Geometrical aspects of Banach spaces and generalized projection methods

Abstract

This Ph. D. thesis is organized as follows: 1. In chapter #1, we introduce a brief history and a motivation for the problem of approximating a xed point, assuming that it exists, for single-valued mappings in Banach and Hilbert spaces and multi-valued mappings in Banach spaces and how this problem was solved by using the notions of both the metric and generalized projection operators. We explain the importance of generalized projection operator of Banach spaces that it was presented analogously to metric projection in Hilbert spaces. 2. In chapter #2, we present almost of the details needed in this thesis and it contains the most important de nitions, examples, theorems and results obtained in various Banach spaces. 3. In chapter #3, we show the basic properties about various types of smoothness and convexity conditions that the norm of a Banach space may(or may not) satisfy. We present the normalized duality mapping of Banach spaces and explain the main role of it to determine the geometric properties of Banach spaces. Also, we introduce the concepts of Birkho orthogonality and J-orthogonality in Banach spaces and study their properties. 4. In chapter #4, we present orthogonality, projection methods in Hilbert spaces and clarify the relations between them. We introduce the metric projection operator and its properties in Banach spaces and explain the main links between metric projection and normalized duality mappings and the relation between metric projection and orthogonality in Banach spaces. We show the generalized projection operator and explain the generalization of it from uniformly convex and uniformly smooth Banach 4