A study on Extended Newton-type Methods for Variational Inclusions

Abstract

In this work, we deal with the two types of variational inclusions. Firstly, we consider the variational inclusion problem of the form 0 ∈ ζ(¯s) + g(¯s) + ξ(¯s), (A) where S and T are Banach spaces, ζ : S → T is differentiable in a neighborhood Υ ⊆ S of a solution s∗ of (A), g : S → T is differentiable at s∗ but may not differentiable in Υ and ξ : S ⇒ 2T is a set-valued mapping with closed graph. This work consists three parts and the main works we have done in this dissertation that are organized as follows. In the first part, particularly in Chapter 3, we study the Newton-type method for solving the variational inclusion problem (A) which is introduced in [2]. Under some suitable assumptions on the Fr´echet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newtontype method and prove that the sequence generated by the method (3.1.3) converges linearly, quadratically and superlinearly to a solution of the variational inclusion (A). Specifically, when the Fr´echet derivative of the differentiable function is continuous, Lipschitz continuous and H¨older continuous, divided difference admissible function admits first order divided difference and the set-valued mapping is pseudo-Lipschitz continuous, we show the linear, quadratic and superlinear convergence by the method (3.1.3). In Chapter 4, we introduce and study the extended Newton-type-----