/ / Article

Research Article | Open Access

Volume 2013 |Article ID 430409 | https://doi.org/10.1155/2013/430409

Kamonrat Sombut, Somyot Plubtieng, "Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems", Abstract and Applied Analysis, vol. 2013, Article ID 430409, 8 pages, 2013. https://doi.org/10.1155/2013/430409

# Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems

Accepted13 Dec 2012
Published26 Feb 2013

#### Abstract

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012) and many others.

#### 1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space . A mapping is said to be nonexpansive if for all . Denote the set of fixed points of by . On the other hand, a mapping is said to be quasi-nonexpansive if and for all and . If is nonexpansive and the set of fixed points of is nonempty, then is quasi-nonexpansive. Fixed point iterations process for nonexpansive mappings and quasi-nonexpansive mappings in Banach spaces including Mann and Ishikawa iterations process have been studied extensively by many authors to solve the nonlinear operator equations (see [14]).

Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that The set of solutions of (1) is denoted by . Numerous problems in physics, optimization, and economics reduce to find a solution of (1) in Hilbert spaces; see, for instance, Blum and Oettli [5], Flam and Antipin [6], and Moudafi [7]. Moreover, Flam and Antipin [6] introduced an iterative scheme of finding the best approximation to the solution of equilibrium problem, when is nonempty, and proved a strong convergence theorem (see also in [811]). Let be two-monotone bifunction and is a constant. Recently, Moudafi [12] considered the following of a system of equilibrium problem, denoting the set of solution of SEP by , for finding such that He also proved the weak convergence theorem of this problem (some related work can be found in [13, 14]).

The split feasibility problem (SFP) in Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction was first introduced by Censor and Elfving [15] (see, e.g. [16, 17]). It has been found that the SFP can also be used to model the intensity-modulated radiation therapy (see [18, 19]). In this work, the SFP is formulated as finding a point with the property where and are the nonempty closed convex subsets of the infinite-dimensional real Hilbert spaces and , respectively, and (i.e., is a bounded linear operator from to ). Very recently, there are related works which we can find in [16, 18, 2026] and the references therein.

A special case of the SFP is called the convex constrained linear inverse problem (see [27]), that is, the problem to finding an element such that In fact, it has been extensively investigated in the literature using the projected Landweber iterative method [27, 28]. Throughout this paper, we assume that the solution set of the is nonempty.

Motivated and inspired by the regularization method and extragradient method due to Ceng et al. [29], we introduce and analyze an extragradient method with regularization for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. Our results represent the improvement, extension, and development of the corresponding results in [14, 29].

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto .

Some important properties of projections are gathered in the following proposition.

Proposition 1 (see [29]). For given and : (i); (ii).

Definition 2 (see [30, 31]). Let be a nonlinear operator with domain and range , and let and be given constants. The operator is called(a)monotone if (b)-strongly  monotone if (c)-inverse  strongly  monotone (ism) if

We can easily see that if is nonexpansive, then is monotone. It is also easy to see that a projection is a 1-ism.

Definition 3 (see [29]). A mapping is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping, that is, where and is nonexpansive. More precisely, when (9) holds, we say that is . It is easly to see that if is an averaged mapping, then is nonexpansive.

Proposition 4 (see [20]). Let be a given mapping. Then consider the following.(i) is nonexpansive if and only if the complement is -ism. (ii) is averaged if and only if the complement is -ism for some . Indeed, for is -averaged if and only if is -ism. (iii)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and , where , then the composite is -averaged, where .

In this paper, we use an equilibrium bifunction for solving the equilibrium problems, let us assume that satisfies the following conditions: (A1); (A2) is monotone, that is, ;(A3)for each is weakly upper semicontinuous; (A4)for each is convex; semicontinuous.

Lemma 5 (see [6]). Assume that satisfies (A1)–(A4). For and , define a mapping as follows: for all . Then, the following hold: (i) is single-valued; (ii) is firmly nonexpansive, that is, for any ;(iii); (iv) is closed and convex.

Lemma 6 (see [32]). Let be a closed convex subset of a real Hilbert space . Let and be two mappings from satisfying (A1)–(A4) and let and are defined as in Lemma 5 associated to and , respectively. For given is a solution of problem (2) if and only if is a fixed point of the mapping defined by where .

Lemma 7 (see [33]). Let and be sequences of nonnegative real numbers satisfying the inequality If and , then exists. If, in addition, has a subsequence which converges to zero, then .

#### 3. Weak Convergence Theorem

In this section, we prove a weak convergence theorem by an extragradient methods for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems in Hilbert spaces. The function is a continuous differentiable function with the minimization problem given by In 2010, Xu [17] considered the following Tikhonov regularized problem: where is the regularization parameter. The gradient given by is -Lipschitz continuous and -strongly monotone (see [29] for the details).

Lemma 8 (see [17, 29]). The following hold: (i) for any , where and denote the set of fixed points of and the solution set of VIP; (ii) is -averaged for each , where .

Theorem 9. Let be a nonempty closed convex subset in a real Hilbert space . Let , and be two bifunctions from satisfying . Let be a quasi-nonexpansive mapping of into itself such that be demiclosed at zero, that is, if and , then , with . Let , , , and be the sequence in generated by the following extragradient algorithm: where , for some and for some . Then, the sequences and converge weakly to an element .

Proof. By Lemma 8, we have that is -averaged for each , where . Hence, by Proposition , is nonexpansive. From and , we have . Without loss of generality, we assume that . Hence, for each is -averaged with This implies that is nonexpansive for all .
Next, we show that the sequence is bounded. Indeed, take a fixed arbitrarily. Let and be defined as in Lemma 5 associated to and , respectively. Thus, we get and . Put and . From (29), we have This implies that . Thus, we obtain . Put for each . Then, by Proposition 1(ii), we have Hence, by Proposition 1(i), we have So, we have Then, from the last inequality we conclude that where and . Since and for some , we conclude that and . Therefore, by Lemma 7, we note that exists for each and hence the sequences , and are bounded. From (22), we also obtain where and . Since exists and , it follows that Similarly, from inequality (22), we have where and . Since exists and , we obtain Moreover, we note that From (26) and , it is implied that Note that . This together with (24) and (28) implies that . Also, from , it follows that . Since is a Lipschitz condition, where is the adjoint of , we have . Since is a bounded sequence, there exists a subsequence of that converges weakly to some .
Next, we show that . Since and , it is known that and . Let be a set value mappings defined by where . Hence, by [34], is maximal monotone and if and only if . Let . Then we have and hence . So, we obtain . On the other hand, from and , we have and hence Therefore, from and , we get By taking , we obtain . Since is maximal monotone, it follows that and hence . Therefore, by Lemma 8, .
Next, we show that . Since and , it follows by the demiclosed principle that . Hence, we have .
Next, we show that . Let be a mapping which is defined as in Lemma 6, thus we have and hence By taking , we have . From and , we obtain . According to demiclosedness and Lemma 6, we have . Therefore, we have . Let be another subsequence of such that . We show that , suppose that . Since exists for all , it follows by the Opial's condition that It is a contradiction. Thus, we have and so . Further, from , it follows that and hence , . This completes the proof.

Theorem 9 extends the extragradient method according to Nadezhkina and Takahashi [35].

Corollary 10. Let be a nonempty closed convex subset in a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 5 associated to and , respectively. Let be a quasi-nonexpansive mapping of into itself such that . Let , , , and be the sequence in generated by the following extragradient algorithm: where , for some and for some . Then, the sequences and converge weakly to an element .

Corollary 11. Let be a nonempty closed convex subset in a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 5 associated to and , respectively. Let be a quasi-nonexpansive mapping of into itself such that . Let , , and be the sequence in generated by the following extragradient algorithm: where , for some and for some . Then, the sequences and converge weakly to an element .

Proof. Setting in Theorem 9, we obtain the desired result.

Corollary 12. Let be a nonempty closed convex subset in a real Hilbert space . Let be a quasi-nonexpansive mapping of into itself such that . Let , , and be the sequence in generated by the following extragradient algorithm: where , for some and for some . Then, the sequences and converge weakly to an element .

Proof. Setting in Theorem 9, we obtain the desired result.

#### Acknowledgments

The authors would like to thank The Commission on Higher Education for financial support. Moreover, K. Sombut is also supported by The Strategic Scholarships Fellowships Frontier Research Networks under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.

#### References

1. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
2. W. G. Dotson Jr., “On the Mann iterative process,” Transactions of the American Mathematical Society, vol. 149, pp. 65–73, 1970.
3. S. Plubtieng, R. Wangkeeree, and R. Punpaeng, “On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 1018–1029, 2006.
4. S. Plubtieng and R. Wangkeeree, “Strong convergence theorems for multi-step Noor iterations with errors in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 10–23, 2006.
5. 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.
6. D. S. Flam and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.
7. A. Moudafi, “Second-order differential proximal methods for equilibrium problems,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 18, 2003.
8. 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.
9. W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008.
10. S. Plubtieng and W. Sriprad, “A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 567147, 20 pages, 2009.
11. S. Plubtieng and T. Thammathiwat, “A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities,” Journal of Global Optimization, vol. 46, no. 3, pp. 447–464, 2010.
12. A. Moudafi, “From alternating minimization algorithms and systems of variational inequalities to equilibrium problems,” Communications on Applied Nonlinear Analysis, vol. 16, no. 3, pp. 31–35, 2012.
13. Y. Yao, Y.-C. Liou, and J.-C. Yao, “New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems,” Optimization, vol. 60, no. 3, pp. 395–412, 2011. View at: Publisher Site | Google Scholar | MathSciNet
14. S. Plubtieng and K. Sombut, “Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space,” Journal of Inequalities and Applications, vol. 2010, Article ID 246237, 12 pages, 2010.
15. Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2–4, pp. 221–239, 1994.
16. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at: Publisher Site | Google Scholar | MathSciNet
17. H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, article 105018, 2010.
18. Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005.
19. Y. Censor, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1244–1256, 2007. View at: Publisher Site | Google Scholar | MathSciNet
20. C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
21. B. Qu and N. Xiu, “A note on the $CQ$ algorithm for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1655–1665, 2005.
22. H.-K. Xu, “A variable KrasnoselskiiMann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006. View at: Publisher Site | Google Scholar | MathSciNet
23. Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261–1266, 2004.
24. J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1791–1799, 2005.
25. Y. Yao, R. Chen, G. Marino, and Y. C. Liou, “Applications of fixed-point and optimization methods to the multiple-set split feasibility problem,” Journal of Applied Mathematics, vol. 2012, Article ID 927530, 21 pages, 2012.
26. X. Yu, N. Shahzad, and Y. Yao, “Implicit and explicit algorithms for solving the split feasibility problem,” Optimization Letters, vol. 6, no. 7, pp. 1447–1462, 2012. View at: Publisher Site | Google Scholar | MathSciNet
27. B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 13, no. 5-6, pp. 413–429, 1992.
28. L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” American Journal of Mathematics, vol. 73, pp. 615–624, 1951.
29. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012. View at: Publisher Site | Google Scholar | MathSciNet
30. D. P. Bertsekas and E. M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem,” Mathematical Programming Study, no. 17, pp. 139–159, 1982.
31. D. Han and H. K. Lo, “Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities,” European Journal of Operational Research, vol. 159, no. 3, pp. 529–544, 2004.
32. 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.
33. M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88, 2002.
34. R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
35. N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006.

Copyright © 2013 Kamonrat Sombut and Somyot Plubtieng. 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.