Common fixed point theorems for weakly increasing mappings on ordered orbitally complete metric spaces

R E S E A R C H

Open Access

Common fixed point theorems for weakly

increasing mappings on ordered orbitally

complete metric spaces

Hui-Sheng Ding

, Zoran Kadelburg

and Hemant Kumar Nashine

* Correspondence: dinghs@mail. ustc.edu.cn

1_{College of Mathematics and}

Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People_’s Republic of China

Full list of author information is available at the end of the article

Abstract

In this article, we prove existence results for common fixed points of two or three relatively asymptotically regular mappings satisfying the orbital continuity of one of the involved maps on ordered orbitally complete metric spaces. We furnish suitable examples to demonstrate the validity of the hypotheses of our results.

Mathematics Subject Classification (2010): 47H10; 54H25.

Keywords:Partially ordered set, asymptotically regular map, orbitally complete metric space, orbital continuity, weakly increasing maps

1 Introduction and preliminaries

Browder and Petryshyn introduced the concept of asymptotic regularity of a self-map at a point in a metric space.

Definition 1[1] A self-map_T on a metric space(X,d)is said to be asymptotically regular at a pointx∈X if limn→∞d(Tnx,Tn+1x) = 0.

Recall that the setO(x0;T) ={Tnx0:n= 0, 1, 2,. . .}is called the orbit of the self-map_T at the point x0∈X.

Definition 2 [2] A metric space(X,d)is said to be_T-orbitally complete if every Cauchy sequence contained inO(x;T)(for somexin_X) converges in_X.

Here, it can be pointed out that every complete metric space is_T-orbitally complete for any_T, but a_T-orbitally complete metric space need not be complete.

Definition 3[1] A self-mapT defined on a metric space(X,d)is said to be orbitally continuous at a pointzinX if for any sequence{xn} ⊂O(x;T)(for somex∈X), xn

®zasn®∞impliesTxn →Tzasn® ∞.

Clearly, every continuous self-mapping of a metric space is orbitally continuous, but not conversely.

Sastry et al. [3] extended the above concepts to two and three mappings and employed them to prove common fixed point results for commuting mappings. In what follows, we collect such definitions for three maps.

Definition 4LetS,T,Rbe three self-mappings defined on a metric space(X,d). 1. If for a point x0∈X, there exits a sequence {xn} in X such that Rx2n+1=Sx2n,Rx2n+2=Tx2n+1,n= 0, 1, 2,. . ., then the set

O(x0;S,T,R) ={Rxn:n= 1, 2,. . .}is called the orbit of(S,T,R)atx0.

2. The space(X,d)is said to be(S,T,R)-orbitally complete at x0 if every Cauchy sequence inO(x0;S,T,R)converges inX.

3. The map R is said to be orbitally continuous at x0 if it is continuous on

O(x0;S,T,R).

4. The pair (S,T) is said to be asymptotically regular (in short a.r.) with respect to _R at x0 if there exists a sequence {xn} in X such that Rx2n+1=Sx2n,Rx2n+2=Tx2n+1,n= 0, 1, 2,. . ., andd(Rxn,Rxn+1)→0as n®∞.

5. If_Ris the identity mapping on_X, we omit_Rin respective definitions.

On the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [4] who presented its applications to matrix equations. Subsequently, Nieto and López [5] extended this result for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed point theorems in ordered metric spaces. For more details, see [6-15] and the references cited therein.

Recently, Nashine and Altun (HK Nashine and I Altun, unpublished work) proved the following ordered version of a result of Zhang [16]:

Theorem 1 Let (X,d,)be a complete partially ordered metric space and let

S,T :X →X be two weakly increasing mappings such that

F(d(Tx,Sy))≤ψ(F([T,S](x,y)))

holds for each comparablex,y∈X,where F,ψ: [0, +∞)®[0, +∞)are functions such that (i) F is nondecreasing, continuous, and F(0) = 0 <F(t)for every t> 0;

(ii) ψis nondecr easing, right continuous, andψ(t) <t for every t> 0,and

[T,S](x,y) = max{d(x,y),d(x,Tx),d(y,Sy),1

2[d(x,Sy) +d(y,Tx)]}.

If_SorT is continuous, then_SandT have a unique common fixed point.

In this article, we generalize this theorem of Nashine and Altun (HK Nashine and I Altun, unpublished work) (and, hence, some other related common fixed point results) in two directions. The first is treated in Section 3, where a pair of asymptotically regu-lar mappings in an orbitally complete ordered metric space is considered. The exis-tence and (under additional assumptions) uniqueness of their common fixed point is obtained under assumptions that these mappings are strictly weakly isotone increasing, one is orbitally continuous and they satisfy a generalized weakly contractive condition.

In Section 4, we consider the case of three self-mappings _S,T,Rwhere the pair

S,T isR-relatively asymptotically regular and relatively weakly increasing, while the contractive condition is given with the help of two control functions.

We furnish suitable examples to demonstrate the validity of the hypotheses of our results.

2 Notation and definitions

Definition 5Let(X,)be a partially ordered set andS,T :X →X. 1. The mappingT is called dominating ifxTxfor eachx∈X[17].

2. The pair(S,T)is called weakly increasing ifSxT Sx and TxSTx for all

x∈X[18,19].

3. The mapping _Sis said to beT-weakly isotone increasing if for all x∈X we have

SxT SxST Sx[18-20].

4. The mapping_S is said to be_T-strictly weakly isotone increasing if, for all x∈X

such that x≺Sx, we have _Sx≺T Sx≺ST Sx(HK Nashine, B Samet, and C Vetro, unpublished work).

5. Let _R:X →X be such that T X _⊆RX and SX _⊆RX, and denote

R−1_{(x) :={u∈X :Ru=x}, for}

x∈X. We say that _T and _S are weakly increasing with respect to_Rif and only if for allx∈X, we have [10]:

TxSy, ∀y∈R−1(Tx)

and

SxTy, ∀y∈R−1(Sx).

Example 1[17] LetX = [0, 1]be endowed with the usual ordering. Let _T :X →X

be defined byTx=√n

x. Sincex≤√n

x=Txfor all x∈X,T is a dominating map. Remark 1(1) None of two weakly increasing mappings need be nondecreasing. There exist some examples to illustrate this fact in [21].

(2) IfS,T :X→X are weakly increasing, then SisT-weakly isotone increasing. (3) Scan beT-strictly weakly isotone increasing, while some of these two map-pings can be not strictly increasing (see the following example).

(4) If Ris the identity mapping (Rx=xfor all x∈X), thenT andS are weakly increasing with respect toRif and only if they are weakly increasing mappings.

Example 2 Let X = [0, +∞)be endowed with the usual ordering and define

S,T :X →X as

Sx=

2x, ifx∈[0, 1],

3x, ifx>1; Tx=

2, ifx∈[0, 1], 2x, ifx>1.

Clearly, we havex≺Sx≺T Sx≺ST Sxfor allx∈X, and so,_S isT-strictly weakly isotone increasing;_T is not strictly increasing.

Definition 6[22,23]. Let(X,d)be a metric space and f,g:X →X.

1. Ifw=fx=gx, for somex∈X, thenxis called a coincidence point offand g, and w is called a point of coincidence off andg.If w=x, thenxis a common fixed point offandg.

2. The mappings fandg are said to be compatible if limn®∞ d(fgxn,gfxn) = 0,

when-ever {xn} is a sequence inX such that limn®∞fxn= limn®∞gxn=tfor somet∈X.

Definition 7 LetX be a nonempty set. Then(X,d,)is called an ordered metric

space if

(i)(X,d)is a metric space,

The space(X,d,)is called regular if the following hypothesis holds: if {zn} is a

non-decreasing sequence in_Xwith respect to≼such thatzn→z∈Xasn®∞, thenzn≼z.

3 Common fixed points for_T-strictly weakly isotone increasing mappings In this section, we improve the results of Nashine and Altun (HK Nashine and I Altun, unpublished work) by considering the following:

1. a pair of asymptotically regular mappings; 2. orbital continuity of one of the involved maps; 3. strictly weakly isotone increasing condition; 4. generalized weakly contractive condition, and 5. an ordered orbitally complete metric space.

We will denote by _F and Ψ the set of functionsF, ψ : [0, +∞)® [0, +∞), respec-tively, such that:

(i) Fis nondecreasing, continuous, andF(0) = 0 <F(t) for everyt> 0; (ii)ψis nondecreasing, right continuous, andψ(0) = 0.

The first main result of this section is as follows:

Theorem 2Let(X,d,)be an ordered metric space. LetS,T :X →X be two map-pings satisfying

F(d(Tx,Sy))≤F([T,S](x,y)))−ψ(F([T,S](x,y))) (3:1)

for all x,y∈O(x0;S,T) (for some x0) such that x and y are comparable, where

F∈F,ψÎΨand

[T,S](x,y) = max{d(x,y),d(x,Tx),d(y,Sy),1

2(d(x,Sy) +d(y,Tx))}. (3:2)

We assume the following hypotheses: (i)(T,S)is a.r. at x0;

(ii)X is(S,T)-orbitally complete at x0; (iii)_S orT is(S,T)-orbitally continuous at x0; (iv)_SisT-strictly weakly isotone increasing; (v) there exists an x0∈X such thatx0≺Sx0.

Then _Sand_T have a common fixed point. Moreover, the set of common fixed points ofS,T in_O(x0;S,T)is well ordered if and only if it is a singleton.

Proof First of all we show that, ifSorT has a fixed point, then it is a common fixed point ofS andT. Indeed, let zbe a fixed point ofS. Now assumed(z,Tz)>0. If we use the inequality (3.1), for x=y=z, we have

F(d(Tz,z)) =F(d(Tz,Sz))≤F([T,S](z,z))−ψ(F([T,S](z,z))) =F(d(Tz,z))−ψ(F(d(Tz,z))),

wherefromψ(F(d(Tz,z))) = 0, which is a contradiction. Thusd(z,Tz) = 0and soz is a common fixed point of_S and_T. Analogously, one can observe that ifzis a fixed point of_T, then it is a common fixed point of_Sand_T.

Since(T,S)is a.r. atx0inX, there exists a sequence {xn} inX such that

and

lim

n→∞d(xn,xn+1) = 0. (3:4) If xn0 =Sxn0orxn0 =Txn0for somen0, then the proof is finished. So assumexn≠xn

+1for all n.

Since SisT-strictly weakly isotone increasing, we have

x1=Sx0≺T Sx0=Tx1=x2≺ST Sx0=STx1=Sx2=x3,

x3=Sx2≺T Sx2=Tx3=x4≺ST Sx2=STx3=Sx4=x5,

and continuing this process we get

x1≺x2≺ · · · ≺xn≺xn+1≺ · · ·. (3:5)

Next, we claim that {xn} is a Cauchy sequence in the metric spaceO(x0;S,T). We proceed by negation and suppose that {xn} is not a Cauchy sequence. That is, there

exists ε> 0 such that d(xn,xm)≥ε for infinitely many values ofm andnwithm <n.

This assures that there exist two sequences {m(k)}, {n(k)} of natural numbers, with m (k) <n(k), such that for eachkÎN

d(x2m(k),x2n(k)+1)> ε. (3:6)

It is not restrictive to suppose that n(k) is the least positive integer exceeding m(k) and satisfying (3.6). We have

ε <d(x2m(k),x2n(k)+1)

≤d(x2m(k),x2n(k)−1) +d(x2n(k)−1,x2n(k)) +d(x2n(k),x2n(k)+1)

≤ε+d(x2n(k)−1,x2n(k)) +d(x2n(k),x2n(k)+1)

and lettingk®∞, we haved(x2m(k),x2n(k)+1)®ε. We note that

dx2m(k),x2n(k)+1

−dx2m(k),x2m(k)+1

−dx2n(k)+2,x2n(k)+1

≤dx2m(k)+1,x2n(k)+2

≤dx2m(k),x2n(k)+1

+dx2m(k),x2m(k)+1

+dx2n(k)+2,x2n(k)+1

,

and thus dx2m(k)+1,x2n(k)+2

→εask®∞. We have

[T,S]x2n(k)+1,x2m(k)

= maxdx2n(k)+1,x2m(k)

,dx2n(k)+1,x2n(k)+2

,dx2m(k),x2m(k)+1

,

1 2

dx2n(k)+1,x2m(k)+1

+dx2m(k),x2n(k)+2

≤maxdx2n(k)+1,x2m(k)

,dx2n(k)+1,x2n(k)+2

,dx2m(k),x2m(k)+1

,

dx2n(k)+1,x2m(k)

+1₂dx2m(k),x2m(k)+1

+dx2n(k)+1,x2n(k)+2

and so letting k ® ∞, we havelimk→∞[T,S]x2n(k)+1,x2m(k)

≤ε. Therefore, we have

Fdx2m(k)+1,x2n(k)+2

=FdSx2m(k),Tx2n(k)+1

≤FΘ[T,S]x2n(k)+1,x2m(k)

−ψFΘ[T,S]x2n(k)+1,x2m(k)

F(ε)≤F(ε)−ψ(F(ε))<F(ε),

a contradiction. Therefore, {xn} is a Cauchy sequence in O(x0;S,T). SinceX is

(T,S)-orbitally complete atx0, there existsz∈X with limn®∞xn=z.

If_S orT is orbitally continuous, then clearlyz=Sz=Tz

Theorem 3 Let(X,d,)andS,T :X →X satisfy all the conditions of Theorem 2,

except that condition(iii)is substituted by

(iii’)_X is regular.

Then the same conclusions as in Theorem 2 hold.

Proof Following the proof of Theorem 2, we have that {xn} is a Cauchy sequence in (X,d)which is(S,T)-orbitally complete atx0. Then, there existsz∈X such that

lim

n→∞xn=z.

Now suppose that d(z,Sz) >0. From regularity of_X, we havex2nzfor allnÎN. Hence, we can apply the considered contractive condition. Then, setting x=x2nandy = zin (3.1), we obtain:

F(d(x2n+2,Sz))=F(d(Tx2n+1,Sz))

≤FΘ[T,S]x2n+1,z

−ψFΘ[T,S]x2n+1,z

,

where

Θ[T,S]x2n+1,z

= maxdx2n+1,z

,d(x2n+1,Tx2n+1),d(z,Sz), 1

2

d(x2n+1,Sz)+d(z,Tx2n+1)

= maxd(x2n+1,z),d(x2n+1,x2n+2),d(z,Sz) , 1

2

d(x2n+1,Sz)+d(z,x2n+2)

.

Letting n®∞in the above inequality and using the continuity of Fand right conti-nuity ofψ, we have

F(d(z,Sz))≤F(d(z,Sz))−ψ(F(d(z,Sz)))<F(d(z,Sz))

a contradiction. Therefore,d(z,Sz)= 0and thusz=Sz. Hence,zis a common fixed point ofT and_S.

Corollary 1Let(X,d,)be an ordered metric space. Let_T :X →X be a mapping

satisfying

F(d(Tx,Ty))≤F(Θ1[T](x,y))−ψ(F(Θ1[T](x,y))) (3:7)

for all x,y∈O(x0;T)(for some x0) such that x and y are comparable, whereF∈F,

ψÎ Ψand

Θ1[T](x,y) = max

d(x,y),d(Tx,x),dTy,y,1 2

dx,Ty+dTx,y .

(ii)X isT -orbitally complete at x0;

(iii)T is orbitally continuous at x0orX is regular.

Also suppose that Tx≺T (Tx)for all x∈X such that x≺Txand there exists an

x0∈X such thatx0≺Tx0. ThenT has a fixed point. Moreover, the set of fixed points of_T in_O(x0;T)is well ordered if and only if it is a singleton.

We also state a corollary of Theorem 2 involving a contraction of integral type.

Corollary 2 Let_S and_T satisfy the conditions of Theorem 2, except that condition

(3.1) is replaced by the following: there exists a positive Lebesgue integrable function u onℝ+such that

_ε

0u(t)dt>0for eachε> 0and that

F(d(Sx,Ty))

0

u(t)dt≤

F(Θ[T,S](x,y))

0

u(t)dt−

_ψ(F(Θ[T,S](x,y)))

0

u(t)dt.

Then, _Sand_T have a common fixed point. Moreover, the set of common fixed points ofSandT inO(x0;S,T)is well ordered if and only if it is a singleton.

We present an example showing how our results can be used.

Example 3 LetX ={0} ∪A∪B, where A={1_n|n∈N}and B =(1, +∞), be equipped with Euclidean metric dand the order≼given by

xy⇔x=yor x,y∈Aandx≥y.

Consider the mappingsS,T :X →X given by

Sx=

⎧ ⎨ ⎩

0, x= 0,

1 n+1,x=

1 n,

3x, x∈B,

n∈N, Tx=

⎧ ⎨ ⎩

0, x= 0,

1 n+1,x=

1 n,

2x, x∈B. n∈N,

It is easy to check thatS andT satisfy conditions (i)-(v) of Theorem 2 withx0= 1₂. Take F∈F defined by

F(t) =

⎧ ⎨ ⎩

0, t= 0, t1/t_{, 0<t<1,}

t, t≥1,

and ψÎΨ,given asψ(t) = 1₂t. In order to check the contractive condition (3.1), take

x,y∈O(x0;S,T)with, sayx ≺y,i.e., x>y (the case x = y is trivial). Then x= 1nand

y= 1

mfor somem,nÎN,m>n. We get thatd

Tx,Sy=_n1₊₁−_m1₊₁ = _(n+1)(m−_mn₊₁₎and

FdTx,Sy=

m−n

(n+ 1)(m+ 1)

(n+ 1)(m+ 1)

m−n

=

m−n

(n+ 1)(m+ 1)

n+m+ 1

m−n nm

(n+ 1)(m+ 1)

nm

m−nm−n nm

nm

m−n

<1 2·1·

m−n nm nm m−n =1 2F

d(x,y)≤1 2F

Θ[T,S](x,y)

=FΘ[T,S](x,y)−ψFΘ[T,S](x,y).

Hence, (3.1) is fulfilled. Applying Theorem 2, we conclude that S and T have a (unique) common fixed point (z= 0).

4 Common fixed points for relatively weakly increasing mappings

In this section, we improve and generalize the results of Nashine and Altun (HK Nashine and I Altun, unpublished work) by taking into account the following for three mapsR,S,T:

1.(S,T)is a pair of asymptotically regular mappings with respect to_R; 2. orbital continuity of one of the involved maps;

3.(S,T)is a pair of weakly increasing mappings with respect toR; 4.(S,T)is a pair of dominating maps;

5.(S,T)is a pair of compatible maps, and

6. the basic space is an ordered orbitally complete metric space.

We will denote by Fthe set of functions: [0 +∞)®[0, +∞), such that is right continuous, (0) = 0 and(t) <tfor everyt> 0.

The first result of this section is the following.

Theorem 4 Let(X,d,)be a regular ordered metric space and letT,S and _Rbe

self-maps on_X satisfying

F(d(Tx,Sy))≤ϕ(F(M[T,S,R](x,y))) (4:1)

for all x,y∈O(x0;S,T,R) (for some x0) such that Rx and Ry are comparable, where F∈F,ÎFand

M[T,S,R](x,y) = maxdRx,Ry,d(Tx,Rx),dSy,Ry, 1

2

dRx,Sy+dTx,Ry. (4:2)

We assume the following hypotheses:

(i)(S,T)is a.r. with respect to_Ratx0∈X; (ii)_X is(S,T,R)-orbitally complete at x0;

(iii)_T and_Sare weakly increasing with respect to_R; (iv)_T and_S are dominating maps;

(v)_Ris monotone and orbitally continuous at x0. Assume either

(a)SandRare compatible; or (b)T andRare compatible.

Then S,T and Rhave a common fixed point. Moreover, the set of common fixed points ofS,T and_Rin_O(x0;S,T,R)is well ordered if and only if it is a singleton.

Proof Since(S,T)is a.r. with respect toRat x0inX, there exists a sequence {xn} in X such that

Rx2n+1=Sx2n, Rx2n+2=Tx2n+1, ∀n∈N0={0, 1, 2, ...}, (4:3)

and

lim

n→∞d(Rxn,Rxn+1)= 0 (4:4)

holds. We claim that

RxnRxn+1, ∀n∈N0. (4:5)

Rx1=Sx0Ty, ∀y∈R−1(Sx0).

Since Rx1=Sx0, then x1∈R−1(Sx0), and we get

Rx1=Sx0Tx1=Rx2.

Again,

Rx2=Tx1Sy, ∀y∈R−1(Tx1). Since x2∈R−1(Tx1), we get

Rx2=Tx1Sx2=Rx3.

Hence, by induction, (4.5) holds. Therefore, we can apply (4.1) forx = xpandy = xq

for all pand q.

Now, we assert that{Rxn}is a Cauchy sequence in the metric spaceO(x0;S,T,R). We proceed by negation and suppose that{Rx2n}is not Cauchy. Then, there existsε> 0 for which we can find two sequences of positive integers {m(k)} and {n(k)} such that for all positive integersk,

n(k)>m(k)>k, dRx2m(k),Rx2n(k)

≥ε, dRx2m(k),Rx2n(k)−2

< ε. (4:6)

From (4.6) and using the triangular inequality, we get

ε≤dRx2m(k),Rx2n(k)

≤dR2m(k),Rx2n(k)−2

+dRx2n(k)−2,Rx2n(k)−1

+dRx2n(k)−1,Rx2n(k)

< ε+dRx2n(k)−2,Rx2n(k)−1

+dRx2n(k)−1,Rx2n(k)

.

Lettingk®∞in the above inequality and using (4.4), we obtain

lim

k→∞d

Rx2m(k),Rx2n(k)

=ε. _(4:7)

Again, the triangular inequality gives us

dRx2n(k),Rx2m(k)−1

−dRx2n(k),Rx2m(k)≤d

Rx2m(k)−1,Rx2m(k)

.

Lettingk®∞in the above inequality and using (4.4) and (4.7), we get:

lim

k→+∞d

Rx2n(k),Rx2m(k)−1

=ε. _(4:8)

On the other hand, we have

dRx2n(k),Rx2m(k)

≤dRx2n(k),Rx2n(k)+1

+dRx2n(k)+1,Rx2m(k)

=dRx2n(k),Rx2n(k)+1

+dSx2n(k),Tx2m(k)−1

.

Letting k®∞in the above inequality and using (4.4), (4.7) and properties ofF∈F, we have

F(ε)≤ lim

k→∞F

dSx2n(k),Tx2m(k)−1

. _(4:9)

Applying (4.1), we get:

FdSx2n(k),Tx2m(k)−1

≤ϕFM[T,S,R]x2m(k)−1,x2n(k)

One can check easily that forklarge enough, we have:

M[T,S,R]x2m(k)−1,x2n(k)

=dRx2n(k),Rx2m(k)−1

+dk,

wheredk≥0 anddk®0 as k® ∞. From (4.10), forklarge enough, we have

FdSx2n(k),Tx2m(k)−1

≤ϕFdRx2n(k),Rx2m(k)−1

+dk

. (4:11)

Lettingk®∞in (4.11) and using properties ofFand, we have

lim

k→+∞F

dSx2n(k),Tx2m(k)−1

≤ϕ (F(ε)) <F(ε). _(4:12)

Combining (4.9) and (4.12), we get F(ε) <F(ε), a contradiction.

Hence, we deduce that {Rxn}is a Cauchy sequence inO(x0;S,T,R). SinceX is

(S,T,R)-orbitally complete atx0, there exists somez∈X such that

Rxn→zasn→ ∞. (4:13)

We will prove thatzis a common fixed point of the three mappingsS,T and_R. We have

Sx2n=Rx2n+1→zasn→ ∞ (4:14)

and

Tx2n+1=Rx2n+2→zasn→ ∞. (4:15)

Suppose that (a) holds, i.e.,_S and_Rare compatible. Then, using condition (v),

lim

n→∞SRx2n+2= limn→∞RSx2n+2=Rz. (4:16)

From (4.13) and the orbitally continuity of_R, we have also

R(Rxn)→Rzasn→ ∞. (4:17)

Now, using (iv), x2n+1Tx2n+1=Rx2n+2and since Ris monotone, Rx2n+1and

RRx2n+2are comparable. Thus, we can apply (4.1) to obtain

F(d(SRx2n+2,Tx2n+1))≤ϕ

FM[T,S,R](Rx2n+2,x2n+1)

, (4:18)

where

M[T,S,R](Rx2n+2,x2n+1)

= max{d(RRx2n+2,Rx2n+1),d(RRx2n+2,SRx2n+2),d(Rx2n+1,Tx2n+1), 1

2[d(RRx2n+2,Tx2n+1) +d(SRx2n+2,Rx2n+1)]}.

Lettingn®∞in (4.18), using (4.13)-(4.17), we obtain

F(d(Rz,z))≤ϕ (F(d(Rz,z))) <F(d(Rz,z)),

unless

Rz=z. (4:19)

Now, x2n+1Tx2n+1andTx2n+1→zas n® ∞, so by the assumption we havex2n+1

F(d(Sz,Tx2n+1))≤ϕ(F(max{d(Rz,Rx2n+1),d(Sz,Rz),d(Tx2n+1,Rx2n+1), 1

2[d(Rz,Tx2n+1) +d(Sz,Rx2n+1)]})).

Passing to the limit asn®∞in the above inequality and using (4.19), it follows that

F(d(Sz,z))≤ϕ(F(max{0,d(Sz,z), 0,1₂d(Sz,z)})) ≤ϕ(F(d(Sz,z)))<F(d(Sz,z)),

which holds unless

Sz=z. (4:20)

Similarly, x2nSx2nandSx2n→zasn ®∞, implies thatx2nz, henceRx2nand

Rzare comparable. From (4.1) we get

F(d(Sx2n,Tz))≤ϕ(F(max{d(Rx2n,Rz),d(Rx2n,Sx2n),d(Rz,Tz), 1

2(d(Rx2n,Tz) +d(Sx2n,Rz))})).

Passing to the limit asn®∞, we have

F(d(z,Tz))≤ϕ (F(max{0, 0,d(z,Tz),d(z,Tz)})) ≤ϕ (F(d(z,Tz))) <F(d(z,Tz)),

which gives that

z=Tz. (4:21)

Therefore, Sz=Tz=Rz=z, hencezis a common fixed point ofR,SandT. Similarly, the result follows when condition (b) holds.

Now, suppose that the set of common fixed points of S,T andRinO₍x0;S,T,R) is well ordered. We claim that there is a unique common fixed point ofS,T and_Rin

O(x0;S,T,R). Assume to the contrary thatSu=Tu=Ru=uandSv=Tv=Rv=v but u≠v.By supposition, we can replacexbyuand ybyvin (4.1) to obtain

F(d(u,v)) =F(d(Su,Tv))

≤ϕ(F(max{d(Ru,Rv),d(Ru,Su),d(Rv,Tv)

1

2[d(Ru,Tv) +d(Su,Rv)]}))

=ϕ(F(max{d(u,v), 0, 0,d(u,v)}))<F(d(u,v)),

a contradiction. Hence, u = v.The converse is trivial. We obtain the following corollaries from Theorem 4.

Corollary 3 Let(X,d,)be a regular ordered metric space and letT andS be self-maps onX satisfying

FdTx,Sy≤ϕFM1[T,S](x,y)

,

for all x,y∈O(x0;S,T) (for some x0) such that x and y are comparable, where

F∈F,ÎFand

M1[T,S](x,y) = max{d(x,y),d(Tx,x),d(Sy,y),

1

2[d(x,Sy) +d

Tx,y]}.

(ii)X is(S,T)-orbitally complete at x0; (iii)T andSare weakly increasing; (iv)T and_S are dominating maps.

ThenT and_Shave a common fixed point. Moreover, the set of common fixed points of_T and_S in_O(x0;S,T)is well ordered if and only if it is a singleton.

Corollary 4Let(X,d,)be a regular ordered metric space and let_T and_Rbe self-maps on_X satisfying

F(d(Tx,Ty))≤ϕ(F(M2[T,R](x,y))),

for all x,y∈O(x0;T,T,R) (for some x0) such that Rx and Ryare comparable, where F∈F,ÎFand

M2[T,R](x,y) = max

d(Rx,Ry),d(Tx,Rx),d(Ty,Ry), 1

2[d(Rx,Ty) +d(Tx,Ry)]}.

We assume the following hypotheses: (i)T is a.r. with respect toRat x0∈X; (ii)X is(T,R)-orbitally complete at x0; (iii)T is weakly increasing with respect toR; (iv)T is a dominating map;

(v)Ris monotone and orbitally continuous at x0.

ThenT and_Rhave a common fixed point. Moreover, the set of common fixed points ofT andRinO₍x0;T,T,R) is well ordered if and only if it is a singleton.

Corollary 5Let(X,d,)be a regular ordered metric space and let _T be a self-map onX satisfying for allx,y∈O(x0;T)such that x and y are comparable,

F(d(Tx,Ty))≤ϕ(F(M3[T](x,y))),

where F∈F,ÎFand

M3[T](x,y) = max{d(x,y),d(Tx,x),d(Ty,y),

1

2[d(x,Ty) +d(Tx,y)]}.

We assume the following hypotheses: (i)_T is a.r. at some point x0ofX; (ii)_X is_T -orbitally complete at x0; (iii)TxT(Tx)for allx∈X; (iv)_T is a dominating map.

ThenT has a fixed point. Moreover, the set of fixed points ofT inO(x0;T)is well ordered if and only if it is a singleton.

We also state a corollary of Theorem 4 involving a contraction of integral type.

Corollary 6 LetS,T andRsatisfy the conditions of Theorem 4, except that

condi-tion (4.1)is replaced by the following: there exists a positive Lebesgue integrable func-tion u on ℝ+ such that

_ε

0u(t)dt>0for eachε> 0and that

F(d(Sx,Ty))

0

u(t)dt≤

_ϕ(F(M[T,S,R](x,y)))

0

Then, S,T and Rhave a common fixed point. Moreover, the set of common fixed points ofS,T and_Rin_O(x0;S,T,R)is well ordered if and only if it is a singleton.

Example 4 Let the set X =[0, +∞)be equipped with the usual metric d and the order defined by

xy⇔x≥y.

Consider the following self-mappings on_X:

Rx= 6x, Sx=

₁

2x, 0≤x≤ 1 2

x, x> 1₂, , Tx=

₁

3x, 0≤x≤ 1 3

x, x> 1₃. ,

Take x0= 1₂. Then it is easy to show that

O(x0;S,T,R)⊂

1

2k_·3l :k,l∈N

and O₍x0;S,T,R)=O(x0;S,T,R)∪ {0}, and all the conditions (i)-(v) and (a)-(b) of Theorem 4 are fulfilled (condition (iii) on O₍x0;S,T,R). Takeψ(t) = 16t and

F∈F of the formF(t) =kt,k> 0. Then contractive condition (4.1) takes the form

1₂x−1

3y

≤ 1

6max

6x−6y,11 2x,

17 3 y,

1 2

6x−1

3y

+6y−1 2x

,

forx,y∈O(x0;S,T,R). Using substitutiony=tx, t≥ 0, the last inequality reduces to

|3−2t| ≤max6|1−t|,11₂,17₃t,1₂6− 1₃t+6t− 1₂,

and can be checked by discussion on possible values for t≥0. Hence, all the condi-tions of Theorem 4 are satisfied and _S,T,Rhave a unique common fixed point in

O(x0;S,T,R)(which is 0).

Remark 2It was shown by examples in [24] that (in similar situations):

(1) if the contractive condition is satisfied just onO(x0;S,T,R), there might not exist a (common) fixed point;

(2) under the given hypotheses (common) fixed point might not be unique in the whole space_X.

Acknowledgements

The authors are highly indebted to the referees for their careful reading of the manuscript and valuable suggestions. H-S Ding acknowledges the support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), the Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. Z. Kadelburg is thankful to the Ministry of Science and Technological Development of Serbia.

Author details

1_{College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s}

Republic of China2_{Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia} 3_{Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri}

Marg, Mandir Hasaud, Raipur-492101, Chhattisgarh, India

Authors’contributions

Competing interests

The authors declare that they have no competing interests.

Received: 22 January 2012 Accepted: 19 May 2012 Published: 19 May 2012

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