Common fixed points of a generalized ordered g-quasicontraction in partially ordered metric spaces

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R E S E A R C H

Open Access

Common ﬁxed points of a generalized

ordered

g

-quasicontraction in partially

ordered metric spaces

Xiaolan Liu

1,2*

_{and Siniša Ješi´c}

3

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Sichuan Normal University, Chengdu, Sichuan 610068, China

2_{School of Science, Sichuan}

University of Science and Engineering, Zigong, Sichuan 643000, China

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

Abstract

The concept of a generalized orderedg-quasicontraction is introduced, and some ﬁxed and common ﬁxed point theorems for ag-nondecreasing generalized ordered

g-quasicontraction mapping in partially ordered complete metric spaces are proved. We also show the uniqueness of the common fixed point in the case of a generalized orderedg-quasicontraction mapping. Finally, we prove fixed point theorems for mappings satisfying the so-called weak contractive conditions in the setting of a partially ordered metric space. Presented theorems are generalizations of very recent fixed point theorems due to Golubovićet al.(Fixed Point Theory Appl. 2012:20, 2012). MSC: 47H10; 47N10

Keywords: G-nondecreasing; generalized orderedg-quasicontraction; coincidence point; common ﬁxed point; comparable mappings

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 andg-quasicontractions in partially ordered metric spaces.

Recall that if (X,) is a partially ordered set andf :X→Xis such that for x,y∈X, xyimpliesfxfy, then a mappingFis said to be non-decreasing. The main result of Golubovićet al.[] is the following 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 thatgxfx;

(vi) if{gxn}is a nondecreasing sequence that converges to somegz∈gX,thengxngz

for eachn∈Nandgzg(gz).

Then f and g have a coincidence point, i.e.,there exists z∈X such that fz=gz.If,in addition,

(vii) f andgare weakly compatible[, ],i.e.,fx=gximpliesfgx=gfxfor eachx∈X, then they have a common ﬁxed point.

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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 generalized orderedg-quasicontractions in partially ordered metric spaces and prove the respective (common) ﬁxed point theorems which generalize the results of Theorem ..

In Section  of this article, the uniqueness of a common fixed point theorem is obtained when for allx,u∈X, there existsa∈Xsuch thatfais comparable tofxandfuin addition to the hypotheses in Theorem . of Section . Our results are an answer to finding suf-ficient conditions for the uniqueness of a common fixed point in the case of an ordered g-quasicontraction in Theorem .. Finally, two examples show that the comparability is a sufficient condition for the uniqueness of a common fixed point in the case of an ordered g-quasicontraction, so our results are extensions of known ones.

In Section  of this article, we consider weak contractive conditions in the setting of a partially ordered metric space and prove respective common ﬁxed point theorems.

2 Common ﬁxed points of a generalized orderedg-quasicontraction

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 will say that the mappingf isg-nondecreasing (resp.,g -nonincreasing) if

gxgy ⇒ fxfy ()

(resp.,gxgy⇒fxfy) holds for eachx,y∈X.

Deﬁnition .(See []) We will say that the mappingf is 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 will say that the mappingfis a generalized orderedg-quasicontraction if there is a continuous and non-decreasing functionψ: [, +∞)→[, +∞) withψ(s+t)≤

ψ(s) +ψ(t) for eachs,t> ,ψ(t)≥tfort≥ and there existsα∈(, )

ψd(fx,fy)≤αmaxψd(gx,gy),ψd(gx,fx),ψd(gy,fy),

ψd(gx,fy),ψd(gy,fx) ()

for allx,y∈Xfor whichgxgy;

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For arbitraryx∈X, one can construct the so-called Jungck sequence{yn}in the follow-ing way: Denotey=fx∈f(X)⊂g(X); there existsx∈Xsuch thatgx=y=fx; now

y=fx∈f(X)⊂g(X) and there existsx∈Xsuch thatgx=y=fx and 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 a generalized orderedg-quasicontraction; (v) there exists anx∈Xwithgxfx;

(vi) {g(xn)} ⊂Xis a non-decreasing 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 thatgxfx. Sincef(X)⊂g(X), we can choosex∈Xso that

gx=fx. Again fromf(X)⊂g(X), we can choosex∈Xsuch thatgx=fx. Continuing

this process, we can construct a Jungck sequence{yn}in X such that

gxn+=fxn=yn, ∀n≥. ()

Sincegxfxandgx=fx, we havegxgx. Then by (),

fxfx. ()

Thus, by (),gxgx. Again by (),

fxfx, ()

that is,gxgx. Continuing this process, we obtain

fxfxfxfx · · · fxnfxn+. ()

LetO(yk,n) ={yk,yk+, . . . ,yk+n}. We will claim that{yn}is a Cauchy sequence. To prove our claim, we follow the arguments of Das and Naik []. Fixk≥ andn∈ {, , . . .}. If diam[O(yk;n)] = , thenyk=yk+, which yields that{yn}is a constant sequence and also a Cauchy sequence. Then our claims holds. Thus we suppose thatdiam[O(yk;n)] > . Now, fori,jwith ≤i<j, by (), we have

ψd(yi,yj)

=ψd(fxi,fxj)

≤αmaxψd(gxi,gxj)

,ψd(gxi,fxi)

,ψd(gxj,fxj)

,ψd(xi,fxj)

,ψd(gxj,fxi)

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=αmaxψd(yi–,yj–)

,ψd(yi–,yi)

,ψd(yj–,yj)

,ψd(yi–,yj)

,ψd(yj–,yi)

≤αψdiamO(yi–;j–i+ )

,

and so

ψd(yi,yj)

. ()

Now, for somei,jwithk≤i<j≤k+n,diam[O(yk;n)] =d(yi,yj). Ifi>kby () and (), then we have

ψdiamO(yk;n)

≤αψdiamO(yk;n)

. ()

It follows that ψ(diam[O(yk;n)]) = , as diam[O(yk;n)]≤ψ(diam[O(yk;n)]) = , then diam[O(yk;n)] = . It is a contradiction! Thus,

diamO(yk;n)

=d(yk,yj) forjwithk<j≤k+n. ()

Also, by () and (), we have

ψdiamO(yk;n)

=ψd(yk,yj)

≤αψdiamO(yk–;j–k+ )

≤αψdiamO(yk–;n+ )

. ()

Using the triangle inequality, by (), () and (), we obtain that

ψdiamO(yl;m)

=ψd(yl,yj)

≤ψd(yl,yl+) +d(yl+,yj)

≤ψd(yl,yl+)

+ψd(yl+,yj)

≤ψd(yl,yl+)

+αψdiamO(yl+;m– )

≤ψd(yl,yl+)

+αψdiamO(yl;m)

, ()

and so

ψdiamO(yl;m)

≤ 

 –αψ

d(yl,yl+)

. ()

As a result, we have

ψdiamO(yk;n)

≤αψdiamO(yk–;n+ )

≤α·αψdiamO(yk–;n+ )

≤αkψdiamO(y;n+k)

≤ αk

 –αψ

d(y,y)

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Now let> , there exists an integernsuch that

αkψd(y,y)

< ( –α) for allk>n. ()

Form>n>n, we have

ψd(ym,yn)

≤ψdiamO(yn;m–n)

≤ αn

 –αψ

d(y,y)

<. ()

Sinceψ(t)≥tast> , thend(ym,yn)≤ψ(d(ym,yn)) <. Therefore,{yn}is a Cauchy se-quence.

Since by () we have{fxn=gxn+} ⊆g(X) andg(X) is closed, then there existsz∈Xsuch that

lim

n→∞gxn=gz. ()

Now we show thatzis a coincidence point off andg. Since from condition (iv) and () we havegxngzfor alln, then by the triangle inequality and (), we have that

ψd(fz,gz)≤ψd(gz,fxn) +d(fxn,fz)

≤ψd(gz,fxn)

+ψd(fxn,fz)

≤ψd(gz,fxn)

+αmaxψd(gxn,gz)

,ψd(gxn,fxn)

,

ψd(gz,fz),ψd(gxn,fz)

,ψd(gz,fxn)

. ()

So, letting n → ∞ yields ψ(d(fz,gz))≤ αψ(d(fz,gz)). Hence ψ(d(fz,gz)) = , hence d(fz,gz) = , which yieldsfz=gz. Thus we have proved thatf andghave a coincidence point.

Suppose now thatf andgcommute atz. Setw=fz=gz. Then

fw=f(gz) =g(fz) =gw. ()

Since from (vi) we have thatgzg(gz) =gwand asfz=gzandfw=gw, from () we have that

ψd(fz,fw)≤αmaxψd(gz,gw),ψd(gz,fz),ψd(gw,fw),

ψd(gz,fw),ψd(gw,fz)

=αψd(gz,gw). ()

Hence,ψ(d(fz,fw)) = , that is,d(w,fw) = . Therefore,

fw=gw=w. ()

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Accordingly, we can also obtain the results similar to Theorem  in [].

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 thatfxandgxare 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 whetherfxgxorgxfx(condition (v)), Lemma  of []

im-plies that the adjacent terms of the Jungck sequence{yn}are comparable. This is again suﬃcient to imply that{yn}is a Cauchy sequence. Hence, it converges to somegz∈gX.

By (vi), there exists a subsequenceynk =fxnk =gxnk+,k∈N, having all the terms

com-parable withgz. Hence, we can apply the contractive condition to obtain

ψd(fz,gz)≤ψd(gz,fxnk)

+ψd(fz,fxnk)

≤ψd(gz,fxnk)

+αmaxψd(gz,gxnk)

,ψd(gz,fz),

ψd(gxnk,fxnk)

,ψd(gz,fxnk)

,ψd(gxnk,fz)

.

Letting k → ∞, it yields that ψ(d(fz,gz))≤αψ(d(gz,fz)), then ψ(d(fz,gz)) = . Thus d(fz,gz) = . It follows that fz=gz=w. The rest of conclusions follow in the same way

as in Theorem ..

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-map 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 non-decreasing sequence withxn→uinX,thenxnu,∀nhold,or

(ii) f is continuous.

If there exists an x∈X with xfx,then f has a ﬁxed point.

(b)The same holds if f is nonincreasing,there exists xcomparable with fxand(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

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(b) Letube the limit of the Picard sequence{fn_x_}_{and let}_fnk_x

be a subsequence having

all the terms comparable withu. Then we can apply the contractivity condition to obtain

ψd(fu,u)≤ψdu,fnk+_x 

+dfu,fnk+_x 

≤ψdu,fnk+_x 

+ψdfu,fnk+_x 

≤ψdu,fnk+_x 

+αmaxψdu,fnk_x 

,ψd(u,fu),

ψdfnk_x

,fnk+x

,ψdu,fnk+_x 

,ψdfu,fnk_x 

.

Lettingk→ ∞, we have that

ψd(fu,u)≤αmax,ψd(u,fu), , ,ψd(u,fu)

=αψd(u,fu).

It follows that ψ(d(fu,u)) = . Thusd(fu,u) =  asd(fu,u)≤ψ(d(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.

3 Uniqueness of a common ﬁxed point offandg

The following theorem gives the suﬃcient condition for the uniqueness of a common ﬁxed point off andg.

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.

Proof Since a set of common ﬁxed points off andg is 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 assumption, there existsa∈Xsuch thatfais comparable tofxandfu. Deﬁne a se-quence{gan}such thata=aand

gan=fan– for alln. ()

Further, setx=xandu=uand in the same way deﬁne{gxn}and{gun}such that

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Sincefx(=gx=gx) is comparable tofa(=fa=ga) andf isg-nondecreasing, it is easy

to show

gxga. ()

Recursively, we can get that

gangx for alln. ()

By (), we have that

ψd(gan+,gx)

=ψd(fan,fx)

≤αmaxψd(gan,gx)

,ψd(gan,fan)

,ψd(gx,fx),

ψd(gan,fx)

,ψd(gx,fan)

. ()

By the proof of Theorem ., we obtain that {gan} is a convergent sequence, and there existsga¯such thatgan→ga¯. Lettingn→ ∞in () andψ is continuous, we can obtain that

lim n→∞ψ

d(gan+,gx)

=ψd(ga¯,gx)

≤αmaxψd(ga¯,gx), , ,ψd(ga¯,fx),ψd(gx,ga¯)

=αψd(ga¯,gx).

Therefore, we obtain

ψd(ga¯,gx)= .

Sinceψ(t)≥tast≥, thend(ga¯,gx) =  and hence

ga¯=gx. ()

Similarly, we can show that

lim n→∞ψ

d(gan+,gu)

=ψd(ga¯,gu)

≤αmaxψd(ga¯,gu), , ,ψd(ga¯,fu),ψd(gu,ga¯)

=αψd(ga¯,gu).

Therefore, we obtain

ψd(ga¯,gu)= .

Sinceψ(t)≥tast≥, thend(ga¯,gu) =  and hence

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Thus, from () and (), we havegx=gu. It follows that

x=fx=gx=gu=fu=u. ()

It means thatxis the unique common ﬁxed point off andg.

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 the orderedg-quasicontraction in []. Whenψ=I(I= the identity mapping), our condition (vi) reduces to the 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– , g(x,y)=x,y–  for all (x,y)∈X.

Letφ,ψ: [,∞)→[,∞) be given by

ψ(t) =

t for allt∈[,∞).

Obviously, for (, ) and (, )∈X, butf((, )) = (, ) is not comparable to g((, )) = (, ). However,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,y,z,w∈X. Letφ,ψ: [,∞)→[,∞) be given by

ψ(t) = t for allt∈[,∞).

It is easy to check that all the conditions of Theorem . are satisﬁed.

ψd(fx,fy)=  |x–y|

≤·α·

|x–y|

≤max

·

|x–y|, ·

_x– x



, ·

y– y 

,

· x–

y 

, ·

y– x 

=maxψd(gx,gy),ψd(gx,fx),ψd(gy,fy),ψd(gx,fy),ψd(gy,fx).

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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 andghave a unique common ﬁxed point. In fact,f andghave only one common ﬁxed point. It isx= .

4 Weak ordered contractions

We denote bythe set of functionsψ: [, +∞)→[, +∞) satisfying the following hy-potheses:

(ψ) ψis continuous and nondecreasing, (ψ) ψ(t) = if and only ift= .

We denote bythe set of functionsφ: [, +∞)→[, +∞) satisfying the following hy-potheses:

(φ) lims→t+φ(s) > for allt> , (φ) φ(t) = if and only ift= .

Let (X,d) be a metric space and letf,g:X→X. In the article [] (in the setting of partially ordered metric spaces), the authors obtained contractive conditions of the form

ψd(fx,fy)≤ψM(x,y)–φM(x,y), ()

where

M(x;y) =max

d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy) +d(gy,fx) 

. ()

We will use here the following more general contractive condition:

M(x,y) =maxd(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx). ()

We begin with the following result.

Theorem . Let (X,d,)be a partially ordered metric space and let f and g be self-mappings of X satisfying the following conditions:

(i) f(X)⊂g(X); (ii) g(X)is complete; (iii) f isg-nondecreasing;

(iv) f andgsatisfy the following condition:

ψd(fx,fy)≤ψM(x,y)–φM(x,y) ()

for allx,y∈Xsuch thatgygx,whereψ∈,φ∈and

M(x,y) =maxd(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx). ()

Suppose that,in addition,

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(vi) ψ(s+t)≤ψ(s) +ψ(t)for eachs,t> ; (vii) limt→+∞φ(t) =∞;

(viii) there existsx∈Xsuch thatgxfx;

(ix) if{gxn}is a nondecreasing sequence that converges to somegz∈gX,thengxngz

for eachn∈Nandgzg(gz).

Then f and g have a coincidence point.If,in addition,

(x) f andgare weakly compatible,then they have a common ﬁxed point.

Further,if

(xi) for arbitraryv,w∈X,there existsy∈Xsuch thatfyis comparable tofvandfw, thenf andghave a unique common ﬁxed point.

Proof As in the proof of Theorem ., we can construct a nondecreasing Jungck sequence

{yn}with

yn=fxn=gxn+

for alln≥. Denote

O(yk,n) ={yk,yk+,yk+, . . . ,yk+n}, ()

O(yk) ={yk,yk+,yk+, . . . ,yk+n, . . .}. ()

We will prove that the Jungck sequence{yn}is bounded, that is,

diamO(y)

=diam{y,y,y, . . . ,yn, . . .}

≤K ()

for someK∈R. Letk<nbe any ﬁxed positive integer and letdiam(O(yk,n)) =d(yi,yj) for somei,jwithk≤i<j≤k+n. We will show that

ψdiamO(yk,n)

≤ψdiamO(yi–,j–i+ )

–φdiamO(yi–,j–i+ )

. ()

Sincediam(O(yk,n)) =d(yi,yj),yi=fxi,yj=fxjandgxigxj, then from () we have

ψdiamO(yk,n)

=ψd(fxi,fxj)

≤ψM(xi,xj)

–φM(xi,xj)

, ()

where

M(xi,xj) =max

d(gxi,gxj),d(gxi,fxi),d(gxj,fxj),d(gxi,fxj),d(gxj,fxi)

=maxd(yi–,yj–),d(yi–,yi),d(yj–,yj),d(yi–,yj),d(yj–,yi)

.

Sinceyi–,yi,yj–,yj∈O(yi–,j–i+ ), then

M(xi,xj)≤diam

{yi–,yi,yj–,yj}

≤diamO(yi–,j–i+ )

.

So, from (v),

ψM(xi,xj)

–φM(xi,xj)

≤ψdiamO(yi–,j–i+ )

–φdiamO(yi–,j–i+ )

.

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Note thatφ(diam(O(yi–,j–i+ ))) > , and so from (),

diamO(yk,n)

<diamO(yi–,j–i+ )

. ()

Now we will show that ifdiam(O(yk,n)) =d(yi,yj), theni=k, that is,

diamO(yk,n)

=d(yk,yj) for somek<j≤k+n. ()

Suppose, to the contrary, thati>k. Then{yi–,yi, . . . ,yj} ⊆ {yk,yk+, . . . ,yi, . . . ,yj}and hence we conclude that

diamO(yk,n)

=d(yi,yj) =diam

O(yi–,j–i+ )

=diamO(yi,j–i)

=diamO(yk,j–k)

.

This contradicts (). Therefore,i=kand so we have proved ().

We will prove that the Jungck sequence {yn} is bounded. From () it follows that diam(O(y,n)) =d(y,yj) for someyj∈ {y,y, . . . ,yn}. By the triangle inequality,

diamO(y,n)

=d(y,yj)≤d(y,y) +d(y,yj).

Now, from (ψ) and (ψ), we get

ψdiamO(y,n)

≤ψd(y,y) +d(y,yj)

≤ψd(y,y)

+ψd(y,yj)

. ()

Sinced(y,yj) =d(fx,fxj) and asgxgxj, from () we have

ψd(y,yj)

≤ψM(x,xj)

–φM(x,xj)

,

where

M(x,xj) =max

d(y,yj–),d(y,y),d(yj–,yj),d(y,yj),d(yj–,y)

.

Clearly,M(x,xj)≤diam{y,y,yj–,yj} ≤diam(O(y,n)). Thus by (v), we get

ψM(x,xj)

–φM(x,xj)

≤ψdiamO(y,n)

–φdiamO(y,n)

.

Now, by (),

ψdiamO(y,n)

≤ψd(y,y)

+ψdiamO(y,n)

–φdiamO(y,n)

.

Hence

φdiamO(y,n)

≤ψd(y,y)

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Sincediam({y,y, . . . ,yn})≤diam({y,y, . . . ,yn+}), the sequence{diam(O(y,n))}∞n= is

nondecreasing, and so there exists its limitdiam(O(y)), which is ﬁnite or inﬁnite.

Sup-pose thatlimn→∞diam(O(y,n)) = +∞. Then (vii) implies that the left-hand side of ()

becomes unbounded whenntends to inﬁnity, but the right-hand side is bounded, a contra-diction. Therefore,lim_n_→∞diam(O(y,n)) =diam(O(y)) < +∞. Thus we have proved ().

Now we show that{yn}is a Cauchy sequence. For alln≥, set similarly as in (),

O(yn) ={yn,yn+, . . .}.

Clearly,O(yn+)⊂O(yn) and sodiam(O(yn+))≤diam(O(yn)). Therefore,{diam(O(yn))}∞n=

is the monotone decreasing sequence of ﬁnite nonnegative numbers and converges to someδ≥.

We will prove thatδ= . Letn≥ ands≥n+ . Sincegxn+gxs, from (),

ψd(yn+,ys)

=ψd(fxn+,fxs)

≤ψM(xn+,xs)

–φM(xn+,xs)

,

where

M(xn+,xs) =max

d(yn,ys–),d(yn,yn+),d(ys–,ys),d(yn,ys),d(ys–,yn+)

.

Sinceyn,yn+,ys–,ys∈ {yn,yn+, . . .}=O(yn), we conclude thatM(xn+,xs)≤diam(O(yn)), and so by (v), we get

ψd(yn+,ys)

≤ψdiamO(yn)

–φdiamO(yn)

. ()

Sincelims→+∞d(yn+,ys) =diam(O(yn+)) andψis continuous, we havelims→+∞ψ(d(yn+,

ys)) =ψ(diam(O(yn+))). Thus, taking the limit in () whens→+∞, we get

ψdiamO(yn+)

≤ψdiamO(yn)

–φdiamO(yn)

. ()

Suppose thatlimn→∞diam(O(yn)) =δ> . Sincediam(O(yn))→δ+ asn→ ∞, then from (φ), we havelimn→∞φ(diam(O(yn))) =q> . Therefore, taking the limits asn→+∞in () and using the continuity ofψ, we get

ψ(δ)≤ψ(δ) –q<ψ(δ),

a contradiction. Therefore,δ=  and so we have proved that

lim n→∞diam

{yn,yn+, . . .}

= .

Hence we conclude that{yn}is a Cauchy sequence.

Sinceyn=fxn=gxn+, by the assumption (ii) thatg(X) is complete, there is somez∈X

such that

lim

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We show thatfz=gz. Suppose, to the contrary, thatd(fz,gz) > . Condition (ix) implies thatgxngzand we can apply the contractive condition () to obtain

ψd(fz,fxn+)

≤ψM(z,xn+)

–φM(z,xn+)

, ()

where

M(z,xn+) =max

d(gz,gxn+),d(gz,fz),d(gxn+,fxn+),d(gz,fxn+),d(gxn+,fz)

=maxd(gz,fxn),d(gz,fz),d(fxn,fxn+),d(gz,fxn+),d(fxn,fz)

.

By the triangle inequality,

d(gz,fz)≤d(gz,fxn+) +d(fz,fxn+).

Now, from (ψ) and (ψ),

ψd(gz,fz)≤ψd(gz,fxn+) +d(fz,fxn+)

≤ψd(gz,fxn+)

+ψd(fz,fxn+)

.

Hence from () we have

ψd(gz,fz)≤ψd(gz,fxn+)

+ψM(z,xn+)

–φM(z,xn+)

. ()

Sincelimn→∞fxn=gz, for large enoughn, we have

M(z,xn+) =max

d(gz,fz),d(fxn,fz)

.

IfM(z,xn+) =d(gz,fz), then from ()

ψd(gz,fz)≤ψd(gz,fxn+)

+ψd(gz,fz)–φd(gz,fz).

Lettingntend to inﬁnity and using the continuity ofψ, we get

ψd(gz,fz)≤ψd(gz,fz)–φd(gz,fz).

Henceφ(d(gz,fz)) = , a contradiction with (φ) and the assumptiond(gz,fz) > .

Similarly, ifM(z,xn+) =d(fxn,fz), then from ()

ψd(gz,fz)≤ψd(gz,fxn)

+ψd(fxn,fz)

–φd(fxn,fz)

.

Lettingntend to inﬁnity and having in mind thatd(fxn,fz)→d(gz,fz)+, we obtain

ψd(gz,fz)≤ψd(gz,fz)– lim d(fxn,fz)→d(gz,fz)+φ

d(fxn,fz)

and hence we get

lim d(fxn,fz)→d(gz,fz)+

φd(fxn,fz)

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a contradiction with (φ). Thus our assumptiond(gz,fz) >  is wrong. Therefore, d(gz,

fz) = . Hencegz=fz, that is,zis a coincidence point off andg.

If the condition (x) is fulfilled, putw=fz=gz. We will show thatwis a common fixed point off andg. Sincefz=gzandf andgare weakly compatible, we obtain, by the defini-tion of weak compatibility, thatfgz=gfz. Thus we havefw=gw. Using the condition (ix) thatgzggz=gw, we can apply the contractive condition () to obtain

ψd(fw,fz)≤ψM(w,z)–φM(w,z),

where

M(w,z) =maxd(gw,gz),d(gw,fw),d(gz,fz),d(gw,fz),d(gz,fw)=d(fw,fz).

Thus

ψd(fw,fz)≤ψd(fw,fz)–φd(fw,fz).

Henceφ(d(fw,fz)) = , and so by (φ),d(fw,fz) = . Hencefw=fz. Therefore

w=fz=fw=ﬀz=gfz=gw.

Thus we showed thatwis a common ﬁxed point off andg.

Suppose now that the condition (xi) is fulfilled. Since a set of common fixed points off andgis not empty, assume thatwandvare two common fixed points off andg,i.e.,

fw=gw=w, fv=gv=v. ()

We claim thatgw=gv.

By assumption, there existsy∈Xsuch thatfyis comparable tofwandfv. Deﬁne a

sequence{gyn}such that

gyn=fyn– for alln. ()

Further, setw=wandv=vand, in the same way, deﬁne{gwn}and{gvn}such that

gwn=fwn–, gvn=fvn– for alln. ()

From () and (), we havefw=gw=gwandfv=gv=gv. Sincefyis comparable

tofwandfv, andf isg-nondecreasing, it is easy to show

gwgy. ()

Recursively, we can get that

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By (), we have that

ψd(gyn+,gw)

=ψd(fyn,fw)

≤ψmaxd(gyn,gw),d(gyn,fyn),d(gw,fw),d(gyn,fw),d(gw,fyn)

–φmaxd(gyn,gw),d(gyn,fyn),d(gw,fw),d(gyn,fw),d(gw,fyn)

. ()

Similarly as in the proof of Theorem ., we can prove that{gyn}is a convergent sequence. Thus there existsy¯∈Xsuch thatgyn→gy¯. Since alsolimn→∞fyn=gy¯, for large enough n, we have

maxd(gyn,gw),d(gyn,fyn),d(gw,fw),d(gyn,fw),d(gw,fyn)

=d(gy¯,gw).

Thus from (), for large enoughn,

ψd(gyn+,gw)

≤ψd(gy¯,gw)–φd(g¯y,gw). ()

Lettingn→ ∞in (), by (ψ) we get

lim n→∞ψ

d(gyn+,gw)

=ψd(gy¯,gw)≤ψd(g¯y,gw)–φd(gy¯,gw).

Hence we obtain

ψd(gy¯,gw)= .

Then by (ψ),d(gy¯,gw) =  and hence

gy¯=gw. ()

Similarly, we can show that

lim n→∞ψ

d(gyn+,gv)

=ψd(gy¯,gv)≤ψd(gy¯,gv)–φd(gy¯,gv),

and hence we obtain

gy¯=gv. ()

Therefore, from () and (), we havegw=gv. It follows that

w=fw=gw=gv=fv=v. ()

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Corollary . Let(X,d,)be a complete partially ordered metric space and let f be a self-mapping of X satisfying the following condition:

d(fx,fy)≤m(x,y) –φm(x,y)

for all x,y∈X such that gygx,where

m(x,y) =maxd(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)

andφ∈.Suppose that,in addition,t–φ(t)is non-decreasing,limt→+∞φ(t) =∞,there exists x∈X such that xfxand if{fxn}is a nondecreasing sequence such that it con-verges to some z∈X,then fxnz.Then f has a unique ﬁxed point.

Proof Takingψ(t) =tandg(t) =tin the proof of Theorem ., we obtain Corollary ..

Remark . Theorem . extends Theorem  due to Berinde [], Theorems . and . due to Beg and Abbas [] and Theorem . due to Song [].

We present an example to show that our result is a real generalization of the recent result of Golubovićet al.[] as well as of the existing results in the literature.

Example . LetX= [,_] be the closed interval with the 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 allx∈X,

φ(t) =t for ≤t≤ ,

φ(t) =

t fort>  .

Letx,yinXbe arbitrary. We say thatxyifx≤y. For anyx,y∈Xsuch thatxy, we have

M(x,y) =maxdg(x),g(y),dg(x),f(x),dg(y),f(y),dg(x),f(y),dg(y),f(x)

=dg(y),f(x),

ψdg(y),f(x)=dg(y),f(x)=y–x –x

=y–x –x.

Sincey_≥_y_–_x_{( –}_x_{) for all}_x_∈_[,

], it follows that

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Thus we have

ψdf(x),f(y)=y–y–x+x=y–x –x–y

≤y–x –x–y–x –x

=dg(y),f(x)–dg(y),f(x)

=ψM(x,y)–φM(x,y).

Therefore,fandgsatisfy (). Also, it is easy to see that the mappingsψ(t) andφ(t) possess all properties (ψ), (ψ) and (φ), (φ) respectively, as well as hypotheses (v), (vi) and (vii)

in Theorem .. Thus we can apply our Theorem . and Corollary ..

On the other hand, forx=  and eachy> , the contractive condition in Theorems  and  of Golubovićet al.[]:

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(;y) =maxdg();g(y);dg();f();dg(y);f(y);dg();f(y),dg(y);f()

=maxy; ;y;y–y,y=y.

Thus, for any ﬁxedλ;  <λ< , we have, forx=  and eachy∈Xwith  <y<√ –λ,

df(),f(y)=y–y= –yy>λ·y

=λ·dg(y),g()=λ·M(,y).

Thus, f does not satisfy (). Therefore, the theorems of Jungck and Hussain [], Al-Thagaﬁ and Shahzad [] and Das and Naik [] also cannot be applied.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610068, China.}2_{School of Science, Sichuan}

University of Science and Engineering, Zigong, Sichuan 643000, China.3_{Department of Applied Mathematics, Faculty of}

Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade, Serbia.

Acknowledgements

Siniša Ješić was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Project grant number 174032. Xiaolan Liu was supported by Scientific Research Fund of Sichuan Provincial Education Department (12ZA098), Scientific Research Fund of Sichuan University of Science and Engineering (2012KY08), and Scientific Research Fund of School of Science SUSE (10LXYB03).

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