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Journal of Applied Mathematics

Volume 2012 (2012), Article ID 812783, 10 pages

http://dx.doi.org/10.1155/2012/812783

## Approximation of Solutions of an Equilibrium Problem in a Banach Space

^{1}School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China^{2}College of Science, Hebei University of Engineering, Handan 056038, China

Received 15 December 2011; Accepted 4 February 2012

Academic Editor: Rudong Chen

Copyright © 2012 Hecai Yuan and Guohong Shi. 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.

#### Abstract

An equilibrium problem is investigated based on a hybrid projection iterative algorithm. Strong convergence theorems for solutions of the equilibrium problem are established in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property.

#### 1. Introduction

Equilibrium problems which were introduced by Fan [1] and Blum and Oettli [2] have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. It has been shown [3–8] that equilibrium, problems include variational inequalities, fixed point, the Nash equilibrium, and game theory as special cases. A number of iterative algorithms have recently been studying for fixed point and equilibrium problems, see [9–26] and the references therein. However, there were few results established in the framework of the Banach spaces. In this paper, we suggest and analyze a projection iterative algorithm for finding solutions of equilibrium in a Banach space.

#### 2. Preliminaries

In what follows, we always assume that is a Banach space with the dual space . Let be a nonempty, closed, and convex subset of . We use the symbol to stand for the normalized duality mapping from to defined by

where denotes the generalized duality pairing of elements between and .

Let be the unit sphere of . is said to be strictly convex if for all with . It is said to be uniformly convex if for any there exists such that for any ,

It is known that a uniformly convex Banach space is reflexive and strictly convex; for details see [27] and the references therein.

Recall that a Banach space is said to have the Kadec-Klee property if a sequence of satisfies that , where denotes the weak convergence, and , where denotes the strong convergence, and then . It is known that if is uniformly convex, then enjoys the Kadec-Klee property; for details see [26] and the references therein.

is said to be smooth provided exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all .

It is well known that if is strictly convex, then is single valued; if is reflexive, and smooth, then is single valued and demicontinuous; for more details see [27, 28] and the references therein.

It is also well known that if is a nonempty, closed, and convex subset of a Hilbert space , and is the metric projection from onto , then is nonexpansive. This fact actually characterizes the Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [29] introduced a generalized projection operator in the Banach spaces which is an analogue of the metric projection in the Hilbert spaces.

Let be a smooth Banach space. Consider the functional defined by Notice that, in a Hilbert space , (2.3) is reduced to for all . The generalized projection is a mapping that is assigned to an arbitrary point , the minimum point of the functional ; that is, , where is the solution to the following minimization problem:

The existence and uniqueness of the operator follow from the properties of the functional and the strict monotonicity of the mapping ; see, for example, [27, 28]. In the Hilbert spaces, . It is obvious from the definition of the function that

Let be a mapping. Recall that 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 asymptotic fixed points of will be denoted by . is said to be relatively nonexpansive if

The asymptotic behavior of a relatively nonexpansive mapping was studied in [27, 29, 30].

Let be a bifunction from to , where denotes the set of real numbers. In this paper, we consider the following equilibrium problem. Find such that We use to denote the solution set of the equilibrium problem (2.3). That is,

Given a mapping , let

Then if and only if is a solution of the following variational inequality. Find such that

To study the equilibrium problem (2.8), we may assume that satisfies the following conditions:

(A1) ;

(A2) is monotone, that is, ;

(A3)

(A4) for each , is convex and weakly lower semicontinuous.

In this paper, we study the problem of approximating solutions of equilibrium problem (2.8) based on a hybrid projection iterative algorithm in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property. To prove our main results, we need the following lemmas.

Lemma 2.1. *Let be a strictly convex and uniformly smooth Banach space and a nonempty, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4). Let and . Then *(a)*(see [2]). There exists such that*(b)*(see [31]). Define a mapping by**
Then the following conclusions hold:*(1)* is single valued;*(2)* is a firmly nonexpansive-type mapping; that is, for all ,*(3)*;
*(4)* is closed and convex;*(5)* is relatively nonexpansive.*

Lemma 2.2 (see [29]). *Let be a reflexive, strictly convex, and smooth Banach space and a nonempty, closed, and convex subset of . Let , and . Then if and only if*

Lemma 2.3 (see [29]). *Let be a reflexive, strictly convex, and smooth Banach space and a nonempty, closed, and convex subset of , and . Then*

Lemma 2.4 (see [27]). *Let be a reflexive, strictly convex, and smooth Banach space. Then one has the following*

#### 3. Main Results

Theorem 3.1. *Let be a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property and a nonempty, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4) such that . Let be a sequence generated by the following manner:
**
where is a real number sequence in , where is some positive real number. Then the sequence converges strongly to .*

*Proof. *In view of Lemma 2.1, we see that is closed and convex. Next, we show that is closed and convex. It is not hard to see that is closed. Therefore, we only show that is convex. It is obvious that is convex. Suppose that is convex for some . Next, we show that is also convex for the same . Let and , where . It follows that
where . From the above two inequalities, we can get that
where . It follows that is closed and convex. This completes the proof that is closed, and convex.

Next, we show that . It is obvious that . Suppose that for some . For any , we see from Lemma 2.1 that
On the other hand, we obtain from (2.6) that
Combining (3.4) with (3.5), we arrive at
which implies that . This shows that . This completes the proof that .

Next, we show that is a convergent sequence and strongly converges to , where . Since , we see from Lemma 2.2 that
It follows from that
By virtue of Lemma 2.3, we obtain that
This implies that the sequence is bounded. It follows from (2.5) that the sequence is also bounded. Since the space is reflexive, we may assume that . Since is closed and convex, we see that . On the other hand, we see from the weakly lower semicontinuity of the norm that
which implies that as . Hence, as . In view of the Kadec-Klee property of , we see that as . Notice that . It follows that
Since and , we arrive at . This shows that is nondecreasing. It follows from the boundedness that exists. It follows that
By virtue of , we find that
It follows that
In view of (2.5), we see that
Since , we find that
It follows that
This implies that is bounded. Note that both and are reflexive. We may assume that . In view of the reflexivity of , we see that there exists an element such that . It follows that
Taking on the 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 from (3.17) that . Since is demicontinuous, we find that . This implies from (3.16) and the Kadec-Klee property of that . This in turn implies that . Since is uniformly norm-to-norm continuous on any bounded sets, we find that
Next, we show that . In view of Lemma 2.1, we find from that
It follows from condition (A2) and (3.20) that
In view of condition (A4), we obtain from (3.17) that
For and , define . It follows that , which yields that . It follows from conditions (A1) and (A4) that
That is,
Letting , we find from condition (A3) that , . This implies that . This shows that .

Finally, we prove that . Letting in (3.8), we see that
In view of Lemma 2.2, we can obtain that . This completes the proof.

In the framework of the Hilbert spaces, we have the following.

Corollary 3.2. *Let be a Hilbert space and a nonempty, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4) such that . Let be a sequence generated by the following manner:
**
where is a real number sequence in , where is some positive real number. Then the sequence converges strongly to .*

#### References

- K. Fan, “A minimax inequality and applications,” in
*Inequalities, III*, O. Shisha, Ed., pp. 103–113, Academic Press, New York, NY, USA, 1972. View at Zentralblatt MATH - E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”
*The Mathematics Student*, vol. 63, no. 1–4, pp. 123–145, 1994. View at Zentralblatt MATH - X. Qin, S. Y. Cho, and S. M. Kang, “Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications,”
*Journal of Computational and Applied Mathematics*, vol. 233, no. 2, pp. 231–240, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. J. Huang, J. Li, and H. B. Thompson, “Implicit vector equilibrium problems with applications,”
*Mathematical and Computer Modelling*, vol. 37, no. 12-13, pp. 1343–1356, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,”
*Mathematical and Computer Modelling*, vol. 48, no. 7-8, pp. 1033–1046, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. N. Fuentes, “Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space,”
*Journal of Mathematical Economics*, vol. 47, pp. 768–776, 2011. View at Publisher · View at Google Scholar - A. Koh, “An evolutionary algorithm based on Nash Dominance for equilibrium problems with equilibrium constraints,”
*Applied Soft Computing Journal*, vol. 12, no. 1, pp. 161–173, 2012. View at Publisher · View at Google Scholar - X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,”
*Journal of Computational and Applied Mathematics*, vol. 225, no. 1, pp. 20–30, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Petrot, K. Wattanawitoon, and P. Kumam, “A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces,”
*Nonlinear Analysis. Hybrid Systems*, vol. 4, no. 4, pp. 631–643, 2010. View at Publisher · View at Google Scholar - S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-
*ϕ*-asymptotically nonexpansive mappings,”*Abstract and Applied Analysis*, vol. 2010, Article ID 357120, 22 pages, 2010. View at Publisher · View at Google Scholar - S. Yang and W. Li, “Iterative solutions of a system of equilibrium problems in Hilbert spaces,”
*Advances in Fixed Point Theory*, vol. 1, no. 1, pp. 15–26, 2011. - X. Qin, S. S. Chang, and Y. J. Cho, “Iterative methods for generalized equilibrium problems and fixed point problems with applications,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 4, pp. 2963–2972, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Qin, S. Y. Cho, and S. M. Kang, “Iterative algorithms for variational inequality and equilibrium problems with applications,”
*Journal of Global Optimization*, vol. 48, no. 3, pp. 423–445, 2010. View at Publisher · View at Google Scholar - J. Ye and J. Huang, “Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces,”
*Journal of Mathematical and Computational Science*, vol. 1, pp. 1–18, 2011. - J. K. Kim, S. Y. Cho, and X. Qin, “Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 312602, 18 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. K. Kim, P. N. Anh, and Y. M. Nam, “Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems,”
*Journal of the Korean Mathematical Society*, vol. 49, no. 1, pp. 187–200, 2012. - X. Qin, S. Y. Cho, and S. M. Kang, “On hybrid projection methods for asymptotically quasi-
*ϕ*-nonexpansive mappings,”*Applied Mathematics and Computation*, vol. 215, no. 11, pp. 3874–3883, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. M. Kang, S. Y. Cho, and Z. Liu, “Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 827082, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,”
*Nonlinear Analysis*, vol. 67, no. 6, pp. 1958–1965, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, H. W. J. Lee, and C. K. Chan, “A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications,”
*Nonlinear Analysis*, vol. 73, no. 7, pp. 2260–2270, 2010. View at Publisher · View at Google Scholar - X. Qin, S. Y. Cho, and S. M. Kang, “Strong convergence of shrinking projection methods for quasi-
*ϕ*-nonexpansive mappings and equilibrium problems,”*Journal of Computational and Applied Mathematics*, vol. 234, no. 3, pp. 750–760, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Lv, “Generalized systems of variational inclusions involving (A,
*η*)-monotone mappings,”*Advanced in Fixed Point Theory*, vol. 1, no. 1, pp. 1–14, 2011. - J. K. Kim, “Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-
*ϕ*-nonexpansive mappings,”*Fixed Point Theory and Applications*, vol. 2011, 2011. View at Publisher · View at Google Scholar - J. K. Kim, S. Y. Cho, and X. Qin, “Some results on generalized equilibrium problems involving strictly pseudocontractive mappings,”
*Acta Mathematica Scientia*, vol. 31, no. 5, pp. 2041–2057, 2011. View at Publisher · View at Google Scholar - X. Qin, Y. J. Cho, and S. M. Kang, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications,”
*Nonlinear Analysis*, vol. 72, no. 1, pp. 99–112, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, “Convergence of a modified Halpern-type iteration algorithm for quasi-
*ϕ*-nonexpansive mappings,”*Applied Mathematics Letters*, vol. 22, no. 7, pp. 1051–1055, 2009. View at Publisher · View at Google Scholar - I. Cioranescu,
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems*, vol. 62 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. - W. Takahashi,
*Nonlinear Functional Analysis*, Yokohama Publishers, Yokohama, Japan, 2000. - Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*, vol. 178, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. View at Zentralblatt MATH - Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,”
*Optimization*, vol. 37, no. 4, pp. 323–339, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,”
*Nonlinear Analysis*, vol. 70, no. 1, pp. 45–57, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH