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Reich type weak contractions on metric spaces endowed with a graph
Fixed Point Theory and Applications volume 2015, Article number: 57 (2015)
Abstract
In this paper, we define a new class of Reich type multivalued contractions on a complete metric space satisfying the ggraph preserving condition and we study the fixed point theorem for such mappings. In addition, we present the existence and uniqueness of the fixed point for at least one of two multivalued mappings. The results of this paper extend and generalize several wellknown results. Some examples illustrate the usability of our results.
Introduction
The classical contraction mapping principle of Banach states that if \((X,d)\) is a complete metric space and \(f:X \to X\) such that \(d(f(x),f(y)) \le\alpha d(x,y)\) for all \(x,y \in X\), where \(\alpha \in[0,1)\), then f has a unique fixed point. The Banach fixed point theorem plays an important role in studying the existence of solutions of nonlinear integral equations, system of linear equations, nonlinear differential equations, and proving the convergence of algorithms in computational mathematics. The Banach fixed point theorem has been extended in many directions; see [1–17]. Fixed point theory of multivalued mappings plays a central role in control theory, optimization, partial differential equations, and economics. For a metric space \((X,d)\), we let \(\operatorname{CB}(X)\) be the set of all nonempty, closed, and bounded subsets of X. A point \(x\in X\) is a fixed point of a multivalued mapping \(T:X \to 2^{X}\) if \(x \in Tx\). Nadler [18] has proved a multivalued version of the Banach contraction principle which we state as the following theorem.
Theorem 1.1
Let \((X,d)\) be a complete metric space and \(T:X \to \operatorname{CB}(X)\). Assume that there exists \(k \in[0,1)\) such that \(H(Tx,Ty) \le kd(x,y)\) for all \(x,y\in X\). Then there exists \(z \in X\) such that \(z \in Tz\).
Reich [19] generalized the Banach fixed point theorem for singlevalued maps and multivalued maps as the two following theorems.
Theorem 1.2
Let \((X,d)\) be a complete metric space and let \(f:X \to X\) be a Reich type singlevalued \((a,b,c)\)contraction, that is, there exist nonnegative numbers a, b, c with \(a+b+c< 1\) such that
for each \(x,y \in X\). Then f has a unique fixed point.
Theorem 1.3
Let \((X,d)\) be a complete metric space and let a mapping \(T:X \to P_{\mathrm{cl}} (X)\), where \(P_{\mathrm{cl}} (X)\) is the set of all nonempty closed subsets of X, be a Reich type multivalued \((a,b,c)\)contraction, that is, there exist nonnegative numbers a, b, c with \(a+b+c< 1\) such that
for each \(x,y \in X\). Then there exists \(z \in X\) such that \(z \in Tz\).
In 2008, Jachymski [20] introduced the concept of a contraction concerning a graph, called a Gcontraction, and proved some fixed point results of the Gcontraction in a complete metric space endowed with a graph and he showed that the results of many authors can be derived by his results.
Definition 1.4
Let \((X,d)\) be a metric space and \(G = (V(G),E(G))\) a directed graph such that \(V(G)=X\) and \(E(G)\) contains all loops, i.e., \(\Delta= \{ (x,x)\mid x \in X\} \subset E(G)\). We say that a mapping \(f: X\to X\) is a Gcontraction if f preserves edges of G, i.e., for every \(x,y\in X\),
and there exists \(\alpha \in(0,1)\) such that, for \(x,y\in X\),
Jachymski showed in [20] that assuming some properties for X, a Gcontraction \(f:X \to X\) has a fixed point if and only if there exists \(x\in X\) such that \((x,f(x)) \in E(G)\). The results of Jachymski were generalized by several authors (see, for example, Bojor [3]; Chifu and Petrusel [6]; Samreen and Kamran [13]; Asl et al. [2]; Abbas and Nazir [1]). Recently, Tiammee and Suantai [21] introduced the concept of ggraph preserving for multivalued mappings and proved their fixed point theorem in a complete metric space endowed with a graph.
Definition 1.5
[21]
Let X be a nonempty set and \(G=(V(G), E(G))\) be a graph such that \(V(G)=X\), and let \(T: X\rightarrow \operatorname{CB}(X)\). T is said to be graph preserving if it satisfies the following:
Definition 1.6
[21]
Let X be a nonempty set and \(G=(V(G), E(G))\) be a graph such that \(V(G)=X\), and let \(T: X\rightarrow \operatorname{CB}(X)\), \(g: X\rightarrow X\). T is said to be ggraph preserving if it satisfies the following: for each \(x, y\in X\),
By using the concept of ‘ggraph preserving’ introduced by Tiammee and Suantai [21] and the concept of a Reich type multivalued contraction defined by Reich [19], we define a new class of Reich type multivalued contraction on a complete metric space satisfying the ggraph preserving condition and then we shall study the fixed point theorem for such mappings. Moreover, we establish some results on common fixed points for two multivalued mappings. The results of this research extend and generalize several wellknown results from previous work.
Main results
Let \((X,d)\) be a metric space. Denote \(\operatorname{CB}(X)\) the set of all nonempty closed and bounded subsets of X. For \(a\in X\) and \(A, B\in \operatorname{CB}(X)\), define
Also, define
The mapping H is said to be a Hausdorff metric induced by d. The next lemma will play central roles in our main results.
Lemma 2.1
Let \((X,d)\) be a metric space. If \(A,B\in \operatorname{CB}(X)\) and \(x\in A\), then for each \(\epsilon>0\), there is \(b\in B\) such that
We start with the new class of Reich type multivalued \((\alpha,\beta ,\gamma)\)contraction on a complete metric space.
Definition 2.2
Let \((X, d)\) be a metric space, \(G= (V(G), E(G))\) be a directed graph such that \(V(G)= X\), \(g: X\rightarrow X\), and \(T: X\rightarrow \operatorname{CB}(X)\). T is said to be a Reich type weak Gcontraction with respect to g or a \((g,\alpha, \beta,\gamma)\)Gcontraction provided that

(1)
T is ggraph preserving;

(2)
there exist nonnegative numbers α, β, γ with \(\alpha+\beta+\gamma< 1\) and
$$ H(Tx, Ty) \leq\alpha d\bigl(g(x),g(y)\bigr)+ \beta D\bigl(g(x), Tx\bigr)+ \gamma D\bigl(g(y),Ty\bigr) $$for all \(x, y\in X\) such that \((g(x), g(y))\in E(G)\).
Example 2.3
Let \(\mathbb{N}\) be a metric space with the usual metric. Consider the directed graph defined by \(V(G)= X\) and \(E(G)\) = \(\{(2n1, 2n+1): n\in\mathbb{N} \}\) ∪ \(\{(2n, 2n+2): n\in\mathbb{N}\{1\}\}\) ∪ \(\{(2n, 2n+4): n\in\mathbb{N}\{1\}\}\) ∪ \(\{(2n, 2n): n\in\mathbb{N}\{1\}\}\) ∪ \(\{(1,1), (6, 4)\}\). Let \(T:X \to \operatorname{CB}(X)\) be defined by
and \(g: \mathbb{N}\rightarrow\mathbb{N}\) be defined by
We will show that T is a \((g, \alpha, \beta, \gamma)\)Gcontraction with \(\alpha=0\), \(\beta=\frac{1}{3}\), \(\gamma=\frac{1}{3}\). Let \((g(x),g(y))\in E(G)\). If \((g(x), g(y))= (2k1, 2k+1)\) for \(k\in\mathbb{N}\), then \((x,y)= (2k+1, 2k+3)\), \(Tx=\{2k+2, 2k+4\}\), and \(Ty=\{2k+4, 2k+6\}\). We obtain \((2k+2, 2k+4), (2k+2, 2k+6),(2k+4, 2k+4),(2k+4, 2k+6)\in E(G)\). Also, \(D(g(x), Tx)=3\), \(D(g(y), Ty)=3\), and
If \((g(x), g(y))= (2k, 2k+2)\mbox{ or }(2k, 2k+4)\mbox{ or }(2k, 2k)\) for \(k\in \mathbb{N}\{1\}\), then \(Tx=Ty=\{1\}\) and \((1,1)\in E(G)\). It follows that
If \((g(x), g(y))= (1, 1)\), then \(x=y=3\), \(Tx=Ty=\{4, 6\}\), and \((4,4), (4, 6), (6, 4), (6, 6)\in E(G)\). It follows that
If \((g(x), g(y))= (6, 4)\), then \(x=8\), \(y=6\), and \(Tx=Ty=\{1\}\) and \((1,1)\in E(G)\). Note that \(d(g(x),g(y))=2\), \(D(g(x), T8)= 5\), \(D(g(y), T6)=3\), and so
It follows that T is a \((g, 0, \frac{1}{3}, \frac{1}{3})\)Gcontraction.
Property A
For any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in X, if \(x_{n}\rightarrow x\) and \((x_{n}, x_{n+1})\in E(G)\) for \(n\in \mathbb{N}\), then there is a subsequence \(\{x_{n_{k}}\}_{n_{k}\in\mathbb {N}}\) such that \((x_{n_{k}}, x)\in E(G)\) for \(n_{k}\in\mathbb{N}\).
Lemma 2.4
Let \((X,d)\) be a metric space with the directed graph G, \(g_{1}, g_{2}:X \to X\) be surjective maps and, for \(i=1, 2\), \(T_{i}: X\to \operatorname{CB}(X)\) be ggraph preserving satisfying the following: there exist nonnegative numbers α, β, γ with \(\alpha+ \beta+ \gamma<1\) such that, for all \(x,y\in X\), if \((g_{1}(x), g_{2}(y))\in E(G)\), then
and if \((g_{2}(x), g_{1}(y))\in E(G)\), then
If

(A)
there exists \(x_{0}\in X\) such that \((g_{1}(x_{0}), u)\in E(G)\) for some \(u\in T_{1}x_{0}\), and

(B)
if \((g_{1}(x), g_{2}(y))\in E(G)\), then \((z,w)\in E(G)\) for all \(z\in T_{1}x\), \(w\in T_{2}y\), and if \((g_{2}(x), g_{1}(y))\in E(G)\), then \((b,r)\in E(G)\) for all \(b\in T_{2}x\), \(r\in T_{1}y\),
then there exists a sequence \(\{x_{k}\}_{k\in\mathbb{N}\cup\{0\}}\) in X such that, for each \(k\in\mathbb{N}\),
and \(\{S(x_{k})\}\) is a Cauchy sequence in X where
Proof
Since \(g_{2}\) is surjective, there exists \(x_{1}\in X\) such that \(g_{2}(x_{1})\in T_{1}x_{0}\) and \((g_{1}(x_{0}), g_{2}(x_{1}))\in E(G)\). By Lemma 2.1, there exists \(x_{2}\in X\) such that \(g_{1}(x_{2})\in T_{2}x_{1}\) and \(d(g_{2}(x_{1}), g_{1}(x_{2}))\leq H(T_{1}x_{0}, T_{2}x_{1})+ (\alpha+\beta)\).
By (B), it follows that \((g_{2}(x_{1}), g_{1}(x_{2}))\in E(G)\). By assumption,
Hence,
It follows that
where \(\eta=\frac{(\alpha+\beta)}{1\gamma}<1\).
Next, by Lemma 2.1, we can choose \(x_{3}\in X\) such that \(g_{2}(x_{3})\in T_{1}x_{2}\) and
By (B) again, it follows that \((g_{1}(x_{2}), g_{2}(x_{3}))\in E(G)\). By assumption again, we have
Hence,
It follows that
By the inequality of (1), we have
Continuing in this fashion, we obtain sequences \(\{x_{k}\}\) and \(\{ S(x_{k})\}\) with the property that
and for each \(k\in\mathbb{N}\),
and
Since \(\eta<1\), we have
It is straightforward to check that \(\{S(x_{k})\}\) is a Cauchy sequence in X. □
Theorem 2.5
Let \((X,d)\) be a complete metric space with the directed graph G, \(g_{1}, g_{2}:X \to X\) be surjective maps and, for \(i=1, 2\), \(T_{i}: X\to \operatorname{CB}(X)\) be ggraph preserving satisfying the following: there exist nonnegative numbers α, β, γ with \(\alpha+ \beta+ \gamma<1\) such that, for all \(x,y\in X\), if \((g_{1}(x), g_{2}(y))\in E(G)\), then
and if \((g_{2}(x), g_{1}(y))\in E(G)\), then
If the following hold:

(1)
there exists \(x_{0}\in X\) such that \((g_{1}(x_{0}), u)\in E(G)\) for some \(u\in T_{1}x_{0}\);

(2)
if \((g_{1}(x), g_{2}(y))\in E(G)\), then \((z,w)\in E(G)\) for all \(z\in T_{1}x\), \(w\in T_{2}y\) and if \((g_{2}(x), g_{1}(y))\in E(G)\), then \((b,r)\in E(G)\) for all \(b\in T_{2}x\), \(r\in T_{1}y\);

(3)
X has Property A,
then there exist \(u,v\in X\) such that \(g_{1}(u)\in T_{1}u\) or \(g_{2}(v)\in T_{2}v\).
Proof
By (1), let \(x_{0}\in X\) be such that \((g_{1}(x_{0}), g_{2}(x_{1}))\in E(G)\) for some \(g_{2}(x_{1})\in T_{1}x_{0}\). By Lemma 2.4, there exists a sequence \(\{x_{k}\}_{k\in\mathbb{N}\cup\{0\} }\) in X such that, for each \(k\in\mathbb{N}\),
and \(\{S(x_{k})\}\) is a Cauchy sequence in X where
Since X is complete, the sequence \(\{S(x_{k})\}\) converges to a point w for some \(w\in X\). Let \(u, v\in X \) be such that \(g_{1}(u)= w= g_{2}(v)\). By Property A in (3), there is a subsequence \(\{S(x_{k_{n}})\}\) such that \((S(x_{k_{n}}), g_{1}(u))\in E(G)\) for any \(n\in\mathbb{N}\). We claim that \(g_{1}(u)\in T_{1}u\) or \(g_{2}(v)\in T_{2}v\). Let \(A= \{k_{n} \mid k_{n} \text{ is even}\}\) and \(B= \{k_{n} \mid k_{n} \text{ is odd}\}\). Since \(A\cup B\) is infinite, at least A or B must be infinite. If A is infinite, for each \(g_{2}(x_{k_{n}+1})\), where \(k_{n}\in A\), we have
We obtain
Since \(\{g_{1}(x_{k_{n}})\}\) and \(\{g_{2}(x_{k_{n}+1})\}\) are subsequences of \(S(x_{m})\), they converge to \(g_{2}(v)\) as \(n\rightarrow\infty\), and hence \(D(g_{2}(v), T_{2}v)= 0\). Since \(T_{2}v\) is closed, we conclude that \(g_{2}(v)\in T_{2}v\). Similarly, if B is infinite, we can show that \(g_{1}(u)\in T_{1}u\), completing the proof. Note that if A and B in Theorem 2.5 are both infinite then \(g_{2}(v)\in T_{2}v\) and \(g_{1}(u)\in T_{1}u\). □
The following example illustrates Theorem 2.5.
Example 2.6
Let \((X,d)\) be a metric space where \(X=[0, 1]\) and d is a usual metric on \(\mathbb{R}\). Consider the directed graph \(G=(V(G), E(G))\) defined by \(V(G)=X\) and
Let \(T,S:X \to \operatorname{CB}(X)\) and \(g_{1}, g_{2} : X\to X\) be defined by
It is clear that S, T are \(g_{1}, g_{2}\)graph preserving, respectively. It is straightforward to check that the conditions (1), (2), (3) of Theorem 2.5 are satisfied. Next, we will show that, for all \(x,y\in X\), if \((g_{1}(x), g_{2}(y))\in E(G)\), then
and if \((g_{2}(x), g_{1}(y))\in E(G)\), then
where \(\alpha= 0\), \(\beta=\gamma=\frac{1}{3}\).
If \((g_{1}(x), g_{2}(y)), (g_{2}(x), g_{1}(y))\in E(G)\setminus\{(1,1)\}\), then \(H(Tx, Sy)=0=H(Sx,Ty)\). So the above inequalities are satisfied.
If \((g_{1}(x), g_{2}(y))=(1,1)\), then \(g_{1}(x)= 1\), \(g_{2}(y)=1\), which implies that \(x= 1\) and \(y= 1\). Thus we have \(Tx=\frac{1}{2}\) and \(Sy=\frac{1}{4}\) and hence
If \((g_{2}(x), g_{1}(y))=(1,1)\), then \(g_{2}(x)= 1\) and \(g_{1}(y)=1\), which yields \(x= 1\) and \(y= 1\). Thus we have \(Sx=\frac{1}{4}\) and \(Ty=\frac {1}{2}\) and hence
By Theorem 2.5, there are \(u,v\in X\) such that \(g_{1}(u)\in Tu\) or \(g_{2}(v)\in Sv\). In this example, choose \(u=\frac{1}{4}\). Since \(g_{2}(u)=u\), we have \(u\in Su=\{u\}\).
From Theorem 2.5 one deduces the next two corollaries.
Corollary 2.7
Let \((X,d)\) be a complete metric space with the directed graph G, \(g:X\to X\) be a surjective map, and \(T_{1}, T_{2}: X\to \operatorname{CB}(X)\) be ggraph preserving satisfying
for all \(x,y\in X\) with \((g(x), g(y))\in E(G)\). If the following hold:

(1)
there exists \(x_{0}\in X\) such that \((g(x_{0}), u)\in E(G)\) for some \(u\in T_{1}x_{0}\);

(2)
X has Property A,
then there exists \(u\in X\) such that \(g(u)\in T_{1}u\) or \(g(u)\in T_{2}u\).
Proof
Set \(g_{1}=g_{2}=g\). Then this corollary follows immediately from Theorem 2.5. □
Corollary 2.8
Let \((X, d)\) be a complete metric space, \(G= (V(G), E(G))\) be a directed graph such that \(V(G)= X\), and let \(g: X\to X\) be a surjective map. If \(T: X\to \operatorname{CB}(X)\) is a multivalued mapping satisfying the following properties:

(1)
T is a \((g, \alpha, \beta, \gamma)\)Gcontraction;

(2)
the set \(X_{T}=\{x\in X \mid (g(x),y)\in E(G) \textit{ for some } y\in Tx\}\neq \emptyset\);

(3)
X has Property A,
then there exists \(u\in X\) such that \(g(u)\in Tu\).
Proof
Set \(T_{1}=T_{2}=T\) and \(g_{1}=g_{2}=g\). Then the result follows directly from Theorem 2.5. □
A partial order is a binary relation ≤ over the set X which satisfies the following conditions:

(1)
\(x\leq x\) (reflexivity);

(2)
if \(x\leq y\) and \(y\leq x\), then \(x=y\) (antisymmetry);

(3)
if \(x\leq y\) and \(y\leq z\), then \(x\leq z\) (transitivity),
for all \(x,y\in X\). A set with a partial order ≤ is called a partially ordered set. We write \(x < y\) if \(x\leq y\) and \(x\neq y\).
Definition 2.9
Let \((X,\leq)\) be a partially ordered set. For each \(A,B \subset X\), \(A\prec B\) if \(a\leq b\) for any \(a\in A, b\in B\).
Definition 2.10
[21]
Let \((X,d)\) be a metric space endowed with a partial order ≤, \(g:X\to X\) a surjective map, and \(T:X\to \operatorname{CB}(X)\). T is said to be gincreasing if for any \(x,y\in X\),
In the case g is the identity map, the mapping T is called an increasing mapping.
Theorem 2.11
Let \((X, d)\) be a metric space endowed with a partial order ≤, \(g: X\to X\) be a surjective map and \(T:X\to \operatorname{CB}(X)\) be a multivalued mapping. Suppose that

(1)
T is gincreasing;

(2)
there exist \(x_{0}\in X\) and \(u\in Tx_{0}\) such that \(g(x_{0})< u\);

(3)
for each sequence \(\{x_{k}\}\) such that \(g(x_{k})< g(x_{k+1})\) for all \(k\in\mathbb{N}\) and \(g(x_{k})\) converges to \(g(x)\) for some \(x\in X\), then \(g(x_{k})< g(x)\) for all \(k\in\mathbb{N}\);

(4)
there exist nonnegative numbers α, β, γ with \(\alpha+\beta+\gamma< 1\) such that
$$H(Tx, Ty) \leq\alpha d\bigl(g(x), g(y)\bigr)+ \beta D\bigl(g(x),Tx\bigr)+\gamma D\bigl(g(y), Ty\bigr) $$for all \(x, y\in X\) such that \(g(x)< g(y)\);

(5)
the metric d is complete.
Then there exists \(u\in X\) such that \(g(u)\in Tu\). If g is injective, then there is a unique \(u\in X\) such that \(g(u)\in Tu\).
Proof
Define \(G=(V(G), E(G))\), where \(V(G)= X\) and \(E(G)= \{ (x,y) \mid x< y\}\). Let \(x, y\in X\) be such that \((g(x), g(y))\in E(G)\). Then \(g(x)< g(y)\) so by (1) it implies that \(Tx\prec Ty\). For each \(u\in Tx\), \(v\in Ty\), we have \(u< v\), thus \((u, v)\in E(G)\). That is, T is ggraph preserving. By assumption (2), there exist \(x_{0}\in X\) and \(u\in Tx_{0}\) such that \(g(x_{0})< u\). So \((g(x_{0}), u)\in E(G)\) and hence the property (1) in Corollary 2.7 is satisfied. Moreover, we obtain the property (2) of Corollary 2.7 from the assumption (3). Set \(T_{1}= T_{2}=T\), then the \(T_{1}\), \(T_{2}\) are ggraph preserving mappings satisfying
for all \(x,y\in X\) with \((g(x), g(y))\in E(G)\). Therefore, from the result of this theorem follows Corollary 2.7.
Assume that g is injective. Let \(u,v\in X\) be such that \(g(u)\in Tu\) and \(g(v)\in Tv\). Suppose that \(g(u)\neq g(v)\). Without loss of generality, assume that \(g(u)< g(v)\). Since \(g(u)\in Tu\) and \(g(v)\in Tv\), it follows that \(D(g(u),Tu)=D(g(v),Tv)=0\) and hence
This leads to a contradiction. Thus \(g(u)=g(v)\). Since g is injective, we have \(u=v\). □
We obtain the following result by considering \(g(x)= x\) for all \(x\in X\).
Corollary 2.12
Let \((X, d)\) be a metric space endowed with a partial order ≤ and \(T:X\rightarrow \operatorname{CB}(X)\) be a multivalued mapping. Suppose that

(1)
T is increasing;

(2)
there exist \(x_{0}\in X\) and \(u\in Tx_{0}\) such that \(x_{0}< u\);

(3)
for each sequence \(\{x_{k}\}\) such that \(x_{k}< x_{k+1}\) for all \(k\in\mathbb{N}\) and \(x_{k}\) converges to x for some \(x\in X\), then \(x_{k}< x\) for all \(k\in\mathbb{N}\);

(4)
there exist nonnegative numbers α, β, γ with \(\alpha+\beta+\gamma< 1\) such that
$$H(Tx, Ty) \leq\alpha d(x, y)+ \beta D(x,Tx)+\gamma D(y, Ty) $$for all \(x, y\in X\) such that \(x< y\);

(5)
the metric d is complete.
Then there is a unique \(u\in X\) such that \(u\in Tu\).
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This paper was supported by the Faculty of Science and Technology, Prince of Songkla University, Pattani Campus.
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Phonon, A., SamaAe, A., Makaje, N. et al. Reich type weak contractions on metric spaces endowed with a graph. Fixed Point Theory Appl 2015, 57 (2015). https://doi.org/10.1186/s1366301503074
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DOI: https://doi.org/10.1186/s1366301503074
MSC
 47H04
 47H10
Keywords
 fixed point theorem
 multivalued mapping
 graph preserving
 Reich type contraction