On (α∗,ψ)-contractive multi-valued mappings

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R E S E A R C H

Open Access

On

(α

∗

,

ψ

)

-contractive multi-valued

mappings

Muhammad Usman Ali

1

_{and Tayyab Kamran}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Quaid-i-Azam University, Islamabad, Pakistan

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

Abstract

In this paper, we generalize the contractive condition for multi-valued mappings given by Asl, Rezapour and Shahzad in 2012. We establish some ﬁxed point theorems for multi-valued mappings from a complete metric space to the space of closed or bounded subsets of the metric space satisfying generalized (α∗,

ψ)-contractive

condition.

MSC: 47H10; 54H25

Keywords:

α-admissible;

α

∗-ψ-contractive mapping; generalized (α∗,

ψ)-contractive mapping

1 Introduction

Sametet al.[] introduced the notion ofα-ψ-contractive self-mappings of a metric space. Recently, Aslet al. [] introduced the notion ofα∗-ψ-contractive mappings to extend the notionα-ψ-contractive mappings. In this paper, we generalize the notion ofα∗-ψ -contractive mappings and prove some ﬁxed point theorems for such mappings.

Let be a family of nondecreasing functions, ψ : [,∞) → [,∞) such that

∞

n=ψn(t) <∞ for eacht> , whereψn is thenth iterate ofψ. It is known that for eachψ ∈, we have ψ(t) <t for all t>  andψ() =  fort=  []. Let (X,d) be a metric space. A mappingG:X→Xis calledα-ψ-contractive if there exist two func-tions α:X×X→[,∞) and ψ ∈ such that α(x,y)d(Gx,Gy)≤ψ(d(x,y)) for each

x,y∈X. A mappingG:X→Xis calledα-admissible [] ifα(x,y)≥⇒α(Gx,Gy)≥. We denote byN(X) the space of all nonempty subsets of X, by B(X) the space of all nonempty bounded subsets ofX and byCL(X) the space of all nonempty closed sub-sets ofX. ForA∈N(X) andx∈X,d(x,A) =inf{d(x,a) :a∈A}. For everyA,B∈B(X),

δ(A,B) =sup{d(a,b) :a∈A,b∈B}. WhenA={x}, we denoteδ(A,B) byδ(x,B). For every

A,B∈CL(X), let

H(A,B) =

⎧ ⎨ ⎩

max{sup_x_∈_Ad(x,B),sup_y_∈_Bd(y,A)} if the maximum exists;

∞ otherwise.

Such a mapH is called generalized Hausdorﬀ metric induced byd. Let (X, ,d) be an ordered metric space andA,B⊆X. We say thatA≺rBif for eacha∈Aandb∈B, we havea b. We give a few deﬁnitions and the result due to Aslet al.[] for convenience.

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Deﬁnition . [] Let (X,d) be a metric space and letα:X×X→[,∞) be a map-ping. A mappingG:X→CL(X) isα∗-admissible ifα(x,y)≥⇒α∗(Gx,Gy)≥, where

α∗(Gx,Gy) =inf{α(a,b) :a∈Gx,b∈Gy}.

Deﬁnition .[] Let (X,d) be a metric space. A mappingG:X→CL(X) is calledα∗-ψ -contractive if there exist two functionsα:X×X→[,∞) andψ∈such that

α∗(Gx,Gy)H(Gx,Gy)≤ψd(x,y) (.) for allx,y∈X.

Theorem .[] Let(X,d)be a complete metric space,letα:X×X→[,∞)be a func-tion,letψ∈ be a strictly increasing map and T be a closed-valued,α∗-admissible and

α∗-ψ-contractive multi-function on X.Suppose that there exist x∈X and x∈Gxsuch

thatα(x,x)≥.Assume that if{xn}is a sequence in X such thatα(xn,xn+)≥for all n

and xn→x,thenα(xn,x)≥for all n.Then G has a ﬁxed point.

2 Main results

We begin this section by introducing the following deﬁnition.

Deﬁnition . Let (X,d) be a metric space and letG:X→CL(X) be a mapping. We say thatGis generalized (α∗,ψ)-contractive if there existsψ∈such that

α∗(Gx,Gy)d(y,Gy)≤ψd(x,y) (.) for eachx∈Xandy∈Gx, whereα∗(Gx,Gy) =inf{α(a,b) :a∈Gx,b∈Gy}.

Note that anα∗-ψ-contractive mapping is generalized (α∗,ψ)-contractive. In case when

ψ ∈ is strictly increasing, generalized (α∗,ψ)-contractive is called strictly generalized (α∗,ψ)-contractive. The following lemma is inspired by [, Lemma .].

Lemma . Let(X,d)be a metric space and B∈CL(X).Then,for each x∈X with d(x,B) > and q> ,there exists an element b∈B such that

d(x,b) <qd(x,B). (.)

Proof It is given thatd(x,B) > . Choose

= (q– )d(x,B).

Then, by using the deﬁnition ofd(x,B), it follows that there existsb∈Bsuch that

d(x,b) <d(x,B) +=qd(x,B).

Lemma . Let(X,d)be a metric space and G:X→CL(X).Assume that there exists a sequence{xn}in X such thatlimn→∞d(xn,Gxn) = and xn→x∈X.Then x is a ﬁxed point

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Proof Supposef(ξ) =d(ξ,Gξ) is lower semi-continuous atx, then

d(x,Gx)≤lim inf

n f(xn) =lim infn d(xn,Gxn) = .

By the closedness ofGit follows thatx∈Gx. Conversely, suppose thatxis a ﬁxed point ofG, thenf(x) = ≤lim infnf(xn).

Theorem . Let (X,d) be a complete metric space and let G : X →CL(X) be an

α∗-admissible strictly generalized (α∗,ψ)-contractive mapping. Assume that there exist x∈X and x∈Gx such thatα(x,x)≥.Then x is a ﬁxed point of G if and only if

f(ξ) =d(ξ,Gξ)is lower semi-continuous at x.

Proof By the hypothesis, there existx∈Xandx∈Gxsuch thatα(x,x)≥. Ifx=x, then we have nothing to prove. Letx=x. Ifx∈Gx, thenxis a ﬁxed point. Letx∈/Gx. SinceGisα∗-admissible, soα∗(Gx,Gx)≥, we have

 <d(x,Gx)≤α∗(Gx,Gx)d(x,Gx). (.) For givenq>  by Lemma ., there existsx∈Gxsuch that

 <d(x,x) <qd(x,Gx). (.)

It follows from (.), (.) and (.) that

 <d(x,x) <qψ

d(x,x)

. (.)

It is clear thatx=xandα(x,x)≥. Thusα∗(Gx,Gx)≥. Sinceψ is strictly increas-ing, by (.), we have

ψd(x,x)

<ψqψd(x,x)

.

Putq=ψ(qψψ(d(x(d(x,x,x))))), thenq> . Ifx∈Gx, thenx is a ﬁxed point. Letx∈/Gx, then

by Lemma ., there existsx∈Gxsuch that

 < d(x,x) <qd(x,Gx)≤qα∗(Gx,Gx)d(x,Gx) ≤qψ

d(x,x)

=ψqψd(x,x)

.

It is clear thatx=x, α(x,x)≥ and ψ(d(x,x)) <ψ(qψ(d(x,x))). Now putq= ψ_(q_ψ_(d(x

,x)))

ψ(d(x,x)) . Thenq> . If x∈Gx, thenx is a ﬁxed point. Letx∈/ Gx. Then by

Lemma . there existsx∈Gxsuch that

 < d(x,x) <qd(x,Gx)≤qα∗(Gx,Gx)d(x,Gx) ≤qψ

d(x,x)

=ψqψd(x,x)

.

By continuing the same process, we get a sequence{xn}inXsuch thatxn+∈Gxn. Also,

xn=xn+,α(xn,xn+)≥ and  <d(xn,xn+) <ψn–(qψ(d(x,x))) or  <d(xn,Gxn) <ψn–

qψd(x,x)

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For eachm>n, we have

d(xn,xm)≤ m–

i=n

d(xi,xi+) < m–

i=n

ψi–qψd(x,x)

.

Sinceψ∈, it follows that{xn}is a Cauchy sequence inX. Thus there isx∈Xsuch that

xn→x. Lettingn→ ∞in (.), we have

lim

n→∞d(xn,Gxn) = . (.)

The rest of the proof follows from Lemma ..

Example . LetX=Rbe endowed with the usual metricd. DeﬁneG:X→CL(X) and

α:X×X→[,∞) by

Gx=

⎧ ⎨ ⎩

[x,∞) ifx≥,

(–∞, –x_] _if_x_{< } (.)

and

α(x,y) =

⎧ ⎨ ⎩

 ifx,y≥,

 otherwise. (.)

Letψ(t) =_t for allt≥. For eachx∈Xandy∈Gx, we have

α∗(Gx,Gy)d(y,Gy) = ≤  d(x,y).

HenceGis a strictly generalized (α∗,ψ)-contractive mapping. Clearly,Gisα∗-admissible. Also, we havex=  andx= ∈Gxsuch thatα(x,x) = . Therefore, all conditions of Theorem . are satisfied andGhas infinitely many fixed points. Note that Theorem . in Section  is not applicable here. For example, takex=  andy= –.

Corollary . Let(X, ,d)be a complete ordered metric space, ψ∈ be a strictly in-creasing map and G:X→CL(X)be a mapping such that for each x∈X and y∈Gx with x y,we have

d(y,Gy)≤ψd(x,y). (.)

Also,assume that

(i) there existx∈Xandx∈Gxsuch thatx x,

(ii) ifx y,thenGx≺rGy.

Then x is a ﬁxed point of G if and only if f(ξ) =d(ξ,Gξ)is lower semi-continuous at x.

Proof Deﬁneα:X×X→[,∞) by

α(x,y) =

⎧ ⎨ ⎩

 ifx y,

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By using condition (i) and the definition ofα, we haveα(x,x) = . Also, from condition (ii), we havex yimpliesGx≺rGy; by using the definitions ofαand≺r, we haveα(x,y) =  impliesα∗(Gx,Gy) = . Moreover, it is easy to check thatGis a strictly generalized (α∗,ψ )-contractive mapping. Therefore, by Theorem ., xis a fixed point of Gif and only if

f(ξ) =d(ξ,Gξ) is lower semi-continuous atx.

Deﬁnition . Let (X,d) be a metric space andG:X→B(X) be a mapping. We say that

Gis a generalized (α∗,ψ,δ)-contractive mapping if there existsψ∈such that

α∗(Gx,Gy)δ(y,Gy)≤ψd(x,y) (.)

for eachx∈Xandy∈Gx, whereα∗(Gx,Gy) =inf{α(a,b) :a∈Gx,b∈Gy}.

Lemma . Let(X,d)be a metric space and G:X→B(X).Assume that there exists a sequence{xn}in X such thatlimn→∞δ(xn,Gxn) = and xn→x∈X.Then{x}=Gx if and

only if the function f(ξ) =δ(ξ,Gξ)is lower semi-continuous at x.

Proof Suppose thatf(ξ) =δ(ξ,Gξ) is lower semi-continuous atx, then

δ(x,Gx)≤lim inf

n f(xn) =lim infn δ(xn,Gxn) = .

Hence,{x}=Gxbecauseδ(A,B) =  impliesA=B={a}. Conversely, suppose that{x}=

Gx. Thenf(x) = ≤lim inf_nf(xn).

Theorem . Let (X,d) be a complete metric space and let G : X → B(X) be an

α∗-admissible generalized(α∗,ψ,δ)-contractive mapping.Assume that there exist x∈X

and x∈Gxsuch thatα(x,x)≥.Then there exists x∈X such that{x}=Gx if and only

if f(ξ) =δ(ξ,Gξ)is lower semi-continuous at x.

Proof By the hypothesis of the theorem, there exist x∈ X and x ∈Gx such that

α(x,x)≥. Assume thatx=x, for otherwise,xis a ﬁxed point. Letx∈/ Gx. AsG isα∗-admissible, we haveα∗(Gx,Gx)≥. Then

δ(x,Gx)≤α∗(Gx,Gx)δ(x,Gx)≤ψ

d(x,x)

. (.)

SinceGx=∅, there isx∈Gx. Then

 <d(x,x)≤δ(x,Gx). (.)

From (.) and (.), we have

 <d(x,x)≤ψ

d(x,x)

. (.)

Sinceψis nondecreasing, we have

ψd(x,x)

≤ψd(x,x)

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Asx∈Gx, we haveα(x,x)≥. SinceGx=∅, there isx∈Gx. Assume thatx=x, for otherwise,xis a ﬁxed point ofG. Then

 <d(x,x)≤δ(x,Gx)≤α∗(Gx,Gx)δ(x,Gx) ≤ψd(x,x)

≤ψd(x,x)

. (.)

Sinceψis nondecreasing, we have

ψd(x,x)

≤ψd(x,x)

. (.)

By continuing in this way, we get a sequence{xn}inXsuch thatxn+∈Gxnandxn=xn+ forn= , , , , . . . . Further we have

 <d(xn,xn+)≤δ(xn,Gxn)≤ψn

d(x,x)

. (.)

For eachm>n, we have

d(xn,xm)≤ m–

i=n

d(xi,xi+)≤ m–

i=n

ψid(x,x)

.

Sinceψ∈, it follows that{xn}is a Cauchy sequence inX. AsXis complete, there exists

x∈Xsuch thatxn→x. Lettingn→ ∞in (.), we have

lim

n→∞δ(xn,Gxn) = . (.)

The rest of the proof follows from Lemma ..

Example . LetX={, , , , , , . . .}be endowed with the usual metric d. Deﬁne

G:X→B(X) andα:X×X→[,∞) by

Gx=

⎧ ⎨ ⎩

{(x– ),x} ifx= , {} ifx= 

and

α(x,y) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 ifx=y= ,

 ifx=y= , 

 otherwise.

Letψ(t) =_t for allt≥. For eachx∈Xandy∈Gx, we have

α∗(Gx,Gy)δ(y,Gy)≤ 

d(x,y).

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Corollary . Let(X, ,d)be a complete ordered metric space,ψ∈and G:X→B(X)

be a mapping such that for each x∈X and y∈Gx with x y,we have

δ(y,Gy)≤ψd(x,y). (.)

Also,assume that

(i) there existsx∈Xsuch that{x} ≺Gx,i.e.,there existsx∈Gxsuch thatx x,

(ii) ifx y,thenGx≺rGy.

Then there exists x∈X such that {x}=Gx if and only if f(ξ) =δ(ξ,Gξ)is lower semi-continuous at x.

Proof Deﬁneα:X×X→[,∞) by

α(x,y) =

⎧ ⎨ ⎩

 ifx y,

 otherwise.

By using condition (i) and the deﬁnition ofα, we haveα(x,x) = . Also, from condition (ii), we havex yimpliesGx≺rGy, by using the deﬁnitions ofαand≺r, we haveα(x,y) =  implies α∗(Gx,Gy) = . Moreover, it is easy to check that Gis a generalized (α∗,ψ,δ )-contractive mapping. Therefore, by Theorem ., there existsx∈Xsuch that{x}=Gxif and only iff(ξ) =δ(ξ,Gξ) is lower semi-continuous atx.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally in this article.

Author details

1_{Centre for Advanced Mathematics and Physics, National University of Sciences and Technology H-12, Islamabad,}

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

Acknowledgements

Authors are grateful to referees for their suggestions and careful reading.

Received: 14 February 2013 Accepted: 10 May 2013 Published: 28 May 2013

References

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3. Kamran, T: Mizoguchi-Takahashi’s type ﬁxed point theorem. Comput. Math. Appl.57, 507-511 (2009)

doi:10.1186/1687-1812-2013-137