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Coincidence point and fixed point theorems for a new type of Gcontraction multivalued mappings on a metric space endowed with a graph
Fixed Point Theory and Applications volume 2015, Article number: 171 (2015)
Abstract
In this paper, a new type of Gcontraction multivalued mappings in a metric space endowed with a directed graph is introduced and studied. This type of mappings is more general than that of Mizoguchi and Takahashi (J. Math. Anal. Appl. 141:177188, 1989), Berinde and Berinde (J. Math. Anal. Appl. 326:772782, 2007), Du (Topol. Appl. 159:4956, 2012), and Sultana and Vetrivel (J. Math. Anal. Appl. 417:336344, 2014). A fixed point and coincidence point theorem for this type of mappings is established. Some examples illustrating our main results are also given. The main results obtained in this paper extend and generalize those in (Tiammee and Suantai in Fixed Point Theory Appl. 2014:70, 2014) and many wellknown results in the literature.
Introduction
Fixed point theory plays a very important role in nonlinear analysis and applications. It is well known that many metric fixed point theorems were motivated from the Banach contraction principle.
Theorem 1.1
Let \((X,d) \) be a complete metric space and \(T : X \to X \) be a selfmap. Assume that there exists a nonnegative number \(k<1 \) such that
Then T has a unique fixed point in X.
In 1969, Nadler [1] extended the Banach contraction principle for multivalued mappings.
Theorem 1.2
([1])
Let \((X, d) \) be a complete metric space and let T be a mapping from X into \(\operatorname{CB}(X) \). Assume that there exists \(k \in[0 ,1) \) such that
where H is the PompeiuHausdorff metric on \(\operatorname{CB}(X) \). Then there exists \(z \in X \) such that \(z \in T(z)\).
Nadler’s fixed point theorem for multivalued contractive mappings has been extended in many directions (see [2–5]). Reich [6] proved the following fixed point theorem for multivalued φcontraction mappings.
Theorem 1.3
([6])
Let \((X, d) \) be a complete metric space and let T be a mapping from X into \(\operatorname{Comp}(X) \). Assume that there exists a function \(\varphi: [0,\infty )\to[0, 1) \) such that \(\limsup_{r\to t^{+}}\varphi(r)<1 \) for each \(t\in[0,\infty) \) and
Then there exists \(z \in X \) such that \(z \in T(z)\).
In 1989, Mizoguchi and Takahashi [7] relaxed the compactness assumption on T to closed and bounded subsets of X. They proved the following theorem, which is a generalization of Nadler’s theorem.
Theorem 1.4
([7])
Let \((X, d) \) be a complete metric space and let T be a mapping from X into \(\operatorname{CB}(X) \). Assume that there exists a function \(\varphi: [0,\infty )\to[0, 1) \) such that \(\limsup_{r\to t^{+}}\varphi(r)<1 \) for each \(t\in[0,\infty) \) and
Then there exists \(z \in X \) such that \(z \in T(z)\).
In 2007, Berinde and Berinde [4] extended Theorem 1.1 to the class of multivalued weak contractions.
Definition 1.5
([4])
Let \((X, d) \) be a metric space and \(T:X \to \operatorname{CB}(X) \) be a multivalued mapping. T is said to be a multivalued weak contraction or a multivalued \((\theta,L) \)weak contraction if there exist two constants \(\theta\in(0,1) \) and \(L\geq0 \) such that
Definition 1.6
([4])
Let \((X, d) \) be a metric space and \(T:X \to \operatorname{CB}(X) \) be a multivalued mapping. T is said to be a generalized multivalued \((\alpha,L) \)weak contraction if there exist \(L\geq0 \) and a function \(\alpha: [0,\infty)\to[0,1) \) satisfying \(\limsup_{r\to t^{+}}\alpha(r)<1 \), for each \(t\in[0,\infty) \) such that
They proved that in a complete metric space, every multivalued \((\theta, L) \)weak contraction has a fixed point. In the same paper, they also proved that every generalized multivalued \((\alpha, L) \)weak contraction has a fixed point (see [4]). This result was generalized by Du [8] in 2012 as in the following theorem.
Theorem 1.7
([8])
Let \((X, d) \) be a complete metric space and \(T:X \to \operatorname{CB}(X) \) be a multivalued mapping, \(g:X\to X \) be a continuous selfmap and \(\alpha : [0,\infty)\to[0,1) \) a mapping satisfying \(\limsup_{r\to t^{+}}\alpha(r)<1 \), for each \(t\in[0,\infty) \). Assume that:

(a)
\(T(x) \) is ginvariant (i.e., \(g(T(x)) \subseteq T(x)\) for each \(x \in X \)),

(b)
there exists a function \(h:X\to[0,\infty) \) such that
$$H\bigl(T(x),T(y)\bigr) \leq\alpha\bigl(d(x,y)\bigr)d(x,y) + h\bigl(g(y)\bigr)d \bigl(g(y),T(x)\bigr),\quad \textit {for all } x, y \in X . $$
Then \(\mathcal{COP}(g,T) \cap\mathcal{F}(T) \neq\emptyset\), where \(\mathcal{COP}(g,T)=\{x\in X : g(x)\in T(x)\} \) and \(\mathcal{F}(T)=\{ x\in X : x\in T(x)\} \).
In 2008, Jachymski [9] introduced the concept of a Gcontraction and proved some fixed point results of Gcontractions in a complete metric space endowed with a directed graph.
Let \((X, d) \) be a metric space and let \(G=(V(G),E(G)) \) be a directed graph such that \(V(G)=X \) and \(E(G) \) contains all loops, i.e., \(\Delta= \{(x,x): x\in X\}\subseteq E(G)\).
Definition 1.8
([9])
We say that a mapping \(f:X\to X \) is a Gcontraction if f preserves edges of G, i.e., for each \(x,y \in X \),
and there exists \(\alpha\in(0,1) \) such that for each \(x,y \in X \),
He showed that in the case that there are certain properties on \((X,d,G) \) a Gcontraction \(f:X\to X \) has a fixed point if and only if \(X_{f}=\{x\in X : (x,f(x))\in E(G)\} \) is nonempty. The mapping \(f : X\to X \) satisfying condition (1.1) is also called a graphpreserving mapping.
In 2010, Beg and Butt [5] introduced the concept of Gcontraction for a multivalued mapping \(T : X\to \operatorname{CB}(X) \) as follows.
Definition 1.9
([5])
We say that a mapping \(T:X\to \operatorname{CB}(X) \) is a Gcontraction if there exists \(k\in(0,1) \) such that for each \((x,y) \in E(G) \),
and if \(u\in T(x) \) and \(v\in T(y) \) are such that for each \(\alpha >0 \),
then \((u,v)\in E(G) \).
Recently, in 2015, Alfuraidan [10] pointed out that the above definition of a Gcontraction is flawed and the argument behind the proof of the main result of [5] fails.
By using the idea of multivalued contraction mappings in [11, 12], Alfuraidan introduced the following concept of a Gcontraction.
Definition 1.10
([10])
A multivalued mapping \(T:X\to2^{X} \) is said to be a monotone increasing Gcontraction if there exists \(\alpha\in[0,1) \) such that, for any \(u,v\in X \) with \((u,v)\in E(G) \) and any \(U\in T(u) \), there exists \(V\in T(v) \) such that \((U,V)\in E(G) \) and \(d(U,V)\leq\alpha d(u,v) \).
He showed that under some properties on a metric space, a monotone increasing Gcontraction multivalued mappings has a fixed point (see [10], Theorem 3.1).
In 2014, Tiammee and Suantai [13] introduced the concept of graphpreserving for multivalued mappings as follows.
Definition 1.11
([13])
Let X be a nonempty set and \(G = (V(G),E(G)) \) be a directed graph such that \(V(G) = X \), and let \(T:X\to \operatorname{CB}(X) \). Then T is said to be graphpreserving if
In the same year, Sultana and Vetrivel [14] introduced a concept of a MizoguchiTakahashi Gcontraction as follows.
Definition 1.12
([14])
A multivalued mapping \(T : X \to \operatorname{CB}(X) \) is said to be a MizoguchiTakahashi Gcontraction if there exists a function \(\alpha: (0,\infty) \to[0,1) \) satisfying \(\limsup_{r \to t^{+}} \alpha(r) < 1 \) for every \(t \in[0,\infty)\), and for every \(x,y \in X\), \(x \neq y \) with \((x,y) \in E(G)\),

(i)
\(H(T(x),T(y)) \leq\alpha(d(x,y))d(x,y) \),

(ii)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G) \).
They showed that if there are some properties on a metric space, a multivalued MizoguchiTakahashi Gcontraction has a fixed point (see [14], Theorem 3).
In this paper, we introduce a new concept of a Gcontraction in a metric space endowed with a directed graph which is more general than the MizoguchiTakahashi Gcontraction for multivalued mappings. We establish some coincidence point and fixed point theorems for this type of mappings and give some examples illustrating our main results.
Preliminaries
Let \((X, d) \) be a metric space, and let \(\operatorname{CB}(X) \) and \(\operatorname{Comp}(X) \) be the set of all nonempty closed bounded subsets of X and the set of all nonempty compact subsets of X, respectively. For each \(x\in X \) and \(A\subseteq X \), let \(d(x,A) = \inf_{y\in A} d(x,y) \). A function \(H:\operatorname{CB}(X)\times \operatorname{CB}(X)\to [0,\infty) \) defined by
is called a PompeiuHausdorff metric on \(\operatorname{CB}(X) \) induced by d on X.
Let \(g:X\to X \) be a selfmap and \(T:X\to2^{X} \) be a multivalued map. A point x in X is a coincidence point of g and T if \(g(x)\in T(x) \). If g is the identity map on X, then \(x=g(x)\in T(x) \) and we call x a fixed point of T. The set of all fixed points of T and the set of all coincidence points of g and T are denoted by \(\mathcal{F}(T)\) and \(\mathcal {COP}(g,T) \), respectively.
The following lemmas are useful for our main results.
Lemma 2.1
([1])
Let \((X,d) \) be a metric space. If \(A,B \in \operatorname{CB}(X) \) and \(a \in A\), then, for each \(\varepsilon> 0\), there exists \(b \in B \) such that \(d(a,b) \leq H(A,B) + \varepsilon\).
Lemma 2.2
([6])
Let \((X,d) \) be a metric space, \(\{A_{k}\} \) be a sequence in \(\operatorname{CB}(X) \) and \(\{x_{k}\} \) be a sequence in X such that \(x_{k} \in A_{k1} \). Let \(\alpha: [0,\infty) \to[0,1) \) be a function satisfying \(\limsup_{r \to t^{+}} \alpha(r) < 1 \) for every \(t \in [0,\infty) \). Suppose that \(\{d(x_{k1},x_{k})\} \) is a nonincreasing sequence such that
where \(n_{1} < n_{2} < \cdots \) and \(k,n_{k} \in\mathbb{N} \). Then \(\{ x_{k}\} \) is a Cauchy sequence in X.
Main results
We first introduce a concept of weak Gcontraction on a metric space endowed with a directed graph.
Definition 3.1
Let \((X,d) \) be a metric space and let \(G = (V(G),E(G)) \) be a directed graph such that \(V(G) = X \). Let \(T : X \to \operatorname{CB}(X) \) and \(g : X \to X\). Then T is said to be a weak Gcontraction with respect to g if there exists a function \(\alpha: (0,\infty) \to[0,1) \) satisfying \(\limsup_{r \to t^{+}} \alpha(r) < 1 \) for every \(t \in[0,\infty) \) and \(h : X \to[0,\infty) \) such that for every \(x,y \in X\), \(x \neq y \) with \((x,y) \in E(G) \),

(i)
\(H(T(x),T(y)) \leq\alpha(d(x,y))d(x,y) + h(g(y))d(g(y),T(x)) \),

(ii)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G) \).
Example 3.2
Let \(X = \{\frac{1}{2^{n}} \mid n \in\mathbb {N}\cup\{0\} \}\cup\{0\} \), \(d(x,y) =  xy \) for \(x,y \in X \). Let \(E(G) = \{(\frac{1}{2^{n}},0),(\frac {1}{2^{2n}},\frac{1}{2^{2n+1}}),(\frac{1}{2^{2n+1}},\frac {1}{2^{2n+2}}) ; n \in\mathbb{N}\cup\{0\} \} \cup \{ (0,0) \}\). Let \(\alpha: (0,\infty) \to[0,1) \) be defined by \(\alpha(t) = \frac{1}{2} \) for all \(t \in(0,\infty)\). Let \(T : X \to \operatorname{CB}(X) \) be defined by
Let \(g : X \to X \) be defined by
Let \(h : X \to[0,\infty) \) be defined by
We show that \(T : X \to \operatorname{CB}(X) \) is a weak Gcontraction with respect to g. Let \((x,y)\in E(G) \).
Case 1. \((x,y)=(\frac{1}{2^{n}},0) \) for some \(n \in \mathbb{N}\cup\{0\} \).
If \(n = 2k \) for some \(k \in\mathbb{N}\cup\{0\} \), we have
If \(n = 2k+1 \) for some \(k \in\mathbb{N}\cup\{0\} \), we have
Since \((x,0) \in E(G) \) for all \(x\in X \) and \(T(0)=\{0\}\), it implies \((u,0) \in E(G) \) for all \(u \in T(\frac{1}{2^{n}})\). So the condition (ii) is satisfied.
Case 2. \((x,y)= (\frac{1}{2^{2n}},\frac {1}{2^{2n+1}}) \) for some \(n \in\mathbb{N}\cup\{0\} \). Then
We see that for each \(u \in T(\frac{1}{2^{2n}}) \) and \(v \in T(\frac{1}{2^{2n+1}}) \), we have
Case 3. \((x,y)= (\frac{1}{2^{2n+1}},\frac {1}{2^{2n+2}}) \) for some \(n \in\mathbb{N}\cup\{0\} \). Then
We also see that for each \(u \in T(\frac{1}{2^{2n+1}}) \) and \(v \in T(\frac{1}{2^{2n+2}}) \), we have
Therefore \(T : X \to \operatorname{CB}(X) \) is a weak Gcontraction with respect to g.
Theorem 3.3
Let \((X,d) \) be a complete metric space and let \(g : X \to X \) be a continuous selfmap and \(T : X \to \operatorname{CB}(X) \) a weak Gcontraction with respect to g. Suppose that:

(1)
there is \(x_{0} \in X \) such that \((x_{0},y) \in E(G) \) for some \(y \in T(x_{0}) \),

(2)
\(T(x) \) is ginvariant (i.e., \(g(T(x)) \subseteq T(x) \) for each \(x \in X \)),

(3)
for any sequence \(\{x_{n}\} \) in X, if \(x_{n} \to x \) and \((x_{n},x_{n+1})\in E(G) \) for all \(n \in\mathbb{N} \), then there is a subsequence \(\{ x_{n_{k}} \} \) of \(\{ x_{n} \} \) with \((x_{n_{k}},x) \in E(G) \) for some \(k \in\mathbb{N} \).
Then \(\mathcal{COP}(g,T) \cap\mathcal{F}(T) \neq\emptyset\).
Proof
Let \(x_{0} \in X \) and \(x_{1} \in T(x_{0}) \) such that \((x_{0},x_{1}) \in E(G) \).
By assumption (2), we have \(g(x_{1}) \in T(x_{0})\). If \(x_{0}=x_{1} \) or \(\alpha(d(x_{0},x_{1}))=0\), then \(x_{0}\in\mathcal{COP}(g,T) \cap\mathcal{F}(T) \). Suppose \(x_{0}\neq x_{1} \) and \(\alpha(d(x_{0},x_{1}))\neq0\). We can choose \(n_{1} \in\mathbb{N} \) such that
This implies by Lemma 2.1 that there exists \(x_{2} \in T(x_{1}) \) such that
Hence \((x_{1},x_{2}) \in E(G)\). By assumption (2), we have \(g(x_{2}) \in T(x_{1})\). If \(x_{1}=x_{2} \) or \(\alpha (d(x_{1},x_{2}))=0\), then we have \(x_{1}\in\mathcal{COP}(g,T) \cap \mathcal{F}(T) \). Suppose \(x_{1}\neq x_{2} \) and \(\alpha (d(x_{1},x_{2}))\neq0\). We can choose \(n_{2} \in\mathbb{N} \) with \(n_{2} > n_{1} \) such that
It follows by Lemma 2.1 that there exists \(x_{3} \in T(x_{2}) \) such that
Hence \((x_{2},x_{3}) \in E(G)\). By assumption (2), we have \(g(x_{3}) \in T(x_{2}) \).
By induction, we obtain a sequence \(\{x_{k}\} \) in X and a sequence of positive integers \(\{n_{k}\}_{k \in\mathbb{N}} \) satisfying the property that for each \(k \in\mathbb{N} \), \(x_{k+1} \in T(x_{k}) \), \(g(x_{k}) \in T(x_{k1}) \), \((x_{k},x_{k+1}) \in E(G) \),
and
From the above inequality, we get \(d(x_{k},x_{k+1}) < d(x_{k1},x_{k}) \), i.e., \(\{d(x_{k},x_{k+1})\} \) is a decreasing sequence. It follows from Lemma 2.2 that \(\{x_{k}\} \) is a Cauchy sequence in X. Since X is complete, there is an \(x \in X \) such that \(x_{k} \to x \) as \(k \to\infty\). Since g is continuous, \(g(x_{k}) \to g(x) \) as \(k \to\infty\). By assumption (3), there is a subsequence \(\{x_{k_{n}}\} \) of \(\{x_{k}\} \) such that \((x_{k_{n}},x) \in E(G) \) for all \(n \in\mathbb{N} \). Since T is a weak Gcontraction with respect to g, we have
By taking \(n \to\infty\) in the above inequality, we get \(d(x,T(x)) = 0 \). Since \(T(x) \) is closed, we have \(x \in T(x) \). By assumption (2), we get \(g(x) \in T(x)\). Therefore \(x \in \mathcal {COP}(g,T) \cap\mathcal{F}(T) \). □
Example 3.4
Let X, \(E(G)\), d, α, T, g, and h be the same as in Example 3.2. Then T is a weak Gcontraction with respect to g, and g is continuous. It is easy to see that the conditions (1)(3) of Theorem 3.3 hold. Hence all conditions of Theorem 3.3 are satisfied and we see that \(\mathcal{COP}(g,T) \cap\mathcal{F}(T)=\{0,1\}\).
Remark 3.5

(i)
In Theorem 3.3, if we take a directed graph G with \(E(G)= X\times X\), we obtain Theorem 2.2 of Du [8] immediately.

(ii)
In Theorem 3.3, if we take a function \(h=0 \), then we obtain the existence result which is similar to Theorem 3 of Sultana and Vetrivel [14].

(iii)
In Theorem 3.3, if we take a directed graph G with \(E(G)= X \times X \) and a function \(h=0 \), we obtain immediately the MizoguchiTakahashi theorem [7].

(iv)
In Theorem 3.3, if we take a directed graph G with \(E(G)= X\times X\), a function g that is the identity mapping on X, and a function \(h=L \), for some \(L\geq0\), we obtain the Berinde and Berinde theorem ([4], Theorem 4).
Example 3.6
Let X, d, α be the same as in Example 3.2. Let
Let \(T : X \to \operatorname{CB}(X) \) be defined by
Let \(g : X \to X \) be defined by
Let \(L=\frac{5}{2} \) and \(h : X \to[0,\frac{5}{2}] \) be defined by
It is easy to see that assumptions (1), (2), and (3) of Theorem 3.3 hold true and g is continuous on X. By using the same calculation as in Example 3.2, it can be shown that T is a weak Gcontraction with respect to g. We note that the function h above is a bounded function on X and \(0\leq h(x)\leq \frac{5}{2} \) for all \(x\in X\). Therefore all conditions of Theorem 3.3 are satisfied and we see that \(\mathcal{COP}(g,T) \cap\mathcal{F}(T) = \{\frac {1}{2^{2n+1}} : n\in\mathbb{N}\cup\{0\} \}\cup \{0 \} \).
The following result is also immediately obtained by Theorem 3.3, by setting \(h(x)=L \) for all \(x\in X \) and some \(L\geq0\).
Corollary 3.7
Let \((X,d) \) be a complete metric space. Let \(T : X \to \operatorname{CB}(X) \) be a multivalued mapping, \(g : X \to X \) be a continuous selfmap, and \(\alpha: (0,\infty) \to[0,1) \) a mapping satisfying \(\limsup_{r \to t^{+}} \alpha(r) < 1 \) for every \(t \in[0,\infty) \). Suppose that the following conditions hold:

(1)
there is \(x_{0} \in X \) such that \((x_{0},y) \in E(G) \) for some \(y \in T(x_{0}) \),

(2)
\(T(x) \) is ginvariant (i.e., \(g(T(x)) \subseteq T(x) \) for each \(x \in X \)),

(3)
for any sequence \(\{x_{n}\}\) in X, if \(x_{n} \to x \) and \((x_{n},x_{n+1}) \in E(G) \) for all \(n \in\mathbb{N} \), then there is a subsequence \(\{ x_{n_{k}} \} \) of \(\{ x_{n} \} \) with \((x_{n_{k}},x) \in E(G) \) for some \(k \in\mathbb{N} \).
If T satisfies the following two conditions:

(4)
there exists \(L\geq0 \) such that
$$H\bigl(T(x),T(y)\bigr) \leq\alpha\bigl(d(x,y)\bigr)d(x,y) + Ld\bigl(g(y),T(x) \bigr) , \quad \textit{for all } x,y \in X, $$\(x \neq y \) with \((x,y) \in E(G) \),

(5)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G)\).
Then \(\mathcal{COP}(g,T) \cap\mathcal{F}(T) \neq\emptyset\).
If we set g in Theorem 3.3 to be the identity map on X, then we obtain the following result.
Corollary 3.8
Let \((X,d) \) be a complete metric space. Let \(T : X \to \operatorname{CB}(X) \) be a multivalued mapping, and \(\alpha: (0,\infty) \to[0,1) \) a mapping satisfying \(\limsup_{r \to t^{+}} \alpha(r) < 1 \) for every \(t \in [0,\infty) \). Suppose that the following conditions hold:

(1)
there is \(x_{0} \in X \) such that \((x_{0},y) \in E(G) \) for some \(y \in T(x_{0}) \),

(2)
for any sequence \(\{x_{n}\}_{n \in\mathbb{N}} \) in X, if \(x_{n} \to x \) and \((x_{n},x_{n+1}) \in E(G) \) for all \(n \in\mathbb {N} \), then there is a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \} \) with \((x_{n_{k}},x) \in E(G) \) for some \(k \in\mathbb{N} \).
If T satisfies the following two conditions:

(3)
there exists a function \(h : X \to[0,\infty) \) such that
$$H\bigl(T(x),T(y)\bigr) \leq\alpha\bigl(d(x,y)\bigr)d(x,y) + h(y)d\bigl(y,T(x) \bigr) ,\quad \textit{for all } x,y \in X, $$\(x \neq y \) with \((x,y) \in E(G) \),

(4)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G)\).
Then \(\mathcal{F}(T) \neq\emptyset\).
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Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions for improving this work and the Thailand Research Fund under the project RTA 5780007 and Chiang Mai University for the financial support. The first author was supported by the Science Achievement Scholarship of Thailand.
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Hanjing, A., Suantai, S. Coincidence point and fixed point theorems for a new type of Gcontraction multivalued mappings on a metric space endowed with a graph. Fixed Point Theory Appl 2015, 171 (2015) doi:10.1186/s1366301504204
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MSC
 47H10
 54H25
Keywords
 fixed point
 multivalued mappings
 MizoguchiTakahashi Gcontraction