- Research Article
- Open Access
Ray's Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces
© Satit Saejung. 2010
- Received: 3 July 2010
- Accepted: 29 September 2010
- Published: 11 October 2010
We prove that every firmly nonexpansive-like mapping from a closed convex subset of a smooth, strictly convex and reflexive Banach pace into itself has a fixed point if and only if is bounded. We obtain a necessary and sufficient condition for the existence of solutions of an equilibrium problem and of a variational inequality problem defined in a Banach space.
- Hilbert Space
- Banach Space
- Convex Subset
- Equilibrium Problem
- Unique Element
Let be a subset of a Banach space . A mapping is nonexpansive if for all . In 1965, it was proved independently by Browder , Göhde , and Kirk  that if is a bounded closed convex subset of a Hilbert space and is nonexpansive, then has a fixed point. Combining the results above, Ray  obtained the following interesting result (see  for a simpler proof).
is single-valued and we do consider the singleton as an element in . If is additionally assumed to be strictly convex, that is, there are no distinct elements such that , then is one-to-one. Let us note here that if is a Hilbert space, then the duality mapping is just the identity mapping.
The following three generalizations of firmly nonexpansive mappings in Hilbert spaces were introduced by Aoyama et al. . For a subset of a (smooth) Banach space , a mapping is of
Recently, Takahashi et al.  successfully proved the following theorem.
The purpose of this short paper is to prove the analogue of these results for mappings of type (P). Let us note that our result is different from the existence theorems obtained recently by Aoyama and Kohsaka . We also obtain a necessary and sufficient condition for the existence of solutions of certain equilibrium problems and of variational inequality problems in Banach spaces.
The following result was proved by Aoyama et al. .
We denote the set of solutions of the equilibrium problem for by . We assume that a bifunction satisfies the following conditions (see ):
It is noted (see ) that if satisfies conditions (C1)–(C3) and the following condition:
Lemma 2.3 (see ).
Let be a subset of a Banach space . We now discuss a variational inequality problem for a mapping , that is, the problem of finding an element such that for all and the set of solutions of this problem is denoted by . Recall that a mapping is said to be
It is known that if is a nonempty weakly compact and convex subset of a reflexive Banach space and is monotone and hemicontinuous, then (see e.g., ).
As a consequence of Theorem 2.4, we obtain a necessary and sufficient condition for the existence of solutions of a variational inequality problem.
For each , we have , that is, is monotone. Moreover, it is proved in [6, Theorem ] that is demicontinuous. Therefore, .
(c) (a) It is a corollary of [13, Theorem ]
We finally discuss an equilibrium problem defined in the dual space of a Banach space. This problem was initiated by Takahashi and Zembayashi . Let be a closed subset of a smooth, strictly convex and reflexive Banach space such that is closed and convex. We assume that a bifunction satisfies the following conditions:
In , a bifunction is assumed to satisfy conditions (D1)–(D3) and
The following lemma was proved by Takahashi and Zembayashi (, Lemma ) where the bifunction satisfies conditions (D1)–(D3) and (D4'). However, it can be proved that the conclusion remains true under the conditions (D1)–(D4). We also note that the uniform smoothness assumption on a space in [16, Lemma ] is a misprint.
Based on the preceding lemma and Theorem 2.4, we obtain the result whose proof is omitted.
The author would like to thank the referee for pointing out information on Theorem of . The author was supported by the Thailand Research Fund, the Commission on Higher Education and Khon Kaen University under Grant RMU5380039.
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