Iteration Methods For Solving Generalized Equations In Banach Spaces

In this work, we deal with two types of generalized equations. Firstly, we consider the following generalized equation problem (P1)0∈f(x)+F(x), where X and Y are Banach spaces,f:Ω(?)Xâ†'Y is single-valued function,Ω is an open subset of X and F:X(?)2Y is set-valued mapping with closed graph, and secondly we consider the generalized equation of the following form (P2)0∈T(x), where X is a Banach space and T:X(?)2x is a set-valued mapping with locally closed graph. This work consists two parts and the main works we have done in this dissertation that are organized as follows.In the first part, particulaly in Chapter3, We introduce and study the Gauss-Newton method for solving the generalized equation (P1) when f is smooth function, that is, when f is Frechet differentiable and it can be expressed as a classical linearization. We provide here semilocal and local convergence of the Gauss-Newton method. Furthermore, we introduce and study the Gauss-Newton method for solving the generalized equation (P1) when f is nonsmooth function, that is, when f doesn’t possess a Frechet derivative and the classical linearization of f is no longer available. We also present here semilocal and local convergence of the Gauss-Newton method.In the second part, specifically in Chapter4, we introduce the Gauss-type proximal point algorithm for solving the generalized equation (P2) by choosing a sequence of bounded constants{λk} which are away from zero. We establish the convergence criteria of the Gauss-type proximal point algorithm, which guarantees the existence and the convergence of any sequence generated by the algorithm under mild conditions. More precisely, semilocal and local convergence of the Gauss-type proximal point method are analyzed when T is metrically regular. We also study the stability properties of this algorithm.