**R E S E A R C H**

**Open Access**

## (α

## ,

*ψ*

## ,

*ξ*

## )

## -contractive multivalued mappings

### Muhammad Usman Ali

1_{, Tayyab Kamran}

2_{and Erdal Karapınar}

3*
*_{Correspondence:}

erdalkarapinar@yahoo.com

3_{Department of Mathematics,}

Atilim University, Incek, Ankara 06836, Turkey

Full list of author information is available at the end of the article

**Abstract**

In this paper, we introduce the notion of (α,

*ψ,*

*ξ)-contractive multivalued mappings*

to generalize and extend the notion of*α-ψ*

-contractive mappings to closed valued
multifunctions. We investigate the existence of ﬁxed points for such mappings. We
also construct an example to show that our result is more general than the results of
*α-ψ-contractive closed valued multifunctions.*

**MSC:** 47H10; 54H25

**Keywords:**

*α*

_{∗}-admissible mappings; (α,

*ψ,ξ*

)-contractive mappings
**1 Introduction and preliminaries**

Recently, Samet*et al.* [] introduced the notions of*α*-*ψ*-contractive and*α*-admissible
self-mappings and proved some ﬁxed-point results for such mappings in complete
met-ric spaces. Karapınar and Samet [] generalized these notions and obtained some
ﬁxed-point results. Asl*et al.*[] extended these notions to multifunctions by introducing the
notions of*α*∗-*ψ*-contractive and*α*∗-admissible mappings and proved some ﬁxed-point
results. Some results in this direction are also given in [–]. Ali and Kamran [] further
generalized the notion of*α*_{∗}-*ψ*-contractive mappings and obtained some ﬁxed-point
the-orems for multivalued mappings. Salimi*et al.*[] modiﬁed the notions of*α*-*ψ*-contractive
and*α*-admissible self-mappings by introducing another function*η*and established some
ﬁxed-point theorems for such mappings in complete metric spaces. N. Hussain*et al.*[]
extended these modiﬁed notions to multivalued mappings. Recently, Mohammadi and
Rezapour [] showed that the results obtained by Salimi*et al.*[] follow from
correspond-ing results for*α*-*ψ*-contractive mappings. More recently, Berzig and Karapinar [] proved
that the ﬁrst main result of Salimi*et al.*[] follows from a result of Karapınar and Samet
[]. The purpose of this paper is to introduce the notion of (*α*,*ψ*,*ξ*)-contractive
multival-ued mappings to generalize and extend the notion of*α*-*ψ*-contractive mappings to closed
valued multifunctions and to provide ﬁxed-point theorems for (*α*,*ψ*,*ξ*)-contractive
mul-tivalued mappings in complete metric spaces.

We recollect the following deﬁnitions, for the sake of completeness. Let (*X*,*d*) be a metric
space. We denote by*CB*(*X*) the class of all nonempty closed and bounded subsets of*X*and
by*CL*(*X*) the class of all nonempty closed subsets of*X*. For every*A*,*B*∈*CL*(*X*), let

*H*(*A*,*B*) =

⎧ ⎨ ⎩

max{sup* _{x∈A}d*(

*x*,

*B*),sup

*(*

_{y∈B}d*y*,

*A*)}, if the maximum exists;

∞, otherwise.

Such a map*H*is called the generalized Hausdorﬀ metric induced by the metric*d*. Letbe
a set of nondecreasing functions,*ψ*: [,∞)→[,∞) such that∞_{n}_{=}*ψn*_{(}_{t}_{) <}_{∞}_{for each}

*t*> , where*ψn*is the*n*th iterate of*ψ*. It is known that for each*ψ*∈, we have*ψ*(*t*) <*t*

for all*t*> and*ψ*() = for*t*= . More details as regards such a function can be found in

*e.g.*[, ].

**Deﬁnition .**[] Let (*X*,*d*) be a metric space and*α*:*X*×*X*→[,∞) be a mapping.
A mapping*G*:*X*→*CL*(*X*) is*α*∗-admissible if

*α*(*x*,*y*)≥ ⇒ *α*∗(*Gx*,*Gy*)≥,

where*α*_{∗}(*Gx*,*Gy*) =inf{*α*(*a*,*b*) :*a*∈*Gx*,*b*∈*Gy*}.

**2 Main results**

We begin this section by considering a familyof functions*ξ*: [,∞)→[,∞) satisfying
the following conditions:

(i) *ξ* is continuous;

(ii) *ξ* is nondecreasing on[,∞);
(iii) *ξ*() = and*ξ*(*t*) > for all*t*∈(,∞);
(iv) *ξ* is subadditive.

**Example .** Suppose that*φ*: [,∞)→[,∞) is a Lebesgue integrable mapping which
is summable on each compact subset of [,∞), for each > ,_{}*φ*(*t*)*dt*> , and for each

*a*,*b*> , we have

*a*+*b*

*φ*(*t*)*dt*≤
*a*

*φ*(*t*)*dt*+

*b*

*φ*(*t*)*dt*.

Deﬁne*ξ*: [,∞)→[,∞) by*ξ*(*t*) =_{}*tφ*(*w*)*dw*for each*t*∈[,∞). Then*ξ*∈.

**Lemma .** *Let*(*X*,*d*)*is a metric space and letξ*∈.*Then*(*X*,*ξ*◦*d*)*is a metric space*.

**Lemma .** *Let*(*X*,*d*)*be a metric space*,*letξ*∈*and let B*∈*CL*(*X*).*Assume that there*
*exists x*∈*X such thatξ*(*d*(*x*,*B*)) > .*Then there exists y*∈*B such that*

*ξd*(*x*,*y*) <*qξd*(*x*,*B*) ,

*where q*> .

*Proof* By hypothesis we have*ξ*(*d*(*x*,*B*)) > . We choose

= (*q*– )*ξd*(*x*,*B*) .

By the deﬁnition of an inﬁmum, since*ξ*◦*d*is a metric space, it follows that there exists

*y*∈*B*such that

**Deﬁnition .** Let (*X*,*d*) be a metric space. A mapping*G*:*X*→*CL*(*X*) is called (*α*,*ψ*,*ξ*
)-contractive if there exist three functions*ψ*∈,*ξ*∈and*α*:*X*×*X*→[,∞) such that

*x*,*y*∈*X*, *α*(*x*,*y*)≥ ⇒ *ξH*(*Gx*,*Gy*) ≤*ψξM*(*x*,*y*) , (.)

where*M*(*x*,*y*) =max{*d*(*x*,*y*),*d*(*x*,*Gx*),*d*(*y*,*Gy*),*d*(*x*,*Gy*)+_{}*d*(*y*,*Gx*)}.

In case when*ψ*∈is strictly increasing, the (*α*,*ψ*,*ξ*)-contractive mapping is called a
strictly (*α*,*ψ*,*ξ*)-contractive mapping.

**Theorem .** *Let*(*X*,*d*)*be a complete metric space and let G*:*X*→*CL*(*X*)*be a strictly*

(*α*,*ψ*,*ξ*)*-contractive mapping satisfying the following assumptions*:

(i) *Gis anα*_{∗}*-admissible mapping*;

(ii) *there existx*∈*Xandx*∈*Gx**such thatα*(*x*,*x*)≥;

(iii) *Gis continuous*.
*Then G has a ﬁxed point*.

*Proof* By hypothesis, there exist*x*∈*X*and*x*∈*Gx*such that*α*(*x*,*x*)≥. If*x*=*x*, then

we have nothing to prove. Let*x*=*x*. If*x*∈*Gx*, then*x* is a ﬁxed point. Let*x*∈/*Gx*.

Then from equation (.), we have

< *ξH*(*Gx*,*Gx*)

≤*ψ*

*ξ*

max

*d*(*x*,*x*),*d*(*x*,*Gx*),*d*(*x*,*Gx*),

*d*(*x*,*Gx*) +*d*(*x*,*Gx*)

=*ψξ*max*d*(*x*,*x*),*d*(*x*,*Gx*)

, (.)

since *d*(*x*_{},*Gx*) ≤max{*d*(*x*,*x*),*d*(*x*,*Gx*)}. Assume that max{*d*(*x*,*x*),*d*(*x*,*Gx*)}=*d*(*x*,

*Gx*). Then from equation (.), we have

< *ξd*(*x*,*Gx*) ≤*ξ*

*H*(*Gx*,*Gx*)

≤*ψξ*max*d*(*x*,*x*),*d*(*x*,*Gx*)

=*ψξd*(*x*,*Gx*)

, (.)

which is a contradiction. Hence,max{*d*(*x*,*x*),*d*(*x*,*Gx*)}=*d*(*x*,*x*). Then from equation

(.), we have

<*ξd*(*x*,*Gx*) ≤*ξ*

*H*(*Gx*,*Gx*) ≤*ψ*

*ξd*(*x*,*x*)

. (.)

For*q*> by Lemma ., there exists*x*∈*Gx*such that

<*ξd*(*x*,*x*) <*qξ*

*d*(*x*,*Gx*). (.)

From equations (.) and (.), we have

<*ξd*(*x*,*x*) <*qψ*

*ξd*(*x*,*x*)

Applying*ψ*in equation (.), we have

<*ψξd*(*x*,*x*)

<*ψqψξd*(*x*,*x*)

. (.)

Put*q*=*ψ*(_{ψ}qψ_{(}* _{ξ}*(

_{(}

*ξ*(

_{d}_{(}

*d*(

_{x}*x*

_{,}

*,*

_{x}*x*

_{)))})))). Then

*q*> . Since

*G*is an

*α*∗-admissible mapping, then

*α*∗(

*Gx*,

*Gx*)≥. Thus we have*α*(*x*,*x*)≥*α*∗(*Gx*,*Gx*)≥. If*x*∈*Gx*, then*x*is a ﬁxed point.

Let*x*∈/*Gx*. Then from equation (.), we have

< *ξH*(*Gx*,*Gx*)

≤*ψ*

*ξ*

max

*d*(*x*,*x*),*d*(*x*,*Gx*),*d*(*x*,*Gx*),

*d*(*x*,*Gx*) +*d*(*x*,*Gx*)

=*ψξ*max*d*(*x*,*x*),*d*(*x*,*Gx*)

, (.)

since *d*(*x*,_{}*Gx*) ≤max{*d*(*x*,*x*),*d*(*x*,*Gx*)}. Assume thatmax{*d*(*x*,*x*),*d*(*x*,*Gx*)}=*d*(*x*,

*Gx*). Then from equation (.), we have

< *ξd*(*x*,*Gx*) ≤*ξ*

*H*(*Gx*,*Gx*)

≤*ψξ*max*d*(*x*,*x*),*d*(*x*,*Gx*)

=*ψξd*(*x*,*Gx*)

, (.)

which is a contradiction. Hence,max{*d*(*x*,*x*),*d*(*x*,*Gx*)}=*d*(*x*,*x*). Then from equation

(.), we have

<*ξd*(*x*,*Gx*) ≤*ξ*

*H*(*Gx*,*Gx*) ≤*ψ*

*ξd*(*x*,*x*)

. (.)

For*q*> by Lemma ., there exists*x*∈*Gx*such that

<*ξd*(*x*,*x*) <*q**ξ*

*d*(*x*,*Gx*) . (.)

From equations (.) and (.), we have

<*ξd*(*x*,*x*) <*q**ψ*

*ξd*(*x*,*x*)

=*ψqψξd*(*x*,*x*)

. (.)

Applying*ψ*in equation (.), we have

<*ψξd*(*x*,*x*)

<*ψ**qψξd*(*x*,*x*)

. (.)

Put*q*=*ψ*

_{(}_{qψ}_{(}_{ξ}_{(}_{d}_{(}_{x}_{,}_{x}_{))))}

*ψ*(*ξ*(*d*(*x*,*x*))) . Then*q*> . Since*G*is an*α*∗-admissible mapping, then*α*∗(*Gx*,

*Gx*)≥. Thus we have*α*(*x*,*x*)≥*α*∗(*Gx*,*Gx*)≥. If*x*∈*Gx*, then*x*is a ﬁxed point.

Let*x*∈/*Gx*. Then from equation (.), we have

<*ξH*(*Gx*,*Gx*)

≤*ψ*

*ξ*

max

*d*(*x*,*x*),*d*(*x*,*Gx*),*d*(*x*,*Gx*),

*d*(*x*,*Gx*) +*d*(*x*,*Gx*)

=*ψξ*max*d*(*x*,*x*),*d*(*x*,*Gx*)

since *d*(*x*,_{}*Gx*) ≤max{*d*(*x*,*x*),*d*(*x*,*Gx*)}. Assume thatmax{*d*(*x*,*x*),*d*(*x*,*Gx*)}=*d*(*x*,

*Gx*). Then from equation (.), we have

< *ξd*(*x*,*Gx*) ≤*ξ*

*H*(*Gx*,*Gx*)

≤*ψξ*max*d*(*x*,*x*),*d*(*x*,*Gx*)

=*ψξd*(*x*,*Gx*)

, (.)

which is a contradiction to our assumption. Hence,max{*d*(*x*,*x*),*d*(*x*,*Gx*)}=*d*(*x*,*x*).

Then from equation (.), we have

<*ξd*(*x*,*Gx*) ≤*ξ*

*H*(*Gx*,*Gx*) ≤*ψ*

*ξd*(*x*,*x*)

. (.)

For*q*> by Lemma ., there exists*x*∈*Gx*such that

<*ξd*(*x*,*x*) <*q**ξ*

*d*(*x*,*Gx*). (.)

From equations (.) and (.), we have

<*ξd*(*x*,*x*) <*q**ψ*

*ξd*(*x*,*x*)

=*ψ**qψξd*(*x*,*x*)

. (.)

Continuing the same process, we get a sequence{*xn*}in*X*such that*xn*+∈*Gxn,xn*+=*xn,*

*α*(*xn,xn*+)≥, and

*ξd*(*xn*+,*xn*+) <*ψn*

*qψξd*(*x*,*x*)

for each*n*∈N∪ {}. (.)

Let*m*>*n*, we have

*ξd*(*xm*,*xn*) ≤
*m*–

*i*=*n*

*ξd*(*xi*,*xi*+) <

*m*–

*i*=*n*

*ψi*–*qψξd*(*x*,*x*)

.

Since*ψ*∈, we have

lim
*n*,*m→∞ξ*

*d*(*xm*,*xn*) = . (.)

This implies that

lim

*n*,*m→∞d*(*xm,xn) = .* (.)

Hence{*xn*}is a Cauchy sequence in (*X*,*d*). By completeness of (*X*,*d*), there exists*x*∗∈*X*
such that*xn*→*x*∗as*n*→ ∞. Since*G*is continuous, we have

*dx*∗,*Gx*∗ = lim

*n→∞d*

*xn*+,*Gx*∗ ≤* _{n→∞}*lim

*H*

*Gxn,Gx*∗ = .

**Theorem .** *Let*(*X*,*d*)*be a complete metric space and let G*:*X*→*CL*(*X*)*be a strictly*

(*α*,*ψ*,*ξ*)*-contractive mapping satisfying the following assumptions*:

(i) *Gis anα*∗*-admissible mapping*;

(ii) *there existx*∈*Xandx*∈*Gx**such thatα*(*x*,*x*)≥;

(iii) *if*{*xn*}*is a sequence inXwithxn*→*xasn*→ ∞*andα*(*xn,xn*+)≥*for each*

*n*∈N∪ {},*then we haveα*(*xn*,*x*)≥*for eachn*∈N∪ {}.

*Then G has a ﬁxed point*.

*Proof* Following the proof of Theorem ., we know that{*xn*}is a Cauchy sequence in*X*
with*xn*→*x*∗as*n*→ ∞and*α*(*xn,xn*+)≥ for each*n*∈N∪ {}. By hypothesis (iii), we

have*α*(*xn*,*x*∗)≥ for each*n*∈N∪ {}. Then from equation (.), we have

*ξHGxn,Gx*∗ ≤*ψ*

*ξ*

max

*dxn,x*∗ ,*d*(*xn,Gxn),dx*∗,*Gx*∗ ,

*d*(*xn*,*Gx*∗) +*d*(*x*∗,*Gxn*)

. (.)

Suppose that*d*(*x*∗,*Gx*∗)= .
We let*xn*→*x*∗. Taking =*d*(*x*

∗_{,}* _{Gx}*∗

_{)}

we can ﬁnd*N*∈Nsuch that

*dx*∗,*xm* <

*d*(*x*∗,*Gx*∗)

for each*m*≥*N*. (.)

Moreover, as{*xn*}is a Cauchy sequence, there exists*N*∈Nsuch that

*d*(*xm*,*Gxm*)≤*d*(*xm*,*xm*+) <

*d*(*x*∗,*Gx*∗)

for each*m*≥*N*. (.)

Furthermore,

*dx*∗,*Gxm* ≤*d*

*x*∗,*xm*+ <

*d*(*x*∗,*Gx*∗)

for each*m*≥*N*. (.)

As*d*(*xm*,*Gx*∗)→*d*(*x*∗,*Gx*∗). Taking =*d*(*x*

∗_{,}* _{Gx}*∗

_{)}

we can ﬁnd*N*∈Nsuch that

*dxm*,*Gx*∗ <

*d*(*x*∗,*Gx*∗)

for each*m*≥*N*. (.)

It follows from equations (.), (.), (.), and (.) that

max

*dxm,x*∗ ,*d*(*xm,Gxm),dx*∗,*Gx*∗ ,*d*(*xm,Gx*

∗_{) +}_{d}_{(}* _{x}*∗

_{,}

_{Gx}_{m)}

=*dx*∗,*Gx*∗ ,

for*m*≥*N*=max{*N*,*N*,*N*}. Moreover, for*m*≥*N*, by the triangle inequality, we have

*ξdx*∗,*Gx*∗ ≤*ξdx*∗,*xm*+

+*ξHGxm*,*Gx*∗

≤*ξdx*∗,*xm*+

+*ψ*

*ξ*

max

*dxm*,*x*∗ ,*d*(*xm*,*Gxm*),*d*

*d*(*xm,Gx*∗) +*d*(*x*∗,*Gxm)*

=*ξdx*∗,*xm*+

+*ψξdx*∗,*Gx*∗ . (.)

Letting*m*→ ∞in the above inequality, we have

*ξdx*∗,*Gx*∗ ≤*ψξdx*∗,*Gx*∗ . (.)

This is not possible if*ξ*(*d*(*x*∗,*Gx*∗)) > . Therefore, we have*ξ*(*d*(*x*∗,*Gx*∗)) = , which implies

that*d*(*x*∗,*Gx*∗) = ,*i.e.*,*x*∗=*Gx*∗.

**Example .** Let*X*=Rbe endowed with the usual metric*d*. Deﬁne*G*:*X*→*CL*(*X*) by

*Gx*=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(–∞, ] if*x*< ,

{,*x*

} if ≤*x*< ,

{} if*x*= ,
[*x*_{,}_{∞}_{)} _{if}_{x}_{> }

and*α*:*X*×*X*→[,∞) by

*α*(*x*,*y*) =

⎧ ⎨ ⎩

if*x*,*y*∈[, ],

otherwise.

Take*ψ*(*t*) = _{}*t* and*ξ*(*t*) =√*t*for each*t*≥. Then*G*is an (*α*,*ψ*,*ξ*)-contractive mapping.
For*x*= and ∈*Gx*we have *α*(, ) = . Also, for each*x*,*y*∈*X* with *α*(*x*,*y*) = , we

have*α*_{∗}(*Gx*,*Gy*) = . Moreover, for any sequence{*xn*}in*X* with*xn*→*x*as*n*→ ∞and
*α*(*xn,xn*+) = for each*n*∈N∪ {}, we have*α*(*xn,x*) = for each*n*∈N∪ {}. Therefore, all

conditions of Theorem . are satisﬁed and*G*has inﬁnitely many ﬁxed points. Note that
Nadler’s ﬁxed-point theorem is not applicable here; see, for example,*x*= . and*y*= .

**3 Consequences**

It can be seen, by restricting*G*:*X*→*X*and taking*ξ*(*t*) =*t*for each*t*≥ in Theorems .
and ., that:

• Theorem . and Theorem . of Samet*et al.*[] are special cases of Theorem . and
Theorem ., respectively;

• Theorem . of Asl*et al.*[] is a special case of Theorem .;
• Theorem . of Amiri*et al.*[] is a special case of Theorem .;

• Theorem . of Salimi*et al.*[] is a special case of Theorems . and ..

Further, it can be seen, by considering*G*:*X*→*CB*(*X*) and*ξ*(*t*) =*t*for each*t*≥, that

• Theorem . and Theorem . of Mohammadi*et al.*[] are special cases of our
results;

• Theorem . of Amiri*et al.*[] is a special case of Theorem ., when*ψ*∈is
sublinear.

**Remark .** Note that in Example .,*ξ*(*t*) =√*t*. Therefore, one may not apply the
afore-mentioned results and, as a consequence, conclude that*G*has a ﬁxed point.

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

All authors contributed equally in writing this article. All authors read and approved the ﬁnal manuscript.

**Author details**

1_{Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12,}

Islamabad, Pakistan. 2_{Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.}3_{Department of}

Mathematics, Atilim University, Incek, Ankara 06836, Turkey.

**Acknowledgements**

The authors are grateful to the reviewers for their careful reviews and useful comments.

Received: 30 September 2013 Accepted: 10 December 2013 Published:06 Jan 2014

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**Cite this article as:**Ali et al.: (*α*,*ψ*,*ξ*)-contractive multivalued mappings.*Fixed Point Theory and Applications*