R E S E A R C H
Open Access
Remarks on contractive type mappings
Jarosław Górnicki
**Correspondence:
Department of Mathematics and Applied Physics, Rzeszów University of Technology, P.O. Box 85, Rzeszów, Poland
Abstract
We present an analog of Banach’s fixed point theorem forCJMcontractions in preordered metric spaces. These results substantially extend theorems of Ran and Reurings’ (Proc. Am. Math. Soc. 132(5): 1435-1443, 2003) and Nieto and
Rodríguez-López (Order 22: 223-239, 2005).
MSC: Primary 47H10; secondary 54H25
Keywords: contractive map; fixed point; preordered space;G-metric space
1 Introduction
In , Ran and Reurings [] established a fixed point theorem that extends the Banach contraction principle (BCP) to the setting of partially ordered metric spaces. In the original version, Ran and Reurings used a continuous function. In , Nieto and Rodríguez-López [] established a similar result replacing the continuity of the nonlinear operator by monotonicity. The key feature in this theorems is that the contractivity condition on the nonlinear map is only assumed to hold on elements that are comparable in the partial order.
As an application the authors obtained a theorem on the existence of a unique solu-tion for periodic boundary problems relative to ordinary differential equasolu-tions. Similar applications for a mixed monotone mapping were given by Gnana Bhaskar and Laksh-mikantham []. Further improvements of the above results were found by Petruşel and Rus [] and Jachymski [].
In this paper we extend fixed point theorems established by Ran and Reurings and Ni-eto and Rodríguez-López toCJMcontractions on preordered metric spaces, where a pre-ordered binary relation is weaker than a partial order.
2 Definitions
We have to introduce many types of contractions.
Definition . LetT be a mapping on a metric space (X,d).
. T is said to be a (usual)contraction(C, for short) [], if there existsλ∈[, )such that d(Tx,Ty)≤λd(x,y) for any x,y∈X.
. Tis said to be aBrowder contraction(Bro, for short) [], if there exists a functionϕ
from(,∞)into itself satisfying the following:
(a) ϕis non-decreasing and right continuous, (b) ϕ(t) <tfor anyt∈(,∞),
(c) d(Tx,Ty)≤ϕ(d(x,y))for anyx,y∈X.
. T is said to be aBoyd-Wong contraction(BoWo, for short) [], if there exists a functionϕfrom(,∞)into itself satisfying the following:
(a) ϕis upper semicontinuous from the right,i.e.,
λi↓λ≥⇒lim supi→∞ϕ(λi)≤ϕ(λ), (b) ϕ(t) <tfor anyt∈(,∞),
(c) d(Tx,Ty)≤ϕ(d(x,y))for anyx,y∈X.
. Tis said to be anRi contraction(Ri, for short) [], if there exists a functionϕfrom
[,∞)into itself satisfying the following: (a) lim sups→t+ϕ(s) <tfor anyt∈(,∞), (b) ϕ(t) <tfor anyt∈(,∞),
(c) d(Tx,Ty)≤ϕ(d(x,y))for anyx,y∈X.
. T is said to be aMeir-Keeler contraction(MeKe, for short) [], if for anyε> , there existsδ> such that, for allx,y∈X,
ε≤d(x,y) <ε+δ implies d(Tx,Ty) <ε.
. Tis said to be aMatkowski contraction(Mat, for short) [], if there exists a function
ϕfrom(,∞)into itself satisfying the following: (a) ϕis non-decreasing,
(b) limn→∞ϕn(t) = for everyt∈(,∞), (c) d(Tx,Ty)≤ϕ(d(x,y))for anyx,y∈X.
. T is said to be aWardowski contraction(War, for short) [], if there exists a function F: (,∞)→Rsatisfying the following:
(a) Fis strictly increasing,
(b) for any sequence{αn}of positive numbers,
lim
n→∞αn= iff nlim→∞F(αn) = –∞,
(c) there existsk∈(, )such thatlimα→+αkF(α) = ,
(d) for somet> , ifTx=Ty, then
t+Fd(Tx,Ty)≤Fd(x,y).
. T is said to be aĆirić-Jachymski-Matkowski contraction(CJM, for short) [–], if the following hold:
(a) for everyε> , there existsδ> such that, for allx,y∈X,
d(x,y) <ε+δ implies d(Tx,Ty)≤ε,
(b) x=y implies d(Tx,Ty) <d(x,y).
We know the following implications:
C⇒Bro⇒Mat⇒CJM.
Recently Suzuki [] proved the following implications:
Ri⇒BoWo,
War⇒CJK,
and
War+Fis continuous ⇒Bro,
War+Fis left-continuous ⇒Mat,
War+Fis right-continuous ⇒MeKe.
Remark . It is not difficult to prove that conditions (a) and (d) of Definition .() and the assumption thatFis continuous or upper semicontinuous from the right is sufficient for the existence and uniqueness of fixed point ofT ifT maps a complete metric space (X,d) into itself (compare [] Theorem .).
3 Fixed point theorems
In this section we extended fixed point theorems established by Ran and Reurings and Nieto and Rodríguez-López toCJMcontractions on preordered metric spaces, where a preordered binary relation is weaker than a partial order.
Definition . LetX=∅be a set. Binary relationonXis
(a) reflexiveifxxfor allx∈X,
(b) transitiveifxzfor allx,y,z∈Xsuch thatxyandyz.
A reflexive and transitive relation onXis apreorderedonX. In such a case (X,) is a
preordered space. Writex≺ywhenxyandx=y.
Example . Letbe the binary relation onRgiven by
xy ⇔ (x=yorx<y≤).
Thenis a partial order (and so preordered) onR, but it is different from≤.
Definition . Anpreordered metric spaceis a triple (X,d,) where (X,d) is a metric space andis a preorder onX.
One of the most important hypotheses that we shall use in this section is the mono-tonicity of the involved mappings.
Definition . Letbe a binary relation onXandT:X→Xbe a mapping. We say that
T is-non-decreasingifTxTyfor allx,y∈Xsuch thatxy.
all sequences{xn} ⊂Asuch that{xn} →x∈Aandxnxn+for alln∈N, we havexnx for alln∈N.
The following result is the extension of Ran and Reurings’ result toCJMcontraction on preordered metric spaces.
Theorem . Let(X,d,)be a preordered metric space and let T:X→X be a mapping.
Suppose that the following conditions hold:
(i) (X,d)is complete, (ii) Tis-non-decreasing, (iii) Tis continuous,
(iv) there existsx∈Xsuch thatxTx,
(v) for allx,y∈Xwithxy,
(a) for everyε> ,there existsδ> such that,
d(x,y) <ε+δ implies d(Tx,Ty)≤ε,
(b) x=y implies d(Tx,Ty) <d(x,y).
Then there exists a fixed point of T,and it is unique,say u,if for every(x,y)∈X×X there exists w∈X such that xw and yw.Moreover,for each x∈X such that xTx,the
sequence{Tnx
}of iterates converges to u.
Proof Letx∈Xbe a point satisfying (iv), that is,xTx. We define a sequence{xn} ⊂X as follows:
xn=Txn–, n≥. ()
Considering thatTis a-non-decreasing mapping together with () we have
xTx=x implies x=TxTx=x.
Inductively, we obtain
xxx· · ·xn–xnxn+· · ·. ()
Assume that there existsnsuch thatxn=xn+. Sincexn=xn+=Txn, soxn is the
fixed point ofT. Suppose thatxn=xn+for alln≥. Then by (),
x≺x≺x≺ · · · ≺xn–≺xn≺xn+≺ · · ·,
andd(xn,xn+) =d(Txn–,Txn) > for alln≥. Using (b) the following holds, for every n≥:
Hence the sequence{d(xn,xn+)}is monotone decreasing and bounded below, thus it is convergent and we letlimn→∞d(xn,xn+) =d≥. Ifd> we obtain a contradiction. In-deed, then there existsj∈Nsuch that
d<d(xn,xn+) <d+δ(d) for n≥j.
It follows from (a) that
d(xn+,xn+)≤d for n≥j,
which contradicts the above inequality, and therefore
lim
n→∞d(xn,xn+) = .
Now, we show that{d(xn,xn+)}is a Cauchy sequence. Fix anε> . Without loss of gener-ality we may assume thatδ=δ(ε) <ε. Sinced(xn,xn+)↓, there existsj∈Nsuch that
d(xn,xn+) <
δ for n≥j. ()
We shall apply induction to show that, for anym∈N,
d(xj,xj+m) <ε+
δ. ()
Obviously, () holds form= ; see (). Assuming () to hold for somem, we shall prove it form+ . By the triangle inequality, we have
d(xj,xj+m+)≤d(xj,xj+) +d(xj+,xj+m+).
Using (), observe that it suffices to show thatd(xj+,xm+j+)≤ε. By the induction hypoth-esis d(xj,xj+m) <ε+δ<ε+δ. So, by (a),d(xj+,xj+m+)≤ε, completing the induction. Obviously, () implies that{xn}is a Cauchy sequence inX.
From the completeness ofXthere existsu∈Xsuch that{xn} →u. The continuity ofT
yields
d(u,Tu) = lim
n→∞d(xn,Txn) =nlim→∞d(xn,xn+) = ,
sou=Tu.
To prove uniqueness, we assume thatv∈Xis another fixed point ofTsuch thatu=v. By hypothesis, there existsw∈Xsuch thatuwandvw.
Let {wn =Twn–} be the Picard sequence of T based on w=w. As T is -non-decreasing,v=TvTw=w andu=TuTw=w. By induction,vwnanduwn for alln≥.
Case. Ifv=wnfor somen≥, thenv=Tv=Twn=wn+and by induction,wn=v
for alln≥n, so{wn} →v.
Thus{wn} →vand{wn} →u. The uniqueness of the limit concludes thatu=v, soT
has a unique fixed point.
Remark . Theorem . remains true in the more general case, on satisfying the follow-ing (see [] the proof of Theorem ):
(v) for allx,y∈Xwithxy,
(a) for everyε> , there existsδ> such that
max
d(x,y),d(x,Tx),d(y,Ty),
d(x,Ty)+d(y,Tx)<ε+δ implies d(Tx,Ty)≤ε,
(b) x=y implies
d(Tx,Ty) <max
d(x,y),d(x,Tx),d(y,Ty),
d(x,Ty) +d(y,Tx).
From Theorem . we get the following corollary.
Corollary .([], Theorem .) Let(X,d,)be a preordered metric space and let T :
X→X be a mapping.Suppose that the following conditions hold:
(i) (X,d)is complete, (ii) Tis-non-decreasing, (iii) Tis continuous,
(iv) there existsx∈Xsuch thatxTx,
(v) there existsλ∈[, )such that,for allx,y∈Xwithxy,
d(Tx,Ty)≤λd(x,y). ()
Then there exists a fixed point of T,and it is unique,say u,if for every(x,y)∈X×X there exists w∈X such that xw and yw.Moreover,for each x∈X such that xTx,the
sequence{Tnx
}of iterates converges to u.
Observe that the BCP is stronger than Corollary ., which only requires the inequality for comparable points, that is, for allx,y∈Xsuch thatxy.
Example . LetX=Rbe the set of all real numbers endowed with the metricd(x,y) =
|x–y|for allx,y∈R. Consider onRthe partial order
xy ⇔ (x=yorx<y≤).
DefineT:R→Rby
Tx=
⎧ ⎨ ⎩ x
ifx≤, x ifx> .
Letx,y∈Xbe such thatxy. Ifx=y, then
trivially holds. Assume thatx=y. Theny<x≤. Hence
d(Tx,Ty) =x –
y
=
|x–y|= d(x,y).
Hence () holds. However, Definition .() is false in this case because ifx= andy= , then
d(T,T) =| – |= = d(, ).
Although the BCP is not applicable, Corollary . guarantees thatT has a unique fixed point, which isu= .
After the appearance of the Ran and Reurings’ result, Nieto and Rodríguez-López ex-changed the continuity of the mapping T with the condition non-decreasing regularity (Definition .). The following result is an extension of Nieto and Rodríguez-López’ the-orem to aCJMcontraction on preordered metric spaces.
Theorem . Let(X,d,)be a preordered metric space and let T:X→X be a mapping.
Suppose that the following conditions hold:
(i) (X,d)is complete, (ii) Tis-non-decreasing,
(iii) (X,d,)is non-decreasing regular, (iv) there existsx∈Xsuch thatxTx,
(v) for allx,y∈Xwithxy,
(a) for everyε> ,there existsδ> such that
d(x,y) <ε+δ implies d(Tx,Ty)≤ε,
(b) x=y implies d(Tx,Ty) <d(x,y).
Then there exists a fixed point of T,and it is unique,say u,if for every(x,y)∈X×X there exists w∈X such that xw and yw.Moreover,for each x∈X such that xTx,the
sequence{Tnx
}of iterates converges to u.
Proof Following the proof of Theorem ., we have a-non-decreasing sequence{xn= Txn–}which is convergent tou∈X. Due to (iii), we havexnufor alln≥. Now, we show thatuis a fixed point ofT. Fix anε> . Without loss of generality we may assume thatδ<ε. Sinced(u,xn)→ asn→ ∞, there existsj∈Nsuch that
d(u,xn) <δ for n≥j.
So, by (a),
d(Tu,Txn)≤ε.
Therefore, by the triangle inequality, takingn≥jand usingxnufor alln, we get
d(u,Tu)≤d(u,xn+) +d(xn+,Tu)
Sinceε> is arbitrary,d(Tu,u) = . Henceu=Tu. Uniqueness ofucan be observed as in
the proof of Theorem ..
Corollary .([], Theorem .) Let(X,d,)be a preordered metric space and let T:
X→X be a mapping.Suppose that the following conditions hold:
(i) (X,d)is complete, (ii) Tis-non-decreasing,
(iii) (X,d,)is non-decreasing regular, (iv) there existsx∈Xsuch thatxTx,
(v) there existsλ∈[, )such that,for allx,y∈Xwithxy, d(Tx,Ty)≤λd(x,y).
Then there exists a fixed point of T,and it is unique,say u,if for every(x,y)∈X×X there exists w∈X such that xw and yw.Moreover,for each x∈X such that xTx,the
sequence{Tnx
}of iterates converges to u.
4 G-metric spaces
We show the natural extension of Ran-Reurings’ and Nieto-Rodríguez-López’ results to the setting ofG-metric spaces.
In , Mustafa and Sims [] introduced a new class of generalized metric spaces which are calledG-metric spaces as a generalization of metric spaces. Subsequently, many fixed point results on such spaces appeared; see []. Here, we present the necessary defi-nitions and results, which will be useful for the rest of the paper. However, for more details, we refer to [].
Definition .([]) LetXbe a nonempty set. A functionG:X×X×X→[, +∞) sat-isfying the following axioms:
(G) G(x,y,z) = ifx=y=z,
(G) G(x,x,y) > for allx,y∈Xwithx=y,
(G) G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithz=y,
(G) G(x,y,z) =G(x,z,y) =G(y,z,x) =· · · (symmetry in all three variables),
(G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X
is called aG-metriconXand the pair (X,G) is called aG-metric space.
Definition .([]) Let (X,G) be aG-metric space, and let{xn}be a sequence of points ofX, therefore, we say that{xn}isG-convergenttox∈Xiflimn,m→∞G(x,xn,xm) = , that
is, for anyε> , there existsN∈Nsuch thatG(x,xn,xm) <εfor alln,m≥N. We callxthe limitof the sequence and writexn→xorlimn→∞xn=x.
Proposition .([]) Let(X,G)be a G-metric space.The following statements are equiv-alent:
Definition .([]) Let (X,G) be aG-metric space. A sequence{xn}inXis called a G-Cauchysequence if for anyε> there isN∈Nsuch thatG(xn,xm,xk) <εfor alln,m,k≥ N, that is,G(xn,xm,xk)→ asn,m,k→ ∞.
Definition .([]) A G-metric space (X,G) is called G-completeif everyG-Cauchy sequence isG-convergent in (X,G).
Definition .([]) Let (X,G) and (X,G) beG-metric spaces andf : (X,G)→(X,G) be a function, thenf is said to beG-continuous at a point a∈Xif and only if, givenε> , there existsδ> such thatx,y∈XandG(a,x,y) <δimpliesG(f(a),f(x),f(y)) <ε. A func-tionf isG-continuous at Xif and only if it isG-continuous at alla∈X.
Definition . ([]) A G-metric space (X,G) is said to be symmetric if G(x,y,y) =
G(y,x,x) for allx,y∈X.
In the symmetric case, many fixed point theorems onG-metric spaces are particular cases of existing fixed point theorems in metric spaces. Also in the non-symmetry case (such spaces have a quasi-metric structure), many fixed point theorems follows directly from existing fixed point theorems on metric spaces. A key role in this case is played by the following theorem (see [, ]).
Theorem . Let(X,G)be a G-metric space.LetδG:X×X→[, +∞)be defined by
δG(x,y) =max
G(x,y,y),G(y,x,x),
for all x,y∈X.Then
() (X,δG)is a metric space,
() {xn} ⊂XisG-convergent tox∈Xif and only if{xn}is convergent toxin(X,δG), () {xn} ⊂XisG-Cauchy if and only if{xn}is Cauchy in(X,δG),
() (X,G)isG-complete if and only if(X,δG)is complete.
Definition . Anpreordered G-metric spaceis a triple (X,G,) where (X,G) is aG -metric space andis a preordered onX.
The following result can be considered as the natural extension of Ran and Reurings’ result toCJMcontractions on preorderedG-metric spaces.
Theorem . Let(X,G,)be a preordered G-metric space and let T:X→X be a map-ping.Suppose that the following conditions hold:
(i) (X,G)isG-complete, (ii) Tis-non-decreasing, (iii) TisG-continuous,
(iv) there existsx∈Xsuch thatxTx,
(v) for allx,y,z∈Xwithxyz,
(a) for everyε> ,there existsδ> such that
(b) G(x,y,z) > implies G(Tx,Ty,Tz) <G(x,y,z).
Then there exists a fixed point of T,and it is unique,say u,if for every x,y,z∈X there exists w∈X such that xw and yw and zw.Moreover,for each x∈X such that xTx,
the sequence{Tnx}of iterates converges to u.
Proof Theorem . is a particular case of Theorem .. Takingz=yin (a) and (b), we get, for allx,y∈Xwithxy,
G(x,y,y) <ε+δ implies G(Tx,Ty,Ty)≤ε
and
G(x,y,y) > implies G(Tx,Ty,Ty) <G(x,y,y).
Similarly, we can write
G(y,x,x) <ε+δ implies G(Ty,Tx,Tx)≤ε
and
G(y,x,x) > implies G(Ty,Tx,Tx) <G(y,x,x).
It follows from the above, and the definition ofδG, that, for allx,y∈Xwithxy,
δG(x,y) <ε+δ implies δG(Tx,Ty)≤ε
and
δG(x,y) > implies δG(Tx,Ty) <δG(x,y).
The existence and uniqueness of the fixed point follow immediately by Theorem . and
Theorem ..
Corollary .([], Theorem ..) Let(X,G,)be a preordered G-metric space and let T:X→X be a mapping.Suppose that the following conditions hold:
(i) (X,G)isG-complete,
(ii) Tis-non-decreasing,
(iii) TisG-continuous,
(iv) there existsx∈Xsuch thatxTx,
(v) there existsλ∈[, )such that
G(Tx,Ty,Ty)≤λG(x,y,y) for all x,y∈Xwithxy. ()
Then there exists a fixed point of T,and it is unique,say u,if for every(x,y)∈X×X there exists w∈X such that xw and yw.Moreover,for each x∈X such that xTx,the
sequence{Tnx
The main advantage of the contractivity condition () versus
’G(Tx,Ty,Ty)≤λG(x,y,y) for all x,y∈X’ ()
is that () only requires the inequality to hold for comparable points, that is, for allx,y∈X
such thatxy. As in Example ., consider theG-metric
G(x,y,z) =max|x–y|,|x–z|,|y–z| for all x,y,z∈X=R.
Letx,y∈Xbe such thatxy. Ifx=y, thenG(Tx,Ty,Ty)≤λG(x,y,y) trivially holds. As-sume thatx=y. Theny<x≤. Hence
G(Tx,Ty,Ty) =x –
y
=
|x–y|=
G(x,y,y),
so () holds. However, () is false in this case, ifx= andy= , then
G(T,T,T) =G(, , ) = = G(, , ).
Definition . Let (X,G) be aG-metric space, letA⊂Xbe a nonempty subset and let
be a binary relation onX. Then the triple (A,G,) is said to benon-decreasing regular
if for all sequences{xn} ⊂Asuch that{xn} →x∈Aandxnxn+for alln∈N, we have
xnxfor alln∈N.
The following result can be considered as the natural extension of Nieto and Rodríguez-López’ result toCJMcontractions on preorderedG-metric spaces.
Theorem . Let(X,G,)be a preordered metric space and let T:X→X be a mapping.
Suppose that the following conditions hold:
(i) (X,G)isG-complete, (ii) Tis-non-decreasing,
(iii) (X,G,)is non-decreasing regular, (iv) there existsx∈Xsuch thatxTx,
(v) for allx,y,z∈Xwithxyz,
(a) for everyε> ,there existsδ> such that
G(x,y,z) <ε+δ implies G(Tx,Ty,Tz)≤ε,
(b) G(x,y,z) > implies G(Tx,Ty,Tz) <G(x,y,z).
Then there exists a fixed point of T,and it is unique,say u,if for every x,y,z∈X there exists w∈X such that xw and yw and zw.Moreover,for each x∈X such that xTx,
the sequence{Tnx}of iterates converges to u.
Proof Theorem . is a particular case of Theorem ..
(i) (X,G)isG-complete, (ii) Tis-non-decreasing,
(iii) (X,G,)is non-decreasing regular, (iv) there existsx∈Xsuch thatxTx,
(v) there existsλ∈[, )such that,for allx,y∈Xwithxy, G(Tx,Ty,Ty)≤λG(x,y,y).
Then there exists a fixed point of T,and it is unique,say u,if for every(x,y)∈X×X there exists w∈X such that xw and yw.Moreover,for each x∈X such that xTx,the
sequence{Tnx
}of iterates converges to u.
Competing interests
The author declares that he has no competing interests.
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Received: 9 December 2016 Accepted: 4 May 2017
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