A Probabilistic Fixed Point Result Using Altering Distance Functions

In [1], Menger introduced the concept of probabilistic metric space as a generalization of metric spaces, in which the distance between points is expressed bymeans 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 [7–10]. 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].


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 [7][8][9][10].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.
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   : R → [0, 1] 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  ∈ (0, 1) 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.
The mappings  ∈ Φ 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  ∈ Φ and  ∈ (0, 1) 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 Mihet ¸ [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  ∈ Φ and  ∈ (0, 1).

Main Results
In order to prove our results, we will need the following lemma from [10].
Proof.Let  be as in the statement of the theorem.We will show that the sequence (  )  ,   =   (), is Cauchy.
Next, we will prove that  * is a fixed point of .Specifically, we will show that for all  > 0.
Proof.We will show that, for every  ∈ , (, ) is probabilistically bounded.To do so, let  ∈  be arbitrary and define (  )  by   =   () for all  ≥ 0. We will prove by induction on  that for all  ≥ 1.The case  = 1 is trivial.Suppose now that the relation holds for some  ≥ 1.Then Given that from the induction hypothesis we obtain which proves our claim.Now, since lim  → ∞   0  1 (() − ()) = 1 and the family {  }  is equicontinuous at 1, it follows that  (,) ∈ D + .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 that for all ,  ∈ ,  > 0. 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  ∈ (0, 1), 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.
Then  has a unique fixed point.
Specifically, let (, ,   ) be a complete Menger space and let  be a Sehgal contraction on  with contraction constant  < 1/3.Then for all ,  ∈ ,  > 0, and  1 ,  2 ,  3 > 0 with  1 +  2 +  3 = .Thus,  satisfies the conditions of Theorem 13 with  =  =  =  and () = .However, a well-known counterexample of Sherwood ([17], Corollary 1 of Theorem 3.5) 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 3.1 in [18], where the class of contractions considered also includes Sehgal contractions.
Finally, we illustrate the applicability of Theorem 9 with the following example.(34) 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: (37) Thus the condition ( 5) is satisfied in this case as well.
for all  > 0; 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  = 0.