 Research
 Open Access
 Published:
Coincidence point theorems for graphpreserving multivalued mappings
Fixed Point Theory and Applications volume 2014, Article number: 70 (2014)
Abstract
In this paper, we introduce the concepts of graphpreserving multivalued mapping and a new type of multivalued weak Gcontraction on a metric space endowed with a directed graph G. We prove some coincidence point theorems for this type of multivalued mapping and a surjective mapping $g:X\to X$ under some conditions. Several examples for these new concepts and some examples satisfying all conditions of our main results are also given. Our main results extend and generalize many coincidence point and fixed point theorems in partially ordered metric spaces.
MSC:47H04, 47H10.
1 Introduction
Fixed point theory of multivalued mappings plays an important role in science and applied science. It has applications in control theory, convex optimization, differential inclusions and economics.
For a metric space $(X,d)$, we let $CB(X)$ and $Comp(X)$ be the set of all nonempty closed bounded subsets of X and the set of all nonempty compact subsets of X, respectively. A point $x\in X$ is a fixed point a multivalued mapping $T:X\to {2}^{X}$ if $x\in Tx$. The first wellknown theorem for multivalued contraction mappings was given by Nadler in 1969 [1].
Theorem 1.1 Let $(X,d)$ be a complete metric space and let T be a mapping from X into $CB(X)$. Assume that there exists $k\in [0,1)$ such that
Then there exists $z\in X$ such that $z\in Tz$.
Nadler’s fixed point theorem for multivalued contractive mappings has been extended in many directions (see [2–6]). Reich [7] proved the following fixed point theorem for multivalued φ contraction mappings.
Theorem 1.2 Let $(X,d)$ be a complete metric space and let T be a mapping from X into $Comp(X)$. Assume that there exists a function $\phi :[0,\mathrm{\infty})\to [0,1)$ such that
and
Then there exists $z\in X$ such that $z\in Tz$.
The multivalued mapping T considered by Reich [7] in Theorem 1.2 has compact values, that is, Tx is a nonempty compact subset of X for all $x\in X$. In 1988, Mizoguchi and Takahashi [8] 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 1.3 Let $(X,d)$ be a complete metric space and let $T:X\to CB(X)$. Assume that there exists a function $\phi :[0,\mathrm{\infty})\to [0,1)$ such that
and
Then there exists $z\in X$ such that $z\in Tz$.
In 2007, Berinde and Berinde [3] extended Theorem 1.1 to the class of multivalued weak contractions.
Definition 1.4 ([3])
Let $(X,d)$ be a metric space and $T:X\to CB(X)$ 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 $L\ge 0$ such that
They proved in [3], Theorem 3 that in a complete metric space every multivalued $(\theta ,L)$weak contraction has a fixed point. In the same paper, they also introduced a class of multivalued mappings which is more general than that of weak contractions.
Definition 1.5 ([3])
Let $(X,d)$ be a metric space and $T:X\to CB(X)$ a multivalued mapping. T is said to be a generalized multivalued $(\alpha ,L)$weak contraction if there exist $L\ge 0$ and a function $\alpha :[0,\mathrm{\infty})\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to {t}^{+}}\alpha (r)<1$, for every $t\in [0,\mathrm{\infty})$, such that
They also showed that in a complete metric space, every generalized multivalued $(\alpha ,L)$weak contraction has a fixed point (see [[3], Theorem 4]).
For the last ten years, many results concerning the existence of fixed points of both singlevalued and multivalued mappings in metric spaces endowed with a partial ordering have been established. The first result in this direction was given by Ran and Reurings [9] and they also presented its applications to linear and nonlinear matrix equations. After that many authors extended those results and studied fixed point theorems in partially ordered metric spaces (see [9–13]).
In 2008, Jachymski [14] introduced the concept of Gcontraction and proved some fixed point results of Gcontractions in a complete metric space endowed with a graph.
Definition 1.6 ([14])
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., $\u25b3=\{(x,x):x\in X\}\subseteq E(G)$.
We say that a mapping $f:X\to X$ is a Gcontraction if f preserves edges of G, i.e.,
and there exists $\alpha \in (0,1)$ such that
He showed in [14] that under some 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.
Recently, Beg and Butt [5] introduced the concept of Gcontraction for a multivalued mapping $T:X\to CB(X)$ and proved some fixed point results of this kind of mappings.
Definition 1.7 ([5])
Let $T:X\to CB(X)$ be a multivalued mapping. The mapping T is said to be a Gcontraction if there exists $k\in (0,1)$ such that
and if $u\in Tx$ and $v\in Ty$ are such that
then $(u,v)\in E(G)$.
They also showed that if $(X,d)$ is a complete metric space and a triple $(X,d,G)$ has Property A [14], then a Gcontraction mapping $T:X\to CB(X)$ has a fixed point if and only if ${X}_{F}:=\{x\in X:(x,y)\in E(G)\text{for some}y\in Tx\}$ is nonempty.
Recently, in 2013, Dinevari and Frigon [6] introduced a new concept of Gcontraction which is weaker than that of Beg and Butt [5].
Definition 1.8 ([6])
Let $T:X\to {2}^{X}$ be a map with nonempty values. We say that T is a Gcontraction (in the sense of Dinevari and Frigon) if there exists $\alpha \in (0,1)$ such that
$({C}_{G})$ for all $(x,y)\in E(G)$ and all $u\in Tx$, there exists $v\in Ty$ such that
They showed that under some properties on a metric space which is weaker than Property A, a multivalued Gcontraction with closed values has a fixed point (see [6], Theorem 2.10 and Corollary 2.11). We note that the concept of Gcontraction for multivalued mappings does not concern the concept of graphpreserving as seen for singlevalued mappings. Motivated by this observation and those previous works, we are interested in introducing the concept of graphpreserving for multivalued mappings and study their fixed point theorem in a complete metric space endowed with a graph.
2 Preliminaries
Let $(X,d)$ be a metric space and $CB(X)$ be the set of all nonempty closed bounded subsets of X. For $x\in X$ and $A,B\in CB(X)$, define
Denote by H the PompeiuHausdorff metric induced by d, see [4], that is,
The following two lemmas, which can be found in [1] or [8], are useful for our main results.
Lemma 2.1 ([1])
Let $(X,d)$ be a metric space. If $A,B\in CB(X)$ and $a\in A$, then, for each $\u03f5>0$, there exists $b\in B$ such that
Lemma 2.2 ([8])
Let $(X,d)$ be a metric space in $CB(X)$, $\{{x}_{k}\}$ be a sequence in X such that ${x}_{k}\in {A}_{k1}$. Let $\alpha :[0,\mathrm{\infty})\to [0,1)$ be a function satisfying ${lim\hspace{0.17em}sup}_{r\to {t}^{+}}\alpha (r)<1$ for every $t\in [0,\mathrm{\infty})$. 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.
Let $G=(V(G),E(G))$ be a directed graph where $V(G)$ is a set of vertices of the graph and $E(G)$ be a set of its edges. Assume that G has no parallel edges. If x and y are vertices in 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 ${x}_{0}=x$, ${x}_{n}=y$, $({x}_{i1},{x}_{i})\in E(G)$ for $i=1,2,\dots ,n$. A graph G is connected if there is a path between any two vertices of G.
A partial order is a binary relation ≤ over the set X which satisfies the followings conditions:

1.
$x\le x$ (reflexivity);

2.
If $x\le y$ and $y\le x$, then $x=y$ (antisymmetry);

3.
If $x\le y$ and $y\le z$, then $x\le z$ (transitivity)
for all $x,y\in X$. A set with a partial order ≤ is called a partially ordered set. We write $x<y$ if $x\le y$ and $x\ne y$.
Definition 2.3 Let $(X,\le )$ be a partially ordered set. For each $A,B\subset X$,
Definition 2.4 Let $(X,d)$ be a metric space endowed with a partial order ≤. Let $g:X\to X$ be surjective and $T:X\to CB(X)$, T is said to be gincreasing if for any $x,y\in X$,
In the case $g={I}_{X}$, the identity map, the mapping T is called an increasing mapping.
Example 2.5 Let $X=\mathbb{N}$ have the usual relation ≤ and $T:\mathbb{N}\to {2}^{\mathbb{N}}$ and $g:X\to X$ be defined by
$g(1)=1$ and $g(x)=x1$ for $x\ne 1$. It is easy to see that T is gincreasing.
Definition 2.6 Let X be a nonempty set and $G=(V(G),E(G))$ be a graph such that $V(G)=X$, and let $T:X\to CB(X)$. Then T is said to be graphpreserving if
Example 2.7 Let $G=(\mathbb{N},E(G))$, where $E(G)$ = $\{(2n1,2n+1):n\in \mathbb{N}\}$ ∪ $\{(2n,2n+2):n\in \mathbb{N}\}$ ∪ $\{(2n,2n+4):n\in \mathbb{N}\}$ ∪ $\{(2n,2n):n\in \mathbb{N}\}$ ∪ $\{(1,1),(4,2)\}$. Define $T:\mathbb{N}\to CB(\mathbb{N})$ by
We will show that T is a graphpreserving mapping. Let $(x,y)\in E(G)$.
If $(x,y)=(2k,2k+2)$ or $(x,y)=(2k,2k+4)$ or $(x,y)=(2k,2k)$ or $(4,2)$, where $k\in \mathbb{N}$, then $Tx=Ty=\{1\}$ and $(1,1)\in E(G)$.
If $(x,y)=(2k1,2k+1)$, $k\in \mathbb{N}$, then $Tx=\{2k,2k+2\}$, $Ty=\{2k+2,2k+4\}$ and $(2k,2k+2)\in E(G)$, $(2k,2k+4)\in E(G)$, $(2k+2,2k+2)\in E(G)$, $(2k+2,2k+4)\in E(G)$. And we see that $(1,1)\in E(G)$, $T1=\{2,4\}$ and $(2,2),(2,4),(4,2),(4,4)\in E(G)$. Hence T is graphpreserving.
Example 2.8 Let $G=(X,E(G))$, where $X=\{1,2,3,4,6,8\}$ and $E(G)=\{(1,1),(1,3)\}\cup \{(2,2),(2,4),(2,6),(2,8),(4,2),(4,4),(4,8),(6,8)\}$. Define $T:X\to CB(X)$ by
It is easy to see that T is graphpreserving but not Gcontraction in the sense of Dinevari and Frigon [6] since $d(u,v)>\alpha d(1,3)$ for all $u\in T1=\{2,4\}$ and $v\in T3=\{6,8\}$ for any $\alpha \in (0,1)$.
Definition 2.9 Let X be a nonempty set and $G=(V(G),E(G))$ be a graph such that $V(G)=X$, $g:X\to X$ and $T:X\to CB(X)$. Then T is said to be ggraphpreserving if for any $x,y\in X$ such that
Example 2.10 Let $G=(\mathbb{N},E(G))$ and $E(G)$ = $\{(2n1,2n+1):n\in \mathbb{N}\}$ ∪ $\{(2n,2n+2):n>1\}$ ∪ $\{(2n,2n+4):n>1\}$ ∪ $\{(2n,2n):n>1\}$ ∪ $\{(1,1)\cup (6,4)\}$. Let $T:\mathbb{N}\to CB(\mathbb{N})$ be defined as in Example 2.7 and let $g:\mathbb{N}\to \mathbb{N}$ be defined by
We will show that T is ggraphpreserving. Let $(g(x),g(y))\in E(G)$.
If $(g(x),g(y))=(2k1,2k+1)$ for $k\in \mathbb{N}$, then $(x,y)=(2k+1,2k+3)$ and $Tx=\{2k+2,2k+4\}$, $Ty=\{2k+4,2k+6\}$ and $(2k+2,2k+4)\in E(G)$, $(2k+2,2k+6)\in E(G)$, $(2k+4,2k+4)\in E(G)$, $(2k+4,2k+6)\in E(G)$.
If $(g(x),g(y))=(2k,2k+2)$ or $(2k,2k+4)$ or $(2k,2k)$, then $Tx=Ty=\{1\}$ and $(1,1)\in E(G)$.
If $(g(x),g(y))=(1,1)$, then $(x,y)=(3,3)$ and $T3=\{4,6\}$ and $(4,4)\in E(G)$, $(4,6)\in E(G)$, $(6,4)\in E(G)$ and $(6,6)\in E(G)$.
If $(g(x),g(y))=(6,4)$, then $(x,y)=(8,6)$ and $T8=T6=\{1\}$ and $(1,1)\in E(G)$. Hence T is ggraphpreserving.
3 Main results
We start with defining a new type of multivalued mappings.
Definition 3.1 Let $(X,d)$ be a metric space, $G=(V(G),E(G))$ be a directed graph such that $V(G)=X$, $g:X\to X$ and $T:X\to CB(X)$. T is said to be a multivalued weak Gcontraction with respect to g or $(g,\alpha ,L)$Gcontraction if there exists a function $\alpha :[0,\mathrm{\infty})\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to {t}^{+}}\alpha (r)<1$ for every $t\in [0,\mathrm{\infty})$ and $L\ge 0$ with
for all $x,y\in X$ such that $(g(x),g(y))\in E(G)$.
Remark 3.2 If $G=(V(G),E(G))$, where $E(G)=X\times X$ and $g(x)=x$, $\mathrm{\forall}x\in X$, then a $(g,\alpha ,L)$Gcontraction is a generalized multivalued $(\alpha ,L)$weak contraction.
Property A ([14])
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 $n\in \mathbb{N}$, then there is a subsequence ${({x}_{{k}_{n}})}_{n\in \mathbb{N}}$ with $({x}_{{k}_{n}},x)\in E(G)$ for $n\in \mathbb{N}$.
Theorem 3.3 Let $(X,d)$ be a complete metric space and $G=(V(G),E(G))$ be a directed graph such that $V(G)=X$, and let $g:X\to X$ be a surjective mapping. If $T:X\to CB(X)$ is a multivalued mapping satisfying the following properties:

(1)
T is a ggraphpreserving mapping;

(2)
there exists ${x}_{0}\in X$ such that $(g({x}_{0}),y)\in E(G)$ for some $y\in T{x}_{0}$;

(3)
X has Property A;

(4)
T is a $(g,\alpha ,L)$Gcontraction;
then there exists $u\in X$ such that $g(u)\in Tu$.
Proof Since g is surjective, there exists ${x}_{1}\in X$ such that $g({x}_{1})\in T{x}_{0}$. By (2) we obtain $(g({x}_{0}),g({x}_{1}))\in E(G)$. We can choose ${n}_{1}\in \mathbb{N}$ such that
By Lemma 2.1, there exists $g({x}_{2})\in T{x}_{1}$ such that
Since $(g({x}_{0}),g({x}_{1}))\in E(G)$, $g({x}_{1})\in T{x}_{0}$, $g({x}_{2})\in T{x}_{1}$ and T is a ggraphpreserving mapping, we have $(g({x}_{1}),g({x}_{2}))\in E(G)$. Moreover, by (3.1) and (3.2), we get
Next, we can choose ${n}_{2}>{n}_{1}$ such that
By Lemma 2.1, there exists $g({x}_{3})\in T{x}_{2}$ such that
By the above two inequalities and $(g({x}_{1}),g({x}_{2}))\in E(G)$, we get
By induction, we obtain a sequence $\{g({x}_{k})\}$ in X and a sequence $\{{n}_{k}\}$ of positive integers with the property that for each $k\in \mathbb{N}$, $g({x}_{k+1})\in T{x}_{k}$, $(g({x}_{k}),g({x}_{k+1}))\in E(G)$ and
and
Therefore $d(g({x}_{k}),g({x}_{k+1}))\le d(g({x}_{k1}),g({x}_{k}))$ for any $k\in \mathbb{N}$, i.e., $\{g({x}_{k})\}$ is a nonincreasing sequence. Thus it follows from Lemma 2.2 that $\{g({x}_{k})\}$ is a Cauchy sequence in X. Since X is complete, there exists $u\in X$ such that ${lim}_{k\to \mathrm{\infty}}g({x}_{k})=g(u)$. By assumption (3), we have a subsequence $g({x}_{{k}_{n}})$ such that $(g({x}_{{k}_{n}}),g(u))\in E(G)$ for any $n\in \mathbb{N}$. Thus we get
Since $g({x}_{{k}_{n}})$ converges to $g(u)$ as $n\to \mathrm{\infty}$, it follows that $D(g(u),Tu)=0$. Since Tu is closed, we conclude that $g(u)\in Tu$. □
Corollary 3.4 Let $(X,d)$ be a metric space endowed with a partial order ≤, $g:X\to X$ be surjective and $T:X\to CB(X)$ be a multivalued mapping. Suppose that

(1)
T is gincreasing;

(2)
there exist ${x}_{0}\in X$ and $u\in T{x}_{0}$ 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 exists $\alpha :[0,\mathrm{\infty})\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to {t}^{+}}\alpha (r)<1$ for every $t\in [0,\mathrm{\infty})$ and $L>0$ such that
$$H(Tx,Ty)\le \alpha \left(d(g(x),g(y))\right)\left(d(g(x),g(y))\right)+LD(g(y),Tx)$$
for any $x,y\in X$ with $g(x)<g(y)$;

(5)
the metric d is complete.
Then there exists $u\in X$ such that $g(u)\in Tu$.
Proof Define $G=(V(G),E(G))$ by $V(G)=X$ and $E(G)=\{(x,y):x<y\}$. Let $x,y\in X$ such that $(g(x),g(y))\in E(G)$. Then $g(x)<g(y)$ so $Tx\prec Ty$. For any $u\in Tx$ and $v\in Ty$, we have $u<v$, i.e., $(u,v)\in E(G)$. So T is graphpreserving. By assumption (2), there exist ${x}_{0}$ and $u\in T{x}_{0}$ such that $g({x}_{0})<u$, so $(g({x}_{0}),u)\in E(G)$. Hence (2) of Theorem 3.3 is satisfied. It is easy to see that (3) and (4) of Theorem 3.3 are also satisfied. Therefore Corollary 3.4 is obtained directly by Theorem 3.3. □
If we put $g(x)=x$ for all $x\in X$ in Corollary 3.4, we obtain the following result.
Corollary 3.5 Let $(X,d)$ be a metric space endowed with a partial order ≤ and $T:X\to CB(X)$ be a multivalued mapping. Suppose that

(1)
T is increasing;

(2)
there exists ${x}_{0}\in X$ such that ${x}_{0}<T{x}_{0}$;

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

(4)
there exist $\alpha :[0,\mathrm{\infty})\to [0,1)$ and $L>0$ such that
$$H(Tx,Ty)\le \alpha (d(x,y))(d(x,y))+LD(y,Tx),$$
for any $x,y\in X$ with $x<y$, where ${lim\hspace{0.17em}sup}_{r\to {t}^{+}}\alpha (r)<1$ for every $t\in [0,\mathrm{\infty})$;

(5)
the metric d is complete.
Then there exists $u\in X$ such that $u\in Tu$.
Remark 3.6 Theorem 4 in [3] is directly obtained from Theorem 3.3 by setting $G=(V(G),E(G))$, where $V(G)=X$, $E(G)=X\times X$ and $g(x)=x$ for all $x\in X$.
Example 3.7 Let $X=\{0,1,3,4,6,7,9,10,11\}$, $d(x,y)=xy$, $x,y\in E(G)$, $E(G)$ = $\{(1,4),(1,7,),(4,4)(4,7),(7,4),(7,7)\}$ ∪ $\{(0,3),(0,6),(3,3),(3,6),(6,3),(6,6)\}$ ∪ $\{(9,10),(10,9),(10,10),(10,11),(11,10),(11,11)\}$ and $T:X\to CB(X)$ be defined by
We note that
This means that T does not satisfy Nadler’s theorem. We will show that T is a weak contraction with $\alpha (x)=\frac{1}{2}$ and $L=2$.
Let $(x,y)\in E(G)$.
If $(x,y)$ ∈ $\{(3,3),(3,6),(6,3),(6,6),(4,4),(4,7),(7,4),(7,7),(9,10),(10,9),(10,10),(10,11),(11,10),(11,11)\}$ ∈ $E(G)$, we have
If $(x,y)=(1,4)$, we have
If $(x,y)=(1,7)$, we have
If $(x,y)=(0,3)$, we have
If $(x,y)=(0,6)$, we have
Hence T is an $(\alpha ,g,L)$Gcontraction. To show that T is graphpreserving, let $(x,y)\in E(G)$. If $(x,y)=(1,4)$ or $(x,y)=(1,7)$, then $Tx=\{0,3\}$, $Ty=\{3,6\}$, and we see that $(0,3),(0,6),(3,3),(3,6)\in E(G)$. If $(x,y)\in \{(4,4),(4,7),(7,4)\}$, then $Tx=\{3,6\}=Ty$ and $(3,3),(3,6),(6,3),(6,6)\in E(G)$. If $(x,y)=(0,3)$ or $(x,y)=(0,6)$, then $Tx=\{1,4\}$, $Ty=\{4,7\}$ and we see that $(1,4),(1,7),(4,4),(4,7)\in E(G)$. If $(x,y)\in \{(3,3),(3,6),(6,3),(6,6)\}$, then $Tx=\{4,7\}=Ty$ and we see that $(4,4),(4,7),(7,4),(7,7)\in E(G)$. If $(x,y)\in \{(9,10),(10,9),(10,11),(11,10)\}$, then $Tx=\{10,11\}=Ty$ and we see that $(10,10),(10,11),(11,10),(11,11)\in E(G)$. Hence T is graphpreserving. By the definition of T and G, we see that $10\in T9=\{10,11\}$ and $(9,10)\in E(G)$, that is, condition (2) of Theorem 3.3 is satisfied. It is easy to see that X has Property A. Therefore all the conditions of Theorem 3.3 are satisfied, so T has a fixed point and we see that $Fix(T)=\{10,11\}$.
Next, we give an example of a map which lacks assumption (2) and has no fixed point.
Example 3.8 Let $X:=\{0,1,3,4,6,7,9,10,11\}$, $d(x,y)=xy$, $x,y\in X$, $E(G):=\{(1,4),(4,7)\}\cup \{(0,3),(3,0),(3,6),(6,3),(0,6),(6,0)\}\cup \{(9,10),(10,9),(10,11)\}$ and $T:X\to CB(X)$ be defined by
and let $g:X\to X$ be an identity map. The same as in Example 3.7, T is a graphpreserving mapping and T is a $(g,\frac{1}{2},2)$Gcontraction. Moreover, we can easily check that condition (2) of Theorem 3.3 does not hold and we note that T has no fixed point.
References
 1.
Nadler S: Multivalued contraction mappings. Pac. J. Math. 1969, 20(2):475–488.
 2.
Alghamdi MA, Berinde V, Shahzad N: Fixed points of multivalued nonself almost contractions. J. Appl. Math. 2013., 2013: Article ID 621614
 3.
Berinde M, Berinde V: On a general class of multivalued weakly Picard mappings. J. Math. Anal. Appl. 2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016
 4.
Berinde V, Pacurar M: The role of PompeiuHausdorff metric in fixed point theory. Creative Math. Inform. 2013, 22(2):143–150.
 5.
Beg I, Butt AR: The contraction principle for set valued mappings on a metric space with graph. Comput. Math. Appl. 2010, 60: 1214–1219. 10.1016/j.camwa.2010.06.003
 6.
Dinevari T, Frigon M: Fixed point results for multivalued contractions on a metric space with a graph. J. Math. Anal. Appl. 2013, 405: 507–517. 10.1016/j.jmaa.2013.04.014
 7.
Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.
 8.
Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022247X(89)90214X
 9.
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2003, 132(5):1435–1443.
 10.
Beg I, Butt AR: Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces. Math. Commun. 2010, 15: 65–75.
 11.
Beg I, Latif A: Common fixed point and coincidence point of generalized contractions in ordered metric spaces. Fixed Point Theory Appl. 2012. 10.1186/168718122012229
 12.
Beg I, Butt AR: Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 2009, 71: 3699–3704. 10.1016/j.na.2009.02.027
 13.
Bhaskar TG, Laskhmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 2006, 65(7):1379–1393. 10.1016/j.na.2005.10.017
 14.
Jachymski J: The contraction principle for mappings on a metric with a graph. Proc. Am. Math. Soc. 2008, 1(136):1359–1373.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. The first author would like to thank Science Achievement Scholarship of Thailand (SAST). This paper was supported by Chiang Mai University, Chiang Mai, Thailand.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tiammee, J., Suantai, S. Coincidence point theorems for graphpreserving multivalued mappings. Fixed Point Theory Appl 2014, 70 (2014). https://doi.org/10.1186/16871812201470
Received:
Accepted:
Published:
Keywords
 fixed point theorems
 multivalued mappings
 partially ordered set
 monotone mappings
 graphpreserving