R E S E A R C H
Open Access
Pata contractions and coupled type fixed
points
Madjid Eshaghi
1, Samad Mohseni
1, Mohsen Rostamian Delavar
2, Manuel De La Sen
3,
Gwang Hui Kim
4*and Asma Arian
1*Correspondence:
[email protected] 4Department of Mathematics,
Kangnam University, Kangnam, Korea
Full list of author information is available at the end of the article
Abstract
A new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered complete metric spaces is established. It is shown that the coupled fixed point can be unique under some extra suitable conditions involving mid point lower or upper bound properties. Also the corresponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point.
MSC: 47H10; 34B15; 54H25
Keywords: partial ordered metric space; mixed monotone property; coupled fixed point
1 Introduction and preliminaries
The well-known Banach contraction principle, which guarantees the existence of a unique fixed point for a mapping defined on a complete metric space satisfying the contraction condition, was introduced in by Banach []. After this a great deal of effort has gone into the theory and application of the Banach contraction theorem. Some authors gener-alized this theorem from the single-valued case to the multivalued [–]. Some extensions are to fixed point theorems for contraction mappings in generalized form of metric spaces, especially a metric space endowed with a partial order. Single, coupled, tripled and other types of fixed point theorems are investigated in many works, for instance, [–] and refer-ences cited therein. Such fixed point theorems are applied to establishing the existence of a unique solution to periodic boundary value problems, matrix equations, ordinary differ-ential equations, and integral equations [–]. Recently Pata in [] introduced a fixed point theorem with weaker hypotheses than those of the Banach contraction principle with an explicit estimate of the convergence rate (see also []). Motivated by [], we es-tablish a new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered metric spaces. We prove that the coupled fixed point can be unique under some suitable conditions. Also the corre-sponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point.
Definition .[] Let (X,≤) be a partially ordered set. The product spaceX×Xcan be endowed with a partial order such that for (x,y), (u,v)∈X×Xwe can define (u,v)≤ (x,y)⇔x≥u,y≤v.
Definition .[] Let (X,≤) be a partially ordered set andF:X×X→X. We say that Fhas the mixed monotone property onXifF(x,y) is monotone non-decreasing inxand is monotone non-increasing iny, that is, for anyx,y∈X,
x,x∈X, x≤x ⇒ F(x,y)≤F(x,y) and
y,y∈X, y≤y ⇒ F(x,y)≥F(x,y).
Definition .[] An element (x,y)∈X×Xis called a coupled fixed point of the map-pingF:X×X→Xif
x=F(x,y), y=F(y,x).
2 Main results
In this section we prove a coupled type fixed point theorem with the convergence rate estimation. Also an example is given as an application of the main theorem.
Definition . Let (X,d) be a metric space. The mappingd:X×X→[,∞) given by
d(x,y), (u,v)=d(x,u) +d(y,v),
for each pair ((x,y), (u,v))∈X×X, defines a metric onX×X, which will be denoted for convenience byd, too.
For a metric space (X,d), selecting an arbitrary (x,y)∈X×X, we denote
x,y =d(x,y), (x,y)
for all (x,y)∈X×X.
Letψ: [, ]→[,∞) be an increasing function vanishing with continuity at zero. Also consider the vanishing sequence depending onα≥,wn(α) = (αn)α
n
k=ψ(
α
k).
Theorem . Let(X,≤)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X be a continuous mapping having the mixed monotone property on X and let≥,α≥,andβ ∈[,α]be fixed constants.Suppose that there exist x,y∈X such that
x≤F(x,y) and y≥F(y,x).
If the inequality
dF(x,y),F(u,v)≤( –ε) d
(x,y), (u,v)+εαψ(ε) + x,y + u,v β (.) is satisfied for everyε∈[, ]and(x,y), (u,v)∈X×X with u≤x,y≤v,then F has a coupled fixed point(x∗,y∗).
Furthermore,we denote Fn=F◦ · · · ◦F(n times),
dx∗,y∗,Fn(x,y),Fn(y,x)
≤Kwn(α) (.)
Proof SetF(x,y) =xandF(y,x) =y. So
x≥x, y≤y.
Now from lettingx=F(x,y),y=F(y,x) and choosing the notations
F(x,y) =F
F(x,y),F(y,x)
=F(x,y) =x,
F(y,x) =F
F(y,x),F(x,y)
=F(y,x) =y,
with the mixed monotone property ofFwe have
x=F(x,y) =F(x,y)≥F(x,y) =x,
y=F(y,x) =F(y,x)≤F(y,x) =y.
So by continuing this argument we obtain two sequences{xn}∞n= and{yn}∞n= such that
x≤x≤ · · · ≤xn≤ · · · andy≥y≥ · · · ≥yn≥ · · ·.
Furthermore forn= , , . . . we set
xn=F(xn–,yn–) =Fn(x,y), yn=F(yn–,xn–) =Fn(y,x) and
Cn=Fn(x,y),Fn(y,x).
Since (.) is true for everyε∈[, ], settingε= we have the following relations:
dFn+(x,y),Fn+(y,x)
,Fn(x,y),Fn(y,x)
≤C, (.)
dFn+(x,y),Fn+(y,x)
, (x,y)
≤dFn+(x,y),Fn+(y,x)
,F(x,y),F(y,x)
+dF(x,y),F(y,x)
, (x,y)
, (.)
dFn(x,y),Fn(y,x)
, (x,y)
≤dFn(x,y),Fn(y,x)
,Fn+(x,y),Fn+(y,x)
+dFn+(x,y),Fn+(y,x)
, (x,y)
. (.)
Using (.), (.), (.), and (.) we have
Cn≤d
Fn+(x,y),Fn+(y,x)
,F(x,y),F(y,x)
+ C
=dFn+(x,y),F(x,y)
+dFn+(y,x),F(y,x)
+ C
=dFFn(x,y),Fn(y,x)
,F(x,y)
+dFFn(y,x),Fn(x,y)
,F(y,x)
+ C
≤( –ε) d
Fn(x,y),Fn(y,x)
, (x,y)
+( –ε) d
Fn(y,x),Fn(x,y)
, (y,x)
+εαψ(ε) +Fn(y,x),Fn(x,y)+ y,x β
+ C.
But from the definition of · we have
+Fn(x,y),Fn(y,x)+ x,y
≤ +Cn≤ +Cn+ d(x,y),
+Fn(y,x),Fn(x,y)+ y,x
≤ +Cn+ d(x,y),
so forα≥βthere are someE,D> such that
Cn≤
( –ε) Cn+ε
αψ(ε) +C
n+ d(x,y) β
+( –ε) Cn+ε
αψ
(ε) +Cn+ d(x,y) β
+ C
= ( –ε)Cn+ εαψ(ε)
+Cn+ d(x,y) β
+ C≤( –ε)Cn+Eεαψ(ε)Cαn+D.
Accordingly,
εCn≤Eεαψ(ε)Cnα+D,
which holds by hypothesis for anyε∈[, ] taken for eachn∈N. If there is a subsequence Cnk→ ∞, then the choiceεnk=min(,
+D
Cnk) leads to the following contradiction:
≤E( +D)αψ(ε
nk)→ asnk→ ∞.
Then the sequence{Cn}∞n=is bounded. Also the sequences{Fn(x,y)}and{Fn(y,x)}are
Cauchy sequences:
dFn+m+(x,y),Fn+(x,y)
+dFn+m+(y,x),Fn+(y,x)
=dFFn+m(x,y),Fn+m(y,x)
,FFn(x,y),Fn(y,x)
+dFFn+m(y,x),Fn+m(x,y)
,FFn(y,x),Fn(x,y)
≤( –ε) d
Fn+m(x,y),Fn+m(y,x)
,Fn(x,y),Fn(y,x)
+εαψ(ε) +Fn+m(x,y),Fn+m(y,x)+Fn(x,y),Fn(y,x)
β
+( –ε) d
Fn+m(y,x),Fn+m(x,y)
,Fn(y,x),Fn(x,y)
+εαψ(ε) +Fn+m(y,x),Fn+m(x,y)+Fn(y,x),Fn(x,y)
β .
But since
Fn+m(x,y),Fn+m(y,x)+Fn(x,y),Fn(y,x)
Fn+m(y,x),Fn+m(x,y)+Fn(y,x),Fn(x,y)≤cn+m+cn+ d(x,y),
we have
dFn+m+(x,y),Fn+(x,y)
+dFn+m+(y,x),Fn+(y,x)
≤( –ε)dFn+m(x,y),Fn(x,y)
+dFn+m(y,x),Fn(y,x)
+ εαψ(ε) +cn+m+cn+ d(x,y) β
.
For fixedm, set
K=sup n∈N
+ cn+ d(x,y) β
, (.)
andε= – (n+n )α≤ α
n+. So
(n+ )αdFn+m+(x,y),Fn+(x,y)
+dFn+m+(y,x),Fn+(y,x)
≤nαdFn+m(x,y),Fn(x,y)
+dFn+m(y,x),Fn(y,x)
+Kααψ
α
n+ .
Settingrn:=nα[d(Fn+m(x,y),Fn(x,y)) +d(Fn+m(y,x),Fn(y,x))] we have
rn+≤rn+Kααψ
α
n+
≤rn–+Kααψ
α
n +Kα αψ
α
n+
≤ · · ·
≤r+Kαα n+ k= ψ α
k =Kα α n+ k= ψ α k . Therefore
dFn+m(x,y),Fn(x,y)
+dFn+m(y,x),Fn(y,x)
≤K
α
n αn
k=
ψ
α
k =Kwn(α). (.)
Taking limits asn→ ∞, we getd(Fn+m(x,y),Fn(x,y)) +d(Fn+m(y,x),Fn(y,x))→.
This implies that{Fn(x,y)}and{Fn(y,x)}are Cauchy sequences inX. SinceXis a
com-plete metric space, there arex∗,y∗∈Xsuch that
lim n→∞F
n(x
,y) =n→∞limxn=x∗ and m→∞lim Fm(y,x) =m→∞lim ym=y∗.
Using (.) and Definition . we have
lim n→∞d
x∗,y∗,F(xn–,yn–),F(yn–,xn–)
= lim n→∞d
= lim n,m→∞d
Fn+m(x,y),Fn+m(y,x)
,Fn(x,y),Fn(y,x)
≤Kwn(α)
and from the continuity ofFwe haveF(x∗,y∗) =x∗andF(y∗,x∗) =y∗. Also, the convergence rate estimate stated in (.) is obtained from the following relations:
dx∗,y∗,Fn(x,y),Fn(y,x)
=dFx∗,y∗,Fy∗,x∗,Fn(x,y),Fn(y,x)
=dFx∗,y∗,Fn(x,y)
+dFy∗,x∗,Fn(y,x)
≤/dx∗,y∗,Fn–(x,y),Fn–(y,x)
+ /dy∗,x∗,Fn–(y,x),Fn–(x,y)
=dx∗,y∗,Fn–(x,y),Fn–(y,x)
.. .
≤dx∗,y∗, (x,y)
=x∗,y∗,
which implies that
cn=d
Fn(x,y),Fn(y,x)
, (x,y)
≤dFn(x,y),Fn(y,x)
,x∗,y∗+dx∗,y∗, (x,y)
≤x∗,y∗.
From the last inequality and (.) we haveK≤[ + x∗,y∗ + d(x,y)]β.
Example . LetX= [–, ] andd:X×X→[,∞) be a metric onXdefined asd(x,y) = |x–y|for allx,y∈Xwhich (X,d) is a complete metric space. Now considerX×Xwith the partial order defined in Definition .. The metric on the product metric spaceX×X is defined by
d(x,y), (x,y)
=|x–x|+|y–y|, for allxi,yi∈X(i= , ).
Consider the mappingFdefined as
F(x,y) = ,
for all fixed constants≥,α≥, andβ∈[,α], and by using (.) we have
≤( –ε) d
(x,y), (u,v)+εαψ(ε) + x,y + u,v β.
Sinced[(x,y), (u,v)] andεαψ(ε)[ + x,y + u,v ]βare positive, we have(–ε)
≥, which
3 Supplementary results
In this section we prove some supplementary results. The first theorem is about replacing the continuity condition of the functionFin Theorem . with a new one. The second is about the uniqueness of the coupled fixed point. At last we show that the hypothesis of Theorem . is weaker than those of Theorem . in [].
Theorem . Let(X,≤)be a partially ordered set and suppose that there is a metric d on
X such that(X,d)is a complete metric space.Let F:X×X→X be a mapping having the mixed monotone property on X such that there are two elements x,y∈X with
x≤F(x,y) and y≥F(y,x).
Let≥,α≥,andβ∈[,α]be fixed constants.Suppose that the inequality
dF(x,y),F(u,v)≤( –ε) d
(x,y), (u,v)+εαψ(ε) + x,y + u,v β (.)
is satisfied for everyε∈[, ]and(x,y), (u,v)∈X×X with u≤x,y≤v.Furthermore, suppose that X has the following properties:
(i) for a non-decreasing sequencexn→x,thenxn≤xfor alln∈N,
(ii) for a non-increasing sequenceyn→y,thenyn≥yfor alln∈N.
Then F has a coupled fixed point(x∗,y∗).
Furthermore,calling Fn=F◦ · · · ◦F(n times),we have
dx∗,y∗,Fn(x
,y),Fn(y,x)
≤Kwn(α) (.)
for some positive constant K≤( + x∗,y∗ + d(x,y))β.
Proof We only need to prove thatx∗=F(x∗,y∗) andy∗=F(y∗,x∗). Consider the following relations:
dx∗,y∗,Fx∗,y∗,Fy∗,x∗
≤dx∗,y∗,Fn+(x,y),Fn+(y,x)
+dFn+(x,y),Fn+(y,x)
,Fx∗,y∗,Fy∗,x∗
=dx∗,y∗,Fn+(x,y),Fn+(y,x)
+dFFn(x,y),Fn(y,x)
,Fx∗,y∗+dFFn(y,x),Fn(x,y)
,Fy∗,x∗
≤dx∗,y∗,Fn+(x,y),Fn+(y,x)
+( –ε) d
Fn(x,y),Fn(y,x)
,x∗,y∗
+εαψ(ε) +Fn(x,y),Fn(y,x)+x∗,y∗
β
+( –ε) d
Fn(y,x),Fn(x,y)
,y∗,x∗
+εαψ(ε) +Fn(y,x),Fn(x,y)+y∗,x∗
Now since
lim n→∞F
n(x
,y) =n→∞limxn=x∗ and m→∞lim Fm(y,x) =m→∞lim ym=y∗,
and since the contractive condition (.) holds for any real constantε∈[, ], we can re-placeε, for eachn∈N, by a sequence [, ]εn→ asn→ ∞. Then by lettingε=εn→
asn→ ∞, we haved[(x∗,y∗), (F(x∗,y∗),F(y∗,x∗))] = .
Definition . Suppose thatXis a partially ordered metric space with a metricd. A pair of (x,y), (x∗,y∗)∈X×Xhas either amid point lower boundor amid point upper bound if there are (z,z)∈X×X comparable to (x,y) and (u,v) such that d[(x,y), (z,z)] +
d[(z,z), (x∗,y∗)] =d[(x,y), (x∗,y∗)]. The space (X,d) has the mid point lower bound or
the mid point upper bound property if any pair inX×Xhas a mid point lower bound or a mid point upper bound.
Theorem . Adding the condition of the above definition to the space(X,d)in the hy-pothesis of Theorem.,we obtain the uniqueness of the coupled fixed point of F.
Proof If (x,y)∈X×X is another coupled fixed point ofFwherex=F(x,y),y=F(y,x), then we have two cases.
Case (): If (x∗,y∗) is comparable to (x,y) then
d(x,y),x∗,y∗=dF(x,y),F(y,x),Fx∗,y∗,Fy∗,x∗
=dF(x,y),Fx∗,y∗+dF(y,x),Fy∗,x∗
≤( –ε) d
(x,y),x∗,y∗+tεψ(ε)
+( –ε) d
(y,x),y∗,x∗+tεψ(ε),
for somet,t> . Puttingt=t+t, we have
εd(x,y),x∗,y∗≤εtψ(ε),
which is valid for everyε∈[, ]. We can replaceε, for eachn∈N, by a sequence [, ]
εn→ asn→ ∞which forcesd[(x,y), (x∗,y∗)] = .
Case (): If (x∗,y∗) is not comparable to (x,y) then there exists a mid upper bound or mid lower bound (u,v)∈X×Xfor (x∗,y∗) and (x,y). So (F(u,v),F(v,u)) is comparable to (x∗,y∗) = (F(x∗,y∗),F(y∗,x∗)) and (x,y) = (F(x,y),F(y,x)). It follows that
d(x,y),x∗,y∗
=dF(x,y),F(y,x),Fx∗,y∗,Fy∗,x∗
≤dF(x,y),F(y,x),F(u,v),F(v,u)+dF(u,v),F(v,u),Fx∗,y∗,Fy∗,x∗
=dF(x,y),F(u,v)+dF(y,x),F(v,u)+dF(u,v),Fx∗,y∗
≤( –ε) d
(x,y), (u,v)+tεψ(ε) +
( –ε) d
(y,x), (v,u)+tεψ(ε)
+( –ε) d
(u,v),x∗,y∗+tεψ(ε) +
( –ε) d
(v,u),y∗,x∗+tεψ(ε),
for somet,t,t,t> . Sett=t+t+t+t. Hence,
d(x,y),x∗,y∗≤( –ε)d(x,u) +d(y,v) +du,x∗+dv,y∗+tεψ(ε). (.) Now by Definition . we have
εd(x,y),x∗,y∗≤εtψ(ε)
for everyε∈[, ]. We can replaceε, for eachn∈N, by the sequence [, ]εn→ as
n→ ∞, which forcesd[(x,y), (x∗,y∗)] = .
Remark . Note that Theorem . is stronger than Theorem . in []. Indeed with the hypothesis of Theorem . in [], for each (x,y), (u,v)∈X×X, we have
dF(x,y),F(u,v)≤λ d
(x,y), (u,v) for allx≥u,y≤v, ≤λ< .
Thus forα=β= ,ψ(ε) =ε, and=(ν,λ) = νν
(+ν)+ν(–λ)ν, for arbitraryν> , we get
dF(x,y),F(u,v)≤( –ε) d
(x,y), (u,v)+ε+ν + x,y + u,v , for everyε∈[, ].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Basic Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran.2Young
Researchers Club, Semnan Branch, Islamic Azad University, Semnan, Iran.3Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa (Bizkaia) - Aptdo. 644- Bilbao, Bilbao, 48080, Spain. 4Department of Mathematics, Kangnam University, Kangnam, Korea.
Received: 22 November 2013 Accepted: 8 May 2014 Published:02 Jun 2014
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Cite this article as:Eshaghi et al.:Pata contractions and coupled type fixed points.Fixed Point Theory and Applications
