Multivalued nonexpansive mappings in Banach spaces

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It is the objective of this paper to prove some xed point theorems for multivalued mappings. Among other things, we extend Theorem 1.3 to nonself-mappings. Also a simple proof of Theorem 1.3 is presented. Moreover, we give an armative answer to a question of Deimling. A negative answer to a question of Downing and Kirk is included as well.

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Nonlinear Analysis 43 (2001) 693 – 706 www.elsevier.nl/locate/na Multivalued nonexpansive mappings in Banach spaces Hong-Kun Xu Department of Mathematics, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa Received 13 November 1998; accepted 9 February 1999 Keywords: Multivalued nonexpansive mapping; Fixed point; Weak inwardness condition; Uniformly convex Banach space 1. Introduction Let X be a Banach space and E a nonempty subset of X . We shall denote by F(E) the family of nonempty closed subsets of E, by CB(E) the family of nonempty closed bounded subsets of E, by K(E) the family of nonempty compact subsets of E, and by KC(E) the family of nonempty compact convex subsets of E. Let H (·; ·) be the Hausdor distance on CB(X ), i.e., H (A; B) = max sup dist(a; B); sup dist(b; A) ; A; B ∈ CB(X ); a∈A b∈B where dist(a; B) = inf {||a − b||: b ∈ B} is the distance from the point a to the subset B. A multivalued mapping T : E → F(X ) is said to be a contraction if there exists a constant k ∈ [0; 1) such that H (Tx; Ty) ≤ k||x − y||; x; y ∈ E: (1.1) If (1.1) is valid when k = 1, then T is called nonexpansive. A point x is a
xed point for a multivalued mapping T if x ∈ Tx. Banach’s Contraction Principle was extended to a multivalued contraction in 1969. (Below is stated in a Banach space setting.) E-mail address: hkxu@pixie.udw.ac.za (H-K. Xu). 0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 2 2 7 - 8
694 H-K. Xu / Nonlinear Analysis 43 (2001) 693 – 706 Theorem 1.1. (Nadler [14]). Let E be a nonempty closed subset of a Banach space X and T : E → CB(E) a contraction. Then T has a ÿxed point. The
xed point theory of multivalued nonexpansive mappings is however much more complicated and dicult than the corresponding theory of single-valued nonexpansive mappings. One breakthrough was achieved by T.C. Lim in 1974 by using Edelstein’s method of asymptotic centers [4]. Theorem 1.2. (Lim [12]). Let E be a nonempty closed bounded convex subset of a uniformly convex Banach space X and T : E → K(E) a nonexpansive mapping. Then T has a ÿxed point. Lim’s original proof was later simpli
ed independently by Lim himself [13] and Goebel [5]. Another important result for multivalued nonexpansive mappings was obtained by W.A. Kirk and S. Massa in 1990. Theorem 1.3. (Kirk and Massa [9]). Let E be a nonempty closed bounded convex subset of a Banach space X and T : E → KC(E) a nonexpansive mapping. Suppose that the asymptotic center in E of each bounded sequence of X is nonempty and compact. Then T has a ÿxed point. Theorem 1.3 applies to all k-uniformly rotund (k-UR) Banach spaces [17]. However, it does not apply to a nearly uniformly convex (NUC) Banach space [6] as in such a space the asymptotic center of a bounded sequence is not necessarily compact (cf. [10]). Also note that Theorem 1.3 requires that T take convex values. In order to study the
xed point theory for nonself-mappings, we must introduce some terminology for boundary conditions. The inward set of E at x ∈ E is de
ned by IE (x) := {x + (y − x): y ∈ E; ≥ 0}; x ∈ E: Let IE (x) = x + TE (x); x∈E with d(x + y; E) TE (x) = y ∈ X : lim inf =0 ; x ∈ E: →0+ Note that for a convex E, we have IE (x) = IE (x), the closure of IE (x). A multivalued mapping T : E → F(X ) is said to be inward on E if Tx ⊂ IE (x) ∀x ∈ E (1.2) and weakly inward on E if Tx ⊂ IE (x) ∀x ∈ E: (1.3) The two following results are now basic in the
xed point theory of multivalued mappings.
H-K. Xu / Nonlinear Analysis 43 (2001) 693 – 706 695 Theorem 1.4. (Deimling [2]). Let E be a nonempty closed subset of a Banach space X and T : E → F(X ) a contraction. Assume that T is weakly inward on E and that each x ∈ E has a nearest point in Tx. Then T has a ÿxed point. Theorem 1.5. (Downing and Kirk [3] and Reich [16]). Let E be a nonempty closed bounded convex subset of a uniformly convex Banach space X and T : E → K(X ) a nonexpansive mapping. Assume T is inward on E. Then T has a ÿxed point. The following consequence of Theorem 11:5 of Deimling [2] will be often used throughout the paper. Theorem 1.6. Let E be nonempty bounded closed convex subset of a Banach space and T : E → KC(X ) a contraction. Assume Tx ∩ IE (x) 6= ∅ for all x ∈ E. Then T has a ÿxed point. It is the objective of this paper to prove some
xed point theorems for multivalued mappings. Among other things, we extend Theorem 1.3 to nonself-mappings. Also a simple proof of Theorem 1.3 is presented. Moreover, we give an armative answer to a question of Deimling. A negative answer to a question of Downing and Kirk [3] is included as well. Fixed point theorems in Banach spaces with Opial’s property [15] can be found in [11]. 2. The method of asymptotic centers Let K be a weakly compact convex subset of a Banach space X and {x n } a bounded sequence in X . De
ne a function f on X by f(x) = lim sup ||x n − x||; x ∈ X: n→∞ Let r ≡ rK ({x n }) := inf {f(x): x ∈ K} and A ≡ AK ({x n }) := {x ∈ K: f(x) = r}: Recall that r and A are, respectively, called the asymptotic radius and center of {x n } relative to K. As K is weakly compact convex, we see that AK ({x n }) is nonempty, weakly compact and convex. Deÿnition 2.1. Let {x n } and K be as above. Then {x n } is called regular w.r.t. K if rK ({x n }) = rK ({x ni }) for all subsequences {x ni } of {x n }; while {x n } is called asymp- totically uniform if AK ({x n }) = AK ({x ni }) for all subsequences {x ni } of {x n }. The method of asymptotic centers plays an important role in the
xed point theory of both single- and multi-valued nonexpansive mappings, due to the fundamental lemma below.
696 H-K. Xu / Nonlinear Analysis 43 (2001) 693 – 706 Lemma 2.1. (Geobel [5] and Lim [13]). Let {x n } and K be as above. Then we have (i) There always exists a subsequence of {x n } which is regular w.r.t. K; (ii) if K is separable; then {x n } contains a subsequence which is asymptotically uniform w.r.t. K Remark 2.1. If X is uniformly convex in every direction (especially uniformly con- vex), then AK ({x n }) consists of exactly one point so every regular sequence in such a space is always asymptotically uniform w.r.t. K. Let now E be a weakly compact convex subset of a Banach space X and T : E → K(E) a nonexpansive self-mapping. For each integer n ≥ 1, the contraction Tn : E → K(E) de
ned by 1 1 Tn (x) := x0 + 1 − Tx; x ∈ E; (2.1) n n where x0 ∈ E is a
xed point, has a
xed point x n ∈ E. Let r and A be the asymptotic radius and center of {x n } w.r.t. E, respectively. It is easily seen that 1 dist(x n ; Tx n ) ≤ diam(E) → 0: n Since T is compact-valued, we can take yn ∈ Tx n such that ||yn − x n || = dist(x n ; Tx n ); n ≥ 1: Since T is a self-mapping, we may assume that E is separable (otherwise, we can construct a closed convex subset of E that is invariant under T , see [10]). Then by Lemma 2.1 we may assume that {x n } is asymptotically uniform. Take any z ∈ A, as Tz is compact, we can
nd zn ∈ Tz satisfying ||yn − zn || = dist(yn ; Tz) ≤ H (Tx n ; Tz): It follows from the nonexpansiveness of T that ||yn − zn || ≤ ||x n − z||: Because of the compactness of Tz, we may also assume that {zn } (strongly) converges to a point z˜ ∈ Tz. It then follows that lim sup ||x n − z|| ˜ = lim sup ||yn − zn || ≤ lim sup ||x n − z||: This shows that z˜ ∈ A. Hence we can de
ne a multivalued self-map T˜ : A → A by setting for each z ∈ A T˜ z := A ∩ Tz: This map T˜ is in general neither nonexpansive, nor lower semicontinuous. However, it is upper semicontinuous, which is observed by Kirk and Massa [9]. With this observa- tion they are able to prove Theorem 1.3 by using the Bohnenblust–Karlin
xed point theorem (cf. [19]) that is of topological rather than metric nature. In addition, if we as- sume that X is uniformly convex (uniformly convex in every direction is enough), the asymptotic center A consists of exactly one point z. Then the above argument shows
H-K. Xu / Nonlinear Analysis 43 (2001) 693 – 706 697 that we must have z˜ = z and therefore z is a