© Kamran et al.; licensee Springer. 2013
Received: 31 May 2013
Accepted: 5 August 2013
Published: 22 August 2013
In this paper we introduce the notion of probabilistic G-contraction and establish some fixed point theorems in such settings. Our results generalize/extend some recent results of Jachymski and Sehgal and Bharucha-Reid. Consequently, we obtain fixed point results for -chainable PM-spaces and for cyclic operators.
In recent years the Banach contraction principle has been widely used to study the existence of solutions for the nonlinear Volterra integral equations, nonlinear integrodifferential equations in Banach spaces and to prove the convergence of algorithms in computational mathematics. It has been extended in many different directions for single- and multi-valued mappings. Recently, Nieto and Rodríguez-López , Ran and Reurings , Petruşl and Rus  established some new results for contractions in partially ordered metric spaces. The following is the main result due to Nieto and Rodríguez-López [1, 4], Ran and Reurings .
f is continuous; or
for every nondecreasing sequence in S such that in S, we have for all .
If there exists with (or ), then f has a fixed point. Furthermore, if is such that every pair of elements of S has an upper or lower bound, then f is a Picard operator (PO).
Many authors undertook further investigations in this direction to obtain some generalizations and extensions of the above main result (see, e.g., [5–7]). In this context, Jachymski  established a generalized and novel version of Theorem 1.1 by utilizing graph theoretic approach. From then on, investigations have been carried out to obtain better and generalized versions by weakening contraction condition and analyzing connectivity of a graph (see [9–11]).
Motivated by the work of Jachymski, we can pose a very natural question: Is it possible to establish a probabilistic version of the result of Jachymski  (see Corollary 3.12)? In this paper, we give an affirmative answer to this question. Our results are substantial generalizations and improvements of the corresponding results of Jachymski  and Sehgal  and others (see, e.g., [1, 2, 4]). Subsequently, we apply our main results to the setting of cyclical contractions and to that of -contractions as well.
In 1942 Menger introduced the notion of probabilistic metric space (briefly, PM space), and since then enormous developments in the theory of probabilistic metric space have been made in many directions [13–15]. The fundamental idea of Menger was to replace real numbers with distribution functions as values of a metric.
Δ is associative and commutative,
for all ,
for all with and .
Typical examples of t-norms are and .
A trivial example of a t-norm of ℋ-type is , but there exist t-norms of ℋ-type with (see, e.g., ).
for all and ;
if and , then for all and for every .
Definition 2.4 A Menger probabilistic metric space (briefly, Menger PM-space) is a triple , where is a PM-space, Δ is a t-norm and instead of (PM3) in Definition 2.3 it satisfies the following triangle inequality:
(PM3)′ for all and .
Remark 2.5 (Sehgal )
Let be a metric space. Define for all and . Then the triple is a Menger PM-space induced by the metric d. Furthermore, is complete iff d is complete.
Furthermore, if is a Menger PM-space with , then the family of neighborhoods determines a Hausdorff topology for S.
A sequence in S converges to an element x in S (we write or ) if for every and there exists a natural number such that , whenever .
A sequence in S is a Cauchy sequence if for every and there exists a natural number such that , whenever .
A Menger PM-space is complete if and only if every Cauchy sequence in S converges to a point in S.
Now we recall some basic notions from graph theory which we need subsequently. Let be a metric space, let Ω be the diagonal of the Cartesian product , and let G be a directed graph such that the set of its vertices coincides with S and the set of its edges contains all loops, i.e., . Assume that G has no parallel edges. Let be a directed graph. By letter we denote the undirected graph obtained from G by ignoring the direction of edges and by we denote the graph obtained by reversing the direction of edges. Equivalently, the graph can be treated as a directed graph having . If x and y are vertices in a graph G, then a path in G from x to y of length l is a sequence of vertices such that , and for . A graph G is called connected if there is a path between any two vertices. G is weakly connected if is connected. For a graph G such that is symmetric and x is a vertex in G, the subgraph consisting of all edges and vertices which are contained in some path beginning at x is called the component of G containing x. In this case , where is the equivalence class of a relation R defined on by the rule: if there is a path in G from y to z. Clearly, is connected. A mapping is called a Banach G-contraction  if ; , i.e., f is edge-preserving and ().
3 Main results
We start with the following definition.
Example 3.2 Let be a metric space endowed with a graph G, and let the mapping be a Banach G-contraction. Then the induced Menger PM space is a probabilistic G-contraction.
Thus f satisfies (3.1).
From Example 3.2 it is inferred that every Banach G-contraction is a probabilistic G-contraction with the same contraction constant.
f is both a probabilistic -contraction and a probabilistic -contraction with the same contraction constant α.
is f-invariant and is a probabilistic -contraction provided that is such that .
It follows from the symmetry of .
Let . Then there is a path between x and . Since f is a probabilistic G-contraction, . Thus .
Suppose . Then since f is a probabilistic G-contraction. But is f invariant, so we conclude that . Condition (3.1) is satisfied automatically, since is a subgraph of G. □
Lemma 3.4 Let be a Menger PM-space under a t-norm Δ satisfying . Assume that the mapping is a probabilistic G-contraction. Let , then as (). Moreover, for , () if and only if ().
Hence, as . □
Every t-norm can be extended in a unique way to an n-ary as follows: , for . Let be a path between two vertices x and y in a graph G. Let us denote with for all t. Clearly the function is monotone nondecreasing.
G is a -graph in S if there exist a subsequence of and a natural number N such that for ;
G is an -graph in S if for and the sequence of functions converges to uniformly as ().
Example 3.6 Let be a Menger PM-space induced by the metric on , and let I be an identity map on S.
Thus is not an -graph.
Since as . Clearly, is not a -graph. But as (). Thus is an -graph.
From the above examples, we note that the notions of -graph and -graph are independent even if f is an identity map.
The following lemma is essential to prove our fixed point results.
Lemma 3.8 (Miheţ )
Then is a Cauchy sequence.
If G is a -graph, then f has a unique fixed point and for any , . Moreover, if G is weakly connected, then f is a Picard operator.
If G is a weakly connected -graph, then f is a Picard operator.
Hence, . Moreover, if G is weakly connected, then f is a Picard operator as .
Hence, we deduce . Finally, let be arbitrary, then from Lemma 3.4, . □
G is weakly connected.
For every probabilistic G-contraction f on S, if there exists such that , then f is a Picard operator.
Proof (i) ⇒ (ii): It is immediate from Theorem 3.9.
Let , then , which implies . Hence , since G contains all loops. Thus the mapping f preserves edges. Also, for and , we have ; thus (3.1) is trivially satisfied. But and are two fixed points of f contradicting the fact that f is a Picard operator. □
Remark 3.11 Taking , Theorem 3.9 improves and extends the result of Sehgal [, Theorem 3] to all Menger PM-spaces with t-norms of ℋ-type. Theorem 3.9 generalizes claim 40 of [, Theorem 3.2], and thus we have the following consequence.
Corollary 3.12 (Jachymski [, Theorem 3.2])
Let be a complete metric space endowed with the graph G. Assume that the mapping is a Banach G-contraction and the following property is satisfied:
If there exists with , then is a Picard operator. Furthermore, if G is weakly connected, then f is a Picard operator.
Proof Let be the Menger PM-space induced by the metric d. Since the mapping f is a Banach G-contraction, then it is a probabilistic G-contraction (see Example 3.2) and property ( ) invokes that G is a -graph. Hence the conclusion follows from Theorem 3.9(i). □
Further assume that X is endowed with a graph G consisting of and . It can be seen that f is a probabilistic G-contraction with and satisfies all the conditions of Theorem 3.9(i).
Hence, one cannot invoke [, Theorem 3].
Definition 3.14 Let be a Menger PM-space under a t-norm Δ of ℋ-type. A mapping is said to be: (i) continuous at point whenever in S implies as ; (ii) orbitally continuous if for all and any sequence of positive integers, implies as ; (iii) orbitally G-continuous if for all and any sequence of positive integers, and imply (see ).
Theorem 3.15 Let be a complete Menger PM-space under a t-norm Δ of ℋ-type. Assume that the mapping is a probabilistic G-contraction such that f is orbitally G-continuous, and let there exist such that . Then f has a unique fixed point and for every , . Moreover, if G is weakly connected, then f is a Picard operator.
Proof Let , by induction for all . By using Lemma 3.8, it follows that . Since f is orbitally G-continuous, then . This gives . From Lemma 3.4 for any , . □
Remark 3.16 We note that in Theorem 3.15 the assumption that f is orbitally G-continuous can be replaced by orbital continuity or continuity of f.
As a consequence of Theorems 3.9 and 3.15, we obtain the following corollary, which is actually a probabilistic version of Theorem 1.1 and thus generalizes and extends the results of Nieto and Rodríguez-López [, Theorems 2.1 and 2.3], Petruşel and Rus [, Theorem 4.3] and Ran and Reurings [, Theorem 2.1].
f is continuous, or
for every nondecreasing sequence in S such that in S, we have for all .
If there exists with , then f has a fixed point. Furthermore, if is such that every pair of elements of S has an upper or lower bound, then f is a Picard operator .
Proof Consider a graph consisting of and . If f is nondecreasing, then it preserves edges w.r.t. graph and condition (3.10) becomes equivalent to (3.1). Thus f is a probabilistic -contraction. In case f is nonincreasing, consider with and a vertex set coincides with S. Actually, and from Proposition 3.3 if f is a probabilistic -contraction, then it is a probabilistic contraction. Now if f is continuous, then the conclusion follows from Theorem 3.15. On the other hand, if (ii) holds, then and are -graphs and conclusions follow from the first part of Theorem 3.9. □
By relaxing ℋ-type condition on a t-norm, our next result deals with a compact Menger PM-space using the following class of graphs as the fixed point property is closely related to the connectivity of a graph.
Definition 3.19 Let be a PM-space endowed with a graph G and . Assume the sequence in S with for and (), we say that the graph G is -graph if for any subsequence , there exists a natural number N such that for all .
Theorem 3.20 Let be a compact Menger PM-space under a t-norm Δ satisfying . Assume that the mapping is a probabilistic G-contraction, and let there exist such that . If G is an -graph, then f has a unique fixed point .
Hence, . Note that is a path in , so that . □
So far it remains to investigate whether Theorem 3.20 can be extended to a complete PM-space?
Definition 3.21 
The PM space is said to be -chainable if for each there exists a finite sequence of elements in S with and such that for .
Theorem 3.22 Let be a complete -chainable Menger PM-space under a t-norm Δ of ℋ-type. Let the mapping be an -contraction. Then f is a Picard operator.
Thus, . Hence, f is a probabilistic G-contraction and the conclusion follows from Theorem 3.15. □
Remark 3.23 Theorem 3.22 has an advantage over Theorem 7 of Sehgal and Bharucha-Reid  which is only restricted to continuous t-norms satisfying . Moreover, the proof of our result is rather simple and easy, which invokes the novelty of Theorem 3.22.
The metric space is ε-chainbale for some if for every , there exists a finite sequence of elements in S with , and for .
Remark 3.25 
If is an ε-chainable metric space, then the induced Menger PM-space is an -chainable space.
Then f is a Picard operator.
Hence the conclusion follows from Theorem 3.22. □
Kirk et al.  introduced the idea of cyclic contractions and established fixed point results for such mappings.
and the operator f is known as a cyclic operator .
In the following, we present the probabilistic version of the main result of , as a last consequence of Theorem 3.9.
is a cyclic representation of Y w.r.t. f;
such that whenever , , where .
Then f has a unique fixed point and for any .
Proof Since is closed, then is complete. Let us consider a graph G consisting of and . By (i) it follows that f preserves edges. Now, let in Y such that for all . Then by (3.13) it is inferred that the sequence has infinitely many terms in each ; . So that one can easily identify a subsequence of converging to in each ; and since ’s are closed, then . Thus, we can easily form a subsequence in some , such that for . It elicits that G is a weakly connected -graph. Hence, by Theorem 3.9 conclusion follows. □
The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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