FIXED POINT THEOREMS FOR SINGLE-VALUED AND MULTI-VALUED DYNAMICAL SYSTEMS IN METRIC TYOE SOACES

Abstract

The theory of fixed point is a growing field of research with several applications in various fields. It is concerned with the results which state that a single-valued dynamical systems z : Q → Q or a multi-valued dynamical system z : Q → P(Q) admits one or more fixed points under particular circumstances. The necessity to prove theorems about the existence of solutions to differential and integral equations drove the further growth in this theory. There are three major topics of theory of fixed points: Metric, Topological and Discrete fixed point theory. Some of the most well-known and significant results in these fields are: Banach, Brouwer and Tarski fixed point theorems respectively. In 1922, Banach was working on integral equations and proved a theorem known as the Banach contraction principle, which guarantee to exists a unique fixed point in a complete metric space. The Banach contraction principle is a very useful tool in nonlinear analysis with many applications to operator equations, fractal theory, optimization theory and other topics. After Banach, many researchers introduced new type of contractions in metric spaces. It has been observed that a Banach contraction z is always a continuous map. This brings up the question whether some contraction conditions exist which guarantee to exists of unique fixed point of discontinuous mappings. In 1968, Kannan and Cheatterja gave positive answer to this question for complete metric spaces. Another important contraction in this perspective which generalizes both Banach and Kannan contractions i for a complete metric spaces was proved by Reich in 1971. Due to the wide range of applications of Banach contraction principle, many authors have refined the contraction condition or changed the metric space to different abstract spaces to generalize/extend this elegant result. Nadler extended first time the Banach contraction for the multi-valued dynamical systems i.e., z : Q → CB(Q), where CB(Q) = {M ⊆ Q : M is bounded and closed}. He proved for a complete metric space (Q, ρ) that if a map z : Q → CB(Q) satisfies the following condition: H(zp, zq) ≤ κρ(p, q) for each p, q ∈ Q where H is a Hausdorff metric and k ∈ [0, 1), then there is a point u0 ∈ Q which is a F ·P of the map z. After Nadler, a number of authors worked in this direction. Some of the refinements of Nadler fixed point theorem are by Reich, where he used H(Q) the collection of all compact subsets of a metric space Q and by Kamran who used Cl(Q), the collection of all non-void closed subsets of Q instead of CB(Q). Due to the importance of fixed point theory in diverse fields, some researchers have extended the idea of metric space in various ways. In 1993, Czerwik introduced the notion of a b-metric space by replacing the triangular property of a metric space with ρ(p, t) ≤ b[ρ(p, q) + ρ(q, t)], where b ≥ 1. Later on, in 2017 Kamran et al. further extended the concept of b-metric space by introducing extended b-metric spaces. They introduced a function θ : Q × Q → [1, ∞) instead of b in triangular inequality condition. In 2018, Mlaiki et al. gave the idea of controlled metric type spaces. They used θ : Q × Q → [1, ∞) instead of b in triangular inequality condition of metric spaces from a different approach from Kamran et al. In 2007, Huang and Zhang initiated the concept of cone metric space over a Banach space as the generalization of metric spaces. They used ordered Banach space E instead of R as the range set of metric ρ, i.e. they used ρ : Q×Q → E. They also discussed Banach type contraction and proved some fixed point results. After that, many researcher ii concentrated to investigate such spaces and proved a number of fixed point theorems. According to rough statistics, by using cone metric spaces, more than six hundred articles have been published. But recently some scholars obtained the equivalent results of usual metric space (Q, d∗ ) and that of cone metric space (Q, ρ). They defined the real valued metric function d ∗ as the non-linear scalarization function ξ. However, Liu and Xu in 2013 introduced cone metric space by using a real Banach algebra instead of Banach space and defined generalized Lipschitz mapping. They presented an example which established that results of fixed point in metric spaces are not equivalent to that of results in cone metric spaces over Banach algebras. The concept of distances in uniform spaces and metric spaces was first time presented by Valyi in 1985. We call it a Valyi-distances. After Valyi, some other researchers introduced different type of distances in metric spaces and in uniform spaces. Some well known distances are Tataru-distances by Tataru in 1992, ω-distance by Kada in 1996 and τ -distance by Suzuki in 2001. Recently in 2010 Wlodarczyk gave an idea of distances which provide a handy research tool to obtain more general results with weaker assumptions in uniform space known as generalized pseudo-distances. He also introduced generalized Hausdorff distances, gauge spaces, quasi-gauge spaces, triangular spaces, quasi-triangular spaces. The main objective of this thesis is to prove some fixed point theorems and proximity fixed point theorems for single-valued and multi-valued dynamical systems in metric type spaces. This thesis has been organized into six chapters. In Chapter 1, We have recollect some fundamental notions, some well-known contractions, abstract spaces and results in such spaces. Also, we present some basic concepts of comparison functions, introduction and basic theory of fractals in metric type spaces. At the end, we gave the theory of proximity fixed point in metric type spaces and generalized distances. In Chapter 2, we introduced a new geometrical structure which is the hybrid of cone metric space over Banach algebra and extended b-metric space. We prove analogues of iii Banach, Kannan and Reich type fixed point theorems in our predefined space. We also furnish with various concrete examples to establish the validity of our results. At the end, we have added some consequences and applications of our results. Recently, this work has been published in the journal of Filomat. Chapter 3, is concerned with the study of a new type of metric type space which we call a controlled cone metric type space over Banach algebra. By using such spaces we proved some fixed point theorems for generalized R-type contraction and generalized lipschitz mappings. We add an example to show the validity of our results. Work of this chapter has been published in the Journal of Inequalities and Applications. The aim of chapter 4 is two fold. Firstly, we produced several results concern with fixed point for the family of multi-valued contractions by using comparison functions in extended b-metric spaces. Then, we constructed some new multi-valued fractals based on a fixed point approach in the framework of extended b-metric spaces. Later on, using the idea of well-posed problem of fixed point is studied. Some of the results of this chapter has been published in the Journal of function spaces. Chapter 5 is intended to the study of theory of proximity points in controlled metric type spaces. We introduced generalized distances in controlled metric type spaces. We proved some global maximality results by using the defined generalized distances. Chapter 6 is the last chapter of this thesis, where we have introduced a new type of space which we named controlled quasi-triangular space. We introduced left(right) families generated by controlled quasi-triangular space. We proved Banach type theorem by using such families in controlled quasi-triangular space. At the end, we gave some concrete examples to validate our definitions and results.