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.
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.
2. Ray's Theorem for Mappings of Type (P) and Equilibrium Problems
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.
- Browder FE: Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar
- Göhde D: Zum Prinzip der kontraktiven Abbildung. Mathematische Nachrichten 1965, 30: 251–258. 10.1002/mana.19650300312MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar
- Ray WO: The fixed point property and unbounded sets in Hilbert space. Transactions of the American Mathematical Society 1980,258(2):531–537. 10.1090/S0002-9947-1980-0558189-1MathSciNetView ArticleMATHGoogle Scholar
- Sine R: On the converse of the nonexpansive map fixed point theorem for Hilbert space. Proceedings of the American Mathematical Society 1987,100(3):489–490. 10.1090/S0002-9939-1987-0891152-1MathSciNetView ArticleMATHGoogle Scholar
- Aoyama K, Kohsaka F, Takahashi W: Three generalizations of firmly nonexpansive mappings: their relations and continuity properties. Journal of Nonlinear and Convex Analysis 2009,10(1):131–147.MathSciNetMATHGoogle Scholar
- Takahashi W, Yao J-C, Kohsaka F: The fixed point property and unbounded sets in Banach spaces. Taiwanese Journal of Mathematics 2010,14(2):733–742.MathSciNetMATHGoogle Scholar
- Honda T, Ibaraki T, Takahashi W: Duality theorems and convergence theorems for nonlinear mappings in Banach spaces and applications. International Journal of Mathematics and Statistics 2010,6(S10):46–64.MathSciNetGoogle Scholar
- Dhompongsa S, Fupinwong W, Takahashi W, Yao J-C: Fixed point theorems for nonlinear mappings and strict convexity of Banach spaces. Journal of Nonlinear and Convex Analysis 2010, 11: 175–183.MathSciNetMATHGoogle Scholar
- Aoyama K, Kohsaka F: Existence of fixed points of firmly nonexpansive-like mappings in Banach spaces. Fixed Point Theory and Applications 2010, -15.Google Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MathSciNetMATHGoogle Scholar
- Takahashi H, Takahashi W: Existence theorems and strong convergence theorems by a hybrid method for equilibrium problems in Banach spaces. In Fixed Point Theory and Its Applications. Yokohama Publ., Yokohama, Japan; 2008:163–174.Google Scholar
- Takahashi W: Nonlinear Functional Analysis, Fixed point theory and its applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
- Kadec MI: Spaces isomorphic to a locally uniformly convex space. Izvestija Vysših Učebnyh Zavedeniĭ Matematika 1959,6(13):51–57.MathSciNetGoogle Scholar
- Troyanski SL: On locally uniformly convex and differentiable norms in certain non-separable Banach spaces. Studia Mathematica 1970/71, 37: 173–180.MathSciNetMATHGoogle Scholar
- Takahashi W, Zembayashi K: A strong convergence theorem for the equilibrium problem with a bifunction defined on the dual space of a Banach space. In Fixed Point Theory and Its Applications. Yokohama Publ., Yokohama, Japan; 2008:197–209.Google Scholar
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