Abstract and Applied Analysis

Volume 2014 (2014), Article ID 468593, 8 pages

http://dx.doi.org/10.1155/2014/468593

Research Article

Algorithmic Approach to the Equilibrium Points and Fixed Points

Lijin Guo,¹ Shin Min Kang,² and Young Chel Kwun³

¹School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300387, China

²Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

³Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 10 March 2014; Accepted 25 March 2014; Published 22 April 2014

Academic Editor: Chong Li

Copyright © 2014 Lijin Guo 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.

Abstract.

The equilibrium and fixed point problems are considered. An iterative algorithm is presented. Convergence analysis of the algorithm is provided.

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . Let be a nonlinear operator and let be a bifunction. The equilibrium problem is formulated as finding such that

The solution set of (1) is denoted by EP. The problem (1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others. For related work, please see, for example, [1–17]. Next, we recall several interesting results where verifies the following usual conditions ()–() which will be used in the sequel:

for all ; is monotone; that is, for all ;for each , ;for each ,   is convex and lower semicontinuous.

Theorem 1. Let be a nonempty closed convex subset of . Let be a bifunction which satisfies conditions (C1)–(C4). Let be a nonexpansive mapping. Let be a contraction. For arbitrarily, let the sequence be generated by

where and satisfy , ,   , and . Then, the sequence converges strongly to provided .

Chuang et al. [18] considered an iteration process of Halpern’s type for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points for a nonexpansive mapping with perturbation in a Hilbert space and they proved a strong convergence theorem for such iterations.

Theorem 2. Let be a nonempty closed convex subset of . Let be a bifunction which satisfies conditions (C1)–(C4). Let be a nonexpansive mapping. Let be a sequence. For arbitrarily, let the sequence be generated by

where and satisfy ,   , and . Then, the sequence converges strongly to provided .

S. Takahashi and W. Takahashi [17] introduced the following iterative algorithm for finding an element of EP:

And they proved that the sequence converges strongly to .

Remark 3. Algorithm (3) is involved in a variant anchor and the parameters are also relaxed. In [17], the authors considered a general equilibrium problem.

In this paper, our main purpose is to introduce a new iteration process for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points for a nonexpansive mapping in a Hilbert space and then we prove a strong convergence theorem for such iterations. Our iterations are very different from (2)–(4). As a special case, we can find the minimum norm solution of .

2. Preliminaries

In the sequel, we assume is a real Hilbert space. Let be a nonempty closed convex set. Recall that a mapping is called -inverse-strongly monotone if there exists a positive real number such that

A mapping is said to be nonexpansive if

We use to denote the set of fixed points of .

The following lemmas are useful for the next section.

Lemma 4 (see [11]). Let be a nonempty closed convex subset of . Let be a bifunction which satisfies conditions (C1)–(C4). Let and . Then, there exists such that

Set . Then the following hold:

(i) is single-valued and is firmly nonexpansive; that is for any ,  ;(ii) is closed and convex and .

Lemma 5. Let , , , and be as in Lemma 4. Then the following holds:

for all and .

Lemma 6 (see [19]). Let be a nonempty closed convex subset of . Let the mapping be -inverse-strongly monotone and let be a constant. Then, we have

In particular, if , then is nonexpansive.

Lemma 7 (see [20]). Let be a closed convex subset of and let be a nonexpansive mapping with . Then, the mapping is demiclosed. That is, if is a sequence in such that weakly and strongly, then .

Lemma 8 (see [21]). Let and be two bounded sequences in . Let be a sequence in satisfying . Suppose that for all and . Then, .

Lemma 9 (see [22]). Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence such that

(1); (2) or .

Then .

3. Main Results

In this section, we will prove our main results.

Theorem 10. Let be a nonempty closed convex subset of and let be a bifunction satisfying conditions (C1)–(C4). Let be an -inverse-strongly monotone mapping and let be a nonexpansive mapping. Suppose that . Let be a sequence in . For arbitrarily, let the sequence be generated by

where is defined as that in Lemma 4 and ,    , and satisfy

(i) and ;(ii); (iii), where and ;(iv).

Then generated by (11) converges strongly to .

We divide our proofs into several conclusions.

Conclusion. The sequence is bounded.

Proof. Let . We have for all . Set for all . By Lemma 4, we know that is nonexpansive. By the convexity of , we derive

Since is -inverse-strongly monotone, we know from Lemma 6 that

It follows that

So, we have that

Note that . Without loss of generality, we can assume that for some .

By induction, we have

Therefore, is bounded.

Conclusion. and .

Proof. Putting for all , we have

It follows that

From Lemma 6, we know that is nonexpansive for all . Thus, we have that is nonexpansive for all due to the fact that . Then, we get

By Lemma 5, we have

From (18)–(20), we obtain

Then,

Therefore,

Since ,   and , we obtain

By Lemma 8, we get

Consequently, we obtain

From (11) and (14), we have

Then, we obtain

Since ,    , and , we have

Next, we show . By using the firm nonexpansivity of , we have

We note that

Thus,

that is,

It follows that

Hence,

Since ,  ,  , and , we deduce

This implies that

Conclusion. Consider

where .

Proof. Since is bounded, there exists a subsequence of such that weakly and

By (25) and (37), we deduce

Hence,

This together with Lemma 7 implies that .

Next we show that . Since for any , we have

From (C2), we have

Put for all and . Then, we have . So, from (43), we have

Since is -inverse-strongly monotone, is -Lipschitzian. By (37), we derive that . Further, from monotonicity of , we have . Letting in (45) and noting , we have

By , , and (45), we deduce

and hence

Letting , we have, for each ,

This implies . Therefore, we have . So,

Setting for all and taking in (29) to get , so, . Since , we have

Conclusion ().  

Proof. From (11), we have

It is clear that and . We can therefore apply Lemma 9 to conclude that . This completes the proof.

Corollary 11. Let be a nonempty closed convex subset of and let be a bifunction satisfying conditions (C1)–(C4). Let be an -inverse-strongly monotone mapping and let be a nonexpansive mapping. Suppose that . For arbitrarily, let the sequence be generated by

where is defined as that in Lemma 4 and ,  , and satisfy

(i) and ;(ii); (iii), where and .

Then generated by (52) converges strongly to which is the minimum norm element in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This study was supported by research funds from Dong-A University.

References

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.

L.-C. Ceng, Q. H. Ansari, and S. Schaible, “Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems,” Journal of Global Optimization. An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering, vol. 53, no. 1, pp. 69–96, 2012.

L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 92, p. 2012, article 19, 2012.

L. C. Ceng, H.-Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1341–1367, 2011.

L. C. Ceng, A. Petruşel, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.

L. C. Ceng, A. Petruşel, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.

L. C. Ceng, S. Schaible, and J. C. Yao, “Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 139, no. 2, pp. 403–418, 2008.

L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

L.-C. Ceng and J.-C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 729–741, 2008.

L.-C. Ceng and J.-C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal A: Theory and Methods, vol. 72, no. 3-4, pp. 1922–1937, 2010.

P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis. An International Journal, vol. 6, no. 1, pp. 117–136, 2005.

A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,” Journal of Nonlinear and Convex Analysis. An International Journal, vol. 9, no. 1, pp. 37–43, 2008.

A. Moudafi and M. Thera, “Proximal and dynamical approaches to equilibrium problems,” in Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201, Springer, 1999.

J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008.

J.-W. Peng and J.-C. Yao, “Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings,” Taiwanese Journal of Mathematics, vol. 13, no. 5, pp. 1537–1582, 2009.

J.-W. Peng and J.-C. Yao, “Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009.

S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal A: Theory and Methods, vol. 69, no. 3, pp. 1025–1033, 2008.

C.-S. Chuang, L.-J. Lin, and W. Takahashi, “Halpern's type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points,” Journal of Global Optimization. An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering, vol. 56, no. 4, pp. 1591–1601, 2013.

W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.

T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103–123, 2005.

H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.