Convergence of a continuation method under Lipschitz continuous derivative in Banach spaces

Abstract

The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the semilocal convergence of a continuation method combining Chebyshev method and Convex acceleration of Newton’s method for solving nonlinear equations in Banach spaces under the assumption that the first Frechet derivative satisfies the Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in both the examples. Further, we observed that for particular values of α, our analysis reduces to those for Chebyshev method (α=0) and Convex acceleration of Newton’s method (α=1) respectively with improved results.