The Approximation Theory Of Fixed Points Of Operators And Its Applications

In this thesis, we discuss mainly the iterative approximation problems of fixed points of several classes of operators, including their algorithm designs, the con-vergence of their algorithms and their applications in variational inclusions in the framework of several different spaces. Also, using the approximation theory and methods, we study the existence of random coincidence points and random fixed points of a class of random multi-valued operators. This thesis is divided into six chapters.In Chapter 1, we introduce the research backgrounds and the main results of this thesis.Chapter 2 is preliminaries, mainly including some definitions and lemmas which will be used in the following.In Chapter 3, we discuss the iterative approximation of fixed points of multi-valuedΦ-hemicontractive mappings in real normed linear spaces. We introduce and study some new Ishikawa-type iterative algorithms with error items and variable coefficients. General results on the iterative approximation of fixed points of uni-formly continuousΦ-hemicontractive mappings without boundedness conditions on the ranges and the domains are given in real normed linear spaces.In Chapter 4, we discuss the iterative approximation of fixed points of several classes of operators in the framework of uniformly smooth real Banach spaces.In§4.1, we introduce and study a new steepest descent approximation algo-rithm with error items and variable coefficients. A result on the convergence of the iterative sequencess of zero points ofΦ-hemiaccretive mappings without continuity assumption is given. Since the zeros of Φ-hemiaccretive mappings and the fixed points of correspondingΦ-hemicontractive mappings may transform mutually, we obtain a new fixed point theorem forΦ-hemicontractive mappings.In§4.2, we discuss the iterative approximation of fixed points of multi-valued generalizedΦ-hemicontractive mappings in the framework of p-uniformly smooth real Banach spaces. We introduce and study some new Ishikawa-type iterative algorithms with error items and variable coefficients. Some new methods are applied. A new fixed point theorem for generalizedΦ-hemicontractive mappings without continuity assumption and without boundedness conditions on the ranges and the domains of the mappings is obtained in p uniformly smooth real Banach spaces. In case of generalizedΦ-hemicontractive mappings with bounded ranges, the result is extended to uniformly smooth real Banach spaces.In§4.3, we discuss the iterative approximation of fixed points of asymptoticallyκ-strictly pseudo-contractive mappings in the framework of real Hilbert spaces. We introduce and study some new CQ-type algorithms with variable coefficients. New methods are applied in the proof of the iterative approximation theorem of fixed point of asymptoticallyκ-strictly pseudo-contractive mappings. Some boundedness conditions for the mappings in recent literatures are dropped in this thesis.In Chapter 5, by using fixed point approximation theory and methods, we discuss the existence of random coincidence points and random fixed points of a class of random multi-valued operators in separable complete metric spaces. First, we prove a selector theorem, then present some new random coincidence point and random fixed point theorems for multifunctions without Compactness conditions. Even in the non-random case, our results also improve and extend some known results.In the last chapter, fixed point approximation theory and methods are applied to discuss two set-valued variational inclusion problems as follows. Two iterative approximation theorems of solutions to set-valued variational inclusion problems are obtained.1,Let T,A : X→X, N(·,·) : X×X→X, g : X→X~*η: X~*×X~*→X~* be five mappings andφ: X~*→R∪{+∞} be a proper convex functional such thatφisη-subdifferentiable. For any given f∈X, finding u∈X such thatwhereα_ηφdenotesη-subgradients ofφ. 2,Let T, A : X→2~X, g : X→X~*,η: X~*×X~*→X~* be four mappings andφ:X~*→R∪{+∞}be a proper convex functional such thatφisη-subdifferentiable. For any given f∈X, finding x~*∈X, u∈Tx~*,v∈Ax~*, denoting (x~*,u,v), such thatwhereα_ηφdenotesη-subgradients ofφ.