Fixed points of multimaps which are not necessarily nonexpansive
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Let C be a nonempty closed bounded convex subset of a Banach space X whose characteristic of noncompact convexity is less than 1 and T a continuous 1-χ-contractive SL map (which is not necessarily nonexpansive) from C to KC(X) satisfying an inwardness condition, where KC(X) is the family of all nonempty compact convex subsets of X. It is proved that T has a fixed point. Some fixed points results for noncontinuous maps are also derived as applications. Our result contains, as a special case, a recent result of Benavides and Ram´ırez (2004). 1. Introduction During the last four decades, various fixed point results for nonexpansive single-valued maps have been extended to multimaps, see, for instance, the works of Benavides and Ram´ırez [2], Kirk and Massa [6], Lami Dozo [7], Lim [8], Markin [10], Xu [12], and the references therein. Recently, Benavides and Ram´ırez [3] obtained a fixed point theorem for nonexpansive multimaps in a Banach space whose characteristic of noncompact convexity is less than 1. More precisely, they proved the following theorem. Theorem 1.1 (see [3]). Let C be a nonempty closed bounded convex subset of a Banach space X such that α (X) < 1 and T : C → KC(X) a nonexpansive 1-χ-contractive map. If T satisfies T(x) ⊂ IC (x)
∀x ∈ C,
(1.1)
then T has a fixed point. Benavides and Ram´ırez further remarked that the assumption of nonexpansiveness in the above theorem can not be avoided. In this paper, we prove a fixed point result for multimaps which are not necessarily nonexpansive. To establish this, we define a new class of multimaps which includes nonexpansive maps. To show the generality of our result, we present an example. As consequences of our main result, we also derive some fixed point theorems for ∗-nonexpansive maps. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 169–176 DOI: 10.1155/FPTA.2005.169
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Fixed points of multimaps
2. Preliminaries Let C be a nonempty closed subset of a Banach space X. Let CB(X) denote the family of all nonempty closed bounded subsets of X and KC(X) the family of all nonempty compact convex subsets of X. The Kuratowski and Hausdorff measures of noncompactness of a nonempty bounded subset of X are, respectively, defined by α(B) = inf {d > 0 : B can be covered by finitely many sets of diameter ≤ d}, χ(B) = inf {d > 0 : B can be covered by finitely many balls of radius ≤ d}.
(2.1)
Let H be the Hausdorff metric on CB(X) and T : C → CB(X) a map. Then T is called (1) contraction if there exists a constant k ∈ [0,1) such that
H T(x),T(y) ≤ kx − y ,
∀x, y ∈ C;
(2.2)
∀x, y ∈ C;
(2.3)
(2) nonexpansive if H T(x),T(y) ≤ x − y ,
(3) φ-condensing (resp., 1-φ-contractive), where φ = α(·) or χ(·) if T(C) is bounded and, for each bounded subset B of C with φ(B) > 0, the following holds:
φ T(B) < φ(B)
resp., φ T(B) ≤ φ(B) ;
(2.4)
here T(B) = x∈B T(x); (4) upper semicontinuous if {x ∈ C : T(x) ⊂ V } is open in C whenever V ⊂ X is open; (5) lower semicontinuous if the set {x ∈
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