Document Type : Research Paper
Author
- Ebrahim Soori ^{}
Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran.
Abstract
This paper introduces an implicit scheme for a continuous representation of nonexpansive mappings on a closed convex subset of a Hilbert space with respect to a sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup. The main result is to prove the strong convergence of the proposed implicit scheme to the unique solution of the variational inequality on the solution of systems of equilibrium problems and the common fixed points of a sequence of nonexpansive mappings and a continuous representation of nonexpansive mappings.
Keywords
- Continuous representation
- Fixed point
- Equilibrium problem
- Nonexpansive mapping
- Variational inequality
Main Subjects
[1] V. Colao, G.L. Acedo, and G. Marino, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal., 71 (2009) 2708-2715.
[2] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (1) (2005) 117-136.
[3] K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Stud. Adv. Math., Cambridge, 1990.
[4] N. Hirano, K. Kido, and W. Takahashi, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal., 12 (1988), 1269-1281.
[5] N. Hussain, M.L. Bami, and E. Soori, An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., (2014), DOI: 10.1186/1687-1812-2014-238.
[6] K. Kido and W. Takahashi, Mean ergodic theorems for semigroups of linear continuous in Banach spaces, J. Math. Anal. Appl., 103 (1984), 387-394.
[7] G. Marino and H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (1) (2006) 43-52.
[8] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (1) (2000) 46-55.
[9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967) 595-597.
[10] S. Plubtieng and R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336 (2007) 455-469.
[11] X. Qin, M. Shang, and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling, 48 (2008) 1033-1046.
[12] S. Saeidi, Comments on relaxed (γ, r)-cocoercive mappings, Int. J. Nonlinear Anal. Appl., 1 (2010) 54-57.
[13] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001)387-404.
[14] E. Soori, Strong convergence for variational inequalities and equilibrium problems and representations, Int. J. Industrial Math. 5(4) (2013) 341-354.
[15] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000.
[16] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., (2006) doi: 10.1016/j.jmaa.2006.08.036.
[17] W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Anal., (2008), doi: 10.1016/j.na.2008.01.005.
[18] R.U. Verma, General convergence analysis for two-step projection methods and application to variational problems, Appl. Math. Lett., 18 (11)(2005) 12861292.