Convergence Theorem of Hybrid Iterative Algorithm for Equilibrium Problems and Fixed Point Problems of Finite Families of Uniformly Asymptotically Nonexpansive Semigroups

Hongbo Liu, Yi Li

School of Science, Southwest University of Science and Technology, Mianyang, China.

DOI: 10.4236/apm.2014.46033                

Abstract

Throughout this paper, we introduce a new hybrid iterative algorithm for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. We then prove the strong convergence theorem with respect to the proposed iterative algorithm. Our results in this paper extend and improve some recent known results.

Keywords

Hybrid Iterative Algorithm, Uniformly Asymptotically Nonexpansive Semigroups, Equilibrium Problem, Common Fixed Point

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Blum, E. and Oettli, W. (1994) From Optimization and Variational Inequalities to Equilibrium Problems. Mathematics Students, 63, 123-145.
[2]	Flam, S.D. and Antipin, A.S. (1997) Equilibrium Programming Using Proximal-Link Algolithms. Mathematical Programming, 78, 29-41. http://dx.doi.org/10.1007/BF02614504
[3]	Moudafi, A. and Thera, M. (1999) Proximal and Dynamical Approaches to Equilibrium Problems. Lecture Note in Economics and Mathematical Systems, 477, 187-201.
[4]	Bauschke, H.H. and Borwein, J.M. (1996) On Projection Algorithms for Solving Convex Feasibility Problems. SIAM Review, 38, 367-426. http://dx.doi.org/10.1137/S0036144593251710
[5]	Butnariu, D., Censor, Y., Gurfil, P. and Hadar, E. (2008) On the Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces. SIAM Journal on Optimization, 19, 786-807. http://dx.doi.org/10.1137/070689127
[6]	Hale, E.T., Yin, W. and Zhang, Y. (2010) Fixed-Point Continuation Applied to Compressed Sensing: Implementation and Numerical Experiments. Journal of Computational Mathematics, 28, 170-194.
[7]	Maruster, S. and Popirlan, C. (2008) On the Mann-Type Iteration and the Convex Feasibility Problem. Journal of Computational and Applied Mathematics, 212, 390-396. http://dx.doi.org/10.1016/j.cam.2006.12.012
[8]	Byrne, C. (2004) A Unified Treatment of Some Iterative Algorithms in Signal Processing and Image Reconstruction. Inverse Problems, 20, 103-120. http://dx.doi.org/10.1088/0266-5611/20/1/006
[9]	Censor, Y., Elfving, T., Kopf, N. and Bortfeld, T. (2005) The Multiple-Sets Split Feasibility Problem and Its Applications for Inverse Problems. Inverse Problems, 21, 2071-2084. http://dx.doi.org/10.1088/0266-5611/21/6/017
[10]	Xu, H.K. (2006) A variable Krasnoselskii-Mann Algorithm and Themultiple-Set Split Feasibility Problem. Inverse Problems, 22, 2021-2034. http://dx.doi.org/10.1088/0266-5611/22/6/007
[11]	Mann, W.R. (1953) Mean Value Methods in Iteration. Proceedings of the American Mathematical Society, 4, 506-510. http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3
[12]	Nakajo, K. and Takahashi, W. (2003) Strong Convergence Theorems for Nonexpansive Mappings and Nonexpansive Semigroups. Journal of Mathematical Analysis and Applications, 279, 372-379. http://dx.doi.org/10.1016/S0022-247X(02)00458-4
[13]	Takahashi, W., Takeuchi, Y. and Kubota, R. (2008) Strong Convergence Theorems by Hybrid Methods for Families of Nonexpansive Mappings in Hilbert Spaces. Journal of Mathematical Analysis and Applications, 341, 276-286. http://dx.doi.org/10.1016/j.jmaa.2007.09.062
[14]	Suzuki, T. (2003) On Strong Convergence to Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces. Proceedings of the American Mathematical Society, 131, 2133-2136. http://dx.doi.org/10.1090/S0002-9939-02-06844-2
[15]	Tada, A. and Takahashi, W. (2007) Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem. Journal of Optimization Theory and Applications, 133, 359-370. http://dx.doi.org/10.1007/s10957-007-9187-z
[16]	He, H. and Chen, R. (2007) Strong Convergence Theorems of the CQ Method for Nonexpansive Semigroups. Fixed Point Theory and Applications, 2007, Article ID 59735.
[17]	Saejung, S. (2008) strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals. Fixed Point Theory and Applications, 2008, Article ID 745010.
[18]	Marino, G. and Xu, H.K. (2007) Weak and Strong Convergence Theorems for Strict Pseudo-Contractions in Hilbert Space. Journal of Mathematical Analysis and Applications, 329, 336-346. http://dx.doi.org/10.1016/j.jmaa.2006.06.055
[19]	Cholamjiak, W. and Suantai, S. (2010) Ahybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems. Discrete Dynamics in Nature and Society, 2010, Article ID: 349158.
[20]	Mohammad, E. (2013) Hybid Method for Equilibrium Problems and Fixed Piont Problems of Finite of Nonexpansive Semigroups. Revista Serie A Matemáticas, 107, 299-307.
[21]	Chang, S.S., Wang, L., Tang, Y.K., Wang, B. and Qin, L.J. (2012) Strong Convergence Theorems for a Countable Family of Quasi-ψ-Asymptotically Nonexpansive Nonself Mappings. Applied Mathematics and Computation, 218, 7864-7870. http://dx.doi.org/10.1016/j.amc.2012.02.002