SOME RESULTS BY USING CLR’s-PROPERTY IN PROBABILISTIC 2-METRIC SPACE V. SRINIVAS1, K. SATYANNA2,*

The aim of this paper is to generate two fixed point theorems in probabilistic 2-metric space by applying CLR’S-property and occasionally weakly compatible mappings (OWC), these two results generalize the theorem proved by V. K. Gupta, Arihant Jain and Rajesh Kumar. Further these results are justified with suitable examples.


INTRODUCTION
Menger [1] pioneered the statistical metric(SM) space theory. One of the major achievements was the translation of probabilistic concepts into geometry. Menger used the notation of new distance distribution function from p to q by a Fpq. B. Schweizer, and A. Sklar [2] introduced a new notion of a probabilistic-norm. This norm naturally generates topology, convergence ,continuity and completeness in SM-space. Mishra [3] used compatible mappings and generated some fixed points in Menger space. Altumn Turkoglu [4] proved some more results of SM-space by utilizing the implicit relation in multivalued mappings. Zhang, Xiaohong, Huacan He, and Yang Xu [5] employed the Schweizer-Sklar t-norm established fuzzy logic system to contribute in development of SM-space. Sehgal, V. M., and A. T. Bharucha-Reid [6] used classical Banach contraction to establish the first result of Menger space for coincidence points. Weakly compatible mappings were generalized by Al-Thagafi and Shahzad [7], by introducing occasionally weakly compatible mappings. Futher Chauhan, Sunny, Wutiphol Sintunavarat, and Poom Kumam [9] proved some more theorems by using CLR'S-property in fuzzy metric space. Further some more results can be witnessed by using the concepts of sub sequentially continuous and semi compatible mappings in Menger space [10].

(iii)
If the cauchy sequence converges in X then it is referred as a complete 2-Menger space.
We notice that the pair (P, S) has two coincidence points 0, (2.6.1) This shows the mappings P, S are OWC but not weakly compatible.
Definition 2.7 [9] "Self maps P and S of a 2-Menger space (X, F, t) are said to satisfy CLR'S -property (common limit range property) if there exists a sequence 〈 〉 ∈ ∋ , → , for some element z ∈ as n → ∞.
This example shows that mappings P, S satisfy CLR'S-property but they do not have closed ranges.
As a result, because Cauchy sequence exists in complete space X, it has a limit z in X and consequently each sub sequence has the same limit z.
Combining all we get Az = Bz = z = Sz = Tz.
Thus is the required common fixed point for these mappings A, B, S and T.

Uniqueness:
Assume z1 is second common fixed point. Now assume z ≠ z1.
By considering y = z1 , x = z in (3.1.4) we obtain Now we justify our theorem as under.

Example
Let us take X  ( * ) for all elements x, y in X and * > 0 r is continuous self-map on [0, 1] such that r( * ) > * for each o < * < 1.
Then A, B, S and T have unique common fixed point in X.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.