Table of Contents
International Journal of Computational Mathematics
Volume 2016, Article ID 7109190, 14 pages
http://dx.doi.org/10.1155/2016/7109190
Research Article

-Tupled Coincidence Point Theorems for Probabilistic -Contractions in Menger Spaces

Department of Pure and Applied Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, Chhattisgarh 495009, India

Received 14 July 2015; Revised 22 September 2015; Accepted 18 October 2015

Academic Editor: Don Hong

Copyright © 2016 Penumarthy Parvateesam Murthy and Uma Devi Patel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduced -tupled coincidence point for a pair of maps and in Menger space. Utilizing the properties of the pseudometric and the triangular norm, we will establish -tupled coincidence point theorems under weak compatibility as well as -tupled fixed point theorems for hybrid probabilistic -contractions with a gauge function. Our main results do not require the conditions of continuity and monotonicity of . At the end of this paper, an example is given to support our main theorem.

Dedicated to late Professor V. Lakshmikantham

1. Introduction and Preliminaries

Probabilistic metric space was introduced by Menger [1] in the year 1942 by generalizing metric spaces in which a distribution function was used instead of nonnegative real number as value of the metric.

Now we present some basic concepts and results which will be used in this paper.

Throughout this paper we will denote as the set of real numbers, as the nonnegative real numbers, and as the set of all positive integers.

If is a function such that , then is called a gauge function. If , then denotes the th iteration of and .

A mapping is called a distribution function if it is nondecreasing and left continuous with , .

We will denote by the set of all distribution functions and by the specific distribution function defined by

Now, we are ready to recall the following definitions and lemmas for our main results in Section 2.

Definition 1 (see [2]). A function is called a triangular norm (in short, -norm) if the following conditions are satisfied for any : (1);(2);(3), for ;(4).Examples of -norms are and for all and for each -norm.

Definition 2 (see [2, 3]). A triplet is called a Menger probabilistic metric space, if is a nonempty set, is a -norm, and is a mapping from into . We will denote the distribution function by and will represent the value of at satisfying the following conditions:(1);(2) for all if and only if ;(3) for all and ;(4) for all and . is called a non-Archimedean Menger PM-space if it is a Menger PM-space satisfying the following condition:(1) for all and .Schweizer and Sklar [4, 5] pointed out that if the -norm of a Menger PM-space satisfies the condition , then is a first countable Hausdorff topological space in the -topology ; that is, the family of setsis the base of neighborhoods of a point for , whereBy virtue of this topology , a sequence in is said to be convergent and converges to (we write or ) if for all ; is a Cauchy sequence in if any given and , and there exists such that whenever . is said to be complete, if every Cauchy sequence in is a convergent sequence in .

Lemma 3 (see [6, 7]). Let be a usual metric space. Define a mapping byThen is a Menger PM-space; it is called the induced Menger PM-space by and it is complete, if is complete.

An arbitrary -norm can be extended [3, Definition  2.1] in a unique way to an -array operation. For (), the value is defined by and . For each , the sequence is defined by and .

Definition 4 (see [8]). A -norm is said to be of -type if the sequence of functions is equicontinuous at .

The -norm is a trivial example of a -norm of -type, but there are -norms of -type with (see [8]). It is easy to see that if is of -type, then satisfies .

Lemma 5 (see [7, 9]). Let be a Menger PM-space. For each , define a function byThen the following statements hold: (1) if and only if ;(2) for all and ;(3) for all if and only if .

Lemma 6 (see [7]). Let be a Menger PM-space and let be a family of pseudometrics on defined by (13). If is a -norm of -type, then, for each , there exist such that, for each ,

Lemma 7 (see [10]). Let be a non-Archimedean Menger PM-space and let be a family of pseudometrics on defined by (13). If is a -norm of -type, then, for each , there exist such that, for each ,

Lemma 8 (see [11]). Suppose that . For each , let be nondecreasing and satisfy for any . If

In this paper we used the new definitions of -tupled coincidence point given by Imdad et al. [12] and -tupled fixed point given by Samet and Vetro [13].

The following definitions are also needed for our main results.

Definition 9 (see [12]). An element is called -tupled common fixed point of the mapping if , , .

Definition 10 (see [12]). An element is called an -tupled coincidence point of the mappings and if , , .

Definition 11 (see [12]). An element is called an -tupled common fixed point of the mappings and if , , .

Now, we are ready to introduce the concept of commutativity, compatibility, and weak compatibility in Menger PM-spaces for -dimensions.

Definition 12. Let be a nonempty set. Let and be two mappings. is said to be commutative with if for all . A point is called a common fixed point of and if .

Definition 13. Let and be two mappings. Then is said to be -compatible ifwhere are the sequences in such thatfor some are satisfied.

Definition 14. The mappings and are called weakly compatible maps ifimplyfor all .

In this paper, we will introduce -tupled coincidence point, -tupled fixed point, commutativity, compatibility, and weak compatibility in Menger space for function of higher dimension. Utilizing the properties of the pseudometric and the triangular norm, we will establish -tupled coincidence point results as well as -tuple fixed point results using weak compatibility of mappings for hybrid probabilistic -contractions with a gauge function in Menger spaces.

2. Main Results

Lemma 15. Let ba a nonempty set. Let and be two mappings. If , then there exist sequences in such that , .

Proof. Let be arbitrary points in , since .
We define such thatAgain, for we can choose such thatContinuing this process, we can construct sequence in such that

Now we will establish the following theorem by using Lemma 15.

Theorem 16. Let be a Menger PM-space such that is a -norm of -type. Let be a gauge function such that , , and for any . Let and be two mappings such thatfor all , and , where (1);(2) is complete;(3)the pair is weakly compatible.Then there exists such that and have -tupled coincidence point in .

Proof. By Lemma 15, we can construct sequences in such that , , .
Let . From (16), we haveSuppose that . Then from the above inequalities, we obtainThis implies thatSince = and for each , using Lemma 8, we haveTherefore, we getNow we claim that, for any ,Assume that (22) holds for some . Since , we have . By (16) and (22), we haveHence, by the monotonicity of , we haveSimilarly, we obtainTherefore by the induction, (22) holds for all . Suppose that and are given. By hypothesis, is a -norm of -type. There exists such thatBy using (21), there exist such that for all . Hence from (22) and (26) we getfor all and .
Therefore are all Cauchy sequence. Since is complete, there exist such that , . Again implies that there exists such thatHenceFrom (16) and , we haveTaking the lim , we getNow again, we haveTaking the lim , we getSimilarly we haveTaking the lim , we getSo, we haveNow we suppose that and are weakly compatible maps, so (36) implies thatHence and have -tuple coincidence point.

By replacing inequality (16) in Theorem 16 by (38), we have the following theorem.

Theorem 17. Let be a Menger PM-space such that is a -norm of -type. Let be a gauge function such that , , and for any . Let and be two mappings such thatfor all , and , where(1);(2) is complete;(3)the pair is weakly compatible.Then there exists such that and have -tupled coincidence point in .

Proof. Suppose . From (38), we haveSuppose that , . Then from the above inequalities, we obtainThis implies thatSinceand for any , we have .
Moreover, we haveHenceThis implies thatIn the next step we show that, for any ,We will prove the above by induction method; for , it is obvious. Assume that (46) holds for some . Since , we haveNow we haveThus, by the monotonicity of , we haveSimilarly we getSuppose that and are given. Since is -norm of -type, there exist such thatBy (45), there exist such that for all . Hence from (50) and (51), we getfor all and .
Therefore are all Cauchy sequence.
Arguing as in Theorem 16, we have and having -tuple coincidence point.

Again replacing inequality (38) by (53), we have the following result.

Theorem 18. Let be a Menger PM-space such that is a -norm of -type and . Let be a gauge function such that and for any . Let and be two mappings such that