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Common fixed points of a generalized ordered gquasicontraction in partially ordered metric spaces
 Xiaolan Liu^{1, 2}Email author and
 Siniša Ješić^{3}
https://doi.org/10.1186/16871812201353
© Liu and Ješić; licensee Springer 2013
 Received: 9 October 2012
 Accepted: 21 February 2013
 Published: 12 March 2013
Abstract
The concept of a generalized ordered gquasicontraction is introduced, and some fixed and common fixed point theorems for a gnondecreasing generalized ordered gquasicontraction 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 ordered gquasicontraction mapping. Finally, we prove fixed point theorems for mappings satisfying the socalled weak contractive conditions in the setting of a partially ordered metric space. Presented theorems are generalizations of very recent fixed point theorems due to Golubović et al. (Fixed Point Theory Appl. 2012:20, 2012).
MSC:47H10, 47N10.
Keywords
 Gnondecreasing
 generalized ordered gquasicontraction
 coincidence point
 common fixed point
 comparable mappings
1 Introduction
The Banach fixed point theorem for contraction mappings has been extended in many directions (cf. [1–15]). Very recently Golubović et al. [16] presented some new results for ordered quasicontractions and gquasicontractions in partially ordered metric spaces.
Recall that if $(X,\u2aaf)$ is a partially ordered set and $f:X\to X$ is such that for $x,y\in X$, $x\u2aafy$ implies $fx\u2aaffy$, then a mapping F is said to be nondecreasing. The main result of Golubović et al. [16] is the following fixed point theorem.
Theorem 1.1 (See [16], Theorem 1)
 (i)
$fX\subset gX$;
 (ii)
gX is complete;
 (iii)
f is gnondecreasing;
 (iv)
f is an ordered gquasicontraction;
 (v)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aaff{x}_{0}$;
 (vi)
if $\{g{x}_{n}\}$ is a nondecreasing sequence that converges to some $gz\in gX$, then $g{x}_{n}\u2aafgz$ for each $n\in \mathbb{N}$ and $gz\u2aafg(gz)$.
Then f and g have a coincidence point, i.e., there exists $z\in X$ such that $fz=gz$. If, in addition,
 (vii)
f and g are weakly compatible [17, 18], i.e., $fx=gx$ implies $fgx=gfx$ for each $x\in X$, 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 ordered gquasicontraction in Theorem 1.1.
In Section 2 of this article, we introduce generalized ordered gquasicontractions in partially ordered metric spaces and prove the respective (common) fixed point theorems which generalize the results of Theorem 1.1.
In Section 3 of this article, the uniqueness of a common fixed point theorem is obtained when for all $x,u\in X$, there exists $a\in X$ such that fa is comparable to fx and fu in addition to the hypotheses in Theorem 2.1 of Section 2. Our results are an answer to finding sufficient conditions for the uniqueness of a common fixed point in the case of an ordered gquasicontraction in Theorem 1.1. 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 gquasicontraction, so our results are extensions of known ones.
In Section 4 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 ordered gquasicontraction
We start this section with the following definitions. Consider a partially ordered set $(X,\u2aaf)$ and two mappings $f:X\to X$ and $g:X\to X$ such that $f(X)\subset g(X)$.
Definition 2.1 (See [19])
(resp., $gx\u2aafgy\Rightarrow fx\u2aaffy$) holds for each $x,y\in X$.
Definition 2.2 (See [16])
for all $x,y\in X$ for which $gx\u2ab0gy$;
It is obvious that if $\psi =I$, then a generalized ordered gquasicontraction reduces to an ordered gquasicontraction.
For arbitrary ${x}_{0}\in X$, one can construct the socalled Jungck sequence $\{{y}_{n}\}$ in the following way: Denote ${y}_{0}=f{x}_{0}\in f(X)\subset g(X)$; there exists ${x}_{1}\in X$ such that $g{x}_{1}={y}_{0}=f{x}_{0}$; now ${y}_{1}=f{x}_{1}\in f(X)\subset g(X)$ and there exists ${x}_{2}\in X$ such that $g{x}_{2}={y}_{1}=f{x}_{1}$ and the procedure can be continued.
 (i)
$f(X)\subset g(X)$;
 (ii)
$g(X)$ is closed;
 (iii)
f is a gnondecreasing mapping;
 (iv)
f is a generalized ordered gquasicontraction;
 (v)
there exists an ${x}_{0}\in X$ with $g{x}_{0}\u2aaff{x}_{0}$;
 (vi)
$\{g({x}_{n})\}\subset X$ is a nondecreasing sequence with $g({x}_{n})\to gz$ in $g(X)$, then $g{x}_{n}\u2aafgz$, $gz\u2aafg(gz)$, ∀n hold.
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.
Since $\psi (t)\ge t$ as $t>0$, then $d({y}_{m},{y}_{n})\le \psi (d({y}_{m},{y}_{n}))<\u03f5$. Therefore, $\{{y}_{n}\}$ is a Cauchy sequence.
So, letting $n\to \mathrm{\infty}$ yields $\psi (d(fz,gz))\le \alpha \psi (d(fz,gz))$. Hence $\psi (d(fz,gz))=0$, hence $d(fz,gz)=0$, which yields $fz=gz$. Thus we have proved that f and g have a coincidence point.
Thus, we have proved that f and g have a common fixed point. □
Accordingly, we can also obtain the results similar to Theorem 2 in [16].
Theorem 2.2 Let the conditions of Theorem 2.1 be satisfied, except that (iii), (v) and (vi) are, respectively, replaced by:

(iii′) f is a gnonincreasing mapping;

(v′) there exists ${x}_{0}\in X$ such that $f{x}_{0}$ and $g{x}_{0}$ are comparable;

(vi′) if $\{g{x}_{n}\}$ is a sequence in $g(X)$ which has comparable adjacent terms and that converges to some $gz\in gX$, then there exists a subsequence $g{x}_{{n}_{k}}$ of $\{g{x}_{n}\}$ having all the terms comparable with gz and gz is comparable with $ggz$. Then all the conclusions of Theorem 2.1 hold.
Proof Regardless of whether $f{x}_{0}\u2aafg{x}_{0}$ or $g{x}_{0}\u2aaff{x}_{0}$ (condition (v′)), Lemma 1 of [16] implies that the adjacent terms of the Jungck sequence $\{{y}_{n}\}$ are comparable. This is again sufficient to imply that $\{{y}_{n}\}$ is a Cauchy sequence. Hence, it converges to some $gz\in gX$.
Letting $k\to \mathrm{\infty}$, it yields that $\psi (d(fz,gz))\le \alpha \psi (d(gz,fz))$, then $\psi (d(fz,gz))=0$. Thus $d(fz,gz)=0$. It follows that $fz=gz=w$. The rest of conclusions follow in the same way as in Theorem 2.1. □
 (i)
$\{{x}_{n}\}\subset X$ is a nondecreasing sequence with ${x}_{n}\to u$ in X, then ${x}_{n}\u2aafu$, ∀n hold, or
 (ii)
f is continuous.
If there exists an ${x}_{0}\in X$ with ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
(b) The same holds if f is nonincreasing, there exists ${x}_{0}$ comparable with $f{x}_{0}$ and (i) is replaced by
(i′) if a sequence $\{{x}_{n}\}$ converging to some $u\in X$ has every two adjacent terms comparable, then there exists a subsequence $\{{x}_{{n}_{k}}\}$ having each term comparable with x.
Proof (a) If (i) holds, then take $\psi =I$ and $g=I$ (I= the identity mapping) in Theorem 2.1.
It follows that $\psi (d(fu,u))=0$. Thus $d(fu,u)=0$ as $d(fu,u)\le \psi (d(fu,u))=0$. Therefore, $fu=u$.
Note also that instead of the completeness of X, its forbitally completeness is sufficient to obtain the conclusion of the corollary. □
3 Uniqueness of a common fixed point of f and g
The following theorem gives the sufficient condition for the uniqueness of a common fixed point of f and g.
Then f and g have a unique common fixed point.
We claim that $gx=gu$.
It means that x is the unique common fixed point of f and g. □
Remark 3.1 Theorem 3.1 can be considered as an answer to Theorem 3 in [16]. We find the sufficient conditions for the uniqueness of the common fixed point in the case of an ordered gquasicontraction. In this paper, condition (vi) in Theorem 2.1 is weaker than the ordered gquasicontraction in [16]. When $\psi =I$ (I= the identity mapping), our condition (vi) reduces to the ordered gquasicontraction in [16].
It holds when $\alpha =\frac{1}{12}$ and $gx\ge gy$, i.e., $\frac{3}{4}x\ge \frac{3}{4}y$, i.e., $x\ge y$.
In addition, $\mathrm{\forall}x,u\in X$, there exists $a\in X$ such that $fa=\frac{a}{16}$ is comparable to $fx=\frac{x}{16}$ and $fu=\frac{u}{16}$. So, all the conditions of Theorem 3.1 are satisfied.
By applying Theorem 3.1, we conclude that f and g have a unique common fixed point. In fact, f and g have only one common fixed point. It is $x=0$.
4 Weak ordered contractions
We denote by Ψ the set of functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying the following hypotheses:
(${\psi}_{1}$) ψ is continuous and nondecreasing,
(${\psi}_{2}$) $\psi (t)=0$ if and only if $t=0$.
We denote by Φ the set of functions $\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying the following hypotheses:
(${\varphi}_{1}$) ${lim}_{s\to t+}\varphi (s)>0$ for all $t>0$,
(${\varphi}_{2}$) $\varphi (t)=0$ if and only if $t=0$.
We begin with the following result.
 (i)
$f(X)\subset g(X)$;
 (ii)
$g(X)$ is complete;
 (iii)
f is gnondecreasing;
 (iv)f and g satisfy the following condition:$\psi (d(fx,fy))\le \psi (M(x,y))\varphi (M(x,y))$(35)for all $x,y\in X$ such that $gy\u2aafgx$, where $\psi \in \mathrm{\Psi}$, $\varphi \in \mathrm{\Phi}$ and$M(x,y)=max\{d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx)\}.$(36)
Suppose that, in addition,
 (v)
$\psi (t)\varphi (t)$ is nondecreasing;
 (vi)
$\psi (s+t)\le \psi (s)+\psi (t)$ for each $s,t>0$;
 (vii)
${lim}_{t\to +\mathrm{\infty}}\varphi (t)=\mathrm{\infty}$;
 (viii)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aaff{x}_{0}$;
 (ix)
if $\{g{x}_{n}\}$ is a nondecreasing sequence that converges to some $gz\in gX$, then $g{x}_{n}\u2aafgz$ for each $n\in N$ and $gz\u2aafg(gz)$.
Then f and g have a coincidence point. If, in addition,
 (x)
f and g are weakly compatible, then they have a common fixed point.
Further, if
 (xi)
for arbitrary $v,w\in X$, there exists ${y}_{0}\in X$ such that $f{y}_{0}$ is comparable to fv and fw, then f and g have a unique common fixed point.
Hence from (41) we obtain (40).
This contradicts (42). Therefore, $i=k$ and so we have proved (43).
Since $diam(\{{y}_{0},{y}_{1},\dots ,{y}_{n}\})\le diam(\{{y}_{0},{y}_{1},\dots ,{y}_{n+1}\})$, the sequence ${\{diam(O({y}_{0},n))\}}_{n=1}^{\mathrm{\infty}}$ is nondecreasing, and so there exists its limit $diam(O({y}_{0}))$, which is finite or infinite. Suppose that ${lim}_{n\to \mathrm{\infty}}diam(O({y}_{0},n))=+\mathrm{\infty}$. Then (vii) implies that the lefthand side of (45) becomes unbounded when n tends to infinity, but the righthand side is bounded, a contradiction. Therefore, ${lim}_{n\to \mathrm{\infty}}diam(O({y}_{0},n))=diam(O({y}_{0}))<+\mathrm{\infty}$. Thus we have proved (39).
Clearly, $O({y}_{n+1})\subset O({y}_{n})$ and so $diam(O({y}_{n+1}))\le diam(O({y}_{n}))$. Therefore, ${\{diam(O({y}_{n}))\}}_{n=0}^{\mathrm{\infty}}$ is the monotone decreasing sequence of finite nonnegative numbers and converges to some $\delta \ge 0$.
Hence we conclude that $\{{y}_{n}\}$ is a Cauchy sequence.
Hence $\varphi (d(gz,fz))=0$, a contradiction with (${\varphi}_{2}$) and the assumption $d(gz,fz)>0$.
a contradiction with (${\varphi}_{1}$). Thus our assumption $d(gz,fz)>0$ is wrong. Therefore, $d(gz,fz)=0$. Hence $gz=fz$, that is, z is a coincidence point of f and g.
Thus we showed that w is a common fixed point of f and g.
We claim that $gw=gv$.
It means that w is the unique common fixed point of f and g. □
and $\varphi \in \mathrm{\Phi}$. Suppose that, in addition, $t\varphi (t)$ is nondecreasing, ${lim}_{t\to +\mathrm{\infty}}\varphi (t)=\mathrm{\infty}$, there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aaff{x}_{0}$ and if $\{f{x}_{n}\}$ is a nondecreasing sequence such that it converges to some $z\in X$, then $f{x}_{n}\u2aafz$. Then f has a unique fixed point.
Proof Taking $\psi (t)=t$ and $g(t)=t$ in the proof of Theorem 4.1, we obtain Corollary 4.1. □
Remark 4.1 Theorem 4.1 extends Theorem 1 due to Berinde [21], Theorems 2.1 and 2.5 due to Beg and Abbas [22] and Theorem 3.1 due to Song [23].
We present an example to show that our result is a real generalization of the recent result of Golubović et al. [16] as well as of the existing results in the literature.
Therefore, f and g satisfy (35). Also, it is easy to see that the mappings $\psi (t)$ and $\varphi (t)$ possess all properties (${\psi}_{1}$), (${\psi}_{2}$) and (${\varphi}_{1}$), (${\varphi}_{2}$) respectively, as well as hypotheses (v), (vi) and (vii) in Theorem 4.1. Thus we can apply our Theorem 4.1 and Corollary 4.1.
Thus, f does not satisfy (60). Therefore, the theorems of Jungck and Hussain [24], AlThagafi and Shahzad [25] and Das and Naik [26] also cannot be applied.
Declarations
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).
Authors’ Affiliations
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