An Efficient Iterative Procedure for Proximally Quasi-Nonexpansive Mappings and a Class of Boundary Value Problems

Abstract

It is well-known in the literature that many analytical techniques are introduced in order to find a solution for problems such as functional, differential, and integral equations. These analytical techniques sometimes fail to solve such problems, thus prompting the proposal of numerical methods for approaching their approximate solutions. This paper suggests a multi-valued version of an efficient iterative procedure called the F iterative procedure in Banach space and establishes its weak and strong convergence to fixed points of certain proximally quasi-nonexpansive operators. To support these results and to suggest the high accuracy of this procedure, we develop an example of a proximally quasi-nonexpansive operator and perform a comparative numerical experiment. As an application, we solve a two-point boundary value problem (BVP) in Banach space. Our results are new and extend some results from the literature for the new setting of mappings.

Keywords

• quasi-nonexpansive operator
• weak convergence
• strong convergence
• speed of convergence
• BVP
• Banach space