Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces
Introduction
Let be a Banach space and a nonempty closed subset of . We shall denote by the family of nonempty closed subsets of , by the family of nonempty closed bounded subsets of , by the family of nonempty compact subsets of , and by the family of nonempty compact convex subsets of . Let be the Hausdorff distance on , that is, for all , where is the distance from the point to the subset . A multivalued mapping is said to be a contraction if there exists a constant such that for all . If (1) is valid when , the is called nonexpansive. A point is a fixed point for a multivalued mapping if . Banach’s Contraction Principle was extended to a multivalued contraction by Nadler [13] in 1969. Given a and a , we can define a contraction by Then is multivalued and hence it has a (non-unique, in general) fixed point (see [13]): that is If is a single valued, we have (Such a sequence is said to be an approximating fixed point of since it possesses the property that if is bounded, then .) The strong convergence of as for a single-valued nonexpansive self- or nonself-mapping was studied in Hilbert space or certain Banach spaces by many authors (see for instance, Browder [2], Halpern [7], Jung and Kim [8], Jung and Kim [9], Kim and Takahashi [10], Reich [17], Singh and Watson [20], Takahashi and Kim [23], Xu [25], and Xu and Yin [28]).
Let be such that Now a natural question arises of whether Browder’s theorem can be extended to the multivalued case. A simple example given by Pietramala [14] shows that the answer is negative even if is Euclidean. Example 1 Let be the square in the real plane and be defined by Then it is easy to see that for any , showing that is nonexpansive. It is also easy to see that the fixed point set of is . Let . Then the mapping defined by (2) has the fixed point set Let Then satisfies (3) but does not converge. This example also shows that the sequence of fixed point sets of ’s does not converge as to the fixed point set of under the Hausdorff metric. However, López Acedo and Xu [12] gave the strong convergence of under the restriction in Hilbert space. Kim and Jung [11] extended the result of López Acedo and Xu [12] to a Banach space with a sequentially continuous duality mapping. Recently, Sahu [19] also studied the multivalued case in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm.
In this paper, we establish the strong convergence of defined by (3) for the multivalued nonexpansive nonself-mapping in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We also study the strong convergence of for the multivalued nonexpansive nonself-mapping satisfying the inwardness condition in a reflexive Banach space with a uniformly Gâteaux differentiable norm. Our results improve and extend the results in [8], [9], [25], [28] to the multivalued case. We also point out that the condition should be included in the main results of Sahu [19].
Section snippets
Preliminaries
Let be a real Banach space with norm and let be its dual. The value of at will be denoted by .
A Banach space is called uniformly convex if for every , where the modulus of convexity of is defined by for every with . It is well known that if is uniformly convex, then is reflexive and strictly convex (cf. [5]).
The norm of is said to be Gâteaux differentiable (and is said to be smooth) if
Main results
In this section, we first prove a strong convergence theorem for multivalued nonexpansive nonself-mappings in a Banach space with a uniformly Gâteaux differentiable norm. Theorem 1 Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of , and a nonexpansive nonself-mapping. Suppose that is a nonexpansive retract of . Suppose that for any fixed point of and that for each and , the contraction
Acknowledgement
The author thanks the anonymous referee for his/her careful reading and helpful comments and suggestions, which improved the presentation of this manuscript.
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