R E S E A R C H
Open Access
Coincidence point theorems for weak graph
preserving multi-valued mapping
Aniruth Phon-on, Areeyuth Sama-Ae
*, Nifatamah Makaje, Pakwan Riyapan and Saofee Busaman
*Correspondence:
Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand
Abstract
In this paper, we prove some coincidence and fixed point theorems for a new type of multi-valued weakG-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
Keywords: fixed point theorem; multi-valued mapping; weak graph preserving; partially ordered set
1 Introduction
The classical contraction mapping principle of Banach states that if (X,d) is a complete metric space andf :X→Xis a contraction mapping,i.e.,d(f(x),f(y))≤αd(x,y) for all
x,y∈X, where α∈[, ), thenf 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 equa-tions, nonlinear differential equations in Banach spaces and to prove the convergence of algorithms in computational mathematics. Because of its usefulness for mathematical the-ory, Banach fixed point theorem has been extended in many directions; see [–]. 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 (X,d), we letCB(X) andComp(X) be the set of all nonempty closed bounded subsets and the set of all nonempty compact subsets ofX, respectively. A pointxinXis a fixed point of a multi-valued mappingT:X→Xifxis inTx.
Nadler [] 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 []. 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 [].
Theorem . Let(X,d)be a complete metric space and T:X→CB(X).Assume that there exists k∈[, )such that
H(Tx,Ty)≤kd(x,y)
for all x,y∈X.Then there exists z∈X such that z∈Tz.
Reich [] extended Nadler’s fixed point theorem as follows.
Theorem . Let(X,d)be a complete metric space and T:X→Comp(X).Assume that there exists a functionϕ: [,∞)→[, )such that
lim sup
r→t+
ϕ(r) <
for each t∈(,∞)and
H(Tx,Ty)≤ϕd(x,y)d(x,y)
for all x,y∈X.Then there exists z∈X such that z∈Tz.
The multi-valued mappingT studied by Reich [] in Theorem . has compact value, that is,Txis a nonempty compact subset ofXfor allxin the spacesX. In , Mizoguchi and Takahashi [] relaxed the compactness assumption on the mapping to closed and bounded subsets ofX. They proved the following theorem as a generalization of Nadler’s theorem.
Theorem . Let(X,d)be a complete metric space and T:X→CB(X).Assume that there exists a functionϕ: [,∞)→[, )such that
lim sup
r→t+
ϕ(r) <
for each t∈[,∞)and
H(Tx,Ty)≤ϕd(x,y)d(x,y)
for all x,y∈X.Then there exists z∈X such that z∈Tz.
In , Berinde and Berinde [] gave the definition of a multi-valued weak contraction stated as follows.
Definition . Let (X,d) be a metric space andT:X→CB(X) a multi-valued mapping.
T is said to be amulti-valued weak contractionor amulti-valued(θ,L)-weak contraction
if there exist two constantsθ∈(, ) andL≥ such that
H(Tx,Ty)≤θd(x,y) +Ld(y,Tx)
Then they extended Theorem . 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 . Let (X,d) be a metric space andT:X→CB(X) a multi-valued mapping.T
is said to be a generalized multi-valued (α,L)-weak contraction if there exist a nonnegative number Land a function α: [,∞)→[, ) satisfyinglim supr→t+α(r) < for eacht∈
[,∞) such that
H(Tx,Ty)≤αd(x,y)d(x,y) +Ld(y,Tx)
for allx,y∈X.
They showed that in a complete metric space, every generalized multi-valued (α,L )-weak contraction has a fixed point.
In , Jachymski [] introduced the concept of ‘contraction concerning a graph’, calledG-contraction and proved some fixed point results ofG-contraction in a complete metric space endowed with a graph.
Definition . Let (X,d) be a metric space andG= (V(G),E(G)) a directed graph such thatV(G) =XandE(G) contains all loops,i.e.,={(x,x)|x∈X} ⊂E(G). We say that a mappingf:X→Xis aG-contractioniff preserves edges ofG,i.e., for everyx,y∈X,
(x,y)∈E(G) ⇒ f(x),f(y)∈E(G) ()
and there existsα∈(, ) such thatx,y∈X,
(x,y)∈E(G) ⇒ df(x),f(y)≤αd(x,y).
The mappingf :X→Xsatisfying condition () is also called agraph-preserving map-ping. Jachymski showed in [] that under some properties onX, aG-contractionf :X→ Xhas a fixed point if and only if there existsx∈Xsuch that (x,f(x))∈E(G).
Recently, Beg and Butt [] introduced the concept of ‘G-contraction’ for a multi-valued mappingT:X→CB(X) defined as follows.
Definition . LetT:X→CB(X) be a multi-valued mapping. The mappingTis said to be aG-contractionif there existsk∈(, ) such that
H(Tx,Ty)≤kd(x,y)
for all (x,y)∈E(G) and ifu∈Txandv∈Tyare such that
d(u,v)≤kd(x,y) +α
They showed that if (X,d) is a complete metric space and (X,d) has Property A [], then
G-contraction mappingT:X→CB(X) has a fixed point if and only if there existx∈X
andy∈Txsuch that (x,y)∈E(G).
In , Nicolae, O’Regan, and Petrusel [] extended the notion of multi-valued con-traction on a metric space with a graph in considering the fixed point shown below.
Theorem . Let F:X→X be a multi-valued map with nonempty closed values.Assume that
() there existsλ∈(, )such thatD(F(x),F(y))≤λd(x,y)for all(x,y)∈E(G);
() for each(x,y)∈E(G),eachu∈F(x)andv∈F(y)satisfyingd(u,v)≤ad(x,y)for some a∈(, ),(u,v)∈E(G)holds;
() Xhas PropertyA.
If there exist x,x∈X such that x∈[x]G∩F(x),then F has a fixed point.
In , Dinevari and Frigon [] introduced a concept of ‘G-contraction’ which is weaker than that of Beg and Butt [] and weaker than that of Nicolae, O’Regan, and Petrusel [].
Definition . LetT:X→X be a map with nonempty values. We say thatT is a G-contraction(in the sense of Dinevari and Frigon) if there existsα∈(, ) such that for all (x,y)∈E(G) and allu∈Tx, there existsv∈Tysuch that (u,v)∈E(G) andd(u,v)≤αd(x,y).
They showed that under some properties, weaker than Property A, on a metric space a multi-valuedG-contraction with the closed value has a fixed point.
Most recently, Tiammee and Suantai [] introduced the concept of ‘graph preserving’ for multi-valued mappings and proved their fixed point theorem in a complete metric space endowed with a graph.
Definition .[] LetXbe a nonempty set,G= (V(G),E(G)) a directed graph such that
V(G) =X, andT:X→CB(X).Tis said to begraph preservingif
(x,y)∈E(G) ⇒ (u,v)∈E(G)
for allu∈Txandv∈Ty.
Definition .[] Let Xbe a nonempty set,G= (V(G),E(G)) a directed graph such
thatV(G) =X,g:X→X, andT:X→CB(X).Tis said to beg-graph preservingif for any
x,y∈X, such that
g(x),g(y)∈E(G) ⇒ (u,v)∈E(G)
for allu∈Txandv∈Ty.
Definition . Let (X,d) be a metric space, G= (V(G),E(G)) a directed graph such that V(G) =X, g :X→X, andT :X→CB(X). T is said to be a multi-valued weak
α: [,∞)→[, ) satisfying
lim sup
r→t+ α(r) <
for everyt∈[,∞) and a nonnegative numberLwith
H(Tx,Ty)≤αdg(x),g(y)dg(x),g(y)+LDg(y),Tx
for allx,y∈Xsuch that (g(x),g(y))∈E(G).
Theorem . [] Let(X,d)be a complete metric space, G= (V(G),E(G))a directed graph such that V(G) =X,and g:X→X a surjective mapping.If T :X→CB(X)is a multi-valued mapping satisfying the following properties:
() Tis ag-graph preserving mapping;
() there existsx∈Xsuch that(g(x),y)∈E(G)for somey∈Tx;
() Xhas PropertyA;
() Tis a(g,α,L)-G-contraction,
then there exists u∈X such that g(u)∈Tu.
The condition ofT in Definition . to beg-graph preserving requires all pairs (u,v) whereu∈Txandv∈Tyhave connecting edges whenever (g(x),g(y))∈E(G). With some modification, we are interested in proposing the new concept of ‘g-graph preserving’ for multi-valued mappings in a complete metric space endowed with a graph and the fixed point theorem is also determined.
2 Preliminaries
Let (X,d) be a metric space. Forx∈XandA,B∈Comp(X), define
d(x,A) =infd(x,y)|y∈A,
D(A,B) =infd(x,B)|x∈A.
For eacha∈A, define
PB(a) =
b∈B|d(a,b) =d(a,B).
Each element inPB(a) is called aprojection pointofaintoB. Note that ifBis compact,
thenPB(a) is always a nonempty set. Also, define
H(A,B) =max
sup
x∈B
d(x,A),sup
x∈A
d(x,B)
.
The mappingHis said to be aHausdorff metricinduced byd. The next two lemmas will play central roles in our main results.
Lemma .[] Let(X,d)be a metric space.If A,B∈Comp(X) (or CB(X))and x∈A,then for each> ,there is b∈B such that
Lemma .[] Let(X,d)be a metric space,{Ak} ⊂Comp(X) (or CB(X)),{xk}a sequence
in X such that xk∈Ak–,andα: [,∞)→[, )a function satisfyinglim supr→t+α(r) < for every t∈[,∞).Suppose that d(xk–,xk)is a non-increasing sequence such that
H(Ak–,Ak)≤α
d(xk–,xk)
d(xk–,xk),
d(xk,xk+)≤H(Ak–,Ak) +
αd(xk–,xk)
nk
,
where{nk}is an increasing sequence and k,nk∈N.Then{xk}is a Cauchy sequence in X.
Next we will give notions and examples of new types of multi-valued mapping with com-pact value which are weaker than that of Tiammee and Suantai [].
Definition . LetXbe a nonempty set andG= (V(G),E(G)) be a directed graph such
thatV(G) =X, andT:X→Comp(X).T is said to beweak graph preservingif it satisfies the following:
for eachx,y∈X, if(x,y)∈E(G), then for eachu∈Txthere isv∈PTy(u)such that
(u,v)∈E(G).
In Example ., we illustrate a mappingTwhich is weak graph preserving but not graph preserving.
Example . LetNbe a metric space with the usual metric andG= (N,E(G)) where
E(G) =(n– , n) :n∈N∪(n, ) :n∈N.
DefineT:N→Comp(X) by
Tx=
⎧ ⎨ ⎩
{k, k+ } ifx= k– ,k∈N; {} ifx= k,k∈N.
We will show thatT is weak graph preserving. If (x,y) = (k– , k) wherek∈N, then Tx={k, k+ }andTy={}. We can see thatPTy(k) ={}=PTy(k+ ) and (k, ), (k+
, )∈E(G).
If (x,y) = (k, ) wherek∈N, thenTx={}andTy={, }. We can see thatPTy() ={}
and (, )∈E(G). It is easy to see that (, ) /∈E(G) and soTis 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 [].
Example . LetX={, , , , , } be a metric space with the usual metric and G=
(X,E(G)) where
DefineT:X→Comp(X) by
Tx=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
{, } ifx= , ;
{, } ifx= ;
{} ifx= , , .
We will show thatT is weak graph preserving. If (x,y) = (, ) thenTx={, }andTy=
{, }. We can see thatPTy() ={},PTy() ={}and (, ), (, )∈E(G).
If (x,y) = (, ) thenTx={, }andTy={, }. We can see thatPTy() ={},PTy() ={},
and (, ), (, )∈E(G).
If (x,y) = (, ) or (x,y) = (, ) thenTx={}andTy={}. We can see thatPTy() ={}
and (, )∈E(G).
If (x,y) = (, ) or (x,y) = (, ) thenTx={, }andTy={}. We can see thatPTy() =
{}=PTy() and (, ), (, )∈E(G).
So,T is weak graph preserving but it is not aG-contraction in the sense of Dinevari and Frigon since for eachu∈Txandv∈Tywith (x,y) = (, ),d(u,v)≥ >αd(, ) for all
α∈(, ).
Definition . LetX be a nonempty set,G= (V(G),E(G)) a directed graph such that
V(G) =X,T:X→Comp(X), andg:X→X.T is said to beweak g-graph preservingif it satisfies the following:
for eachx,y∈X, if(g(x),g(y))∈E(G), then for eachu∈Txthere isv∈PTy(u)such
that(u,v)∈E(G).
Example . LetNbe a metric space with the usual metric,G= (N,E(G)) where
E(G) =(n– , n) :n∈N∪(n, ) :n∈N.
DefineT:N→Comp(X) by
Tx=
⎧ ⎨ ⎩
{k, k+ } ifx= k– ,k∈N;
{} ifx= k,k∈N,
andg:X→Xby
g(x) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
k ifx= k+ ,k∈N;
k– ifx= k+ ,k∈N;
ifx= ;
ifx= .
We will show thatTis weakg-graph preserving.
If (g(x),g(y)) = (, ), then (x,y) = (, ) andTx={, } andTy={}. We can see that
PTy() ={}=PTy() and (, ), (, )∈E(G).
If (g(x),g(y)) = (, ), then (x,y) = (, ) andTx={}andTy={, }. We can see that
If (g(x),g(y)) = (k– , k) wherek∈N\ {}, then (x,y) = (k+ , k+ ) andTx={k+ , k+ }andTy={}. We can see thatPTy(k+ ) ={}=PTy(k+ ) and (k+ , ), (k+
, )∈E(G).
If (g(x),g(y)) = (k, ) wherek∈N\ {}, then (x,y) = (k+ , ) andTx={}andTy=
{, }. We can see thatPTy() ={}and (, )∈E(G). HenceT is weakg-graph preserving.
3 Main results
We first recall Property A before the main theorem is proved.
Property A For any sequence{xn}inX, ifxn→xand (xn,xn+)∈E(G) forn∈N, then
there is a subsequence{xnk}such that (xnk,x)∈E(G) fork∈N.
Theorem . Let(X,d)be a complete metric space, G= (V(G),E(G))a directed graph such that V(G) =X,and g:X→X a surjective map.If T:X→Comp(X)is a multi-valued mapping satisfying the following properties:
() Tis weakg-graph preserving;
() XT={x∈X|(g(x),y)∈E(G)for somey∈Tx} =∅;
() Xhas PropertyA;
() Tis a(g,α,L)-G-contraction,
then there exists u∈X such that g(u)∈Tu.
Proof By (), letx∈XT. Then there existsy∈Txsuch that (g(x),y)∈E(G). Sincegis
surjective, there existsx∈Xsuch thaty=g(x). Thus we have (g(x),g(x))∈E(G). Since
[α(d(g(x),g(x)))]n→, there existsn∈Nsuch that
αdg(x),g(x)
n≤
–αdg(x),g(x)
dg(x),g(x)
.
SinceTx is compact, it follows thatPTx(g(x))=∅. SinceT is weakg-graph
preserv-ing, there existsy∈PTx(g(x)) such that (g(x),y)∈E(G) andd(g(x),y) =d(g(x),Tx).
Again sinceg is surjective, there isx∈X such thatg(x) =y. By Lemma ., there is
y∈Txsuch that
dg(x),y
≤H(Tx,Tx) +
αdg(x),g(x)
n
.
It follows that
dg(x),g(x)
=dg(x),Tx
≤dg(x),y
≤H(Tx,Tx) +
αdg(x),g(x)
n
.
SinceTis a (g,α,L)-G-contraction and (g(x),g(x))∈E(G), we have
dg(x),g(x)
≤H(Tx,Tx) +
αdg(x),g(x)
n ≤αdg(x),g(x)
dg(x),g(x)
+LDg(x),Tx
+αdg(x),g(x)
≤αdg(x),g(x)
dg(x),g(x)
+αdg(x),g(x)
n ≤dg(x),g(x)
.
Next, since [α(d(g(x),g(x)))]n→, there existsn∈Nsuch thatn>nand
αdg(x),g(x)
n≤
–αdg(x),g(x)
dg(x),g(x)
.
SinceTxis compact, it follows thatPTx(g(x))=∅. SinceT is weakg-graph
preserv-ing, there existsy∈PTx(g(x)) such that (g(x),y)∈E(G) andd(g(x),y) =d(g(x),Tx).
Again sinceg is surjective, there isx∈X such thatg(x) =y. By Lemma ., there is
y∈Txsuch that
dg(x),y
≤H(Tx,Tx) +
αdg(x),g(x)
n
.
It follows that
dg(x),g(x)
=dg(x),Tx
≤dg(x),y
≤H(Tx,Tx) +
αdg(x),g(x)
n
.
SinceTis a (g,α,L)-G-contraction and (g(x),g(x))∈E(G), we have
dg(x),g(x)
≤H(Tx,Tx) +
αdg(x),g(x)
n ≤αdg(x),g(x)
dg(x),g(x)
+LDg(x),Tx
+αdg(x),g(x)
n ≤αdg(x),g(x)
dg(x),g(x)
+αdg(x),g(x)
n ≤dg(x),g(x)
.
Continuing in this process, we produce a sequence{g(xk)}inXand an increasing
se-quence{nk}inNsuch that for eachk∈N,g(xk+)∈Txk, (g(xk),g(xk+))∈E(G),
αdg(xk–),g(xk)
nk≤
–αdg(xk–),g(xk)
dg(xk–),g(xk)
,
H(Txk–,Txk)≤α
dg(xk–),g(xk)
dg(xk–),g(xk)
,
and
dg(xk),g(xk+)
≤H(Txk–,Txk) +
αdg(xk–),g(xk)
nk
.
We note that
dg(xk),g(xk+)
≤H(Txk–,Txk) +
αdg(xk–),g(xk)
nk ≤αdg(xk–),g(xk)
dg(xk–),g(xk)
+αdg(xk–),g(xk)
nk ≤dg(xk–),g(xk)
.
Hence,d(g(xk),g(xk+))≤d(g(xk–),g(xk)) and so{d(g(xk),g(xk+))}is a non-increasing
sequence. By Lemma .,{g(xk)}is a Cauchy sequence inX. SinceXis complete, the
se-quence{g(xk)}converges to a pointg(u) for someu∈X. By Property A in (), there is a
subsequence{g(xkn)}such that (g(xkn),g(u))∈E(G) for anyn∈N. Claim thatg(u)∈Tu.
Note that for eachg(xkn+),
Dg(u),Tu≤dg(u),g(xkn+)
+Dg(xkn+),Tu
≤dg(u),g(xkn+)
+H(Txkn,Tu) ≤dg(u),g(xkn+)
+αdg(xkn),g(u)
dg(xkn),g(u)
+LDg(u),Txkn
≤dg(u),g(xkn+)
+αdg(xkn),g(u)
dg(xkn),g(u)
+Ldg(u),g(xkn+)
.
Sinceg(xkn) converges to g(u) asn→ ∞, it follows thatD(g(u),Tu) = . SinceTuis
compact, we conclude thatg(u)∈Tu, completing the proof.
Remark . Theorem . is an extension of Theorem . in the case of a mappingT :
X→CB(X) having compact values.
A partial order is a binary relation≤over the setXwhich satisfies the following condi-tions:
() x≤x(reflexivity);
() ifx≤yandy≤x, thenx=y(antisymmetry); () ifx≤yandy≤z, thenx≤z(transitivity),
for allx,y∈X. A set with a partial order≤is called a partially ordered set. We writex<y
ifx≤yandx=y.
Definition . Let (X,≤) be a partially ordered set. For eachA,B⊂X,A≺Bifa≤bfor anya∈A,b∈B.
Definition .[] Let (X,d) be a metric space endowed with a partial order≤,g:X→X
a surjective map, andT:X→Comp(X).T is said to beg-increasingif for anyx,y∈X,
g(x) <g(y) ⇒ Tx≺Ty.
In the casegis the identity map, the mappingTis called an increasing mapping.
Corollary . Let(X,d)be a metric space endowed with a partial order≤,g:X→X a surjective map and T:X→Comp(X)a multi-valued mapping.Suppose that
() Tisg-increasing;
() for each sequence{xk}such thatg(xk) <g(xk+)for allk∈Nandg(xk)converges to
g(x)for somex∈X,theng(xk) <g(x)for allk∈N;
() there exist a functionα: [,∞)→[, )satisfyinglim supr→t+α(r) < for every t∈[,∞)and a nonnegative numberLwith
H(Tx,Ty)≤αdg(x),g(y)dg(x),g(y)+LDg(y),Tx
for allx,y∈Xsuch thatg(x) <g(y); () the metricdis complete.
Then there exists u∈X such that g(u)∈Tu.
Proof DefineG= (V(G),E(G)) whereV(G) =XandE(G) ={(x,y)|x<y}. Letx,y∈Xbe such that (g(x),g(y))∈E(G). Theng(x) <g(y) so by () it implies thatTx≺Ty. For eachu∈ Tx,v∈Ty,u<v. SinceTyis compact, it follows thatPTy(x)=∅andPTy(x)⊂Tyfor allx∈
Tx. Thus (x,v)∈E(G) for allv∈PTy(x) and allx∈Tx. That is,Tis weakg-graph preserving.
By assumption (), there existx∈Xandu∈Txsuch thatg(x) <u. So (g(x),u)∈E(G)
and hence the condition () in Theorem . is satisfied. Moreover, the conditions () and () in Theorem . are also satisfied. Therefore the result of this corollary is followed by
Theorem ..
Theorem . Let(X,d)be a metric space endowed with a partial order≤and T:X→
Comp(X)a multi-valued mapping.Suppose that
() Tis increasing;
() there existx∈Xandu∈Txsuch thatx<u;
() for each sequence{xk}such thatxk<xk+for allk∈Nandxkconverges toxfor some
x∈X,thenxk<xfor allk∈N;
() there exist a functionα: [,∞)→[, )satisfyinglim supr→t+α(r) < for every t∈[,∞)and a nonnegative numberLwith
H(Tx,Ty)≤αd(x,y)d(x,y) +LD(y,Tx)
for allx,y∈Xwithx<y; () the metricdis complete.
Then there exists u∈X such that u∈Tu.Furthermore,ifsup{α(r) :r∈[,∞)}< and L< –sup{α(r) :r∈[,∞)},then T has a unique fixed point.
Proof Settingg(x) =x, by Corollary . we haveFix(T)=∅. With conditionssup{α(r) :r∈
[,∞)}< andL< –sup{α(r) :r∈[,∞)}, we will show thatT has a unique fixed point. Letu,v∈Fix(T). Suppose to a contrary thatu=v. Without loss of generality, assume that
u<v. By the condition (), we have
d(u,v)≤H(Tu,Tv)
≤αd(u,v)d(u,v) +LD(v,Tu)
≤supα(r) :r∈[,∞)d(u,v) +Ld(u,v)
Sincesup{α(r) :r∈[,∞)}+L< , this yieldsd(u,v) <d(u,v), a contradiction. Therefore
u=v, which implies thatThas a unique fixed point.
Next we give an example such that T has a unique fixed point but T is neither a graph-preserving nor a multi-valuedG-contraction in the sense of Nicolae, O’Regan, and Petrusel [].
Example . LetX={} ∪ {n :n∈N∪ {}}be a metric space with the usual metricd.
Consider the directed graph defined byV(G) =Xand
E(G) =
n,
n+
,
n,
:n∈N∪ {}
∪,
whereis the diagonal inX×X. LetT:X→Comp(X) be defined by
Tx= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
{, } ifx= ;
{
n+, } ifx=n,n∈N; {
} ifx= .
Then:
() Thas a fixed point;
() Tis not graph preserving as defined by Tiammee and Suantai [];
() Tis not aG-contraction in the sense of Nicolae, O’Regan, and Petrusel [].
Proof () We will show thatT is a (α,g,L)-G-contraction withα≥ ,L≥, andg=iX.
Let (x,y)∈E(G).
If (x,y) = (,), thenTx={},Ty={, }, and
H(Tx,Ty) = =
d(x,y)≤α
d(x,y)d(x,y) +LD(y,Tx).
If (x,y) = (, ), thenTx={},Ty={, }, and
H(Tx,Ty) = =
d(x,y)≤α
d(x,y)d(x,y) +LD(y,Tx).
If (x,y) = (n,n+) for alln∈N, thenTx={n+, }andTy={n+, }. It is easy to check
that
H(Tx,Ty) =max
sup
a∈Ty
d(a,Tx),sup
b∈Tx
d(b,Ty)
=d
n+,
n+
= d
n,
n+
= d(x,y)
If (x,y) = (n, ) for alln∈N, thenTx={n+, }andTy={, }. We have
H(Tx,Ty) =max
sup
a∈Ty
d(a,Tx),sup
b∈Tx
d(b,Ty)
=
n+
= d
n,
= d(x,y)
≤αd(x,y)d(x,y) +LD(y,Tx).
If (x,y) = (, ) and (x,y) = (n,n), thenTx=Tywhich obviously implies thatH(Tx,Ty) =
. So,
H(Tx,Ty)≤αd(x,y)d(x,y) +LD(y,Tx).
HenceTis a (α,g,L)-G-contraction. Next, we will show thatTis weak graph preserving. Let (x,y)∈E(G).
If (x,y) = (,), thenTx={},Ty={, }, andPTy() ={}and (,)∈E(G).
If (x,y) = (, ), thenTx={},Ty={, }, andPTy() ={, }and (, )∈E(G).
If (x,y) = (n,n+) for alln∈N, thenTx={n+, }andTy={n+, }. It is easy to see that PTy(n+) ={n+},PTy() ={}, and (n+,n+), (, )∈E(G).
If (x,y) = (
n, ) for alln∈N, thenTx={n+, }andTy={, }. We havePTy(n+) ={}, PTy() ={}, and (n+, ), (, )∈E(G).
If (x,y) = (, ) thenTx={, }=Ty. We havePTy() ={},PTy() ={}, and (, ), (, )∈
E(G).
If (x,y) = (n,n), thenTx=Ty={n+, }which obviously implies thatPTy(n+) ={n+}, PTy() ={}and (n+,n+), (, )∈E(G). So,T is weak graph preserving. Next, we can see
that n+ ∈T(n) and (n,n+)∈E(G). So the condition () of Theorem . is satisfied.
Also, it is obvious that the condition () of Theorem . is satisfied. Thus all conditions of Theorem . are obtained. Therefore we can conclude thatThas a fixed point and the fixed point setFix(T) ={}.
()Tis not graph preserving since
∈T, ∈T but (
, ) /∈E(G).
()T is not a multi-valued contraction in the sense of Nicolae, O’Regan, and Petrusel since (, )∈E(G),∈T, ∈T andd(, ) <αd(, ) withα> but (, ) /∈E(G).
Remark . As a consequence of Example ., we can neither use Theorem . nor The-orem . to check whether or notThas a fixed point.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Acknowledgements
This paper was supported by the Faculty of Science and Technology, Prince of Songkla University, Pattani Campus.
Received: 20 July 2014 Accepted: 8 December 2014 Published:18 Dec 2014
References
1. Bojor, F: Fixed point theorems for Reich type contraction on metric spaces with a graph. Nonlinear Anal.75, 3895-3901 (2012)
2. Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. J. Math. Anal. Appl.29, 531-536 (2002)
3. Browder, FE: The fixed point theory of multi-valued mapping in topological spaces. Math. Ann.177, 283-301 (1968) 4. Du, WS: New cone fixed point theorems for nonlinear multi-valued maps with applications. Appl. Math. Lett.24(2),
172-178 (2011)
5. Feng, Y, Liu, S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl.317, 103-112 (2006)
6. Khaojasten, F, Abbas, M, Costache, S: Two new types of fixed point theorems in complete metric spaces. Abstr. Appl. Anal.2014, Article ID 325840 (2014)
7. Laza, T, Mot, G, Petrusel, G, Szentesi, S: The theory of Reich’s fixed point theorem for multi-valued operators. Fixed Point Theory Appl. (2010). doi:10.1155/2010/178421
8. Minak, G, Acar, O, Altun, I: Multi-valued pseudo-Picard operators and fixed point results. J. Funct. Spaces Appl.2013, Article ID 827458 (2013)
9. Petrusel, A, Petrusel, G: Selection theorems for multi-valued generalized contractions. Math. Morav.9, 43-52 (2005) 10. Sitthithakerngkiet, K, Plubtieng, S: Existence solutions of vector equilibrium problems and fixed point of multi-valued
mappings. Abstr. Appl. Anal.2013, Article ID 952021 (2013)
11. Ume, JS, Lee, BS, Cho, SJ: Some results on fixed point theorems for multi-valued mappings in complete metric spaces. Int. J. Math. Math. Sci.30(6), 319-325 (2002)
12. Nadler, SB: Multi-valued contraction mappings. Pac. J. Math.30, 475-488 (1969)
13. Kirk, WA, Massa, S: Remarks on asymptotic and Chebyshev centers. Houst. J. Math.16, 465-468 (1990) 14. Reich, S: Fixed points of contractive functions. Boll. Unione Mat. Ital.5, 26-42 (1972)
15. Mizoguchi, N, Takahashi, W: Fixed point theorems for multi-valued mapping on complete metric spaces. J. Math. Anal. Appl.141, 177-188 (1989)
16. Berinde, M, Berinde, V: On a general class multi-valued weakly Picard mappings. J. Math. Anal. Appl.326, 772-782 (2007)
17. Jachymski, J: Fixed the contraction principle for mappings on a metric with a graph. Proc. Am. Math. Soc.139, 1359-1373 (2008)
18. Beg, I, Butt, AR: Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal.71, 3699-3704 (2009)
19. Nicolae, A, O’Regan, D, Petrusel, A: Fixed point theorems for single-valued and multi-valued generalized contractions in metric spaces endowed with a graph. Georgian Math. J.18, 307-327 (2011)
20. Dinevari, T, Frigon, T: Fixed point results for multi-valued contractions on a metric space with a graph. J. Math. Anal. Appl.405, 507-517 (2013)
21. Tiammee, J, Suantai, S: Coincidence point theorems for graph-preserving multi-valued mappings. Fixed Point Theory Appl. (2014). doi:10.1186/1687-1812-2014-70
10.1186/1687-1812-2014-248
