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Product Code: 00006519

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In this project, we state and proved a generalization of Ciric fixed point theorem in generalized metric space by using a quasi-contractive map. Result presented in this project generalize and extend well known fundamental metrical fixed-point theorems in the literature in the setting of generalized metric space.



1.1 Introduction 1
1.2 Generalized Metric Space 3
1.3 Fixed Point Theorem 3
1.4 Quasi Metric Space 3
1.5 Aims and Objectives 4
1.6 Scope of the Study 4
1.7 Motivation 4

2.0 Historical Review 5

3.1 Basic definitions 7
3.1.1 Generalized metric space 7
3.1.2 Fixed Point Theorem 8
3.1.3 Quasi Metric space 9
3.1.4 Convergence of a Sequence 10
3.1.5 Left Cauchy 10
3.1.6 Right Cauchy 10
3.1.7 Cauchy Sequence 10
3.1.8 Completeness of Cauchy Sequence 11
3.1.9 Symmetric Metric Space 12
3.2 Propositions 12
3.3 Theorems 13
Banach Fixed Point Theorem 14
Brouwer Fixed Point Theorem 14

Fixed-Point Theorem in Generalized Metric Space
4.1 Application in Banach Space 16
4.2 Application in Cantor and Fractal Sets 18
4.3 Application on G-Metric space 19
4.4 Application in Quasi-Metric 20
4.5 Application in Economics 23

5.1 SUMMARY 25


1.1 Introduction
A generalized fixed-point theorem is a fundamental result in mathematics that extends the classical concept of fixed points to a broader class of functions and spaces. Fixed-point theorems provide powerful tools for establishing the existence of solutions to various equations and problems in diverse mathematical disciplines, including topology, analysis, and optimization. These theorems have far-reaching applications in fields such as economics, physics, and computer science.

The essence of a generalized fixed-point theorem lies in the idea of finding points within a space that remain unchanged under the action of a function. In its simplest form, a fixed point of a function f on set x is a point x∈X such that f(x)=x. However, in many practical situations, we encounter functions that may not have such simple fixed points, or we may need to work in more general spaces than just the real numbers.

A generalized fixed-point theorem relaxes the requirement of strict equality between f(x) and x and instead considers situations where f(x) is "close" to x in some sense. This closeness is typically defined using a topological or metric structure on the space x, and the theorem provides conditions under which such "approximate" fixed points exist.

The formal statement of a generalized fixed-point theorem involves several key elements:

1. The choice of a topological or metric space, denoted as (X,d), where x  is the underlying set, and d is the distance or metric function defining the space's topology.

2. A function f: X →X that maps elements of the space to itself.

3. The specification of a suitable property or condition, such as continuity or contractiveness, that  f must satisfy.

4. A statement that guarantees the existence of a point x∈X for which d(f(x),x) is sufficiently small, implying that f(x) is close to x according to the chosen metric or topological structure.

A metric is a function d: X × X → R (the real numbers) that assigns a non-negative real number to every pair of points in the set X. This function satisfies the following properties for all ,x, y, and z in X:
   a. Non-negativity:d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y.
   b. Symmetry: d(x,y) = d(y,x ) for all x, y in X.
   c. Triangle inequality: d(x,y) + d(y,z) ≥ d(x,z).

Generalized fixed-point theorems come in various forms, each tailored to specific mathematical contexts and requirements. Prominent examples include the Banach Fixed-Point Theorem, Brouwer's Fixed-Point Theorem, and Kakutani's Fixed-Point Theorem, among others. These theorems have been instrumental in solving problems across multiple disciplines and have paved the way for the development of many mathematical theories and applications.

In 2006, Mustafa and Sims introduced a new class of generalized metric spaces, which are called G-metric spaces, to overcome fundamental flaws in Dhage’s theory of generalized metric space. Subsequently, many fixed point results on such spaces appeared.
Brouwer was the first to prove a fixed point theorem which states that a continuous mapping of a closed unit ball in n-dimensional Euclidean space has atleast one fixed point. Several proofs of this basic result can be found in the existing literature.

Fixed point theory is one of the most powerful and fruitful tools of modern mathematics and may be considered a core subject of nonlinear analysis. In the last 50 years, fixed point theory has been a flourishing area of research for many mathematicians. The origins of the theory, which date to the later part of the nineteenth century, rest in the use of successive approximations to establish the existence and uniqueness of solutions, particularly to differential equations. This method is associated with many celebrated mathematicians like Cauchy, Fredholm, Liouville, Lipschitz, Peano, and Picard. It is worth noting that the abstract formulation of Banach is credited as the starting point to metric fixed point theory. But the theory did not gain enough impetus till Felix Browder’s major contribution to the development of the nonlinear functional analysis as an active and vital branch of mathematics.

Fixed point theory is an important area in the fast growing fields of non-linear analysis and non-linear operators. It is relatively young and fully developed area for research. There are several domains like classical analysis, functional analysis, operator theory, topology, algebraic topology, etc. Where the study of existence of fixed points falls. Fixed point theorems are mainly useful in the existence theory of differential equations, integral equations, partial differential equations, random differential equations, non-linear oscillations, fluid flows, approximation theory, chemical reactions, steady state temperature distribution, economic theories and in other related areas. Fixed point theory has a very fruitful application in eigon value problems as well as in boundary value problems.

In summary, generalized fixed-point theorems are essential tools in mathematics that provide a framework for finding approximate solutions to equations and problems when strict equality is not necessary or feasible. They have broad applications and are a cornerstone of many mathematical investigations and practical applications.

1.2 Generalized Metric Space
Let X be a non empty set. Suppose that  G:X×X×X→[0,+∞) is a function satisfying the following conditions: 

(G1). G(x,y,z)=0 if and only if x=y=z;

(G2) 0

(G3) G(x,x,y)≤G(x,y,z)  ∀ x ,y,z∈X with y≠z;

(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)=•••(symmetry in all three variables);

(G5)  G(x,y,z)≤G(x,a,a)+G(a,y,z)  ∀ x,y,z,a∈X .

Then the function G is called a generalized metric or more specifically G -metric on X and the pair (X,G)  is called a G -metric space.

1.3 Fixed point theorem
Let X be a topological space. We define a mapping T from  X into itself. If for every continuous mapping T from X into itself there exists a point x ∈X such that Tx = x, then we say that the topological space X has a fixed point property 

1.4 Quasi-metric space
Let X be a non empty set and d:X×X→[0,∞)  be a given function which satisfies 

(1) d(x,y)=0 if and only if x=y;

(2) d(x,y)≤d(x,z)+d(z,y)  for any points x,y,z∈X.

Then d is called a quasi-metric and the pair (X,d)  is called a quasi-metric space. 

1.5 Aim and Objectives
The aim of this project is to study fixed point theorem In generalized metric space. This will be achieved through the following objective:

To investigate some basic fixed-point principles on a metric space.

To investigate some necessary results in G -metric

1.6 Scope of the Study
The purpose of this project is to provide a broad view of fixed-point theorem in generalized metric spaces.

1.7 Motivation
Generalized fixed point theorems are powerful mathematical tools that have applications in various fields, including mathematics, economics, engineering, and computer science.

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