Generalized Viscosity Implicit Iterative Process for Asymptotically Non-Expansive Mappings in Banach Spaces

Abstract

In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium problems. Further, a numerical example is given to illustrate our convergence analysis. The results generalize and improve corresponding results in the literature.

A New Algorithm for the Common Solutions of a Generalized Variational Inequality System and a Nonlinear Operator Equation in Banach Spaces

“…Eventually, we show an application for solving the standard constrained problem of convex optimization to illustrate the efficiency of our main theorem. Some other outcomes proposed by other authors are also improved, see [9,10,[17][18][19][20][21][22].…”

“…We study the common elements of the set of solutions of an asymptotically non-expansive operation equation with the L mapping defined by (5) and the solution set of the generalized proposed system problem (4). The convergence analysis of a new way by using the generalized semi-closure principle supplied by us in Wang et al [9] for finding the propose common elements in Banach spaces are investigated. Under the suitable conditions imposed on parameters, some strong convergence theorems are attained.…”

“…The set of the solutions of the variational inequality (1) is denoted by V I(C, A). Lots of problems in physics, optimization, differential equation (inclusion), finance and minimax problem reduce to find an element of (1) and relevant numerical analysis methods can be considered to solve the problems, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”

Multi-Step Inertial Regularized Methods for Hierarchical Variational Inequality Problems Involving Generalized Lipschitzian Mappings

“…On the other hand, in order to accelerate the convergence, the inertial methods have been studied extensively by many scholars [5][6][7][8][9][10][11][12]. One of the important results is the inertial Mann algorithm which is introduced by Maingé [5] in 2007:…”

Viscosity Approximation Methods for a General Variational Inequality System and Fixed Point Problems in Banach Spaces

“…Variational inequality theory has played a significant role in nonlinear analysis and the optimization problem. Many iterative methods have been used to solve variational inequality problems due to the applications in some branches of applied science, convex optimization, mathematical physics, and operator studies, see [1][2][3][4][5][6][7][8][9] and the references therein. In fact, the classical variational inequality problem in Banach spaces is to find q ∈ E such that Aq, j(x − q) ≥ 0, ∀x ∈ E.…”