## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2013 (2013), 8 pages.

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

Kamonrat Sombut and Somyot Plubtieng

**Full-text: Access denied (no subscription detected)** We're sorry, but we are unable to provide you with the
full text of this article because we are not able to identify you as
a subscriber. If you have a personal subscription to this journal,
then please login. If you are already logged in, then you may need
to update your profile to register your subscription. Read more about accessing full-text

#### 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.

#### Article information

**Source**

Abstr. Appl. Anal. Volume 2013 (2013), 8 pages.

**Dates**

First available in Project Euclid: 18 April 2013

**Permanent link to this document**

http://projecteuclid.org/euclid.aaa/1366306763

**Digital Object Identifier**

doi:10.1155/2013/430409

**Mathematical Reviews number (MathSciNet)**

MR3035187

#### Citation

Sombut, Kamonrat; Plubtieng, Somyot. Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems. Abstr. Appl. Anal. 2013 (2013), 1--8. doi:10.1155/2013/430409. http://projecteuclid.org/euclid.aaa/1366306763.

#### References

- W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953.Mathematical Reviews (MathSciNet): MR0054846

Zentralblatt MATH: 0050.11603

Digital Object Identifier: doi: 10.1090/S0002-9939-1953-0054846-3 - W. G. Dotson Jr., “On the Mann iterative process,”
*Transactions of the American Mathematical Society*, vol. 149, pp. 65–73, 1970.Mathematical Reviews (MathSciNet): MR0257828

Zentralblatt MATH: 0203.14801

Digital Object Identifier: doi: 10.1090/S0002-9947-1970-0257828-6 - 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.Mathematical Reviews (MathSciNet): MR2250633

Zentralblatt MATH: 1097.47057

Digital Object Identifier: doi: 10.1016/j.jmaa.2005.09.078 - 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.Mathematical Reviews (MathSciNet): MR2236536

Zentralblatt MATH: 1095.47042

Digital Object Identifier: doi: 10.1016/j.jmaa.2005.08.029 - 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. - D. S. Flam and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,”
*Mathematical Programming*, vol. 78, no. 1, pp. 29–41, 1997.Mathematical Reviews (MathSciNet): MR1454787

Zentralblatt MATH: 0890.90150

Digital Object Identifier: doi: 10.1016/S0025-5610(96)00071-8 - 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. - 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.Mathematical Reviews (MathSciNet): MR2306020

Zentralblatt MATH: 1122.47056

Digital Object Identifier: doi: 10.1016/j.jmaa.2006.08.036 - 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.Mathematical Reviews (MathSciNet): MR2395311

Zentralblatt MATH: 1187.47054

Digital Object Identifier: doi: 10.1155/2008/528476 - 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. - 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.Mathematical Reviews (MathSciNet): MR2593241

Zentralblatt MATH: 1203.47064

Digital Object Identifier: doi: 10.1007/s10898-009-9448-5 - 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. - 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.Mathematical Reviews (MathSciNet): MR2780926

Digital Object Identifier: doi: 10.1080/02331930903196941 - 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. - 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.Mathematical Reviews (MathSciNet): MR1309222

Zentralblatt MATH: 0828.65065

Digital Object Identifier: doi: 10.1007/BF02142692 - C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,”
*Inverse Problems*, vol. 18, no. 2, pp. 441–453, 2002.Mathematical Reviews (MathSciNet): MR1910248

Digital Object Identifier: doi: 10.1088/0266-5611/18/2/310 - H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,”
*Inverse Problems*, vol. 26, no. 10, article 105018, 2010.Mathematical Reviews (MathSciNet): MR2719779

Zentralblatt MATH: 1213.65085

Digital Object Identifier: doi: 10.1088/0266-5611/26/10/105018 - 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.Mathematical Reviews (MathSciNet): MR2183668

Zentralblatt MATH: 1089.65046

Digital Object Identifier: doi: 10.1088/0266-5611/21/6/017 - 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.Mathematical Reviews (MathSciNet): MR2280001

Digital Object Identifier: doi: 10.1016/j.jmaa.2006.05.010 - 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.Mathematical Reviews (MathSciNet): MR2044608

Zentralblatt MATH: 1051.65067

Digital Object Identifier: doi: 10.1088/0266-5611/20/1/006 - 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.Mathematical Reviews (MathSciNet): MR2173415

Zentralblatt MATH: 1080.65033

Digital Object Identifier: doi: 10.1088/0266-5611/21/5/009 - H.-K. Xu, “A variable KrasnoselskiiMann algorithm and the multiple-set split feasibility problem,”
*Inverse Problems*, vol. 22, no. 6, pp. 2021–2034, 2006.Mathematical Reviews (MathSciNet): MR2277527

Digital Object Identifier: doi: 10.1088/0266-5611/22/6/007 - Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,”
*Inverse Problems*, vol. 20, no. 4, pp. 1261–1266, 2004.Mathematical Reviews (MathSciNet): MR2087989

Zentralblatt MATH: 1066.65047

Digital Object Identifier: doi: 10.1088/0266-5611/20/4/014 - J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,”
*Inverse Problems*, vol. 21, no. 5, pp. 1791–1799, 2005.Mathematical Reviews (MathSciNet): MR2173423

Zentralblatt MATH: 1080.65035

Digital Object Identifier: doi: 10.1088/0266-5611/21/5/017 - 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. - 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.Mathematical Reviews (MathSciNet): MR2980555

Digital Object Identifier: doi: 10.1007/s11590-011-0340-0 - 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.Mathematical Reviews (MathSciNet): MR1187903

Zentralblatt MATH: 0769.65026

Digital Object Identifier: doi: 10.1080/01630569208816489 - L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,”
*American Journal of Mathematics*, vol. 73, pp. 615–624, 1951.Mathematical Reviews (MathSciNet): MR0043348

Zentralblatt MATH: 0043.10602

Digital Object Identifier: doi: 10.2307/2372313 - 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.Mathematical Reviews (MathSciNet): MR2948609 - 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.Mathematical Reviews (MathSciNet): MR654697

Zentralblatt MATH: 0478.90071

Digital Object Identifier: doi: 10.1007/BFb0120965 - 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.Mathematical Reviews (MathSciNet): MR2078853

Zentralblatt MATH: 1065.90015

Digital Object Identifier: doi: 10.1016/S0377-2217(03)00423-5 - 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. - 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. - R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”
*Transactions of the American Mathematical Society*, vol. 149, pp. 75–88, 1970.Mathematical Reviews (MathSciNet): MR0282272

Zentralblatt MATH: 0222.47017

Digital Object Identifier: doi: 10.1090/S0002-9947-1970-0282272-5 - 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.Mathematical Reviews (MathSciNet): MR2201895

Zentralblatt MATH: 1130.90055

Digital Object Identifier: doi: 10.1007/s10957-005-7564-z

### More like this

- Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

Wangkeeree, Rabian and Boonkong, Uraiwan, Abstract and Applied Analysis, 2012 - Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

Kong, Zhao-Rong, Ceng, Lu-Chuan, and Wen, Ching-Feng, Abstract and Applied Analysis, 2012 - Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium
Problems, a General System of Variational Inequalities and Fixed-Point Problems of a
Countable Family of Nonexpansive Mappings

Dong, Qiao-Li, Guo, Yan-Ni, and Su, Fang, Journal of Applied Mathematics, 2012

- Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

Wangkeeree, Rabian and Boonkong, Uraiwan, Abstract and Applied Analysis, 2012 - Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

Kong, Zhao-Rong, Ceng, Lu-Chuan, and Wen, Ching-Feng, Abstract and Applied Analysis, 2012 - Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium
Problems, a General System of Variational Inequalities and Fixed-Point Problems of a
Countable Family of Nonexpansive Mappings

Dong, Qiao-Li, Guo, Yan-Ni, and Su, Fang, Journal of Applied Mathematics, 2012 - Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems,
Variational Inequality Problems and Common Fixed Point Problems, of
Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach
Spaces

Zhao, Jing, Journal of Applied Mathematics, 2012 - A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a
Fixed Point for a Totally Quasi-
ϕ
-Asymptotically Nonexpansive Mapping in a Banach Space

Kanjanasamranwong, Preedaporn, Kumam, Poom, and Saewan, Siwaporn, Journal of Applied Mathematics, 2012 - A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems,
Variational Inequality Problems, and Fixed Point Problems of Infinite Family
of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

Che, Haitao and Pan, Xintian, Abstract and Applied Analysis, 2013 - Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi-
ϕ
-Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

Tan, J. F. and Chang, S. S., Abstract and Applied Analysis, 2012 - Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators

Dang, Yazheng and Gao, Yan, Abstract and Applied Analysis, 2013 - A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

Sunthrayuth, Pongsakorn and Kumam, Poom, Abstract and Applied Analysis, 2012 - A System of Mixed Equilibrium Problems, a General System of Variational
Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for
Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

Kumam, Poom and Katchang, Phayap, Journal of Applied Mathematics, 2012