A Third Order Newton-Like Method and Its Applications

D Sahu,Ravi Agarwal,Vipin Singh

doi:10.3390/math7010031

Abstract

In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature.

Highlights

IntroductionOur purpose of this paper is to compute solution of nonlinear operator equation of the form
Our purpose of this paper is to compute solution of nonlinear operator equation of the form F ( x ) = 0, (1)where F : D ⊂ X → Y is a nonlinear operator defined on an open convex subset D of a Banach spaceX with values into a Banach space Y.A lot of challenging problems in physics, numerical analysis, engineering, and applied mathematics are formulated in terms of finding roots of the equation of the form Equation (1).In order to solve such problems, we often use iterative methods
Where F : D ⊂ X → Y is a nonlinear operator defined on an open convex subset D of a Banach space

Summary

Introduction

Our purpose of this paper is to compute solution of nonlinear operator equation of the form. Parhi and Gupta [21] have studied the semilocal convergence analysis of Equation (8) for computing a solution of the operator Equation (5), where G : D ⊂ X → X is a nonlinear Fréchet differentiable operator defined on an open convex subset D under the condition:. We derive the Stirling-like iterative method for computing a solution of the fixed point problem Equation (5), where (Ω) does not hold and it gives an affirmative answer to Question 1 and generalizes the results of Parhi and Gupta [21,36] in the context of the condition (Ω).

Preliminary

Fixed Points of Smooth Operators

Fredholm Integral Equations

Conclusions