International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 491357 | 13 pages | https://doi.org/10.1155/2010/491357

Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

Academic Editor: Petru Jebelean
Received21 Aug 2009
Revised01 Dec 2009
Accepted13 Jan 2010
Published22 Feb 2010

Abstract

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).

1. Introduction

Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction from to , where is the set of real numbers. The equilibrium problem for is to find such that

The set of solutions of (1.1) is denoted by . Given a mapping , let for all . Then, if and only if for all . Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); for more details, see [1, 2].

Recall that a self-mapping of a closed convex subset of is nonexpansive [3] if there holds that We denote the set of fixed points of by . There are some methods for approximation of fixed points of a nonexpansive mapping. In 2000, Moudafi [4] introduced the viscosity approximation method for nonexpansive mappings (see [5] for further developments in both Hilbert and Banach spaces). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1, 2, 6, 7]. Recently, Combettes and Hirstoaga [6] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. S. Takahashi and W. Takahashi [7] introduced a Mann iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem.

On the other hand, Ishikawa [8] introduced the following iterative process defined recursively by

where the initial guess is taking in arbitrarily, and are sequences in the interval .

In this paper, motivated by the ideas in [48], we introduce an Ishikawa iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space.

Starting with an arbitrary , define sequences , and by

where and .

We will prove in Section 3 that if the sequences , and of parameters satisfy appropriate conditions, then the sequences , and generated by (1.4) converge strongly to . The results in this paper extend and generalize S. Takahashi and W. Takahashi's results [7].

2. Preliminaries

Let be a real Hilbert space with inner product , and norm and let be a nonempty closed convex subset of . implies that converges strongly to and means that converges weakly to . In a real Hilbert space , we have

for all and ; see [9].

For any , there exists a unique nearest point in , denoted by , such that for all . Such a is called the metric projection of onto . It is also known that is equivalent to for all .

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:

(A1) for all ;(A2) is monotone, that is, for any ;(A3)for each , (A4)for each is convex and lower semicontinuous.

We recall some lemmas needed later.

Lemma 2.1 (see [2]). Let be a nonempty closed convex subset of and let be a bifunction from to satisfying (A1)–(A4). Let and . Then, there exists such that

Lemma 2.2 (see [5]). Let be a nonempty closed convex subset of , and let be a bifunction from to satisfying (A1)–(A4). For and define a mapping as follows: for all . Then, the following statements hold: (1) is single-valued;(2) is firmly nonexpansive, that is, for any (3)(4) is closed and convex.

Lemma 2.3 (see [10]). Let be a sequence of nonnegative real numbers such that where is a sequence of real numbers and is a sequence in such that(i)(ii) or .Then,

3. Strong Convergence Theorem

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space.

Theorem 3.1. Let be a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A4) and let be a nonexpansive mapping of into such that Let be a contraction of into itself and let , and be sequences generated by and (1.4). If and satisfy the following conditions: then, , and converge strongly to where

Proof. Let Then is a contraction of into itself. In fact, there exists such that for all . So, we have that for all Since is complete, there exists a unique element such that . Such a is an element of .
Let . Then from we have
for all Put It is obvious that Suppose Then, we have On the other hand Putting (3.5) into (3.4), we have So, we have that for any . And hence is bounded. We also obtain that and are bounded. Next, we show that In fact, and hence where and .
On the other hand, from and , we have

Putting in (3.9) and in (3.10), we have
So, from the monotonicity of , we get and hence Without loss of generality, let us assume that there exists a real number such that for all Then, we have and hence where So from (3.8), we have Using Lemma in [10], we obtain From (3.15) and we have It follows from (3.7) that Since we have From we have For we have and hence Therefore, from the convexity of , we have and hence So, we have Without loss of generality, let us assume that there exists two real numbers and such that for all Hence, It follows that We also have It follows that Hence, Since we also have . Next, we show that where . To show this inequality, we choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . From we obtain Let us show By we have
From (A2), we also have
and hence, Since and , from (A4), we have For with and , let Since and , we obtain and hence . So we have Dividing by , we get Letting and from (A3), we get for all and hence . We shall show that Assume Since and , from the Opial theorem [11] we have This is a contradiction. So, we get . Therefore, Since , we have From , we have It follows that Hence From , we know that there exists a positive integer , such that for all . Then Putting above inequality into (3.44), we get where , and .
It follows from Lemma 2.3 that
It follows from and (3.42) that and .

By Theorem 3.1, we can obtain the following new result.

Corollary 3.2. Let be a nonempty closed convex subset of . Let be a nonexpansive mapping of into such that Let be a contraction of into itself and let and be sequences generated by and If satisfy the following conditions: then, and converge strongly to where

Proof. Put for all and for all in Theorem 3.1. Then, we get . So from Theorem 3.1, the sequences and converge strongly to , where .

Remark 3.3. Theorem 3.1 and Corollary 3.2, respectively, extend and generalize Theorem and Corollary in [7] from the Mann iterative form to the Ishikawa iterative form.

Acknowledgments

The authors would like to express their thanks to the referees for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05).

References

  1. S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. 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: Google Scholar | Zentralblatt MATH | MathSciNet
  3. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at: MathSciNet
  4. A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  7. S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009. View at: MathSciNet
  10. H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2010 Jian-Wen Peng and Yan Wang. 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.

460 Views | 294 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder