Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

Abstract

Under the hypothesis that the first derivative satisfies some kind of weak Lipschitz conditions, a new semilocal convergence theorem for inexact Newton method is presented. Unified convergence criteria ensuring the convergence of inexact Newton method are also established. Applications to some special cases such as the Kantorovich type conditions andγ-Conditions are provided and some well-known convergence theorems for Newton's method are obtained as corollaries.

Inexact Newton’s method for generalized operator equations in Banach spaces

“…Since our assumptions on F and G in generalized operator equation 9are fairly general, our main result covers a wide variety of nonlinear operator equations. In fact, our main result provides an affirmative answer of Question 1 and also significantly improves the corresponding results of [7,8,10].…”

“…(2) Corollary 1 is an improvement over the [8,Theorem 3.2] in the sense of larger convergence domain and tighter error bounds. (4) For the choice of G = 0, Corollary 2 reduces to the well known Newton Kantorovich theorem which was already discussed by Wang [11] and Tapia [12].…”

“…In this paper we improved and extend the inexact Newton-like methods [7,8,10] in the context of differentiability of involved operator and introduced an inexact Newton-like algorithm for solving the generalized operator equations. We discussed the semilocal convergence analysis of our algorithm under the weak Lipschitz condition.…”

“…To avoid the first disadvantage of Newton's method, a number of Newton-like methods have been developed in literature. To overcome the second disadvantage of Newton's method, linear iterative methods have been extensively developed to find an approximate solution of Newton equation (2) (see [3][4][5][6][7][8]). These methods have been initially introduced by Dembo et al [9] and are known as inexact Newton methods.…”

“…Assuming the residual controls (7) with k = 1, the γ-condition and the Smale's α-theory have been established by Shen and Li in [7] for the inexact Newton method (3). Recently, Xu et al [8], Argyros and Santosh George [10] have studied semilocal convergence analysis of the inexact Newton method (3) under the residual controls (7) with k = 1 and F −1…”