R E S E A R C H
Open Access
Common fixed points of ordered
g
-contractions in partially ordered metric
spaces
Xiao-lan Liu
**Correspondence: [email protected] School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China
Abstract
The concept of orderedg-contraction is introduced, and some fixed and common fixed point theorems forg-nondecreasing orderedg-contraction mapping in partially ordered metric spaces are proved. We also show the uniqueness of the common fixed point in the case of an orderedg-contraction mapping. The theorems presented are generalizations of very recent fixed point theorems due to Golubovi´cet al.(Fixed Point Theory Appl. 2012:20, 2012).
MSC: 47H10; 47N10
Keywords: orderedg-contraction;g-nondecreasing; common fixed point; coincidence fixed point; partially ordered metric spaces
1 Introduction
The Banach fixed point theorem for contraction mappings has been extended in many directions (cf.[–]). Very recently, Golubovićet al.[] presented some new results for ordered quasicontractions and orderedg-quasicontractions in partially ordered metric spaces.
Recall that if (X,) is a partially ordered set andf :X→Xis such that, forx,y∈X, xyimpliesfxfy, then a mappingf is said to be nondecreasing. The main result of Golubovićet al.[] is the following common fixed point theorem.
Theorem .(See [], Theorem ) Let(X,d,)be a partially ordered metric space and let f,g:X→X be two self-maps on X satisfying the following conditions:
(i) fX⊂gX; (ii) gXis complete; (iii) f isg-nondecreasing;
(iv) f is an orderedg-quasicontraction; (v) there existsx∈Xsuch thatgxfx;
(vi) if{gxn}is a nondecreasing sequence that converges to somegz∈gX,thengxngzfor eachn∈Nandgzg(gz).
Then f and g have a coincidence point,i.e.,there exists z∈X such that fz=gz.If,in addi-tion,
(vii) f andgare weakly compatible[, ],i.e.,fx=gximpliesfgx=gfx,for eachx∈X,
then they have a common fixed point.
An open problem is to find sufficient conditions for the uniqueness of the common fixed point in the case of an orderedg-quasicontraction in Theorem ..
In Section of this article, we introduce orderedg-contractions in partially ordered metric spaces and prove the respective (common) fixed point results, which generalizes the results of Theorem ..
In Section of this article, a theorem on the uniqueness of a common fixed point is obtained when for allx,u∈X, there existsa∈X such that fais comparable to fxand fu, in addition to the hypotheses in Theorem . of Section . Our result is an answer to finding sufficient conditions for the uniqueness of the common fixed point in the case of orderedg-contractions in Theorem .. Finally, two examples show that the comparability is a sufficient condition for the uniqueness of common fixed point in the case of ordered g-contractions, so our results are extensions of known ones.
2 Common fixed points of orderedg-contractions
We start this section with the following definitions. Consider a partially ordered set (X,) and two mappingsf :X→Xandg:X→Xsuch thatf(X)⊂g(X).
Definition . (See []) We shall say that the mapping f is g-nondecreasing (resp., g-nonincreasing) if
gxgy ⇒ fxfy ()
(resp.,gxgy⇒fxfy) holds for eachx,y∈X.
Definition .(See []) We shall say that the mappingfis an orderedg-quasicontraction if there existsα∈(, ) such that for eachx,y∈Xsatisfyinggygx, the inequality
d(fx,fy)≤α·M(x,y)
holds, where
M(x,y) =maxd(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx).
Definition . We shall say that the mappingf is an orderedg-contraction if there is a continuous and nondecreasing functionψ: [, +∞)→[, +∞) withψ() = and if there existsα∈(, ), the inequality
ψd(fx,fy)≤maxψαd(gx,gy),ψαd(gx,fx),ψαd(gy,fy),
ψαd(gx,fy),ψαd(gy,fx) ()
holds for allx,y∈Xfor whichgygx.
It is obviously that ifψ=I, then orderedg-contraction reduces to orderedg -quasicon-traction.
For arbitraryx∈Xone can construct a so-called Jungck sequence{yn}in the following
fx∈f(X)⊂g(X) and there existsx∈Xsuch thatgx=y=fxand the procedure can be
continued.
Theorem . Let(X,)be a partially ordered set and suppose there is a metric d on X such that(X,d)is a complete metric space.Let f,g:X→X be two self-maps on X satisfying the following conditions:
(i) f(X)⊂g(X); (ii) g(X)is closed;
(iii) f is ag-nondecreasing mapping; (iv) f is an orderedg-contraction;
(v) there exists anx∈Xwithgxfx;
(vi) {g(xn)} ⊂Xis a nondecreasing sequence withg(xn)→gzing(X),thengxngz,
gzg(gz),∀nhold.
Then f and g have a coincidence point.Further,if f and g are weakly compatible,then f and g have a common fixed point.
Proof Letx∈Xbe such thatgxfx. Sincef(X)⊂g(X), we can choosex∈Xsuch that
gx=fx. Again fromf(X)⊂g(X), we can choosex∈Xsuch thatgx=fx. Continuing
this process we can choose a sequence{yn}inXsuch that
gxn+=fxn=yn, ∀n≥. ()
Sincegxfxandgx=fx, we havegxgx. Then by (),
fxfx. ()
Thus, by (),gxgx. Again by (),
fxfx, ()
that is,gxgx. Continuing this process, we obtain
fxfxfxfx · · · fxnfxn+. ()
Now, we will claim that{yn}is a Cauchy sequence. In what follows, we will suppose that
d(fxn,fxn+) > for alln, since iffxn=fxn+for somen, by (),
fxn+=gxn+, ()
that is,f andghave a coincidence atx=xn+, and so we have finished the proof. Thus we
assume thatd(fxn,fxn+) > for alln. We will show that
From () and (), it follows thatgxngxn+ for alln> . Then apply the contractivity
condition () withx=xnandy=xn+,
ψd(fxn,fxn+)
≤maxψαd(gxn,gxn+)
,ψαd(gxn,fxn)
,ψαd(gxn+,fxn+)
,
ψαd(gxn,fxn+)
,ψαd(gxn+,fxn)
. ()
Thus by (),
ψd(fxn,fxn+)
≤maxψαd(fxn–,fxn)
,ψαd(fxn–,fxn)
,ψαd(fxn,fxn+)
,
ψαd(fxn–,fxn+)
,ψαd(fxn,fxn)
=maxψαd(fxn–,fxn)
,ψαd(fxn,fxn+)
,
ψαd(fxn–,fxn+)
. ()
We divide the proof of () into three cases in the following:
(I) If max{ψ(αd(fxn–,fxn)),ψ(αd(fxn,fxn+)),ψ(αd(fxn–,fxn+))} = ψ(αd(fxn–,fxn)),
from (), then
ψd(fxn,fxn+)
≤ψαd(fxn–,fxn)
. ()
Sinceψ is nondecreasing,d(fxn,fxn+)≤αd(fxn–,fxn). By virtue ofα∈(, ), it follows
thatd(fxn,fxn+)≤d(fxn–,fxn). Thus () holds.
(II) If max{ψ(αd(fxn–,fxn)),ψ(αd(fxn,fxn+)),ψ(αd(fxn–,fxn+))} = ψ(αd(fxn,fxn+)),
from (), then
ψd(fxn,fxn+)
≤ψαd(fxn,fxn+)
. ()
Sinceψis nondecreasing,d(fxn,fxn+)≤αd(fxn,fxn+). By virtue ofα∈(, ),d(fxn,fxn+) =
, and it is a contraction with the assumption thatd(fxn,fxn+) > for alln!
(III) Ifmax{ψ(αd(fxn–,fxn+)),ψ(αd(fxn,fxn+)),ψ(αd(fxn–,fxn+))}=ψ(αd(fxn–,fxn+)),
from () and the triangle inequality, we have
ψd(fxn,fxn+)
≤ψαd(fxn–,fxn+)
≤ψαd(fxn–,fxn) +αd(fxn,fxn+)
. ()
Sinceψis nondecreasing,
d(fxn,fxn+)≤αd(fxn–,fxn) +αd(fxn,fxn+).
Then it follows that
d(fxn,fxn+)≤ α
–αd(fxn–,fxn). ()
Thus () trivially holds whenα∈(,). Hence
Taking into account the previous considerations, we proved that () holds. From (), it follows that the sequenced(fxn,fxn+) of real non-negative numbers is monotone
nonin-creasing. Therefore, there exists someσ≥ such that
lim
n→∞d(fxn,fxn+) =σ. ()
Next we will prove thatσ= . We suppose thatσ> . By the triangle inequality,
d(fxn–,fxn+)≤d(fxn–,fxn) +d(fxn,fxn+). ()
Hence, by (),
d(fxn–,fxn+)≤d(fxn–,fxn). ()
Taking the upper limit asn→ ∞, we get
lim sup n→∞
d(fxn–,fxn+)≤nlim→∞d(fxn–,fxn). ()
Set
lim sup n→∞
d(fxn–,fxn+) =ρ. ()
Then it follows that ≤ρ≤σ. Now, taking the upper limit on the both sides of () and
ψ(t) being continuous, we get
lim n→∞ψ
d(fxn,fxn+)
≤maxψ
lim
n→∞αd(fxn–,fxn)
,ψ
lim
n→∞αd(fxn,fxn+)
,
ψ
lim
n→∞αd(fxn–,fxn+) . ()
From () and (),
ψ(σ)≤maxψ(ασ),ψ(αρ). ()
Ifmax{ψ(ασ),ψ(αρ)}=ψ(αρ), from (), it yieldsψ(σ)≤ψ(αρ). Sinceψis nonde-creasing, thenσ≤αρ. Whenα∈(,), thenσ≤αρ<ρ, it is a contradiction! When
α=, then σ≤ρ, it is a contradiction! Whenα∈(, ), thenσ≤αρ≤ασ, it yields ( – α)σ≤. Since – α< andσ> , it is also a contradiction!
Ifmax{ψ(ασ),ψ(αρ)}=ψ(ασ), then from (), it yieldsψ(σ)≤ψ(ασ). Sinceψis non-decreasing, thenσ≤ασ. By virtue ofα∈(, ), it yieldsσ. It is a contradiction with the assumption thatσ> !
Taking into account the previous consideration,σ= . Therefore, we proved that
lim
n→∞d(fxn,fxn+) = . ()
Now, we prove that{fxn}is a Cauchy sequence. Suppose, to the contrary, that{fxn}is not a
m(k) >n(k)≥kwith
tk=d(fxn(k),fxm(k))≥ fork= , , , . . . . ()
We may also assume
d(fxn(k),fxm(k)–) < ()
by choosingm(k) to be the smallest number which satisfiesm(k) >n(k), and () holds. From (), (), and by the triangle inequality,
≤tk≤d(fxn(k),fxm(k)–) +d(fxm(k)–,fxm(k)) <+d(fxm(k)–,fxm(k)). ()
Hence, by (),
lim
n→∞tk=. ()
Since from () and (), we havegxn(k)+=fxn(k)≤fxm(k)=gxm(k)+, from () and () with
x=xm(k)+andy=xn(k)+, we get
ψd(fxn(k)+,fxm(k)+)
≤maxψαd(gxm(k)+,gxn(k)+)
,ψαd(gxm(k)+,fxm(k)+)
,
ψαd(gxn(k)+,fxn(k)+)
,ψαd(gxm(k)+,fxn(k)+)
,
ψαd(gxn(k)+,fxm(k)+)
=maxψαd(fxm(k),fxn(k))
,ψαd(fxm(k),fxm(k)+)
,
ψαd(fxn(k),fxn(k)+)
,ψαd(fxm(k),fxn(k)+)
,
ψαd(fxn(k),fxm(k)+)
. ()
Denoteσn=d(fxn,fxn+). Then we have
ψd(fxn(k)+,fxm(k)+)
≤maxψ(αtk),ψ(ασm(k)),ψ(ασn(k)),ψ
αd(fxm(k),fxn(k)+)
,
ψαd(fxn(k),fxm(k)+)
=ψmaxαtk,ασm(k),ασn(k),αd(fxm(k),fxn(k)+),
αd(fxn(k),fxm(k)+)
=ψαmaxtk,σm(k),σn(k),d(fxm(k),fxn(k)+),
d(fxn(k),fxm(k)+)
, ()
where the first equality holds, since ψ is nondecreasing, and ψ(max(s,s, . . . ,sn)) = max(ψ(s),ψ(s), . . . ,ψ(sn)) for alls,s, . . . ,sn∈[, +∞). Again, sinceψ is nondecreasing,
by (), it follows that
d(fxn(k)+,fxm(k)+)≤αmax
tk,σm(k),σn(k),d(fxm(k),fxn(k)+),d(fxn(k),fxm(k)+)
Therefore, since
tk≤d(fxn(k),fxn(k)+) +d(fxn(k)+,fxm(k)+) +d(fxm(k)+,fxm(k))
=σn(k)+σm(k)+d(fxn(k)+,fxm(k)+), ()
we have
≤tk
≤σn(k)+σm(k)+αmax
tk,σm(k),σn(k),d(fxm(k),fxn(k)+),
d(fxn(k),fxm(k)+)
. ()
By the triangle inequality, (), and (),
≤tk
≤d(fxn(k),fxm(k)+) +d(fxm(k)+,fxm(k))
=d(fxn(k),fxm(k)+) +σm(k)
≤d(fxn(k),fxm(k)–) +d(fxm(k)–,fxm(k)) +d(fxm(k),fxm(k)+) +σm(k)
≤+σm(k)–+ σm(k). ()
From the equality of () and the last inequality of (), it yields
–σm(k)≤d(fxn(k),fxm(k)+)
≤+σm(k)–+σm(k). ()
Similarly, we obtain
≤tk
≤d(fxn(k),fxn(k)+) +d(fxn(k)+,fxm(k))
=σn(k)+d(fxn(k)+,fxm(k)).
And it follows that
d(fxn(k)+,fxm(k))≤d(fxn(k)+,fxn(k)) +d(fxn(k),fxm(k)–) +d(fxm(k)–,fxm(k))
=σn(k)+d(fxn(k),fxm(k)–) +σm(k)–
≤+σn(k)+σm(k)–. ()
Adding the two inequalities above,
–σn(k)≤d(fxn(k)+,fxm(k))
From () and (), we have
–σn(k)+σm(k)
≤
d(fxn(k),fxm(k)+) +d(fxn(k)+,fxm(k))
≤+σm(k)–+
σn(k)+σm(k)
. ()
Thus from () and (), we get
lim n→∞
d(fxn(k),fxm(k)+) +d(fxn(k)+,fxm(k))
=. ()
Lettingn→ ∞in (), then by (), (), and (), we get, asψis continuous,
≤αmax{, , ,,}=α<, ()
it is a contradiction! Thus our assumption () is wrong. Therefore,{fxn}is a Cauchy
se-quence. Since by (), we have{fxn=gxn+⊆g(X)}andg(X) is closed, there existsz∈X
such that
lim
n→∞gxn=gz. ()
Now we show thatzis a coincidence point off andg. Since from condition (vi) and (), we havegxngzfor alln, then by the triangle inequality and (), we have
ψd(fxn,fz)
≤maxψαd(gxn,gz)
,ψαd(gxn,fxn)
,ψαd(gz,fz),
ψαd(gxn,fz)
,ψαd(gz,fxn)
. ()
So lettingn→ ∞, andψ being continuous, we have
ψd(fz,gz)≤max, ,ψαd(fz,gz),ψαd(fz,gz),
=ψαd(fz,gz).
Since ψ is nondecreasing, then d(fz,gz)≤αd(fz,gz). Since α ∈ (, ), it follows that d(fz,gz) = . Hencefz=gz. Thus we proved thatf andghave a coincidence point.
Suppose now thatf andgcommute atz. Setw=fz=gz. Sincef andgare weakly com-patible,
fw=f(gz) =g(fz) =gw. ()
Since from condition (vi), we havegzg(gz) =gwand asfz=gzandfw=gw, from (), we have
ψd(fz,fw)≤maxψαd(gz,gw),ψαd(gz,fz),ψαd(gw,fw),
ψαd(gz,fw),ψαd(gw,fz)
Sinceψis nondecreasing,d(fz,fw)≤αd(gz,gw),i.e.,d(fz,fw)≤αd(fz,fw). Again fromα∈
(, ),d(fz,fw) = , that is,d(w,fw) = . Therefore,
fw=gw=w. ()
Thus, we have proved thatf andghave a common fixed point. The proof is completed.
If we replace some conditions in Theorem ., then we can obtain the following con-clusions. Note that the way followed in Theorem . is different from that in the proof of Theorem .. In fact, we can use the way in Theorem . to prove the conclusions in Theorem .. Similarly, we can also use the way in Theorem . to prove Theorem .. Here, our aim is to show two different methods of proof. Comparing Theorem . with Theorem ., we can find that the conclusions cover Theorem .; in other words, the condition of Theorem . is more extensive than that in Theorem .. Now, let us treat the following theorem.
Theorem . Let the conditions of Theorem.be satisfied,except that(iii), (v)and(vi) are,respectively,replaced by
(iii) f is ag-nonincreasing mapping;
(v) there existsx∈Xsuch thatfxandgxare comparable;
(vi) if{gxn}is a sequence ing(X)which has comparable adjacent terms and that converges to somegz∈gX,then there exists a subsequencegxnkof{gxn}having all the terms com-parable withgzandgzis comparable withggz.Then all the conclusions of Theorem.
hold.
Proof Regardless of whetherfxgxorgxfx(condition (v)), Lemma of []
im-plies that two arbitrary adjacent terms of the Jungck sequence{yn}are comparable. This
is again sufficient to imply that{yn}is a Cauchy sequence. In the following, we assume the
other case to prove the conclusions of Theorem ..
Letx∈Xbe such thatfxgx, where it is different fromgxfxin Theorem ..
Sincef(X)⊂g(X), we can choosex∈Xsuch thatgx=fx. Again fromf(X)⊂g(X), we
can choosex∈Xsuch thatgx=fx. Continuing this process, we can choose a sequence
{yn}inXsuch that
gxn+=fxn=yn, ∀n≥. ()
Since fxgx and gx =fx, we have gx gx. Then by condition (iii), f is a
g-nonincreasing mapping,
fxfx. ()
Thus, by (), it follows thatgxgx. Again by condition (iii),
that is,gxgx. Continuing this process, we obtain the result that two arbitrary adjacent
terms of the Jungck sequence{yn}are comparable.
LetO(yk,n) ={yk,yk+, . . . ,yk+n}. We will show that{yn}is a Cauchy sequence. To prove
our claim, we follow the arguments of Das and Naik [] again. Fixk≥ andn∈ {, , . . .}. Ifdiam[O(yk;n)] = , then{yn}is also a Cauchy sequence. Thus our claims holds. Now we
suppose thatdiam[O(yk;n)] > . Now fori,jwith ≤i<j, by (), we have
ψd(yi,yj)
=ψd(fxi,fxj)
≤maxψαd(gxi,gxj)
,ψαd(gxi,fxi)
,ψαd(gxj,fxj)
,
ψαd(gxi,fxj)
,ψαd(gxj,fxi)
=maxψαd(yi–,yj–)
,ψαd(yi–,yi)
,ψαd(yj–,yj)
,
ψαd(yi–,yj)
,ψαd(yj–,yi)
=ψmaxαd(yi–,yj–),αd(yi–,yi),αd(yj–,yj),
αd(yi–,yj),αd(yj–,yi)
=ψαmaxd(yi–,yj–),d(yi–,yi),d(yj–,yj),
d(yi–,yj),d(yj–,yi)
≤ψαdiamO(yi–;j–i+ )
,
where the third equality holds, since ψ is nondecreasing, and ψ(max(s,s, . . . ,sn)) = max(ψ(s),ψ(s), . . . ,ψ(sn)) for alls,s, . . . ,sn∈[, +∞). Sinceψis nondecreasing,
d(yi,yj)≤αdiam
O(yi–;j–i+ )
. ()
Now for somei,jwithk≤i<j≤k+n,diam[O(yk;n)] =d(yi,yj). Ifi>k, by () and (),
then we have
diamO(yk;n)
≤αdiamO(yi–;j–i+ )
≤αdiamO(yk;n)
, ()
where the inequality () holds asdiam[O(yi–;j–i+ )]≤diam[O(yk;n)]. Then from ()
andα∈(, ), we havediam[O(yk;n)] = . It is a contradiction with the assumption that diam[O(yk;n)] > ! Thus,
diamO(yk;n)
=d(yk,yj) forjwithk<j≤k+n. ()
Also, by () and (), we have
diamO(yk;n)
=d(yk,yj)
≤αdiamO(yk–;j–k+ )
≤αdiamO(yk–;n+ )
Using the triangle inequality, by (), (), and (), we obtain
diamO(yl;m)
=d(yl,yj)
≤d(yl,yl+) +d(yl+,yj)
≤d(yl,yl+) +αdiam
O(yl+;m– )
≤d(yl,yl+) +αdiam
O(yl;m)
, ()
and so
diamO(yl;m)
≤
–αd(yl,yl+). ()
As a result, we have
diamO(yk;n)
≤αdiamO(yk–;n+ )
≤α·αdiamO(yk–;n+ )
· · ·
≤αkdiamO(y;n+k)
≤ αk
–αd(y,y), ()
where the first inequality holds by the expression () and the last inequality holds by (). Now let> ; there exists an integernsuch that
αkd(y,y) < ( –α) for allk>n. ()
Form>n>n, we have
d(ym,yn)≤diam
O(yn;m–n)
≤ αn
–αd(y,y)
<. ()
Therefore,{yn}is a Cauchy sequence. Sinceg(X) is closed, it converges to somegz∈g(X).
By condition (vi), there exists a subsequenceynk =fxnk=gxnk+,k∈N, having all the
terms comparable withgz. Hence, we can apply the contractivity condition to obtain
ψd(fz,fxnk)
≤maxψαd(gz,gxnk)
,ψαd(gz,fz),ψαd(gxnk,fxnk)
,
ψαd(gz,fxnk)
,ψαd(gxnk,fz)
. ()
So lettingn→ ∞, and asψ is continuous, we have
ψd(fz,gz)≤max,ψαd(fz,gz), , ,ψαd(fz,gz)
Sinceψis nondecreasing,d(fz,gz)≤αd(fz,gz). Sinceα∈(, ), it follows thatd(fz,gz) = . Hencefz=gz. Thus we proved thatf andghave a coincidence point.
Suppose now thatf andgcommute atz. Setw=fz=gz. Sincef andgare weakly com-patible,
fw=f(gz) =g(fz) =gw. ()
Since from condition (vi), we havegzg(gz) =gwand asfz=gzandfw=gw, from (), we have
ψd(fz,fw)≤maxψαd(gz,gw),ψαd(gz,fz),ψαd(gw,fw),
ψαd(gz,fw),ψαd(gw,fz)
=ψαd(fz,fw). ()
Sinceψis nondecreasing,d(fz,fw)≤αd(gz,gw),i.e.,d(fz,fw)≤αd(fz,fw). Again fromα∈
(, ), we haved(fz,fw) = , that is,d(w,fw) = . Therefore,
fw=gw=w. ()
Thus, we have proved thatf andghave a common fixed point. The proof is completed.
Corollary . (a)Let(X,)be a partially ordered set and suppose there is a metric d on X such that(X,d)is a complete metric space.Let f :X→X be a nondecreasing self-mapping such that for someα∈(, )
d(fx,fy)≤αmaxd(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)
for all x,y∈X for which xy.Suppose also that either
(i) {xn} ⊂Xis a nondecreasing sequence withxn→uinX,thenxnu,∀nholds,or
(ii) f is continuous.
If there exists an x∈X with xfx,then f has a fixed point.
(b)The same holds if f is nonincreasing;there exists xcomparable with fxand(i)is
replaced by
(i) if a sequence{xn}converging to someu∈Xhas every two adjacent terms comparable, then there exists a subsequence{xnk}having each term comparable withx.
Proof (a) If (i) holds, then takeψ=Iandg=I(I= the identity mapping) in Theorem .. If (ii) holds, then from () withg=I, we get
z= lim
n→∞xn+=nlim→∞fxn=f
lim n→∞xn
=fz. ()
(b) Letube the limit of the Picard sequence{fnx}and letfnkxbe a subsequence having
term to obtain
d(fu,u)≤du,fnk+x
+dfu,fnk+x
≤du,fnk+x
+αmaxdu,fnkx
,d(u,fu),
dfnkx
,fnk+x
,du,fnk+x
,dfnkx
,fu
.
Lettingk→ ∞, we have
d(fu,u)≤αmax,d(u,fu), , ,d(u,fu)
=αd(u,fu).
It follows thatd(fu,u) = . Therefore,fu=u.
Note also that instead of the completeness ofX, itsf-orbitally completeness is sufficient to obtain the conclusion of the corollary. The proof is completed.
3 Uniqueness of common fixed point offandg
The following theorem gives the sufficient condition for the uniqueness of the common fixed point off andg in the case of orderedg-contractions in partially ordered metric spaces.
Theorem . In addition to the hypotheses of Theorem.,suppose that for all x,u∈X, there exists a∈X such that
fa is comparable to fx and fu. ()
Then f and g have a unique common fixed point x such that
x=fx=gx. ()
Proof Since a set of common fixed points off andgis not empty due to Theorem ., assume now thatxanduare two common fixed points off andg,i.e.,
fx=gx=x, fu=gu=u. ()
We claim thatgx=gu.
By the assumption, there existsa∈Xsuch thatfais comparable tofxandfu. Define a sequence{gan}such thata=aand
gan=fan– for alln. ()
Further, setx=xandu=uand in the same way, define{gxn}and{gun}such that
Sincefx(=gx=gx) is comparable tofa(=fa=ga), without loss of generality, we assume
thatfafx,i.e.,gagx; then it is easy to show
gagx. ()
Sincef isg-nondecreasing, we obtainfafx. Sincegaga, it follows thatgafx,i.e.,
gagx. Recursively, we get
gangx for alln. ()
By (), we have
ψd(gan+,gx)
=ψd(fan,fx)
≤maxψαd(gan,gx)
,ψαd(gan,fan)
,ψαd(gx,fx),
ψαd(gan,fx)
,ψαd(gx,fan)
. ()
By the proof of Theorem ., we find that{gan}is a convergent sequence, and there exists
ga¯such thatgan→ga. Letting¯ n→ ∞in (), we obtain
lim n→∞ψ
d(gan+,gx)
=ψd(ga,¯ gx)
≤maxψαd(ga,¯ gx), , ,ψαd(ga,¯ fx),ψαd(gx,ga)¯
=ψαd(ga,¯ gx).
Therefore, it yields
d(ga,¯ gx) = .
Hence
ga¯=gx. ()
Similarly, we can also show that
gangu for alln.
Apply the contractivity condition, we obtain
lim n→∞ψ
d(gan+,gu)
=ψd(ga,¯ gu)
≤maxψαd(ga,¯ gu), , ,ψαd(ga,¯ fx),ψαd(gx,ga)¯
=ψαd(ga,¯ gu).
Therefore, it yields
Hence
ga¯=gu. ()
Combining () with (), we obtaingx=gu. It follows that
x=fx=gx=gu=fu=u. ()
The proof is completed.
Remark . Theorem . can be considered as an answer to Theorem in []. We find the sufficient conditions for the uniqueness of the common fixed point in the case of an orderedg-quasicontraction. In this paper, condition (vi) in Theorem . is weaker than that orderedg-quasicontraction in []. Whenψ=I(I= the identity mapping), our con-dition (vi) reduces to an orderedg-quasicontraction in [].
Example . LetX={(, –), (, )}, let (a,b)(c,d) if and only ifa≤candb≥d, and let dbe the Euclidean metric. We define the functions as follows:
f(x,y) =x,y– y+ , g(x,y) =x+x–x,y– for all (x,y)∈X.
Letφ,ψ: [,∞)→[,∞) be given by
ψ(t) =
t for allt∈[,∞).
The only comparable pairs of points inXare (x,x) forx∈Xand then the contractivity condition () reduces tod(fx,fx) = , and condition (iv) of Theorem . is trivially fulfilled. The other conditions of Theorem . are also satisfied. It is obvious that for (, –) and (, )∈X,f(, –) = (, –) is not comparable tof(, ) = (, ),i.e., comparability in The-orem . is not satisfied. In fact,f andghave two common fixed points (, –) and (, ), since
f(, –) =g(, –) = (, –), f(, ) =g(, ) = (, ).
Example . LetX= (–∞, +∞) with the usual metricd(x,y) =|x–y|, for allx,y∈X. Let f :X→Xandg:X→Xbe given by
f(x) = x
, g(x) =
x for allx∈X.
Letφ: [,∞)→[,∞) be given by
ψ(t) =
t for allt∈[,∞).
It is easy to check that all the conditions of Theorem . are satisfied. We have
ψd(fx,fy)= ·
|x–y|= |x–y|
≤α ·
=max α
· |x–y|,
α
· x– x
,α
· y– y
, α · x– y
,α
· y– x
=maxψαd(gx,gy),ψαd(gx,fx),ψαd(gy,fy),
ψαd(gx,fy),ψαd(gy,fx),
and this holds whenα≥ andgx≥gy,i.e, x≥y,i.e.,x≥y. This means that the con-tractivity condition () holds whenα∈[, ).
In addition,∀x,u∈X, there existsa∈Xsuch thatfa= a
is comparable tofx= x and
fu=u. So, all the conditions of Theorem . are satisfied.
By applying Theorem ., we conclude thatf andghas a unique common fixed point. In fact,f andghas only one common fixed point. It isx= .
Example . LetX= [,] be the closed interval with usual metric and letf,g:X→X andψ,φ: [, +∞)→[, +∞) be mappings defined as follows:
f(x) =x–x for allx∈X,
g(x) =x for allx∈X,
ψ(t) =t for ≤t≤ ,
ψ(t) =
t fort> .
Letx,yinXbe arbitrary. We say thatyxify≤x. For anyx,y∈Xsuch thatyx, we have
d(fx,fy) =x–x–y–y.
Sincefmax=f(
√
) =
andfmin=f() = ,f is nondecreasing at [,
], thenx
–x– (y–
y)∈[,]. By the definition ofψ, we have
ψd(fx,fy)=x–x–y–y=x–x–y–y
and
maxψαd(gx,gy),ψαd(gx,fx),ψαd(gy,fy),ψαd(gx,fy),ψαd(gy,fx)
=maxαx–y,αx–x–x,αy–y–y,
αx–y–y,αy–x–x
=αx–y–y.
Sincex≥x–y( –y) for allx∈[,
], it follows that
Thus we have
ψd(fx,fy)=x–x–y–y
≤x–x–y –y–y–y
=x–y–y– ·x–y–y·x–y –y
+x–y –y
=x–y–y– x–y–y+x–y–y
≤αx–y–y,
where the last inequality holds wheneverα∈(, ). Therefore,f andgsatisfy (). Also it is easy to see that the mappingsψ(t) possess all properties in Definition ., as well as hypotheses (v), (vi), and (vii) in Theorem .. Thus we can apply Theorem . and Theo-rem ..
On the other hand, fory= and eachx> the contractive condition in Theorem and Theorem of Golubović, Kadelburg and Radenović []:
d(fx,fy)≤λ·M(x,y), ()
where <λ< and
M(x,y) =maxd(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx)
is not satisfied. Indeed,
M(x; ) =maxdg(x),g(),dg(x),f(x),dg(),f(),dg(x),f(),dg(),f(x)
=maxx,x, ,x,x–x=x.
Thus, for any fixedλ; <λ< , we have, fory= and eachx∈Xwith <x<√ –λ,
df(x),f()=x–x= –xx>λ·x
=λ·dg(x),g()=λ·M(x, ).
Thus,f does not satisfy the contractive condition in Definition .. Therefore, the theo-rems of Jungck and Hussain [], Al-Thagafi and Shahzad [] and Das and Naik [] also cannot be applied.
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
This work is partially supported by National Natural Science Foundation of China (Grant No. 11126336), the Scientific Research Fund of Sichuan Provincial Education Department (14ZB0208), Scientific Research Fund of Sichuan University of Science and Engineering (2012KY08).
References
1. ´Ciri´c, L, Caki´c, N, Rajovic, M, Ume, JS, Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl.2008, Article ID 131294 (2008). doi:10.1155/2008/131294
2. Bhashar, TG, Lakshmikantham, V: Fixed point theory in partially ordered spaces and applications. Nonlinear Anal.65, 1379-1393 (2006)
3. Lakshmikantham, V, ´Ciri´c, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.70, 4341-4349 (2009)
4. Luong, NV, Thuan, NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal.74, 983-992 (2011)
5. Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.74(12), 4508-4517 (2010)
6. Karapınar, E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 59(12), 3656-3668 (2010)
7. Karapınar, E: Couple fixed point on cone metric spaces. G.U. J. Sci.24(1), 51-58 (2011)
8. Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal.73, 2524-2531 (2010)
9. Choudhury, BS, Metiya, N, Kundu, A: Coupled coincidence point theorems in ordered metric spaces. Ann. Univ. Ferrara, Sez. 7: Sci. Mat.57, 1-16 (2011)
10. Abbas, M, Khan, MA, Radenovi´c, S: Common coupled fixed point theorems in cone metric space forw-compatible mappings. Appl. Math. Comput.217, 195-203 (2010)
11. Nashine, HK, Shatanawi, W: Coupled common fixed point theorems for pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. (2011). doi:10.1016/j.camwa.2011.06.042
12. Das, KM, Naik, KV: Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc.77, 369-373 (1979)
13. Sabetghadam, F, Masiha, HP, Sanatpour, AH: Some coupled fixed point theorems in cone metric spaces. Fixed Point Theory Appl.2009, Article ID 125426 (2009)
14. Shatanawi, W: Partially ordered cone metric spaces and coupled fixed point results. Comput. Math. Appl.60, 2508-2515 (2010)
15. Shatanawi, W, Samet, B, Abbas, M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. (2011). 10.1016/j.mcm.2011.08.042
16. Shatanawi, W: Some common coupled fixed point results in cone metric spaces. Int. J. Math. Anal.4, 2381-2388 (2010)
17. Shatanawi, W: Fixed point theorems for nonlinear weaklyC-contractive mappings in metric spaces. Math. Comput. Model.54, 2816-2826 (2011)
18. Nieto, JJ, Rodriguez-Lopez, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order22, 223-239 (2005)
19. Berinde, V, Borcut, M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal.74(15), 4889-4897 (2011)
20. Aydi, H, Karapınar, E, Postolache, M: Tripled coincidence point theorems for weakφ-contractions in partially ordered metric spaces. Fixed Point Theory Appl.2012, Article ID 44 (2012). doi:10.1186/1687-1812-2012-44
21. Samet, B, Vetro, C: Coupled fixed point,f-invariant set and fixed point ofN-order. Ann. Funct. Anal.1(2), 46-56 (2010) 22. Karapınar, E: Quartet fixed point for nonlinear contraction. arXiv:1106.5472
23. Karapınar, E, Luong, NV: Quadruple fixed point theorems for nonlinear contractions. Comput. Math. Appl.64, 1839-1848 (2012)
24. Karapınar, E: Quadruple fixed point theorems for weakφ-contractions. ISRN Math. Anal.2011, Article ID 989423 (2011)
25. Karapınar, E, Berinde, V: Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. Banach J. Math. Anal.6(1), 74-89 (2012)
26. Karapınar, E: A new quartet fixed point theorem for nonlinear contractions. J. Fixed Point Theory Appl.6(2), 119-135 (2011)
27. Mustafa, Z, Aydi, H, Karapınar, E: Mixedg-monotone property and quadruple fixed point theorems in partially ordered metric spaces. Fixed Point Theory Appl.2012, Article ID 71 (2012)
28. Roshan, JR, Parvaneh, V, Altun, I: Some coincidence point results in orderedb-metric spaces and applications in a system of integral equations. Appl. Math. Comput.226, 725-737 (2014)
29. Shatanawi, W, Postolache, M: Common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces. Fixed Point Theory Appl.2013, Article ID 60 (2013)
30. Xiao, JZ, Zhu, XH, Shen, ZM: Common coupled fixed point results for hybrid nonlinear contractions in metric spaces. Fixed Point Theory14, 235-250 (2013)
31. Abbas, M, Nazir, T, Radenovi´c, S: Common coupled fixed points of generalized contractive mappings in partially ordered metric spaces. Positivity17, 1021-1041 (2013)
32. Nieto, JJ, Pouso, RL, Rodríguez-López, R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc.135, 2505-2517 (2007)
33. Roshan, JR, Parvaneh, V, Sedghi, S, Shobkolaei, N, Shatanawi, W: Common fixed points of almost generalized (ψ,ϕ)s-contractive mappings in orderedb-metric spaces. Fixed Point Theory Appl.2013, Article ID 159 (2013)
34. Haghi, RH, Postolache, M, Rezapour, S: OnT-stability of the Picard iteration for generalizedϕ-contraction mappings. Abstr. Appl. Anal.2012, Article ID 658971 (2012)
35. Olatinwo, MO, Postolache, M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput.218(12), 6727-6732 (2012)
36. Choudhury, BS, Metiya, N, Postolache, M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl.2013, Article ID 152 (2013)
38. Chandok, S, Mustafa, Z, Postolache, M: Coupled common fixed point theorems for mixedg-monotone mappings in partially orderedG-metric spaces. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys.75(4), 13-26 (2013) 39. Shatanawi, W, Postolache, M: Common fixed point results of mappings for nonlinear contractions of cyclic form in
ordered metric spaces. Fixed Point Theory Appl.2013, Article ID 60 (2013)
40. Shatanawi, W, Postolache, M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl.2013, Article ID 54 (2013)
41. Shatanawi, W, Postolache, M: Some fixed point results for aG-weak contraction inG-metric spaces. Abstr. Appl. Anal. 2012, Article ID 815870 (2012)
42. Shatanawi, W, Pitea, A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013, Article ID 153 (2013)
43. Shatanawi, W, Pitea, A: Omega-distance and coupled fixed point inG-metris spaces. Fixed Point Theory Appl.2013, Article ID 208 (2013)
44. Shatanawi, W, Pitea, A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl.2013, Article ID 275 (2013)
45. Aydi, H, Shatanawi, W, Postolache, M, Mustafa, Z, Tahat, N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal.2012, Article ID 359054 (2012)
46. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for (ψ,φ)-weakly contractive mappings in ordered
G-metric spaces. Comput. Math. Appl.63(1), 298-309 (2012)
47. 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)
48. Miandaragh, MA, Postolache, M, Rezapour, S: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl.2013, Article ID 255 (2013)
49. Golubovi´c, Z, Kadelburg, Z, Radenovi´c, S: Common fixed points of orderedg-quasicontractions and weak contractions in ordered metric spaces. Fixed Point Theory Appl.2012, Article ID 20 (2012)
50. Jungck, G: Commuting mappings and fixed points. Am. Math. Mon.83(4), 261-263 (1976). doi:10.2307/2318216 51. Jungck, G: Compatible mappings and common fixed points. Int. J. Math. Sci.9, 771-779 (1986).
doi:10.1155/S0161171286000935
52. Jungck, G, Hussain, N: Compatible maps and invariant approximations. J. Math. Anal. Appl.325, 1003-1012 (2007) 53. Al-Thagafi, MA, Shahzad, N: Banach operator pairs, common fixed points, invariant approximations and
∗-nonexpansive multimaps. Nonlinear Anal.69, 2733-2739 (2008)
54. Das, KM, Naik, KV: Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc.77, 369-373 (1979)
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