Generalized fixed point theorems for multi valued α ψ contractive mappings

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R E S E A R C H

Open Access

Generalized ﬁxed point theorems for

multi-valued

α

-ψ

-contractive mappings

Nawab Hussain

1*

_{, Jamshaid Ahmad}

2

_{and Akbar Azam}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

The aim of this paper is to establish certain new ﬁxed point results for multi-valued as well as single-valued maps satisfying an

α

-

ψ

-contractive conditions in complete metric space. As an application, we derive some new ﬁxed point theorems for

ψ

-graphic contractions deﬁned on a metric space endowed with a graph as well as an ordered metric space. The presented results complement and extend some very recent results proved by Aslet al.(Fixed Point Theory Appl. 2012:212, 2012) and Samet

et al.(Nonlinear Anal. 75:2154-2165, 2012) as well as other theorems given by Hussain

et al.(Fixed Point Theory Appl. 2013:212, 2013). Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature.

MSC: 46S40; 47H10; 54H25

Keywords: metric space; ﬁxed point;

α

-

ψ

-contraction

1 Introduction

In metric fixed point theory the contractive conditions on underlying functions play an im-portant role for finding solutions of fixed point problems. The Banach contraction princi-ple [] is a fundamental result in metric fixed point theory. Over the years, it has been gen-eralized in different directions by several mathematicians (see [–]). In particular, there has been a number of studies involving altering distance functions which alter the distance between two points in a metric space. In , Sametet al.[] introduced the concepts of α-ψ-contractive andα-admissible mappings and established various fixed point theorems

for such mappings in complete metric spaces.

Denote withthe family of nondecreasing functionsψ: [, +∞)→[, +∞) such that

∞

n=ψn(t) < +∞for allt> , whereψnis thenth iterate ofψ.

The following lemma is well known.

Lemma  Ifψ∈,then the following hold:

(i) (ψn(t))n∈Nconverges toasn→ ∞for allt∈(, +∞);

(ii) ψ(t) <tfor allt> ; (iii) ψ(t) = iﬀt= .

Sametet al.[] deﬁned the notion ofα-admissible mappings as follows.

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Deﬁnition  LetT be a self-mapping onXandα:X×X→[, +∞) be a function. We say thatTis aα-admissible mapping if

x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.

Theorem [] Let(X,d)be a complete metric space and T beα-admissible mapping.

Assume that

α(x,y)d(Tx,Ty)≤ψd(x,y) (.)

for all x,y∈X,whereψ∈.Also,suppose that (i) there existsx∈Xsuch thatα(x,Tx)≥;

(ii) eitherTis continuous or for any sequence{xn}inXwithα(xn,xn+)≥for alln∈N

andxn→xasn→+∞,we haveα(xn,x)≥for alln∈N.

Then T has a ﬁxed point.

Afterwards, Aslet al.[] generalized these notions by introducing the concepts ofα∗ -ψ-contractive multifunctions, and ofα∗-admissibility, and they obtained some ﬁxed point results for these multifunctions.

Deﬁnition [] Let (X,d) be a metric space,T:X→Xbe a given closed-valued mul-tifunction. We say thatTis calledα∗-ψ-contractive multifunction if there exist two func-tionsα:X×X→[, +∞) andψ∈such that

α∗(Tx,Ty)H(Tx,Ty)≤ψd(x,y)

for allx,y∈X, whereH is the Hausdorﬀ generalized metric,α_∗(A,B) =inf{α(a,b) :a∈ A,b∈B}and X_{denotes the family of all nonempty subsets of}_X_.

Deﬁnition [] Let (X,d) be a metric space,T:X→X_{be a given closed-valued} mul-tifunction and α:X×X→[, +∞). We say that T is called α_∗-admissible whenever α(x,y)≥ implies thatα_∗(Tx,Ty)≥.

Very recently Hussain et al. [] modiﬁed the notions of α∗-admissible and α∗-ψ -contractive mappings as follows:

Deﬁnition  LetT :X→X be a multifunction,α,η:X×X→R+ be two functions

whereηis bounded. We say thatTisα∗-admissible mapping with respect toηif α(x,y)≥η(x,y) implies α∗(Tx,Ty)≥η∗(Tx,Ty), x,y∈X,

where

α∗(A,B) = inf

x∈A,y∈Bα(x,y) and η∗(A,B) =_x_∈sup_A_,_y_∈_Bη(x,y).

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Hussainet al.[] proved following generalization of the above mentioned results of [].

Theorem  Let(X,d)be a complete metric space and T:X→X_{be a}_α

∗-admissible with

respect toηand the closed-valued multifunction on X.Assume that forψ∈,

∀x,y∈X, α_∗(Tx,Ty)≥η_∗(Tx,Ty) ⇒ H(Tx,Ty)≤ψd(x,y). (.)

Also suppose that the following assertions hold:

(i) there existx∈Xandx∈Txsuch thatα(x,x)≥η(x,x);

(ii) for a sequence{xn} ⊂Xconverging tox∈Xandα(xn,xn+)≥η(xn,xn+)for all n∈N,we haveα(xn,x)≥η(xn,x)for alln∈N.

For more details onα-ψ-contractions and ﬁxed point theory, we refer the reader to [, , , , , , , –].

The aim of this paper is to unify the concepts ofα-ψ-contractive type mappings and establish some new ﬁxed point theorems in complete metric spaces for such mappings.

Let (X,d) be a complete metric space,x∈Xandr> . We denote byB(x,r) ={x∈X: d(x,x) <r}the open ball with centerxand radiusrand byB(x,r) ={x∈X:d(x,x)≤r}

the closed ball with centerxand radiusr.

The following lemmas of Nadler will be needed in the sequel.

Lemma [] Let A and B be nonempty,closed and bounded subsets of a metric space

(X,d)and <h∈R.Then,for every b∈B,there exists a∈A such that d(a,b)≤H(A,B) +h.

Lemma [] Let(X,d)be a metric space and B be nonempty,closed subsets of X and q> .

Then,for each x∈X with d(x,B) > and q> ,there exists b∈B such that d(x,b) <qd(x,B).

2 Main result

The following result, regarding the existence of the ﬁxed point of the mapping satisfying anα-ψ-contractive condition on the closed ball, is very useful in the sense that it requires the contractiveness of the mapping only on the closed ball instead of the whole space.

Theorem  Let(X,d)be a complete metric space and T:X→X_{be an}_α

∗-admissible

and closed-valued multifunction on X.Assume that forψ∈,

α∗(Tx,Ty)H(Tx,Ty)≤ψd(x,y) (.)

for all x,y∈B(x,r)and for x∈X,there exists x∈Txsuch that

n

i=

ψid(x,x)

<r (.)

for all n∈Nand r> .Also suppose that the following assertions hold:

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(ii) for a sequence{xn}inB(x,r)converging tox∈B(x,r)andα(xn,xn+)≥for all n∈N,we haveα(xn,x)≥for alln∈N.

Proof Sinceα(x,x)≥ andTisα∗-admissible, soα∗(Tx,Tx)≥. From (.), we get

d(x,x) <

n

i=

ψid(x,x)

<r.

It follows that

x∈B(x,r).

Ifx=x, then

α∗(Tx,Tx)H(Tx,Tx)≤ψ

d(x,x)

= 

implies that

Tx=Tx,

and we have ﬁnished. Assume thatx=x. By Lemmas  and , we takex∈Txandh> 

ash=ψ₍_d₍_x

,x)). Then

 < d(x,x)≤H(Tx,Tx) +h

≤ψd(x,x)

+ψd(x,x)

=



i=

ψid(x,x)

.

Note thatx∈B(x,r), since

d(x,x)≤d(x,x) +d(x,x)

≤d(x,x) +ψ

d(x,x)

+ψd(x,x)

=



i=

ψid(x,x)

<r.

By repeating this process, we can construct a sequencexnof points inB(x,r) such that xn+∈Txn,xn=xn–,α(xn,xn+)≥ with

d(xn,xn+)≤

n+

i=

ψid(x,x)

. (.)

Now, for eachn,m∈Nwithm>nusing the triangular inequality, we obtain

d(xn,xm)≤ m–

k=n

d(xk,xk+)≤

m

k=n

ψkd(x,x)

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Thus we proved that{xn}is a Cauchy sequence. SinceB(x,r) is closed. So there exists x∗∈B(x,r) such thatxn→x∗asn→ ∞. Now we prove thatx∗∈Tx∗. Sinceα(xn,x∗)≥ for allnandT isα_∗-admissible with respect toη, soα_∗(Txn,Tx∗)≥ for alln. Then

dx∗,Tx∗≤α∗Txn,Tx∗HTxn,Tx∗+dxn,x∗ ≤ψdxn,x∗+dxn,x∗

≤ψdxn,x∗+dxn,x∗. (.)

Taking the limit asn→ ∞in (.), we getd(x∗,Tx∗) = . Thusx∗∈Tx∗. Example  LetX= [,∞) andd(x,y) =|x–y|. Deﬁne the multi-valued mappingT:X→

X_by

Tx=

[,x_] ifx∈[, ], [_x,_x] ifx∈(,∞).

Considering,x=_andx=_,r=_, thenB(x,r) = [, ] and

α(x,y) =

 ifx,y∈[, ],



 otherwise.

ClearlyT is anα-ψ-contractive mapping withψ(t) = _t. Now

d(x,x) =

 , n

i=

ψnd(x,x)

=

 n

i=

 n< 

 

=

=r.

We prove that all the conditions of our Theorem  are satisﬁed only forx,y∈B(x,r).

Without loss of generality, we suppose thatx≤y. The contractive condition of theorem is trivial for the case whenx=y. So we suppose thatx<y. Then

α_∗(Tx,Ty)H(Tx,Ty) = 

|y–x|=ψ

d(x,y).

Putx=_ andx=_. Thenα(x,x)≥. ThenThas a ﬁxed point .

Now we prove that the contractive condition is not satisﬁed forx,y∈/B(x,r). We

sup-posex=_andy= , then

α∗(Tx,Ty)H(Tx,Ty) =

 ≥  =ψ

d(x,y).

Similarly we can deduce the following corollaries.

Corollary  Let(X,d)be a complete metric space and T:X→X_{be an}_α

∗-admissible

and closed-valued multifunction on X.Assume that forψ∈,we have

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for all x,y∈B(x,r)and for x∈X,there exists x∈Txsuch that

n

i=

ψid(x,x)

<r

(i) α(x,x)≥forx∈Xandx∈Tx;

Corollary  Let(X,d)be a complete metric space and T:X→Xbe anα∗-admissible and closed-valued multifunction on X.Assume that forψ∈,we have

H(Tx,Ty) +lα∗(Tx,Ty)≤ψd(x,y)+l

for all x,y∈B(x,r)and l> and for x∈X,there exists x∈Txsuch that

n

i=

ψid(x,x)

<r

Theorem  Let(X,d)be a complete metric space and T:X→X be anα∗-admissible and closed-valued multifunction on X.Assume that forψ∈,we have

α∗(Tx,Ty)H(Tx,Ty)≤ψ

max d(x,y),d(x,Tx),d(y,Ty),d(x,Tx)d(y,Ty)  +d(x,y)

(.)

for all x,y∈X.Also suppose that the following assertions hold:

(i) there existx∈Xandx∈Txwithα(x,x)≥;

(ii) for a sequence{xn}inXconverging tox∈Xandα(xn,xn+)≥for alln∈N,we have

α(xn,x)≥for alln∈N. Then T has a ﬁxed point.

Proof Sinceα(x,x)≥ andT isα∗-admissible, soα∗(Tx,Tx)≥. Ifx=x, then we

have nothing to prove. Letx=x. Ifx∈Tx, thenxis a ﬁxed point ofT. Assume that x∈/Tx, then from (.), we get

 < d(x,Tx)

≤H(Tx,Tx)

≤ψ

max d(x,x),d(x,Tx),d(x,Tx),

d(x,Tx)d(x,Tx)

 +d(x,x)

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≤ψ

max d(x,x),d(x,x),d(x,Tx),

d(x,x)d(x,Tx)

 +d(x,x)

=ψmaxd(x,x),d(x,Tx)

.

Ifmax{d(x,Tx),d(x,x)}=d(x,Tx), thend(x,Tx)≤ψ(d(x,Tx)). Sinceψ(t) <tfor all t> . Then we get a contradiction. Hence, we obtainmax{d(x,Tx),d(x,x)}=d(x,x).

So

d(x,Tx)≤ψ

d(x,x)

.

Letq> , then from Lemma  we takex∈Txsuch that

d(x,x) <qd(x,Tx)≤qψ

d(x,x)

. (.)

It is clear that x=x. Put q =ψ(ψqψ(d((dx(,x,x))x))). Then q >  andα(x,x)≥. SinceT is

α_∗-admissible, soα_∗(Tx,Tx)≥. Ifx∈Tx, thenx is ﬁxed point ofT. Assume that x∈/Tx. Then from (.), we get

 < d(x,Tx)≤α∗(Tx,Tx)H(Tx,Tx) ≤ψ

max d(x,x),d(x,Tx),d(x,Tx),

d(x,Tx)d(x,Tx)

 +d(x,x)

≤ψ

max d(x,x),d(x,x),d(x,Tx),

d(x,x)d(x,Tx)

 +d(x,x)

=ψmaxd(x,x),d(x,Tx)

.

If max{d(x,Tx),d(x,x)} = d(x,Tx), we get contradiction to the fact d(x,Tx) < d(x,Tx). Hence we obtain

maxd(x,Tx),d(x,x)

=d(x,x).

Sod(x,Tx)≤ψ(d(x,x)). Sinceq> , so by Lemma  we can ﬁndx∈Txsuch that

d(x,x) <qd(x,Tx)≤qψ

d(x,x)

,

d(x,x) <qψ

d(x,x)

≤qψ

d(x,x)

=ψqψd(x,x)

. (.)

It is clear that x=x. Putq =ψ

₍_qψ₍_d₍_x_,_x_)))

ψ(d(x,x)) . Thenq>  andα(x,x)≥. SinceT is

α_∗-admissible, soα_∗(Tx,Tx)≥. Ifx∈Tx, thenx is ﬁxed point ofT. Assume that x∈/Tx. From (.), we have

 < d(x,Tx)≤α∗(Tx,Tx)H(Tx,Tx) ≤ψ

max d(x,x),d(x,Tx),d(x,Tx)

d(x,Tx)d(x,Tx)

 +d(x,x)

≤ψ

max d(x,x),d(x,x),d(x,Tx)

d(x,x)d(x,Tx)

 +d(x,x)

=ψmaxd(x,x),d(x,Tx)

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Ifmax{d(x,Tx),d(x,x)}=d(x,Tx). Then we get a contradiction. Somax{d(x,Tx), d(x,x)}=d(x,x). Thus

d(x,Tx)≤ψ

d(x,x)

.

Sinceq> , so by Lemma  we can ﬁndx∈Txsuch that

d(x,x) <qd(x,Tx)≤qψ

d(x,x)

=ψqψd(x,x)

. (.)

Continuing in this way, we can generate a sequence{xn}inXsuch thatxn∈Txn–and xn=xn–, and

d(xn,xn+)≤ψn–

qψd(x,x)

(.)

for alln. Now, for eachm>n, we have

d(xn,xm)≤ m–

i=n

d(xi,xi+)≤

m–

i=n

ψi–qψd(x,x)

.

This implies that{xn}is a Cauchy sequence inX. SinceXis complete, there existsx∗∈X such thatxn−→x∗asn−→ ∞. We now show thatx∗∈Tx∗. Sinceα(xn,x∗)≥ for alln andTisα∗-admissible, soα∗(Txn,Tx∗)≥ for alln. Then

dx∗,Tx∗≤α_∗Txn,Tx∗HTx∗,Txn

+dxn,x∗ ≤ψ

max dxn,x∗,d(xn,Txn),dx∗,Tx∗,d(xn,Txn)d(x

∗_,_Tx∗₎

 +d(xn,x∗)

+dxn,x∗

≤ψ

max dxn,x∗

,d(xn,xn+),d

x∗,Tx∗,d(xn,xn+)d(x

∗_,_Tx∗₎

 +d(xn,x∗)

+dxn,x∗,

and taking the limit asn→ ∞, we getd(x∗,Tx∗) = . Thusx∗∈Tx∗. Example  LetX= [, ] andd(x,y) =|x–y|. DeﬁneT :X→X byTx= [,_x] for all

x∈Xand

α(x,y) =



|x–y| ifx=y,  ifx=y.

Then α(x,y)≥⇒α∗(Tx,Ty) =inf{α(a,b) :a∈Tx,b∈Ty} ≥. Then clearlyT isα∗ -admissible. Now, forx,yandx≤y, it is easy to check that

α_∗(Tx,Ty)H(Tx,Ty)≤ψ

,

whereψ(t) =_t, for allt≥. Putx=  andx=_. Thenα(x,x) =  > . ThenThas ﬁxed

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Corollary  Let(X,d)be a complete metric space and T:X→X_{be an}_α

∗-admissible

α∗(Tx,Ty) + H(Tx,Ty)≤ψ(R(x,y)),

where

R(x,y) =max d(x,y),d(x,Tx),d(y,Ty),d(x,Tx)d(y,Ty)  +d(x,y)

Corollary  Let(X,d)be a complete metric space and T:X→X_{be an}_α

∗-admissible

H(Tx,Ty) +lα∗(Tx,Ty)≤ψR(x,y)+l,

where

for all x,y∈X and l> .Also suppose that the following assertions hold:

IfT is single-valued in Theorem , we obtain the following ﬁxed point results.

Theorem  Let(X,d)be a complete metric space and T:X→X be anα-admissible

mapping.Assume that forψ∈,we have

α(Tx,Ty)d(Tx,Ty)≤ψ

(.)

(i) there existsx∈Xwithα(x,Tx)≥;

α(xn,x)≥for alln∈N.

Corollary  Let(X,d)be a complete metric space and T:X→X be anα-admissible mapping.Assume that forψ∈,we have

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where

(i) α(x,Tx)≥for somex∈X;

α(xn,x)≥for alln∈N.

Now, we give the following result about a ﬁxed point of self-maps on complete metric spaces.

Theorem  Let(X,d)be a complete metric space,α:X×X→[, +∞)be a mapping,

ψ∈and T be a self-mapping on X such that

α(x,y)d(Tx,Ty)≤

ψ(max{d(x,Tx_d₍_x)d_,_y(₎y,Ty),d(x,y)}) for x=y,

 for x=y, (.)

for all x,y∈X.Suppose that T isα-admissible and there exist x∈X and x∈Txwith

α(x,Tx)≥.If T is continuous.Then T has a unique ﬁxed point.

Proof Takex∈Xsuch thatα(x,Tx)≥, and deﬁne the sequence{xn}inXbyxn+=Txn for alln≥. Ifxn=xn+for somen, thenx∗=xnis a ﬁxed point ofT. Assume thatxn=xn+

for alln. SinceT isα-admissible, so it is easy to check thatα(xn,xn+)≥ for all natural

numbersn. Thus for each natural numbern, we have

d(xn+,xn) =d(Txn,Txn–)

≤α(xn,xn–)d(Txn,Txn–)

≤ψ

max d(xn,Txn)d(xn–,Txn–)

d(xn,xn–)

,d(xn,xn–)

≤ψ

max d(xn,xn+)d(xn–,xn)

d(xn,xn–)

,d(xn,xn–)

≤ψmaxd(xn,xn+),d(xn,xn–)

.

If max{d(xn,xn+),d(xn,xn–)}=d(xn,xn+), thend(xn+,xn)≤ψ(d(xn+,xn)) a

contradic-tion. So we getd(xn+,xn)≤ψ(d(xn,xn–)). Sinceψis nondecreasing, so we have

d(xn+,xn)≤ψ

d(xn,xn–)

≤ψd(xn–,xn–)

≤ · · · ≤ψnd(x,x)

(.)

for alln. It is easy to check that{xn}is a Cauchy sequence. SinceXis complete, so there existsx∗∈Xsuch thatxn→x∗. Further the continuity ofTimplies that

Tx∗=T

lim n→∞xn

= lim

n→∞T(xn) =x

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Thereforex∗is a ﬁxed point ofTinX. Now, if there exists another pointu=x∗inXsuch thatTu=u, then

dx∗,u=dTx∗,Tu≤αx∗,udTx∗,Tu ≤ψ

max d(x

∗_,_Tx∗₎_d₍_u_,_Tu₎

d(x∗,u) ,d

x∗,u ≤ψmax,dx∗,u=ψdx∗,u,

a contradiction. Hencex∗is a unique ﬁxed point ofTinX.

Example  LetX= [,∞) andd(x,y) =|x–y|. DeﬁneT:X→XbyTx=x+  whenever

x,y∈[, ],Tx=_ wheneverx,y∈(, ) andTx=x+ x+  wheneverx∈[,∞). Also deﬁne the mappingsψ: [,∞)→[,∞) byψ(t) = t

 and

α(x,y) =

 ifx,y∈(, ),  otherwise.

By a routine calculation one can easily show that

α(x,y)d(Tx,Ty)≤ψ

max d(x,Tx)d(y,Ty)

d(x,y) ,d(x,y)

for allx,y∈Xand_ is unique ﬁxed point of the mappingT.

3 Fixed point results for graphic contractions

Consistent with Jachymski [], let (X,d) be a metric space anddenote the diagonal of the Cartesian productX×X. Consider a directed graphGsuch that the setV(G) of its vertices coincides withX, and the setE(G) of its edges contains all loops,i.e.,E(G)⊇. We assumeGhas no parallel edges, so we can identifyGwith the pair (V(G),E(G)). More-over, we may treatGas a weighted graph (see []) by assigning to each edge the distance between its vertices. Ifxandyare vertices in a graphG, then a path inGfromxtoy

of lengthN (N∈N) is a sequence{xi}Ni=ofN+  vertices such thatx=x,xN =yand (xn–,xn)∈E(G) fori= , . . . ,N. A graphGis connected if there is a path between any two

vertices.Gis weakly connected ifG˜ is connected (see for details [, , , ]).

Deﬁnition [] We say that a mappingT:X→Xis a BanachG-contraction or simply

G-contraction ifTpreserves edges ofG,i.e.,

∀x,y∈X (x,y)∈E(G) ⇒ T(x),T(y)∈E(G)

andTdecreases weights of edges ofGin the following way:

∃k∈[, ),∀x,y∈X (x,y)∈E(G) ⇒ dT(x),T(y)≤kd(x,y)

Deﬁnition [] A mappingT :X→Xis calledG-continuous, if givenx∈Xand the

sequence{xn}

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Theorem  Let(X,d)be a complete metric space endowed with a graph G and T be a self-mapping on X.Suppose the following assertions hold:

(i) ∀x,y∈X,(x,y)∈E(G)⇒(T(x),T(y))∈E(G); (ii) there existsx∈Xsuch that(x,Tx)∈E(G); (iii) there existsψ∈such that

d(Tx,Ty)≤ψR(x,y)

for all(x,y)∈E(G)where

;

(iv) if{xn}is a sequence inXsuch that(xn,xn+)∈E(G)for alln∈Nandxn→xas

n→+∞,then(xn,x)∈E(G)for alln∈N.

Proof Deﬁne,α:X_→_{[, +}_∞_{) by}_α₍_x_,_y_{) =}, if(x,y)∈E(G),

, otherwise. First we prove thatT is anα

-admissible mapping. Let,α(x,y)≥, then (x,y)∈E(G). From (i), we have (Tx,Ty)∈E(G). That is,α(Tx,Ty)≥. ThusT is anα-admissible mapping. From (ii) there existsx∈X

such that (x,Tx)∈E(G). That is,α(x,Tx)≥. If (x,y)∈E(G), then (Tx,Ty)∈E(G) and

henceα(Tx,Ty) = . Thus, from (iii) we haveα(Tx,Ty)d(Tx,Ty) =d(Tx,Ty)≤ψ(M(x,y)). Condition (iv) implies condition (ii) of Theorem . Hence, all conditions of Theorem 

are satisﬁed andThas a ﬁxed point.

Corollary  Let(X,d)be a complete metric space endowed with a graph G and T be a self-mapping on X.Suppose the following assertions hold:

(i) Tis a BanachG-contraction;

(ii) there existsx∈Xsuch that(x,Tx)∈E(G);

(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈E(G)for alln∈Nandxn→xas

n→+∞,then(xn,x)∈E(G)for alln∈N. Then T has a ﬁxed point.

As an application of Theorem , we obtain;

Theorem  Let(X,d)be a complete metric space endowed with a graph G and T be a

self-mapping on X.Suppose the following assertions hold:

(i) ∀x,y∈X,(x,y)∈E(G)⇒(T(x),T(y))∈E(G); (ii) there existsx∈Xsuch that(x,Tx)∈E(G); (iii) there existsψ∈such that

d(Tx,Ty)≤

ψ(max{d(x,Tx_d₍_x)d_,_y(₎y,Ty),d(x,y)}) for all(x,y)∈E(G)withx=y,

 forx=y;

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Let (X,d,) be a partially ordered metric space. Deﬁne the graphGby

E(G) =(x,y)∈X×X:xy.

For this graph, condition (i) in Theorem  meansTis nondecreasing with respect to this order []. From Theorems - we derive the following important results in partially ordered metric spaces.

Theorem  Let(X,d,)be a complete partially ordered metric space and T be a self-mapping on X.Suppose the following assertions hold:

(i) Tis nondecreasing map;

(ii) there existsx∈Xsuch thatxTx; (iii) there existsψ∈such that

d(Tx,Ty)≤ψR(x,y)

for allxywhere

;

(iv) if{xn}is a sequence inXsuch thatxnxn+for alln∈Nandxn→xasn→+∞, thenxnxfor alln∈N.

Corollary [] Let(X,d,)be a complete partially ordered metric space and T:X→ X be nondecreasing mapping such that

d(Tx,Ty)≤rd(x,y)

for all x,y∈X with xy where≤r< .Suppose that the following assertions hold:

(i) there existsx∈Xsuch thatxTx;

(ii) if{xn}is a sequence inXsuch thatxnxn+for alln∈Nandxn→xasn→+∞, thenxnxfor alln∈N.

Theorem  Let(X,d,)be a complete partially ordered metric space and T be a self-mapping on X.Suppose the following assertions hold:

(i) Tis nondecreasing map;

(ii) there existsx∈Xsuch thatxTx; (iii) there existsψ∈such that

d(Tx,Ty)≤

ψ(max{d(x,Tx_d₍_x)d_,_y(₎y,Ty),d(x,y)}) for allxywithx=y,

 forx=y;

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal paper.

Author details

1_{Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.}2_{Department of}

Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad, 44000, Pakistan.

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.

Received: 14 May 2014 Accepted: 9 August 2014 Published:04 Sep 2014

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10.1186/1029-242X-2014-348

Cite this article as:Hussain et al.:Generalized ﬁxed point theorems for multi-valuedα-ψ-contractive mappings.