International Journal of Computational Mathematics
Volume 2016 (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

Penumarthy Parvateesam Murthy and Uma Devi Patel

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 sets

is the base of neighborhoods of a point for , where

By 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 by

Then 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 by

Then 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 if

where are the sequences in such that

for some are satisfied.

Definition 14. The mappings and are called weakly compatible maps if

imply

for 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 that

Again, for we can choose such that

Continuing 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 that

for 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 have

Suppose that ⋯. Then from the above inequalities, we obtain

This implies that

Since = ⋯ and for each , using Lemma 8, we have

Therefore, we get

Now we claim that, for any ,

Assume that (22) holds for some . Since , we have . By (16) and (22), we have

Hence, by the monotonicity of , we have

Similarly, we obtain

Therefore by the induction, (22) holds for all . Suppose that and are given. By hypothesis, is a -norm of -type. There exists such that

By using (21), there exist such that for all . Hence from (22) and (26) we get

for all and .
Therefore are all Cauchy sequence. Since is complete, there exist such that , . Again implies that there exists such that

Hence

From (16) and , we have

Taking the lim , we get

Now again, we have

Taking the lim , we get

Similarly we have

Taking the lim , we get

So, we have

Now we suppose that and are weakly compatible maps, so (36) implies that

Hence 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 that

for 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 have

Suppose that , . Then from the above inequalities, we obtain

This implies that

Since

and for any , we have .
Moreover, we have

Hence

This implies that

In 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 have

Now we have

Thus, by the monotonicity of , we have

Similarly we get

Suppose that and are given. Since is -norm of -type, there exist such that

By (45), there exist such that for all . Hence from (50) and (51), we get

for 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

for 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 obtain

This implies that

In the next step we show that is Cauchy sequence. For each , suppose that . Then, . From (56) we see that . By Lemma 5, we have

By Lemma 6, for each , there exists such that

Suppose 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 that

for 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 have

This implies that

Thus, we have

Suppose that and are given. By (60), there exists such that

for all and . Hence, from (64) it follows that

This 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 that

Hence

We have

Taking the lim , we get

Now again, we have

Taking lim , we get

Similarly, we have

Taking the lim , we get

The above implies that

but and are weakly compatible, so that (76) implies that

Hence 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 that

for 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 have

Then, from the above it follows that . we have

In 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 that

Using Lemma 7 we obtain that, for each , there exists such that

Suppose 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 that

for 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 that

for 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 that

for 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 by

We 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 and

Then we have , and so . It follows that

Hence 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); then

Now

and , , .

Hence .

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

And

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

Now

Hence 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)).

References

K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.

B. Schweizer and A. Sklar, Pobabilistic Metric Spaces, North-Holland, Amsterdam, The Netherlands, 1983.

Y. Liu and Z. Li, “Coincidence point theorems in probabilistic and fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 158, no. 1, pp. 58–70, 2007.

B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.

B. Schweizer, A. Sklar, and E. Thorp, “The metrization of statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 673–675, 1960.

V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces,” Mathematical Systems Theory, vol. 6, no. 1, pp. 97–102, 1972.

J.-X. Fang, “Common fixed point theorems of compatible and weakly compatible maps in Menger spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1833–1843, 2009.

O. Hadzic and E. Pap, Fixed Point Theory in Complete Metic Spaces, Kluwer Academic, Dordrecht, The Netherlands, 2001.

J. X. Fang, “Fixed point theorems of local contraction mappings on Menger spaces,” Applied Mathematics and Mechanics, vol. 12, no. 4, pp. 363–372, 1991.

J. Z. Xiao, X. H. Zhu, and Y. F. Cao, “Common coupled fixed point results for probabilistic ψ-contractions in Menger spaces,” Nonlinear Analysis, vol. 74, pp. 4589–4600, 2011.

J. Jachymski, “On probabilistic -contractions on Menger spaces,” Nonlinear Analysis—Theory, Methods & Applications, vol. 73, no. 7, pp. 2199–2203, 2010.

M. Imdad, A. H. Soliman, B. S. Choudhury, and P. Das, “On n-tupled coincidence and common fixed points results in metric spaces,” Journal of Operators, vol. 2013, Article ID 532867, 8 pages, 2013.

B. Samet and C. Vetro, “Coupled fixed point, f-invarient set and fixed point of N-order,” Annals of Functional Analysis, vol. 1, no. 2, pp. 4656–4662, 2010.

M. Imdad, A. Sharma, and K. P. R. Rao, “n-Tupled coincidence and common fixed point results for weakly contractive mappings in compatible metric spaces,” Bulletin of Mathematical Analysis and Applications, vol. 5, no. 4, pp. 19–39, 2013.