An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces

Abstract

Let C be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E. Let T : C → C be relatively nonexpansive mapping and let A i : C → E * be L i -Lipschitz monotone mappings, for i = 1, 2. In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a relatively nonexpansive mapping and the solution set of variational inequality problems for A 1 and A 2 . Under some mild assumptions, we show that the proposed algorithm converges strongly to a point inOur theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. Mathematics Subject Classification

Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces

“…Our results extend and generalize many results in the literature. In particular, Theorem 1.11 extends the results in [3,4,7,13,16,17,38] from real Hilbert spaces to real reflexive Banach spaces. Moreover, Theorem 1.11 extends the classes of mappings in Theorem 3.1 of Tufa and Zegeye [17] and Theorem 3.2 of Wega and Zegeye [18] from Lipschitz monotone mapping to continuous pseudomonotone mappings in reflexive real Banach spaces.…”

“…In space, more general than Hilbert spaces, Tufa and Zegeye [17] introduced an iterative algorithm for approximating a common solution of VIP and fixed point problem of Lipschitz monotone and relatively nonexpansive mappings, respectively in real 2-uniformly convex and uniformly smooth Banach spaces. They proved that the sequence generated by their algorithm converges strongly to a common solution of the problems.…”

Cyclic subgradient extragradient methods for equilibrium problems

“…In 2005, Combettes and Hirstoaga [14] introduced a general procedure for solving CSEPs. After that, many methods were also proposed for solving CSVIPs and CSEPs, see for instance [4,21,30,[32][33][34][35] and the references therein. However, the general procedure in [14] and the most existing methods are frequently based on the proximal point method (PPM) [22,28], i.e., at the current step, given x n , the next approximation x n+1 is the solution of the following regularized equilibrium problem (REP).…”