On the Solution of Generalized Banach Space Valued Equations

Ramandeep Behl,Ioannis K Argyros

doi:10.3390/math10010132

Abstract

We develop a class of Steffensen-like schemes for approximating solution of Banach space valued equations. The sequences generated by these schemes are, converging to the solution under certain hypotheses that are weaker than in earlier studies. Hence, extending the region of applicability of these schemes without additional hypotheses. Benefits include: more choices for initial points; the computation of fewer iterates to reach a certain accuracy in the error distances, and a more precise knowledge of the solution. Technique is applicable on other schemes our due to its generality.

Highlights

A plethora of problems from optimal control, variational inequations, mathematical programming and other disciplines can be converted to finding a solution x∗ of the generalized equation
We need the auxiliary result on fixed points
Suppose that the following hypotheses hold for σ0 ∈ B1, ρ ≥ 0 and p ∈ [0, 1)

Summary

Introduction

A plethora of problems from optimal control, variational inequations, mathematical programming and other disciplines can be converted to finding a solution x∗ of the generalized equation. We extend the region of applicability of SLS by finding more initial point x0 ; providing fewer iterates for achieving a certain accuracy and we give a better information related to the where about x∗. These are obtained without additional hypothesis by developing the center-Lipchitz idea. This technique may be utilized to extend the applicability of other schemes with the same benefits. Another novelty of our idea is that all the benefits are obtained without additional to the previous works hypotheses.

Background and Auxiliary Results

Convergence for SLS

Inexact Scheme

Convergence of Higher Order Schemes

Numerical Examples

Conclusions