R E S E A R C H
Open Access
Approximate fixed points of generalized
convex contractions
Mehdi Amir Miandaragh
1, Mihai Postolache
2*and Shahram Rezapour
1*Correspondence:
[email protected] 2Department of Mathematics,
University Politehnica of Bucharest, 313 Splaiul Independen¸tei, Bucharest, 060042, Romania Full list of author information is available at the end of the article
Abstract
In this paper, we introduce the concept of generalized convex contractions and give some results about approximate fixed points of the contractions on metric spaces. By providing some examples, we show that our results are real generalization of the main results of Ghorbanianet al.(Comput. Math. Appl. 63:1361-1368, 2012) and Istratescu (I. Ann. Mat. Pura Appl. 130(4):89-104, 1982).
1 Introduction
Istratescu [] introduced the notion of convex contraction. He proved that each convex contraction has a unique fixed point on a complete metric space, see also []. Recently Ghorbanian, Rezapour and Shahzad generalized his results to complete ordered metric spaces, []. In recent years, there have appeared some works on approximate fixed point results (see, for example, [–] and the references therein). In this paper, by considering the key work [] and using the main idea of [], we introduce the concept of generalized convex contractions and generalize the main results of [] and [].
2 Preliminaries
Let (X,d) be a metric space,T be a selfmap onXandα:X×X→[,∞) be a mapping. In accordance with [], we say that T isα-admissible whenever α(x,y)≥ implies α(Tx,Ty)≥. Also, we say thatX has the property(H) whenever for eachx,y∈X, there existsz∈Xsuch thatα(x,z)≥ andα(y,z)≥; also see [].
The selfmapT onXis called ageneralized convex contractionwhenever there exist a mappingα:X×X→[,∞) anda,b∈[, ), witha+b< , such that
α(x,y)dTx,Ty≤ad(Tx,Ty) +bd(x,y)
for allx,y∈X.
We say thatαis the based mapping. Also, we say that the selfmapTonXis ageneralized convex contraction of order whenever there exist a mappingα:X×X→[,∞) and
a,a,b,b∈[, ) witha+a+b+b< such that
α(x,y)dTx,Ty≤a
d(x,Tx) +ad
Tx,Tx+b
d(y,Ty) +bd
Ty,Ty
for allx,y∈X. Other useful references: [–].
Letε> be given.x∈Xis anε-fixed pointof the selfmapTonXwheneverd(x,Tx) <
ε, see []. Denote the set of allε-fixed points ofTbyFε(T). We say thatThas an
approx-imate fixed point (orT has the approximate fixed point property) wheneverT has an ε-fixed point for allε> , see []. It is known that there are selfmaps which have approx-imate fixed points while have no fixed points.
We need the following result in our main results.
Lemma .([]) Let(X,d)be a metric space and T be an asymptotic regular selfmap on X,that is,d(Tn(x),Tn+(x))→for all x∈X.Then T has the approximate fixed point property.
3 The main results
Now, we are ready to state and prove our main results.
Theorem . Let(X,d)be a metric space and T be a generalized convex contraction on X with the based mappingα.Suppose that T isα-admissible and there exists x∈X such thatα(x,Tx)≥.
Then T has an approximate fixed point.
Moreover,T has a fixed point whenever T is continuous and(X,d)is a complete metric space,and also T has a unique fixed point whenever X has the property(H).
Proof Letx∈Xbe such thatα(x,Tx)≥. Define the sequence{xn}byxn+=Tn+xfor
alln≥.
Ifxn=xn+for somen, then we have nothing to prove.
Assume that xn =xn+ for all n≥ . Since T is α-admissible, it is easy to check
that α(xn,xn+)≥ for all n. Let v=d(Tx,Tx) +d(x,Tx) and λ= a+b. Then d(Tx,Tx)≤v. Now, putx=Txandy=x. Then
dTx,Tx
≤α(Tx,x)d
Tx,Tx
≤adTx,Tx
+bd(x,Tx)≤λv.
By continuing this process and using a similar technique to that in the proof of Theo-rem in [], it is easy to see thatd(Tm+x
,Tmx)≤λl–v, wherem= lorm= l– for
alll≥. This implies thatd(Tm+x,Tmx)→.
By using Lemma .,T has an approximate fixed point.
Also following arguments analogous to those in Theorem in [], it is easy to see that
d(Tmx
,Tnx)≤ λ
l
–λvfor alln>m. This shows that{xn}is a Cauchy sequence.
IfT is continuous and (X,d) is a complete metric, then there existsx∗∈Xsuch that
xn→x∗. Thus,Txn→Tx∗and soTx∗=x∗.
Now, suppose thatXhas also the property (H). We show thatThas a unique fixed point. Letx∗andy∗be fixed points ofT. Choosez∈Xsuch thatα(x∗,z)≥ andα(y∗,z)≥. SinceTisα-admissible,α(x∗,Tmz)≥ andα(y∗,Tmz)≥ for allm≥. Putλ=a+band v=d(x∗,Tz) +d(x∗,Tz). Then we have
dx∗,Tz=dTx∗,T(Tz)≤αx∗,TzdTx∗,T(Tz)
and
dx∗,Tz=dTx∗,TTz≤αx∗,TzdTx∗,TTz
≤adx∗,Tz+bdx∗,Tz=adx∗,Tz+abdx∗,Tz+bdx∗,Tz
≤adx∗,Tz+bdx∗,Tz+adx∗,Tz+bdx∗,Tz=λv.
Also, we have
dx∗,Tz=dTx∗,TTz≤αx∗,TzdTx∗,TTz
≤adx∗,Tz+bdx∗,Tz
≤adx∗,Tz+abdx∗,Tz+ abdx∗,Tz+bdx∗,Tz
=a+ abdx∗,Tz+ab+bdx∗,Tz≤λv
and one can easily get thatd(x∗,Tz)≤λv.
By continuing this process, we obtaind(x∗,Tmz)≤λl–v, wherem= lorm= l– for
alll≥. Hence,Tmz→x∗. Similarly, we can show thatTmz→y∗. Thus, we getx∗=y∗
and soThas a unique fixed point.
In , Haghi, Rezapour and Shahzad proved that some fixed point generalizations are not real generalizations []. But the following examples show that the notion of generalized convex contractions is a real generalization for the notions of convex contractions and ordered convex contractions which were provided, respectively, in [] and [].
Example . LetX={, , },d(x,y) =|x–y|andTbe a selfmap onXdefined byT = ,
T = andT = . Then, by puttinga=,b=,x= andy= , we have =d(T,T) > ad(T,T) +bd(, ) = . Thus,T is not a convex contraction, while by puttingα(x,y) = whenever x≤yandα(x,y) = otherwise,a= andb= , it is easy to see thatT is a generalized convex contraction.
Example . LetX={, , },d(x,y) =|x–y|. Define the order≤onXby≤={(, ), (, ), (, ), (, ), (, ), (, )}and define the selfmapTonXbyT = ,T = andT = . Then, by puttinga=andb=,x= andy= , we have
=dT,T>ad(T,T) +bd(, ) = .
Ifx= andy= orx= andy= , then we have
=dT,T>ad(T,T) +bd(, ) = .
and
=dT,T>ad(T,T) +bd(, ) = ..
Thus,T does not satisfy the condition of Theorem . in []. If we puta= andb= and defineα(x,y) = wheneverx≤yandα(x,y) = otherwise, then it is easy to see that
Theorem . Let(X,d)be a metric space and T be a generalized convex contraction of
orderon X with the based mappingα.Suppose that T isα-admissible and there exists x∈X such thatα(x,Tx)≥.
Then T has an approximate fixed point.
Moreover,T has a fixed point whenever T is continuous and(X,d)is a complete metric space,and also T has a unique fixed point whenever X has the property(H).
Proof Letx∈Xbe such thatα(x,Tx)≥. Define the sequence{xn}byxn+=Tn+xfor
alln≥.
Ifxn=xn+for somen, then we have nothing to prove.
Assume that xn=xn+ for alln≥. Since T isα-admissible, it is easy to check that
α(xn,xn+)≥ for alln. Letv=d(Tx,Tx) +d(x,Tx),β= –bandλ=a+a+b.
Then we have
dTx ,Tx
≤α(Tx,x)d
Tx ,Tx
≤ad(x,Tx) +ad
Tx,Tx
+bd
Tx,Tx
+bd
Tx,Tx
≤av+ (a+b)v+bd
Tx,Tx
.
Hence,d(Tx
,Tx)≤(λβ)v.
Now, putx=Txandy=Tx. Then
dTx,Tx
≤αTx,Tx
dTx,Tx
≤ad
Tx,Tx
+ad
Tx,Tx
+bd
Tx,Tx
+bd
Tx,Tx
≤av+ (a+b)
a+a+b
–b
v+bd
Tx,Tx
.
Hence,d(Tx
,Tx)≤(βλ)v.
Similarly, we obtaind(Tx
,Tx)≤(βλ)vandd(Tx,Tx)≤( λ β)v.
By continuing this process and following an argument similar to that in Theorem in [] (see also []), it is easy to see thatd(Tm+x
,Tmx)≤(βλ)lv, wherem= lorm= l+
for l ≥ or d(Tm+x
,Tmx)≤(λβ)l–v, where m= l or m= l– for l ≥. Thus,
d(Tm+x,Tmx)→.
By using Lemma .,Thas an approximate fixed point.
Now, suppose thatTis continuous and (X,d) is a complete metric space. Then, by using a similar technique to that in the proof of Theorem in [] (see also []), it is easy to see that{xn}is a Cauchy sequence. Choosex∗∈Xsuch thatxn→x∗. SinceTis continuous, Txn→Tx∗and soTx∗=x∗. IfXhas the property (H), then by using a similar technique to
that in the proof of Theorem ., we can prove uniqueness of the fixed point ofT.
Example . LetX={, , },d(x,y) =|x–y|andTbe a selfmap onXdefined byT = ,
T = andT = . Then, by puttinga=a=b=b=,x= andy= , we have
=dT,T>ad(,T) +ad
T,T+bd(,T) +bd
T,T= ..
Thus,Tis not a convex contraction of order , while by puttingα(x,y) = wheneverx≤y
andα(x,y) = otherwise anda=a=b=b=, it is easy to see thatTis a generalized
convex contraction of order .
Example . LetX={, , },≤={(, ), (, ), (, ), (, )},d(x,y) =|x–y|andTbe a self-map onXdefined byT = ,T = andT = . Then, by puttinga=a=b=b=, x= andy= , we have
=dT,T>ad(,T) +ad
T,T+bd(,T) +bd
T,T= ..
Thus,T is not an ordered convex contraction of order which has been used in Theo-rem . of [], while by puttingα(x,y) = wheneverx≤yandα(x,y) = otherwise and
a=a=b=b=, it is easy to check that the selfmapTis a generalized convex
contrac-tion of order .
Recently, the notion of weakly Zamfirescu mappings was provided in [] (see also Zam-firescu []).
Definition . Let (X,d) be a metric space andT be a selfmap onX. ThenT is called
weakly Zamfirescuwhenever there existsγ:X×X→[, ] with
θ(a,b) :=supγ(x,y) :a≤d(x,y)≤b<
for all <a≤b, such thatd(T(x),T(y))≤γ(x,y)MT(x,y) for allx,y∈X, where
MT(x,y) =max
d(x,y),
dx,T(y)+dy,T(x),
dx,T(x)+dy,T(y) .
Now, by using the main idea of this paper, we defineα-weakly Zamfirescu selfmaps as follows.
Let (X,d) be a metric space,α:X×X→[,∞) be a function andT be a selfmap onX. ThenTis calledα-weakly Zamfirescuwhenever there existsγ :X×X→[, ] with θ(a,b) :=sup{γ(x,y) :a≤d(x,y)≤b}< for all <a≤bsuch thatα(x,y)d(T(x),T(y))≤ γ(x,y)MT(x,y) for allx,y∈X.
Theorem . Let(X,d)be a metric space,α:X×X→[,∞)be a function and T be an
α-weakly Zamfirescu selfmap on X.Suppose that T isα-admissible and there exists x∈X such thatα(x,Tx)≥.
Then T has an approximate fixed point.
Proof Letx∈Xbe such thatα(x,Tx)≥. Define the sequence{xn}byxn+=Tn+x
for all n≥. We show thatd(xn,xn+)≤γ(xn–,xn)d(xn–,xn) for all n≥. SinceT is
α-admissible, it is easy to check thatα(xn,xn+)≥ for alln. But, for eachn, we have
MT(xn–,xn) =max
d(xn–,xn),
dxn–,T(xn–)
+dxn,T(xn)
,
dxn–,T(xn)
+dxn,T(xn–)
=max
d(xn–,xn),
d(xn–,xn) +d(xn,xn+)
,
d(xn–,xn+) +d(xn,xn)
=max
d(xn–,xn),
d(xn–,xn) +d(xn,xn+)
,
d(xn–,xn+) .
IfMT(xn–,xn) =d(xn–,xn), then
d(xn,xn+)≤α(xn–,xn)d
T(xn–),T(xn)
≤γ(xn–,xn)d(xn–,xn).
IfMT(xn–,xn) =[d(xn–,xn) +d(xn,xn+)], then
α(xn–,xn)d(xn,xn+)≤γ(xn–,xn)
[d(xn–,xn) +d(xn,xn+)]
,
d(xn,xn+)≤
γ(xn–,xn)
α(xn–,xn) –γ(xn–,xn)
d(xn–,xn)
≤γ(xn–,xn)d(xn–,xn).
IfMT(xn–,xn) =d(xn–,xn+), then
α(xn–,xn)d(xn,xn+)≤
γ(xn–,xn)
d(xn–,xn+)
≤γ(xn–,xn)
d(xn–,xn) +d(xn,xn+)
,
and so
d(xn,xn+)≤
γ(xn–,xn)
α(xn–,xn) –γ(xn–,xn)
d(xn–,xn)≤γ(xn–,xn)d(xn–,xn).
Thus, the claim is proved.
This implies that the sequence{d(xn,xn+)}is non-increasing and so it converges to the
real numberd=infn≥d(xn–,xn).
We have to show thatd= .
Letd> . Since <d≤d(xn,xn+)≤d(x,x) for alln,γ(xn–,xn)≤θ for alln, where
θ=θ(d,d(x,x)). Hence,
for alln. But this is impossible becaused> and ≤θ < . Therefore,Thas an approxi-mate fixed point.
Now, suppose that (X,d) is a complete metric space andTis continuous. Following argu-ments similar to those in Theorem of [], we can show that{xn}is a Cauchy sequence.
This implies easily thatThas a fixed point.
The following examples show that there existα-weakly Zamfirescu mappings which are not weakly Zamfirescu.
Example . LetX= [, ],d(x,y) =|x–y|, and let the selfmapT onX be defined by
T(x) =xfor allx∈X. Sinceγ(, )≥for each existent mapγin the definition of weakly Zamfirescu mapping,Tis not weakly Zamfirescu. Now, by puttingα(x,y) =
andγ(x,y) =
for allx,y∈X, it is easy to check thatTisα-weakly Zamfirescu.
Example . LetX= [,∞),d(x,y) =|x–y|, and let the selfmapT onXbe defined by
T(x) =
xwheneverx∈[, ] andT(x) =
wheneverx> . Sinceγ(, )≥
for each
exis-tent mapγ in the definition of weakly Zamfirescu mappings,Tis not weakly Zamfirescu. Now, by puttingα(x,y) = andγ(x,y) = for allx,y∈X, it is easy to check thatT is α-weakly Zamfirescu.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper together. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Azarbaijan University of Shahid Madani, Azarshahr, Tabriz, Iran.2Department of
Mathematics, University Politehnica of Bucharest, 313 Splaiul Independen¸tei, Bucharest, 060042, Romania.
Acknowledgements
Research of the first and third authors was supported by Azarbaidjan Shahid Madani University.
Received: 12 July 2013 Accepted: 16 October 2013 Published:07 Nov 2013
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Cite this article as:Miandaragh et al.:Approximate fixed points of generalized convex contractions.Fixed Point Theory
