G-Prešić operators on metric spaces endowed with a graph and fixed point theorems
© Shukla and Shahzad; licensee Springer. 2014
Received: 11 March 2014
Accepted: 21 May 2014
Published: 27 May 2014
The purpose of this paper is to introduce the Prešić type contraction in metric spaces endowed with a graph and to prove some fixed point results for the G-Prešić operators in such spaces. The results proved here generalize, extend, and unify several comparable results in the existing literature. Some examples are included which illustrate the results proved herein.
1 Introduction and preliminaries
The famous Banach contraction mapping principle is one of the most simple and useful tools in the theory of fixed points. There are several generalizations of this famous principle. In 1965, Prešić [1, 2] gave a generalization of Banach contraction principle in product spaces which ensures the convergence of a particular sequence. Prešić proved the following theorem.
for every , where are nonnegative constants such that . Then there exists a unique point such that . Moreover, if are arbitrary points in X and for , , then the sequence is convergent and .
A mapping T satisfying (1.1) is called a Prešić operator. Note that the above theorem in the case reduces to the well-known Banach contraction mapping principle. Therefore Theorem 1.1 is a generalization of the Banach fixed point theorem.
Prešić type operators have applications in solving nonlinear difference equations and in the study of convergence of sequences; for example, see [1–4]. In the recent years, several authors generalized and extend the result of Prešić in different directions. For more on the generalizations of Prešić type operators the reader is referred to [5–19].
The existence of fixed point in metric spaces endowed with a partial order was investigated by Ran and Reurings  and then by Nieto and Rodríguez-Lopez [21, 22]. For the related results see, for instance, [23–27] and references therein. In  Malhotra et al. (see also [15, 19]) considered the Prešić type mappings in partially ordered sets and proved the ordered version of theorem of Prešić. This result is further generalized by Shukla et al. .
Kirk et al.  introduced the notion of cyclic operators and generalized the Banach contraction principle by proving the fixed point results for cyclic operators. Since then, many authors have made their contribution in this area; see, for example, [29–33] and the references cited therein. Very recently, Shukla and Abbas  generalized both the notions of cyclic operators and of Prešić operators by introducing the notion of cyclic-Prešić operators.
On the other hands, Jachymski  initiate the study of fixed point theorems in metric spaces endowed with graphs. Jachymski generalized the Banach contraction principle and unified the results of Ran and Reurings , Nieto and Rodríguez-Lopez [21, 22] and Edelstein . For other related results, see, for instance, [23, 36–41].
In this paper, we generalize the result of Prešić in the metric spaces endowed with a graph. The notion of G-Prešić operators is introduced and fixed point results for such operators are proved. The results of this paper generalize and unify the results of Prešić [1, 2], Luong and Thuan  and Shukla and Abbas  for the spaces endowed with a graph, also these results generalize the results of Ran and Reurings , Nieto and Rodríguez-Lopez [21, 22] and Kirk et al.  in product spaces. Some examples are provided which illustrate the results proved herein.
First we recall some definitions and results which will be needed in the sequel.
Let be a metric space. Let Δ denote the diagonal of the Cartesian product . Consider a directed graph G such that the set of its vertices coincides with X, and the set of its edges contains all loops, that is, . We assume that G has no parallel edges, so we can identify G with the pair . Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.
If x and y are vertices in a graph G, then a path in G from x to y of length l is a sequence of vertices such that , and for . A graph G is called connected if there is a path between any two vertices of G. G is weakly connected if is connected.
Throughout this paper we assume that X is a nonempty set, G is a directed graph such that and .
Now we can state our main results.
2 Main results
First we define some notions which will be useful in the sequel.
Now we define a G-Prešić operator on a metric space endowed with a graph.
Definition 2.2 Let be a metric space endowed with a graph G and k be a positive integer. Let be a mapping satisfying the following conditions:
(GP1) if be any path in G then ;
where ’s are nonnegative reals such that .
Then T is called a G-Prešić operator.
Example 2.3 Any constant function is a G-Prešić operator since contains all loops.
Example 2.4 Let T be any Prešić operator, that is, T satisfies (1.1); then T is a -Prešić operator where the graph is defined by .
Remark 2.5 In the case , G-Prešić operator reduces into a G-contraction (see Jachymski ). Proposition 2.1 of Jachymski  shows that if T is a G-contraction then it is both -contraction and -contraction. But in the case a G-Prešić operator need not be a -Prešić operator or a -Prešić operator, as shown in the following example.
For , define a mapping by , , , , and for all other values of x and y. Then T is a G-Prešić operator with , .
and so , where , , is a path in . Now and , , so there are no nonnegative constants , such that and . Because we have , where , , is also a path in and so T is not a -Prešić operator.
Remark 2.7 Let be a metric space endowed with a graph G, k a positive integer and be a G-Prešić operator. If then .
Similarly, if we obtain the same result. □
Theorem 2.8 Let be a metric space endowed with a graph G and k be a positive integer. Suppose be a G-Prešić operator and . Then for every path in , the PP-sequence with initial values is a Cauchy sequence.
Then by (2.1), (2.2) we have . Therefore by (2.1), (2.2) and (GP1) we have and is a path of vertices in G. In a similar way, we find that, for any fix , is a path of n vertices in G.
be the induction hypothesis.
Since , it follows from the above inequality that , that is, the sequence is a Cauchy sequence. □
Note that the above theorem cannot be considered as an existence theorem for G-Prešić operator even when the space is complete. The following example verifies this fact.
Then it is easy to see that T is a G-Prešić operator with . For any pair with , we have , that is, so . Thus, all the conditions of Theorem 2.8 are satisfied and is a complete metric space, but T has no fixed point.
The following is an existence theorem for a G-Prešić operator and provides a sufficient condition for a G-Prešić operator to be a WPP-operator.
Theorem 2.10 Let be a complete metric space endowed with a graph G and k be a positive integer. Suppose be a G-Prešić operator and . Then for every path in , the PP-sequence with initial values is a Cauchy sequence. In addition, if has the property (P) then is a WPP-operator.
Since , letting in the above inequality we obtain , that is, . Thus u is a fixed point of T. □
In the above theorem the fixed point of the mapping T may not be unique; moreover, T may not be a PP-operator and may have infinitely many fixed points as illustrated in the following example.
Now it is easy to see that T is a G-Prešić operator with . For all pairs , we have , that is, so . Also the property (P) is satisfied trivially in this case. Thus, all the conditions of Theorem 2.10 are satisfied and T has infinitely many fixed points, precisely . Therefore T is not a PP-operator but WPP-operator on .
In the next theorem a condition for the uniqueness of a fixed point of a G-Prešić operator is provided.
Theorem 2.12 Let be a complete metric space endowed with a graph G, k be a positive integer and be a G-Prešić operator such that all the conditions of Theorem 2.10 are satisfied, then is a WPP-operator. In addition, if the subgraph is weakly connected, where and , then is a PP-operator.
Proof The existence of a fixed point follows from Theorem 2.8. Suppose is weakly connected and with . Since is weakly connected, there exists a path of vertices with , and for .
Note that T is also -Prešić operator and so by Remark 2.7 we have . Thus, the fixed point of T is unique and is a PP-operator. □
Remark 2.13 In Example 2.11 the fixed point of the operator is not unique. Note that is not weakly connected. Indeed, for all with . Therefore, Example 2.11 shows that when considering the uniqueness, the additional condition ‘ is weakly connected’ in Theorem 2.12 cannot be relaxed.
Definition 2.14 
Let a nonempty set X is equipped with a partial order ‘⊑’ such that is a metric space, then is called an ordered metric space. A sequence in X is said to be nondecreasing with respect to ‘⊑’ if . Let k be a positive integer and be a mapping, then T is said to be nondecreasing with respect to ‘⊑’ if for any finite nondecreasing sequence we have . T is said to be an ordered Prešić type contraction if:
(OP1) T is nondecreasing with respect to ‘⊑’;
for all with .
Definition 2.15 
, are nonempty sets;
, , … , , … , where for all .
is called a cyclic-Prešić operator if the following conditions are met:
(CP1) is a cyclic representation of Y with respect to T;
for all ( where for all ).
Now we give some consequences of our main results.
The following corollary is the fixed point result for the ordered Prešić type contractions (for details, see ).
T is an ordered Prešić type contraction;
there exist such that ;
if a nondecreasing sequence converges to , then for all .
Then T has a fixed point . In addition, is well-ordered if and only if the fixed point of T is unique.
Then (OP1) implies (GP1) and (OP2) implies (GP2) therefore T is a G-Prešić operator. By condition (B) it follows that and the path . Condition (C) insures that has the property (P) and well-orderedness of implies that is weakly connected. By Theorem 2.12, T has a unique fixed point. □
Corollary 2.17 Let be closed subsets of a complete metric space , k a positive integer, and . Let be a cyclic-Prešić operator, then T has a fixed point . In addition, fixed point of T is unique if and only if .
where for all . Note that Y is a complete subspace of X. Since is a cyclic representation of Y with respect to T, therefore (GP1) holds and (CP2) implies (GP2) and so T is a G-Prešić operator. As , are nonempty, therefore . Proposition 2.1 of  shows that the subspace has the property (P). Finally, note that if then is weakly connected, therefore proof follows from Theorem 2.12. □
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support. The authors would like to thank the reviewers for their valuable suggestions.
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