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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 498487, 12 pages
A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces
1Leshan College of Profession and Technology, Sichuan, Leshan 614000, China
2Department of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China
3College of Business, City University of Hong Kong, Hong Kong
Received 3 July 2012; Revised 24 August 2012; Accepted 24 August 2012
Academic Editor: Yeong-Cheng Liou
Copyright © 2012 Mei Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalized f-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.
It is well known that metric projection operator in Hilbert and Banach spaces is widely used in different areas of mathematics such as functional analysis and numerical analysis, theory of optimization and approximation, and also for the problems of optimal control and operations research, nonlinear and stochastic programming and game theory.
Let be a real Banach space with its dual , and let be a nonempty, closed, and convex subset of . In 1994, Alber  introduced the generalized projections and in uniformly convex and uniformly smooth Banach spaces based on the function defined on p.3 and studied their properties in detail. In 2005, Li  extended the definition of the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator. Recently, Wu and Huang  introduced a new generalized -projection operator in a Banach space. By making use of (2.5), they extended the definition of the generalized projection operators introduced by Abler  and proved some properties of the generalized -projection operator. Wu and Huang  studied a relation between the generalized projection operator and the resolvent operator for the subdifferential of a proper convex and lower semicontinuous functional in reflexive and smooth Banach spaces (see [5–9]). Very recently, Li et al.  studied some properties of the generalized -projection operator, and proved the strong convergence theorems for relatively nonexpansive mappings in Banach spaces.
On the other hand, equilibrium problem was introduced by Blum and Oettli , in 1994. It is a hot topic of intensive research efforts, because it has a great impact and influence in the development of several branches of pure and applied sciences. It has been shown that equilibrium problem theory provides a novel and unified treatment of a wide class of problems arisen in economics, finance, physics, image reconstruction, ecology, transportation, network, elasticity, and optimization problems. Numerous issues in physics, optimization, and economics reduce to finding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems (see, e.g, [12–14] and the references therein).
In this paper, motivated and inspired by the work mentioned above, we introduce a new hybrid projection algorithm based on the shrinking projection method for relatively nonexpansive mapping and equilibrium problem. Using the new algorithm, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces. The result presented in this paper extends and improves the main result of Li et al. .
Let be a real Banach space with its dual and . We denote the duality between and by , and the norms of Banach space and by and , respectively. A Banach space is said to be strictly convex if for all with and . It is also said to be uniformly convex if for any two sequences , in , such that and . The function is called the modulus of convexity of .
A Banach space is said to be smooth provided that exists for all with . It is also said to be uniformly smooth if the limit is attained uniformly for . The function is called the modulus of smoothness of .
When is a sequence in , we denote the strong convergence of with a cluster by and the weak convergence of with a weak cluster by . A Banach space is said to have the Kadec-Klee property if a sequence of satisfies that and , then . It is known that if is uniformly convex, then has the Kadec-Klee property.
The normalized duality mapping from to is defined by for any . We list some properties of mapping as follows. (i) If is a smooth Banach space (with Gâteaux differential norm), then is single-valued and demicontinuous. If is a smooth reflexive Banach space, then is single-valued and hemicontinuous. If is a strongly smooth Banach space (with Fréchet differential norm), then is single-valued and continuous. (ii) is uniformly continuous on every bounded set of a uniformly smooth Banach space. (iii) If is a reflexive, smooth and strictly convex Banach space, is the duality mapping of , then .
Let be a smooth Banach space and be a nonempty, closed and convex subset of . The function is defined by for all .
Next, we recall the concept of the generalized -projector operator, together with its properties. Let be a functional defined as follows: where , is a positive number and is proper, convex, and lower semicontinuous.
From the definitions of and , it is easy to have the following properties: (i) is convex and continuous with respect to when is fixed; (ii) is convex and lower semicontinuous with respect to when is fixed.
Definition 2.1. Let be a real smooth Banach space and be a nonempty, closed and convex subset of . We say that is a generalized -projection operator if
In order to obtain our results, the following lemmas are crucial to us.
Lemma 2.2 (see ). Let be a real Banach space and be a lower semicontinuous convex functional. Then there exist and such that
Lemma 2.3 (see ). Let be a uniformly convex and smooth Banach space and let , be two sequences of . If and either or is bounded, then .
Let be a closed subset of a real Banach space , and let be a mapping from to . We denote by the set of all fixed points of . A point in is said to be an asymptotic fixed point of , if contains a sequence which converges weakly to such that . The set of all asymptotic fixed points of will be denoted by . is called nonexpansive if for all , and relatively nonexpansive if and for all and . Obviously, the definition of relatively nonexpansive mapping is equivalent to and for all and .
Lemma 2.4 (see ). Let be a strictly convex and smooth Banach space, let be a closed, and convex subset of , and let be a relatively nonexpansive mapping from into itself. Then is closed, and convex.
Lemma 2.5 (see ). Let be a real reflexive and smooth Banach space and let be a nonempty, closed, and convex subset of . The following statements hold: (i) is a nonempty, closed, and convex subset of for all ;(ii) for all if and only if (iii) if is strictly convex, then is a single-valued mapping.
Lemma 2.6 (see ). Let be a real reflexive and smooth Banach space, let be a nonempty, closed, and convex subset of , and let , . Then
Lemma 2.7 (see ). Let be a Banach space and . Let be a proper, convex and lower semicontinuous functional with convex domain . If is a sequence in such that and , then .
Let be a closed and convex subset of a real Banach space and be a bifunction. The equilibrium problem for is as follows. Find such that The set of all solutions for the above equilibrium problem is denoted by . For solving the equilibrium problem, one always assumes that the bifunction satisfies the following conditions:(A1), for all ; (A2) is monotone, that is, , for all ; (A3) for all ; (A4) for all , is convex and lower semicontinuous.
In order to prove our results, we present several necessary lemmas.
Lemma 2.8 (see ). Let be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space , and be a bifunction from satisfying the conditions (A1)–(A4). For all and , define the mappingas follows. Then, the following statements hold: (B1) is single-valued; (B2) is a firmly nonexpansive-type mapping, that is, for all , (B3); (B4) is closed and convex.
Lemma 2.9 (see ). Let be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space , be a bifunction from to satisfying the conditions (A1)–(A4), and . Then, for any and ,
3. The Main Result
In this section, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces.
Theorem 3.1. Let be a uniformly convex and uniformly smooth Banach space, and be two nonempty, closed and convex subsets of such that . Let be a relatively nonexpansive mapping and a convex and lower semicontinuous mapping with . Let be a bifunction from , which satisfies the conditions (A1)–(A4). Assume that is a sequence in such that , and for some . Define a sequence in by the following algorithm: If is nonempty, then converges strongly to .
Proof . The proof is divided into the following four steps.
(I) First, we prove the following conclusion: is a closed convex set and for all .
It is obvious that is a closed convex set and . Thus, we only need to show that is a closed convex set and for all .
Since and are respectively equivalent to it follows that is closed and convex for all . Thus, we know that is well defined. Further, for any and , we have On the other hand, it follows from the definition of and Lemma 2.8 that . From Lemma 2.9, we obtain which implies that Therefore, for all .
(II) Second, we show that is bounded and exists.
Since is a convex and lower semicontinuous mapping, a direct application of Lemma 2.2 yields that there exist and such that It follows that Since , it follows from (3.7) that which implies that is bounded and so is . By the fact that and Lemma 2.6, we obtain It is obvious that and so is nondecreasing. Therefore, we know that exists.
(III) Third, we prove that, if , then , where is an arbitrarily weakly convergent subsequence of .
It follows from the definition of and that Taking in (3.11), we get Applying Lemma 2.3, we obtain
Next, we show that .
From the fact that is uniformly norm-to-norm continuous on bounded sets, we have Note that and thus From (3.14) and , we get Since is uniformly norm-to-norm continuous on bounded sets, we have Since we have and so Now, we show . Since we get Since is a uniformly norm-to-norm continuous on bounded sets, we have From the assumption that , we get It follows from that From (A2), we obtain Since , we obtain For any , it follows that and so . From (3.27) and (A4), we know that Letting we have It follows from (A1) that and thus Taking the limit as in (3.34) and from (A3), we have and so .
(IV) Last, we prove that .
Since is a closed convex set, from Lemma 2.5, we know that is single-valued and denote that . Since and , we have For each given , is convex and lower semicontinuous with respect to , it is easy to see that is weakly lower semicontinuous with respect to and so From the definition of and , we know that and so . It follows from Lemma 2.7 that . The Kadec-Klee property of implies that converges strongly to . Since is an arbitrarily weakly convergent sequence of , we conclude that converges strongly to . This completes the proof.
This work was supported by the Key Program of NSFC (Grant no. 70831005) and the National Natural Science Foundation of China (11171237, 11101069). The work is also supported by HK CityU, CTTFS (9360142).
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