R E S E A R C H
Open Access
The class of
(α
,
ψ
)
-type contractions in
b-metric spaces and fixed point theorems
Bessem Samet
**Correspondence: bsamet@ksu.edu.sa Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
Abstract
We study the existence and uniqueness of fixed points for self-operators defined in a b-metric space and belonging to the class of (
α
,ψ
)-type contraction mappings. The obtained results generalize and unify several existing fixed point theorems in the literature.MSC: 47H10
Keywords:
α
-ψ
-type contraction;b-metric space; fixed point; existence; uniqueness1 Introduction and preliminaries
Very recently, we studied in [] the existence and uniqueness of fixed points for self-operators defined in a metric space and belonging to the class of (α,ψ)-type contraction mappings (see [–] for some works in this direction). We proved that the class ofα-ψ -type contractions includes large classes of contraction--type operators, whose fixed points can be obtained by means of the Picard iteration. The aim of this paper is to extend the obtained results in [] to self-operators defined in ab-metric space.
We start by recalling the following definition.
Definition .([]) LetXbe a nonempty set. A mappingd:X×X→[,∞) is called b-metric if there exists a real numberb≥ such that for everyx,y,z∈X, we have
(i) d(x,y) = if and only ifx=y; (ii) d(x,y) =d(y,x);
(iii) d(x,z)≤b[d(x,y) +d(y,z)].
In this case, the pair (X,d) is called ab-metric space.
There exist many examples in the literature (see [–]) showing that the class of b -metrics is effectively larger than that of metric spaces.
The notions of convergence, compactness, closedness and completeness in b-metric spaces are given in the same way as in metric spaces. For works on fixed point theory inb-metric spaces, we refer to [–] and the references therein.
Definition .([]) Letψ: [,∞)→[,∞) be a given function. We say thatψis a com-parison function if it is increasing andψn(t)→,n→ ∞, for anyt≥, whereψnis the nth iterate ofψ.
In [, ], several results regarding comparison functions can be found. Among these we recall the following.
Lemma . Ifψ: [,∞)→[,∞)is a comparison function,then
(i) each iterateψkofψ,k≥,is also a comparison function;
(ii) ψis continuous at zero; (iii) ψ(t) <tfor anyt> ; (iv) ψ() = .
The following concept was introduced in [].
Definition . Letb≥ be a real number. A mappingψ: [,∞)→[,∞) is called a b-comparison function if
(i) ψis monotone increasing;
(ii) there existk∈N,a∈(, )and a convergent series of nonnegative terms
∞
k=vk
such that
bk+ψk+(t)≤abkψk(t) +vk
fork≥kand anyt≥.
The following lemma has been proved.
Lemma .([, ]) Letψ: [,∞)→[,∞)be a b-comparison function.Then
(i) the series∞k=bkψk(t)converges for anyt≥; (ii) the functionsb: [,∞)→[,∞)defined by
sb(t) =
∞
k=
bkψk(t), t≥
is increasing and continuous at.
Lemma .([]) Any b-comparison function is a comparison function.
Throughout this paper, forb≥, we denote bybthe set ofb-comparison functions.
Definition . Let (X,d) be ab-metric space with constantb≥, and letT:X→Xbe a given mapping. We say thatTis anα-ψcontraction if there exist ab-comparison function
ψ∈band a functionα:X×X→Rsuch that
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X. (.)
2 Main results
LetT:X→Xbe a given mapping. We denote byFix(T) the set of its fixed points; that is,
Forb≥ andψ∈b, letbψbe the set defined by
ψb =
σ∈(,∞) :σ ψ∈b
.
We have the following result.
Proposition . Let(X,d)be a b-metric space with constant b≥,and let T:X→X be a given mapping.Suppose that there existα:X×X→Randψ∈bsuch that T is anα-ψ
contraction.Suppose that there existsσ∈b
ψ and for some positive integer p,there exists
a finite sequence{ξi}pi=⊂X such that
ξ=x, ξp=Tx, α
Tnξi,Tnξi+
≥σ–, n∈N,i= , . . . ,p– ,x∈X. (.)
Then{Tnx
}is a Cauchy sequence in(X,d).
Proof Letϕ=σ ψ. By the definition ofb
ψ, we haveϕ∈b. Let{ξi}pi=be a finite sequence inXsatisfying (.). Consider the sequence{xn}n∈NinXdefined byxn+=Txn,n∈N. We
claim that
dTrξi,Trξi+
≤ϕrd(ξi,ξi+)
, r∈N,i= , . . . ,p– . (.)
Leti∈ {, , . . . ,p– }. From (.), we have
σ–d(Tξi,Tξi+)≤α(ξi,ξi+)d(Tξi,Tξi+)≤ψ
d(ξi,ξi+)
,
which implies that
d(Tξi,Tξi+)≤ϕ
d(ξi,ξi+)
. (.)
Again, we have
σ–dTξi,Tξi+
≤α(Tξi,Tξi+)d
T(Tξi),T(Tξi+)
≤ψd(Tξi,Tξi+)
,
which implies that
dTξi,Tξi+
≤ϕd(Tξi,Tξi+)
. (.)
Sinceϕis an increasing function (from Lemma .), from (.) and (.), we obtain
dTξi,Tξi+
≤ϕd(ξi,ξi+)
.
Continuing this process, by induction we obtain (.).
Now, using the property (iii) of ab-metric and (.), for everyn∈N, we have
d(xn,xn+) =d
Tnx,Tn+x
≤bdTnξ,Tnξ
+bdTnξ,Tnξ
+· · ·+bpdTnξp–,Tnξp
=
p–
i=
bi+dTnξ i,Tnξi+
≤
p–
i=
bi+ϕnd(ξi,ξi+)
.
Thus we proved that
d(xn,xn+)≤
p–
i=
bi+ϕnd(ξi,ξi+)
, n∈N,
which implies that forq≥,
d(xn,xn+q)≤ n+q–
j=n
bj–n+d(xj,xj+)
≤
n+q–
j=n
bj–n+
p–
i=
bi+ϕjd(ξi,ξi+)
= bn–
p–
i= bi+
n+q–
j=n
bjϕjd(ξi,ξi+)
≤
bn–
p–
i= bi+
∞
j=n
bjϕjd(ξi,ξi+)
.
Sinceb≥, using Lemma .(i), we obtain
bn–
p–
i= bi+
∞
j=n
bjϕjd(ξi,ξi+)
→ asn→ ∞.
This proves that{xn}is a Cauchy sequence in theb-metric space (X,d).
Our first main result is the following fixed point theorem which requires the continuity of the mappingT.
Theorem . Let(X,d)be a complete b-metric space with constant b≥,and let T:X→ X be a given mapping.Suppose that there existα:X×X→Randψ ∈bsuch that T
is anα-ψ contraction.Suppose also that(.)is satisfied.Then{Tnx
}converges to some x∗∈X.Moreover,if T is continuous,then x∗∈Fix(T).
Proof From Proposition ., we know that{Tnx
}is a Cauchy sequence. Since (X,d) is a completeb-metric space, there existsx∗∈Xsuch that
lim
n→∞d
Tnx,x∗
= .
The continuity ofTyields
lim
n→∞d
Tn+x,Tx∗
= .
In the next theorem, we omit the continuity assumption ofT.
Theorem . Let(X,d)be a complete b-metric space with constant b≥,and let T:X→ X be a given mapping.Suppose that there existα:X×X→Randψ ∈bsuch that T
is anα-ψ contraction.Suppose also that(.)is satisfied.Then{Tnx
}converges to some x∗∈X.Moreover,if there exists a subsequence{Tγ(n)x
}of{Tnx}such that
maxαTγ(n)x ,x∗
,αx∗,Tγ(n)x
≥∈(,∞), n large enough,
then x∗∈Fix(T).
Proof From Proposition . and the completeness of theb-metric space (X,d), we know that{Tnx
}converges to somex∗∈X.
Suppose now that there exists a subsequence{Tγ(n)x
}of{Tnx}such that
maxαTγ(n)x,x∗
,αx∗,Tγ(n)x
≥∈(,∞), nlarge enough. (.)
SinceTis anα-ψcontraction, we have
αTγ(n)x,x∗
dTγ(n)+x,Tx∗
≤ψdTγ(n)x,x∗
, n∈N
and
αx∗,Tγ(n)x
dTγ(n)+x,Tx∗
≤ψdTγ(n)x,x∗
, n∈N.
Thus we have
maxαTγ(n)x,x∗
,αx∗,Tγ(n)x
dTγ(n)+x,Tx∗
≤ψdTγ(n)x,x∗
, n∈N.
From (.), we get
dTγ(n)+x,Tx∗
≤ψdTγ(n)x,x∗
, nlarge enough. (.)
On the other hand, using the property (iii) of ab-metric, we get
dTγ(n)+x,Tx∗
≥
bd
x∗,Tx∗–dx∗,Tγ(n)+x
, n∈N. (.)
Now, (.) and (.) yield
bd
x∗,Tx∗–dx∗,Tγ(n)+x ≤ψ
dTγ(n)x,x∗
, nlarge enough.
Lettingn→ ∞in the above inequality, using Lemma . and Lemma .(ii) and (iv), we obtain
≤ bd
x∗,Tx∗≤ψ() = ,
We provide now a sufficient condition for the uniqueness of the fixed point.
Theorem . Let(X,d)be a b-metric space with constant b≥,and let T:X→X be a given mapping.Suppose that there existα:X×X→Randψ∈bsuch that T is anα-ψ
contraction.Suppose also that
(i) Fix(T)=∅;
(ii) for every pair(x,y)∈Fix(T)×Fix(T)withx=y,ifα(x,y) < ,then there exists
η∈bψand for some positive integerq,there is a finite sequence{ζi(x,y)}qi=⊂Xsuch that
ζ(x,y) =x, ζq(x,y) =y, α
Tnζi(x,y),Tnζi+(x,y)
≥η–
forn∈Nandi= , . . . ,q– .
Then T has a unique fixed point.
Proof Let ϕ =ηψ ∈b. Suppose that u,v∈ X are two fixed points of T such that
d(u,v) > . We consider two cases.
Case:α(u,v)≥. SinceTis anα-ψcontraction, we have
d(u,v)≤α(u,v)d(Tu,Tv)≤ψd(u,v).
On the other hand, from Lemma . and Lemma .(iii), we have
ψd(u,v)<d(u,v).
The two above inequalities yield a contradiction.
Case:α(u,v) < . By assumption, there exists a finite sequence{ζi(u,v)}qi=inXsuch that
ζ(u,v) =u, ζq(u,v) =v, α
Tnζi(u,v),Tnζi+(u,v)
≥η–
forn∈Nandi= , . . . ,q– . As in the proof of Proposition ., we can establish that
dTrζi(u,v),Trζi+(u,v)
≤ϕrdζi(u,v),ζi+(u,v)
, r∈N,i= , . . . ,q– . (.)
On the other hand, we have
d(u,v) =dTnu,Tnv
≤
q–
i=
bi+dTnζi(u,v),Tnζi+(u,v)
≤
q–
i=
bi+ϕndζi(u,v),ζi+(u,v)
→ asn→ ∞(by Lemma .).
3 Particular cases
In this section, we deduce from our main theorems several fixed point theorems in b-metric spaces.
3.1 The class of
ψ
-type contractions inb-metric spacesDefinition . Let (X,d) be ab-metric space with constantb≥. A mappingT:X→X is said to be aψ-contraction if there existsψ∈bsuch that
d(Tx,Ty)≤ψd(x,y) for allx,y∈X. (.) Theorem . Let(X,d)be a b-metric space with constant b≥,and let T :X→X be a given mapping.Suppose that there existsψ∈bsuch that T is aψ-contraction.Then
there existsα:X×X→Rsuch that T is anα-ψ contraction. Proof Consider the functionα:X×X→Rdefined by
α(x,y) = for allx,y∈X. (.)
Clearly, from (.),Tis anα-ψcontraction.
Corollary .([]) Let(X,d)be a complete b-metric space with constant b≥,and let T :X→X be a given mapping.If T is aψ-contraction for someψ∈b,then T has a
unique fixed point.Moreover,for any x∈X,the Picard sequence{Tnx}converges to this fixed point.
Proof From Lemma ., we have
d(Tx,Ty)≤d(x,y) for allx,y∈X,
which implies thatT is a continuous mapping. From Theorem .,Tis anα-ψ contrac-tion, whereα is defined by (.). Clearly, for anyx∈X, (.) is satisfied withp= and
σ = . By Theorem .,{Tnx}converges to a fixed point ofT. The uniqueness follows
immediately from (.) and Theorem ..
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T:X→ X be a given mapping.Suppose that
d(Tx,Ty)≤kd(x,y) for all x,y∈X
for some constant k∈(, /b).Then T has a unique fixed point.Moreover,for any x∈X, the Picard sequence{Tnx
}converges to this fixed point.
Proof It is an immediate consequence of Corollary . withψ(t) =kt.
3.2 The class of rational-type contractions inb-metric spaces
.. Dass-Gupta-type contraction in b-metric spaces
that
d(Tx,Ty)≤μd(y,Ty) +d(x,Tx)
+d(x,y) +λd(x,y) for allx,y∈X. (.) Theorem . Let(X,d)be a b-metric space with constant b≥,and let T:X→X be a given mapping.Suppose that T is a Dass-Gupta contraction.Then there existψ∈band
α:X×X→Rsuch that T is anα-ψ contraction. Proof From (.), for allx,y∈X, we have
d(Tx,Ty) –μd(y,Ty) +d(x,Tx)
+d(x,y) ≤λd(x,y), which yields
–μd(y,Ty)( +d(x,Tx))
( +d(x,y))d(Tx,Ty) d(Tx,Ty)≤λd(x,y), x,y∈X,Tx=Ty. (.) Consider the functionsψ: [,∞)→[,∞) andα:X×X→Rdefined by
ψ(t) =λt, t≥ (.)
and
α(x,y) =
⎧ ⎨ ⎩
–μ(+d(y,dTy(x)(+,y))dd((Txx,Tx,Ty))), ifTx=Ty,
, otherwise.
(.)
Since ≤λb< , thenψ∈b. On the other hand, from (.) we have
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X.
ThenT is anα-ψcontraction.
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T:X→ X be a given mapping.If T is a Dass-Gupta contraction with parametersλ,μ≥such that
λb+μ< ,then T has a unique fixed point.Moreover,for any x∈X,the Picard sequence
{Tnx
}converges to this fixed point.
Proof Letxbe an arbitrary point inX. If for somer∈N,Trx=Tr+x, thenTrxwill be a fixed point ofT. So we can suppose thatTrx=Tr+xfor allr∈N. From (.), for all n∈N, we have
αTnx,Tn+x
= –μd(T
n+x
,Tn+x)( +d(Tnx,Tn+x)) ( +d(Tnx
,Tn+x))d(Tn+x,Tn+x) = –μ> .
On the other hand, from (.) we have
( –μ)–ψ(t) = λ
From the condition λb+μ< , clearly we have ( –μ)–ψ ∈
b, which is equivalent to
( –μ)–∈b
ψ. Then (.) is satisfied withp= andσ= ( –μ)–. From the first part of
Theorem ., the sequence{Tnx
}converges to somex∗∈X.
Suppose thatx∗is not a fixed point ofT, that is,d(x∗,Tx∗) > . Then
Tn+x=Tx∗, nlarge enough.
From (.), we have
αx∗,Tnx
= –μd(T
nx
,Tn+x)( +d(x∗,Tx∗)) ( +d(Tnx
,x∗))d(Tn+x,Tx∗)
, nlarge enough.
On the other hand, using the property (iii) of ab-metric, we have
dTn+x,Tx∗
≥
bd
x∗,Tx∗–dx∗,Tn+x
> , nlarge enough.
Thus we have
αx∗,Tnx
≥ –μ d(T
nx
,Tn+x)( +d(x∗,Tx∗)) ( +d(Tnx
,x∗))(bd(x∗,Tx∗) –d(x∗,Tn+x))
, nlarge enough.
Since
lim
n→∞ –μ
d(Tnx
,Tn+x)( +d(x∗,Tx∗)) ( +d(Tnx
,x∗))(bd(x∗,Tx∗) –d(x∗,Tn+x)) = ,
we have
αx∗,Tnx
>
, nlarge enough.
By Theorem ., we deduce thatx∗∈Fix(T), which is a contradiction. ThusFix(T)=∅. For the uniqueness, observe that for every pair (x,y)∈Fix(T)×Fix(T) withx=y, we haveα(x,y) = . By Theorem .,x∗is the unique fixed point ofT.
Ifb= , Corollary . recovers the Dass-Gupta fixed point theorem [].
.. Jaggi-type contraction in b-metric spaces
Definition . Let (X,d) be ab-metric space with constantb≥. A mappingT:X→X is said to be a Jaggi contraction if there exist constantsλ,μ≥ withλb+μ< such that
d(Tx,Ty)≤μd(x,Tx)d(y,Ty)
d(x,y) +λd(x,y) for allx,y∈X,x=y. (.) Theorem . Let(X,d)be a b-metric space with constant b≥,and let T :X→X be a given mapping.Suppose that T is a Jaggi contraction.Then there existψ∈b andα:
X×X→Rsuch that T is anα-ψcontraction. Proof From (.), for allx,y∈Xwithx=y, we have
d(Tx,Ty) –μd(x,Tx)d(y,Ty)
which yields
–μd(x,Tx)d(y,Ty)
d(x,y)d(Tx,Ty) d(Tx,Ty)≤λd(x,y), x,y∈X,Tx=Ty. (.) Consider the functionsψ: [,∞)→[,∞) andα:X×X→Rdefined by
ψ(t) =λt, t≥ (.)
and
α(x,y) =
⎧ ⎨ ⎩
–μdd((xx,,Txy)d)(dTx(y,,TyTy)), ifTx=Ty,
, otherwise.
(.)
Sinceλb< , we haveψ∈b. From (.), we have
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X.
ThenT is anα-ψcontraction.
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T : X→X be a continuous mapping.If T is a Jaggi contraction with parametersλ,μ≥such that λb+μ< ,then T has a unique fixed point. Moreover,for any x∈X,the Picard sequence{Tnx
}converges to this fixed point.
Proof Letxbe an arbitrary point inX. Without loss of generality, we can suppose that Trx
=Tr+xfor allr∈N. From (.), for alln∈N, we have
αTnx,Tn+x
= –μd(T
nx
,Tn+x)d(Tn+x,Tn+x) d(Tnx
,Tn+x)d(Tn+x,Tn+x)
= –μ> .
On the other hand, from (.), for allt≥, we have
( –μ)–ψ(t) = λ –μt.
Sinceλb+μ< , we have ( –μ)–ψ ∈
b, that is, ( –μ)–∈bψ. Then (.) is satisfied
withp= andσ = ( –μ)–. By the first part of Theorem .,{Tnx}converges to some x∗∈X. SinceT is continuous, by the second part of Theorem .,x∗is a fixed point ofT. Moreover, for every pair (x,y)∈Fix(T)×Fix(T) withx=y, we haveα(x,y) = . Then, by
Theorem .,x∗is the unique fixed point ofT.
Ifb= , Corollary . recovers the Jaggi fixed point theorem [].
3.3 The class of Berinde-type mappings inb-metric spaces
Definition . Let (X,d) be ab-metric space with constantb≥. A mappingT:X→X is said to be a Berinde-type contraction if there existλ∈(, /b) andL≥ such that
Theorem . Let(X,d)be a b-metric space with constant b≥,and let T:X→X be a given mapping.If T is a Berinde-type contraction,then there existα:X×X→Rand
ψ∈bsuch that T is anα-ψ contraction.
Proof From (.), we have
d(Tx,Ty) –Ld(y,Tx)≤λd(x,y) for allx,y∈X, which yields
–L d(y,Tx)
d(Tx,Ty) d(Tx,Ty)≤λd(x,y), x,y∈X,Tx=Ty. (.) Consider the functionsψ: [,∞)→[,∞) andα:X×X→Rdefined by
ψ(t) =λt, t≥ and
α(x,y) =
⎧ ⎨ ⎩
–Ldd((Txy,Tx,Ty)), ifTx=Ty,
, otherwise. (.)
Sinceλb< , thenψ∈b. From (.), we have
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X.
ThenT is anα-ψcontraction.
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T : X→X be a given mapping.If T is a Berinde-type contraction with parametersλ,L≥ such that <λb< ,then for any x∈X,the Picard sequence{Tnx}converges to a fixed point of T.
Proof Letxbe an arbitrary point inX. Without loss of generality, we can suppose that Trx
=Tr+xfor allr∈N. From (.), for alln∈N, we have
αTnx,Tn+x
= –Ld(T
n+x
,Tn+x) d(Tn+x
,Tn+x) = .
Then (.) holds withσ= andp= . From the first part of Theorem ., the sequence
{Tnx
}converges to somex∗∈X.
Suppose now thatx∗is not a fixed point ofT, that is,d(x∗,Tx∗) > . Then
Tn+x=Tx∗, nlarge enough.
From (.), we have
αTnx,x∗
= –L d(x ∗,Tn+x
) d(Tn+x
,Tx∗)
Using the property (iii) of ab-metric, we have
dTn+x,Tx∗
≥
bd
x∗,Tx∗–dx∗,Tn+x
> , nlarge enough.
Thus we have
αTnx,x∗
≥ –L d(x ∗,Tn+x
)
bd(x∗,Tx∗) –d(x∗,Tn+x)
, nlarge enough.
Since
lim
n→∞ –L
d(x∗,Tn+x )
bd(x∗,Tx∗) –d(x∗,Tn+x)
= ,
then
αTnx,x∗
>
, nlarge enough.
By Theorem ., we deduce thatx∗∈Fix(T), which is a contradiction.
Thusx∗is a fixed point ofT.
Ifb= , Corollary . recovers the Berinde fixed point theorem [].
Note that a Berinde mapping need not have a unique fixed point (see [], Exam-ple .).
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T : X→X be a given mapping.Suppose that there exists a constant k∈(, /b(b+ ))such that
d(Tx,Ty)≤kd(x,Tx) +d(y,Ty) for all x,y∈X. (.)
Then,for any x∈X,the Picard sequence{Tnx}converges to a fixed point of T.
Proof At first, observe that from (.), for allx,y∈X, we have
d(Tx,Ty)≤λd(x,y) +Ld(y,Tx),
where
λ= kb
–kb and L= kb –kb·
With the conditionk∈(, /b(b+ )), we have <λ< /bandL≥. ThenTis a Berinde-type contraction. From Corollary ., ifx∈X, then{Tnx}converges to a fixed point
ofT.
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T : X→X be a given mapping.Suppose that there exists a constant k∈(, /b)such that
d(Tx,Ty)≤kd(x,Ty) +d(y,Tx) for all x,y∈X. (.)
Then,for any x∈X,the Picard sequence{Tnx}converges to a fixed point of T.
Proof From (.), we have
d(Tx,Ty)≤λd(x,y) +Ld(y,Tx),
where
λ= kb
–kb and L=
k(b+ ) –kb ·
With the conditionk∈(, /b), we have <λ< /bandL≥. ThenTis a Berinde-type contraction. From Corollary ., ifx∈X, then{Tnx}converges to a fixed point ofT.
Ifb= , Corollary . recovers the Chatterjee fixed point theorem [].
3.4 ´Ciri´c-type mappings inb-metric spaces
Definition . Let (X,d) be ab-metric space with constantb≥. A mappingT:X→X is said to be a Ćirić-type mapping if there existsλ∈(, /b) such that for allx,y∈X, we have
mind(Tx,Ty),d(x,Tx),d(y,Ty)–mind(x,Ty),d(y,Tx)≤λd(x,y). (.)
Theorem . Let(X,d)be a b-metric space with constant b≥,and let T:X→X be a given mapping.If T is a Ćirić-type mapping with parameterλ∈(, /b),then there exist
α:X×X→Randψ∈bsuch that T is anα-ψ contraction.
Proof Consider the functionsψ: [,∞)→[,∞) andα:X×X→Rdefined by
ψ(t) =λt, t≥ (.)
and
α(x,y) =
⎧ ⎨ ⎩
min{,dd((Txx,Tx,Ty)),dd(Tx(y,Ty,Ty))}–min{dd((Txx,Ty,Ty)),dd((Txy,Tx,Ty))}, ifTx=Ty,
, otherwise. (.)
From (.), we have
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X, (.)
Corollary . Let(X,d)be a complete b-metric space with constant b≥,and let T : X→X be a continuous mapping.If T is a Ćirić-type mapping with parameterλ∈(, /b), then for any x∈X,the Picard sequence{Tnx}converges to a fixed point of T.
Proof Letx∈Xbe an arbitrary point. Without loss of generality, we can suppose that Trx=Tr+xfor allr∈N. From (.), for alln∈N, we have
αTnx,Tn+x
=min
, d(T
nx
,Tn+x) d(Tn+x
,Tn+x) ,d(T
n+x
,Tn+x) d(Tn+x
,Tn+x)
–min
d(Tnx
,Tn+x) d(Tn+x
,Tn+x) ,d(T
n+x
,Tn+x) d(Tn+x
,Tn+x)
=min
, d(T
nx
,Tn+x) d(Tn+x
,Tn+x)
.
Suppose that for somen∈N, we have
αTnx,Tn+x
= d(T
nx
,Tn+x) d(Tn+x
,Tn+x) .
In this case, from (.) and (.), we have
dTnx,Tn+x
≤λdTnx,Tn+x
.
This implies (from the assumptionTrx=Tr+xfor allr∈N) thatλ≥, which is a con-tradiction. Then
αTnx,Tn+x
= for alln∈N.
Then (.) is satisfied withp= andσ= . By Theorem ., we deduce that the sequence
{Tnx
}converges to a fixed point ofT.
Ifb= , Corollary . recovers Ćirić’s fixed point theorem [].
3.5 Edelstein fixed point theorem inb-metric spaces
Another consequence of our main results is the following generalized version of Edelstein fixed point theorem [] inb-metric spaces.
Corollary . Let (X,d) be a complete b-metric space with constant b ≥ , and
ε-chainable for someε> ;i.e.,given x,y∈X,there exist a positive integer N and a se-quence{xi}Ni=⊂X such that
x=x, xN =y, d(xi,xi+) <ε for i= , . . . ,N– . (.) Let T:X→X be a given mapping such that
Proof It is clear from (.) that the mappingTis continuous. Now, consider the function
α:X×X→Rdefined by
α(x,y) =
⎧ ⎨ ⎩
, ifd(x,y) <ε,
, otherwise. (.)
From (.), we have
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X.
Letx∈X. Forx=xandy=Tx, from (.) and (.), for some positive integerp, there exists a finite sequence{ξi}pi=⊂Xsuch that
x=ξ, ξp=Tx, α(ξi,ξi+)≥ fori= , . . . ,p– . Now, leti∈ {, . . . ,p– }be fixed. From (.) and (.), we have
α(ξi,ξi+)≥ ⇒ d(ξi,ξi+) <ε
⇒ d(Tξi,Tξi+)≤ψ
d(ξi,ξi+)
≤d(ξi,ξi+) <ε
⇒ α(Tξi,Tξi+)≥.
Again,
α(Tξi,Tξi+)≥ ⇒ d(Tξi,Tξi+) <ε
⇒ dTξi,Tξi+
≤ψd(Tξi,Tξi+)
≤d(Tξi,Tξi+) <ε
⇒ αTξi,Tξi+
≥.
By induction, we obtain
αTnξi,Tn+ξi+
≥ for alln∈N.
Then (.) is satisfied withσ= . From Theorem ., the sequence{Tnx
}converges to a fixed point ofT. Using a similar argument, we can see that condition (ii) of Theorem . is satisfied, which implies thatThas a unique fixed point.
3.6 Contractive mapping theorems inb-metric spaces with a partial order
Let (X,d) be ab-metric space with constantb≥, and letbe a partial order onX. We denote
=(x,y)∈X×X:xyoryx.
Corollary . Let T:X→X be a given mapping.Suppose that there existsψ∈bsuch
that
Suppose also that
(i) T is continuous;
(ii) for some positive integerp,there exists a finite sequence{ξi}pi=⊂Xsuch that
ξ=x, ξp=Tx,
Tnξi,Tnξi+
∈, n∈N,i= , . . . ,p– . (.) Then{Tnx
}converges to a fixed point of T.
Proof Consider the functionα:X×X→Rdefined by
α(x,y) =
⎧ ⎨ ⎩
, if (x,y)∈,
, otherwise. (.)
From (.), we have
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X.
Then the result follows from Theorem . withσ= . Corollary . Let T:X→X be a given mapping.Suppose that
(i) there existsψ∈bsuch that(.)holds;
(ii) condition(.)holds.
Then{Tnx
}converges to some x∗∈X.Moreover,if
(iii) there exist a subsequence{Tγ(n)x
}of{Tnx}andN∈Nsuch that
Tγ(n)x,x∗
∈, n≥N,
then x∗is a fixed point of T.
Proof We continue to use the same functionαdefined by (.). From the first part of Theorem ., the sequence{Tnx
} converges to somex∗∈X. From (iii) and (.), we have
αTγ(n)x,x∗
= , n≥N.
By the second part of Theorem . (with = ), we deduce that x∗ is a fixed point
ofT.
The next result follows from Theorem . withη= .
Corollary . Let T:X→X be a given mapping.Suppose that
(i) there existsψ∈bsuch that(.)holds;
(ii) Fix(T)=∅;
(iii) for every pair(x,y)∈Fix(T)×Fix(T)withx=y,if(x,y) /∈,there exist a positive integerqand a finite sequence{ζi(x,y)}qi=⊂Xsuch that
ζ(x,y) =x, ζq(x,y) =y,
Tnζi(x,y),Tnζi+(x,y)
∈
forn∈Nandi= , . . . ,q– .
Observe that in our results we do not suppose thatTis monotone orTpreserves order as it is supposed in many papers (see [–] and others).
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (12-MAT 2895-02).
Received: 11 March 2015 Accepted: 3 June 2015
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