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Approximation of solutions to an equilibrium problem in a nonuniformly smooth Banach space
Journal of Inequalities and Applications volume 2013, Article number: 387 (2013)
Abstract
An equilibrium problem based on a projection algorithm is investigated. A strong convergence theorem for solutions of the equilibrium problems is established in a nonuniformly smooth Banach space.
1 Introduction
Recently, equilibrium problems have been studied as an effective and powerful tool for studying a wide class of real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and networks; see [1–16] and the references therein. It is well known that equilibrium problems include many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, fixed point problems, saddle point problems, and game theory. For the solutions of equilibrium problems, there are several algorithms to solve the problem; see [17–28] and the references therein. However, most of these results are obtained in the framework of Hilbert spaces or uniformly smooth Banach spaces.
The purpose of this paper is to study solution problems of an equilibrium problem based on a projection algorithm in a nonuniformly smooth Banach space. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a projection algorithm is introduced and the convergence analysis is given. A strong convergence theorem is established in a nonuniformly smooth Banach space. Applications of the main results are also discussed in this section.
2 Preliminaries
Let E be a real Banach space, and let be the dual space of E. We denote by J the normalized duality mapping from E to defined by
where denotes the generalized duality pairing. A Banach space E is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in E such that and . Let be the unit sphere of E. Then the Banach space E is said to be smooth provided
exists for each . It is also said to be uniformly smooth if the above limit is attained uniformly for . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if is uniformly convex.
Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence and with and , then as . For more details on the Kadec-Klee property, the readers can refer to [29] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property. In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
Let C be a nonempty subset of E. Recall that a mapping is said to be monotone iff
is said to be α-inverse-strongly monotone iff there exists a positive real number α such that
Recall also that a monotone mapping Q is said to be maximal iff its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping Q is maximal iff for , for every implies . An operator Q from C into E is said to be hemi-continuous if for all , the mapping f of into E defined by is continuous with respect to the weak∗ topology of . Let f be a bifunction from to ℝ, where ℝ denotes the set of real numbers. In this paper, we investigate the following equilibrium problem. Find such that
We use to denote the solution set of equilibrium problem (2.1). That is,
Given a mapping , let
Then iff p is a solution of the following variational inequality. Find p such that
In order to study the solution problem of equilibrium problem (2.1), we assume that f satisfies the following conditions:
(A1) , ;
(A2) f is monotone, i.e., , ;
(A3)
(A4) for each , is convex and weakly lower semi-continuous.
As we all know, if C is a nonempty closed convex subset of a Hilbert space H and is the metric projection of H onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently it is not available in more general Banach spaces. In this connection, Alber [30] recently introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that in a Hilbert space H, the equality is reduced to , . The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping J; see, for example, [29] and [30]. In Hilbert spaces, . It is obvious from the definition of a function ϕ that
and
Let be a mapping. In this paper, we use to denote the fixed point set of T. A point p in C is said to be an asymptotic fixed point of T iff C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by . T is said to be relatively nonexpansive iff and for all and . T is said to be quasi-ϕ-nonexpansive iff and for all and .
Remark 2.1 The class of quasi-ϕ-nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the restriction: .
Remark 2.2 The class of quasi-ϕ-nonexpansive mappings is a generalization of quasi-nonexpansive mappings in Hilbert spaces.
In this paper, we investigate the solution problem of equilibrium problem (2.1) based on a projection algorithm. A strong convergence theorem for solutions of the equilibrium problems is established in a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property.
In order to give our main results, we need the following lemmas.
Lemma 2.3 [30]
LetCbe a nonempty, closed, and convex subset of a smooth Banach spaceE, and. Thenif and only if
Lemma 2.4 [30]
LetEbe a reflexive, strictly convex, and smooth Banach space, letCbe a nonempty, closed, and convex subset ofE, and. Then
Lemma 2.5 [30]
LetEbe a reflexive, strictly convex, and smooth Banach space. Then
The following lemma can be obtained from [15] and [31].
Lemma 2.6LetCbe a closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE. Letfbe a bifunction fromto ℝ satisfying (A1)-(A4). Letand. Then
-
(a)
There exists such that
-
(b)
Define a mapping by
Then the following conclusions hold:
-
(1)
;
-
(2)
is a firmly nonexpansive-type mapping, i.e., for all,
-
(3)
is single-valued;
-
(4)
-
(5)
is closed and convex;
-
(6)
is quasi-ϕ-nonexpansive.
3 Main results
Theorem 3.1LetEbe a reflexive, strictly convex, and smooth Banach space such that bothEandhave the Kadec-Klee property. LetCbe a nonempty, closed, and convex subset of E. Letfbe a bifunction fromto ℝ satisfying (A1)-(A4) such thatis nonempty. Letbe a sequence generated in the following manner:
whereis a real sequence such that. Then the sequenceconverges strongly to, whereis the generalized projection fromEonto.
Proof First, we show that is closed and convex, that the projection on it is well defined. It is obvious that is closed and convex. Suppose that is closed and convex for some . We next prove that is also closed and convex for the same m. Let For , we see that . It follows that , where . Notice that
The above inequalities are equivalent to
and
Multiplying t and on both sides of (3.1) and (3.2), respectively, yields that
That is,
This gives that is closed and convex. Then is closed and convex. Now, we are in a position to prove that . is obvious. Suppose that for some . Fix . It follows that
which implies that . This proves that . In the light of , from Lemma 2.3, we find that for any . It follows from that
It follows from Lemma 2.4 that
This implies that the sequence is bounded. It follows from (2.3) that the sequence is also bounded. Since the space is reflexive, we may assume that . Since is closed and convex, we find that . On the other hand, we see from the weak lower semicontinuity of the norm that
which implies that . Hence, we have . In view of the Kadec-Klee property of E, we find that as . Next, we show that . In view of construction of , we find that
Since and , we arrive at . This shows that is nondecreasing. It follows from the boundedness that exists. This implies from (3.4) that
Since , we arrive at . It follows that
In view of (2.3), we see that . This in turn implies that . It follows that . This implies that is bounded. Note that both E and are reflexive. We may assume that . In view of the reflexivity of E, we see that . This shows that there exists an element such that . It follows that
Taking on both sides of the equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain that . Notice that
It follows that . From the restriction on the sequence , we find that
In view of , we see that
It follows from (A2) that
By taking the limit as in the above inequality, from (A4) we obtain that
For and , define . It follows that , which yields that . It follows from (A1) and (A4) that
That is,
Letting , we obtain from (A3) that , . This implies that .
Finally, we prove that . Letting in (3.3), we see that
In the light of Lemma 2.3, we find that . This completes the proof. □
We remark that , where is a space which satisfies the restriction in Theorem 3.1. Since every uniformly convex and uniformly smooth Banach space is a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property, we find from Theorem 3.1 the following result.
Corollary 3.2LetEbe a uniformly convex and uniformly smooth Banach space. LetCbe a nonempty, closed, and convex subset ofE. Letfbe a bifunction fromto ℝ satisfying (A1)-(A4) such thatis nonempty. Letbe a sequence generated in the following manner:
whereis a real sequence such that. Then the sequenceconverges strongly to, whereis the generalized projection fromEonto.
In the framework of Hilbert spaces, we find from Theorem 3.1 the following result.
Corollary 3.3LetEbe a Hilbert space. LetCbe a nonempty, closed, and convex subset of E. Letfbe a bifunction fromto ℝ satisfying (A1)-(A4) such thatis nonempty. Letbe a sequence generated in the following manner:
whereis a real sequence such that. Then the sequenceconverges strongly to, whereis the metric projection fromEonto.
Proof Notice that . The generalized metric projection is reduced to the metric projection and the normalized duality mapping J is reduced to the identity mapping I in Hilbert spaces. The result can be obtained from Theorem 3.1 immediately. □
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Zhao, J. Approximation of solutions to an equilibrium problem in a nonuniformly smooth Banach space. J Inequal Appl 2013, 387 (2013). https://doi.org/10.1186/1029-242X-2013-387
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DOI: https://doi.org/10.1186/1029-242X-2013-387