الفهرس 
Chapter 1 IntroductionOnly 14 pages are availabe for public view
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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 fixed 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 impor- tance 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 definitions, 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 Ba- nach space may(or may not) satisfy. We present the normalized duality mapping of Banach spaces and explaining the main role of it to determine the geometric properties of Banach spaces. Also, we introduce the concepts of Birkhoff orthogonality and J-orthogonality in Banach spaces and study their properties.
4. In chapter #4, we study orthogonality and projection methods in Hilbert spaces and clarifying the relations between them. We introduce the metric projection operator and its proper- ties in Banach spaces and explain the main links between met- ric projection and normalized duality mappings and the rela- tion between metric projection and orthogonality in Banach spaces. We study the generalized projection operator and ex- plain the generalization of it from uniformly convex and uni-
CONTENTS 5
formly smooth Banach spaces to reflexive Banach spaces. We also give the basic properties of generalized projection opera- tors in uniformly convex and uniformly smooth Banach spaces and in reflexive Banach spaces.
5. In chapter #5, we introduce our new results, the first new result is published in Journal of inequalities and applications see [36], through this paper we generalize the concepts of nor- malized duality mappings, J-orthogonality and Birkhoff or- thogonality from normed spaces to smooth countably normed spaces. Also, We give some basic properties of J-orthogonality in smooth countably normed spaces and we show the relation between J-orthogonality and metric projection on smooth uni- formly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and we prove some basic results about the J-dual cone and the J-orthogonal complement of S. The second new result is sub- mitted to journal of fixed point theory and applications and to be accepted soon see [37], through this paper we extend the concept of generalized projection operator “ΠK : E K” from uniformly convex uniformly smooth Banach spaces to uni- formly convex uniformly smooth countably normed spaces and study its properties. Also, we show the relation between J- orthogonality and generalized projection operator ΠK and give examples to clarify this relation. Moreover, we introduce a comparison between metric projection operator PK and gener- alized projection operator ΠK in uniformly convex uniformly
smooth complete countably normed spaces in addition to ex- tend the generalized projection operator “πK : E∗ K” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and showing its properties.