1. Introduction and Preliminaries

In mathematics, analysis plays an important role in the development of mathematics. Among several branches of analysis, functional analysis which deals with the study of several functions, come under Functional Analysis. It describes two types of functional analysis one is linear and another is non-linear functional analysis.

Fixed point theory is one of the most important topics of non-linear functional analysis since 1960. It has wide applications to the numerous fields of mathematics as well as outside mathematics such as differential equations, integral equations, variational problems, optimization problems, game theory, graph theory, image and signal processing, economics, and many more.

The notion of distance later known as metric space, introduced by M. Frechet in 1906, furnishes the common idealization of a large number of mathematical, physical and other scientific constructs in which the distance of a 'distance' appears. The objects under consideration may be most varied. They may be points, functions, sets, and even the subjective experiences of sensation. What matters is the possibility of associating a non-negative real number with each ordered pair of elements of a certain set, and that the number associated with pairs and triples of such elements satisfy certain conditions. However, in numerous instances in which the theory of metric spaces is applied, this very association of a single number with a pair of elements is, realistically speaking, an over idealization. This is so even in the measurement of an ordinary length, where the number given as the distance between two points is often not the result of a single measurement, but the average of a series of measurements. Indeed, in this and many similar situations, it is appropriate to look upon the distance concept as a statistical rather a determinate one. More precisely, instead of associating a number - the distance - with every pair of elements , one should associate a distribution function any for any positive number , interpret as the probability that the distance from to less than When this is done one obtains a generalization of the concept of metric space - a generalization which was first introduced by Austrian Mathematicians Karl Menger in 1942 and following him, is called a statistical metric space [8].

In this paper, we analyze the different contraction conditions in probabilistic metric space and their inter-relationships.

Definition 1.1: Metric space is a pair , where is a non-empty set and is a distance function or metric of the space defined by , satisfies the following conditions:

Example 1.1: Let be a non-empty set. For we define

Then, is discrete metric and the space is discrete metric space.

Definition 1.2: Let be a map. Then, an element is said to be fixed point of if

Example 1.2: Let , cubic equation.

Then, it can be transferred to as

Here,

So, by definition are fixed points of .

Definition 1.3: Let be a metric space and let be a mapping. Then, is called contraction if there exists a fixed constant such that

Example 1.3: Let be defined by,

Then, for all . So, is a contraction on . But is not continuous and thus not a contraction map.

Definition 1.4: For the set of real numbers, a function is called a distribution function if

(i) is non-decreasing,

(ii) is left continuous, and

(iii) and

If is a non-empty set, is called probabilistic distance on and is usually denoted by . We will denote by the family of all distribution function on and on

Example 1.4: Let is a maximal element for then, distribution function is defined by

Figure 1. Distribution Function

Definition 1.5: [13] A probabilistic metric space (brief, PM-space) is an order pair where is a non-empty set and is a function defined by (the set of all distribution functions) that is associates a distribution function with every pair of points in . The distribution function is denoted by whence the symbol will represent the value of at And the function are assumed to satisfy following conditions:

(i) ; (ii) , (iii) for every

(iv) For every and for every

The interpretation of as the probability that the distance from to is less than it is clear that PM condition (iii), (i) and (ii) are straight forward generalizations of the corresponding metric space conditions (i), (ii) and (iii). The PM condition (iv) is a 'minimal' generalization of the triangle inequality of metric space condition (iv). If it is certain that the distance of and is less than and like wise certain that the distance of and is less than , then it is certain that the distance of and is less than The PM condition (iv) is always satisfied in metric spaces, where it reduces to the ordinary triangle inequality.

Definition 1.6: [6] A mapping is called a triangular norm (shortly t-norm) if for all the following conditions are satisfied:

Example 1.5 of t-norms

The four basic standard t-norms are:

(i) The minimum t-norm, , is defined by

(ii) The product t-norm, is defined by

(iii) The Lukasiewicz t-norm, , is defined by

(iv) The weakest t-norm, the drastic product, , is defined by

With references to the point wise ordering, we have the following inequalities

Definition 1.7: [8] A Menger probabilistic metric space (briefly, Menger PM-space) is a triple where is a probabilistic metric space, is a triangular norm and also satisfies the following conditions, for all and . This is the extension of triangle inequality. This inequality is called Menger's triangle inequality.

Example 1.6: Let and

where

then is Menger Space.

Definition 1.8: [4] Let be a Menger Space and be a continuous t-norm (1) A sequence in is said to be converge to a point in (written ) iff for every and there exists an integer such that for all

(2) A sequence in is called a Cauchy if for every and , there exists an integer such that for all

(3) A Menger space in which every Cauchy sequence is convergent is said to be Complete Menger Space.

Banach Contraction Condition in Metric Space: The most basic fixed-point theorem is analysis known as the Banach Contraction Principle (BCP). It is due to S. Banach [1] and appeared in his Ph.D. thesis (1920, published in 1922). The BCP was first stated and proved by Banach for the Contraction maps in setting of complete normed linear spaces. At about the same time the concept of an abstract metric space was introduced by Hausdorff for the set valued mappings, which then provided the general framework for the principle for contraction mappings in a complete metric space. The BCP can be applied to mappings which are differentiable, or more generally, Lipschitz continuous.

Theorem 1.1: Let be a complete metric space, then each contraction map has a unique fixed point.

Example 1.7: Obviously is a Banach contraction and

where denotes the fixed point of the mapping .

2. Contraction Conditions in Probabilistic Metric Space

2.1. V.M. Seghal and A.T. Bharucha-Reid (B) Contraction Conditions in PM Space

The following definition of a contraction mapping was suggested and studied by V.M. Seghal and A.T. Bharucha-Reid in 1972, which is very natural probabilistic version of the notion of Banach contraction in metric space.

Definition 2.1.1: [12] The following definition of a contraction mapping was suggested and studied by V.M. Seghal and A.T. Bharucha-Reid in 1972, which is very natural probabilistic version of the notion of Banach contraction in metric space.

Let be a probabilistic metric space. A mapping is a contraction mapping (or a SB - Contraction mapping or B-contraction) on if and only if there is a such that

(2.1)

where and It is also known as probabilistic k-contraction.

The geometrical interpretation expression (2.1) is that the probability that the distance between the image points being less than is at least equal to the probability that the distance between that is less than

Dentition 2.1.2: [2] Let be a probabilistic metric space. A mapping is a probabilistic q-contraction if

(2.2)

for every and every

It is obvious that is a probabilistic q-contraction if and only if for every and every the following implication holds

(2.3)

The inequality (2.2) is a generalization of inequality.

where and is a metric space. In order to prove that (2.3) implies (2.2) recall that every metric space is also a Menger space , if is defined in the following way:

(2.4)

Suppose that is such that (2.3) holds and prove that (2.2) is satisfied i.e.,

that for every we have

If then and (2.3) implies

which means that

2.2. Hick’s Contraction (C) in PM Space

Definition 2.2.1: [7] T.L. Hicks in 1996, defined the following C-contraction mapping in PM space.

Let be a probabilistic metric space and . The mapping is called Hicks C-contraction (or, C-contraction) if there exists such that the following implication holds for every : and for every

Definition 2.2.2: [9] D.Mihet in 2005, introduced the weak- hicks contraction in PM Space as follows:

Let be a nonempty set and be a probabilistic distance on . A mapping is said to be weak - Hicks contraction (w-H contraction) if there exists such that, for all

Example 2.2.1: Let and

It is known ([10], [11]) that is a complete Menger space under the triangular norm . Also, it can easily be seen that the mapping ,

is a w-H contraction for every

2.3. Generalization of Bharucha (B)-Contraction

As a generalization of the notion of a probabilistic B-contraction, we shall introduce the notion of a probabilistic (m,k) - B-contraction where and

Definition 2.3.1: [6] If is a PM - space, and , a function is called probabilistic (m,k)-B-contraction if for any there is an with such that for every

If and then a probabilistic is a probabilistic

B-contraction.

As a generalization of C-contraction, we have

Definition 2.3.2: [6] If is a PM - space, and , a function is called a (m,k)-C-contraction if for any there is an with such that for every

If and then a probabilistic is a probabilistic C-contraction.

2.4. Probabilistic G-contraction Mapping

Definition 2.5.1: [5] g-contraction mapping is the generalization of Hick’s C-contraction in Probabilistic Metric Space. Let be two mappings defined on a Menger space with values into itself and let us suppose that is bijective. The mapping is called a probabilistic g-contraction with a constant if

and impies

The notion of g-contraction is justified because the images of two points under the function are nearer than images of the same points under the function

3. Conclusions [3]

The Probabilistic g-contraction is Hicks C-contraction when g = I, an identity mapping. Since H-contraction need not be B-contraction. So, Probabilistic g-contraction need not be B-contraction. Moreover, C-contraction is an extension of Banach contraction in Probabilistic Metric Space.

It is clear that

(i) (m-k) contraction C-contraction B-contraction Banach contraction

(ii) g-contraction C-contraction B-contraction

(iii) C-contraction (w-H) contraction