Probabilistic G-contractions

  • Tayyab Kamran1,

    Affiliated with

    • Maria Samreen2 and

      Affiliated with

      • Naseer Shahzad3Email author

        Affiliated with

        Fixed Point Theory and Applications20132013:223

        DOI: 10.1186/1687-1812-2013-223

        Received: 31 May 2013

        Accepted: 5 August 2013

        Published: 22 August 2013

        Abstract

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq1_HTML.gif -chainable PM-spaces and for cyclic operators.

        MSC: 47H10, 54H25.

        Keywords

        fixed point Menger PM-space directed graph Picard operator

        1 Introduction

        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 [1], Ran and Reurings [2], Petruşl and Rus [3] 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 [2].

        Theorem 1.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif be a complete metric space endowed with the partial order ‘. Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is nondecreasing (or nonincreasing) with respect to the partial order ‘’ on S and there exists a real number α, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq4_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ1_HTML.gif
        (1.1)
        Also suppose that either
        1. (i)

          f is continuous; or

           
        2. (ii)

          for every nondecreasing sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif in S such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq6_HTML.gif in S, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq7_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq8_HTML.gif .

           

        If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq9_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq10_HTML.gif (or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq11_HTML.gif ), then f has a fixed point. Furthermore, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq12_HTML.gif 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., [57]). In this context, Jachymski [8] 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 [911]).

        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 [8] (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 [8] and Sehgal [12] and others (see, e.g., [1, 2, 4]). Subsequently, we apply our main results to the setting of cyclical contractions and to that of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq1_HTML.gif -contractions as well.

        2 Preliminaries

        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 [1315]. The fundamental idea of Menger was to replace real numbers with distribution functions as values of a metric.

        A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq13_HTML.gif is called a distribution function if it is nondecreasing, left continuous and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq15_HTML.gif . In addition, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq16_HTML.gif , then F is called a distance distribution function. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq17_HTML.gif denote the set of all distance distribution functions satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq18_HTML.gif . The space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq17_HTML.gif is partially ordered with respect to the usual pointwise ordering of functions, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq19_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq20_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq21_HTML.gif . The element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq22_HTML.gif acts as the maximal element in the space and is defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ2_HTML.gif
        (2.1)
        Definition 2.1 A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq23_HTML.gif is called a triangular norm (briefly t-norm) if the following conditions hold:
        1. (i)

          Δ is associative and commutative,

           
        2. (ii)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq24_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq25_HTML.gif ,

           
        3. (iii)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq26_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq29_HTML.gif .

           

        Typical examples of t-norms are http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq31_HTML.gif .

        Definition 2.2 (Hadzić [16], Hadzić and Pap [14])

        A t-norm Δ is said to be of ℋ-type if the family of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq32_HTML.gif is equicontinuous at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq33_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq34_HTML.gif is recursively defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equa_HTML.gif

        A trivial example of a t-norm of ℋ-type is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq35_HTML.gif , but there exist t-norms of ℋ-type with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq36_HTML.gif (see, e.g., [16]).

        Definition 2.3 A probabilistic metric space (briefly, PM-space) is an ordered pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq37_HTML.gif , where S is a nonempty set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq38_HTML.gif if the following conditions are satisfied ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq40_HTML.gif ):
        • (PM1) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq42_HTML.gif ;

        • (PM2) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq43_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq44_HTML.gif ;

        • (PM3) if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq46_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq47_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq48_HTML.gif and for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq49_HTML.gif .

        Definition 2.4 A Menger probabilistic metric space (briefly, Menger PM-space) is a triple http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq37_HTML.gif is a PM-space, Δ is a t-norm and instead of (PM3) in Definition 2.3 it satisfies the following triangle inequality:

        (PM3)′ http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq51_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq49_HTML.gif .

        Remark 2.5 (Sehgal [12])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif be a metric space. Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq53_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif . Then the triple http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq56_HTML.gif is a Menger PM-space induced by the metric d. Furthermore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq57_HTML.gif is complete iff d is complete.

        Schweizer et al.[17] introduced the concept of neighborhood in PM-spaces. For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq59_HTML.gif , the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq60_HTML.gif -neighborhood of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq61_HTML.gif is denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq62_HTML.gif and is defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equb_HTML.gif

        Furthermore, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq63_HTML.gif is a Menger PM-space with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq64_HTML.gif , then the family of neighborhoods http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq65_HTML.gif determines a Hausdorff topology for S.

        Definition 2.6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a Menger PM-space.
        1. (1)

          A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif in S converges to an element x in S (we write http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq6_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq66_HTML.gif ) if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq67_HTML.gif there exists a natural number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq68_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq69_HTML.gif , whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq70_HTML.gif .

           
        2. (2)

          A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif in S is a Cauchy sequence if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq67_HTML.gif there exists a natural number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq68_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq71_HTML.gif , whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq72_HTML.gif .

           
        3. (3)

          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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif be a metric space, let Ω be the diagonal of the Cartesian product http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq73_HTML.gif , and let G be a directed graph such that the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq74_HTML.gif of its vertices coincides with S and the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq75_HTML.gif of its edges contains all loops, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq76_HTML.gif . Assume that G has no parallel edges. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq77_HTML.gif be a directed graph. By letter http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif we denote the undirected graph obtained from G by ignoring the direction of edges and by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq79_HTML.gif we denote the graph obtained by reversing the direction of edges. Equivalently, the graph http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif can be treated as a directed graph having http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq80_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq81_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq82_HTML.gif vertices such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq85_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq86_HTML.gif . A graph G is called connected if there is a path between any two vertices. G is weakly connected if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif is connected. For a graph G such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq75_HTML.gif is symmetric and x is a vertex in G, the subgraph http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq87_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq88_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq89_HTML.gif is the equivalence class of a relation R defined on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq74_HTML.gif by the rule: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq90_HTML.gif if there is a path in G from y to z. Clearly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq87_HTML.gif is connected. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq91_HTML.gif is called a Banach G-contraction [8] if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq92_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq93_HTML.gif , i.e., f is edge-preserving and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq94_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq92_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq95_HTML.gif ).

        3 Main results

        We start with the following definition.

        Definition 3.1 A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is said to be a probabilistic G-contraction if f preserves edges and there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq96_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ3_HTML.gif
        (3.1)

        Example 3.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif be a metric space endowed with a graph G, and let the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif be a Banach G-contraction. Then the induced Menger PM space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq57_HTML.gif is a probabilistic G-contraction.

        To see this, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq97_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq98_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq99_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq100_HTML.gif . Now, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equc_HTML.gif

        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.

        Proposition 3.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif be a probabilistic G-contraction with contraction constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq96_HTML.gif . Then
        1. (i)

          f is both a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif -contraction and a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq79_HTML.gif -contraction with the same contraction constant α.

           
        2. (ii)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq101_HTML.gif is f-invariant and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq102_HTML.gif is a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq103_HTML.gif -contraction provided that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq9_HTML.gif is such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq104_HTML.gif .

           
        Proof
        1. (i)

          It follows from the symmetry of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq105_HTML.gif .

           
        2. (ii)

          Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq106_HTML.gif . Then there is a path http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq107_HTML.gif between x and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq108_HTML.gif . Since f is a probabilistic G-contraction, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq109_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq110_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq111_HTML.gif .

           

        Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq112_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq113_HTML.gif since f is a probabilistic G-contraction. But http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq114_HTML.gif is f invariant, so we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq115_HTML.gif . Condition (3.1) is satisfied automatically, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq116_HTML.gif is a subgraph of G. □

        Lemma 3.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq117_HTML.gif be a Menger PM-space under a t-norm Δ satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq118_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq119_HTML.gif . Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is a probabilistic G-contraction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq120_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq121_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif ). Moreover, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq124_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ) if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq125_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ).

        Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq126_HTML.gif , then there exists a path http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq128_HTML.gif , in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif from x to y with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq129_HTML.gif . From Proposition 3.3, f is a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif -contraction. By induction, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq131_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq133_HTML.gif . Thus we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equd_HTML.gif
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq67_HTML.gif be given. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq134_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq135_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq136_HTML.gif . Choose a natural number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq137_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq138_HTML.gif we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq140_HTML.gif . We get, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq138_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Eque_HTML.gif
        so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq141_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif ). Continuing recursively, one can easily show that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equf_HTML.gif
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq142_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq67_HTML.gif be given. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq134_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq143_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq136_HTML.gif . Choose a natural number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq144_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq145_HTML.gif we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq147_HTML.gif . So that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq145_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equg_HTML.gif

        Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq125_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif . □

        Every t-norm can be extended in a unique way to an n-ary as follows: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq149_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq150_HTML.gif  . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq151_HTML.gif be a path between two vertices x and y in a graph G. Let us denote with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq152_HTML.gif for all t. Clearly the function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq153_HTML.gif is monotone nondecreasing.

        Definition 3.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a PM-space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif . Suppose that there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq154_HTML.gif in S such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq156_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif . We say that:
        1. (i)

          G is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph in S if there exist a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq158_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq154_HTML.gif and a natural number N such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq159_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq160_HTML.gif ;

           
        2. (ii)

          G is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq161_HTML.gif -graph in S if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq162_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq8_HTML.gif and the sequence of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq163_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq164_HTML.gif uniformly as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq165_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif ).

           

        Example 3.6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a Menger PM-space induced by the metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq166_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq167_HTML.gif , and let I be an identity map on S.

        Consider the graph http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq168_HTML.gif consisting of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq169_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equh_HTML.gif
        We note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq170_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif . Also, it is easy to see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq168_HTML.gif is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq171_HTML.gif -graph. But since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq172_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equi_HTML.gif

        Thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq168_HTML.gif is not an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq173_HTML.gif -graph.

        Example 3.7 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a Menger PM space induced by the metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq166_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq174_HTML.gif , and let I be an identity map on S. Consider the graph http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq175_HTML.gif consisting of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq176_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equj_HTML.gif

        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq170_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq177_HTML.gif is not a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq171_HTML.gif -graph. But http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq178_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif ). Thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq175_HTML.gif is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq173_HTML.gif -graph.

        From the above examples, we note that the notions of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq161_HTML.gif -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 [18])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq179_HTML.gif be a Menger PM-space under a t-norm Δ of ℋ-type. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif be a sequence in S, and let there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq96_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equk_HTML.gif

        Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif is a Cauchy sequence.

        Theorem 3.9 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq117_HTML.gif be a complete Menger PM-space under a t-norm Δ of ℋ-type. Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is a probabilistic G-contraction and there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq180_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif , then the following assertions hold.
        1. (i)

          If G is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph, then f has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq182_HTML.gif and for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq183_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq184_HTML.gif . Moreover, if G is weakly connected, then f is a Picard operator.

           
        2. (ii)

          If G is a weakly connected http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq161_HTML.gif -graph, then f is a Picard operator.

           
        Proof Since f is a probabilistic G-contraction and there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq9_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif . By induction http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq185_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq8_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ4_HTML.gif
        (3.2)
        (i) Since the t-norm Δ is of ℋ-type, then from Lemma 3.8 it can be inferred that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq186_HTML.gif is a Cauchy sequence in S. From completeness of the Menger PM-space S, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq187_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ5_HTML.gif
        (3.3)
        Now we prove that ϱ is a fixed point of f. Let G be a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph. Then there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq188_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq186_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq189_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq190_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq160_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq191_HTML.gif is a path in G and so in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq108_HTML.gif to ϱ, thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq192_HTML.gif . Since f is a probabilistic G-contraction and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq193_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq160_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq160_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equl_HTML.gif
        We obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ6_HTML.gif
        (3.4)
        Hence, we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq194_HTML.gif . Now, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq195_HTML.gif , then from Lemma 3.4 we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ7_HTML.gif
        (3.5)
        Next to prove the uniqueness of a fixed point, suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq196_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq197_HTML.gif . Then from Lemma 3.4, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ8_HTML.gif
        (3.6)

        Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq198_HTML.gif . Moreover, if G is weakly connected, then f is a Picard operator as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq199_HTML.gif .

        (ii) Let G be a weakly connected http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq161_HTML.gif -graph. By using the same arguments as in the first part of the proof, we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq200_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq201_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq202_HTML.gif be a path in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq203_HTML.gif to ϱ with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq204_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq205_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq206_HTML.gif .
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ9_HTML.gif
        (3.7)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq207_HTML.gif .

        Since G is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq161_HTML.gif -graph and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq185_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq208_HTML.gif , then the sequence of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq209_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq164_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif ) uniformly. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq67_HTML.gif be given. Since the family http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq210_HTML.gif is equicontinuous at point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq33_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq211_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq212_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq213_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq214_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq145_HTML.gif we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq216_HTML.gif . So that in view of (3.7), for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq145_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ10_HTML.gif
        (3.8)

        Hence, we deduce http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq217_HTML.gif . Finally, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq218_HTML.gif be arbitrary, then from Lemma 3.4, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq219_HTML.gif . □

        Corollary 3.10 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a complete Menger PM-space under a t-norm Δ of ℋ-type. Assume that S is endowed with a graph G which is either http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq161_HTML.gif -graph. Then the following statements are equivalent:
        1. (i)

          G is weakly connected.

           
        2. (ii)

          For every probabilistic G-contraction f on S, if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq9_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif , then f is a Picard operator.

           

        Proof (i) ⇒ (ii): It is immediate from Theorem 3.9.

        (ii) ⇒ (i): Suppose that G is not weakly connected. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif is disconnected, i.e., there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq220_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq222_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq223_HTML.gif , we construct a self-mapping f by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equm_HTML.gif

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq97_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq224_HTML.gif , which implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq225_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq98_HTML.gif , since G contains all loops. Thus the mapping f preserves edges. Also, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq226_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq227_HTML.gif ; thus (3.1) is trivially satisfied. But http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq229_HTML.gif are two fixed points of f contradicting the fact that f is a Picard operator. □

        Remark 3.11 Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq230_HTML.gif , Theorem 3.9 improves and extends the result of Sehgal [[12], Theorem 3] to all Menger PM-spaces with t-norms of ℋ-type. Theorem 3.9 generalizes claim 40 of [[8], Theorem 3.2], and thus we have the following consequence.

        Corollary 3.12 (Jachymski [[8], Theorem 3.2])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif be a complete metric space endowed with the graph G. Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is a Banach G-contraction and the following property is satisfied:

        ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq231_HTML.gif ) For any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif in S, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq232_HTML.gif in S and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq233_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq8_HTML.gif , then there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq234_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq235_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq236_HTML.gif .

        If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq9_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq102_HTML.gif is a Picard operator. Furthermore, if G is weakly connected, then f is a Picard operator.

        Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq237_HTML.gif 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 ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq231_HTML.gif ) invokes that G is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph. Hence the conclusion follows from Theorem 3.9(i). □

        Example 3.13 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq238_HTML.gif be a Menger PM-space where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq240_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq238_HTML.gif is complete. Define a self-mapping f on X by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ11_HTML.gif
        (3.9)

        Further assume that X is endowed with a graph G consisting of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq241_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq242_HTML.gif . It can be seen that f is a probabilistic G-contraction with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq243_HTML.gif and satisfies all the conditions of Theorem 3.9(i).

        Note that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq244_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq245_HTML.gif and for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq99_HTML.gif , we can easily set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq246_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equn_HTML.gif
        or
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equo_HTML.gif

        Hence, one cannot invoke [[12], Theorem 3].

        Definition 3.14 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq63_HTML.gif be a Menger PM-space under a t-norm Δ of ℋ-type. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is said to be: (i) continuous at point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq61_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq6_HTML.gif in S implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq247_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ; (ii) orbitally continuous if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq42_HTML.gif and any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq248_HTML.gif of positive integers, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq249_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq250_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq122_HTML.gif ; (iii) orbitally G-continuous if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq42_HTML.gif and any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq248_HTML.gif of positive integers, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq252_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq253_HTML.gif imply http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq250_HTML.gif (see [8]).

        Theorem 3.15 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq117_HTML.gif be a complete Menger PM-space under a t-norm Δ of ℋ-type. Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is a probabilistic G-contraction such that f is orbitally G-continuous, and let there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq254_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif . Then f has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq187_HTML.gif and for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq183_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq184_HTML.gif . Moreover, if G is weakly connected, then f is a Picard operator.

        Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif , by induction http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq185_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif . By using Lemma 3.8, it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq255_HTML.gif . Since f is orbitally G-continuous, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq256_HTML.gif . This gives http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq257_HTML.gif . From Lemma 3.4 for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq195_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq258_HTML.gif . □

        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.

        Remark 3.17 Theorem 3.15 generalizes and extends claims 20 and 30 [[8], Theorem 3.3] and claim 30 [[8], Theorem 3.4].

        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 [[4], Theorems 2.1 and 2.3], Petruşel and Rus [[3], Theorem 4.3] and Ran and Reurings [[2], Theorem 2.1].

        Corollary 3.18 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq12_HTML.gif be a partially ordered set, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a complete Menger PM-space under a t-norm Δ of ℋ-type. Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is nondecreasing (nonincreasing) with respect to the order ‘’ on S and there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq96_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ12_HTML.gif
        (3.10)
        Also suppose that either
        1. (i)

          f is continuous, or

           
        2. (ii)

          for every nondecreasing sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq5_HTML.gif in S such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq6_HTML.gif in S, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq7_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq8_HTML.gif .

           

        If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq9_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq10_HTML.gif , then f has a fixed point. Furthermore, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq12_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq259_HTML.gif consisting of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq260_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq261_HTML.gif . If f is nondecreasing, then it preserves edges w.r.t. graph http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq259_HTML.gif and condition (3.10) becomes equivalent to (3.1). Thus f is a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq259_HTML.gif -contraction. In case f is nonincreasing, consider http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq168_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq262_HTML.gif and a vertex set coincides with S. Actually, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq263_HTML.gif and from Proposition 3.3 if f is a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq259_HTML.gif -contraction, then it is a probabilistic http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq168_HTML.gif contraction. Now if f is continuous, then the conclusion follows from Theorem 3.15. On the other hand, if (ii) holds, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq259_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq168_HTML.gif are http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq264_HTML.gif be a PM-space endowed with a graph G and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif . Assume the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq154_HTML.gif in S with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq265_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq266_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif ), we say that the graph G is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq267_HTML.gif -graph if for any subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq268_HTML.gif , there exists a natural number N such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq269_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq160_HTML.gif .

        Theorem 3.20 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq117_HTML.gif be a compact Menger PM-space under a t-norm Δ satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq134_HTML.gif . Assume that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is a probabilistic G-contraction, and let there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq254_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif . If G is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq267_HTML.gif -graph, then f has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq270_HTML.gif .

        Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq181_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq271_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq272_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equp_HTML.gif
        From compactness, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq188_HTML.gif be a subsequence such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq273_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq67_HTML.gif be given. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq134_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq274_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq136_HTML.gif choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq275_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq276_HTML.gif we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq277_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq278_HTML.gif . Then we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equq_HTML.gif

        Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq279_HTML.gif .

        Choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq280_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq281_HTML.gif we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq282_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq283_HTML.gif . Since G is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq267_HTML.gif -graph, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq284_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq285_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq286_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq287_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq288_HTML.gif we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equr_HTML.gif

        Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq217_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq289_HTML.gif is a path in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq78_HTML.gif , so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq192_HTML.gif . □

        So far it remains to investigate whether Theorem 3.20 can be extended to a complete PM-space?

        Definition 3.21[12]

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq37_HTML.gif be a PM-space, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq290_HTML.gif be fixed real numbers. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq60_HTML.gif -contraction if there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq96_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq291_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ13_HTML.gif
        (3.11)

        The PM space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq37_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq292_HTML.gif -chainable if for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq54_HTML.gif there exists a finite sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq293_HTML.gif of elements in S with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq294_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq295_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq296_HTML.gif .

        It is important to note that every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq292_HTML.gif -contraction mapping is continuous. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq6_HTML.gif in S, then there exists a natural number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq68_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq297_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq70_HTML.gif . Thus, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq70_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equs_HTML.gif

        Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq298_HTML.gif .

        Theorem 3.22 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif be a complete http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq292_HTML.gif -chainable Menger PM-space under a t-norm Δ of ℋ-type. Let the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif be an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq292_HTML.gif -contraction. Then f is a Picard operator.

        Proof Consider the graph G consisting of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq299_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq74_HTML.gif coinciding with S. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq54_HTML.gif . Since the PM-space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq37_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq60_HTML.gif -chainable, there exists a finite sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq300_HTML.gif in S with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq294_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq301_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq302_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq303_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq304_HTML.gif , which yields that G is connected. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq97_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq305_HTML.gif . Since the mapping f is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq60_HTML.gif -contraction, thus (3.1) is satisfied. Finally we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equt_HTML.gif

        Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq98_HTML.gif . 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 [12] which is only restricted to continuous t-norms satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq306_HTML.gif . Moreover, the proof of our result is rather simple and easy, which invokes the novelty of Theorem 3.22.

        Definition 3.24 (Edelstein [19, 20])

        The metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif is ε-chainbale for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq58_HTML.gif if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq307_HTML.gif , there exists a finite sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq308_HTML.gif of elements in S with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq294_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq309_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq304_HTML.gif .

        Remark 3.25[12]

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif is an ε-chainable metric space, then the induced Menger PM-space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq237_HTML.gif is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq292_HTML.gif -chainable space.

        Corollary 3.26 (Edelstein [19, 20])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif be a complete ε-chainable metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq3_HTML.gif and let there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq96_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ14_HTML.gif
        (3.12)

        Then f is a Picard operator.

        Proof Since the metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq2_HTML.gif is ε-chainable, then the induced Menger PM-space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq50_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq292_HTML.gif -chainable for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq290_HTML.gif . We only need to show that the self-mapping f on S is an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq60_HTML.gif -contraction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq54_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq310_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq311_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq312_HTML.gif . The definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq313_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq314_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq100_HTML.gif . Now, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq55_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equu_HTML.gif

        Hence the conclusion follows from Theorem 3.22. □

        Kirk et al.[21] introduced the idea of cyclic contractions and established fixed point results for such mappings.

        Let S be a nonempty set, let m be a positive integer, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq315_HTML.gif be nonempty closed subsets of S, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq316_HTML.gif be an operator. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq317_HTML.gif is known as a cyclic representation of S w.r.t. f if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_Equ15_HTML.gif
        (3.13)

        and the operator f is known as a cyclic operator [21].

        In the following, we present the probabilistic version of the main result of [21], as a last consequence of Theorem 3.9.

        Theorem 3.27 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq117_HTML.gif be a complete Menger PM-space under a t-norm Δ of ℋ-type. Let m be a positive integer, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq315_HTML.gif be nonempty closed subsets of S, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq318_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq319_HTML.gif . Assume that
        1. (i)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq320_HTML.gif is a cyclic representation of Y w.r.t. f;

           
        2. (ii)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq321_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq322_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq323_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq324_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq325_HTML.gif .

           

        Then f has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq184_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq327_HTML.gif .

        Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq320_HTML.gif is closed, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq328_HTML.gif is complete. Let us consider a graph G consisting of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq329_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq330_HTML.gif . By (i) it follows that f preserves edges. Now, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq331_HTML.gif in Y such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq332_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq132_HTML.gif . Then by (3.13) it is inferred that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq154_HTML.gif has infinitely many terms in each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq333_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq334_HTML.gif . So that one can easily identify a subsequence of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq154_HTML.gif converging to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq228_HTML.gif in each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq333_HTML.gif ; and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq333_HTML.gif ’s are closed, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq335_HTML.gif . Thus, we can easily form a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq158_HTML.gif in some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq336_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq337_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq338_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq339_HTML.gif . It elicits that G is a weakly connected http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_556_IEq157_HTML.gif -graph. Hence, by Theorem 3.9 conclusion follows. □

        Declarations

        Acknowledgements

        The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Quaid-i-Azam University
        (2)
        Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12
        (3)
        Department of Mathematics, King Abdulaziz University

        References

        1. Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23:2205–2212.MathSciNetView ArticleMATH
        2. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132:1435–1443.MathSciNetView ArticleMATH
        3. Petrusel A, Rus IA: Fixed point theorems in ordered L -spaces. Proc. Am. Math. Soc. 2006, 134:411–418.MathSciNetView ArticleMATH
        4. Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22:223–239.MathSciNetView ArticleMATH
        5. O’Regan D, Petruşel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341:1241–1252.MathSciNetView ArticleMATH
        6. Agarwal RP, El-Gebeily MA, O’Regan E: Generalized contraction in partially ordered metric spaces. Appl. Anal. 2008, 87:1–8.MathSciNetView Article
        7. Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65:1379–1393.MathSciNetView ArticleMATH
        8. Jachymski J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 2007, 136:1359–1373.MathSciNetView Article
        9. Bojor F: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 2012, 75:3895–3901.MathSciNetView ArticleMATH
        10. Samreen M, Kamran T: Fixed point theorems for integral G -contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 149
        11. Aleomraninejad SMA, Rezapour S, Shahzad N: Some fixed point results on a metric space with a graph. Topol. Appl. 2012, 159:659–663.MathSciNetView ArticleMATH
        12. Sehgal VM, Bharucha-Reid AT: Fixed points of contraction mappings on probabilistic metric spaces. Math. Syst. Theory 1972, 6:97–102.MathSciNetView ArticleMATH
        13. Schweizer B, Sklar A: Probabilistic Metric Spaces. Elsevier/North-Holland, New York; 1983.MATH
        14. Hadzić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.View Article
        15. Cho Y, Park KS, Chang SS: Fixed point theorems in metric spaces and probabilistic metric spaces. Int. J. Math. Math. Sci. 1996, 19:243–252.MathSciNetView ArticleMATH
        16. Hadzić O: Fixed point theorems for multivalued mapping in probabilistic metric spaces. Fuzzy Sets Syst. 1997, 88:219–226.View ArticleMATH
        17. Schweizer B, Sklar A, Thorp E: The metrization of statistical metric spaces. Pac. J. Math. 1960, 10:673–675.MathSciNetView ArticleMATH
        18. Miheţ D: A generalization of a contraction principle in probabilistic metric spaces. Part II. Int. J. Math. Math. Sci. 2005, 5:729–736.
        19. Edelstein M: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 1962, 37:74–79.MathSciNetView ArticleMATH
        20. Edelstein M: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 1961, 12:7–10.MathSciNetMATH
        21. Kirk WA, Srinivasan PS, Veeranmani P: Fixed points for mappings satisfying cyclical contractive condition. Fixed Point Theory 2003,4(1):79–89.MathSciNetMATH

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        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.