# Coincidence point theorems for weak graph preserving multi-valued mapping

## Abstract

In this paper, we prove some coincidence and fixed point theorems for a new type of multi-valued weak G-contraction mapping with compact values. The results of this paper extend and generalize several known results from a complete metric space endowed with a graph. Some examples are given to illustrate the usability of our results.

MSC:47H04, 47H10.

## 1 Introduction

The classical contraction mapping principle of Banach states that if $\left(X,d\right)$ is a complete metric space and $f:Xâ†’X$ is a contraction mapping, i.e., $d\left(f\left(x\right),f\left(y\right)\right)â‰¤\mathrm{Î±}d\left(x,y\right)$ for all $x,yâˆˆX$, where $\mathrm{Î±}âˆˆ\left[0,1\right)$, then f has a unique fixed point. Banach fixed point theorem plays an important role in several branches of mathematics. For instance, it has been used to study the existence of solutions of nonlinear integral equations, system of linear equations, nonlinear differential equations in Banach spaces and to prove the convergence of algorithms in computational mathematics. Because of its usefulness for mathematical theory, Banach fixed point theorem has been extended in many directions; see [1â€“11]. Several well-known fixed point theorems of single-valued mappings such as Banach and Schauder have been extended to multi-valued mappings in Banach spaces.

Fixed point theory of multi-valued mappings plays an important role in control theory, optimization, partial differential equations, economics, and applied science. For a metric space $\left(X,d\right)$, we let $CB\left(X\right)$ and $Comp\left(X\right)$ be the set of all nonempty closed bounded subsets and the set of all nonempty compact subsets of X, respectively. A point x in X is a fixed point of a multi-valued mapping $T:Xâ†’{2}^{X}$ if x is in Tx.

Nadler [12] has proved a multi-valued version of the Banach contraction principle which states that each closed and bounded value contraction map on a complete metric space has a fixed point. One of the most general fixed point theorems for multi-valued nonexpansive self-mappings was studied by Kirk and Massa in 1990 [13]. They proved the existence of fixed points in Banach spaces for which the asymptotic center of a bounded sequence in a closed convex subset is nonempty and compact.

The following theorem is the first well-known theorem of multi-valued contractions studied by Nadler in 1969 [12].

Theorem 1.1 Let $\left(X,d\right)$ be a complete metric space and $T:Xâ†’CB\left(X\right)$. Assume that there exists $kâˆˆ\left[0,1\right)$ such that

$H\left(Tx,Ty\right)â‰¤kd\left(x,y\right)$

for all $x,yâˆˆX$. Then there exists $zâˆˆX$ such that $zâˆˆTz$.

Reich [14] extended Nadlerâ€™s fixed point theorem as follows.

Theorem 1.2 Let $\left(X,d\right)$ be a complete metric space and $T:Xâ†’Comp\left(X\right)$. Assume that there exists a function $\mathrm{Ï†}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ such that

$\underset{râ†’{t}^{+}}{limâ€‰sup}\mathrm{Ï†}\left(r\right)<1$

for each $tâˆˆ\left(0,\mathrm{âˆž}\right)$ and

$H\left(Tx,Ty\right)â‰¤\mathrm{Ï†}\left(d\left(x,y\right)\right)d\left(x,y\right)$

for all $x,yâˆˆX$. Then there exists $zâˆˆX$ such that $zâˆˆTz$.

The multi-valued mapping T studied by Reich [14] in Theorem 1.2 has compact value, that is, Tx is a nonempty compact subset of X for all x in the spaces X. In 1989, Mizoguchi and Takahashi [15] relaxed the compactness assumption on the mapping to closed and bounded subsets of X. They proved the following theorem as a generalization of Nadlerâ€™s theorem.

Theorem 1.3 Let $\left(X,d\right)$ be a complete metric space and $T:Xâ†’CB\left(X\right)$. Assume that there exists a function $\mathrm{Ï†}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ such that

$\underset{râ†’{t}^{+}}{limâ€‰sup}\mathrm{Ï†}\left(r\right)<1$

for each $tâˆˆ\left[0,\mathrm{âˆž}\right)$ and

$H\left(Tx,Ty\right)â‰¤\mathrm{Ï†}\left(d\left(x,y\right)\right)d\left(x,y\right)$

for all $x,yâˆˆX$. Then there exists $zâˆˆX$ such that $zâˆˆTz$.

In 2007, Berinde and Berinde [16] gave the definition of a multi-valued weak contraction stated as follows.

Definition 1.4 Let $\left(X,d\right)$ be a metric space and $T:Xâ†’CB\left(X\right)$ a multi-valued mapping. T is said to be a multi-valued weak contraction or a multi-valued $\left(\mathrm{Î¸},L\right)$ -weak contraction if there exist two constants $\mathrm{Î¸}âˆˆ\left(0,1\right)$ and $Lâ‰¥0$ such that

$H\left(Tx,Ty\right)â‰¤\mathrm{Î¸}d\left(x,y\right)+Ld\left(y,Tx\right)$

for all $x,yâˆˆX$.

Then they extended Theorem 1.2 to the class of multi-valued weak contraction and showed that in a complete metric space, every multi-valued weak contraction has a fixed point. In the same paper, they also introduced a class of multi-valued mappings which is more general than that of weak contraction defined as follows.

Definition 1.5 Let $\left(X,d\right)$ be a metric space and $T:Xâ†’CB\left(X\right)$ a multi-valued mapping. T is said to be a generalized multi-valued $\left(\mathrm{Î±},L\right)$-weak contraction if there exist a nonnegative number L and a function $\mathrm{Î±}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ satisfying ${limâ€‰sup}_{râ†’{t}^{+}}\mathrm{Î±}\left(r\right)<1$ for each $tâˆˆ\left[0,\mathrm{âˆž}\right)$ such that

$H\left(Tx,Ty\right)â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+Ld\left(y,Tx\right)$

for all $x,yâˆˆX$.

They showed that in a complete metric space, every generalized multi-valued $\left(\mathrm{Î±},L\right)$-weak contraction has a fixed point.

In 2008, Jachymski [17] introduced the concept of â€˜contraction concerning a graphâ€™, called G-contraction and proved some fixed point results of G-contraction in a complete metric space endowed with a graph.

Definition 1.6 Let $\left(X,d\right)$ be a metric space and $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$ and $E\left(G\right)$ contains all loops, i.e., $\mathrm{â–³}=\left\{\left(x,x\right)âˆ£xâˆˆX\right\}âŠ‚E\left(G\right)$. We say that a mapping $f:Xâ†’X$ is a G-contraction if f preserves edges of G, i.e., for every $x,yâˆˆX$,

$\left(x,y\right)âˆˆE\left(G\right)\phantom{\rule{1em}{0ex}}â‡’\phantom{\rule{1em}{0ex}}\left(f\left(x\right),f\left(y\right)\right)âˆˆE\left(G\right)$
(1)

and there exists $\mathrm{Î±}âˆˆ\left(0,1\right)$ such that $x,yâˆˆX$,

$\left(x,y\right)âˆˆE\left(G\right)\phantom{\rule{1em}{0ex}}â‡’\phantom{\rule{1em}{0ex}}d\left(f\left(x\right),f\left(y\right)\right)â‰¤\mathrm{Î±}d\left(x,y\right).$

The mapping $f:Xâ†’X$ satisfying condition (1) is also called a graph-preserving mapping. Jachymski showed in [17] that under some properties on X, a G-contraction $f:Xâ†’X$ has a fixed point if and only if there exists $xâˆˆX$ such that $\left(x,f\left(x\right)\right)âˆˆE\left(G\right)$.

Recently, Beg and Butt [18] introduced the concept of â€˜G-contractionâ€™ for a multi-valued mapping $T:Xâ†’CB\left(X\right)$ defined as follows.

Definition 1.7 Let $T:Xâ†’CB\left(X\right)$ be a multi-valued mapping. The mapping T is said to be a G-contraction if there exists $kâˆˆ\left(0,1\right)$ such that

$H\left(Tx,Ty\right)â‰¤kd\left(x,y\right)$

for all $\left(x,y\right)âˆˆE\left(G\right)$ and if $uâˆˆTx$ and $vâˆˆTy$ are such that

$d\left(u,v\right)â‰¤kd\left(x,y\right)+\mathrm{Î±}$

for each $\mathrm{Î±}>0$, then $\left(u,v\right)âˆˆE\left(G\right)$.

They showed that if $\left(X,d\right)$ is a complete metric space and $\left(X,d\right)$ has Property A [18], then G-contraction mapping $T:Xâ†’CB\left(X\right)$ has a fixed point if and only if there exist $xâˆˆX$ and $yâˆˆTx$ such that $\left(x,y\right)âˆˆE\left(G\right)$.

In 2011, Nicolae, Oâ€™Regan, and Petrusel [19] extended the notion of multi-valued contraction on a metric space with a graph in considering the fixed point shown below.

Theorem 1.8 Let $F:Xâ†’X$ be a multi-valued map with nonempty closed values. Assume that

1. (1)

there exists $\mathrm{Î»}âˆˆ\left(0,1\right)$ such that $D\left(F\left(x\right),F\left(y\right)\right)â‰¤\mathrm{Î»}d\left(x,y\right)$ for all $\left(x,y\right)âˆˆE\left(G\right)$;

2. (2)

for each $\left(x,y\right)âˆˆE\left(G\right)$, each $uâˆˆF\left(x\right)$ and $vâˆˆF\left(y\right)$ satisfying $d\left(u,v\right)â‰¤ad\left(x,y\right)$ for some $aâˆˆ\left(0,1\right)$, $\left(u,v\right)âˆˆE\left(G\right)$ holds;

3. (3)

X has Property  A.

If there exist ${x}_{0},{x}_{1}âˆˆX$ such that ${x}_{1}âˆˆ{\left[{x}_{0}\right]}_{G}^{1}âˆ©F\left({x}_{0}\right)$, then F has a fixed point.

In 2013, Dinevari and Frigon [20] introduced a concept of â€˜G-contractionâ€™ which is weaker than that of Beg and Butt [18] and weaker than that of Nicolae, Oâ€™Regan, and Petrusel [19].

Definition 1.9 Let $T:Xâ†’{2}^{X}$ be a map with nonempty values. We say that T is a G-contraction (in the sense of Dinevari and Frigon) if there exists $\mathrm{Î±}âˆˆ\left(0,1\right)$ such that for all $\left(x,y\right)âˆˆE\left(G\right)$ and all $uâˆˆTx$, there exists $vâˆˆTy$ such that $\left(u,v\right)âˆˆE\left(G\right)$ and $d\left(u,v\right)â‰¤\mathrm{Î±}d\left(x,y\right)$.

They showed that under some properties, weaker than Property A, on a metric space a multi-valued G-contraction with the closed value has a fixed point.

Most recently, Tiammee and Suantai [21] introduced the concept of â€˜graph preservingâ€™ for multi-valued mappings and proved their fixed point theorem in a complete metric space endowed with a graph.

Definition 1.10 [21]

Let X be a nonempty set, $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$, and $T:Xâ†’CB\left(X\right)$. T is said to be graph preserving if

$\left(x,y\right)âˆˆE\left(G\right)\phantom{\rule{1em}{0ex}}â‡’\phantom{\rule{1em}{0ex}}\left(u,v\right)âˆˆE\left(G\right)$

for all $uâˆˆTx$ and $vâˆˆTy$.

Definition 1.11 [21]

Let X be a nonempty set, $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$, $g:Xâ†’X$, and $T:Xâ†’CB\left(X\right)$. T is said to be g-graph preserving if for any $x,yâˆˆX$, such that

$\left(g\left(x\right),g\left(y\right)\right)âˆˆE\left(G\right)\phantom{\rule{1em}{0ex}}â‡’\phantom{\rule{1em}{0ex}}\left(u,v\right)âˆˆE\left(G\right)$

for all $uâˆˆTx$ and $vâˆˆTy$.

Definition 1.12 Let $\left(X,d\right)$ be a metric space, $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$, $g:Xâ†’X$, and $T:Xâ†’CB\left(X\right)$. T is said to be a multi-valued weak G-contraction with respect to g or a $\left(g,\mathrm{Î±},L\right)$ -G-contraction if there exists a function $\mathrm{Î±}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ satisfying

$\underset{râ†’{t}^{+}}{limâ€‰sup}\mathrm{Î±}\left(r\right)<1$

for every $tâˆˆ\left[0,\mathrm{âˆž}\right)$ and a nonnegative number L with

$H\left(Tx,Ty\right)â‰¤\mathrm{Î±}\left(d\left(g\left(x\right),g\left(y\right)\right)\right)d\left(g\left(x\right),g\left(y\right)\right)+LD\left(g\left(y\right),Tx\right)$

for all $x,yâˆˆX$ such that $\left(g\left(x\right),g\left(y\right)\right)âˆˆE\left(G\right)$.

Theorem 1.13 [21]

Let $\left(X,d\right)$ be a complete metric space, $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$, and $g:Xâ†’X$ a surjective mapping. If $T:Xâ†’CB\left(X\right)$ is a multi-valued mapping satisfying the following properties:

1. (1)

T is a g-graph preserving mapping;

2. (2)

there exists ${x}_{0}âˆˆX$ such that $\left(g\left({x}_{0}\right),y\right)âˆˆE\left(G\right)$ for some $yâˆˆT{x}_{0}$;

3. (3)

X has Property  A;

4. (4)

T is a $\left(g,\mathrm{Î±},L\right)$-G-contraction,

then there exists $uâˆˆX$ such that $g\left(u\right)âˆˆTu$.

The condition of T in Definition 1.11 to be g-graph preserving requires all pairs $\left(u,v\right)$ where $uâˆˆTx$ and $vâˆˆTy$ have connecting edges whenever $\left(g\left(x\right),g\left(y\right)\right)âˆˆE\left(G\right)$. With some modification, we are interested in proposing the new concept of â€˜g-graph preservingâ€™ for multi-valued mappings in a complete metric space endowed with a graph and the fixed point theorem is also determined.

## 2 Preliminaries

Let $\left(X,d\right)$ be a metric space. For $xâˆˆX$ and $A,BâˆˆComp\left(X\right)$, define

$\begin{array}{c}d\left(x,A\right)=inf\left\{d\left(x,y\right)âˆ£yâˆˆA\right\},\hfill \\ D\left(A,B\right)=inf\left\{d\left(x,B\right)âˆ£xâˆˆA\right\}.\hfill \end{array}$

For each $aâˆˆA$, define

${P}_{B}\left(a\right)=\left\{bâˆˆBâˆ£d\left(a,b\right)=d\left(a,B\right)\right\}.$

Each element in ${P}_{B}\left(a\right)$ is called a projection point of a into B. Note that if B is compact, then ${P}_{B}\left(a\right)$ is always a nonempty set. Also, define

$H\left(A,B\right)=max\left\{\underset{xâˆˆB}{sup}d\left(x,A\right),\underset{xâˆˆA}{sup}d\left(x,B\right)\right\}.$

The mapping H is said to be a Hausdorff metric induced by d. The next two lemmas will play central roles in our main results.

Lemma 2.1 [12]

Let $\left(X,d\right)$ be a metric space. If $A,BâˆˆComp\left(X\right)$ (or $CB\left(X\right)$) and $xâˆˆA$, then for each $\mathrm{Ïµ}>0$, there is $bâˆˆB$ such that

$d\left(a,b\right)â‰¤H\left(A,B\right)+\mathrm{Ïµ}.$

Lemma 2.2 [15]

Let $\left(X,d\right)$ be a metric space, $\left\{{A}_{k}\right\}âŠ‚Comp\left(X\right)$ (or $CB\left(X\right)$), $\left\{{x}_{k}\right\}$ a sequence in X such that ${x}_{k}âˆˆ{A}_{kâˆ’1}$, and $\mathrm{Î±}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ a function satisfying ${limâ€‰sup}_{râ†’{t}^{+}}\mathrm{Î±}\left(r\right)<1$ for every $tâˆˆ\left[0,\mathrm{âˆž}\right)$. Suppose that $d\left({x}_{kâˆ’1},{x}_{k}\right)$ is a non-increasing sequence such that

$\begin{array}{c}H\left({A}_{kâˆ’1},{A}_{k}\right)â‰¤\mathrm{Î±}\left(d\left({x}_{kâˆ’1},{x}_{k}\right)\right)d\left({x}_{kâˆ’1},{x}_{k}\right),\hfill \\ d\left({x}_{k},{x}_{k+1}\right)â‰¤H\left({A}_{kâˆ’1},{A}_{k}\right)+{\left(\mathrm{Î±}\left(d\left({x}_{kâˆ’1},{x}_{k}\right)\right)\right)}^{{n}_{k}},\hfill \end{array}$

where $\left\{{n}_{k}\right\}$ is an increasing sequence and $k,{n}_{k}âˆˆ\mathbb{N}$. Then $\left\{{x}_{k}\right\}$ is a Cauchy sequence in X.

Next we will give notions and examples of new types of multi-valued mapping with compact value which are weaker than that of Tiammee and Suantai [21].

Definition 2.3 Let X be a nonempty set and $G=\left(V\left(G\right),E\left(G\right)\right)$ be a directed graph such that $V\left(G\right)=X$, and $T:Xâ†’Comp\left(X\right)$. T is said to be weak graph preserving if it satisfies the following:

for each $x,yâˆˆX$, if $\left(x,y\right)âˆˆE\left(G\right)$, then for each $uâˆˆTx$ there is $vâˆˆ{P}_{Ty}\left(u\right)$ such that $\left(u,v\right)âˆˆE\left(G\right)$.

In Example 2.4, we illustrate a mapping T which is weak graph preserving but not graph preserving.

Example 2.4 Let â„• be a metric space with the usual metric and $G=\left(\mathbb{N},E\left(G\right)\right)$ where

$E\left(G\right)=\left\{\left(2nâˆ’1,2n\right):nâˆˆ\mathbb{N}\right\}âˆª\left\{\left(2n,1\right):nâˆˆ\mathbb{N}\right\}.$

Define $T:\mathbb{N}â†’Comp\left(X\right)$ by

We will show that T is weak graph preserving. If $\left(x,y\right)=\left(2kâˆ’1,2k\right)$ where $kâˆˆ\mathbb{N}$, then $Tx=\left\{2k,2k+2\right\}$ and $Ty=\left\{1\right\}$. We can see that ${P}_{Ty}\left(2k\right)=\left\{1\right\}={P}_{Ty}\left(2k+2\right)$ and $\left(2k,1\right),\left(2k+2,1\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(2k,1\right)$ where $kâˆˆ\mathbb{N}$, then $Tx=\left\{1\right\}$ and $Ty=\left\{2,4\right\}$. We can see that ${P}_{Ty}\left(1\right)=\left\{2\right\}$ and $\left(1,2\right)âˆˆE\left(G\right)$. It is easy to see that $\left(1,4\right)âˆ‰E\left(G\right)$ and so T is not graph preserving.

Next we will give an another example of a weak graph-preserving mapping that is not a G-contraction in the sense of Dinevari and Frigon [20].

Example 2.5 Let $X=\left\{1,2,3,4,6,8\right\}$ be a metric space with the usual metric and $G=\left(X,E\left(G\right)\right)$ where

$E\left(G\right)=\left\{\left(1,1\right),\left(1,3\right),\left(2,2\right),\left(2,6\right),\left(4,2\right),\left(4,4\right),\left(4,6\right)\right\}.$

Define $T:Xâ†’Comp\left(X\right)$ by

We will show that T is weak graph preserving. If $\left(x,y\right)=\left(1,1\right)$ then $Tx=\left\{2,4\right\}$ and $Ty=\left\{2,4\right\}$. We can see that ${P}_{Ty}\left(2\right)=\left\{2\right\}$, ${P}_{Ty}\left(4\right)=\left\{4\right\}$ and $\left(2,2\right),\left(4,4\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(1,3\right)$ then $Tx=\left\{2,4\right\}$ and $Ty=\left\{6,8\right\}$. We can see that ${P}_{Ty}\left(2\right)=\left\{6\right\}$, ${P}_{Ty}\left(4\right)=\left\{6\right\}$, and $\left(2,6\right),\left(4,6\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(2,2\right)$ or $\left(x,y\right)=\left(2,6\right)$ then $Tx=\left\{2\right\}$ and $Ty=\left\{2\right\}$. We can see that ${P}_{Ty}\left(2\right)=\left\{2\right\}$ and $\left(2,2\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(4,2\right)$ or $\left(x,y\right)=\left(4,6\right)$ then $Tx=\left\{2,4\right\}$ and $Ty=\left\{2\right\}$. We can see that ${P}_{Ty}\left(2\right)=\left\{2\right\}={P}_{Ty}\left(4\right)$ and $\left(2,2\right),\left(4,2\right)âˆˆE\left(G\right)$.

So, T is weak graph preserving but it is not a G-contraction in the sense of Dinevari and Frigon since for each $uâˆˆTx$ and $vâˆˆTy$ with $\left(x,y\right)=\left(1,3\right)$, $d\left(u,v\right)â‰¥2>\mathrm{Î±}d\left(1,3\right)$ for all $\mathrm{Î±}âˆˆ\left(0,1\right)$.

Definition 2.6 Let X be a nonempty set, $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$, $T:Xâ†’Comp\left(X\right)$, and $g:Xâ†’X$. T is said to be weak g-graph preserving if it satisfies the following:

for each $x,yâˆˆX$, if $\left(g\left(x\right),g\left(y\right)\right)âˆˆE\left(G\right)$, then for each $uâˆˆTx$ there is $vâˆˆ{P}_{Ty}\left(u\right)$ such that $\left(u,v\right)âˆˆE\left(G\right)$.

Example 2.7 Let â„• be a metric space with the usual metric, $G=\left(\mathbb{N},E\left(G\right)\right)$ where

$E\left(G\right)=\left\{\left(2nâˆ’1,2n\right):nâˆˆ\mathbb{N}\right\}âˆª\left\{\left(2n,1\right):nâˆˆ\mathbb{N}\right\}.$

Define $T:\mathbb{N}â†’Comp\left(X\right)$ by

and $g:Xâ†’X$ by

We will show that T is weak g-graph preserving.

If $\left(g\left(x\right),g\left(y\right)\right)=\left(1,2\right)$, then $\left(x,y\right)=\left(1,2\right)$ and $Tx=\left\{2,4\right\}$ and $Ty=\left\{1\right\}$. We can see that ${P}_{Ty}\left(2\right)=\left\{1\right\}={P}_{Ty}\left(4\right)$ and $\left(2,1\right),\left(4,1\right)âˆˆE\left(G\right)$.

If $\left(g\left(x\right),g\left(y\right)\right)=\left(2,1\right)$, then $\left(x,y\right)=\left(2,1\right)$ and $Tx=\left\{1\right\}$ and $Ty=\left\{2,4\right\}$. We can see that ${P}_{Ty}\left(1\right)=\left\{2\right\}$ and $\left(1,2\right)âˆˆE\left(G\right)$.

If $\left(g\left(x\right),g\left(y\right)\right)=\left(2kâˆ’1,2k\right)$ where $kâˆˆ\mathbb{N}âˆ–\left\{1\right\}$, then $\left(x,y\right)=\left(2k+1,2k+2\right)$ and $Tx=\left\{2k+2,2k+4\right\}$ and $Ty=\left\{1\right\}$. We can see that ${P}_{Ty}\left(2k+2\right)=\left\{1\right\}={P}_{Ty}\left(2k+4\right)$ and $\left(2k+2,1\right),\left(2k+4,1\right)âˆˆE\left(G\right)$.

If $\left(g\left(x\right),g\left(y\right)\right)=\left(2k,1\right)$ where $kâˆˆ\mathbb{N}âˆ–\left\{1\right\}$, then $\left(x,y\right)=\left(2k+2,1\right)$ and $Tx=\left\{1\right\}$ and $Ty=\left\{2,4\right\}$. We can see that ${P}_{Ty}\left(1\right)=\left\{2\right\}$ and $\left(1,2\right)âˆˆE\left(G\right)$. Hence T is weak g-graph preserving.

## 3 Main results

We first recall Property A before the main theorem is proved.

Property A For any sequence $\left\{{x}_{n}\right\}$ in X, if ${x}_{n}â†’x$ and $\left({x}_{n},{x}_{n+1}\right)âˆˆE\left(G\right)$ for $nâˆˆ\mathbb{N}$, then there is a subsequence $\left\{{x}_{{n}_{k}}\right\}$ such that $\left({x}_{{n}_{k}},x\right)âˆˆE\left(G\right)$ for $kâˆˆ\mathbb{N}$.

Theorem 3.1 Let $\left(X,d\right)$ be a complete metric space, $G=\left(V\left(G\right),E\left(G\right)\right)$ a directed graph such that $V\left(G\right)=X$, and $g:Xâ†’X$ a surjective map. If $T:Xâ†’Comp\left(X\right)$ is a multi-valued mapping satisfying the following properties:

1. (1)

T is weak g-graph preserving;

2. (2)

;

3. (3)

X has Property  A;

4. (4)

T is a $\left(g,\mathrm{Î±},L\right)$-G-contraction,

then there exists $uâˆˆX$ such that $g\left(u\right)âˆˆTu$.

Proof By (2), let ${x}_{0}âˆˆ{X}_{T}$. Then there exists ${y}_{1}âˆˆT{x}_{0}$ such that $\left(g\left({x}_{0}\right),{y}_{1}\right)âˆˆE\left(G\right)$. Since g is surjective, there exists ${x}_{1}âˆˆX$ such that ${y}_{1}=g\left({x}_{1}\right)$. Thus we have $\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)âˆˆE\left(G\right)$. Since ${\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{n}â†’0$, there exists ${n}_{1}âˆˆ\mathbb{N}$ such that

${\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{{n}_{1}}â‰¤\left[1âˆ’\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right).$

Since $T{x}_{1}$ is compact, it follows that . Since T is weak g-graph preserving, there exists ${y}_{2}âˆˆ{P}_{T{x}_{1}}\left(g\left({x}_{1}\right)\right)$ such that $\left(g\left({x}_{1}\right),{y}_{2}\right)âˆˆE\left(G\right)$ and $d\left(g\left({x}_{1}\right),{y}_{2}\right)=d\left(g\left({x}_{1}\right),T{x}_{1}\right)$. Again since g is surjective, there is ${x}_{2}âˆˆX$ such that $g\left({x}_{2}\right)={y}_{2}$. By Lemma 2.1, there is ${y}_{1}^{â€²}âˆˆT{x}_{1}$ such that

$d\left(g\left({x}_{1}\right),{y}_{1}^{â€²}\right)â‰¤H\left(T{x}_{0},T{x}_{1}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{{n}_{1}}.$

It follows that

$\begin{array}{rl}d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)& =d\left(g\left({x}_{1}\right),T{x}_{1}\right)\\ â‰¤d\left(g\left({x}_{1}\right),{y}_{1}^{â€²}\right)\\ â‰¤H\left(T{x}_{0},T{x}_{1}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{{n}_{1}}.\end{array}$

Since T is a $\left(g,\mathrm{Î±},L\right)$-G-contraction and $\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)âˆˆE\left(G\right)$, we have

$\begin{array}{rl}d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)â‰¤& H\left(T{x}_{0},T{x}_{1}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{{n}_{1}}\\ â‰¤& \mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)+LD\left(g\left({x}_{1}\right),T{x}_{0}\right)\\ +{\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{{n}_{1}}\\ â‰¤& \mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right)\right)\right]}^{{n}_{1}}\\ â‰¤& d\left(g\left({x}_{0}\right),g\left({x}_{1}\right)\right).\end{array}$

Next, since ${\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{n}â†’0$, there exists ${n}_{2}âˆˆ\mathbb{N}$ such that ${n}_{2}>{n}_{1}$ and

${\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{{n}_{2}}â‰¤\left[1âˆ’\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right).$

Since $T{x}_{2}$ is compact, it follows that . Since T is weak g-graph preserving, there exists ${y}_{3}âˆˆ{P}_{T{x}_{2}}\left(g\left({x}_{2}\right)\right)$ such that $\left(g\left({x}_{2}\right),{y}_{3}\right)âˆˆE\left(G\right)$ and $d\left(g\left({x}_{2}\right),{y}_{3}\right)=d\left(g\left({x}_{2}\right),T{x}_{2}\right)$. Again since g is surjective, there is ${x}_{3}âˆˆX$ such that $g\left({x}_{3}\right)={y}_{3}$. By Lemma 2.1, there is ${y}_{2}^{â€²}âˆˆT{x}_{2}$ such that

$d\left(g\left({x}_{1}\right),{y}_{2}^{â€²}\right)â‰¤H\left(T{x}_{1},T{x}_{2}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{{n}_{2}}.$

It follows that

$\begin{array}{rl}d\left(g\left({x}_{2}\right),g\left({x}_{3}\right)\right)& =d\left(g\left({x}_{2}\right),T{x}_{2}\right)\\ â‰¤d\left(g\left({x}_{2}\right),{y}_{2}^{â€²}\right)\\ â‰¤H\left(T{x}_{1},T{x}_{2}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{{n}_{2}}.\end{array}$

Since T is a $\left(g,\mathrm{Î±},L\right)$-G-contraction and $\left(g\left({x}_{2}\right),g\left({x}_{3}\right)\right)âˆˆE\left(G\right)$, we have

$\begin{array}{rl}d\left(g\left({x}_{2}\right),g\left({x}_{3}\right)\right)â‰¤& H\left(T{x}_{1},T{x}_{2}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{{n}_{2}}\\ â‰¤& \mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)+LD\left(g\left({x}_{2}\right),T{x}_{1}\right)\\ +{\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{{n}_{2}}\\ â‰¤& \mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right)\right)\right]}^{{n}_{2}}\\ â‰¤& d\left(g\left({x}_{1}\right),g\left({x}_{2}\right)\right).\end{array}$

Continuing in this process, we produce a sequence $\left\{g\left({x}_{k}\right)\right\}$ in X and an increasing sequence $\left\{{n}_{k}\right\}$ in â„• such that for each $kâˆˆ\mathbb{N}$, $g\left({x}_{k+1}\right)âˆˆT{x}_{k}$, $\left(g\left({x}_{k}\right),g\left({x}_{k+1}\right)\right)âˆˆE\left(G\right)$,

$\begin{array}{c}{\left[\mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)\right]}^{{n}_{k}}â‰¤\left[1âˆ’\mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)\right]d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right),\hfill \\ H\left(T{x}_{kâˆ’1},T{x}_{k}\right)â‰¤\mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right),\hfill \end{array}$

and

$d\left(g\left({x}_{k}\right),g\left({x}_{k+1}\right)\right)â‰¤H\left(T{x}_{kâˆ’1},T{x}_{k}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)\right]}^{{n}_{k}}.$

We note that

$\begin{array}{rl}d\left(g\left({x}_{k}\right),g\left({x}_{k+1}\right)\right)â‰¤& H\left(T{x}_{kâˆ’1},T{x}_{k}\right)+{\left[\mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)\right]}^{{n}_{k}}\\ â‰¤& \mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\\ +{\left[\mathrm{Î±}\left(d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)\right)\right]}^{{n}_{k}}\\ â‰¤& d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right).\end{array}$

Hence, $d\left(g\left({x}_{k}\right),g\left({x}_{k+1}\right)\right)â‰¤d\left(g\left({x}_{kâˆ’1}\right),g\left({x}_{k}\right)\right)$ and so $\left\{d\left(g\left({x}_{k}\right),g\left({x}_{k+1}\right)\right)\right\}$ is a non-increasing sequence. By Lemma 2.2, $\left\{g\left({x}_{k}\right)\right\}$ is a Cauchy sequence in X. Since X is complete, the sequence $\left\{g\left({x}_{k}\right)\right\}$ converges to a point $g\left(u\right)$ for some $uâˆˆX$. By Property A in (3), there is a subsequence $\left\{g\left({x}_{{k}_{n}}\right)\right\}$ such that $\left(g\left({x}_{{k}_{n}}\right),g\left(u\right)\right)âˆˆE\left(G\right)$ for any $nâˆˆ\mathbb{N}$. Claim that $g\left(u\right)âˆˆTu$. Note that for each $g\left({x}_{{k}_{n}+1}\right)$,

$\begin{array}{rl}D\left(g\left(u\right),Tu\right)â‰¤& d\left(g\left(u\right),g\left({x}_{{k}_{n}+1}\right)\right)+D\left(g\left({x}_{{k}_{n}+1}\right),Tu\right)\\ â‰¤& d\left(g\left(u\right),g\left({x}_{{k}_{n}+1}\right)\right)+H\left(T{x}_{{k}_{n}},Tu\right)\\ â‰¤& d\left(g\left(u\right),g\left({x}_{{k}_{n}+1}\right)\right)+\mathrm{Î±}\left(d\left(g\left({x}_{{k}_{n}}\right),g\left(u\right)\right)\right)d\left(g\left({x}_{{k}_{n}}\right),g\left(u\right)\right)\\ +LD\left(g\left(u\right),T{x}_{{k}_{n}}\right)\\ â‰¤& d\left(g\left(u\right),g\left({x}_{{k}_{n}+1}\right)\right)+\mathrm{Î±}\left(d\left(g\left({x}_{{k}_{n}}\right),g\left(u\right)\right)\right)d\left(g\left({x}_{{k}_{n}}\right),g\left(u\right)\right)\\ +Ld\left(g\left(u\right),g\left({x}_{{k}_{n}+1}\right)\right).\end{array}$

Since $g\left({x}_{{k}_{n}}\right)$ converges to $g\left(u\right)$ as $nâ†’\mathrm{âˆž}$, it follows that $D\left(g\left(u\right),Tu\right)=0$. Since Tu is compact, we conclude that $g\left(u\right)âˆˆTu$, completing the proof.â€ƒâ–¡

Remark 3.2 Theorem 3.1 is an extension of Theorem 1.13 in the case of a mapping $T:Xâ†’CB\left(X\right)$ having compact values.

A partial order is a binary relation â‰¤ over the set X which satisfies the following conditions:

1. (1)

$xâ‰¤x$ (reflexivity);

2. (2)

if $xâ‰¤y$ and $yâ‰¤x$, then $x=y$ (antisymmetry);

3. (3)

if $xâ‰¤y$ and $yâ‰¤z$, then $xâ‰¤z$ (transitivity),

for all $x,yâˆˆX$. A set with a partial order â‰¤ is called a partially ordered set. We write $x if $xâ‰¤y$ and .

Definition 3.3 Let $\left(X,â‰¤\right)$ be a partially ordered set. For each $A,BâŠ‚X$, $Aâ‰ºB$ if $aâ‰¤b$ for any $aâˆˆA$, $bâˆˆB$.

Definition 3.4 [21]

Let $\left(X,d\right)$ be a metric space endowed with a partial order â‰¤, $g:Xâ†’X$ a surjective map, and $T:Xâ†’Comp\left(X\right)$. T is said to be g-increasing if for any $x,yâˆˆX$,

$g\left(x\right)

In the case g is the identity map, the mapping T is called an increasing mapping.

Corollary 3.5 Let $\left(X,d\right)$ be a metric space endowed with a partial order â‰¤, $g:Xâ†’X$ a surjective map and $T:Xâ†’Comp\left(X\right)$ a multi-valued mapping. Suppose that

1. (1)

T is g-increasing;

2. (2)

there exist ${x}_{0}âˆˆX$ and $uâˆˆT{x}_{0}$ such that $g\left({x}_{0}\right);

3. (3)

for each sequence $\left\{{x}_{k}\right\}$ such that $g\left({x}_{k}\right) for all $kâˆˆ\mathbb{N}$ and $g\left({x}_{k}\right)$ converges to $g\left(x\right)$ for some $xâˆˆX$, then $g\left({x}_{k}\right) for all $kâˆˆ\mathbb{N}$;

4. (4)

there exist a function $\mathrm{Î±}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ satisfying ${limâ€‰sup}_{râ†’{t}^{+}}\mathrm{Î±}\left(r\right)<1$ for every $tâˆˆ\left[0,\mathrm{âˆž}\right)$ and a nonnegative number L with

$H\left(Tx,Ty\right)â‰¤\mathrm{Î±}\left(d\left(g\left(x\right),g\left(y\right)\right)\right)d\left(g\left(x\right),g\left(y\right)\right)+LD\left(g\left(y\right),Tx\right)$

for all $x,yâˆˆX$ such that $g\left(x\right);

1. (5)

the metric d is complete.

Then there exists $uâˆˆX$ such that $g\left(u\right)âˆˆTu$.

Proof Define $G=\left(V\left(G\right),E\left(G\right)\right)$ where $V\left(G\right)=X$ and $E\left(G\right)=\left\{\left(x,y\right)âˆ£x. Let $x,yâˆˆX$ be such that $\left(g\left(x\right),g\left(y\right)\right)âˆˆE\left(G\right)$. Then $g\left(x\right) so by (1) it implies that $Txâ‰ºTy$. For each $uâˆˆTx$, $vâˆˆTy$, $u. Since Ty is compact, it follows that and ${P}_{Ty}\left(x\right)âŠ‚Ty$ for all $xâˆˆTx$. Thus $\left(x,v\right)âˆˆE\left(G\right)$ for all $vâˆˆ{P}_{Ty}\left(x\right)$ and all $xâˆˆTx$. That is, T is weak g-graph preserving. By assumption (2), there exist ${x}_{0}âˆˆX$ and $uâˆˆT{x}_{0}$ such that $g\left({x}_{0}\right). So $\left(g\left({x}_{0}\right),u\right)âˆˆE\left(G\right)$ and hence the condition (2) in Theorem 3.1 is satisfied. Moreover, the conditions (3) and (4) in Theorem 3.1 are also satisfied. Therefore the result of this corollary is followed by Theorem 3.1.â€ƒâ–¡

Theorem 3.6 Let $\left(X,d\right)$ be a metric space endowed with a partial order â‰¤ and $T:Xâ†’Comp\left(X\right)$ a multi-valued mapping. Suppose that

1. (1)

T is increasing;

2. (2)

there exist ${x}_{0}âˆˆX$ and $uâˆˆT{x}_{0}$ such that ${x}_{0};

3. (3)

for each sequence $\left\{{x}_{k}\right\}$ such that ${x}_{k}<{x}_{k+1}$ for all $kâˆˆ\mathbb{N}$ and ${x}_{k}$ converges to x for some $xâˆˆX$, then ${x}_{k} for all $kâˆˆ\mathbb{N}$;

4. (4)

there exist a function $\mathrm{Î±}:\left[0,\mathrm{âˆž}\right)â†’\left[0,1\right)$ satisfying ${limâ€‰sup}_{râ†’{t}^{+}}\mathrm{Î±}\left(r\right)<1$ for every $tâˆˆ\left[0,\mathrm{âˆž}\right)$ and a nonnegative number L with

$H\left(Tx,Ty\right)â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right)$

for all $x,yâˆˆX$ with $x;

1. (5)

the metric d is complete.

Then there exists $uâˆˆX$ such that $uâˆˆTu$. Furthermore, if $sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}<1$ and $L<1âˆ’sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}$, then T has a unique fixed point.

Proof Setting $g\left(x\right)=x$, by Corollary 3.5 we have . With conditions $sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}<1$ and $L<1âˆ’sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}$, we will show that T has a unique fixed point. Let $u,vâˆˆFix\left(T\right)$. Suppose to a contrary that . Without loss of generality, assume that $u. By the condition (4), we have

$\begin{array}{rl}d\left(u,v\right)& â‰¤H\left(Tu,Tv\right)\\ â‰¤\mathrm{Î±}\left(d\left(u,v\right)\right)d\left(u,v\right)+LD\left(v,Tu\right)\\ â‰¤sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}d\left(u,v\right)+Ld\left(u,v\right)\\ =\left(sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}+L\right)d\left(u,v\right).\end{array}$

Since $sup\left\{\mathrm{Î±}\left(r\right):râˆˆ\left[0,\mathrm{âˆž}\right)\right\}+L<1$, this yields $d\left(u,v\right), a contradiction. Therefore $u=v$, which implies that T has a unique fixed point.â€ƒâ–¡

Next we give an example such that T has a unique fixed point but T is neither a graph-preserving nor a multi-valued G-contraction in the sense of Nicolae, Oâ€™Regan, and Petrusel [19].

Example 3.7 Let $X=\left\{0\right\}âˆª\left\{\frac{1}{{2}^{n}}:nâˆˆ\mathbb{N}âˆª\left\{0\right\}\right\}$ be a metric space with the usual metric d. Consider the directed graph defined by $V\left(G\right)=X$ and

$E\left(G\right)=\left\{\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n+1}}\right),\left(\frac{1}{{2}^{n}},0\right):nâˆˆ\mathbb{N}âˆª\left\{0\right\}\right\}âˆª\mathrm{Î”},$

where Î” is the diagonal in $XÃ—X$. Let $T:Xâ†’Comp\left(X\right)$ be defined by

Then:

1. (1)

T has a fixed point;

2. (2)

T is not graph preserving as defined by Tiammee and Suantai [21];

3. (3)

T is not a G-contraction in the sense of Nicolae, Oâ€™Regan, and Petrusel [19].

Proof (1) We will show that T is a $\left(\mathrm{Î±},g,L\right)$-G-contraction with $\mathrm{Î±}â‰¥\frac{1}{2}$, $Lâ‰¥2$, and $g={i}_{X}$. Let $\left(x,y\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(1,\frac{1}{2}\right)$, then $Tx=\left\{\frac{1}{2}\right\}$, $Ty=\left\{\frac{1}{4},1\right\}$, and

$H\left(Tx,Ty\right)=\frac{1}{4}=\frac{1}{2}d\left(x,y\right)â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right).$

If $\left(x,y\right)=\left(1,0\right)$, then $Tx=\left\{\frac{1}{2}\right\}$, $Ty=\left\{0,1\right\}$, and

$H\left(Tx,Ty\right)=\frac{1}{2}=\frac{1}{2}d\left(x,y\right)â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right).$

If $\left(x,y\right)=\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n+1}}\right)$ for all $nâˆˆ\mathbb{N}$, then $Tx=\left\{\frac{1}{{2}^{n+1}},1\right\}$ and $Ty=\left\{\frac{1}{{2}^{n+2}},1\right\}$. It is easy to check that

$\begin{array}{rl}H\left(Tx,Ty\right)& =max\left\{\underset{aâˆˆTy}{sup}d\left(a,Tx\right),\underset{bâˆˆTx}{sup}d\left(b,Ty\right)\right\}\\ =d\left(\frac{1}{{2}^{n+1}},\frac{1}{{2}^{n+2}}\right)\\ =\frac{1}{2}d\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n+1}}\right)\\ =\frac{1}{2}d\left(x,y\right)\\ â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right).\end{array}$

If $\left(x,y\right)=\left(\frac{1}{{2}^{n}},0\right)$ for all $nâˆˆ\mathbb{N}$, then $Tx=\left\{\frac{1}{{2}^{n+1}},1\right\}$ and $Ty=\left\{0,1\right\}$. We have

$\begin{array}{rl}H\left(Tx,Ty\right)& =max\left\{\underset{aâˆˆTy}{sup}d\left(a,Tx\right),\underset{bâˆˆTx}{sup}d\left(b,Ty\right)\right\}\\ =\frac{1}{{2}^{n+1}}\\ =\frac{1}{2}d\left(\frac{1}{{2}^{n}},0\right)\\ =\frac{1}{2}d\left(x,y\right)\\ â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right).\end{array}$

If $\left(x,y\right)=\left(0,0\right)$ and $\left(x,y\right)=\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n}}\right)$, then $Tx=Ty$ which obviously implies that $H\left(Tx,Ty\right)=0$. So,

$H\left(Tx,Ty\right)â‰¤\mathrm{Î±}\left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right).$

Hence T is a $\left(\mathrm{Î±},g,L\right)$-G-contraction. Next, we will show that T is weak graph preserving. Let $\left(x,y\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(1,\frac{1}{2}\right)$, then $Tx=\left\{\frac{1}{2}\right\}$, $Ty=\left\{\frac{1}{4},1\right\}$, and ${P}_{Ty}\left(\frac{1}{2}\right)=\left\{\frac{1}{4}\right\}$ and $\left(\frac{1}{2},\frac{1}{4}\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(1,0\right)$, then $Tx=\left\{\frac{1}{2}\right\}$, $Ty=\left\{0,1\right\}$, and ${P}_{Ty}\left(\frac{1}{2}\right)=\left\{0,1\right\}$ and $\left(\frac{1}{2},0\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n+1}}\right)$ for all $nâˆˆ\mathbb{N}$, then $Tx=\left\{\frac{1}{{2}^{n+1}},1\right\}$ and $Ty=\left\{\frac{1}{{2}^{n+2}},1\right\}$. It is easy to see that ${P}_{Ty}\left(\frac{1}{{2}^{n+1}}\right)=\left\{\frac{1}{{2}^{n+2}}\right\}$, ${P}_{Ty}\left(1\right)=\left\{1\right\}$, and $\left(\frac{1}{{2}^{n+1}},\frac{1}{{2}^{n+2}}\right),\left(1,1\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(\frac{1}{{2}^{n}},0\right)$ for all $nâˆˆ\mathbb{N}$, then $Tx=\left\{\frac{1}{{2}^{n+1}},1\right\}$ and $Ty=\left\{0,1\right\}$. We have ${P}_{Ty}\left(\frac{1}{{2}^{n+1}}\right)=\left\{0\right\}$, ${P}_{Ty}\left(1\right)=\left\{1\right\}$, and $\left(\frac{1}{{2}^{n+1}},0\right),\left(1,1\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(0,0\right)$ then $Tx=\left\{0,1\right\}=Ty$. We have ${P}_{Ty}\left(0\right)=\left\{0\right\}$, ${P}_{Ty}\left(1\right)=\left\{1\right\}$, and $\left(0,0\right),\left(1,1\right)âˆˆE\left(G\right)$.

If $\left(x,y\right)=\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n}}\right)$, then $Tx=Ty=\left\{\frac{1}{{2}^{n+1}},1\right\}$ which obviously implies that ${P}_{Ty}\left(\frac{1}{{2}^{n+1}}\right)=\left\{\frac{1}{{2}^{n+1}}\right\}$, ${P}_{Ty}\left(1\right)=\left\{1\right\}$ and $\left(\frac{1}{{2}^{n+1}},\frac{1}{{2}^{n+1}}\right),\left(1,1\right)âˆˆE\left(G\right)$. So, T is weak graph preserving. Next, we can see that $\frac{1}{{2}^{n+1}}âˆˆT\left(\frac{1}{{2}^{n}}\right)$ and $\left(\frac{1}{{2}^{n}},\frac{1}{{2}^{n+1}}\right)âˆˆE\left(G\right)$. So the condition (2) of Theorem 3.1 is satisfied. Also, it is obvious that the condition (3) of Theorem 3.1 is satisfied. Thus all conditions of Theorem 3.1 are obtained. Therefore we can conclude that T has a fixed point and the fixed point set $Fix\left(T\right)=\left\{0\right\}$.

(2) T is not graph preserving since $\frac{1}{2}âˆˆT1$, $1âˆˆT0$ but $\left(\frac{1}{2},1\right)âˆ‰E\left(G\right)$.

(3) T is not a multi-valued contraction in the sense of Nicolae, Oâ€™Regan, and Petrusel since $\left(1,0\right)âˆˆE\left(G\right)$, $\frac{1}{2}âˆˆT1$, $1âˆˆT0$ and $d\left(\frac{1}{2},1\right)<\mathrm{Î±}d\left(1,0\right)$ with $\mathrm{Î±}>\frac{1}{2}$ but $\left(\frac{1}{2},1\right)âˆ‰E\left(G\right)$.â€ƒâ–¡

Remark 3.8 As a consequence of Example 3.7, we can neither use Theorem 1.13 nor Theorem 1.8 to check whether or not T has a fixed point.

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## Acknowledgements

This paper was supported by the Faculty of Science and Technology, Prince of Songkla University, Pattani Campus.

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Phon-on, A., Sama-Ae, A., Makaje, N. et al. Coincidence point theorems for weak graph preserving multi-valued mapping. Fixed Point Theory Appl 2014, 248 (2014). https://doi.org/10.1186/1687-1812-2014-248