A Method for Solving the Variational Inequality Problem and Fixed Point Problems in Banach Spaces

Abstract

The purpose of this research is to modify Halpern iteration’s process for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a strictly pseudo contractive mapping in q-uniformly smooth Banach space. We also introduce a new technique to prove a strong convergence theorem for a finite family of strictly pseudo contractive mappings in q-uniformly smooth Banach space. Moreover, we give a numerical result to illustrate the main theorem.

Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces

“…We note that the following inclusions hold for the classes of the mappings: firmly nonexpansive ⊂ nonexpansive ⊂ quasi-nonexpansive ⊂ k-strictly pseudo-contractive ⊂ k-demi-contractive. Demicontractive mappings constitute one of the most general classes of nonexpansive type mappings with important applications for which the fixed points can be obtained by iterative schemes, there was and still is a great interest in studying their properties, see [15,19,20] and most of the references therein. Construction of fixed points of nonlinear mappings is an important subject in the theory of nonlinear mappings and find applications in a variety of applied areas, in particular, in inverse problems, partial differential equations, image recovery, and signal processing (see, [1,3]).…”