We prove a general fixed point theorem in Menger spaces for mappings satisfying a contractive condition of Ćirić type, formulated by means of altering distance functions. Thus, we extend some recent results of Choudhury and Das, Miheţ, and Babačev and also clarify some aspects regarding a theorem of Choudhury, Das, and Dutta.

1. Introduction and Preliminaries

In [1], Menger introduced the concept of probabilistic metric space as a generalization of metric spaces, in which the distance between points is expressed by means of distribution functions. This idea has made probabilistic metric spaces suitable for modeling phenomena when the uncertainty regarding measurements is assumed as inherent to the measuring process, as, for instance, in the investigation of certain physical quantities and physiological thresholds [2]. Probabilistic metric space theory has become a very active field of research. In particular, fixed point theory in probabilistic structures has found relevant applications in studying the existence and uniqueness of solutions of random equations [3], as well as algorithm complexity analysis [4, 5], and convergence analysis for stochastic optimization algorithms [6].

In the present paper, we establish a fixed point result for probabilistic contractions of Ćirić type, with the contractive condition stated by means of an altering distance function. Our theorem is obtained under very weak hypotheses, and thus it generalizes or improves several known results [710]. We also discuss the connections with a related theorem given by Choudhury et al. in [11], in order to explain the role of our assumptions.

We begin by recalling some fundamental concepts of probabilistic metric space theory. For a comprehensive exposition on this topic we refer the reader to the monographs [2, 3].

Definition 1. A triangular norm (or -norm) is a mapping which is associative, commutative, and nondecreasing in each variable and satisfies for all .

Some basic examples are , , and (the minimum, product, and Łukasiewicz -norm, resp.). Another important class is that of -norms of Hadžić type [12], that is, -norms whose family of iterates defined by , , is equicontinuous at .

We will denote by the set of functions which are nondecreasing and left continuous on , such that and .

Definition 2. A Menger space is a triple where is a nonempty set, is a mapping from to , and is a -norm, such that the following conditions are satisfied:(i) for all iff .(ii), for all , .(iii), for all , .
(Here and in the following, will be denoted by .)

Let be a Menger space such that . The family , where is a base for a Hausdorff uniformity on , named strong uniformity. The corresponding, strong topology on is introduced by the family of neighbourhoods of , where , and this topology is metrizable [2].

Definition 3. Let be a Menger space. A sequence in is said to be (i)Cauchy if, for any , , there exists such that , for all ;(ii)convergent to if, for any , , there exists such that for all .
is said to be complete if every Cauchy sequence is convergent in .

If the -norm is continuous, and the sequences , converge, respectively, to and , then converges to , for each continuity point of [2].

Definition 4. Given a set , the probabilistic diameter of is the mapping defined by is said to be probabilistically bounded if .

The notion of contraction in a Menger space was introduced by Sehgal in [13].

Definition 5 (see [13]). Let be a Menger space. A mapping is said to be a probabilistic contraction (or Sehgal contraction) if there exists such that

Many significant contributions to the development of fixed point theory in probabilistic structures can be found in monograph [14]. It should be pointed out that the triangular norm by which the space is endowed plays a key role in the existence of fixed points of probabilistic contractions. It was shown by Radu [15] that the largest class of continuous -norms with the property that every Sehgal contraction on a complete Menger space has a unique fixed point is that of -norms of Hadžić type.

The idea of using altering distance functions in order to obtain more general contractive conditions first appears in [16], in the setting of metric spaces. The corresponding concept of generalized probabilistic contraction was introduced by Choudhury and Das in [8] as follows.

Definition 6 (see [8]). A mapping is said to belong to the class if it satisfies (i) iff ;(ii) is strictly increasing and ;(iii) is continuous at and left continuous on .

The mappings will be called altering distance functions.

Definition 7 (cf. [8]). Let be a Menger space. The mapping is said to be a generalized probabilistic contraction of Choudhury-Das type if there exist and such that

It was proved in [8] that such contractions on a complete Menger space endowed with the strongest -norm have a unique fixed point. The result was subsequently generalized by Miheţ [9] for the case of arbitrary continuous -norms, under the supplementary assumption that the orbit of the mapping at some is probabilistically bounded.

Our aim is to prove a fixed point result for mappings satisfying the more general contractive condition for some and .

2. Main Results

In order to prove our results, we will need the following lemma from [10].

Lemma 8 (see [10]). Let be a Menger space; and . If are such that then .

For each , we will denote by the orbit of the mapping at ; that is, .

Theorem 9. Let be a complete Menger space with a continuous -norm. Suppose is a mapping satisfying the contractive condition (5), for some and . If there exists such that is probabilistically bounded, then has a unique fixed point in .

Proof. Let be as in the statement of the theorem. We will show that the sequence , , is Cauchy.
From (5) it follows thatBy replacing with above we getThereforeand, inductively, for all and for any positive integer . Since is strictly increasing, we obtain for all . By letting it follows that Consequently, for all and , Next, let be a positive integer. We prove by induction on thatfor all . The case is immediate. Suppose now that inequality (14) holds for some . Then for all . By the monotonicity of , it follows that , for all , whence As such, we conclude that inequality (14) holds for all , or equivalently Now, let and . Given that is continuous at , there exists with . Also, since , there exists such that , for all .
Thus, for all ,  , and therefore is a Cauchy sequence. Accordingly, there exists , .
Next, we will prove that is a fixed point of . Specifically, we will show that for all .
By the contractive condition (5), for all and . If is such that is continuous at , then (19) follows by letting in the above inequality and taking into account relation (13). If is not continuous at , let be a strictly increasing sequence converging to such that is continuous at , for all . As above, we infer that , , whence, for , we obtain (19). By Lemma 8 we conclude that .
Finally, we prove that is the only fixed point of in . To that end, let be such that . Then, using (5), we getOnce again, by Lemma 8, it follows that .

Corollary 10. If is a complete Menger space with a continuous -norm of Hadžić type, and satisfies condition (5) for some and some such that , then has a unique fixed point in .

Proof. We will show that, for every , is probabilistically bounded. To do so, let be arbitrary and define by for all . We will prove by induction on that for all . The case is trivial. Suppose now that the relation holds for some . Then Given that from the induction hypothesis we obtainwhich proves our claim.
Now, since and the family is equicontinuous at , it follows that .

By setting in the above corollary we get the following.

Corollary 11. Let be a complete Menger space with being a continuous -norm of Hadžić type and let be a mapping such thatfor all , . Then has a unique fixed point in .

Remark 12. In paper [10], Babačev proved a fixed point result for mappings satisfying the contractive condition for some altering distance function and some , in Menger spaces with the -norm . We note that, by applying the triangle inequality, so this condition essentially reduces to (5). Therefore Theorem 9 improves the result in [10], as well as Ćirić’s result in [7] (which can be obtained from that of Babačev for ).

Also, in [11], Choudhury et al. gave the following related theorem.

Theorem 13 (see [11]). Let be a complete Menger space with continuous -norm , and let be positive numbers with , and . Suppose that satisfies the inequalityfor all , , and with . Then has a unique fixed point.

As indicated in [11], a mapping satisfying the contractive condition (29) must also verify our condition (5). Namely, suppose that (29) holds. Let and let , , . It follows that Due to the monotonicity of , the above relation implies that for all , where .

Note that Theorem 13 only requires that the -norm by which the space is endowed is continuous. Unfortunately, we can show that this assumption alone is not sufficient to guarantee the existence of fixed points for contractions of this type.

Specifically, let be a complete Menger space and let be a Sehgal contraction on with contraction constant . Thenfor all , , and with . Thus, satisfies the conditions of Theorem 13 with and . However, a well-known counterexample of Sherwood ([17], Corollary of Theorem ) shows that there exist Sehgal contractions on complete Menger spaces endowed with the -norm having no fixed point.

It should be mentioned that a similar observation regarding continuity can be made with respect to Theorem in [18], where the class of contractions considered also includes Sehgal contractions.

Finally, we illustrate the applicability of Theorem 9 with the following example.

Example 14. Let and . Define for all and . is a complete Menger space. We will only show that the triangle inequality is verified.

Assume that and . Since the function is increasing, it holds that

Let for all , , and One can easily check that is an altering distance function and that is probabilistically bounded for every .

We will prove that condition (5) of Theorem 9 is satisfied. The following three cases are possible:(1)If , then for all we have (2)If , then for all .(3)If and , then, for all , Thus the condition (5) is satisfied in this case as well.

However, note that by setting and we obtain for all ; therefore, does not satisfy the stronger condition (4).

By applying Theorem 9 we conclude that the function has a unique fixed point. It is easy to see that this point is .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.