R E S E A R C H
Open Access
(α
,
ψ
,
ξ
)
-contractive multivalued mappings
Muhammad Usman Ali
1, Tayyab Kamran
2and Erdal Karapınar
3**Correspondence:
3Department 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 ofα-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 ofα-ψ-contractive closed valued multifunctions.
MSC: 47H10; 54H25
Keywords:
α
∗-admissible mappings; (α,ψ,ξ
)-contractive mappings1 Introduction and preliminaries
Recently, Sametet al. [] introduced the notions ofα-ψ-contractive andα-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. Aslet al.[] extended these notions to multifunctions by introducing the notions ofα∗-ψ-contractive andα∗-admissible mappings and proved some fixed-point results. Some results in this direction are also given in [–]. Ali and Kamran [] further generalized the notion ofα∗-ψ-contractive mappings and obtained some fixed-point the-orems for multivalued mappings. Salimiet al.[] modified the notions ofα-ψ-contractive andα-admissible self-mappings by introducing another functionηand established some fixed-point theorems for such mappings in complete metric spaces. N. Hussainet al.[] extended these modified notions to multivalued mappings. Recently, Mohammadi and Rezapour [] showed that the results obtained by Salimiet al.[] follow from correspond-ing results forα-ψ-contractive mappings. More recently, Berzig and Karapinar [] proved that the first main result of Salimiet 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 ofα-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 byCB(X) the class of all nonempty closed and bounded subsets ofXand byCL(X) the class of all nonempty closed subsets ofX. For everyA,B∈CL(X), let
H(A,B) =
⎧ ⎨ ⎩
max{supx∈Ad(x,B),supy∈Bd(y,A)}, if the maximum exists;
∞, otherwise.
Such a mapHis called the generalized Hausdorff metric induced by the metricd. Letbe a set of nondecreasing functions,ψ: [,∞)→[,∞) such that∞n=ψn(t) <∞for each
t> , whereψnis thenth iterate ofψ. It is known that for eachψ∈, we haveψ(t) <t
for allt> andψ() = fort= . More details as regards such a function can be found in
e.g.[, ].
Definition .[] Let (X,d) be a metric space andα:X×X→[,∞) be a mapping. A mappingG:X→CL(X) isα∗-admissible if
α(x,y)≥ ⇒ α∗(Gx,Gy)≥,
whereα∗(Gx,Gy) =inf{α(a,b) :a∈Gx,b∈Gy}.
2 Main results
We begin this section by considering a familyof functionsξ: [,∞)→[,∞) satisfying the following conditions:
(i) ξ is continuous;
(ii) ξ is nondecreasing on[,∞); (iii) ξ() = andξ(t) > for allt∈(,∞); (iv) ξ is subadditive.
Example . Suppose thatφ: [,∞)→[,∞) 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.
Defineξ: [,∞)→[,∞) byξ(t) =tφ(w)dwfor eacht∈[,∞). Thenξ∈.
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 haveξ(d(x,B)) > . We choose
= (q– )ξd(x,B) .
By the definition of an infimum, sinceξ◦dis a metric space, it follows that there exists
y∈Bsuch that
Definition . Let (X,d) be a metric space. A mappingG:X→CL(X) is called (α,ψ,ξ )-contractive if there exist three functionsψ∈,ξ∈andα:X×X→[,∞) such that
x,y∈X, α(x,y)≥ ⇒ ξH(Gx,Gy) ≤ψξM(x,y) , (.)
whereM(x,y) =max{d(x,y),d(x,Gx),d(y,Gy),d(x,Gy)+d(y,Gx)}.
In case whenψ∈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∈Gxsuch thatα(x,x)≥;
(iii) Gis continuous. Then G has a fixed point.
Proof By hypothesis, there existx∈Xandx∈Gxsuch thatα(x,x)≥. Ifx=x, then
we have nothing to prove. Letx=x. Ifx∈Gx, thenx is a fixed point. Letx∈/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)
=ψξmaxd(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)
≤ψξmaxd(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)
. (.)
Forq> by Lemma ., there existsx∈Gxsuch that
<ξd(x,x) <qξ
d(x,Gx). (.)
From equations (.) and (.), we have
<ξd(x,x) <qψ
ξd(x,x)
Applyingψin equation (.), we have
<ψξd(x,x)
<ψqψξd(x,x)
. (.)
Putq=ψ(ψqψ(ξ((ξd((dx(x,x,x))))))). Thenq> . SinceGis anα∗-admissible mapping, thenα∗(Gx,
Gx)≥. Thus we haveα(x,x)≥α∗(Gx,Gx)≥. Ifx∈Gx, thenxis a fixed point.
Letx∈/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)
=ψξmaxd(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)
≤ψξmaxd(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)
. (.)
Forq> by Lemma ., there existsx∈Gxsuch that
<ξd(x,x) <qξ
d(x,Gx) . (.)
From equations (.) and (.), we have
<ξd(x,x) <qψ
ξd(x,x)
=ψqψξd(x,x)
. (.)
Applyingψin equation (.), we have
<ψξd(x,x)
<ψqψξd(x,x)
. (.)
Putq=ψ
(qψ(ξ(d(x,x))))
ψ(ξ(d(x,x))) . Thenq> . SinceGis anα∗-admissible mapping, thenα∗(Gx,
Gx)≥. Thus we haveα(x,x)≥α∗(Gx,Gx)≥. Ifx∈Gx, thenxis a fixed point.
Letx∈/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)
=ψξmaxd(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)
≤ψξmaxd(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)
. (.)
Forq> by Lemma ., there existsx∈Gxsuch that
<ξd(x,x) <qξ
d(x,Gx). (.)
From equations (.) and (.), we have
<ξd(x,x) <qψ
ξd(x,x)
=ψqψξd(x,x)
. (.)
Continuing the same process, we get a sequence{xn}inXsuch thatxn+∈Gxn,xn+=xn,
α(xn,xn+)≥, and
ξd(xn+,xn+) <ψn
qψξd(x,x)
for eachn∈N∪ {}. (.)
Letm>n, we have
ξd(xm,xn) ≤ m–
i=n
ξd(xi,xi+) <
m–
i=n
ψi–qψξd(x,x)
.
Sinceψ∈, 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 existsx∗∈X such thatxn→x∗asn→ ∞. SinceGis 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∈Gxsuch 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 inX withxn→x∗asn→ ∞andα(xn,xn+)≥ for eachn∈N∪ {}. By hypothesis (iii), we
haveα(xn,x∗)≥ for eachn∈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 thatd(x∗,Gx∗)= . We letxn→x∗. Taking =d(x
∗,Gx∗)
we can findN∈Nsuch that
dx∗,xm <
d(x∗,Gx∗)
for eachm≥N. (.)
Moreover, as{xn}is a Cauchy sequence, there existsN∈Nsuch that
d(xm,Gxm)≤d(xm,xm+) <
d(x∗,Gx∗)
for eachm≥N. (.)
Furthermore,
dx∗,Gxm ≤d
x∗,xm+ <
d(x∗,Gx∗)
for eachm≥N. (.)
Asd(xm,Gx∗)→d(x∗,Gx∗). Taking =d(x
∗,Gx∗)
we can findN∈Nsuch that
dxm,Gx∗ <
d(x∗,Gx∗)
for eachm≥N. (.)
It follows from equations (.), (.), (.), and (.) that
max
dxm,x∗ ,d(xm,Gxm),dx∗,Gx∗ ,d(xm,Gx
∗) +d(x∗,Gxm)
=dx∗,Gx∗ ,
form≥N=max{N,N,N}. Moreover, form≥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∗ . (.)
Lettingm→ ∞in the above inequality, we have
ξdx∗,Gx∗ ≤ψξdx∗,Gx∗ . (.)
This is not possible ifξ(d(x∗,Gx∗)) > . Therefore, we haveξ(d(x∗,Gx∗)) = , which implies
thatd(x∗,Gx∗) = ,i.e.,x∗=Gx∗.
Example . LetX=Rbe endowed with the usual metricd. DefineG:X→CL(X) by
Gx=
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(–∞, ] ifx< ,
{,x
} if ≤x< ,
{} ifx= , [x,∞) ifx>
andα:X×X→[,∞) by
α(x,y) =
⎧ ⎨ ⎩
ifx,y∈[, ],
otherwise.
Takeψ(t) = t andξ(t) =√tfor eacht≥. ThenGis an (α,ψ,ξ)-contractive mapping. Forx= and ∈Gxwe have α(, ) = . Also, for eachx,y∈X with α(x,y) = , we
haveα∗(Gx,Gy) = . Moreover, for any sequence{xn}inX withxn→xasn→ ∞and α(xn,xn+) = for eachn∈N∪ {}, we haveα(xn,x) = for eachn∈N∪ {}. Therefore, all
conditions of Theorem . are satisfied andGhas infinitely many fixed points. Note that Nadler’s fixed-point theorem is not applicable here; see, for example,x= . andy= .
3 Consequences
It can be seen, by restrictingG:X→Xand takingξ(t) =tfor eacht≥ in Theorems . and ., that:
• Theorem . and Theorem . of Sametet al.[] are special cases of Theorem . and Theorem ., respectively;
• Theorem . of Aslet al.[] is a special case of Theorem .; • Theorem . of Amiriet al.[] is a special case of Theorem .;
• Theorem . of Salimiet al.[] is a special case of Theorems . and ..
Further, it can be seen, by consideringG:X→CB(X) andξ(t) =tfor eacht≥, that
• Theorem . and Theorem . of Mohammadiet al.[] are special cases of our results;
• Theorem . of Amiriet al.[] is a special case of Theorem ., whenψ∈is sublinear.
Remark . Note that in Example .,ξ(t) =√t. Therefore, one may not apply the afore-mentioned results and, as a consequence, conclude thatGhas 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
1Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12,
Islamabad, Pakistan. 2Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.3Department 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
