A FIXED POINT THEOREM FOR A-CONTRACTIONS ON A CLASS OF GENERALIZED METRIC SPACES.
ABSTRACT
This project explains the concept of fixed point theory and contractive mappings in the context of generalized metric spaces, a broader framework that encompasses classical metric spaces. The study of fixed point theorems has far-reaching implications in various fields of mathematics, and this work contributes to the understanding of these theorems when applied to a class of generalized metric spaces. The primary objective of this research is to establish fixed point theorems for a specific type of contraction mapping defined on generalized metric spaces. We begin by providing a comprehensive review of classical fixed point theorems, such as the Banach Fixed Point Theorem and the Brouwer fixed point theorem and their relevance in metric spaces. Subsequently, we introduce the concept of generalized metric spaces and explore their fundamental properties. The core of this project revolves around the development and proof of fixed point theorems for contractions in generalized metric spaces. We derive necessary conditions for the existence of fixed points under these contractions and investigate the convergence behavior of iterative methods designed to approximate these fixed points. Our analysis extends beyond Euclidean spaces, allowing for a broader application in various mathematical and scientific disciplines. In conclusion, this project not only provides novel fixed point theorems for contractions in generalized metric spaces but also illustrates their relevance in solving complex problems across different domains of mathematics and science. The findings presented here open up new avenues for research and offer valuable insights into the study of fixed points in a broad mathematical context.
TABLE OF CONTENTS
DECLARATION ii
CERTIFICATION iii
DEDICATION iv
ACKNOWLEDGEMENTS v
ABSTRACT vi
TABLE OF CONTENTS vii
CHAPTER ONE
1.1 Background of the study 1
1.2 Statement of the problem 2
1.3 Aims and Objectives of the Study 2
1.4 Scope of the study 3
1.5 Significance of the Study 3
1.6 Definitions of Terms 3
1.6 Organization of the project 6
CHAPTER TWO
2.1 INTRODUCTION 8
2.2 A Review on the Study of Fixed Point Theorem 12
2.2 Overview of Generalized Metric Spaces: 13
CHAPTER THREE
3.1 BANACH FIXED POINT THEOREM 15
CHAPTER FOUR
4.0 INTRODUCTION 22
4.1 Generalized metric spaces 22
4.2 A-contractions 24
CHAPTER FIVE
5.1 Summary 28
5.2 Conclusion 28
5.3 Recommendations 29
REFERENCES 30
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background of the study
The Banach Contraction Principle, formulated by Stefan Banach in the early 1920s, holds a special place in the pantheon of mathematical theorems. At its core, it provides a rigorous framework for the existence and uniqueness of fixed-points for certain types of functions defined on metric spaces. Its elegance lies in its simplicity and generality, making it a cornerstone of mathematical analysis with far-reaching consequences. The Banach Contraction Principle, also known as the Contraction Mapping Theorem, stands as a fundamental result in mathematical analysis with profound implications in various branches of mathematics and applications in science and engineering.
Brinciari (2000) introduced the notion of generalized metric space and obtained a generalization of the Banach contraction principle, where after many authors proved various fixed point results in such spaces by replacing the triangular inequality by a more general using simple work.
The Banach Contraction Principle remains a cornerstone of mathematical analysis. Its elegant formulation and far-reaching consequences make it a testament to the beauty and utility of mathematics. As we continue to explore and expand the frontiers of mathematical theory and its applications, this fundamental theorem continues to shape our understanding of fixed-points, convergence, and the beauty of mathematical reasoning.
1.2 Statement of the problem
Das (2002) studied fixed point for Kannan’s type maps of generalized metric spaces investigating the existence and properties of points that remain unchanged under a given mapping or transformation. While this field of study has many applications and is widely used in various branches of mathematics, it also presents several challenges and problems.. One of the fundamental questions in the study of fixed-points is whether a given mapping or transformation has a fixed-point in a given metric space. In some cases, it can be challenging to determine the existence of fixed-points, especially when dealing with complex or nonlinear mappings. Even if a mapping has a fixed-point, it may not be unique. Multiple fixed-points can exist, and determining their number and properties can be a difficult task. In some cases, there may be an infinite number of fixed-points, making the analysis even more complex.
Fixed-point theorems often rely on the completeness of the metric space. However, many interesting metric spaces encountered in practice, such as function spaces or infinite-dimensional spaces, are non-compact. Dealing with non-compact metric spaces requires the development of specialized techniques and theorems. Fixed-point theorems are often applied to nonlinear mappings, which can introduce additional difficulties. Nonlinear mappings may exhibit complex behavior, such as bifurcations, chaos, or multiple fixed-points. Analyzing the properties of fixed-points in nonlinear mappings can be challenging and may require advanced mathematical tools.
1.3 Aim and objectives of the study
The aim of this research work is to study the existence of fixed point for A- Contraction on a specific class of generalized metric spaces. This aim can be achieved through the following objectives:
1. To define and explain the concept of a fixed point in a metric space.
2. To study various fixed-point theorems and their proofs, such as the Banach fixed-point theorem and its application.
3.We apply Banach fixed point theorem to proof the existence of A-contraction on a specific class of generalized metric spaces.
1.4 Scope of the study
The scope of this research work is to demonstrate the existence of fixed-points theorem for A-contractions defined on a specific class of generalized metric spaces. .A-Contractions: A-contractions are generalization of contractions in metric spaces. In this research work, we analyse some properties of Banach contraction principle in a generalized metric space with some specific conditions.
1.5 Significance of the study
The study of a Fixed Point Theorem for A-Contractions in a Class of Generalized Metric Spaces" is poised to make a substantial contribution to the field of mathematics, with broad-reaching significance in both theoretical and practical contexts. This research work significance lies in its potential to advance mathematical theory. Fixed point theorems are fundamental in mathematics, providing a bedrock for various mathematical disciplines. By extending the theory to A-contractions in generalized metric spaces, this research expands the toolbox available to mathematicians, enabling the exploration of more complex mathematical structures.
1.6 Definitions of terms
The definition, examples and theorem in this section could be found in (Akram M. Zafar A.A and A.A Siddiqui, A general class of contract ions: A- contractions, submitted (1972).
1.6.1 Metric spaces
Let X be non-empty set, then the function d∶ X × X → R such that for all x,y ∈ X:
(1) d(x,y) ≥ 0 and d(x,y) = 0 if and only if x = y
(2) d(x,y) = d(y,x) (symmetry)
(3) d(x,y) ≤ d(x,z) + d(z, y) (triangle inequality) d is a metric defined on X and the pair (X,d) is called a metric space.
1.6.2 Lipschitz continuity
A mapping T: X→X is said to satisfy the Lipschitz condition with constant α if d(Tx,Ty) ≤αd(x,y) holds for all x,y∈X
1.6.3 Contraction
A mapping T on a metric space X is a contraction if there exists a constant 0≤k<1 such that for all x,y ∈X, the following condition hold, d (Tx ,Ty)≤ k d (x,y).Where d is the metric or distance function on the metric space.
1.6.4 Cauchy sequence
A sequence (xn)=(x1,x2,….), where xn ∈X for every n, is called a cauchy sequence in a metric space (X,d) if and only if d(xn,xm)→0 (m,n→∞),i.e for every ε>0 there exist N0=N0(ε) such that d(xn,xm)<ε for all n,m >N0.
1.6.5 Complete metric space
A metric space (X,d) is called complete if and only if every Cauchy sequence converges (to a point of X). Explicitly, we require that if d(xn,xm)→0(m,n→∞), then there exist x∈X such that d(xn, x)→0(n→∞ ).
1.6.6 Fixed-point
Let S be a set and F a function from S to S. A fixed-point of F is simply a point a∈S such that F (a)=a . In other words, a fixed-point of F is nothing but a solution of function equation, F (x)=x,for all x∈S.
1.7 Organization of the project
This project will provide a comprehensive background on A-contractions and generated metric spaces. A-contractions are a class of mappings that have applications in diverse fields, including computer science, economics, and physics. Generalized metric spaces, on the other hand, are an important subset of metric spaces with wide-ranging practical implications. The motivation behind this research lies in the need to bridge the gap between theory and application by developing a fixed theorem that characterizes A-contractions in a specific class of generalized metric spaces.
The project's methodology will be organized meticulously, outlining the steps involved in proving the fixed theorem for A-contractions in the chosen class of generated metric spaces. This section will provide a clear roadmap for conducting the research, including the choice of mathematical tools, techniques, and data analysis methods.
The literature review used in this project begins by thoroughly surveying the existing body of research on A-contractions and generated metric spaces. This includes academic papers, books, and relevant publications. The goal is to provide an in-depth understanding of what has already been accomplished in these areas, it also critically analyzes the methodologies employed in previous research. This involves assessing the mathematical tools, techniques, and data analysis methods used in existing studies. It is essential to evaluate the strengths and weaknesses of these methodologies to guide the project's approach.
The methodology used in the project involves a comprehensive theoretical analysis of A-contractions in the chosen class of generalized metric spaces. This includes studying relevant theorems, definitions, and existing mathematical structures. Theoretical analysis is crucial for understanding the fundamental properties and behaviors of A-contractions. In the concluding section, the project's findings will be summarized, highlighting the significance of the fixed theorem for A-contractions in the specified class of generated metric spaces.
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