Lipschitz-Like Mapping and Its Application to Convergence Analysis of a Variant of Newton’s Method
Mohammed Harunor RASHİD · Numerical Analysis and Applications · 2021
Let $$X$$ and $$Y$$ be Banach spaces. Let $$f:\Omega\to Y$$ be a Fréchet differentiable function on an open subset $$\Omega$$ 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 and analyze semilocal and local convergence of the this method under weaker conditions than those associated by Jean-Alexis and Piétrus [13]. In fact, we show that the variant of Newton’s method is superlinearly convergent when the Fréchet 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 presented, which illustrate the theoretical results.