الفهرس 
APPROVAL SHEET
Metric and S??mpacetric sesOnly 14 pages are availabe for public view
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Abstract
The work presented in this thesis is based on fixed points, coupled fixed points and multi-dimensional fixed points of a variety of single valued and multi-valued mappings satisfying different generalized contractive conditions and defined on different types of spaces. Also, random fixed point results for two mappings defined on S-metric spaces are investigated in a new setting with applications.
The thesis is divided into five chapters:
Chapter 1: Introduction
This chapter is introductory in nature and provides the historical development of the subject under research and necessary background to the rest of the chapters in the thesis.
Chapter 2: Coupled Fixed Point Theorems for Rational Type Contractions in Metric and b-Metric Spaces
This chapter is divided into three sections.
In Section 2.1, we study coupled coincidence points for rational type contractions involving generalized altering distance functions in metric spaces. Our results unify and generalize various known comparable results from the current literature, Gupta et. al. [50], Nashine and Aydi [77], Rashwan and Saleh [92] and many known results. An example is also given to support our main results.
The main results of this section are published in [99].
In Section 2.2, we consider b-metric space with a partial order and prove the existence of coupled coincidence and common coupled fixed points for two single valued mappings.
In Section 2.3, we study the existence and uniqueness of fuzzy coupled coincidence and fuzzy common coupled fixed points for single valued and fuzzy mappings under contractive condition of rational type in b-metric spaces and obtain the corresponding results for hybrid pair of single valued and multi-valued mappings.
The main results of Sections 2.2 and 2.3 are published in [96].
Chapter 3: Altering Distances and Multidimensional Fixed Point Results in Partially Ordered Metric Spaces
This chapter is divided into three sections.
In Section 3.1, some coupled coincidence and common coupled fixed point theorems for two self-mappings have been derived which satisfy certain inequality involving a function of two variables that measures the distance between points in ordered metric spaces. For particular choices of this function, several generalizations of many fixed point theorems which contain altering distance functions may be obtained. Our results can be applied directly to study multidimensional fixed point theorems which cover the concepts of coupled , tripled, quadruple fixed point etc.
The main results of this section are published in [100].
In Section 3.2, we establish a common fixed point theorem for two pairs of mappings satisfying an almost generalized contractive condition for comparable elements in a partially ordered metric space. Our results generalize and extend the main results in [5] to multi-valued setting.
The main results of this section are published in [93].
In Section 3.3, some existence and uniqueness fixed point theorems for g-mixed monotone mapping in any number of variables under general contractive conditions in partially ordered metric spaces by using the condition of weak compatibility were derived. Our results are different, more natural and generalizations of many results on multidimensional fixed points. For illustration of the effectiveness of our generalizations, some examples are equipped.
The main results of this section are published in [94].
Chapter 4: Multidimensional Fixed Points for Hybrid pairs of Mappings in Partially Ordered Metric Spaces
This chapter contains three sections.
In Section 4.1, we introduce the concepts of g-mixed monotone property for multi-valued mapping under any number of variables, where g is single valued mapping. Then we apply this concept to prove the existence of n-coincidence point for this hybrid pair under general contractive condition in partially ordered metric spaces. Also we give a supporting example for our theorem.
In Section 4.2, we obtain a n-coincidence point results for hybrid pair of non-linear contractions in partially ordered metric space by using ∆_g-symmetric property instead of g-mixed monotone property.
The main results of Sections 4.1 and 4.2 are published in [95].
In Section 4.3, we introduce the notion of weakly g-mixed monotone property for two multi-valued mapping with n-variables defined on partially ordered metric space and then prove coincidence and common fixed point for two hybrid pairs under two different contractive conditions. These theorems extend and generalize very recent results that can be found in [68] and many others.
The main results of this section are published in [97].
Chapter 5: Some Coupled Fixed Point Theorems in S-Metric Spaces with Applications
This chapter contains three sections.
Section 5.1, mainly presents a multi-valued version of the concept of weakly mixed monotone property and then presents results on existence and uniqueness common coupled fixed point for two multi-valued mappings satisfying certain contractive condition in partially ordered S-metric spaces.
The main results of Section 5.1 are accepted for publication in Assiut University Journal of Mathematics and Computer Sciences.
Section 5.2, is mainly devoted to give a random version of the concept of weakly mixed monotone property and prove some common and common coupled random fixed point theorems for two random operators satisfying some rational type contraction in partially ordered S-metric spaces. Also, an example is given to support our results.
Section 5.3, deals with the existence and uniqueness of a solution of some random functional equation, as an application of common random fixed point theorem mentioned in Section 5.2.
The main results of Sections 5.2 and 5.3 are published in [98].