R E S E A R C H
Open Access
Some results on fixed points of
α
-
ψ
-Ciric
generalized multifunctions
B Mohammadi
1, S Rezapour
2and N Shahzad
3**Correspondence:
3Department of Mathematics, King
Abdulaziz University, P.O. Box 80203, Jeddah, 21859, Saudi Arabia Full list of author information is available at the end of the article
Abstract
In 2012, Samet, Vetro and Vetro introduced
α-ψ-contractive mappings and gave
some results on a fixed point of the mappings (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012). In fact, their technique generalized some ordered fixed point results (see (Alikhani et al. in Filomat, 2012, to appear) and (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012)). By using the main idea of (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012), we give some new results forα-ψ-Ciric generalized
multifunctions and some related self-maps. Also, we give an affirmative answer to a recent open problem which was raised by Haghi, Rezapour and Shahzad in 2012.
Keywords:
α-ψ-Ciric generalized multifunction; fixed point; quasi-contractive
multifunction1 Introduction
In , Samet, Vetro and Vetro introducedα-ψ-contractive mappings and gave fixed
point results for such mappings []. Their results generalized some ordered fixed point results (see []). Immediately, using their idea, some authors presented fixed point results
in the field (see, for example, []). Denote by the family of nondecreasing functions
ψ: [, +∞)→[, +∞) such thatn+=∞ψn(t) < +∞for eacht> . It is well known that
ψ(t) <tfor allt> . Let (X,d) be a metric space,β: X×X→[, +∞) be a mapping
andψ∈. A multivalued operatorT:X→X is said to beβ-ψ-contractive whenever
β(Tx,Ty)H(Tx,Ty)≤ψ(d(x,y)) for allx,y∈X, whereHis the Hausdorff distance (see []).
Alikhani, Rezapour and Shahzad proved fixed point results forβ-ψ-contractive
multi-functions []. Let (X,d) be a metric space,α:X×X→[, +∞) be a mapping andψ∈.
We say thatT:X→Xis anα-ψ-Ciric generalized multifunction if
α(x,y)H(Tx,Ty)≤ψ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for allx,y∈X. One can find an idea of this notion in []. Also, we say that the self-mapF
onXisα-admissible wheneverα(x,y)≥ impliesα(Fx,Fy)≥ []. In this paper, we give
fixed point results forα-ψ-Ciric generalized multifunctions.
In , Haghi, Rezapour and Shahzad proved that some fixed point generalizations are not real ones []. Here, by presenting a result and an example, we are going to show that obtained results in this new field are real generalizations in respect to old similar results in the literature (compare the next result and the example with main results of [] and []). The following result has been proved in [].
Lemma . Let(X,d)be a complete metric space,α:X×X→[,∞)be a function,ψ∈
and T be a self-map on X such that
α(x,y)d(Tx,Ty)≤ψ
max
d(x,y),d(x,Tx),d(y,Ty),
d(x,Ty) +d(y,Tx)
for all x,y∈X. Suppose that T isα-admissible and there exists x∈X such thatα(x,
Tx)≥.Assume that if {xn}is a sequence in X such that α(xn,xn+)≥for all n and
xn→x,thenα(xn,x)≥for all n.Then T has a fixed point.
Here, we give the following example which shows that Lemma . holds while we cannot use some similar old results, for example, Theorem of [].
Example . LetM={mn :m= , , , , . . . andn= k+ (k≥)},d(x,y) =|x–y|,M= {m
n :m= , , , , . . . andn= k+ (k≥)}andM={k:k≥}. Now, putM=M∪
M∪Mand define the self-mapT:M→MbyTx=xwheneverx∈M,Tx=x
when-ever x∈M andTx= xwhenever x∈M. PutM(x,y) =max{d(x,y),d(x,Tx),d(y,Ty),
d(x,Ty)+d(y,Tx)
}for allx,y∈M. Ifx∈M andy∈M, thend(Tx,Ty) =|
x
–
y
|= |x–
y
|.
Now, we consider two cases. Ifx>y, then we have
x–y
=
x–y
≤
x–y
=
d(x,Ty).
Hence,d(Tx,Ty)≤d(x,Ty)≤M(x,y). Ifx<y, then
x–y
= y
–x
≤(y–x)
=
d(x,y)≤ M(x,y).
Hence,d(Tx,Ty)≤
M(x,y). Ifx,y∈M, then
d(Tx,Ty) = x
–
y
=
d(x,y)≤ M(x,y).
If x,y∈M, thend(Tx,Ty) =|x – y|= d(x,y)≤ M(x,y). Defineα(x,y) = whenever
x,y∈M∪Mandα(x,y) = otherwise. Also, putψ(t) =t for allt≥. Then it is easy
to see thatα(x,y)d(Tx,Ty)≤ψ(M(x,y)) for allx,y∈M. Also,α(,T) =α(, /)≥. If
α(x,y)≥, thenx,y∈M∪M and soα(Tx,Ty)≥. Thus,T isα-admissible. It is easy
to show that if {xn} is a sequence inMsuch thatα(xn,xn+)≥ for allnandxn→x,
then α(xn,x)≥ for all n. Now, by using Theorem .,T has a fixed point. Now, we
show that Theorem in [] does not apply here. Putx= andy= . Thend(T,T) =
| – |=
,d(, ) = ,d(,T) =d(, ) =
,d(,T) =d(, ) = ,d(,T) =d(, ) =
, d(,T) =d(,) = . Thus,N(, ) = and so d(T,T) = > =N(, ), where
N(x,y) =max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}.
2 An affirmative answer for an open problem
normality []. In , Haghi, Rezapour and Shahzad proved same results without the
ad-ditional assumption and forλ∈(, ) []. Then Amini-Harandi proved a result on the
existence of fixed points of set-valued quasi-contraction maps in metric spaces by using
the technique of [] (see []). But similar to [], he could prove it only forλ∈(,)
[]. In , Haghi, Rezapour and Shahzad proved the main result of [] by using a sim-ple method []. Also, they introduced quasi-contraction type multifunctions and showed that the main result of [] holds for quasi-contraction type multifunctions. They raised an open problem about the difference between quasi-contraction and quasi-contraction type
multifunctions []. In this section, we give a positive answer to the question. Let (X,d)
be a metric space. Recall that the multifunctionT:X→X is called quasi-contraction
whenever there existsλ∈(, ) such that
H(Tx,Ty)≤λmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)
for all x,y∈X []. Also, a multifunctionT :X→X is called quasi-contraction type
whenever there existsλ∈(, ) such that
H(Tx,Ty)≤λmaxd(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)
for allx,y∈X[]. It is clear that each contraction type multifunction is a
quasi-contractive multifunction. In , Haghi, Rezapour and Shahzad raised this question that (see []): Is there any quasi-contractive multifunction which is not quasi-contraction type? By providing the following example, we give a positive answer to the problem.
Example . LetX= [, ] and d(x,y) =|x–y|. Define T :X→CB(X) byTx= [,]
wheneverx∈[, /],Tx= [,] wheneverx∈(,) andTx= [, ] wheneverx∈[, ].
We show thatTis a quasi-contraction while it is not a quasi-contractive type
multifunc-tion. PutN(x,y) =max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}for allx,y∈X. First, we
show thatTis a quasi-contraction. It is easy to check thatH(Tx,Ty)≤N(x,y) whenever
x,y∈[,] orx,y∈(,) orx,y∈[, ]. Ifx∈[,] andy∈[, ], thenH(Tx,Ty) = and
d(x,y)≥–=. Hence,H(Tx,Ty) =≤d(x,y)≤N(x,y). Ifx∈[,] andy∈(,),
thenH(Tx,Ty) = andd(x,Tx)≥. Hence,
H(Tx,Ty) =
≤
d(x,Tx)≤ N(x,y).
Ifx∈[, ] andy∈(,), thenH(Tx,Ty) =H([, ], [,]) = andd(x,Tx)≥. Hence,
H(Tx,Ty) = ≤ d(x,Tx)≤ N(x,y). Therefore, T is a quasi-contraction with λ=.
Now, putx=andy=. Then we haved(,T) =d(, [,]) =,d(,T) =d(, [, ]) =
,d(
,T
) =d(
, [
, ]) = andd(
,T
) =d(
, [,
]) = . Hence,N(
,
) =max{d( ,
T),d(,T),d(,T),d(,T)}=max{, }=. Also, we haveH(T,T) =H([,], [, ]) =. Thus,H(T,T) >λN(,) for allλ∈(, ). This shows thatTis not a
quasi-contractive type multifunction.
3 Main results
Now, we are ready to state and prove our main results. Let (X,d) be a metric space,
α:X×X→[,∞) be a mapping andT:X→CB(X) be a multifunction. We say thatX
andxn→x, there exists a subsequence{xnk}of{xn}such thatα(xnk,x)≥ for allk(see []
for the idea of this notion). Recall thatTis continuous wheneverH(Txn,Tx)→ for all
se-quence{xn}inXwithxn→x. Also, we say thatTisα-admissible whenever for eachx∈X
andy∈Txwithα(x,y)≥, we haveα(y,z)≥ for allz∈Ty. Note that this notion is
differ-ent from the notion ofα∗-admissible multifunctions which has been provided in []. But,
by providing a similar proof to that of Theorem . in [], we can prove the following result.
Theorem . Let(X,d)be a complete metric space,α:X×X→[,∞)be a function,ψ∈ be a strictly increasing map and T:X→CB(X)be anα-admissible multifunction such thatα(x,y)H(Tx,Ty)≤ψ(d(x,y))for all x,y∈X and there exist x∈X and x∈Txwith
α(x,x)≥.If T is continuous or X satisfies the condition(Cα),then T has a fixed point.
Corollary . Let(X,d)be a complete metric space,ψ∈be a strictly increasing map,
x∗∈X and T:X→CB(X)be a multifunction such that H(Tx,Ty)≤ψ(d(x,y))for all x,y∈ X with x∗∈Tx∩Ty.Suppose that there exist x∈X and x∈Txsuch that x∗∈Tx∩Tx.
Assume that for each x∈X and y∈Tx with x∗∈Tx∩Ty,we have x∗∈Ty∩Tz for all z∈Ty.
If T is continuous or X satisfies the condition(Cα),then T has a fixed point.
Proof Defineα:X×X→[, +∞) byα(x,y) = wheneverx∗∈Tx∩Tyandα(x,y) =
otherwise. Then, by using Theorem .,Thas a fixed point.
Let (X,) be an ordered set andA,B⊆X. We say thatABwhenever for eacha∈A
there existsb∈Bsuch thatab.
Corollary . Let(X,,d)be a complete ordered metric space,ψ ∈ be a strictly in-creasing map and T:X→CB(X)be a multifunction such that H(Tx,Ty)≤ψ(d(x,y))for all x,y∈X with TxTy or TyTx.Suppose that there exist x∈X and x∈Txsuch that
TxTxor TxTx.Assume that for each x∈X and y∈Tx with TxTy or TyTx,
we have TyTz or TzTy for all z∈Ty.If T is continuous or X satisfies the condition
(Cα),then T has a fixed point.
Now, we give the following result.
Theorem . Let(X,d)be a complete metric space,α:X×X→[,∞)be a function,
ψ ∈ be a strictly increasing map and T :X→CB(X)be anα-admissibleα-ψ-Ciric generalized multifunction,and let there exist x∈X and x∈Txwithα(x,x)≥.If T is
continuous,then T has a fixed point.
Proof Ifx=x, then we have nothing to prove. Letx=x. Then we have
d(x,Tx)≤α(x,x)H(Tx,Tx)
≤ψ
max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
=ψ
max
d(,x),d(x,Tx),
d(x,Tx)
≤ψ
max
d(x,x),d(x,Tx),
d(x,x) +d(x,Tx)
=ψmaxd(x,x),d(x,Tx)
If max{d(x,x),d(x,Tx)} =d(x,Tx), then d(x,Tx)≤ ψ(d(x,Tx)) and so we get
d(x,Tx) = . Thus,d(x,x) = , which is a contradiction. Hence, we obtainmax{d(x,x),
d(x,Tx)}=d(x,x) and sod(x,Tx)≤ψ(d(x,x)). Ifx∈Tx, thenxis a fixed point
ofT. Letx∈/Txandq> . Then
<d(x,Tx) <qψ
d(x,x)
.
Putt=d(x,x). Thent> andd(x,Tx) <qψ(t). Hence, there existsx∈Txsuch that
d(x,x) <qψ(t) and soψ(d(x,x)) <ψ(qψ(t)). It is clear thatx=x. Putq=ψψ((dqψ(x,(tx)))).
Thenq> and we have
d(x,Tx)≤α(x,x)H(Tx,Tx)
≤ψ
max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
=ψ
max
d(x,x),d(x,Tx),
d(x,Tx)
≤ψmaxd(x,x),d(x,Tx)
.
Similarly, we should havemax{d(x,x),d(x,Tx)}=d(x,x) and so we get
d(x,Tx)≤ψ
d(x,x)
.
Ifx∈Tx, thenxis a fixed point ofT. Letx∈/Tx. Then <d(x,Tx) <qψ(d(x,x)).
Hence, there existsx∈Txsuch thatd(x,x) <qψ(d(x,x)) =ψ(qψ(t)). It is clear that
x=xandψ(d(x,x)) <ψ(qψ(t)). Putq=ψ (qψ(t))
ψ(d(x,x)). Thenq> . Also, we have
d(x,Tx)≤α(x,x)H(Tx,Tx)
≤ψ
max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
=ψ
max
d(x,x),d(x,Tx),
d(x,Tx)
≤ψmaxd(x,x),d(x,Tx)
.
By continuing this process, we obtain a sequence{xn}inXsuch thatxn∈Txn–,xn=xn–
andd(xn,xn+)≤ψn–(qψ(t)) for alln. Letm>n. Then
d(xn,xm)≤ m–
i=n
d(xi,xi+)≤
m–
i=n
ψi–qψ(t)
and so{xn}is a Cauchy sequence inX. Hence, there existsx∈Xsuch thatxn→x. IfT
is continuous, then
dx,Tx= lim
n→∞d
xn+,Tx
≤ lim
n→∞H
Txn,Tx
=
Corollary . Let(X,d)be a complete metric space,ψ∈be a strictly increasing map,
x∗∈X and T:X→CB(X)be a multifunction such that
H(Tx,Ty)≤ψ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for all x,y∈X with x∗∈Tx∩Ty.Suppose that there exist x∈X and x∈Txsuch that
x∗∈Tx∩Tx.Assume that for each x∈X and y∈Tx with x∈Tx∩Ty,we have x∈
Ty∩Tz for all z∈Ty.If T is continuous,then T has a fixed point.
Corollary . Let(X,,d)be a complete ordered metric space, ψ∈ be a strictly in-creasing map and T:X→CB(X)be a multifunction such that
H(Tx,Ty)≤ψ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for all x,y∈X with TxTy or TyTx.Suppose that there exist x∈X and x∈Txsuch
that TxTxor TxTx.Assume that for each x∈X and y∈Tx with TxTy or Ty
Tx,we have TyTz or TzTy for all z∈Ty.If T is continuous,then T has a fixed point.
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,
φ: [,∞)→[,∞)be a continuous and nondecreasing map such thatφ(t) <t for all t>
and T be a self-map on X such that
α(x,y)d(Tx,Ty)≤φ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for all x,y∈X.Assume that there exists x∈X such thatα(Tix,Tjx)≥for all i,j≥
with i=j.Suppose that T is continuous orα(Tix
,x)≥for all i≥whenever Tix→x.
Then T has a fixed point.
Proof It is easy to check thatlimn→∞φn(t) = for allt> . Letxn=Txn–for alln≥. If
xn=xn–for somen, thenxn–is a fixed point ofT. Suppose thatxn=xn–for alln. Then
we have
d(xn+,xn) =d(Txn,Txn–)≤α(xn,xn–)d(Txn,Txn–)
≤φ
max
d(xn,xn–),d(xn,Txn),d(xn–,Txn–),
d(xn–,Txn) +d(xn,Txn–)
=φ
max
d(xn–,xn),d(xn,xn+),
d(xn–,xn+)
≤φ
max
d(xn–,xn),d(xn,xn+),
d(xn–,xn) +d(xn,xn+)
=φmaxd(xn–,xn),d(xn,xn+)
Sinced(xn+,xn)≤φ(d(xn+,xn)) does not hold, we getd(xn+,xn)≤φ(d(xn,xn–)). Sinceφ
is nondecreasing, we have
d(xn+,xn)≤φ
d(xn,xn–)
≤φd(xn–,xn–)
≤ · · · ≤φnd(x,x)
for alln. Hence,d(xn+,xn)→. If{xn}is not a Cauchy sequence, then there existsε>
and subsequences{xni}and{xmi}of{xn}withni<misuch thatd(xni,xmi) >εfor alli. For eachni, putki=min{mi|d(xni,xmi) >ε}. Then we have
ε<d(xni,xki)≤d(xni,xki–) +d(xki–,xki)≤ε+d(xki–,xki)
and sod(xni,xki)→ε. But we have
d(xni,xki) –d(xni,xni+) –d(xki,xki+)
≤d(xni+,xki+)≤d(xni,xki) +d(xni,xni+) +d(xki,xki+)
and sod(xni+,xki+)→ε. Thus, we get
max
d(xni,xki),d(xni+,xni),d(xki+,xki),
d(xni+,xki) +d(xki+,xni)
≤max
d(xni,xki),d(xni+,xni),d(xki+,xki),
d(xni+,xki+) +d(xki+,xki) +d(xki+,xki) +d(xki,xni)
≤max
d(xni,xki),d(xni+,xni),d(xki+,xki),
d(xni,xki) + d(xki+,xki) +d(xni+,xni)
≤d(xni,xki) +
d(xki+,xki) +d(xni+,xni)
and soM(xni,xki)→ε. On the other hand, we have
d(xni+,xki+) =d(Txni,Txki)≤α(xni,xki)d(Txni,Txki)≤φ
M(xni,xki)
and soε≤φ(ε). This contradiction shows that{xn}is a Cauchy sequence. SinceXis
com-plete, there existsx∈Xsuch thatxn→x. IfT is continuous, then we get
d(x,Tx) = lim
n→∞d(xn+,Tx) =n→∞limd(Txn,Tx) = .
Now, suppose thatα(Tix
,x)≥ for alli≥ wheneverTix→x. Then
d(x,Tx) = lim
n→∞d(xn+,Tx) =n→∞lim d(Txn,Tx)≤n→∞lim α(xn,x)d(Txn,Tx)
≤ lim
n→∞φ
maxd(xn,x),d(xn,xn+),d(x,Tx),
d(xn,Tx) +d(x,xn+)
/
=φd(x,Tx).
Here, we give the following example to show that there are discontinuous mappings satisfying the conditions of Theorem ..
Example . LetX= [,∞) andd(x,y) =|x–y|. DefineT:X→XbyTx=x+ whenever
x∈[, ],Tx=wheneverx∈(, ) andTx=x+ x+ wheneverx∈[,∞). Also, define
the mappingsφ: [,∞)→[,∞) andα:X×X→[,∞) byφ(t) = t,α(x,y) = whenever
x,y∈(, ) andα(x,y) = otherwise. An easy calculation shows us that
α(x,y)d(Tx,Ty)≤φ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for allx,y∈X. Putx=. SinceTix=x for alli≥,α(Tix,Tjx)≥ for alli,j≥
withi=jandα(Tix,x)≥ for alli≥ wheneverTix→x. Thus, the mapTsatisfies the
conditions of Theorem .. Note thatx=is a fixed point ofT.
Corollary . Let(X,,d)be a complete ordered metric space,φ: [,∞)→[,∞)be a continuous and nondecreasing map such thatφ(t) <t for all t> and T be a self-map on X such that
d(Tx,Ty)≤φ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for all comparable elements x,y∈X.Assume that there exists x∈X such that Tixand
Tjxare comparable for all i,j≥with i=j.Suppose that T is continuous or Tixand x
are comparable for all i≥whenever Tix→x.Then T has a fixed point.
Proof Define the mappingα:X×X→[, +∞) byα(x,y) = wheneverxandyare
com-parable andα(x,y) = otherwise. Then, by using Theorem .,Thas a fixed point.
Corollary . Let(X,,d)be a complete ordered metric space,z∈X,φ: [,∞)→[,∞)
be a continuous and nondecreasing map such thatφ(t) <t for all t> and T be a self-map on X such that
d(Tx,Ty)≤φ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
for all x,y∈X which are comparable with z.Assume that there exists x∈X such that Tix
and Tjx
are comparable with z for all i,j≥with i=j.Suppose that T is continuous or
Tix
and x are comparable with z for all i≥whenever Tix→x.Then T has a fixed
point.
Proof Define the mappingα:X×X→[, +∞) byα(x,y) = wheneverxandyare
com-parable withzandα(x,y) = otherwise. Then, by using Theorem .,Thas a fixed point.
Proposition . Let(X,d)be a complete metric space,α:X×X→[,∞)be a mapping,
ψ : [,∞)→[,∞)be a lower semi-continuous function,η: [,∞)→[,∞)be a map
such thatη–({}) ={}andlim inft→∞η(t) > for all t> and T be a self-map on X such that
ψα(x,y)d(Tx,Ty)≤ψM(x,y)–ηM(x,y)
for all x,y∈X.Assume that there exists x∈X such thatα(Tix,Tjx)≥for all i,j≥
with i=j.Suppose that T is continuous orα(Tix
,x)≥for all i≥whenever Tix→x.
Then T has a fixed point.
Proposition . Let(X,d)be a complete metric space, α:X×X→[,∞)be a map-ping, ψ ∈ be a nondecreasing map which is continuous from right at each point, φ : [,∞)→[,∞)be a map such thatφ(t) <t for all t> and T be a self-map on X such thatψ(α(x,y)d(Tx,Ty))≤φ(ψ(M(x,y))) for all x,y∈X.Assume that there exists x∈X
such that α(Tix
,Tjx)≥ for all i,j≥ with i=j. Suppose that T is continuous or
α(Tix
,x)≥for all i≥whenever Tix→x.Then T has a fixed point.
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 manuscript.
Author details
1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.2Department of
Mathematics, Azarbaijan Shahid Madani University, Tabriz, Azarshahr, Iran. 3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21859, Saudi Arabia.
Acknowledgements
The authors express their gratitude to the referees for their helpful suggestions concerning the final version of this paper.
Received: 26 July 2012 Accepted: 13 January 2013 Published: 6 February 2013
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