Algorithms of common solutions for a fixed point of hemicontractive-type mapping and a generalized equilibrium problem

Abstract

In this paper, we introduce and study an iterative algorithm for finding a common element of the set of fixed points of a Lipschitz hemicontractive-type multi-valued mapping and the set of solutions of a generalized equilibrium problem in the framework of Hilbert spaces. Our results improve and extend most of the results that have been proved previously by many authors in this research area.

Convergence and stability of modified multi-step Noor iterative procedure with errors for strictly hemicontractive-type mappings in Banach spaces

“…In the last few decades, fixed-point theorem-based iterative procedures whose convergence established on the strictly hemicontractive-type mappings earn a great attention for its rigorous applications in the diverse fields of various mathematical problems; see for instance [2][3][4][5] and the references cited therein. Application of strictly hemicontractive-type mapping was initiated by Chidume and Osilike [4] for improving the consequence of Chidume [5].…”

“…Application of strictly hemicontractive-type mapping was initiated by Chidume and Osilike [4] for improving the consequence of Chidume [5]. After Chidume and Osilike [4], several researchers studied strictly hemicontractive-type mapping in many directions; see for instance [1][2][3][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references cited therein. Among the articles cited in [1][2][3][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], Hussain et al [1] studied Lipschitz strictly hemicontractive-type mapping in arbitrary Banach spaces to extend and improve the equivalent consequences of the monographs [4,5,[12][13][14]…”

“…In 2006, Plubtieng and Wangkeeree [24] introduced and studied the following multistep Noor iterative procedure with errors for some special type of asymptotically nonexpansive mappings (asymptotically non-expansive mapping in the intermediate sense and asymptotically quasi-non-expansive mapping) in Banach spaces: For a given u 1 ∈ B, and a fixed m ∈ N (set of all positive integers), the iterative sequences {u (1) n }, {u (2) n }, . .…”

“…. , {u (m) n } defined by u (1) n = a (1) n T n u n + b (1) n u n + c (1) n v (1) n , u (2) n = a (2) n T n u (1) n + b (2) n u n + c (2) n v (2) n ,…”

Approximation of common solutions for a fixed point problem of asymptotically nonexpansive mapping and a generalized equilibrium problem in Hilbert space

“…If we define F(x, y) = Ax, y − x for all x, y ∈ C, then z ∈ EP(F) if and only if Az, y − z ≥ 0 for all y ∈ C and hence z ∈ VI(C, A). The problem (2) is very general in the sense that it includes many special cases such as optimization problems, variational inequalities, minimax problems, and the Nash equilibrium problem in noncooperative games; see Blum and Oettli [2], Kazmi and Rizvi [3], Meche et al [4], Moudafi and Théra [5], Zegeye et al [6], and the references therein.…”