Common fixed point theorem of two mappings in menger PM: spaces

By using the concept of contraction of the two mappings in menger PM-spaces, the paper introduces coincidence and common fixed points in menger PM- spaces. The paper extends the result of the paper “Marwan Amin Kutbi” [5]

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Introduction

The concept of Probabilistic metric space (PM- Space) was first studied by Menger [1] in 1942 See also [2-4] There after some fixed point results were given by Schgal and Bharucha-Reid [6, 7] By using contractive condition in probabilistic metric space, they proved a unique fixed point

Result

which was an extension of Banach’s work [8] regarding fixed point theorem in metric space. Many fixed point result were proved in the space; see [13-20] In particular, Dutta et al. [21] Nonlinear – contractive mapping in Menger PM-space and proved the result using this contractive mapping in G-complete Menger PM-Spaces. Weaking this – contractive mapping, Marian Amin Kutbi, Dhananjag Gopal, Calogera Vetro and Wutiphol Sintumaverat [5] gave some fixed point results in G-complete and M-complete Menger PM-spaces. On the basis of studying the result [5]; we defined – contraction of one mapping with respect to f and proved the coincidence and fixed points of the two mappings in the Menger PM- space.

Here, we state some definitions which are needed to prove our result. We denote the set of real numbers by by , the set of non-negative real numbers and by , the set of positive integers. Definition 1.1 [9, 22] A mapping is called a distribution function if it is non- decreasing and left continuous with and We denote by the set of all distribution functions, while will always be denote the specific distribution function defined by Definition 1.2: [22] A binary operation is a continuous if the following conditions hold: (a) is commutative and associative, (b) is continuous. Maths 2016; 1(3): 52-57 Vinod Kumar Department of mathematics, Govt P.G.

College, Hisar, Haryana, India Ankush Department of mathematics, Govt P.G. College, Hisar, Haryana, India. Correspondence: Vinod Kumar Department of mathematics, Govt P.G. College, Hisar, Haryana, India. (c) T (a, 1) = a for all ,1] (d) T(a, b), whenever and , for The following are three basic continuous from the literature : i.

The minimum , say defined by . ii. The product , say defined by iii. The Lukasiewicz defined by These are related in the following way : Definition 1.3: [5] A menger space is a triple where is a nonempty set, is a continuous and is a mapping from into such that, if denoted the value of at the pair , the following conditions hold if and only if for all for all, for all . Definition 1.4: [5] Let be a Menger PM-space, then (i) A sequence in is said to be convergent to if, every and, there exists a positive integer such that whenever . (ii) A sequence in is called Cauchy sequence if, for every and, there exists a positive integer such that whenever . (iii) A Menger PM-space is said to be M-complete if every Cauchy sequence in is convergent to a point in . (iv) A sequence is called G-Cauchy if for each and . (v) The space is called G-complete if every G-Cauchy sequence in is convergent.

The following class of functions was introduced in [10] and will be used in proving our results in the next section. Definition 1.5: [11] A function is said to be function if it satisfies the following conditions: (i) if and only if . (ii) is strictly increasing and as (iii) is left continuous in (iv) is continuous at 0 Definition 1.6: Let be a Manger PM- space and , be the self mappings. A point in is called a coincidence point (common fixed point ) if .Also the pair , of mappings are weakly compatible if they commute on the set of coincidence points. Definition 1.7: ) Let be a Manager PM-space.

The probabilistic metric is triangular if it satisfies the condition For every and each In the sequel, the class of all – functions will be denoted by .Also we denoted the class of all continuous non-decreasing functions such that and as Theorem 1.1 [5] Let be a -complete Menger space and be a mapping satisfying the following inequality: Where and such that then f has a unique fixed point. A mapping satisfying condition is usually called contractive mapping. However for some discussion on this notion and theorem 1.1 the reader can refer to the recent paper of Gopal et al [23] where analogous result are proved by using some control function 2. The Main Results Theorem 2.1 Let be a menger space and be the mapping satisfying the following inequality Where and such that If the range of is a - complete subspace of, then and have coincidence point.

Further if the pair of mapping ) is weakly compatible, then and have a common fixed point. Proof Let . Take a point in such that . This is possible as the range of contain the range of .

Continuing in this way, for every in . One can find a such that Without loss of generality assume the for all ; otherwise and have a coincidence point and there is nothing to prove. In case From (2.2) we deduce that and Again by applying we get that is On using and the hypothesis that is non – decresing the above inequality becomes Repeating the above procedure successively times, we obtain If we change with in the previous inequality then for all , We get (2.5) Since whenever therefore the above inequality implies that Now let be given, then by using the properties (i) to (iv) of a function we can find such that . It follows from (2.6) that for every (2.7) Hence is a G-Cauchy sequence in and is G-complete.

Therefore as , for some . Consequently we obtain a point in such that Now we show that is coincidence point of and . Since, By using the properties (i) and (iv) of a function , we can find such that . Again since as , then there exists such that, for all we have …… .

Therefore, for every we obtain …. Since is continuous at 0 and , we obtain From (2.8) and (2.9), we get for every which implies that . This proves that u is coincidence point of and T. Further suppose .

Since and T are compatible, then Obviously, Now we show that . (2.10) Since is continuous at 0 and , we obtain From (2.8) and (2.10), we get for every , which implies that Hence is fixed point of T and . Further suppose that w is another fixed pint of T and , then Since is continuous at 0 and , we obtain From (2.8) and (2.10), we get =1 for every which implies that Thus we have guarantee of uniqueness of fixed point.

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