.5. Let (A, B) be a nonempty convex proximal pair in a normed linear space M. Let A be compact and F : A B be a multimap such that (1) F (x) is convex for each x A; and (2) F 1 (y) is unionly open for each y B. Then there is a w A such that d(w, F (w)) = d(a, B). Theorem 2.5 is the case that (A, B) is a proximal pair. existence theorem is for A A 0 or B B 0 ; The following Theorem 2.6. Let A and B be nonempty compact convex subsets of a normed linear space M, A 0 and F : A B be a multimap such that (1) F (x) B 0 for each x A 0 ; (2) F (x) is convex for each x A 0 ; and (3) F 1 (y) is transfer open for each y B 0. Then there is a w A such that d(w, F (w)) = d(a, B). Proof. The first part of this proof is proved by [2]. Let x 1, x 2 A 0. Then there exist y 1, y 2 B 0 such that x 1 y 1 = x 2 y 2 = d(a, B). For λ (0, 1), let w = λx 1 + (1 λ)x 2 and z = λy 1 + (1 λ)y 2. Then w A, z B and z w λ x 1 y 1 + (1 λ) x 2 y 2 = d(a, B). So w A 0 and hence A 0 is convex. In order to prove A 0 is compact, it is enough to show that A 0 is closed. Let {x n } n N be a sequence in A 0 which converges to x A. Then there exists {y n } B 0 B such that x n y n = d(a, B). Since B is compact, there exists a convergent subsequence {y nk } of {y n } which converges to y B. Then d(a, B) x y x x nk + x nk y nk + y nk y. Since x x nk and y nk y converges to 0, d(a, B) = x y. Therefore x A 0 and so A 0 is compact. Similarly the convexity and compactness of B 0 can be proved. For each x A 0, there exists a y B 0 such that x y = d(a, B). Since d(a, B) d(a 0, B 0 ) x y, x y = d(a, B) = d(a 0, B 0 ). Put C = A 0 and D = B 0. The above argument shows that C = C 0. By the same method we can show that D = D 0. Therefore (A 0, B 0 ) is a proximal pair of M. Define S : A 0 B 0 by S(x) = F (x) B 0 for each x A 0. By Lemma 2.1, S 1 (y) is transfer open for each y B 0 and S has convex values. By Theorem 2.5, there is a w A 0 such that d(w, S(w)) = d(a 0, B 0 ). Since d(w, S(w)) d(w, F (w)) d(a, B) = d(a 0, B 0 ), d(w, F (w)) = d(a, B). Theorem 2.7. For each i I = {1,, n}, let A and B i be nonempty compact convex subsets of a normed linear space M, and F i : A B i be a multimap such that (1) C := i I A 0i where A 0i := {x A : x y = d(a, B i ) for some y B i };

6 608 Hoonjoo Kim (2) F i (x) B i0 for each x C; (3) F i (x) is convex for each x C; (4) F 1 i (y i ) is open for each y i B i0 ; and (5) i I P A(y i ) for each (y 1,, y n ) B i0. Then there is a w A such that d(w, F i (w)) = d(a, B i ) for each i I. Proof. Let i I. According to the proof of Theorem 2.6, A 0i and B i0 are compact and convex. Therefore, C is compact and convex. For each x C, there exists a y i B i0 such that x y i = d(a, B i ) by (1). And d(a, B i ) d(c, B i0 ) x y i, so x y i = d(c, B i0 ), that is, C = {x C : x y i = d(c, B i0 ) for some y i B i0 } = C 0. Note that d(c, B i0 ) = d(a, B i ). For (y 1,, y n ) B i0, there exists an x A such that x y i = d(a, y i ) for each i I, by condition (5). Since y i B i0, d(a, y i ) = d(a, B i ). And x y i = d(c, B i0 ) as d(c, B i0 ) = d(a, B i ). That is, x i I P C(y i ) for each (y 1,, y n ) B i0. And B i = {y i B i : x y i = d(c, B i0 ) for some x C} for each i I. Hence (C, B i0 ) is a proximity pair of M for each i I. Define S i : C B i0 by S i (x) = F i (x) B i0 for each x C. Since S i satisfies all the conditions of Theorem 2.4, there is a w C such that d(w, S i (w)) = d(c, B i0 ). So d(a, B i ) d(w, F i (w)) d(w, S i (w)) = d(c, B i0 ) = d(a, B i ) for each i I. Therefore the conclusion holds. Acknowledgements. This work was supported by the Sehan University Research Fund in References [1] H. Kim, Generalized KKM-type theorems for best proximity points, Bull. Korean Math. Soc., 53 (2016), [2] W. K. Kim and K. H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl., 316 (2006), [3] D. T. Luc, E. Sarabi and A. Soubeyran, Existence of solutions in variational relation problems without convexity, J. Math. Anal. Appl., 364 (2010), [4] V. Sankar Raj and S. Somasundaram, KKM-type theorems for best proximal points, Applied Math. Letters, 25 (2012), [5] G. Q. Tian, Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium,

7 KKM-type theorems for best proximal points in normed linear space 609 and complementarity, J. Math. Anal. Appl., 170 (1992), Received: November 1, 2018; Published: December 5, 2018