Open Access

Probabilistic G-contractions

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 ( ϵ , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif be a complete metric space endowed with the partial order ‘. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is nondecreasing (or nonincreasing) with respect to the partial order ‘’ on S and there exists a real number α, 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq4_HTML.gif, such that
d ( f x , f y ) α d ( x , y ) for all  x , y S , x y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ1_HTML.gif
(1.1)
Also suppose that either
  1. (i)

    f is continuous; or

     
  2. (ii)

    for every nondecreasing sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif in S such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq6_HTML.gif in S, we have x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq7_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq8_HTML.gif.

     

If there exists x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq9_HTML.gif with x 0 f x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq10_HTML.gif (or x 0 f x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq11_HTML.gif), then f has a fixed point. Furthermore, if ( S , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( ϵ , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 F : R [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq13_HTML.gif is called a distribution function if it is nondecreasing, left continuous and inf t R F ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq14_HTML.gif, sup t R F ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq15_HTML.gif. In addition, if F ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq16_HTML.gif, then F is called a distance distribution function. Let D + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq17_HTML.gif denote the set of all distance distribution functions satisfying lim t F ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq18_HTML.gif. The space D + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq17_HTML.gif is partially ordered with respect to the usual pointwise ordering of functions, i.e., F G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq19_HTML.gif if and only if F ( t ) G ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq20_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq21_HTML.gif. The element ϵ 0 D + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq22_HTML.gif acts as the maximal element in the space and is defined by
ϵ 0 ( t ) = { 0 if  t 0 , 1 if  t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ2_HTML.gif
(2.1)
Definition 2.1 A mapping Δ : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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)

    Δ ( a , 1 ) = a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq24_HTML.gif for all a [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq25_HTML.gif,

     
  3. (iii)

    Δ ( a , b ) Δ ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq26_HTML.gif for all a , b , c , d [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq27_HTML.gif with a c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq28_HTML.gif and b d https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq29_HTML.gif.

     

Typical examples of t-norms are Δ M ( a , b ) = min { a , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq30_HTML.gif and Δ P ( a , b ) = a b https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq31_HTML.gif.

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

A t-norm Δ is said to be of -type if the family of functions { Δ n ( t ) } n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq32_HTML.gif is equicontinuous at t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq33_HTML.gif, where Δ n : [ 0 , 1 ] [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq34_HTML.gif is recursively defined by
Δ 1 ( t ) = Δ ( t , t ) , and Δ n ( t ) = Δ ( Δ n 1 ( t ) , t ) ; t [ 0 , 1 ] , n = 2 , 3 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equa_HTML.gif

A trivial example of a t-norm of -type is Δ M : = min https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq35_HTML.gif, but there exist t-norms of -type with Δ Δ M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq36_HTML.gif (see, e.g., [16]).

Definition 2.3 A probabilistic metric space (briefly, PM-space) is an ordered pair ( S , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq37_HTML.gif, where S is a nonempty set and F : S × S D + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq38_HTML.gif if the following conditions are satisfied ( F ( p , q ) = F p , q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq39_HTML.gif, ( p , q ) S × S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq40_HTML.gif):
  1. (PM1)

    F x , y ( t ) = ϵ 0 ( t ) x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq41_HTML.gif and x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq42_HTML.gif;

     
  2. (PM2)

    F x , y ( t ) = F y , x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq43_HTML.gif for all x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq42_HTML.gif and t R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq44_HTML.gif;

     
  3. (PM3)

    if F x , y ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq45_HTML.gif and F y , z ( s ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq46_HTML.gif, then F x , z ( t + s ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq47_HTML.gif for all x , y , z S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq48_HTML.gif and for every t , s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq49_HTML.gif.

     

Definition 2.4 A Menger probabilistic metric space (briefly, Menger PM-space) is a triple ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif, where ( S , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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)′ F x , z ( t + s ) Δ ( F x , y ( t ) , F y , z ( s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq51_HTML.gif for all x , y , z S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq52_HTML.gif and t , s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq49_HTML.gif.

Remark 2.5 (Sehgal [12])

Let ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif be a metric space. Define F x y ( t ) = ϵ 0 ( t d ( x , y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq53_HTML.gif for all x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq54_HTML.gif and t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif. Then the triple ( S , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq56_HTML.gif is a Menger PM-space induced by the metric d. Furthermore, ( S , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq57_HTML.gif is complete iff d is complete.

Schweizer et al. [17] introduced the concept of neighborhood in PM-spaces. For ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq58_HTML.gif and δ ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq59_HTML.gif, the ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq60_HTML.gif-neighborhood of x S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq61_HTML.gif is denoted by N x ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq62_HTML.gif and is defined by
N x ( ε , δ ) = { y S : F x , y ( ε ) > 1 δ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equb_HTML.gif

Furthermore, if ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq63_HTML.gif is a Menger PM-space with sup 0 < a < 1 Δ ( a , a ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq64_HTML.gif, then the family of neighborhoods { N x ( ε , δ ) : x S , ε > 0 , δ ( 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq65_HTML.gif determines a Hausdorff topology for S.

Definition 2.6 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a Menger PM-space.
  1. (1)

    A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif in S converges to an element x in S (we write x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq6_HTML.gif or lim n x n = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq66_HTML.gif) if for every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq58_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq67_HTML.gif there exists a natural number N ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq68_HTML.gif such that F x n , x ( ε ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq69_HTML.gif, whenever n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq70_HTML.gif.

     
  2. (2)

    A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif in S is a Cauchy sequence if for every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq58_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq67_HTML.gif there exists a natural number N ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq68_HTML.gif such that F x n , x m ( ε ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq71_HTML.gif, whenever n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif be a metric space, let Ω be the diagonal of the Cartesian product S × S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq73_HTML.gif, and let G be a directed graph such that the set V ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq74_HTML.gif of its vertices coincides with S and the set E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq75_HTML.gif of its edges contains all loops, i.e., E ( G ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq76_HTML.gif. Assume that G has no parallel edges. Let G = ( V ( G ) , E ( G ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq77_HTML.gif be a directed graph. By letter G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif we denote the undirected graph obtained from G by ignoring the direction of edges and by G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq79_HTML.gif we denote the graph obtained by reversing the direction of edges. Equivalently, the graph G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif can be treated as a directed graph having E ( G ˜ ) : = E ( G ) E ( G 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( x i ) i = 0 l https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq81_HTML.gif of l + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq82_HTML.gif vertices such that x 0 = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq83_HTML.gif, x l = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq84_HTML.gif and ( x i 1 , x i ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq85_HTML.gif for i = 1 , , l https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq86_HTML.gif. A graph G is called connected if there is a path between any two vertices. G is weakly connected if G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif is connected. For a graph G such that E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq75_HTML.gif is symmetric and x is a vertex in G, the subgraph G x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 V ( G x ) = [ x ] G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq88_HTML.gif, where [ x ] G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq89_HTML.gif is the equivalence class of a relation R defined on V ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq74_HTML.gif by the rule: y R z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq90_HTML.gif if there is a path in G from y to z. Clearly, G x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq87_HTML.gif is connected. A mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq91_HTML.gif is called a Banach G-contraction [8] if x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq92_HTML.gif; ( x , y ) E ( G ) ( f x , f y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq93_HTML.gif, i.e., f is edge-preserving and α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq94_HTML.gif x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq92_HTML.gif ( ( x , y ) E ( G ) d ( f x , f y ) α d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq95_HTML.gif).

3 Main results

We start with the following definition.

Definition 3.1 A mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is said to be a probabilistic G-contraction if f preserves edges and there exists α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq96_HTML.gif such that
x , y S , ( x , y ) E ( G ) F f x , f y ( α t ) F x , y ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ3_HTML.gif
(3.1)

Example 3.2 Let ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif be a metric space endowed with a graph G, and let the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif be a Banach G-contraction. Then the induced Menger PM space ( S , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq57_HTML.gif is a probabilistic G-contraction.

To see this, let ( x , y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq97_HTML.gif, then ( f x , f y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq98_HTML.gif and there exists α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq99_HTML.gif such that d ( f x , f y ) α d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq100_HTML.gif. Now, for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif, we have
F f x , f y ( α t ) = ϵ 0 ( α t d ( f x , f y ) ) ϵ 0 ( α t α d ( x , y ) ) = F x , y ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif be a probabilistic G-contraction with contraction constant α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq96_HTML.gif. Then
  1. (i)

    f is both a probabilistic G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif-contraction and a probabilistic G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq79_HTML.gif-contraction with the same contraction constant α.

     
  2. (ii)

    [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq101_HTML.gif is f-invariant and f | [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq102_HTML.gif is a probabilistic G ˜ x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq103_HTML.gif-contraction provided that x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq9_HTML.gif is such that f x 0 [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq104_HTML.gif.

     
Proof
  1. (i)

    It follows from the symmetry of F x , y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq105_HTML.gif.

     
  2. (ii)

    Let x [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq106_HTML.gif. Then there is a path x = z 0 , z 1 , , z l = x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq107_HTML.gif between x and x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq108_HTML.gif. Since f is a probabilistic G-contraction, ( f z i 1 , f z i ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq109_HTML.gif i = 1 , 2 , , l https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq110_HTML.gif. Thus f x [ f x 0 ] G ˜ = [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq111_HTML.gif.

     

Suppose ( x , y ) E ( G ˜ x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq112_HTML.gif. Then ( f x , f y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq113_HTML.gif since f is a probabilistic G-contraction. But [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq114_HTML.gif is f invariant, so we conclude that ( f x , f y ) E ( G ˜ x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq115_HTML.gif. Condition (3.1) is satisfied automatically, since G ˜ x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq116_HTML.gif is a subgraph of G. □

Lemma 3.4 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq117_HTML.gif be a Menger PM-space under a t-norm Δ satisfying sup a < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq118_HTML.gif Δ ( a , a ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq119_HTML.gif. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is a probabilistic G-contraction. Let y [ x ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq120_HTML.gif, then F f n x , f n y ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq121_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif ( t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif). Moreover, for z S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq123_HTML.gif, f n x z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq124_HTML.gif ( n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif) if and only if f n y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq125_HTML.gif ( n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif).

Proof Let x S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq61_HTML.gif and y [ x ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq126_HTML.gif, then there exists a path ( x i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq127_HTML.gif, i = 0 , 1 , 2 , , l https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq128_HTML.gif, in G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif from x to y with x 0 = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq83_HTML.gif, x l = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq84_HTML.gif and ( x i 1 , x i ) E ( G ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq129_HTML.gif. From Proposition 3.3, f is a probabilistic G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif-contraction. By induction, for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif, we have ( f n x i 1 , f n x i ) E ( G ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq130_HTML.gif and F f n x i 1 , f n x i ( α t ) F f n 1 x i 1 , f n 1 y i ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq131_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif and i = 1 , , l https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq133_HTML.gif. Thus we obtain
F f n x i 1 , f n x i ( t ) F f n 1 x i 1 , f n 1 x i ( t α ) F f n 2 x i 1 , f n 2 x i ( t α 2 ) F x i 1 , x i ( t α n ) 1 ( as  n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equd_HTML.gif
Let t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq67_HTML.gif be given. Since sup a < 1 Δ ( a , a ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq134_HTML.gif, then there exists λ ( δ ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq135_HTML.gif such that Δ ( 1 λ , 1 λ ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq136_HTML.gif. Choose a natural number n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq137_HTML.gif such that for all n n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq138_HTML.gif we have F f n x 0 , f n x 1 ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq139_HTML.gif and F f n x 1 , f n x 2 ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq140_HTML.gif. We get, for all n n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq138_HTML.gif,
F f n x 0 , f n x 2 ( t ) Δ ( F f n x 0 , f n x 1 ( t 2 ) , F f n x 1 , f n x 2 ( t 2 ) ) Δ ( 1 λ , 1 λ ) > 1 δ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Eque_HTML.gif
so that F f n x 0 , f n x 2 ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq141_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif ( t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif). Continuing recursively, one can easily show that
F f n x 0 , f n x l ( t ) 1 as  n ( t > 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equf_HTML.gif
Let f n x z S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq142_HTML.gif. Let t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq67_HTML.gif be given. Since sup a < 1 Δ ( a , a ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq134_HTML.gif, then there exists λ ( δ ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq143_HTML.gif such that Δ ( 1 λ , 1 λ ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq136_HTML.gif. Choose a natural number n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq144_HTML.gif such that for all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq145_HTML.gif we have F f n x , f n y ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq146_HTML.gif and F z , f n x ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq147_HTML.gif. So that for all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq145_HTML.gif, we have
F z , f n y ( t ) Δ ( F z , f n x ( t 2 ) , F f n x , f n y ( t 2 ) ) Δ ( 1 λ , 1 λ ) > 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equg_HTML.gif

Hence, f n y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq125_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif. □

Every t-norm can be extended in a unique way to an n-ary as follows: Δ i = 1 0 x i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq148_HTML.gif, Δ i = 1 n x i = Δ ( Δ i = 1 n 1 x i , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq149_HTML.gif for n = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq150_HTML.gif . Let ( x i ) i = 1 l https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq151_HTML.gif be a path between two vertices x and y in a graph G. Let us denote with L x , y ( t ) = Δ i = 1 l F x i 1 , x i ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq152_HTML.gif for all t. Clearly the function L x , y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq153_HTML.gif is monotone nondecreasing.

Definition 3.5 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a PM-space and f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif. Suppose that there exists a sequence { f n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq154_HTML.gif in S such that f n x x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq155_HTML.gif and ( f n x , f n + 1 x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq156_HTML.gif for n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif. We say that:
  1. (i)

    G is a ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graph in S if there exist a subsequence { f n k x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq158_HTML.gif of { f n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq154_HTML.gif and a natural number N such that ( f n k x , x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq159_HTML.gif for k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq160_HTML.gif;

     
  2. (ii)

    G is an ( H f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq161_HTML.gif-graph in S if f n x [ x ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq162_HTML.gif for n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq8_HTML.gif and the sequence of functions { L f n x , x ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq163_HTML.gif converges to ϵ 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq164_HTML.gif uniformly as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq165_HTML.gif ( t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif).

     

Example 3.6 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a Menger PM-space induced by the metric d ( x , y ) = | x y | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq166_HTML.gif on S = { 1 n : n N } { 0 } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq167_HTML.gif, and let I be an identity map on S.

Consider the graph G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq168_HTML.gif consisting of V ( G 2 ) = S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq169_HTML.gif and
E ( G 2 ) = { ( 1 n , 1 n + 1 ) , ( 1 n + 1 , n ) , ( n , 0 ) , ( 1 5 n , 0 ) ; n N } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equh_HTML.gif
We note that x n = 1 n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq170_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif. Also, it is easy to see that G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq168_HTML.gif is a ( C I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq171_HTML.gif-graph. But since Δ ( a , b ) = min { a , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq172_HTML.gif, then
L x n , 0 ( t ) = Δ ( Δ ( ϵ 0 ( t | 1 n 1 n + 1 | ) , ϵ 0 ( t | 1 n + 1 n | ) ) , ϵ 0 ( t n ) ) = ϵ 0 ( t n ) ϵ 0 ( t ) as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equi_HTML.gif

Thus G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq168_HTML.gif is not an ( H I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq173_HTML.gif-graph.

Example 3.7 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a Menger PM space induced by the metric d ( x , y ) = | x y | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq166_HTML.gif on S = { 1 n : n N } { 5 n + 1 : n N } { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq174_HTML.gif, and let I be an identity map on S. Consider the graph G 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq175_HTML.gif consisting of V ( G 3 ) = S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq176_HTML.gif and
E ( G 3 ) = { ( 1 n , 1 n + 1 ) , ( 1 n , 5 n + 1 ) , ( 5 n + 1 , 0 ) ; n N } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equj_HTML.gif

Since x n = 1 n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq170_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif. Clearly, G 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq177_HTML.gif is not a ( C I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq171_HTML.gif-graph. But L x n , 0 ( t ) = ϵ 0 ( t 5 n + 1 ) ϵ 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq178_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif ( t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif). Thus G 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq175_HTML.gif is an ( H I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq173_HTML.gif-graph.

From the above examples, we note that the notions of ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graph and ( H f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq179_HTML.gif be a Menger PM-space under a t-norm Δ of -type. Let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif be a sequence in S, and let there exist α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq96_HTML.gif such that
F x n , x n + 1 ( α t ) F x n 1 , x n ( t ) for all  n N , t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equk_HTML.gif

Then { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif is a Cauchy sequence.

Theorem 3.9 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq117_HTML.gif be a complete Menger PM-space under a t-norm Δ of -type. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is a probabilistic G-contraction and there exists x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq180_HTML.gif such that ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif, then the following assertions hold.
  1. (i)

    If G is a ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graph, then f has a unique fixed point ϱ [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq182_HTML.gif and for any y [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq183_HTML.gif, f n y ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq184_HTML.gif. Moreover, if G is weakly connected, then f is a Picard operator.

     
  2. (ii)

    If G is a weakly connected ( H f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq161_HTML.gif-graph, then f is a Picard operator.

     
Proof Since f is a probabilistic G-contraction and there exists x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq9_HTML.gif such that ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif. By induction ( f n x 0 , f n + 1 x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq185_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq8_HTML.gif and
F f n x 0 , f n + 1 x 0 ( α t ) F f n 1 x 0 , f n x 0 ( t ) for all  n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ4_HTML.gif
(3.2)
(i) Since the t-norm Δ is of -type, then from Lemma 3.8 it can be inferred that { f n x 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq186_HTML.gif is a Cauchy sequence in S. From completeness of the Menger PM-space S, there exists ϱ S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq187_HTML.gif such that
lim n f n x 0 = ϱ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ5_HTML.gif
(3.3)
Now we prove that ϱ is a fixed point of f. Let G be a ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graph. Then there exists a subsequence { f n k x 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq188_HTML.gif of { f n x 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq186_HTML.gif and N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq189_HTML.gif such that ( f n k x 0 , ϱ ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq190_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq160_HTML.gif. Note that ( x 0 , f x 0 , f 2 x 0 , , f n 1 x 0 , , f n N x 0 , ϱ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq191_HTML.gif is a path in G and so in G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif from x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq108_HTML.gif to ϱ, thus ϱ [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq192_HTML.gif. Since f is a probabilistic G-contraction and ( f n k x 0 , ϱ ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq193_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq160_HTML.gif. For t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq160_HTML.gif, we get
F f n k + 1 x 0 , f ϱ ( t ) F f n k + 1 x 0 , f ϱ ( α t ) F f n k x 0 , ϱ ( t ) 1 as  k . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equl_HTML.gif
We obtain
lim k f n k + 1 x 0 = f ϱ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ6_HTML.gif
(3.4)
Hence, we conclude that f ϱ = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq194_HTML.gif. Now, let y [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq195_HTML.gif, then from Lemma 3.4 we get
lim n f n y = ϱ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ7_HTML.gif
(3.5)
Next to prove the uniqueness of a fixed point, suppose ϱ [ x 0 ] G ˜ = [ ϱ ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq196_HTML.gif such that f ϱ = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq197_HTML.gif. Then from Lemma 3.4, for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif, we have
F ϱ , ϱ ( t ) = F f n ϱ , f n ϱ ( t ) 1 , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ8_HTML.gif
(3.6)

Hence, ϱ = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq198_HTML.gif. Moreover, if G is weakly connected, then f is a Picard operator as [ x 0 ] G ˜ = S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq199_HTML.gif.

(ii) Let G be a weakly connected ( H f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq161_HTML.gif-graph. By using the same arguments as in the first part of the proof, we obtain lim n f n x 0 = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq200_HTML.gif. For each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif let ( z i n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq201_HTML.gif; i = 0 , , M n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq202_HTML.gif be a path in G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif from f n x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq203_HTML.gif to ϱ with z 0 n = f n x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq204_HTML.gif, z M n = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq205_HTML.gif and ( z i 1 n , z i n ) E ( G ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq206_HTML.gif.
F ϱ , f ϱ ( t ) F ϱ , f ϱ ( α t ) Δ ( F ϱ , f n + 1 x 0 ( α t 2 ) , F f n + 1 x 0 , f ϱ ( α t 2 ) ) Δ ( F ϱ , f n + 1 x 0 ( α t 2 ) , Δ i = 1 M n F f z i 1 n , f z i n ( α t 2 M n ) ) Δ ( F ϱ , f n + 1 x 0 ( α t 2 ) , Δ i = 1 M n F z i 1 n , z i n ( t 2 M n ) ) Δ ( F ϱ , f n + 1 x 0 ( α t 2 ) , L f n x 0 , ϱ ( t 2 M ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ9_HTML.gif
(3.7)

where M = max { M n : n N } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq207_HTML.gif.

Since G is an ( H f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq161_HTML.gif-graph and ( f n x 0 , f n + 1 x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq185_HTML.gif for n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif with lim n f n x 0 = ϱ S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq208_HTML.gif, then the sequence of functions { L f n x 0 , ϱ ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq209_HTML.gif converges to ϵ 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq164_HTML.gif ( t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif) uniformly. Let t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq67_HTML.gif be given. Since the family { Δ p ( t ) } p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq210_HTML.gif is equicontinuous at point t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq33_HTML.gif, there exists λ ( δ ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq211_HTML.gif such that Δ p ( 1 λ ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq212_HTML.gif for every p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq213_HTML.gif. Choose n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq214_HTML.gif such that for all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq145_HTML.gif we have F ϱ , f n + 1 x 0 ( α t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq215_HTML.gif and L f n x 0 , ϱ ( t 2 M ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq216_HTML.gif. So that in view of (3.7), for all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq145_HTML.gif, we have
F ϱ , f ϱ ( t ) Δ ( 1 λ , 1 λ ) = Δ 1 ( 1 λ ) > 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ10_HTML.gif
(3.8)

Hence, we deduce f ϱ = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq217_HTML.gif. Finally, let y S = [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq218_HTML.gif be arbitrary, then from Lemma 3.4, lim n f n y = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq219_HTML.gif. □

Corollary 3.10 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a complete Menger PM-space under a t-norm Δ of -type. Assume that S is endowed with a graph G which is either ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graph or ( H f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq9_HTML.gif such that ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif is disconnected, i.e., there exists x S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq220_HTML.gif such that [ x ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq221_HTML.gif and S [ x ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq222_HTML.gif. Let y S [ x ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq223_HTML.gif, we construct a self-mapping f by
f x = { x if  x [ x ] G ˜ , y if  x S [ x ] G ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equm_HTML.gif

Let ( x , y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq97_HTML.gif, then [ x ] G ˜ : = [ y ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq224_HTML.gif, which implies f x = f y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq225_HTML.gif. Hence ( f x , f y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq98_HTML.gif, since G contains all loops. Thus the mapping f preserves edges. Also, for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq226_HTML.gif, we have F f x , f y ( α t ) = 1 F x , y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq227_HTML.gif; thus (3.1) is trivially satisfied. But x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq228_HTML.gif and y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq229_HTML.gif are two fixed points of f contradicting the fact that f is a Picard operator. □

Remark 3.11 Taking G = ( S , S × S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 -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 ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif be a complete metric space endowed with the graph G. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is a Banach G-contraction and the following property is satisfied:

( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq231_HTML.gif ) For any sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif in S, if x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq232_HTML.gif in S and ( x n , x n + 1 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq233_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq8_HTML.gif, then there exists a subsequence { x n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq234_HTML.gif with ( x n k , x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq235_HTML.gif for all k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq236_HTML.gif.

If there exists x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq9_HTML.gif with ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif, then f | [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq102_HTML.gif is a Picard operator. Furthermore, if G is weakly connected, then f is a Picard operator.

Proof Let ( S , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq231_HTML.gif ) invokes that G is a ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graph. Hence the conclusion follows from Theorem 3.9(i). □

Example 3.13 Let ( X , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq238_HTML.gif be a Menger PM-space where X = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq239_HTML.gif and F x , y ( t ) = t t + | x y | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq240_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif. Then ( X , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq238_HTML.gif is complete. Define a self-mapping f on X by
f x = { x 2 p if  x = 1 n  and  p 3  is a fixed integer , 0 otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ11_HTML.gif
(3.9)

Further assume that X is endowed with a graph G consisting of V ( G ) : = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq241_HTML.gif and E ( G ) : = Ω { ( 1 n , 1 m ) : n , m N  and  n | m } { ( x , 0 ) : x 1 n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq242_HTML.gif. It can be seen that f is a probabilistic G-contraction with α = 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq243_HTML.gif and satisfies all the conditions of Theorem 3.9(i).

Note that for x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq244_HTML.gif and y = 5 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq245_HTML.gif and for each α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq99_HTML.gif, we can easily set 0 < t < 1 6 ( 1 α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq246_HTML.gif such that
α t α t + | 1 3 0 | < t t + | 1 5 6 | , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equn_HTML.gif
or
F f 1 , f 5 6 ( α t ) < F 1 , 5 6 ( t ) for  0 < t < 1 6 ( 1 α ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equo_HTML.gif

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

Definition 3.14 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq63_HTML.gif be a Menger PM-space under a t-norm Δ of -type. A mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is said to be: (i) continuous at point x S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq61_HTML.gif whenever x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq6_HTML.gif in S implies f x n f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq247_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif; (ii) orbitally continuous if for all x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq42_HTML.gif and any sequence { k n } n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq248_HTML.gif of positive integers, f k n x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq249_HTML.gif implies f ( f k n x ) f y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq250_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq122_HTML.gif; (iii) orbitally G-continuous if for all x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq42_HTML.gif and any sequence { k n } n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq248_HTML.gif of positive integers, f k n x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq251_HTML.gif and ( f k n x , f k n + 1 x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq252_HTML.gif n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq253_HTML.gif imply f ( f k n x ) f y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq250_HTML.gif (see [8]).

Theorem 3.15 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq117_HTML.gif be a complete Menger PM-space under a t-norm Δ of -type. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is a probabilistic G-contraction such that f is orbitally G-continuous, and let there exist x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq254_HTML.gif such that ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif. Then f has a unique fixed point ϱ S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq187_HTML.gif and for every y [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq183_HTML.gif, f n y ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq184_HTML.gif. Moreover, if G is weakly connected, then f is a Picard operator.

Proof Let ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif, by induction ( f n x 0 , f n + 1 x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq185_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif. By using Lemma 3.8, it follows that f n x 0 ϱ S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq255_HTML.gif. Since f is orbitally G-continuous, then f ( f n x 0 ) f ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq256_HTML.gif. This gives ϱ = f ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq257_HTML.gif. From Lemma 3.4 for any y [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq195_HTML.gif, f n y ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq12_HTML.gif be a partially ordered set, and let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a complete Menger PM-space under a t-norm Δ of -type. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is nondecreasing (nonincreasing) with respect to the order ‘’ on S and there exists α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq96_HTML.gif such that
F f x , f y ( α t ) F x , y ( t ) for all  x , y S , x y ( t > 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ12_HTML.gif
(3.10)
Also suppose that either
  1. (i)

    f is continuous, or

     
  2. (ii)

    for every nondecreasing sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq5_HTML.gif in S such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq6_HTML.gif in S, we have x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq7_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq8_HTML.gif.

     

If there exists x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq9_HTML.gif with x 0 f x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq10_HTML.gif, then f has a fixed point. Furthermore, if ( S , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq259_HTML.gif consisting of V ( G 1 ) = S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq260_HTML.gif and E ( G 1 ) = { ( x , y ) S × S : x y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq261_HTML.gif. If f is nondecreasing, then it preserves edges w.r.t. graph G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq259_HTML.gif and condition (3.10) becomes equivalent to (3.1). Thus f is a probabilistic G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq259_HTML.gif-contraction. In case f is nonincreasing, consider G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq168_HTML.gif with E ( G 2 ) = { ( x , y ) S × S : x y  or  x y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq262_HTML.gif and a vertex set coincides with S. Actually, G 2 : = G 1 ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq263_HTML.gif and from Proposition 3.3 if f is a probabilistic G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq259_HTML.gif-contraction, then it is a probabilistic G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 G 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq259_HTML.gif and G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq168_HTML.gif are ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq157_HTML.gif-graphs and conclusions follow from the first part of Theorem 3.9. □

By relaxing -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 ( S , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq264_HTML.gif be a PM-space endowed with a graph G and f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif. Assume the sequence { f n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq154_HTML.gif in S with ( f n x , f n + 1 x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq265_HTML.gif for n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif and F f n x , f n + 1 x ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq266_HTML.gif ( t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif), we say that the graph G is ( E f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq267_HTML.gif-graph if for any subsequence f n k x z S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq268_HTML.gif, there exists a natural number N such that ( f n k x , z ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq269_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq160_HTML.gif.

Theorem 3.20 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq117_HTML.gif be a compact Menger PM-space under a t-norm Δ satisfying sup a < 1 Δ ( a , a ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq134_HTML.gif. Assume that the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is a probabilistic G-contraction, and let there exist x 0 S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq254_HTML.gif such that ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif. If G is an ( E f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq267_HTML.gif-graph, then f has a unique fixed point ϱ [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq270_HTML.gif.

Proof Since ( x 0 , f x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq181_HTML.gif, then ( f n x 0 , f n + 1 x 0 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq271_HTML.gif for n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq272_HTML.gif and
F f n x 0 , f n + 1 x 0 ( t ) F f n 1 x 0 , f n x 0 ( t α ) F x 0 , f x 0 ( t α n ) 1 as  n ( t > 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equp_HTML.gif
From compactness, let { f n k x 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq188_HTML.gif be a subsequence such that f n k x 0 ϱ S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq273_HTML.gif. Let t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq67_HTML.gif be given. Since sup a < 1 Δ ( a , a ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq134_HTML.gif, then there exists λ ( δ ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq274_HTML.gif such that Δ ( 1 λ , 1 λ ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq136_HTML.gif choose n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq275_HTML.gif such that for all k n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq276_HTML.gif we have F f n k x 0 , ϱ ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq277_HTML.gif and F f n k x 0 , f n k + 1 x 0 ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq278_HTML.gif. Then we obtain
F f n k + 1 x 0 , ϱ ( t ) Δ ( F f n k + 1 x 0 , f n k x 0 ( t 2 ) , F f n k x 0 , ϱ ( t 2 ) ) Δ ( 1 λ , 1 λ ) > 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equq_HTML.gif

Thus, f n k + 1 x 0 ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq279_HTML.gif.

Choose n 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq280_HTML.gif such that for all k n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq281_HTML.gif we have F f n k + 1 x 0 , ϱ ( t 2 ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq282_HTML.gif and F f n k x 0 , ϱ ( t 2 α ) > 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq283_HTML.gif. Since G is an ( E f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq267_HTML.gif-graph, there exists n 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq284_HTML.gif such that ( f n k x 0 , ϱ ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq285_HTML.gif for all k n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq286_HTML.gif. Let n 0 = max { n 1 , n 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq287_HTML.gif, then for k n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq288_HTML.gif we get
F f ϱ , ϱ ( t ) Δ ( F f n k + 1 x 0 , f ϱ ( t 2 ) , F f n k + 1 x 0 , ϱ ( t 2 ) ) Δ ( F f n k x 0 , ϱ ( t 2 α ) , F f n k + 1 x 0 , ϱ ( t 2 ) ) Δ ( 1 λ , 1 λ ) > 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equr_HTML.gif

Hence, f ϱ = ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq217_HTML.gif. Note that { x 0 , f x 0 , , f n 1 x 0 , , f n N x 0 , ϱ } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq289_HTML.gif is a path in G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq78_HTML.gif, so that ϱ [ x 0 ] G ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq37_HTML.gif be a PM-space, and let ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq58_HTML.gif and 0 < δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq290_HTML.gif be fixed real numbers. A mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif is said to be ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq60_HTML.gif-contraction if there exists a constant α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq96_HTML.gif such that for x S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq61_HTML.gif and y N x ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq291_HTML.gif we have
F f x , f y ( α t ) F x , y ( t ) for all  t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ13_HTML.gif
(3.11)

The PM space ( S , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq37_HTML.gif is said to be ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq292_HTML.gif-chainable if for each x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq54_HTML.gif there exists a finite sequence ( x n ) n = 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq293_HTML.gif of elements in S with x 0 = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq83_HTML.gif and x N = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq294_HTML.gif such that x i + 1 N x i ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq295_HTML.gif for i = 0 , 1 , , N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq296_HTML.gif.

It is important to note that every ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq292_HTML.gif-contraction mapping is continuous. Let x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq6_HTML.gif in S, then there exists a natural number N ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq68_HTML.gif such that x n N x ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq297_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq70_HTML.gif. Thus, for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif and for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq70_HTML.gif, we obtain
F f x n , f x ( t ) F f x n , f x ( α t ) F x n , x ( t ) 1 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equs_HTML.gif

Hence, f x n f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq298_HTML.gif.

Theorem 3.22 Let ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif be a complete ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq292_HTML.gif-chainable Menger PM-space under a t-norm Δ of -type. Let the mapping f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif be an ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq292_HTML.gif-contraction. Then f is a Picard operator.

Proof Consider the graph G consisting of E ( G ) = { ( x , y ) S × S : F x , y ( ε ) > 1 δ } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq299_HTML.gif and V ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq74_HTML.gif coinciding with S. Let x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq54_HTML.gif. Since the PM-space ( S , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq37_HTML.gif is ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq60_HTML.gif-chainable, there exists a finite sequence ( x i ) i = 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq300_HTML.gif in S with x 0 = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq83_HTML.gif and x N = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq294_HTML.gif such that F x i , x i + 1 ( ε ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq301_HTML.gif for i = 0 , 1 , N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq302_HTML.gif. Hence, ( x i , x i + 1 ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq303_HTML.gif for i = 0 , 1 , , N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq304_HTML.gif, which yields that G is connected. Let ( x , y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq97_HTML.gif, then y N x ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq305_HTML.gif. Since the mapping f is an ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq60_HTML.gif-contraction, thus (3.1) is satisfied. Finally we have
F f x , f y ( ε ) F f x , f y ( α ε ) F x , y ( ε ) > 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equt_HTML.gif

Thus, ( f x , f y ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 Δ ( t , t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif is ε-chainbale for some ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq58_HTML.gif if for every x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq307_HTML.gif, there exists a finite sequence ( x i ) n = 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq308_HTML.gif of elements in S with x 0 = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq83_HTML.gif, x N = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq294_HTML.gif and d ( x i , x i + 1 ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq309_HTML.gif for i = 0 , 1 , , N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq304_HTML.gif.

Remark 3.25 [12]

If ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif is an ε-chainable metric space, then the induced Menger PM-space ( S , F , Δ M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq237_HTML.gif is an ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq292_HTML.gif-chainable space.

Corollary 3.26 (Edelstein [19, 20])

Let ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif be a complete ε-chainable metric space. Let f : S S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq3_HTML.gif and let there exist α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq96_HTML.gif such that
x , y S { d ( x , y ) < ε d ( f x , f y ) α d ( x , y ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_Equ14_HTML.gif
(3.12)

Then f is a Picard operator.

Proof Since the metric space ( S , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq2_HTML.gif is ε-chainable, then the induced Menger PM-space ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq50_HTML.gif is ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq292_HTML.gif-chainable for each 0 < δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq290_HTML.gif. We only need to show that the self-mapping f on S is an ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq60_HTML.gif-contraction. Let x , y S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq54_HTML.gif be such that y N x ( ε , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq310_HTML.gif, i.e., F x , y ( ε ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq311_HTML.gif or ϵ 0 ( ε d ( x , y ) ) > 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq312_HTML.gif. The definition of ϵ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq313_HTML.gif implies d ( x , y ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq314_HTML.gif and thus d ( f x , f y ) α d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq100_HTML.gif. Now, for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq55_HTML.gif, we get
F f x , f y ( α t ) = ϵ 0 ( α t d ( f x , f y ) ) ϵ 0 ( t d ( x , y ) ) = F x , y ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 { A i } i = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq315_HTML.gif be nonempty closed subsets of S, and let f : i = 1 m A i i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq316_HTML.gif be an operator. Then S : = i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq317_HTML.gif is known as a cyclic representation of S w.r.t. f if
f ( A 1 ) A 2 , , f ( A m 1 ) A m , f ( A m ) A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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 ( S , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq117_HTML.gif be a complete Menger PM-space under a t-norm Δ of -type. Let m be a positive integer, let { A i } i = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq315_HTML.gif be nonempty closed subsets of S, Y : = i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq318_HTML.gif and f : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq319_HTML.gif. Assume that
  1. (i)

    i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq320_HTML.gif is a cyclic representation of Y w.r.t. f;

     
  2. (ii)

    α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq321_HTML.gif such that d ( f x , f y ) α d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq322_HTML.gif whenever x A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq323_HTML.gif, y A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq324_HTML.gif, where A m + 1 = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq325_HTML.gif.

     

Then f has a unique fixed point ϱ i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq326_HTML.gif and f n y ϱ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq184_HTML.gif for any y i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq327_HTML.gif.

Proof Since i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq320_HTML.gif is closed, then ( Y , F , Δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq328_HTML.gif is complete. Let us consider a graph G consisting of V ( G ) : = Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq329_HTML.gif and E ( G ) : = Ω { ( x , y ) Y × Y : x A i , y A i + 1 ; i = 1 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq330_HTML.gif. By (i) it follows that f preserves edges. Now, let f n x x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq331_HTML.gif in Y such that ( f n x , f n + 1 x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq332_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq132_HTML.gif. Then by (3.13) it is inferred that the sequence { f n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq154_HTML.gif has infinitely many terms in each A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq333_HTML.gif; i { 1 , 2 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq334_HTML.gif. So that one can easily identify a subsequence of { f n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq154_HTML.gif converging to x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq228_HTML.gif in each A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq333_HTML.gif; and since A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq333_HTML.gif’s are closed, then x i = 1 m A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq335_HTML.gif. Thus, we can easily form a subsequence { f n k x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq158_HTML.gif in some A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq336_HTML.gif, j { 1 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq337_HTML.gif such that ( f n k x , x ) E ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq338_HTML.gif for k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_556_IEq339_HTML.gif. It elicits that G is a weakly connected ( C f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-223/MediaObjects/13663_2013_Article_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
(3)
Department of Mathematics, King Abdulaziz University

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© Kamran et al.; licensee Springer. 2013

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