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

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 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 (53), we have Suppose that , . Then, operating by -norm on the above equations, from the condition , we obtainThis implies thatIn the next step we show that is Cauchy sequence. For each , suppose that . Then, . From (56) we see that . By Lemma 5, we haveBy Lemma 6, for each , there exists such thatSuppose that and are given. Since , there exists such that for all . Thus, by (57) and (58), we have . Using Lemma 5, we obtain that for all ; that is, is a Cauchy sequence. Similarly, we can show that are also Cauchy sequence.
In a similar manner of Theorem 16, we can find and having -tuple coincidence point.

3. -Tupled Coincidence Point Results in Non-Archimedean Menger Spaces

In this section, we are going to prove two coincidence point theorems in non-Archimedean Menger space.

Theorem 19. Let be a non-Archimedean Menger PM-space such that and . 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.If there exist such that for any then there exists such that and have -tupled coincidence point in .

Proof. From (59), we have supposed that ,Suppose that , . Then operating -norm above with , we haveThis implies thatThus, we haveSuppose that and are given. By (60), there exists such thatfor all and . Hence, from (64) it follows thatThis shows that is a Cauchy sequence. Similarly we get the following:Hence are all Cauchy sequence. Since is complete, there exist such that , . Again implies the existence of so thatHenceWe haveTaking the lim , we getNow again, we haveTaking lim , we getSimilarly, we haveTaking the lim , we getThe above implies thatbut and are weakly compatible, so that (76) implies thatHence and have -tuple coincidence point.

In the following theorem we are going to replace inequality (59) by (78).

Theorem 20. Let be a non-Archimedean 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 that . From (78), we haveThen, from the above it follows that . we haveIn the next step, we will show that are Cauchy sequences. For this , suppose that . Then, . From (80) we see that . It follows from Lemma 5 thatUsing Lemma 7 we obtain that, for each , there exists such thatSuppose that and are given. Since , there exists such that for all . Thus, by (81) and (82), we have . Furthermore, by Lemma 5 we have for all ; that is, is a Cauchy sequence. Similarly, we can show that are all Cauchy sequence.
In a similar manner of Theorem 19 we can find and having -tuple coincidence point.

4. Corollaries and Examples

Now we are ready to deduce some of the main theorems to obtain the following corollaries.

In this section, we will obtain some corollaries and an example of our main results proven in Sections 2 and 3.

By putting (identity map on ) in Theorem 16 and as an immediate consequence, we have the following.

Corollary 21. Let be a Menger PM-space such that is a -norm of -type. Let be a gauge function such that , , and for any . Let be a mapping such thatfor all , and . Then there exists such that has -tupled fixed point in .

Since each hybrid contraction with a gauge function includes the case of linear contraction as a special case, if we set or , , in theorems of Section 2, then we have the following -tupled coincidence points for the mappings and corollaries as follows.

Corollary 22. Let be a Menger PM-space such that is a -norm of -type and . 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 .

If we take the mapping as the identity mapping on in Corollary 22, we get the following -tupled fixed point theorems for the mapping .

Corollary 23. Let be a complete Menger PM-space such that is a -norm of -type and . Let be a mapping such thatfor all , and . Then there exists such that has -tupled coincidence point in .

Now we are ready to give an illustrative example to support our main Theorem 16. One can see further example of this nature in [12, 14].

Example 24. Suppose that . Then is a -norm of -type and . Define byWe claim that is a Menger PM-space. Conditions (1), (2), and (3) of Theorem 16 are very easy to check. To prove condition (4), we assume that for all andThen we have , and so . It follows thatHence condition (4) holds. It is clear that is complete.

Suppose that .

For , define and as follows:respectively.

Note that is the point of coincidence of and . It is clear that the air is weakly compatible on .

Also we will show that the pair is not compatible. Let us consider the sequences

, , (when is even); thenNowand , , .

Hence .

Similarly, we can check when is odd. For this we can consider the sequence , , (when is odd); thenAnd

Hence pair is not compatible. Then, for each and for each , we have

NowHence all the conditions of Theorem 18 are satisfied and is the point of coincidence of and .

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgment

The First author (P. P. Murthy) would like to thank University Grants Commission, New Delhi, India, for the financial support through the Major Research Project (MRP) (file number 42-32/2013 (SR)).