Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions

Ioannis Argyros,Stepan Shakhno

doi:10.3390/math7020207

Abstract

We present a local convergence of the combined Newton-Kurchatov method for solving Banach space valued equations. The convergence criteria involve derivatives until the second and Lipschitz-type conditions are satisfied, as well as a new center-Lipschitz-type condition and the notion of the restricted convergence region. These modifications of earlier conditions result in a tighter convergence analysis and more precise information on the location of the solution. These advantages are obtained under the same computational effort. Using illuminating examples, we further justify the superiority of our new results over earlier ones.

Highlights

IntroductionWhere F is a Fréchet-differentiable nonlinear operator on an open convex subset D of a Banach space
Consider the nonlinear equationF ( x ) + Q( x ) = 0, (1)where F is a Fréchet-differentiable nonlinear operator on an open convex subset D of a Banach spaceE1 with values in a Banach space E2, and Q : D → E2 is a continuous nonlinear operator.Let x, y be two points of D
In [17], we introduced a generalized Lipschitz condition for the divided differences of the second order, and we have studied the local convergence of the Kurchatov method (5)

Summary

Introduction

Where F is a Fréchet-differentiable nonlinear operator on an open convex subset D of a Banach space. In [17], we introduced a generalized Lipschitz condition for the divided differences of the second order, and we have studied the local convergence of the Kurchatov method (5). Let us suppose, that: (1) H ( x ) ≡ F ( x ) + Q( x ) = 0 has a solution x∗ ∈ D, for which there exists a Fréchet derivative H 0 ( x∗ ) and it is invertible; (2) F has the Fréchet derivative of the first order, and Q has divided differences of the first and second order on B( x∗ , 3r ) ⊂ D, so that for each x, y, u, v ∈ D. where x θ = x∗ + θ ( x − x∗ ), $( x ) = k x − x∗ k, L01 , L02 , N0 L1 , L2 and N are positive nondecreasing integrable functions and r > 0 satisfies the equation r. The convergence rate of sequence { xn }n≥0 to x∗ is quadratic

Uniqueness Ball of the Solution

Corollaries

Numerical Examples

Conclusions