الفهرس 
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Abstract
It is possible to say that the topics and concepts of mathematics
arose to solve the problems of life. Also these concepts evolved to
the emergence of new issues that do not suffice with the topics that
have been solved, from the emergence of engineering on the banks
of the Nile to solve the problems of measuring the earth, and even
recent issues related to non-specific concepts in politics, economics
and medicine. Functional analysis is an important branch of pure
mathematics that has attracted the attention of many research teams
around the world because of its great scientific importance and wide
applications in many fields such as applied mathematics (quantum
mechanics and dynamic systems), as well as in physics (waves),
geology, engineering, economics and other fields of life.
An important subject in functional analysis is convergence theorems
to fixed point in some metric spaces. Metric spaces can be described
as a general framework for measuring similarities and differences
between elements of a group of things by distance function between the elements of this grouping, the mappings on them are changes to this group and the fixed point theorems study these changes in which one element remains unchanged.
An example of this is the X group of students in a study hall measured
by the GPA. If we ask students to change their places except to change the place of the first student who has the highest cumulative rate, then this student is a fixed point. There are many researchers interested in the study of convergence theorems in some metric spaces. For example Ahmed (see [2], [3]), ((Ahmed, Zeyada) (see[[4]- [6]])) and Kirk(see [19]).
The main objective of this thesis which consists of five chapters is to
study of convergence theorems to fixed point of of single-valued mappings and multi-valued mappings in some metric spaces under conditions like quasi-nonexpansive, weakly quasi-nonexpansive and similar maps, as well as some applications.
The thesis consists of five chapters organized as follows:
In Chapter 1 is presentation for all previous results in the last years without proofs but with many references and it contains some of basic concepts and definitions which used in this thesis. So, the aim of this introductory chapter is twofold: Firstly and principally , it serves as independent survey readable in some properties and theorems in different metric spaces. Secondly, it prepares from a historical point of view what follows and it emphasizes the main purpose of the thesis, that is, to clear how we can prove the convergence
for a mappings in some metric spaces.
This chapter divided into five parts, firstly basic concepts, complex-valued metric space, multiplicative metric space, some fixed point theorems and finally convergence theorems.
In Chapter 2 we introduce our new results on convergence theorems
in metric spaces. In this chapter improve, generalize and extend some
recent results (see e.g. [[2] - [5]]).
In Chapter 3 we give the concept property of left quasi-metric space,
then we use this property to prove convergence theorems in this space, and show that a sequence of this space converges to a unique fixed point for any mappings which satisfy some conditions.
Examples are also given to support our some idea. Finally, we give
applications that motivated by the paper of Ahmed and Zeyada [6].
Chapter 4 is devoted to prove some convergence theorems in complex-valued metric spaces, we give some theorems as applications on complex-valued metric spaces .
In Chapter 5 we establish some convergence theorems to a unique
fixed point in a multiplicative metric space.