Lipschitz-like mapping and its application to convergence analysis of a variant of Newton's method
M.H. Rashid1,2
1Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun east road, Haidian district, Beijing-100190, China
2University of Rajshahi, Rajshahi-6205, Bangladesh
Keywords: set-valued mappings, lipschitz-like mappings, generalized equations, variant of Newton's method, semilocal convergence
Abstract
Let X and Y be Banach spaces. Let f: Ω → Y be a Frèchet differentiable function on an open subset Ω of X and F be a set-valued mapping with closed graph. Consider the following generalized equation problem: 0 in f(x)+ F(x). In the present paper, we study a variant of Newton's method for solving generalized equation (1) and analyze semilocal and local convergence of this method under weaker conditions than those considered by Jean-Alexis and Piètrus [13]. In fact, we show that the variant of Newton's method is superlinearly convergent when the Frèchet derivative of f is (L,p)-Hölder continuous and (f+F)-1 is Lipzchitz-like at a reference point. Moreover, applications of this method to a nonlinear programming problem and a variational inequality are given. Numerical experiments are provided which illustrate the theoretical results.
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