Journal of Function Spaces
Volume 2015 (2015), Article ID 470574, 11 pages
http://dx.doi.org/10.1155/2015/470574

Research Article

Some Results on Best Proximity Points of Cyclic Contractions in Probabilistic Metric Spaces

Manuel De la Sen¹ and Erdal Karapınar2,3

¹Institute of Research and Development of Processes IIDP, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644 de Bilbao Barrio Sarriena, Leioa, 48940 Bizkaia, Spain
²Department of Mathematics, ATILIM University, Incek, 06836 Ankara, Turkey
³Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 12 October 2014; Accepted 30 December 2014

Academic Editor: Calogero Vetro

Copyright © 2015 Manuel De la Sen and Erdal Karapınar. 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.

This paper investigates properties of convergence of distances of -cyclic contractions on the union of the subsets of an abstract set defining probabilistic metric spaces and Menger probabilistic metric spaces as well as the characterization of Cauchy sequences which converge to the best proximity points. The existence and uniqueness of fixed points and best proximity points of -cyclic contractions defined in induced complete Menger spaces are also discussed in the case when the associate complete metric space is a uniformly convex Banach space. On the other hand, the existence and the uniqueness of fixed points of the -composite mappings restricted to each of the subsets in the cyclic disposal are also investigated and some illustrative examples are given.

1. Introduction

Fixed point theory in the framework of probabilistic metric spaces [1–4] is receiving important research attention. See, for instance, [2–11]. On the other hand, Menger probabilistic metric spaces are a special case of the wide class of probabilistic metric spaces which are endowed with a triangular norm [2, 3, 5, 7, 9, 12, 13]. In probabilistic metric spaces, the deterministic notion of distance is considered to be probabilistic in the sense that, given any two points and of a 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 a real value   , [3].

Fixed point theorems in complete Menger probabilistic metric spaces for probabilistic concepts of and -contractions can be found in [2] together with a new notion of contraction, referred to as a -contraction. Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of -contraction. On the other hand, -cyclic -contractions on intersecting subsets of complete Menger spaces were discussed in [5] for contractions based on control -functions. See also [6]. It was found that fixed points are unique. Also, -contractions in complete probabilistic Menger spaces have been also studied in [9] through the use of altering distances. On the other hand, probabilistic Banach spaces versus fixed point theory were discussed in [8]. The concept of probabilistic complete metric space was adapted to the formalism of Banach spaces defined with norms being defined by triangular functions and under a suitable ordering in the considered space. In parallel, mixed monotone operators in such Banach spaces were discussed while the existence of coupled minimal and maximal fixed points for these operators was analyzed and discussed in detail. Further extensions to contractive mappings in complete fuzzy metric spaces by using generalized distribution functions have been studied in [6, 7] and some references therein. The concept of altering distances was exploited in a very general context to derive fixed point results in [14], and extended later on in [12] to Menger probabilistic metric spaces. On the other hand, general fixed point theorems have been very recently obtained in [13] for two new classes of contractive mappings in Menger probabilistic metric spaces. The results have been established for contractive mappings and for a generalized -type one. It has also to be pointed out that the parallel background literature related to best proximity points and fixed points in cyclic mappings in metric and Banach spaces is exhaustive. See, for instance, [15–28] and references therein. Fixed point theory has also been widely applied to stability and equilibrium problems since, even based on intuition, the convergence of trajectory-solutions of differential or difference equations or dynamic systems to a point can be typically associated to the convergence of sequences to fixed points; see, for instance, [27, 29, 30] and references therein, and to ergodic processes [31].

This paper investigates properties of convergence of distances of -cyclic contractions on the union of the subsets of the abstract set defining the probabilistic metric spaces and the Menger probabilistic metric spaces as well as the characterization of Cauchy sequences which converge to best proximity points. The existence and the uniqueness of fixed points and best proximity points of -cyclic contractions is also discussed in induced complete Menger probabilistic metric spaces in the case that the associate complete metric space is a uniformly convex Banach space. The fixed points of the -composite mappings restricted to each of the subsets in the cyclic framework disposal are also investigated. Finally, some examples are discussed.

2. Main Results

Denote , , , , and . Denote by the set of distribution functions , which are nondecreasing and left continuous, such that and . Let be a nonempty abstract set of elements and let be a mapping from into the set of distribution functions , which are symmetric functions of elements for every . Then, the ordered pair is a probabilistic metric space (PM), [1–4], if

(1),(2); , ,(3).

Note that an interpretation of the PM-space is that is a set of distribution functions. A particular distribution function is a probabilistic metric (or distance) which takes values , identified with the values of a mapping in the set of all the distribution functions , so that two points are (probabilistically) identical if the probabilistic metric in-between them gives probabilistic certainty (namely, probability equal to one) for a mutual distance being smaller than any given positive real number.

A Menger PM-space is a triplet , where is a PM-space which satisfies

under which is a -norm (or triangular norm) belonging to the set of -norms which satisfy the properties

A property which is a consequence of the above ones is . The probabilistic diameter of a subset of is a function from to defined by and which is probabilistically bounded if [2]. Note that the diameter of a set refers to the real interval length where the argument of the probabilistic metric is nonzero while the probabilistic diameter is a measure of boundedness or unboundedness of such a set. The (probabilistic) distance in-between the subsets and of defines the argument interval length of zero probability distance in-between points of two subsets and of and it is defined as

The concept of -cyclic contraction in a PM-space is recalled below (see, e.g., [1–6]).

Definition 1. A mapping is a -cyclic contraction in a PM-space, where are nonempty subsets of with being the distance between the adjacent subsets and , ; if , and there exists a real constant such that, for each pair , , the following constraint holds:

equivalently, ; , and

Note that, if is -cyclic contraction in a metric space , then there is such that, for each , ,

The space is the induced PM-space of the metric space if ; ; .

The definitions of convergent sequence and Cauchy sequence in a PM-space follows.

Definition 2. A sequence converges to if, for any given real constants , with , there is such that for .

Definition 3. A sequence is a Cauchy sequence in if for any given real constants , with , there is such that for .

Note that if converges to some then , , and, if is a Cauchy sequence in , then , . A Menger PM-space is complete if every Cauchy sequence is convergent in .

Definition 4. and are adjacent best proximity points so that the pair is an adjacent pair of best proximity points for any given if ; and , where , .

Since is nondecreasing and left continuous with then if are adjacent best proximity points for . If , that is, all subsets , , pair-wise intersect, so that , then all the best proximity points are coincident at a unique fixed point in and for which implies from the first property of the probabilistic space. The subsequent result addresses the fact that the sequences built by iterations through -cyclic contractions in complete Menger PM-spaces lead to convergent Cauchy sequences on each of the subsets , , if such subsets are closed, the limits of the subsequences being in the closures of the subsets of the cyclic disposal in the general case. The convergence points of such subsequences are fixed points of the composite mappings restricted to each one of the subsets , , and best proximity points located at each pair of the closures of the adjacent subsets and , , and at the sets themselves if the subsets are closed.

Theorem 5. Assume that is a -cyclic contraction in a Menger PM-space endowed with a -norm , where are nonempty subsets of such that , , under a probability density function such that for and , , for any , .
Then, the following property holds:

Assume, furthermore, that the Menger PM-space is complete and that is continuous and for each .Then, the subsequent additional property holds:

with for some , and, for any given and , there is such that , , so that the sequences , are Cauchy sequences; then convergent to adjacent best proximity points,    if is closed and    if is closed, , , which are fixed points of the composite mapping on and , respectively.

Proof. Note that and , , . Thus, one gets from (5) that

, , . It turns out that we can fix the argument of the first left-hand side to and, correspondingly, its first right-hand side to . In the same way, we can use for both arguments with the corresponding pairs and . Furthermore, it is obvious that

and then, for any given and , there is such that for , , . Hence, property (i) follows. On the other hand, note from property (i) for and any that

and that, for any given and , there is such that for . Then, one gets from (1), (5), the second and third conditions of (2), and the condition for each and :

Then, , , since is nondecreasing with supremum over equalizing unity. Then, for any given and , there is such that , so that is a Cauchy sequence which is convergent to a point in , since is complete, if for any . Now, if we take for some arbitrary given , one obtains in a similar way that

and then , . Then, for any given and , there is such that , and so that for any is a Cauchy sequence, and a subsequence of , which is convergent to a point in , since is complete. Since and are convergent to and for any , any , then one gets by taking into account property (i) that and are adjacent best proximity points; that is, . Assume the contrary, so that , , and for some and some so that:

and then thus , , . It also turns out that those points are also fixed points of the restricted composite self-map to and, respectively, to since, otherwise, the respective sequences and would not be convergent. Property (ii) has been proved.

Note that Theorem 5 does not address the uniqueness of the best proximity points in the subsets , . Their existence is guaranteed in the closures of the subsets since all sequences of the form are Cauchy convergent sequences to two respective limits at distance with probability one allocated in adjacent sets and for any given initial point for any . The subsequent result addresses the existence and the uniqueness of best proximity points if is a uniformly convex Banach space and is a complete Menger PM-space provided that the subsets , , are closed and convex.

The following Corollary to Theorem 5 follows for the case when , by making .

Corollary 6. Assume that is a contraction in a Menger PM-space endowed with a -norm , where is a nonempty subset of under a probability density function such that for and for for any given .
Then, the following property holds:

If, furthermore, the Menger PM-space is complete and that is continuous and for each .Then, the subsequent additional property holds:

with , and, for any given and , there is such that , , so that the sequence is a Cauchy sequence, then convergent to a fixed point if is closed which is also a fixed point of the composite mapping on .

A second main result of the paper and a corresponding corollary for the case when all the subsets of the cyclic disposal coincide follow below.

Theorem 7. Assume that is a -cyclic contraction in a PM-space , where are nonempty subsets of such that , , endowed with a probability density function such that for and for for any , . Then, the following properties hold:

Assume, furthermore, that is a uniformly convex Banach space under a complete norm-induced metric , so that is a complete Menger PM-space under the distribution function , , for any subject to

Assume also that the subsets of are closed and convex, . Then, the following two further properties hold.

(ii)Consider sequences for any and for some arbitrarily given such that

(1) as ,(2) for all and each given and some .

Then, for each given , there exists such that for all ; then .

(iii), , and are Cauchy sequences for any , , and any given and then bounded, and convergent to unique fixed points and of the composite mapping restricted to each into itself, . Also, and are the unique best proximity points in the adjacent subsets and of , .

Proof. Since is a -cyclic contraction in a PM-space , one gets from (4) for any , , that

Thus, since is nondecreasing and left-continuous,

Hence, property (i) has been proved. To prove property (ii), first note that , , for any since and then and also for .
We now proceed by contradiction. Assume the contrary. Then, such that, for each , there are for which . Choose such that and choose . For such an , there is and such that, since as , , then for . Since, furthermore, for , , and some since is a Menger PM-space. Then, if , since are closed and convex, , and is a uniformly convex (and hence reflexive) Banach space with modulus of convexity for and is strictly increasing, one has that, for any , and ,

and then, for any , and any arbitrary given ,

and there is such that the following contradiction is got if is not a Cauchy sequence in the Menger PM-space so that for and some :

Hence, property (ii) follows. To prove property (iii), note from property (ii) that, for any and initial points , one has that for and some and any , then so that is a Cauchy sequence for any and any and then bounded, which is convergent to some since the Menger PM-space is complete. For the other two Cauchy sequences, the same proof applies. Since the sequence is convergent to (since is closed) then . Assume that for some . Then, since does not converge to then so that for such an , a contradiction. Thus, , . It is now proved by contradiction that is unique, . Assume that for some . Then,

which contradicts for some . Thus, the fixed points of the restricted composite mapping to each are all unique, . Now, assume that , , is false; that is, for and is some for some . Then, since is a -cyclic contraction in , one has the following contradiction for some , positive reals:

Then, , .

Corollary 8. Assume that is a contraction in a PM-space , where is nonempty subsets of , endowed with a probability density function such that for and for for any . Then, the following properties hold:

Assume, furthermore, that is a uniformly convex Banach space under a complete norm-induced metric , so that is a complete Menger PM-space under the distribution function , for any subject to

Assume also that is closed and convex. Then consider the following.

(ii)Consider sequences for any such that

(1) as ;(2) for all and each given and some .

Then, for each given , there exists such that for all and .

(iii) is a Cauchy sequence for any and then bounded, and convergent to a unique fixed point .

Remark 9. Note that if is a -cyclic contraction in then the composite mappings from into , , satisfy from (4) for each ; that:

with . However, each composite mapping is not, in general, a contraction on each , , although any iterated sequences are Cauchy sequences, , in a complete Menger PM-space .

The following result is a direct consequence of Theorem 7 if the subsets of the cyclic disposal intersect.

Corollary 10. Assume that all the hypothesis of Theorem 7 hold while ; that is, the sets , , are not disjoint. Then, the following properties hold:

Assume, furthermore, that is a uniformly convex Banach space under a norm-induced metric , so that is a complete Menger PM-space under the distribution function , , for any subject to

Assume also that the subsets of are closed and convex, . Then, the following two further properties hold.

(ii)Consider the sequences for any and for some arbitrarily given such that:

(1) as ,(2) for all and each given and some .

Then, for each given , there exists such that for all ; then .

(iii), , and are Cauchy sequences for any , , and any given , then bounded, and convergent to a unique fixed point of and , .

Proof. The proof of Theorem 7(i) and the fact that the sequences of Theorem 7(iii) are Cauchy sequences are particular cases of Theorem 11 and do not need a proof. Now, assume that two adjacent best proximity points are distinct, namely, for some . Then, in the same way as in the last part of the proof of Theorem 7, one gets the contradiction

together with the parallel contradiction to for some :

Thus, , , which is in , since is nonempty and closed.

The subsequent result formulates a connection between metric spaces and their induced Menger PM-spaces [3] concerning some basic properties of -cyclic contractions whose contractive condition is defined on the metric space rather than on the Menger PM-space.

Theorem 11. Let be a metric space, let be an induced PM-space, where , in being defined by , , . Let be a -cyclic contraction in ; that is,

for some real constant , where are nonempty subsets of , , such that , .
Then, the following properties hold:

(ii)Assume, in addition, that is a uniformly convex Banach space under a norm-induced metric, so that is an induced complete probabilistic Menger PM-space under the -norm of the minimum. Assume also that the subsets of are closed and convex, . Then, is a Cauchy sequence and then bounded and convergent to a unique fixed point of the composite mapping restricted to into itself, , while and are the unique best proximity points in the adjacent subsets and of , . There is a unique fixed point which is coincident with all the best proximity points if the subsets , , have a nonempty intersection.

Proof. If is the mapping induced by the metric and is a -cyclic contraction then there is a real constant such that

and is an induced PM-space by the metric space . Thus, one gets from (35) in accordance with the definition , , , for any , and since is nondecreasing and left-continuous and that

, , . Then, by proceeding recursively with (36) since , , one gets

, , . Then, there exists the limit , , . In the same way, and since ; , one gets

Note from (1) that if then

so that for and property (i) has been proved. Property (ii) follows if is now a Menger PM-space under the triangular norm of the minimum induced by the metric space . Note that is a -cyclic contraction, is a uniformly convex Banach space, and then a complete metric space and the subsets of are closed and convex, . Thus, from Theorem 7 (see also [4] for the deterministic framework), is a complete Menger PM-space under the -norm of the minimum defined by . Then, for any given and there is , , , and , , some unique , any given , and . Thus, is a Cauchy sequence, then bounded, and convergent to a unique fixed point of the composite mapping restricted to into itself, , while and are the unique best proximity points in the adjacent subsets and of , .

3. Examples

Example 1. Assume that the PM-space is induced by a metric space and that is a -cyclic contraction in with contractive constant , then in with being the distance in-between adjacent subsets and , . If then for any and any given and any given ,

for all integer and some integer . Since , the condition implies that and then for . Assume that for some . If the probability of the distance in-between and is less than for some real then the probability of the distance between and being less than in the PM-space , for some and any , holds for any integer . Such an integer satisfies uniquely the constraints for the given integers ; that is, . The (deterministic) distances in the metric space for any integer provided that . For any and some arbitrary given , the subsequences and of , which are Cauchy convergent sequences, that is and , satisfy the same constraints if , respectively, for any (see Theorem 5). The points and are adjacent to best proximity points of (being identical and a unique fixed point of if the subsets intersect) and fixed points of the composite map restricted to and of restricted to , respectively.
Also, if is a complete Menger PM-space, then the subsequences , and are Cauchy and convergent sequences. Note that

for all integer and some integer , for any given and . Then,

such that which holds for with

This can be refined with Theorem 7(ii) (see (12)) leading to the particular case for :

implying and , for any given arbitrary , , all integer and a sufficiently large integer number . This implies that the subsequences , and are Cauchy and convergent and contained in -neighborhoods for which are centred at some for .

Example 2. Assume that the PM- space is induced by a metric space and that are possible sequences of -cyclic mappings with in with an associate sequence of constants with being the distance in-between adjacent subsets and , .
Furthermore, assume that where and are disjoint and is nonempty. It is assumed that for , with at least one satisfying for some prefixed real constant and for . Note that there is at least one member of which is also a member of the set of value in .

Now, construct sequences defined by for and all subject to

equivalently,

Assume that the objective is to construct admissible sequences of cyclic self-mappings of the given class so as to keep a distance for some prefixed real constant with guaranteed probability in-between consecutive sequence points of value at least for some given real constants . It follows from the theoretical framework that for composite -cyclic mappings , of the form for with contractive constants in the set , that is, of values less than one, one has that , . Thus, there is such that , , .

Then, a composite mapping for is built with any sequence of -cyclic mappings that satisfies the property , , , if the following steps are followed for its construction.

(a) are taken with contractive constants in the set for and some . Then, , . It suffices that since the distribution function is nondecreasing.(b) are taken with constants in the set as being contractive, expansive or nonexpansive depending on the values of their respective constants in the set for any integer such that the inequalities below hold:

for any . If the above chain of inequalities fails for , some tentative self-mapping and some , then is changed to with contractive constant so that to guarantee

provided from (46) that for ; that is, is chosen to satisfy , .

(c)For , repeat Step (a) for some such that .

Example 3. Let be a Menger PM-space, where with subsets and , is the minimum -norm and define the -cyclic mapping as follows: , , and let be defined by

Note that so that these two sets intersect; then , is a -cyclic contraction for any real constant , since , , and

for any real . Thus, is a complete Menger PM-space and is the unique fixed point of to which any sequences generated through converge in accordance with Theorem 5. A close example has been discussed in [6] related to a cyclic contraction in a -Menger PM-space.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors are grateful to the Spanish Government for its support of this research with Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. Finally, the authors thank the referees for their useful comments.

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