On Probabilistic Alpha-Fuzzy Fixed Points and Related Convergence Results in Probabilistic Metric and Menger Spaces under Some Pompeiu-Hausdorff-Like Probabilistic Contractive Conditions

De la Sen, M.

doi:https://doi.org/10.1155/2015/213174

Journal of Function Spaces

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Research Article | Open Access

Volume 2015 | Article ID 213174 | https://doi.org/10.1155/2015/213174

On Probabilistic Alpha-Fuzzy Fixed Points and Related Convergence Results in Probabilistic Metric and Menger Spaces under Some Pompeiu-Hausdorff-Like Probabilistic Contractive Conditions

M. De la Sen¹

Academic Editor: Pasquale Vetro

Received10 Aug 2015

Revised24 Sept 2015

Accepted13 Oct 2015

Published31 Dec 2015

Abstract

In the framework of complete probabilistic metric spaces and, in particular, in probabilistic Menger spaces, this paper investigates some relevant properties of convergence of sequences to probabilistic α-fuzzy fixed points under some types of probabilistic contractive conditions.

1. Introduction

Fixed point theory is an important tool to investigate the convergence of sequences to limits and unique limits in metric spaces and normed spaces. See, for instance, [1–34] and the wide list of references cited in those papers. In particular, fixed point theory is also a relevant tool to investigate iterative schemes and stability theory of continuous-time and discrete-time dynamic systems, boundedness of the trajectory solutions, stability of equilibrium points, convergence to stable equilibrium points, and the existence oscillatory solution trajectories [15–17, 23, 35]. See also references therein. On the other hand, fixed point theory is nowadays receiving important research attention in the framework of probabilistic metric spaces. See, for instance, [25, 26, 28, 33, 34, 36–38] and references therein. It has also to be pointed out that Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces which are endowed with a triangular norm [25, 26, 28, 33, 34, 36, 37] and which are very useful in the context of fixed point theory. Note that the triangular norm plays a close role to that of the norm in normed spaces. In probabilistic metric spaces, the deterministic notion of distance is revisited as being probabilistic in the sense that, given any two points and of a certain metric space, a measure of the distance between them is a probabilistic metric , rather than the deterministic distance , which is interpreted as the probability of the distance between and being less than [33, 34].

Fixed point theorems in complete Menger spaces for probabilistic concepts of and -contractions can be found in [33] together with a new notion of contraction, referred to as -contraction. Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of -contraction. On the other hand, fuzzy metric spaces have been investigated more recently and some ad hoc versions of fixed point theorems have been obtained in that framework. See, for instance, [4–6, 32] and some references therein.

This paper investigates some relevant properties of convergence of sequences to the so-called and defined probabilistic -fuzzy fixed points under some types of probabilistic contractive conditions. The concept of probabilistic -fuzzy fixed point is defined as an “ad hoc” conceptual extension of that of -fuzzy fixed points of [6, 32] and it is oriented to the derivation of convergence properties of fuzzy mappings defined on probabilistic metric spaces and, in particular, in probabilistic Menger spaces.

Notation, Preliminaries, and Some Basic Concepts. Denote by , , , , , and denote by (a common used name for this class being ) the set of probability distribution functions , [1], which are nondecreasing and left-continuous such that and . Let be a nonempty set and let the probabilistic metric (or distance) be a symmetric mapping from to , where is an abstract set, to the set of distance distribution functions of the form which are functions of elements for every . Then, the ordered pair is a probabilistic metric space (PM) [16, 28, 33, 34, 36–38] if the following constraints hold:(1), equivalently, where is defined by (2); , (3) A particular distance distribution function is a probabilistic metric (or distance) which takes values identified with a probability distance density function in the set of all the distance distribution functions .

A Menger PM-space is a triplet , where is a PM-space which satisfiesunder which is a -norm (or triangular norm) belonging to the set of -norms which satisfy the following properties:(1),(2),(3) if , ,(4) A property which follows from the above ones is for . Typical continuous -norms are the minimum -norm defined by , the product -norm defined by , and the Lukasiewicz -norm defined by which are related by the inequalities .(i)The triplet is a Menger space, where is a PM-space and is a triangular norm, which satisfies the inequality ; ; .(ii) is the minimum triangular norm defined by .(iii)A sequence in a probabilistic space is said to be(1)convergent to a point , denoted by (a.s.), if for every and , there exists some such that (2)Cauchy if for every and , there exists some such that A PM-space is complete if every Cauchy sequence is convergent.

2. Concepts and Results on Probabilistic -Fuzzy Fixed Points

Let be nonempty subsets of an abstract nonempty set . Then the probabilistic point-to-set distance mapping from to , denoted by and the probabilistic set-to-set distance mapping from to are, respectively, defined byThe Pompeiu-Hausdorff-like probabilistic set-to-set distance is defined by Note that and since . If , where is the set of all nonempty closed bounded subsets of , then .

A fuzzy set in is a function from to whose grade of membership of in is the function-value . The -level set of is denoted by defined bywhere denotes the closure of . Let be the collection of all fuzzy sets in a PM-space . Let be a fuzzy mapping from an arbitrary set to , which is a fuzzy subset in , and the grade of membership of in is .

For , means , . Note also that if and then . If there exists such that , , then defineThe collection of all the approximate quantities in a metric linear space is denoted by . is a fuzzy mapping from an arbitrary set to , which is a fuzzy subset in , and the grade of membership of in is .

The notation means that the domain of the function from to is restricted to the subset of .

The next definition characterizes probabilistic fuzzy fixed points in an appropriate way to establish some results of this paper.

Definition 1. If is the collection of all fuzzy sets in the PM-space , where is a nonempty abstract set and is a fuzzy mapping, then is a probabilistic -fuzzy fixed point of if, for some , and ; that is, .
Note that if is a nonempty abstract set, is a PM-space, , and for some , , then (1); ,(2)if is a probabilistic -fuzzy fixed point of then ; ,(3)if and is not a probabilistic -fuzzy fixed point of then , equivalently, , and ; for some .

The following result holds.

Theorem 2. Let be the collection of all fuzzy sets in a PM-space , where is a nonempty abstract set and let be a fuzzy mapping. Assume that the following conditions hold:(1)for each , there exists such that is a nonempty closed bounded subset of , and each sequence of the form , with ; which satisfies the following contractive constraint for some real constant : , where for ; ,(2) (3) Then, a sequence may be built for any given arbitrary satisfying , with satisfying ; .
If, in addition, is endowed with the minimum triangular norm and is a complete Menger space then each of such sequences is a Cauchy sequence which is convergent to a probabilistic -fuzzy fixed point of .

Proof. Take arbitrary points , for some given existing such that is nonempty and take also some existing such that is nonempty. Note that since , for any and ; then . Thus, one gets from the contractive condition (12) that for any given since for all since . Then, again since , the following cases can arise for each .
Case (a). One hasfor some given . Thus, from (15) and (16), one getsand one gets for the given that since is nondecreasing and left-continuous, then and since , one gets from (17) thatand then since is closed and nonempty, there exists such that from (19) and the fact that ; ,and, equivalently, where ; .
Case (b). One hasand some and can be chosen for the previously taken so that . Thus,and since , one gets for the given Then, one gets from (24) that which implies (21). So, from Cases (a)-(b), for each given , and , there exist and points and in nonempty level sets and such that (22) holds. Proceeding recursively, one gets that a sequence may be built for any arbitrary and with ; which satisfies the recursionwhere Note that sincethen ; , , where and ; are discrete measures of the subsequent sets where and are Dirac measures defined byThen, one gets from (26), (29), and ; , since , that ; with ; since Since ; and any given then for any given and , there is such that ; . Assume on the contrary that there exist such that for and . Then, one has the following contradiction for the subsequence of :Then, ; so that is a Cauchy sequence. Since is complete, one gets and .
It is now proved that . Assume on the contrary that ; that is, and, for some given , there is such that for . Then, since ; and since , If is chosen to fulfill , thena contradiction. Then, with ; that is, it is a probabilistic -fuzzy fixed point of .

Corollary 3. Let be the collection of all fuzzy sets in a PM-space , where is a nonempty abstract set, and let be a fuzzy mapping. Assume that the contractive condition (12) holds for some real constant subject to condition (2) of Theorem 2 and the specific particular form of condition (3):there exists and strictly increasing sequence of nonnegative integers which satisfies

Proof. It is direct from that of Theorem 2 since condition (3′) guarantees condition (3) of Theorem 2 for any finite first element of a strictly increasing sequence subject to .

Corollary 4. Theorem 2 and Corollary 3 also hold if the contractive condition (1) is modified as follows:

Proof. It is direct from that of Theorem 2 since the above contraction condition follows from (12) as a lower-bound which has been used in the proof of Theorem 2.

Corollary 5. Let be the collection of all fuzzy sets in a PM-space , where is a nonempty abstract set, and let be a fuzzy mapping. Assume that the following conditions are fulfilled:(1)for each , there exists such that is a nonempty closed bounded subset of , and each sequence of the form , ; which satisfies for some real constant the contractive constraint: , where for
Then, a sequence may be built for any given arbitrary satisfying , with and ; .
If, in addition, is endowed with the minimum triangular norm and is a complete Menger space then each above sequence is a Cauchy sequence which is convergent to a probabilistic -fuzzy fixed point of.

Proof. Define the indicator function sequence ; aswith defined as in Theorem 2. Then, even if for some and all , because ; , it follows from (38)-(39) and (26) that ; with ; . The rest of the proof is close to that of Theorem 2.

Example 6. It is claimed through this example to revisit the idea of fuzzy fixed point addressed in [4–6, 32] to that of probabilistic fuzzy fixed point according to the definition and the formalism given above. Consider and let be a probabilistic fuzzy mapping defined as follows:The -level sets areNote that , for any and and some . The probability density functions are as follows with ; , : Assume that the contractive condition of Theorem 2 holds under the form for sequences with initial points , , where ; , satisfies for some , some , and some . Note that with ; is a probabilistic -fuzzy fixed point of to which the sequences , converge.

3. Further Results

Note that the uniqueness of probabilistic fuzzy fixed points is not an interesting property to study in the context of probabilistic fuzzy fixed point since distinct level sets associated with mappings of the form can easily have intersections of cardinal greater than one in many problems in the fuzzy context. The subsequent result gives conditions for the case when several distinct probabilistic fuzzy fixed points, if they exist, are in the intersections of their respective level sets under slightly extended contractive conditions of those given in Section 2. The extended conditions are of direct applicability.

Theorem 7. Consider a complete probabilistic metric space under all the assumptions of Theorem 2 with an extended contractive condition (12) such that it also holds for any being probabilistic -fuzzy fixed points and of . Then, the set is nonempty and .

Proof. If the result is obvious. Assume that there exist a probabilistic -fuzzy fixed point and a probabilistic -fuzzy fixed point . Consider two convergent sequences and in . Then,Assume that ; . Then, from the extended contractive condition (12) and since ; , one gets for some and since and are members of , that is, nonempty, closed and bounded sets,so thatand then , so that is nonempty. We can now assume that ; . It is direct to prove in a close way to the above proof that .

Remark 8. Note that a direct consequence of Theorem 7, from the definition of the level sets, is that if and are any probabilistic and -fuzzy fixed points.

Remark 9. Note that corollaries to Theorem 7 can also be stated “mutatis-mutandis” under extended contractive conditions for probabilistic fuzzy fixed points to those given in Corollaries 3–5.

Theorem 10. Consider a complete probabilistic metric space under all the assumptions of Theorem 2 and conditions (1)–(3) where the contractive condition is extended to hold for any with for ; .

Proof. Note, from Theorem 2, a sequence can be built being a (convergent) Cauchy sequence such that , ; for any given . Then, for any given , , , and such thatsince ; , from property (1) of (3) for PM-spaces.

Theorem 11. Let be the collection of all fuzzy sets in a PM-space , where is a nonempty abstract set, and let be a fuzzy mapping. Assume that the following conditions are fulfilled:(1)for each , there exists such that is a nonempty closed bounded subset of , and each sequence of the form , with ; which satisfies the following contractive constraint for some real constant : , where for ; ,(2) (3) Then, a sequence may be constructed for any given arbitrary such that , with satisfying ; .
If, in addition, is endowed with the minimum triangular norm and is a complete Menger space then each of such sequences is a Cauchy sequence which is convergent to a probabilistic -fuzzy fixed point of .

Proof. Since , Now for any given . Then, the following cases can occur:(a)If , , and ; then so that (b)If , , and ; then so that(c)If , , and ; then so that (d)If , , and ; then(e)If , , and ; then(f)If , , and ; then(g)If , , and ; then(h)If , , and ; thenNow, take , , and any such that . Then, one gets for Cases (a)–(h)and we can construct a sequence with arbitrary, for some sequence , such that since , one gets proceeding recursively:so that ; .

Remark 12. Extensions of Theorem 11 are direct for the case when the subsequent contractive condition holds instead of (48):Extensions of Theorem 11 and its variant of Remark 12 to the light of Theorems 2 and 10 are direct concerning the case when the coefficients of the respective contractive condition are functions for ; . Also, close examples to Example 6 can be given for the more general contractive conditions of this section.

Conflict of Interests

The author declares that he has no conflict of interests.

Acknowledgments

The author is very grateful to the Spanish Government for Grant DPI2012-30651 and to the Basque Government and UPV/EHU for Grants IT378-10, SAIOTEK S-PE13UN039, and UFI 2011/07.

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Copyright © 2015 M. De la Sen. 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.

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