-Tuplet Coincidence Point Theorems in Partially Ordered Probabilistic Metric Spaces

Abstract

In this paper, we investigate existence of -tuplet coincidence point theorems in partially ordered probabilistic metric spaces. Also, we gave uniqueness of -tuplet fixed point theorems in this space.

1. Introduction

Probabilistic metric spaces were introduced by Menger in his fundamental paper [1] in 1942. In Menger’s theory, the notion of distance between two points and is replaced by a distribution function . The value of at any point is represented by . is interpreted as probability that the distance between and is less than . Sehgal first introduced the notion of contraction mapping on a probabilistic metric space in [2, 3]. In fact the study of such spaces received an impetus with the pioneering works of Schewizer and Sklar [4]. Then several authors have studied probabilistic spaces; see [5–9].

Guo and Lakshmikantham initiated the concept of coupled fixed point [10] in 1987. After that, Bhaskar and Lakshmikantham introduced the notion of mixed monotone property and gave some coupled fixed point theorems in ordered metric spaces in 2006 [11]. In 2012, the extention of coupled fixed point theorems to tripled fixed point theorems for nonlinear mapping in partially ordered metric space was introduced by Berinde and Borcut [12]. Then, some coupled and tripled fixed point results were obtained by many authors [13–19]. In 2013, Ertürk and Karakaya gave the concept of -tuplet fixed point theorems in partially ordered metric spaces [20]. Alam, Imdad, and Ali unified -tuplet fixed point results in ordered metric space in 2016 [21]. Their survey article is recommended to someone who wants to have details about this theory.

Hu and Ma studied couple coincidence point theorems in partially ordered probabilistic metric spaces in [22]. Recently, Binayak S. et al. [23] gave tripled coincidence point results in partially ordered probabilistic metric spaces.

Inspired by the above studies, we introduce -tuplet fixed point theorems in partially ordered probabilistic metric spaces. This paper is organized as follows. Section 2 is devoted to giving some preliminaries. In Section 3, we obtain existence of -tuplet fixed point theorems. Finally, Section 4 concerns the uniqueness of fixed point. These are the extensions of coupled and tripled fixed points in partially ordered probabilistic metric spaces.

2. Preliminaries

Definition 1 (see [4]). A triangular norm is a binary operation on the interval , that satisfies the following conditions:(a), ,(b),(c) for , ,(d)For all , .Principal examples of are (i),(ii),(iii),(iv)Above have the following relations:

Definition 2 (see [4]). A function is called a distribution function if it is nondecreasing, left continuous with In addition, if , then is called a distance distribution function. We denote the set of all distance distribution functions by and is a specific distance distribution function (also known as Heaviside function) defined by

Definition 3 (see [1]). Let be a nonempty set, be a mapping defined on into , and be a . If the following conditions are satisfied, is called Menger probabilistic metric spaces: (1) for all if and only if ;(2), for all and ;(3)

Example 4. Let be an arbitrary , , andfor every Then is a Menger Probabilistic Metric (for short PM) space given in [24].

Definition 5 (see [4, 25]). Let be a Menger space.(i)A sequence in is said to be convergent to a point if, for every , .(ii)A sequence in is called Cauchy sequence if, for each and , there exists for each (iii)A Menger space in which every Cauchy sequence is convergent is said to be complete.

Lemma 6 (see [26]). If be a Menger space where is continuous -norm, then for every fixed , if , then

Lemma 7 (see [23]). If is continuous and are such that for all for some and is bounded, then,

Let and . Each satisfies the following conditions:(1) is strict increasing,(2) is upper semicontinuous from the right,(3), .

If for all .

Lemma 8 (see [27]). Let be sequence in a Menger space , where is a minimum -norm. If there exists a function and , Then is a Cauchy sequence in .

Lemma 9 (see [27]). Let be a Menger space. If there exists a function and ,
, then .

3. Main Results

Definition 10 (see [20]). Let be a partially ordered set and . is said to have mixed monotone property if is monotone nondecreasing in its odd arguments and it is monotone nonincreasing in its even argument. That is, for any ,

Definition 11 (see [20]). Let be a partially ordered set and . We say that has the mixed -monotone property if is monotone -nondecreasing in its odd arguments and it is monotone -nonincreasing in its even argument. That is,
for any

If is taken identity mapping Definition 11 reduces to Definition 10.

Definition 12 (see [28]). Let . An element is defined as a tuplet fixed point of the mapping if

Definition 13 (see [28]). Let . An element is defined as a tuplet coincidence point of the mapping and if

When is taken identity mapping Definition 13 reduces to Definition 12.

Definition 14 (see [20]). Let be a partially ordered set and and is called commutative if for all

Definition 15. Let be a Menger space. and are said to be compatible if for all whenever sequences in such that

Theorem 16. Let be a partially ordered set and be a complete Menger space, where is a minimum -norm. Let be two mappings such that has the mixed -monotone property. Suppose there exist and such thatfor all , with Assume that is continuous, monotonic increasing, compatible with such that and suppose either(a) is continuous or(b) has the following property:(i)if nondecreasing sequence , then for all ,(ii)if nondecreasing sequence , then for all .If there exist such thatthen there exist such that That is, and have a -tuplet coincidence point.

Proof. Let satisfy condition (16). Since , we can define sequences as follows:
for ,Next step, we show that, for all ,To prove this claim, we will use the inductive method for mathematics. Because of the inequalities in (16), (19) holds for . We have Let us assume that (19) is true for ; that is,Since has the mixed -monotone property and from (21), Similar way we get So, inequalities in (19) are true for all .Now, we will show that are Cauchy sequences.
For all , By Lemma 8, we obtain that is a Cauchy sequence. We obtain that is a Cauchy sequence.
Using same way we conclude that are also Cauchy sequences. Since is complete metric space, there exist such thatSince is continuous, we can writeAs is compatible and is continuous,We will indicate that Suppose that (a) holds. From (27), (28), and (29), Now assume that (b) holds. From (19) and (29), we have for all For all , we have If , .
When we apply limit to both parts of above inequality, we get Because is arbitrary , by taking , and left continuous property of distribution function , we have is increasing and also is monotone increasing. Accordingly, From Lemma 9, .
By the same way, we can find Consequently, and have -tuplet coincidence point in .

Taking in Theorem 16, we obtain Corollary 17.

Corollary 17. Let be a partially ordered set and be a complete Menger space, where is minimum -norm. Let be two mappings such that has the mixed -monotone property. Let exist such thatfor all , with Also is continuous, monotonic increasing, compatible with and And suppose either(a) is continuous or(b) has the following properties:(i)if a nondecreasing sequence , then ,(ii)if a nonincreasing sequence , then .If there are such that then and have -tuplet coincidence point in .
That is, there exist such that

When we take , in Theorem 16, we obtain Corollary 18.

Corollary 18. Let be a partially ordered set and be a complete Menger space, where is minimum -norm. Let be two mappings such that has the mixed -monotone property. Let and exist such thatfor all , with Also is continuous, monotonic increasing, compatible with and And suppose either(a) is continuous or(b) has the following properties:(i)if a nondecreasing sequence , then ,(ii)if a nonincreasing sequence , then .If there are such that then and have -tuplet coincidence point in . That is, there exist such that

4. Uniqueness of -Tuplet Fixed Point

For all , is equal to iff

Theorem 19. In addition to the hypotheses of Theorem 16, suppose that for all there exist such that is comparable to and Then and have a unique -tuplet common fixed point; that is, there exist such that

Proof. The set of -tuplet coincidence points is nonempty due to Theorem 16. We shall show that if and are -tuplet coincidence points, that is, if and thenBy assumption there is such thatis comparable toandWe can determine sequences such that and From comparability of (56) and (55) with (54), we suppose that Using (24), we have Also from (46), we getfor each If we take lower limit when from Lemmas 8 and 9, we haveLikewise, one can show thatUsing (61), (62), and triangle inequality property, So, we have
Thus, we showed (53).
Using commutativity of and ,Indicate From (64),Therefore, is a -tuplet coincidence point. Then from the assumption with it follows ; that is,By (65) and (66), Hence, is a -tuplet common fixed point of and . To prove the uniqueness, assume that is another -tuplet common fixed point. Then by assumption we have

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

A part of the results of this paper were presented in International Conference on Operators in Morrey-Type Spaces and Applications, OMSTA 2017 [29].

Conflicts of Interest

The authors have no conflicts of interest regarding this paper.

Acknowledgments

First author was supported by the Scientific and Technological Research Council of Turkey (TUBITAK Programme, 2211-A). The authors would like to thank Müzeyyen Ertürk for her useful comments and suggestions that allowed them to improve this paper.