Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

Abstract

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of k-strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three above described sets. We give an application of our results. Our results extend important recent results from the current literature.

The hybrid algorithm for the system of mixed equilibrium problems, the general system of finite variational inequalities and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings

“…Then, we prove strong convergence theorem under some mind conditions. The results presented in this article extend and improve the results of Shehu [21] and many authors.…”

“…In 2011, Shehu [21] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of an infinite family of k-strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solution of variational inequality problems in Hilbert spaces. Starting with an arbitrary…”

“…In the last two decades, many articles have appeared in the literature on the existence of solutions of equilibrium problems; see, for example [13] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, (see [11][12][13][14][16][17][18][19][20][21][22][23][24]) and references therein.…”

A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems

“…Inspired and motivated by above works and the work of other studies 10,11,17,19,[29][30][31] in this paper, we modify CQ methods for finding a common solution set of system found in equilibrium problem and the fixed-point set of a finite family of demicontractive mappings. Further, we prove a strong convergence theorem for the sequences generated by iterative method and derive some consequences which are new and interesting.…”

Strong Convergence of Strict Pseudo-Contractions in Q-Uniformly Smooth Banach Spaces

“…Among nonlinear mappings, nonexpansive mappings and strict pseudo-contractions are two kinds of the most important nonlinear mappings. The study of them has a very long history (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][19][20][21][22][23][24][25][26][27][28][29][30][31] and the references therein). Recall that a mapping T : C → E is nonexpansive if ∥T x − T y∥ ≤ ∥x − y∥ for all x, y ∈ C.…”