R E S E A R C H
Open Access
The
(
α
,
β
)
-generalized convex contractive
condition with approximate fixed point
results and some consequence
Abdul Latif
1, Aphinat Ninsri
2*and Wutiphol Sintunavarat
2* *Correspondence:[email protected]; [email protected]
2Department of Mathematics and
Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand
Full list of author information is available at the end of the article
Abstract
The aim of this work is to introduce some new notions of generalized convex
contraction mappings and establish some approximate fixed point theorems for such mappings in the setting of complete metric spaces. Examples and application to approximate fixed point results for cyclic mappings are also given in order to illustrate the effectiveness of the obtained results.
MSC: 47H09; 47H10
Keywords: approximate fixed point; contraction mapping; (
α
,β
)-generalized convex contraction mapping; cyclic generalized convex contraction mappings1 Introduction
It is well known that fixed point theory is one of the important tools for solving various problems in nonlinear analysis and various fields of applied mathematical analysis. The Banach contraction mapping principle presented by Banach [] in his thesis is one of the cornerstones in the development of fixed point theory. This principle has been used to solve several problems such as the existence and uniqueness problems for a solution of nonlinear integral equations and nonlinear differential equations. Furthermore, it can be applied to the convergence theorem for solving some problems in computational math-ematics. Hence, a large number of researchers have focused on the development of this topic. For instant, one of an interesting directions is the extension of fixed point results to approximate fixed point results. Indeed, in various practical situations, some conditions in the fixed point results are too strong and thus the existence of a fixed point is not guaran-teed. In this case, one can consider points close to fixed points, which we call approximate fixed points.
On the other hand, the concept of convex contractions was introduced by Istratescu [] in . He also proved that each convex contraction self mapping on a complete metric space has a unique fixed point. Recently, Miandaraghet al.[, ] introduced two more general concepts of convex contractions, which are called generalized convex contractions and generalized convex contractions of order and they also discussed some approximate fixed point results for such mappings.
The purpose of this paper is to formulate the concepts of generalized convex contrac-tions and generalized convex contraccontrac-tions of order in general terms and prove the
tence results of approximate fixed points for these mappings on a complete metric space by using the idea of cyclic (α,β)-admissible mappings due to Alizadehet al.[]. We fur-nish an illustrative example to demonstrate the validity of the hypotheses and the degree of utility of our results. Our result extends, unifies, and generalizes various well-known fixed point and approximate fixed point results such as the Banach contraction mapping prin-ciple [], Kannan’s fixed point results [], fixed point and approximate fixed point results for convex contraction mappings due to Istratescu [], and many results in the literature. As a consequence of the presented results, the approximate fixed point results for cyclic mappings are also given in order to illustrate the effectiveness of the obtained results.
2 Preliminaries
In this section, we give some definitions, examples, and remarks which are useful for the main results in this paper. Throughout this paper,Z+denotes the set of positive integers andRdenotes the set of real numbers.
Definition .([]) Let (X,d) be a metric space,T:X→Xbe a mapping andε> be a given real number. A pointx∈Xis said to be anε-fixed point(approximate fixed point)
ofTif
d(x,Tx) <ε.
Remark . We observe that a fixed point is an ε-fixed point, whereεis an arbitrary positive real number. However, the converse is not true.
For a metric space (X,d) and a givenε> , the set of allε-fixed points of a mapping
T:X→Xis denoted by
Fε(T) :=
x∈X|d(x,Tx) <ε.
Definition .([]) Let (X,d) be a metric space andT:X→Xbe a mapping. We say that
Thas theapproximate fixed point propertyif for allε> , there exists anε-fixed point ofT, that is,
∀ε> , Fε(T)=∅,
or, equivalently,
inf
x∈Xd(x,Tx) = .
Example . LetX= (, )\{ln() }and a metricdonXbe defined byd(x,y) =|x–y|for allx,y∈X. Define a mappingT:X→XbyTx=xexfor allx∈X. ThenTdoes not have
a fixed point butThas the approximate fixed point property. Indeed,
inf
x∈Xd(x,Tx) =xinf∈X x–x
e
Figure 1 Geometry for Example 2.4.
Figure 2 Geometry for Example 2.5.
For instance, forε= there isx= . such that
d(x,Tx) =
. –. e
(.)< =ε
(see Figure ).
Example . LetX= (, ) and a metricdonXbe defined byd(x,y) =|x–y|for allx,y∈X. Define a mappingT:X→XbyTx=xsin(x) for allx∈X. ThenTdoes not have a fixed
point butT has the approximate fixed point property. Indeed,
inf
x∈Xd(x,Tx) =xinf∈X
x–xsinx= .
For instance, forε= there isx= . such that
d(x,Tx) =. – (.)sin
(.)< =ε
(see Figure ).
Definition .([]) A self mappingTon a metric space (X,d) is said to beasymptotically regularat a pointx∈Xif
dTnx,Tn+x→ asn→ ∞,
whereTnxdenotes thenth iterate ofTatx.
It is not hard to prove the following results.
Lemma . Let(X,d)be a metric space and T:X→X be an asymptotically regular at some point z∈X.Then T has the approximate fixed point property.
In , Alizadehet al.[] introduced the notions of cyclic (α,β)-admissible mappings as follows.
Definition . LetXbe a nonempty set andα,β:X→[,∞) be two given functions. A mapping T:X→Xis said to be acyclic(α,β)-admissible mapping if the following conditions hold:
(i) α(x)≥for somex∈Ximpliesβ(Tx)≥; (ii) β(y)≥for somey∈Ximpliesα(Ty)≥.
Example . LetT:R→Rbe defined by
Tx=
⎧ ⎨ ⎩
–x, x∈[,∞),
–x, otherwise.
Assume thatα,β:R→[,∞) are defined by
α(x) =
⎧ ⎨ ⎩
ex, x∈[,∞),
, otherwise,
and
β(y) =
⎧ ⎨ ⎩
e–y, y∈(–∞, ],
, otherwise.
If α(x) =ex≥, thenx≥, which impliesTx≤. Therefore,β(Tx) =e–Tx≥. Also, if β(y) =e–y≥, theny≤, which impliesTy≥. So,α(Ty) =eTy≥. ThenT is a cyclic (α,β)-admissible mapping.
3 Main results
Definition . Let (X,d) be a metric space. The mappingT:X→Xis called an (α,β) -generalized convex contractionif there exist mappingsα,β:X→[,∞) anda,b∈[,∞) witha+b< , satisfying the following condition:
for allx,y∈X, α(x)β(y)≥ ⇒ dTx,Ty≤ad(Tx,Ty) +bd(x,y). (.) Now, we establish a new approximate fixed point theorem for (α,β)-generalized convex contraction mappings in complete metric spaces.
Theorem . Let(X,d)be a metric space and T:X→X be an(α,β)-generalized convex contraction mapping.Assume that T is a cyclic(α,β)-admissible mapping and there exists x∈X such thatα(x)≥andβ(x)≥.Then T has the approximate fixed point property.
Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point.
Proof First of all, letx∈Xsuch thatα(x)≥ andβ(x)≥. Define the sequence{xn}
inXbyxn+=Tn+xfor alln∈Z+∪ {}. Ifxn=xn+for somen∈Z+∪ {}, thenxn is
a fixed point ofT. So, we assume thatxn=xn+for alln∈Z+∪ {}. By the definition of a
cyclic (α,β)-admissible mapping, we have
α(x)≥ ⇒ β(x)≥ ⇒ α(x)≥ ⇒ · · ·,
β(x)≥ ⇒ α(x)≥ ⇒ β(x)≥ ⇒ · · ·.
Therefore,α(xn)≥ andβ(xn)≥ for alln∈Z+∪ {}. Let us denote
v:=dTx,Tx
+d(x,Tx)
and
γ :=a+b.
Sinceα(x)β(x)≥, we get
dTx,Tx
≤adTx,Tx
+bd(x,Tx)
=γv. Also, sinceα(x)β(x)≥, we get
dTx,Tx
≤adTx,Tx
+bdTx,Tx
≤adTx,Tx
+bd(x,Tx) +bd
Tx,Tx
≤ad(x,Tx) +ad
Tx,Tx
+bd(x,Tx) +bd
Tx,Tx
=γv.
By continuing this process, we get
dTmx,Tm+x
≤
⎧ ⎨ ⎩
Thus it follows thatd(Tmx
,Tm+x)→ asm→ ∞. So,T is an asymptotically regular
at a point x∈X. By using Lemma ., we see thatT has the approximate fixed point
property.
Next, we will show thatThas a fixed point provided that (X,d) is a complete metric space andTis continuous. Now, we will prove that{xn}is a Cauchy sequence inX. Without loss
of generality, we may assume thatm,n∈Z+such thatn>m> . We distinguish two cases
as follows.
Case: Letmbe an even number. That is,m= l, wherel∈Z+. Therefore,
dTmx,Tnx
≤dTmx,Tm+x
+dTm+x,Tm+x
+· · ·+dTn–x,Tnx
≤γlv+γlv+γl+v+γl+v+γl+v+· · · ≤γlv+ γl+v+ γl+v+· · ·
= γ
lv
–γ.
Case: Letmbe an odd number. That is,m= l+ , wherel∈Z+. Therefore,
dTmx,Tnx
≤dTmx,Tm+x
+dTm+x,Tm+x
+· · ·+dTn–x,Tnx
≤γlv+γl+v+γl+v+γl+v+γl+v+· · · ≤γlv+ γl+v+ γl+v+· · ·
= γ
lv
–γ.
It follows that
dTmx,Tnx
≤
⎧ ⎨ ⎩
γm/v
–γ ifm= , , , . . . ,
γ(m–)/v
–γ ifm= , , , . . . . Therefore,d(Tmx
,Tnx)→ asm,n→ ∞, that is,{xn}is a Cauchy sequence inX. By
using the completeness ofX, there existsx∗∈Xsuch thatxn→x∗asn→ ∞. SinceTis
continuous, we obtain
x∗= lim
n→∞xn+=nlim→∞Txn=Tx
∗
and thusThas a fixed point. This completes the proof. Next we give an example to illustrate the usability of Theorem ..
Example . LetX= [,∞) andd:X×X→Rbe defined byd(x,y) =|x–y|for allx,y∈X. DefineT:X→Xandα,β:X→[,∞) by
Tx=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x+
, ifx∈[, ],
√
x, ifx∈(, ),
x– x+ , otherwise,
α(x) =
⎧ ⎨ ⎩
x+
, ifx∈[, ],
and
β(x) =
⎧ ⎨ ⎩
x+
, ifx∈[, ],
, otherwise.
It is easy to see that (X,d) is a complete metric space andT is continuous.
Now, we show that Theorem . can guarantee the existence of fixed point ofT. First of all, we will show thatTis an (α,β)-generalized convex contraction mapping witha= andb=.
Forx,y∈Xwithα(x)β(y)≥, we havex,y∈[, ] and thus
dTx,Ty=T
x+
–T
y+
=x+ –
y+
= |x–y| ≤
|x–y| =
x+ –y+
+
|x–y| =ad(Tx,Ty) +bd(x,y).
This shows thatTis an (α,β)-generalized convex contraction mapping witha=andb=
. Clearly,Tis a cyclic (α,β)-admissible mapping. It is easy to see that there isx= ∈X
such that
α(x) =α() = .≥ and β(x) =β() = /≥. (.)
By using Theorem ., we see thatThas a fixed point inX.
Corollary . Let(X,d)be a metric space,α,β:X→[,∞)be two mappings and T :
X→X be a mapping such that
α(x)β(y)dTx,Ty≤ad(Tx,Ty) +bd(x,y) (.)
for all x,y∈X,where a,b∈[, )with a+b< .Assume that T is a cyclic(α,β)-admissible mapping and there exists x∈X such thatα(x)≥andβ(x)≥.Then T has the
ap-proximate fixed point property.Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point.
Proof We will show thatTis an (α,β)-generalized convex contraction mapping. Suppose thatx,y∈Xwithα(x)β(y)≥ and then
dTx,Ty≤α(x)β(y)dTx,Ty
This implies thatTis an (α,β)-generalized convex contraction mapping. By Theorem .,
we get the desired result.
Corollary . Let(X,d)be a metric space, α,β:X→[,∞)be two mappings and T:
X→X be a mapping such that
dTx,Ty+τα(x)β(y)≤ad(Tx,Ty) +bd(x,y) +τ (.)
for all x,y∈X,where a,b∈[, )with a+b< andτ ≥.Assume that T is an(α,β) -admissible mapping and there exists x∈X such thatα(x)≥andβ(Tx)≥.Then
T has the approximate fixed point property.Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point.
Proof We will show thatTis an (α,β)-generalized convex contraction mapping. Suppose thatx,y∈Xwithα(x)β(y)≥ and hence
dTx,Ty+τ ≤dTx,Ty+τα(x)β(y) ≤ad(Tx,Ty) +bd(x,y) +τ.
This implies that
dTx,Ty≤ad(Tx,Ty) +bd(x,y),
that is,T is an (α,β)-generalized convex contraction mapping. By Theorem ., we get
the desired result.
Corollary . Let(X,d)be a metric space,α,β :X→[,∞)be two mappings and T :
X→X be a mapping such that
τ– +α(x)β(y)d(Tx,Ty)≤τad(Tx,Ty)+bd(x,y) (.)
for all x,y∈X,where a,b∈[, )with a+b< andτ> .Assume that T is a cyclic(α,β) -admissible mapping and there exists x∈X such thatα(x)β(Tx)≥.Then T has the
approximate fixed point property.Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point.
Proof We will show thatTis an (α,β)-generalized convex contraction mapping. Suppose thatx,y∈Xwithα(x)β(y)≥ and hence
τd(Tx,Ty)≤τ– +α(x)β(y)d(Tx,Ty) ≤τad(Tx,Ty)+bd(x,y).
This implies that
that is,T is an (α,β)-generalized convex contraction mapping. By Theorem ., we get
the desired result.
We observe that the Banach contractive condition due to Banach [] implies the con-tractive condition (.) wheneverα,β:X→[,∞) defined byα(x) =β(x) = for allx∈X. From the previous observation, we get the following result.
Corollary . Let(X,d)be a metric space and T:X→X be a Banach contraction map-ping,i.e.,there exists k∈[, )such that
d(Tx,Ty)≤kd(x,y) (.)
for all x,y∈X.Then T has the approximate fixed point property.Moreover,if(X,d)is a complete metric space,then T has a fixed point.
Next, we introduce the concept of an (α,β)-generalized convex contraction of order and also establish a new approximate fixed point theorem for such mappings in complete metric spaces.
Definition . Let (X,d) be a metric space. The mappingT:X→Xis called an (α,β) -generalized convex contraction of order if there exist mappingsα,β :X→[,∞) and
a,a,b,b∈[, ) witha+a+b+b< , satisfying the following condition:
for allx,y∈X, α(x)β(y)≥
⇒ dTx,Ty≤ad(x,Tx) +ad
Tx,Tx+bd(y,Ty) +bd
Ty,Ty. (.)
Theorem . Let(X,d)be a metric space and T:X→X be an(α,β)-generalized convex contraction mapping of order.Assume that T is a cyclic(α,β)-admissible mapping and there exists x∈X such thatα(x)≥andβ(x)≥.Then T has the approximate fixed
point property.Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point.
Proof First of all, letx∈Xsuch thatα(x)≥ andβ(x)≥. Define the sequence{xn}
inXbyxn+=Tn+xfor alln∈Z+∪ {}. Ifxn=xn+for somen∈Z+∪ {}, thenxnis a
fixed point ofT. So, we may assume thatxn=xn+ for alln∈Z+∪ {}. It follows fromT
being a cyclic (α,β)-admissible mapping that
α(x)≥ ⇒ β(x)≥ ⇒ α(x)≥ ⇒ · · ·,
β(x)≥ ⇒ α(x)≥ ⇒ β(x)≥ ⇒ · · ·.
Therefore,α(xn)≥ andβ(xn)≥ for alln∈Z+∪ {}. Let us denote
w:=dTx,Tx
+d(x,Tx),
and
:=a+a+b.
Sinceα(x)β(x)≥, we get
dTx,Tx
≤ad(x,Tx) +ad
Tx,Tx
+bd
Tx,Tx
+bd
Tx,Tx
≤aw+ (a+b)w+bd
Tx,Tx
.
This implies thatd(Tx
,Tx)≤δw. Also, sinceα(x)β(x)≥, we get
dTx,Tx
≤ad
Tx,Tx
+ad
Tx,Tx
+bd
Tx ,Tx
+bd
Tx ,Tx
≤aw+ (a+b)
δw+bd
Tx,Tx
≤aw+ (a+b)w+bd
Tx,Tx
.
It means thatd(Tx
,Tx)≤δw. By continuing this process, we get
dTmx,Tm+x ≤ ⎧ ⎨ ⎩ ( δ)
(m–)/v ifm= , , , . . . ,
(δ)m/v ifm= , , , . . . .
This follows thatd(Tmx
,Tm+x)→ asm→ ∞and thusTis an asymptotically regular
at a point x∈X. By using Lemma ., we see thatT has the approximate fixed point
property.
Next, we show thatT has a fixed point provided that (X,d) is a complete metric space andTis continuous. First, we will claim that{xn}is a Cauchy sequence inX. Letm,n∈Z+
such thatn>m> . Now, we distinguish the following cases.
Case:mis even number such thatm= l, wherel∈Z+. Therefore,
dTmx,Tnx
≤dTmx,Tm+x
+dTm+x,Tm+x
+· · ·+dTn–x,Tnx ≤ δ l w+ δ l w+ δ l+ w+ δ l+ w+ δ l+
w+· · ·
≤
δ
l
w+
δ
l+
w+
δ
l+
w+· · ·
= ( δ)
lw
–δ .
Case:mis odd number such thatm= l+ , wherel∈Z+. Therefore,
dTmx,Tnx
≤dTmx,Tm+x
+dTm+x,Tm+x
+· · ·+dTn–x,Tnx ≤ δ l w+ δ l+ w+ δ l+ w+ δ l+ w+ δ l+
≤
δ
l
w+
δ
l+
w+
δ
l+
w+· · ·
= ( δ)
lw
–δ .
Therefore, we can conclude that
dTmx,Tnx
≤
⎧ ⎨ ⎩
(δ)m/w
–δ ifm= , , , . . . , (δ)(m–)/w
–δ ifm= , , , . . . .
This implies that{xn}is a Cauchy sequence inX. By the completeness ofX, there exists
x∗∈Xsuch thatxn→x∗asn→ ∞. As follows fromT being continuous, we getxn+=
Txn→Tx∗asn→ ∞. By the uniqueness of the limit{xn}, we obtainTx∗=x∗and thusT
has a fixed point. This completes the proof.
Corollary . Let(X,d)be a metric space,α,β:X→[,∞)be two mappings and T:
X→X be a mapping such that
α(x)β(y)dTx,Ty≤ad(x,Tx) +ad
Tx,Tx+bd(y,Ty) +bd
Ty,Ty (.)
for all x,y∈X,where a,a,b,b∈[, )with a+a+b+b< .Assume that T is a cyclic
(α,β)-admissible mapping and there exists x∈X such thatα(x)≥andβ(x)≥.Then
T has the approximate fixed point property.Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point.
Corollary . Let(X,d)be a metric space,α,β:X→[,∞)be two mappings and T :
X→X be a mapping such that
dTx,Ty+τα(x)β(y) ≤ad(x,Tx) +ad
Tx,Tx+bd(y,Ty) +bd
Ty,Ty+τ (.)
for all x,y∈X,where a,a,b,b∈[, )with a+a+b+b< andτ≥.Assume that
T is a cyclic(α,β)-admissible mapping and there exists x∈X such thatα(x)≥and
β(x)≥.Then T has the approximate fixed point property.Moreover,if T is continuous
and(X,d)is a complete metric space,then T has a fixed point.
Corollary . Let(X,d)be a metric space,α,β:X→[,∞)be two mappings and T:
X→X be a mapping such that
τ– +α(x)β(y)d(Tx,Ty)≤τad(x,Tx)+ad(Tx,Tx)+bd(y,Ty)+bd(Ty,Ty) (.)
for all x,y∈X,where a,a,b,b∈[, )with a+a+b+b< andτ> .Assume that
β(x)≥.Then T has the approximate fixed point property.Moreover,if T is continuous
and(X,d)is a complete metric space,then T has a fixed point.
The following result is a special case of Theorem . because the Kannan contractive condition due to Kannan [] implies the contractive condition (.) ifα,β:X→[,∞) defined byα(x) =β(x) = for allx∈X.
Corollary . Let(X,d)be a metric space and T:X→X be a Kannan contraction map-ping,i.e.,there exists k∈[, /)such that
d(Tx,Ty)≤kd(x,Tx) +d(y,Ty) (.)
for all x,y∈X.Then T has the approximate fixed point property.Moreover,if T is contin-uous and(X,d)is a complete metric space,then T has a fixed point.
Corollaries . and ., are interesting for defining the concepts of other classes of non-linear mappings which are generalizations of several well-known mappings due to Chat-terjea [], Ćirić [], Geraghty [], Meir and Keeler [], Mizoguchi and Takahashi [], Suzuki [],etc.
4 Some cyclic contractions via cyclic(
α
,β
)-admissible mappingsIn this section, we introduce the concept of cyclic generalized convex contraction map-pings and prove some approximate fixed point results for such mapmap-pings in complete met-ric spaces.
Definition . LetAandBbe two nonempty closed subsets of a metric space (X,d). The mappingT:A∪B→A∪Bis called acyclic generalized convex contractionif the following conditions hold:
(i) T(A)⊆BandT(B)⊆A,
(ii) there exista,b∈[,∞)witha+b< , satisfying the following condition:
dTx,Ty≤ad(Tx,Ty) +bd(x,y) (.)
for allx∈A,y∈B.
Example . LetX=Randd:X×X→Rbe defined byd(x,y) =|x–y|for allx,y∈X. DefineA= [–, ],B= [, ], andT:A∪B→A∪BbyTx= –x. We will show that the mappingT is a cyclic generalized convex contraction witha= andb=. Assume that
x∈Aandy∈B. Then we have
dTx,Ty=T
–x
–T
–y
=x –
y
= |x–y| ≤
=
x–y
+
|x–y| =ad(Tx,Ty) +bd(x,y).
Therefore,Tis a cyclic generalized convex contraction mapping witha=andb=. Now, we establish approximate fixed point theorems for cyclic generalized convex con-traction mappings as follows.
Theorem . Let A and B be two nonempty closed subsets of a metric space(X,d)such that A∩B=∅and T:A∪B→A∪B be a cyclic generalized convex contraction mapping.
Then T has the approximate fixed point property.Moreover,if T is continuous and(X,d)
is a complete metric space,then T has a fixed point in A∩B.
Proof Define two mappingsα,β:X→[,∞) by
α(x) =
⎧ ⎨ ⎩
, ifx∈A, , otherwise,
and
β(x) =
⎧ ⎨ ⎩
, ifx∈B, , otherwise.
Next, we will show thatTis an (α,β)-generalized convex contraction mapping. Assume thatx,y∈A∪Bsuch thatα(x)β(y)≥ and thenx∈Aandy∈B. Therefore, we have
dTx,Ty≤ad(Tx,Ty) +bd(x,y)
and so the contractive condition (.) holds.
Now we will claim thatT is a cyclic (α,β)-admissible mapping. Assume thatα(x)≥ for somex∈Xand thenx∈A. By condition (i) in Definition ., we getTx∈Band so β(Tx)≥. On the other hand, we may assume thatβ(y)≥ for somey∈Xand soy∈B. Again, by using the condition (i) in Definition ., we get Tx∈A and thenα(Tx)≥. ThereforeT is a cyclic (α,β)-admissible mapping. SinceA∩Bis nonempty, there exists
x∈A∩Bsuch thatα(x)≥ andβ(x)≥. From Theorem ., we can conclude thatT
has the approximate fixed point property.
Finally, we will show thatThas a fixed point provided thatTis continuous and (X,d) is a complete metric space. SinceAandBare two closed subsets of complete metric space (X,d), we see thatA∪Bis also a complete metric space. From Theorem ., we see that
Thas a fixed point inA∪B, sayz. Ifz∈A, then we havez=Tz∈B. Also, ifz∈B, we have
Definition . LetAandBbe two closed subsets of a metric space (X,d). The mapping
T:A∪B→A∪Bis called acyclic generalized convex contraction of order if the following conditions hold:
(i) T(A)⊆BandT(B)⊆A,
(ii) there exista,a,b,b∈[, )witha+a+b+b< , satisfying the following
condition:
dTx,Ty≤ad(x,Tx) +ad
Tx,Tx+bd(y,Ty) +bd
Ty,Ty (.)
for allx∈A,y∈B.
By a similar technique to the proof of Theorem ., we get the following result.
Theorem . Let A and B be two nonempty closed subsets of a metric space(X,d)such that A∩B=∅and T:A∪B→A∪B be a cyclic generalized convex contraction mapping of order.Then T has the approximate fixed point property.Moreover,if T is continuous and(X,d)is a complete metric space,then T has a fixed point in A∩B.
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, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.2Department of
Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand.
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support. The third author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
Received: 9 October 2015 Accepted: 22 April 2016
References
1. Banach, S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math.3, 133-181 (1922)
2. Istratescu, VI: Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters, I. Ann. Mat. Pura Appl.130(4), 89-104 (1982)
3. Miandaragh, MA, Postolache, M, Rezapour, S: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl.2013, 255 (2013)
4. Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalizedα-contractive mappings. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys.75(2), 3-10 (2013)
5. Alizadeh, S, Moradloub, F, Salimic, P: Some fixed point results for (α,β), (ψ,φ)-contractive mappings. Filomat28(3), 635-647 (2014)
6. Kannan, R: Some results on fixed points II. Am. Math. Mon.76, 405-408 (1969)
7. Reich, S: Approximate selections, best approximations, fixed points, and invariant sets. J. Math. Anal. Appl.62, 104-113 (1978)
8. Kohlenbach, U, Leustean, L: The approximate fixed point property in product spaces. Nonlinear Anal.66, 806-818 (2007)
9. Browder, FE, Petrysyn, WV: The solution by iteration of nonlinear functional equation in Banach spaces. Bull. Am. Math. Soc.72, 571-576 (1966)
10. Chatterjea, SK: Fixed point theorems. C. R. Acad. Bulgare Sci.25, 727-730 (1972) 11. ´Ciri´c, LB: A generalization of Banach principle. Proc. Am. Math. Soc.45, 267-273 (1974) 12. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc.40, 604-608 (1973)
13. Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl.28, 326-329 (1969)
14. Mizoguchi, N, Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl.141, 177-188 (1989)
