Lipschitzlike mapping and its application to convergence analysis of a variant of Newton's method
M.H. Rashid^{1,2}
^{1}Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun east road, Haidian district, Beijing100190, China ^{2}University of Rajshahi, Rajshahi6205, Bangladesh
Keywords: setvalued mappings, lipschitzlike 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 setvalued 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 JeanAlexis 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 Lipzchitzlike 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.
