Set-valued G-Prešić operators on metric spaces endowed with a graph and fixed point theorems
- Naseer Shahzad^{1}Email author and
- Satish Shukla^{2}
https://doi.org/10.1186/s13663-015-0262-0
© Shahzad and Shukla; licensee Springer. 2015
Received: 16 September 2014
Accepted: 8 January 2015
Published: 13 February 2015
Abstract
In this paper, we consider the set-valued contractions defined on product spaces when the underlying space is a complete metric space endowed with a graph. Some fixed point results for the so-called set-valued G-Prešić operators are established. Our theorems extend and generalize some known results in product spaces of the recent literature. As an application of our main result, fixed point results for various types of set-valued contractions on product spaces are derived, and a sufficient condition for the existence of a weakly asymptotically stable and global attractor equilibrium point of a kth order nonlinear difference inclusion is established.
Keywords
MSC
1 Introduction
In 1965, Prešić [1, 2] extended the famous Banach contraction principle to the product spaces and obtained some convergence results for some particular sequences. Prešić proved the following theorem.
Theorem 1.1
(Prešić)
A point \(x\in X\) such that \(T(x,x,\ldots,x)=x\) is called a fixed point of T. A mapping T satisfying condition (1.1) is called a Prešić type operator. Note that (1.2) represents a nonlinear difference equation of order k and the fixed point of T is an equilibrium point of (1.2) (see [3]). Therefore the Prešić’s theorem ensures the existence and uniqueness of an equilibrium point of difference equation (1.2). Prešić type operators have applications in solving nonlinear difference equations, cyclic systems and in the study of convergence of sequences; for example, see [1–6]. This interesting result of Prešić has been further extended and generalized by several authors in different directions; see, for instance, [7–20].
Remark 1.2
Let \(A,B\in \operatorname{CB}(X)\) and \(h\in(1,\infty)\) be given. Then, for \(a\in A\), there exists \(b\in B\) such that \(d(a,b)\leq hH(A,B)\).
Nadler [21] extended the Banach contraction principle for the set-valued mappings, that is, for the mappings defined from the space X into the set \(\operatorname{CB}(X)\). Nadler [21] proved the following fixed point theorem.
Theorem 1.3
(Nadler)
Shukla et al. [22] unified the results of Prešić and Nadler and studied the fixed point results for set-valued Prešić type mappings. The results of Shukla et al. [22] are generalized in the recent papers [13, 15, 18].
In 2004, Ran and Reurings [23] initiated the fixed point theory in complete metric spaces endowed with a partial order. Luong and Thuan [20] and Shukla and Radenović [16] considered the Prešić type mappings in partially ordered sets and proved the ordered version of Prešić theorem. These results generalize the result of Ran and Reurings [23] in product spaces.
In 2008, Jachymski [24] presented a nice unification of most of the previous results on a fixed point in metric spaces endowed with a graph. Very recently, Shukla and Shahzad [14] extended, generalized and unified the result of Jachymski [24], Prešić [1, 2], Luong and Thuan [20] by proving fixed point results for G-Prešić type operators in the spaces endowed with a graph. Related results can also be found in [25–32].
In this paper, we introduce the notion of set-valued G-Prešić operators on the product spaces when the underlying space is endowed with a graph and prove some fixed point results for these operators which extend the result of Shukla et al. [22] in spaces endowed with a graph. An example which shows that this extension is proper is given. Our results generalize and unify the results of Jachymski [24], Prešić [1, 2], Luong and Thuan [20], Shukla and Shahzad [14] and several other existing results in the literature. By applying our main results, we derive several fixed point results for various set-valued Prešić type operators. A sufficient condition for the existence of a weakly asymptotically stable and global attractor equilibrium point of a kth order nonlinear difference inclusion is provided.
2 Preliminaries
In this section we recall some definitions and facts about the graphs which will be useful in the sequel.
Let \((X,d)\) be a metric space. Let Δ denote the diagonal of the Cartesian product \(X\times X\). Consider a directed graph G such that the set \(V(G)\) of its vertices coincides with X, and the set \(E(G)\) of its edges contains all loops, that is, \(E(G)\supseteq\Delta\). We assume that G has no parallel edges, so we can identify G with the pair \((V(G),E(G))\). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.
Now we recall a few basic notions concerning connectivity of graphs (see [33]). If x and y are vertices in a graph G, then a path in G from x to y of length \(N\in\mathbb{N}\cup\{0\}\) is a sequence \(\{x_{i}\}_{i=0}^{N}\) of \(N+1\) vertices such that \(x_{0}=x\), \(x_{N}=y\) and \((x_{i-1},x_{i})\in E(G)\) for \(i=1,\ldots,N\). A graph G is called connected if there is a path between any two vertices of G. G is weakly connected if \(\widetilde{G}\) is connected. A sequence \(\{x_{n}\}\) in X is called a termwise connected sequence if \((x_{n},x_{n+1})\in E(G)\) for all \(n\in\mathbb{N}\).
Throughout this paper, we assume that X is a nonempty set, G is a directed graph such that \(V(G)=X\) and \(E(G)\supseteq\Delta\). For the mapping \(T\colon X^{k}\to \operatorname{CB}(X)\), a point \(x\in X\) is called a fixed point of T if \(x\in T(x,\ldots,x)\). We denote the set of all fixed points of T by \(\operatorname{Fix}(T)\).
3 Main results
First we define the set-valued G-Prešić operators on the metric spaces endowed with a graph.
Definition 3.1
- (GP1)There exist nonnegative constants \(\alpha_{i}\)’s such that \(\sum_{i=1}^{k}\alpha_{i}<1\) and$$H\bigl(T(x_{1},x_{2},\ldots,x_{k}),T(x_{2},x_{3}, \ldots,x_{k+1})\bigr)\leq\sum_{i=1}^{k} \alpha_{i}d(x_{i},x_{i+1}). $$
- (GP2)
If \(x_{k+1}\in T(x_{1},x_{2},\ldots,x_{k})\) and \(x_{k+2}\in T(x_{2},x_{3},\ldots,x_{k+1})\) are such that \(d(x_{k+1},x_{k+2})<\max\{ d(x_{i},x_{i+1})\colon i=1,2,\ldots,k\}\), then \((x_{k+1},x_{k+2})\in E(G)\).
Remark 3.2
For \(E(G)=X\times X\), a set-valued G-Prešić operator reduces into a set-valued Prešić type contraction (see Shukla et al. [22]).
Definition 3.3
Let \((X,d)\) be a metric space, k be a positive integer and \(T\colon X^{k}\to \operatorname{CB}(X)\) be a mapping. Define a mapping \(\mathcal{T}\colon X\to \operatorname{CB}(X)\) by \(\mathcal{T}(x)=T(x,x,\ldots,x)\) for all \(x\in X\). Then the mapping \(\mathcal{T}\) is called the associate operator of T.
The following remark will be useful in proving some consequences of our main result.
Remark 3.4
Proof
Now we prove an existence theorem for a set-valued G-Prešić operator.
Theorem 3.5
- (a)
There exists a path \(\{x_{i}\}_{i=1}^{k+1}\) of \(k+1\) vertices in G such that \(x_{k+1}\in T(x_{1},x_{2},\ldots,x_{k})\).
- (b)
For any termwise connected sequence \(\{x_{n}\}\) in X if \(x_{n}\to x\) and \(x_{n+k}\in T(x_{n},x_{n+1},\ldots,x_{n+k-1})\) for all \(n\in \mathbb{N}\), then there exists a subsequence \(\{x_{n_{j}}\}\) such that \((x_{n_{j}},x)\in E(G)\) for all \(j\in\mathbb{N}\).
Proof
We shall show that the sequence \(\{x_{n}\}\) is a Cauchy sequence.
The next example illustrates the above theorem; also, it shows the case when similar results from Shukla et al. [22] are not applicable but the new results are applicable.
Example 3.6
If we define the graph G by \(V(G)=X\) and \(E(G)=X\times X\) in Theorem 3.5, then \(E(G)\supseteq \Delta \) and G has no parallel edges, and so we obtain the following corollary, which is an existence theorem for the set-valued Prešić type contraction (for the related definitions and results, see [22]).
Corollary 3.7
4 Applications to some fixed point results in product spaces
In this section, we apply the results of the previous section and establish some fixed point results for Prešić type operators in various settings.
First, we give the following theorem for set-valued Prešić type operators in ε-chainable spaces (for related definitions, see [34]) which is new even for a single-valued case.
Theorem 4.1
Proof
The following corollary is an extension and generalization of the result of Prešić [1, 2] on ε-chainable spaces, and it extends the result of Edelstein [34] in product spaces.
Corollary 4.2
Proof
Let \((X,\sqsubseteq)\) be a partially ordered set such that d is a metric on X, then the triple \((X,\sqsubseteq,d)\) is called an ordered metric space. A subset \(A\subseteq X\) is called well-ordered if, for all \(x,y\in X\), either \(x\sqsubseteq y\) or \(y\sqsubseteq x\). A sequence \(\{x_{i}\}_{i=1}^{n}\) is called nondecreasing with respect to ⊑ if \(x_{i}\sqsubseteq x_{i+1}\), \(i=1,2,\ldots,n-1\). Next, we define set-valued ordered Prešić operators on an ordered metric space.
Definition 4.3
- (OP1)for a nondecreasing sequence \(\{x_{i}\}_{i=1}^{k+1}\) with respect to ⊑, we havewhere \(\alpha_{i}\)’s are nonnegative constants such that \(\sum_{i=1}^{k}\alpha_{i}<1\);$$H\bigl(T(x_{1},x_{2},\ldots,x_{k}),T(x_{2},x_{3}, \ldots,x_{k+1})\bigr)\leq\sum_{i=1}^{k} \alpha_{i}d(x_{i},x_{i+1}), $$
- (OP2)
if \(\{x_{i}\}_{i=1}^{k+1}\) is a nondecreasing sequence with respect to ⊑, \(x_{k+1}\in T(x_{1},\ldots,x_{k})\) and \(x_{k+2}\in T(x_{2},\ldots,x_{k+1})\) are such that \(d(x_{k+1},x_{k+2})<\max\{ d(x_{i},x_{i+1})\colon i=1,2,\ldots,k\}\), then \(x_{k+1}\sqsubseteq x_{k+2}\).
The following theorem is a fixed point result for a set-valued ordered Prešić operator on an ordered metric space and it extends the results of Malhotra et al. [9] and Luong and Thuan [20] for set-valued mappings.
Theorem 4.4
- (a)
There exists a nondecreasing sequence \(\{x_{i}\}_{i=1}^{k+1}\) such that \(x_{k+1}\in T(x_{1},\ldots,x_{k})\).
- (b)
For any sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\), \(x_{n}\sqsubseteq x_{n+1}\) and \(x_{n+k}\in T(x_{n},x_{n+1},\ldots,x_{n+k-1})\) for all \(n\in\mathbb{N}\), then there exists a subsequence \(\{x_{n_{j}}\}\) such that \(x_{n_{j}}\sqsubseteq x\) for all \(j\in\mathbb{N}\).
Proof
5 Weak stability and weak asymptotic stability
In this section, we consider the weak stability and attractivity of the equilibrium point of a kth order nonlinear difference inclusion.
In the further discussion, we assume that \(\mathfrak{B}\) is a Banach space with the norm \(\|\cdot\|\) and A, a nonempty subset of \(\mathfrak{B}\).
Theorem 5.1
Let A be a closed subset of the Banach space \(\mathfrak{B}\) and \(T\colon A^{k}\to \operatorname{CB}(A)\) be a set-valued Prešić type contraction, then for every set of initial conditions \(x_{1},x_{2},\ldots,x_{k}\in A\) the difference inclusion (5.1) has an equilibrium point \(u\in A\). Furthermore, the equilibrium point u is weakly asymptotically stable and a global attractor.
Proof
Define the graph G by \(V(G)=A\) and \(E(G)=A\times A\). Then, by Corollary 3.7, T has a fixed point in A, and this fixed point is an equilibrium point of the kth order nonlinear difference inclusion (5.1). Furthermore, since \(E(G)=A\times A\), for arbitrary \(x_{1},x_{2},\ldots,x_{k}\in A\), the sequence \(\{x_{n}\}\) defined by \(x_{n+k}\in T(x_{n},x_{n+1},\ldots,x_{n+k-1})\) for all \(n\in\mathbb{N}\) converges to u, therefore, u is weakly asymptotically stable and a global attractor. □
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR for financial support. The authors would like to thank the reviewers for their valuable suggestions.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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