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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 178053, 9 pages
http://dx.doi.org/10.1155/2013/178053
Research Article

A New Iterative Method for Equilibrium Problems and Fixed Point Problems

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran

Received 31 October 2013; Accepted 15 December 2013

Academic Editor: Ljubomir B. Ćirić

Copyright © 2013 Abdul Latif and Mohammad Eslamian. 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

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

“Dedicated to Professor Miodrag Mateljevi’c on the occasion of his 65th birthday”

1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of , and let be a nearest point projection of into ; that is, for , is the unique point in with the property . It is well known that iff for all .

Let be a bifunction from into , such that for all . Consider the Fan inequality [1]: find a point such that where is convex and subdifferentiable on for every . The set of solutions of this problem is denoted by . In fact, the Fan inequality can be formulated as an equilibrium problem. Such problems arise frequently in mathematics, physics, engineering, game theory, transportation, economics, and network. Due to importance of the solutions of such problems, many researchers are working in this area and studying the existence of the solutions of such problems; for example, see, [24]. Further, if for every , where is a mapping from into , then the Fan inequality problem (equilibrium problem) becomes the classical variational inequality problem which is formulated as finding a point such that

Variational inequalities were introduced and studied by Stampacchia [5]. It is well known that this area covers many branches of mathematics, such as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance; see [611].

Here we recall some useful notions.

A mapping is said to be -Lipschitz on if there exists a constant such that for each , In particular, if , then is called a contraction on ; if , then is called a nonexpansive mapping on . The set of fixed points of is denoted by .

A family of mappings on a closed convex subset of a Hilbert space is called a nonexpansive semigroup if it satisfies the following: (i) for all ; (ii) for all ; (iii) for all , is continuous; (iv) for all and .

We use to denote the common fixed point set of the semigroup ; that is, . It is well known that is closed and convex [12]. A nonexpansive semigroup on is said to be uniformly asymptotically regular (in short, u.a.r.) on if for all and any bounded subset of , For each , define . Then Provided that is closed bounded convex subset of . It is known that the set is a u.a.r. nonexpansive semigroup; see [13]. The other examples of u.a.r. operator semigroup can be found in [14].

A bifunction is said to be (i) strongly monotone on with if , ; (ii) monotone on if , ; (iii) pseudomonotone on if , ; (iv) Lipschitz-type continuous on with constants and if , .

Note that if is -Lipschitz on , then for each , the function is a Lipschitz-type continuous with constants .

An operator on is called strongly positive if there is a constant such that

Recently, iterative methods for nonexpansive mappings have been applied to solve convex minimization problems. In [15], Xu defined an iterative sequence in which converges strongly to the unique solution of the minimization problem under some suitable conditions. A well-known typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping on a real Hilbert space : where is a given point in and is strongly positive operator.

For solving the variational inequality problem, Marino and Xu [16] introduced the following general iterative process for nonexpansive mapping based on the viscosity approximation method (see [17]): where is strongly positive bounded linear operator on , is contraction on , and . They proved that, under some appropriate conditions on the parameters, the sequence generated by (8) converges strongly to the unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., , ).

Iterative process for approximating common fixed points of a nonexpansive semigroup has been investigated by various authors (see [13, 14, 1821]). Recently, Li et al. [19] introduced the following iterative procedure for the approximation of common fixed points of a nonexpansive semigroup on a closed convex subset of a Hilbert space : where is a strongly positive bounded linear operator on and is a contraction on . Imposing some appropriate conditions on the parameters, they proved that the iterative sequence generated by (11) converges strongly to the unique solution of the variational inequality , .

For obtaining a common element of and the set of fixed points of a nonexpansive mapping , S. Takahashi and W. Takahashi [9] first introduced an iterative scheme by the viscosity approximation method. They proved that under certain conditions the iterative sequences converge strongly to .

During last few years, iterative algorithms for finding a common element of the set of solutions of Fan inequality and the set of fixed points of nonexpansive mappings in a real Hilbert space have been studied by many authors (see, e.g., [2, 4, 2228]). Recently, Anh [22] studied the existence of a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of Fan inequality for monotone and Lipschitz-type continuous bifunctions. He introduced the following new iterative process: where , is nonempty, closed convex subset of a real Hilbert space , is monotone, continuous, and Lipschitz-type continuous bifunction, is self-contraction on with constant , and is self nonexpansive mapping on . He proved that, under some appropriate conditions over positive sequences , , , and , the sequences , , and converge strongly to which is a solution of the variational inequality , .

In this paper, we introduce a new iterative scheme based on the viscosity method and study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative scheme for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh [22], Cianciaruso et al. [18], and many others.

2. Preliminaries

In this section we collect some lemmas which are crucial for the proofs of our results.

Let be a sequence in and . In the sequel, denotes that weakly converges to and denotes that weakly converges to .

Lemma 1. Let be a real Hilbert space. Then the following inequality holds:

Lemma 2 (see [15]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i), (ii) or .Then .

Lemma 3 (see [16]). Let be a strongly positive linear bounded self-adjoint operator on with coefficient and . Then .

Lemma 4 (see [29]). Let be a Hilbert space and , . Then for any given with and for any positive integer with ,

Lemma 5 (see [30]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers : In fact

Lemma 6 (see [2]). Let be a nonempty closed convex subset of a real Hilbert space and let be a pseudomonotone and Lipschitz-type continuous bifunction with constants . For each , let be convex and subdifferentiable on . Let and let , , and be sequences generated by Then for each ,

3. Main Results

In this section, we prove the main strong convergence result which solves the problem of finding a common element of three sets , , and for finite family of monotone, continuous, and Lipschitz-type continuous bifunctions in a real Hilbert space .

Theorem 7. Let be a nonempty closed convex subset of a real Hilbert space and let be a finite family of monotone, continuous, and Lipschitz-type continuous bifunctions with constants and . Let and be two u.a.r. nonexpansive self-mapping semigroups on such that . Assume that is a -contraction self-mapping of and is a strongly positive bounded linear self-adjoint operator on with coefficient and . Let and let , , and be sequences generated by where and , , , , , , and satisfy the following conditions: (i), (ii), , and ,(iii), where ,(iv), , , , and for each .Then, the sequence converges strongly to which solves the variational inequality:

Proof. Since , , and are closed and convex, is well-defined. We claim that is a contraction from into itself. Indeed, for each , we have
Therefore, by the Banach contraction principle, there exists a unique element such that . We show that is bounded. Since , we can assume, with no loss of generality, that , for all . By Lemma 6, for each , we have
This implies that It follows from Lemma 3 that This implies that is bounded and so are , , , and . Next, we show that . Indeed, by Lemma 6, for each , we have Applying Lemma 4 and inequality (26) for we have that We now compute Therefore, In order to prove that as , we consider the following two cases.

Case 1. Assume that is a monotone sequence. In other words, for large enough , is either nondecreasing or nonincreasing. Since is bounded, it is convergent. Since and and are bounded, from (29), we have and by assumption we get By similar argument we can obtain that Further, for all and , we see that Since is u.a.r. nonexpansive semigroup and , we have Similarly, for all and , we obtain that Next, we show that 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 . Consider Thus, we have By the Opial property of the Hilbert space we obtain for all . Similarly we have that for all . This implies that . Now we show that . For each , since is convex on for each , we see that if and only if where is the (outward) normal cone of at . This follows that where and . By the definition of the normal cone we have Since is subdifferentiable on , by the well-known Moreau-Rockafellar theorem [31] (also see [6]), for , we have Combining this with (43), we have In particular, we have Since , it follows from (32) that . And thus we have This implies that and hence . Since and , we have From Lemma 1, it follows that This implies that where It is easy to see that , , and . Hence, by Lemma 2 the sequence converges strongly to . From (32) we have that and converge strongly to .

Case 2. Assume that is not a monotone sequence. Then, we can define an integer sequence for all (for some large enough ) by Clearly, is a nondecreasing sequence such that as , and for all , From (33) we obtain that Following an argument similar to that in Case 1 we have where , , and . Hence, by Lemma 2, we obtain and . Now Lemma 5 implies that Therefore, converges strongly to . This completes the proof.

4. Application

In this section, we consider a particular Fan inequality corresponding to the function defined by the following: for every where . Then, we obtain the classical variational inequality as follows. The set of solutions of this problem is denoted by . In that particular case, the solution of the minimization problem can be expressed as Let be -Lipschitz continuous on . Then Therefore, hence, satisfies Lipschitz-type continuous condition with .

Using Theorem 7 we obtain the following convergence theorem.

Theorem 8. Let be a nonempty closed convex subset of a real Hilbert space and let    be functions such that, for each , is monotone and -Lipschitz continuous on . Let and be two u.a.r. nonexpansive self-mapping semigroups on such that . Assume that is a -contraction of into itself and is a strongly positive bounded linear self-adjoint operator on with coefficient and . Let and let , , and be sequences generated by where and , , , , , and satisfy the following conditions: (i), (ii), , and ,(iii),(iv), , , , and for each .Then, the sequence converges strongly to which solves the variational inequality

In [32], Baillon proved a strong mean convergence theorem for nonexpansive mappings, and it was generalized in [33]. It follows from the above proof that Theorems 7 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorems for nonexpansive mappings in a Hilbert space.

Theorem 9. Let be a nonempty closed convex subset of a real Hilbert space and let    be functions such that for each , is monotone and -Lipschitz continuous on . Let and be two nonexpansive mappings on such that . Assume that is a -contraction of into itself and is a strongly positive bounded linear self-adjoint operator on with coefficient and . Let , , and be sequences generated by and by where and , , , , , and satisfy the following conditions:(i), , and ,(ii), where ,(iii), , , , and for each .Then, the sequence converges strongly to which solves the variational inequality

Conflict of Interests

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

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, for technical and financial support.

References

  1. K. Fan, “A minimax inequality and applications,” in Inequality III, pp. 103–113, Academic Press, New York, NY, USA, 1972. View at Zentralblatt MATH · View at MathSciNet
  2. P. N. Anh, “A hybrid extragradient method extended to fixed point problems and equilibrium problems,” Optimization, vol. 62, no. 2, pp. 271–283, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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 · View at MathSciNet
  4. L.-C. Ceng, N. Hadjisavvas, and N.-C. Wong, “Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems,” Journal of Global Optimization, vol. 46, no. 4, pp. 635–646, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH · View at MathSciNet
  6. P. Daniele, F. Giannessi, and A. Maugeri, Equilibrium Problems and Variational Models, vol. 68, Kluwer Academic, Norwell, Mass, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J.-W. Peng and J.-C. Yao, “Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems,” Optimization Methods & Software, vol. 25, no. 4–6, pp. 677–698, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. R. Sahu, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for semigroups of asymptotically nonexpansive mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 202095, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Wang and B. Guo, “New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2620–2630, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Yao, Y.-C. Liou, and Y.-J. Wu, “An extragradient method for mixed equilibrium problems and fixed point problems,” Fixed Point Theory and Applications, vol. 2009, Article ID 632819, 15 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. Aleyner and Y. Censor, “Best approximation to common fixed points of a semigroup of nonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 137–151, 2005. View at Zentralblatt MATH · View at MathSciNet
  15. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. F. Cianciaruso, G. Marino, and L. Muglia, “Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 146, no. 2, pp. 491–509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. R. Sahu, N. C. Wong, and J. C. Yao, “A unified hybrid iterative method for solving variational inequalities involving generalized pseudocontractive mappings,” SIAM Journal on Control and Optimization, vol. 50, no. 4, pp. 2335–2354, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. N. Anh, “Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 303–320, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators, and relatively nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 31, no. 7-9, pp. 763–797, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems,” Journal of Global Optimization, vol. 43, no. 4, pp. 487–502, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112–1123, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. 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. 2012, article 92, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. M. Eslamian, “Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems,” Optimization Letters, vol. 7, no. 3, pp. 547–557, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. M. Eslamian, “Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups,” RACSAM, vol. 107, pp. 299–307, 2013. View at Publisher · View at Google Scholar
  29. M. Eslamian and A. Abkar, “One-step iterative process for a finite family of multivalued mappings,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 105–111, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. J.-B. Baillon, “Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert,” Comptes Rendus de l'Académie des Sciences, vol. 280, no. 22, pp. A1511–A1514, 1975. View at Zentralblatt MATH · View at MathSciNet
  33. W. Kaczor, T. Kuczumow, and S. Reich, “A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense,” Journal of Mathematical Analysis and Applications, vol. 246, no. 1, pp. 1–27, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet