Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators

Abstract

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included.

A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces

“…where λ ∈ (0, 1 L ). The subgradient extragradient-type algorithm extend (6) in which the second orthogonal projection onto some constructible set in Euclidean space for solving VI(F, C) in real Hilbert space. Their method is of the following form:…”

“…Under appropriate conditions, there are two general approaches for solving the variational inequality problem, one is the regularized method and the other is the projection method. Many projection-type algorithms for solving the variational inequalities problem have been proposed and analyzed by many authors [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The gradient method is the simplest algorithm in which only one projection on feasible set is performed, and the convergence of the method requires a strongly monotonicity.…”

“…The subgradient extragradient-type algorithm was introduced by Censor et al in [5] for solving variational inequalities in real Hilbert space. Yao et al in [6] proposed an iterative algorithm for solving a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators in Hilbert spaces.…”

Strong Convergence Results of Split Equilibrium Problems and Fixed Point Problems

“…Numerical iterative algorithms have been proposed for finding a split problem of the set of solutions of equilibrium problems and the set of fixed points of nonexpansive operators; see, for example, [35][36][37][38][39] and the references therein. Recently, Yao et al [40] proposed an iterative scheme for solving the split problem (2) and they obtained the weak convergence of the suggested scheme.…”