Study of Some Coupled Fixed Point Theorems in Partially Ordered S-metric Space
Chapter One
Aim and Objectives
The aim of this dissertation is to present some coupled common fixed point results with mixed weakly monotone mappings in the setting of partially ordered S-metric space. The aim is achieved through the following objectives: to
- review the theory of the new generalized metric space called S-metric space;
- study some recently established results of coupled fixed point for mappings with the mixed monotone property and mixed weakly monotone property in partially ordered metric space.
CHAPTER TWO
LITERATURE REVIEW
Introduction
Fixed point theory is an exciting branch of mathematics. It is a mixture of Analy- sis, Topology and Geometry. Over the last 50 years or so the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. It has numerous applications in almost all areas of mathematical sci- ences. For example, in Functional Analysis, it is used for proving the existence of solutions of ordinary and partial differential equations, integral equations, system of linear equations, closed orbit of dynamical systems. In particular fixed point tech- niques have been applied in such diverse fields as Biology, Chemistry, Economics, Engineering and Physics. The concept of fixed point plays a key role in Analysis. Also, fixed point theorems are mainly used in existence theory of random differential equations, numerical methods like Newton-Rapshon method and Picard’s Existence Theorem and in other related areas. It plays a major role in many applications, such as variational and linear inequalities, optimization and applications in the field of approximation theory.
Brief Historical Development
Metric fixed point theory started from the late 19th century, when successive ap- proximations were used to establish the existence and uniqueness of solutions to equations, and especially differential equations. This approach is particularly asso- ciated with the work of Picard, although it was Stefan Banach who in 1922 developed the ideas involved in an abstract setting. The fixed point theorem generally known as
Banach’s contraction mapping principle or Banach’s fixed point theorem, appeared in explicit form in Banach’s thesis in 1922, and states that every self contraction mapping defined on a complete metric space has a unique fixed point. That is if
(X, d) is a complete metric space and T ∶ X → X satisfies d(Tx, Ty) c kd(x, y) for all x, y ‹ X, for some k ‹ (0, 1), then T has a unique fixed point.
Banach’s contraction principle ensures under appropriate conditions the existence and uniqueness of a fixed point. It is one of the most important results in fixed point theory. The principle is based on an iterative scheme called Picard iterative scheme. Many authors have extended, improved and generalized Banach’s theorem in several ways. For example Agarwal et al. (2008), Ariza-Ruiz (2009), Chatterjea (1972), Choudhury (2009), Harjani and Sadarangani (2009) and Rhoades (1977).
Gahler (1963) introduced the notion of a 2-metric space as follows:
Definition 2.2.1 Let X be a nonempty set. A function d ∶ X3 → R is said to be a b2-metric on X if the following conditions hold:
(d1) For any distinct point x, y ‹ X, there exist z ‹ X d(x, y, z) ≠ 0
(d2) d(x, y, z) = 0 if any of the two elements of the set {x, y, z} in X are equal. (d3) d(x, y, z) = d(x, z, y) = d(y, x, z) = d(z, x, y) = d(y, z, x) = d(z, y, x)
(d4) d(x, y, z) c d(x, y, a) + d(x, a, z) + d(a, y, z) for all x, y, z, a ‹ X
The pair (X, d) is called a 2-metric space.
Gahler (1963) claimed that 2-metric space is a generalization of an ordinary metric space. He mentioned that d(x, y, z) geometrically represents the area of a triangle formed by the points x, y, z ‹ X as its vertices. However Sharma (1980) found some mathematical flaws in these claims. It was demonstrated that d(x, y, z) does not always represent the area of a triangle formed by the points x, y, z ‹ X (Sharma, 1980).
CHAPTER THREE
THEORY OF METHODS
Introduction
In this chapter, some properties of S-metric space are studied which are needed to prove the main results.
Properties of S-metric space
Definition 3.2.1 (Sedghi et al. (2012)). Let (X, S) be an S-metric space. For r > 0 and x ‹ X, we define the open ball BS(x, r) and the closed ball BS[x, r] with centre x and radius r as
BS(x, r) = {y ‹ X ∶ S(y, y, x) c r}
BS[x, r] = {y ‹ X ∶ S(y, y, x) c r}
Definition 3.2.2 (Sedghi et al. (2012)). Let (X, S) be an S-metric space and A ⊂ X.
A sequence {xn} in X is said to converge to x ‹ X if and only if S(xn, xn, x) → 0
as n → ∞. That is for each ϵ > 0 there exists nє ‹ N such that ∀n ≥ nє, S(xn, xn, x) c ϵ. Whenever {xn}n≥1 converges to x, we write limn→∞ xn = x.
A sequence {xn} in X is called a Cauchy sequence if and only if for each ϵ > 0,
∃ nє ‹ N such that for all n, m ≥ nє S(xn, xn, xm) c ϵ.
CHAPTER FOUR
COUPLED FIXED POINT IN S-METRIC SPACE
Introduction
In this chapter, we establish some coupled fixed point theorems in the framework of
S-metric space.
Main Results
Theorem 4.2.1 Let (X,c,S) be a partially ordered complete S-metric space. Let f,g: X × X → X be the mappings such that a pair (f,g) has the mixed weakly monotone property on X. Let x0, y0 ‹ X be such that x0 c f (x0, y0), y0 k f (y0.x0) or x0 c g(x0, y0), y0 k g(y0.x0). Suppose that there exist p,q,r,s ≥ 0 with p+q+r+2s c 1 such that.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
In this dissertation we considered the notion of S-metric space as a generalized metric in 3-tuples and studied some of its properties and existing results.
We reviewed some coupled fixed point theorems in partially ordered metric space and used the notion of a mixed weakly monotone pair of mappings to establish some coupled common fixed point theorems in partially ordered S-metric space.
The results presented in chapter four (theorems 4.2.1 and 4.2.2) generalize the results of Gordji et al. (2012) (theorems 2.3.3 and 2.3.4) in the framework of S-metric space.
Conclusion
In this study, we used a similar method for proving the existing results of coupled fixed point theorems for mixed weakly monotone mappings in partially ordered metric space have been used to prove our results in the framework of S-metric space.
The results reveal that though depending on the contractive mapping or contractive condition used for an existing result in metric space, one may generalize such results in the setting of S-metric space.
Recommendations
Various fixed point results for different contractive mappins in metric space have been established. Some of these results have not been (or cannot be) generalized in the framework of S-metric space.
We therefore recommend that the existence of fixed point for different contractive
type mappings in the setting of S-metric space should be investigated.
Also a pair of mixed weakly monotone mappings may be extended to a more general setting of multivalued mixed weakly monotone mappings.
References
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