- Research Article
- Open Access

# Ray's Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces

- Satit Saejung
^{1}Email author

**2010**:806837

https://doi.org/10.1155/2010/806837

© Satit Saejung. 2010

**Received:**3 July 2010**Accepted:**29 September 2010**Published:**11 October 2010

## Abstract

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.

## Keywords

- Hilbert Space
- Banach Space
- Convex Subset
- Equilibrium Problem
- Unique Element

## 1. Introduction

Let
be a subset of a Banach space
. A mapping
is *nonexpansive* if
for all
. In 1965, it was proved independently by Browder [1], Göhde [2], and Kirk [3] 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 [4] obtained the following interesting result (see [5] for a simpler proof).

Theorem 1.1.

Let be a closed and convex subset of a Hilbert space. Then the following statements are equivalent:

(i) for every nonexpansive mapping ;

*firmly nonexpansive*, that is, the following inequality is satisfied by all :

In this case, . We can restate Ray's theorem in the following form.

Theorem 1.2.

Let be a closed and convex subset of a Hilbert space. Then the following statements are equivalent:

(i) for every firmly nonexpansive mapping ;

*smooth*, that is, the limit exists for all norm one elements . This implies that the

*duality mapping*from to defined by

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. [6]. For a subset of a (smooth) Banach space , a mapping is of

Recently, Takahashi et al. [7] successfully proved the following theorem.

Theorem 1.3.

Let be a closed and convex subset of a smooth, strictly convex and reflexive Banach space. Then the following statements are equivalent:

(i) for every mapping which is of type (Q);

As a direct consequence of the duality theorem [8], we obtain the following result (see also [9]).

Theorem 1.4.

Let be a closed subset of a smooth, strictly convex and reflexive Banach space such that is closed and convex. Then the following statements are equivalent:

(i) for every mapping which is of type (R);

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 [10]. 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.

## 2. Ray's Theorem for Mappings of Type (P) and Equilibrium Problems

The following result was proved by Aoyama et al. [6].

Theorem 2.1.

Let be a smooth, strictly convex and reflexive Banach space, and let be a bounded, closed and convex subset of . If a mapping is of type (P), then has a fixed point.

*equilibrium problem*for a bifunction is the problem of finding an element such that

We denote the set of solutions of the equilibrium problem for by . We assume that a bifunction satisfies the following conditions (see [11]):

(C3) is convex and lower semicontinuous for all ;

(C4)
is *maximal monotone*, that is, for each
and
,

Remark 2.2.

It is noted (see [12]) that if satisfies conditions (C1)–(C3) and the following condition:

then satisfies condition (C4).

Lemma 2.3 (see [12]).

Employing the methods in [5, 7], we obtain the following result.

Theorem 2.4.

Let be a smooth, strictly convex and reflexive Banach space and a closed and convex subset of . The following statements are equivalent.

(a) for every mapping which is of type (P);

(b) for every bifunction satisfying conditions (C1)–(C4);

(c) for every bifunction satisfying conditions (C1)–(C3) and (C4');

Proof.

In particular, the restriction of to the closed and convex subset is of type (P). It then follows from (a) that .

(b) (c) It follows directly from Remark 2.2.

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

(ii)*hemicontinuous* if for each
the mapping
, where
, is continuous with respect to the weak* topology of
;

(iii)*demicontinuous* if
converges to
with respect to the weak* topology of
whenever
is a sequence in
such that it converges strongly to
.

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., [13]).

As a consequence of Theorem 2.4, we obtain a necessary and sufficient condition for the existence of solutions of a variational inequality problem.

Corollary 2.5.

Let be a reflexive Banach space and a nonempty, closed and convex subset of . Then the following statements are equivalent:

(a) for every monotone and hemicontinuous mapping ;

(b) for every monotone and demicontinuous mapping ;

Proof.

(a) (b) It is clear since demicontinuity implies hemicontinuity.

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 [16]. 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:

(D3) is convex and lower semicontinuous for all ;

In [16], a bifunction is assumed to satisfy conditions (D1)–(D3) and

and the set of solutions of this problem is denoted by .

The following lemma was proved by Takahashi and Zembayashi ([16], 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.

Lemma 2.6.

Moreover, if is defined by where is given above, then is of type (R).

Based on the preceding lemma and Theorem 2.4, we obtain the result whose proof is omitted.

Theorem 2.7.

Let be a smooth, strictly convex and reflexive Banach space, and let be a closed subset of such that is closed and convex. The following statements are equivalent:

(i) for every mapping which is of type (R);

(ii) for every bifunction satisfying conditions (D1)–(D4);

(iii) for every bifunction satisfying conditions (D1)–(D3) and (D4');

## Declarations

### Acknowledgments

The author would like to thank the referee for pointing out information on Theorem of [13]. The author was supported by the Thailand Research Fund, the Commission on Higher Education and Khon Kaen University under Grant RMU5380039.

## Authors’ Affiliations

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