# Common fixed-point results for nonlinear contractions in ordered partial metric spaces

- Bessem Samet
^{1}Email author, - Miloje Rajović
^{2}, - Rade Lazović
^{3}and - Rade Stojiljković
^{4}

**2011**:71

https://doi.org/10.1186/1687-1812-2011-71

© Samet et al; licensee Springer. 2011

**Received: **19 January 2011

**Accepted: **31 October 2011

**Published: **31 October 2011

## Abstract

In this paper, a new class of a pair of generalized nonlinear contractions on partially ordered partial metric spaces is introduced, and some coincidence and common fixed-point theorems for these contractions are proved. Presented theorems are twofold generalizations of very recent fixed-point theorems of Altun and Erduran (Fixed Point Theory Appl 2011(Article ID 508730):10, 2011), Altun et al. (Topol Appl 157(18):2778-2785, 2010), Matthews (Proceedings of the 8th summer conference on general topology and applications, New York Academy of Sciences, New York, pp. 183-197, 1994) and many other known corresponding theorems.

**2000 Mathematics Subject Classifications:** 54H25; 47H10.

## Keywords

## 1 Introduction

It is well known that the Banach contraction principle is a very useful, simple and classical tool in nonlinear analysis. There exist a vast literature concerning its various generalizations and extensions (see [1–45]). In [22], Matthews extended the Banach contraction mapping theorem to the partial metric context for applications in program verification. After that, fixed-point results in partial metric spaces have been studied [4, 8, 28, 31, 34, 45]. The existence of several connections between partial metrics and topological aspects of domain theory has been pointed by many authors (see [8, 9, 16, 23, 31, 33, 36–38, 41, 42, 46, 47]).

First, we recall some definitions of partial metric spaces and some their properties.

**Definition 1.1** *A partial metric on a set X is a function p* : *X* × *X* → ℝ^{+} *such that for all x*, *y*, *z* ∈ *X:*

*(p1) x* = *y* ⇔ *p*(*x*, *x*) = *p*(*x*, *y*) = *p*(*y*, *y*),

*(p2) p*(*x*, *x*) ≤ *p*(*x*, *y*),

*(p3) p*(*x*, *y*) = *p*(*y*, *x*),

*(p4) p*(*x*, *y*) ≤ *p*(*x*, *z*) + *p*(*z*, *y*) - *p*(*z*, *z*).

Note that the self-distance of any point need not be zero, hence the idea of generalizing metrics so that a metric on a non-empty set *X* is precisely a partial metric *p* on *X* such that for any *x* ∈ *X*, *p*(*x*, *x*) = 0.

Similar to the case of metric space, a partial metric space is a pair (*X*, *p*) consisting of a non-empty set *X* and a partial metric *p* on *X*.

**Example 1.1** *Let a function p* : ℝ^{+} × ℝ^{+} → ℝ^{+} *be defined by p*(*x*, *y*) = max{*x*, *y*} *for any x*, *y* ∈ ℝ^{+}. *Then*, (ℝ^{+} , *p*) *is a partial metric space where the self-distance for any point x* ∈ ℝ^{+} *is its value itself*.

**Example 1.2** *Consider a function p* : ℝ^{-} × ℝ^{-} → ℝ^{+} *defined by p*(*x*, *y*) = *-* min(*x*, *y*) *for any x*, *y* ∈ ℝ^{-}. *The pair* (ℝ^{-}, *p*) *is a partial metric space for which p is called the usual partial metric on* ℝ^{-} *and where the self-distance for any point x* ∈ ℝ^{-} *is its absolute value*.

**Example 1.3** *If X*: = {[*a*, *b*] | *a*, *b* ∈ ℝ, *a* ≤ *b*}, *then p* : *X* × *X* → ℝ^{+} *defined by p*([*a*, *b*], [*c*, *d*]) *=* max{*b*, *d*} - min{*a*, *b*} *defines a partial metric on X*.

*p*on

*X*generates a

*T*

_{0}topology

*τ*

_{ p }on

*X*, which has as a base the family of open

*p*-balls {

*B*

_{ p }(

*x*,

*ε*),

*x*∈

*X*,

*ε*> 0}, where

*p*is a partial metric on

*X*, then the function

*p*

^{ s }:

*X*×

*X*→ ℝ

^{+}defined by

is a metric on *X*.

**Definition 1.2** *Let* (*X*, *p*) *be a partial metric space and* {*x*_{
n
}} *be a sequence in X. Then*,

(*i*) {*x*_{
n
}} *converges to a point x* ∈ *X if and only if p*(*x*, *x*) = lim_{n→+∞}*p*(*x*, *x*_{
n
}),

(*ii*) {*x*_{
n
}} *is a Cauchy sequence if there exists (and is finite)* lim_{n,m→+∞}*p*(*x*_{
n
}, *x*_{
m
}).

**Definition 1.3** *A partial metric space* (*X*, *p*) *is said to be complete if every Cauchy sequence* {*x*_{
n
}} *in X converges, with respect to τ*_{
p
}*, to a point x* ∈ *X, such that p*(*x*, *x*) = lim_{n,m→+∞}*p*(*x*_{
n
}, *x*_{
m
}).

**Remark 1.1** *It is easy to see that every closed subset of a complete partial metric space is complete*.

**Lemma 1.1** ([22, 28]) *Let* (*X*, *p*) *be a partial metric space. Then*

(*a*) {*x*_{
n
}} *is a Cauchy sequence in* (*X*, *P*) *if and only if it is a Cauchy sequence in the metric space* (*X*, *P*^{
s
}),

*b*) (

*X*,

*p*)

*is complete if and only if the metric space*(

*X*,

*p*

^{ s })

*is complete. Furthermore*, lim

_{n→+∞}

*p*

^{ s }(

*x*

_{ n },

*x*) = 0

*if and only if*

Matthews [22] obtained the following Banach fixed-point theorem on complete partial metric spaces.

**Theorem 1.1**(Matthews [22])

*Let f be a mapping of a complete partial metric space*(

*X*,

*p*)

*into itself such that there is a constant c*∈ [0,1)

*satisfying for all x, y*∈

*X*:

*Then, f has a unique fixed point*.

Recently, Altun et al. [4] obtained the following nice result, which generalizes Theorem 1.1 of Matthews.

**Theorem 1.2**(Altun et al. [4])

*Let*(

*X*,

*p*)

*be a complete partial metric space and let T*:

*X*→

*X be a map such that*

*for all x*, *y* ∈ *X*, *where φ* : [0, +∞) → [0, +∞) *satisfies the following conditions:*

*(i) φ is continuous and non-decreasing*,

*(ii)* ${\sum}_{n\ge 1}{\phi}^{n}\left(t\right)$*is convergent for each t* > 0.

*Then*, *T has a unique fixed point*.

On the other hand, existence of fixed points in partially ordered sets has been considered recently in [32], and some generalizations of the result of [32] are given in [1–3, 5–7, 11, 12, 14, 15, 17, 19, 24–27, 29, 30, 39, 40, 43] in partial ordered metric spaces. Also, in [32], some applications to matrix equations are presented, and in [15] and [26], some applications to ordinary differential equations are given. In [29], O'Regan and Petruşel established some fixed-point results for self-generalized contractions in ordered metric spaces. Jachymski [19] established a geometric lemma [19, Lemma 1], giving a list of equivalent conditions for some subsets of the plane. Using this lemma, he proved that some very recent fixed-point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [15] and Amini-Harandi and Emami [5] do follow from an earlier result of O'Regan and Petruşel [29, Theorem 3.6].

Very recently, Altun and Erduran [3] generalized Theorem 1.2 to partially ordered complete partial metric spaces and established the following new fixed-point theorems, involving a function *φ* : [0, +∞) → [0, +∞) satisfying the conditions (i)-(ii) in Theorem 1.2.

**Theorem 1.3**(Altun and Erduran [3]).

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X*,

*p*)

*is a complete partial metric space. Suppose F*:

*X*→

*X is a continuous and non-decreasing mapping (with respect to*≼

*) such that*

*for all x*, *y* ∈ *X with y* ≼ *x*, *where φ* : [0, +∞) → [0, +∞) *satisfies conditions (i)*-*(ii) in Theorem* 1.2. *If there exists x*_{0} ∈ *X such that x*_{0} ≼ *Fx*_{0}, *then there exists x* ∈ *X such that Fx* = *x*. *Moreover*, *p* (*x*, *x*) = 0.

**Theorem 1.4**(Altun and Erduran [3])

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X*,

*p*)

*is a complete partial metric space. Suppose F*:

*X*→

*X is a non-decreasing mapping such that*

*for all x*,

*y*∈

*X with y*≺

*x (y*≼

*x and y*≠

*x)*,

*where φ*: [0, +∞) → [0, +∞)

*satisfies conditions (i)-(ii) in Theorem*1.2.

*Suppose also that the condition*

*holds. If there exists x*_{0} ∈ *X such that x*_{0} ≼ *Fx*_{0}, *then there exists x* ∈ *X such that Fx* = *x*. *Moreover*, *p*(*x*, *x*) = 0.

**Theorem 1.5**(Altun and Erduran [3])

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X*,

*p*)

*is a complete partial metric space. Suppose F*:

*X*→

*X is a continuous and non-decreasing mapping such that*

*for all x*, *y* ∈ *X with y* ≼ *x*, *where φ* : [0, +∞) → [0, +∞) *satisfies conditions (i)-(ii) in Theorem* 1.2. *If there exists x*_{0} ∈ *X such that x*_{0} ≼ *Fx*_{0}, *then there exists x* ∈ *X such that Fx* = *x*. *Moreover*, *p*(*x*, *x*) = 0. *If we suppose that for all x*, *y* ∈ *X there exists z* ∈ *X*, *which is comparable to x and y*, *we obtain uniqueness of the fixed point of F*.

Altun et al. [4], Altun and Erduran [3] and many authors have obtained fixed-point theorems for contractions under the assumption that a comparison function *φ* : [0, +∞) → [0, +∞) is non-decreasing and such that ${\sum}_{n=1}^{\infty}{\phi}^{n}\left(t\right)<\infty $ for each *t* > 0 (see, e.g., [13] and the references in [11, 18]-Added in proof). However, the latter condition is strong and rather hard to verify in practice, though some examples and general criteria for this convergence are known (see, e.g., [3, 44]). So a natural question arises whether this strong condition can be omitted in partial metric fixed-point theory.

The aims of this paper is to establish coincidence and common fixed-point theorems in ordered partial metric spaces with a function *φ* satisfying the condition *φ*(*t*) < *t* for all *t* > 0, which is weaker than the condition ${\sum}_{n=1}^{\infty}{\phi}^{n}\left(t\right)<\infty .$ Presented theorems generalize and extend to a pair of mappings the results of Altun and Erduran [3], Altun et al. [4], Matthews [22] and many other known corresponding theorems.

## 2 Main results

We start this section by some preliminaries.

**Definition 2.1** (Altun and Erduran [3]) *Let* (*X*, *p*) *be a partial metric space, F* : *X* → *X be a given mapping. We say that F is continuous at x*_{0} ∈ *X, if for every ε >* 0, *there exists δ >* 0 *such that F*(*B*_{
p
}(*x*_{0}, *δ*)) ⊆ *B*_{
p
}(*Fx*_{0}, *ε*).

The following result is easy to check.

**Lemma 2.1**

*Let*(

*X*,

*p*)

*be a partial metric space, F*:

*X*→

*X be a given mapping. Suppose that F is continuous at x*

_{0}∈

*X. Then, for all sequence*{

*x*

_{ n }} ⊂

*X, we have*

**Definition 2.2**(Ćirić et al. [11])

*Let*(

*X*, ≼)

*be a partially ordered set and F*,

*g*:

*X*→

*X are mappings of X into itself. One says F is g-non-decreasing if for x*,

*y*∈

*X, we have*

We introduce the following definition.

**Definition 2.3** *Let* (*X*, *p*) *be a partial metric space and F*, *g: X* → *X are mappings of X into itself. We say that the pair* {*F*, *g*} *is partial compatible if the following conditions hold:*

*(b1) p*(*x*, *x*) = 0 ⇒ *p*(*gx*, *gx*) = 0,

*(b2)* lim_{n→+∞}*p*(*Fgx*_{
n
}, *gFx*_{
n
}) = 0, *whenever* {*x*_{
n
}} *is a sequence in X such that Fx*_{
n
} → *t and gx*_{
n
} → *t for some t* ∈ *X*.

It is clear that Definition 2.3 extends and generalizes the notion of compatibility introduced by Jungck [21].

Define by ϕ the set of functions *φ* : [0, +∞) → [0, +∞) satisfying the following conditions:

**(c1)** *φ* is continuous and non-decreasing,

**(c2)** *φ*(*t*) < *t* for each *t* > 0.

Now, we are ready to state and prove our first result.

**Theorem 2.1**

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X*,

*p*)

*is a complete partial metric space. Let F*,

*g*:

*X*→

*X be two continuous self-mappings of X such that FX*⊆

*gX, F is a g-non-decreasing mapping, the pair*{

*F*,

*g*}

*is partial compatible, and*

*for all x*, *y* ∈ *X for which gy* ≼ *gx, where a function φ* ∈ ϕ. *If there exists x*_{0} ∈ *X with gx*_{0} ≼ *Fx*_{0}, *then F and g have a coincidence point, that is, there exists x* ∈ *X such that Fx = gx*. *Moreover, we have p*(*x*, *x*) *= p*(*Fx*, *Fx*) *= p(gx*, *gx*) = 0.

*Proof*. Let

*x*

_{0}∈

*X*such that

*gx*

_{0}≼

*Fx*

_{0}. Since

*FX*⊆

*gX*, we can choose

*x*

_{1}∈

*X*so that

*gx*

_{1}=

*Fx*

_{0}. Again, from

*FX*⊆

*gX*, there exists

*x*

_{2}∈

*X*such that

*gx*

_{2}=

*Fx*

_{1}. Continuing this process, we can choose a sequence {

*x*

_{ n }} ⊂

*X*such that

*gx*

_{0}≼

*Fx*

_{0}and

*Fx*

_{0}=

*gx*

_{1}, then

*gx*

_{0}≼

*gx*

_{1}. Since

*F*is a

*g*-non-decreasing mapping, we have

*Fx*

_{0}≼

*Fx*

_{1}, that is,

*gx*

_{1}≼

*gx*

_{2}. Again, using that

*F*is a

*g*-non-decreasing mapping, we have

*Fx*

_{1}≼

*Fx*

_{2}, that is,

*gx*

_{2}≼

*gx*

_{3}. Continuing this process, we get

*n*∈

*N*such that

*p*(

*Fx*

_{ n },

*Fx*

_{n+1}) = 0. This implies that

*Fx*

_{ n }=

*Fx*

_{n+1}, that is,

*gx*

_{n+1}=

*Fx*

_{n+1}. Then,

*x*

_{n+1}is a coincidence point of

*F*and

*g*, and so we have finished the proof. Thus, we can assume that

*x*=

*x*

_{ n }and

*y*=

*x*

_{n+1}, we get

*φ*is non-decreasing, we have

*φ*(

*t*) <

*t*for all

*t*> 0, we have

Thus, we proved (4).

*φ*is non-decreasing, repeating the inequality (4)

*n*times, we get

*n*→ +∞ in the inequality (6) and using the fact that

*φ*

^{ n }(

*t*) → 0 as

*n*→ +∞ for all

*t*> 0, we obtain

*n*→ +∞ in this inequality, by (7), we get

*Fx*

_{ n }} is a Cauchy sequence in the metric space (

*X*,

*p*

^{ s }). Suppose, to the contrary, that {

*Fx*

_{ n }} is not a Cauchy sequence in (

*X*,

*p*

^{ s }). Then, there exists

*ε*> 0 such that for each positive integer

*k*, there exist two sequences of positive integers {

*m*(

*k*)} and {

*n*(

*k*)} such that

*p*

^{ s }(

*x*,

*y*) ≤ 2

*p*(

*x*,

*y*) for all

*x*,

*y*∈

*X*, from (9), for all positive integer

*k*, we have

*k*→ +∞ and using (7), we get

*k*→ +∞ in this inequality, and using (11) and (7), we get

*x*=

*x*

_{ n }and

*y*=

*x*

_{n+1}, we get

*φ*is a non-decreasing function, we get

*k*→ +∞ in the above inequality, using (7), (11), (12) and the continuity of

*φ*, we have

*Fx*

_{ n }} is not a Cauchy sequence was wrong. Therefore, {

*Fx*

_{ n }} is a Cauchy sequence in the metric space (

*X*,

*p*

^{ s }), and so we have

*X*,

*p*) is complete, then from Lemma 1.1, (

*X*,

*p*

^{ s }) is a complete metric space. Therefore, the sequence {

*Fx*

_{ n }} converges to some

*x*∈

*X*, that is,

*b*) in Lemma 1.1, we have

*n*→ +∞ in the above inequality and using (7), we obtain

*p*

^{ s }and using (14), we get lim

_{m,n→+∞}

*p*(

*Fx*

_{ n },

*Fx*

_{ m }) = 0. Thus, from (15), we have

*F*is continuous, from (16) and using Lemma 2.1, we get

*n*→ +∞ in the above inequality, using (17), (15), (16), the partial compatibility of {

*F*,

*g*}, the continuity of

*g*and Lemma 2.1, we have

*p*(

*Fx*,

*gx*) > 0. Then, from (1) with

*x*=

*y*, we get

a contradiction. Thus, we have *p*(*Fx*, *gx*) = 0, which implies that *Fx* = *gx*, that is, *x* is a coincidence point of *F* and *g*. Moreover, from (16) and since the pair {*F*, *g*} is partial compatible, we have *p*(*x*, *x*) = 0 = *p*(*gx*, *gx*) = *p*(*Fx*, *Fx*). This completes the proof. ■

An immediate consequence of Theorem 2.1 is the following result.

**Theorem 2.2**

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X*,

*p*)

*is a complete partial metric space. Suppose F*:

*X*→

*X is a continuous and non-decreasing mapping (with respect to*≼

*) such that*

*for all x*, *y* ∈ *X with y* ≼ *x*, *where φ* : [0, +∞) → [0, +∞) *is continuous non-decreasing and φ*(*t*) < *t for all t* > 0. *If there exists x*_{0} ∈ *X such that x*_{0} ≼ *Fx*_{0}, *then there exists x* ∈ *X such that Fx = x*. *Moreover*, *p*(*x*, *x*) = 0.

*Proof*. Putting *gx* = *Ix* = *x* in Theorem 2.1, we obtain Theorem 2.2. ■

Now we shall present an example in which *F*: *X* → *X* and *φ* : [0, +∞) → [0, +∞) satisfy all hypotheses of our Theorem 2.2, but not the hypotheses of Theorems of Altun et al. [4], Altun and Erduran [3] with *φ* given in an illustrative example in [3], Matthews [22] and of many other known corresponding theorems.

Before giving our example, we need the following result.

**Lemma 2.2** *Consider X =* [0, +∞) *endowed with the partial metric p* : *X* × *X* → [0, +∞) *defined by p*(*x*, *y*) *=* max{*x*, *y*} *for all x*, *y* ≥ 0. *Let F* : *X* → *X be a non-decreasing function. If F is continuous with respect to the standard metric d*(*x*, *y*) *=* |*x - y*| *for all x*, *y* ≥ 0, *then F is continuous with respect to the partial metric p*.

*Proof*. Let {*x*_{
n
}} be a sequence in *X* such that lim_{n→+∞}*p*(*x*_{
n
}, *x*) = *p*(*x*, *x*) for some *x* ∈ *X*, that is, lim_{n→+∞}max{*x*_{
n
}, *x*} = *x*. Using Lemma 2.1, we have to prove that lim_{n→+∞}*p*(*Fx*_{
n
}, *Fx*) = *p*(*Fx*, *Fx*), that is, lim_{n→+∞}max{*Fx*_{
n
}, *Fx*} = *Fx*.

*F*is a non-decreasing mapping, we have

*F*is continuous with respect to the standard metric, we have

This makes end to the proof. ■

**Example 2.1**

*Let X =*[0, +∞)

*and*(

*X*,

*p*)

*be a complete partial metric space, where p*:

*X*×

*X*→ ℝ

^{ + }

*is defined by p*(

*x*,

*y*)

*=*max{

*x*,

*y*}.

*Let us define a partial order*≼

*on X as follows:*

*Define F*:

*X*→

*X by*

*and let φ*: [0, +∞) → [0, +∞)

*be defined by*

*Clearly the function φ* ∈ ϕ, *that is*, *φ is continuous non-decreasing and φ*(*t*) < *t for each t* > 0. *On the other hand, using Lemma* 2.2, *since F is non-decreasing (with respect to the usual order) and continuous in X with respect to the standard metric, then it is continuous with respect to the partial metric p. The function F is also non-decreasing with respect to the partial order* ≼.

*We now show that F satisfies the nonlinear contractive condition*(20)

*for all x*,

*y*∈

*X with y*≼

*x*.

*By definition of F, we have*

*Thus*,

*Therefore, the contractive condition* (20) *is satisfied for all x*, *y* ∈ *X for which y* ≼ *x*.

*Also, for x*_{0} = 0, *we have x*_{0} ≼ *Fx*_{0}.

*Therefore, all hypotheses of Theorem* 2.2 *are satisfied and F has a fixed point. Note that it is easy to see that the hypothesis* (23) *as well as all other hypotheses in Theorems* 2.3 *and* 2.4 *below is also satisfied*.

*Observe that in this example*,

*φ does not satisfy the condition*${\sum}_{n=1}^{\infty}{\phi}^{n}\left(t\right)<\infty $

*for each t*> 0

*of Theorems in*[3, 4].

*Indeed, let t*

_{0}∈ (0, 1]

*be arbitrary. Then, it is easy to show by induction that φ*

^{ n }(

*t*

_{0}) =

*t*

_{0}/(1 +

*nt*

_{0}).

*Thus*,

*Note that F does not satisfy the contractive condition*(20)

*in Theorem*2.2

*with a function*

*This function is given by Altun and Erduran in their illustrative example in*[3].

*It is easy to show that for y*≼

*x*,

Now, we will prove the following result.

**Theorem 2.3**

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X, p*)

*is a complete partial metric space. Let F,g*:

*X*→

*X be two self-mappings of X such that FX*⊆

*gX, F is a g-non-decreasing mapping and*,

*for all x*,

*y*∈

*X for which gx*≻

*gy, whereφ*∈ ϕ.

*Also suppose*

*holds*. *Also suppose gX is closed. If there exists x*_{0} ∈ *X with gx*_{0} ≼ *Fx*_{0}, *then F and g have a coincidence point x* ∈ *X such that p*(*Fx*, *Fx*) *= p*(*gx*, *gx*) = 0. *Further, if F and g commute at their coincidence points, then F and g have a common fixed point*.

*Proof*. Denote

for all *x*, *y* ∈ *X*.

*x*

_{ n }} in

*X*by

*gx*

_{n+1}=

*Fx*

_{ n }for all

*n*≥ 0. Also, we can assume that

*Fx*

_{ n }≠

*Fx*

_{n+1}for all

*n*≥ 0; otherwise, we are finished. Therefore, we have

*Fx*

_{ n }} is a Cauchy sequence in the complete metric space (

*X*,

*p*

^{ s }), and therefore, there exists

*y*∈

*X*such that

*Fx*

_{ n }} ⊂

*gX*and

*gX*is closed, there exists

*x*∈

*X*such that

*y*=

*gx*. From (24) and hypothesis (23), we have

*x*is a coincidence point of

*F*and

*g*. Using the triangular inequality, we have

*φ*is a non-decreasing function, using (25), the above inequality and

*n*→ +∞ in (27), we get

If *p*(*gx*, *Fx*) > 0, we obtain *p*(*gx*, *Fx*) ≤ *φ*(*p*(*gx*, *Fx*)) < *p*(*gx*, *Fx*): a contradiction. We deduce that *p*(*gx*, *Fx*) = 0, which implies that *gx* = *Fx*, that is, *x* is a coincidence point of *F* and *g*.

*F*and

*g*commute at

*x*. Set

*w*=

*Fx*=

*gx*. Then,

*gx*≼

*g*(

*gx*) =

*gw*. If

*gx*=

*gw*, we get

*w*=

*gw*=

*Fw*, and the proof is finished. Then, suppose that

*gx*≺

*gw*. Applying the considered contraction, we get

*p*(

*Fw*,

*Fx*) > 0, From (29), we get

which is a contradiction. Thus, we have *p*(*Fw*, *Fx*) = 0, which implies that *Fw* = *Fx* = *w*. Therefore, from (28), we have *w* = *Fw* = *gw*, and *w* is a common fixed point of *F* and *g*. This completes the proof. ■

**Remark 2.1**

*The result given by Theorem*2.3

*is also valid if the contraction condition*(22)

*is satisfied for all x, y*∈

*X with gx*≽

*gy and*(23)

*is replaced by*

An immediate consequence of Theorem 2.3 is the following.

**Theorem 2.4**

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a partial metric p on X such that*(

*X, p*)

*is a complete partial metric space. Suppose F*:

*X*→

*X is a non-decreasing mapping such that*

*for all x, y*∈

*X with y*≺

*x*,

*where φ*: [0, +∞) → [0, +∞)

*is continuous non-decreasing and φ*(

*t*)

*< t for all t*> 0.

*Suppose also that the condition*

*holds*. *If there exists x*_{0} ∈ *X such that x*_{0} ≼ *Fx*_{0}, *then there exists x* ∈ *X such that Fx* = *x*. *Moreover*, *p*(*x*, *x*) = 0.

Now, we give a simple example to show that our result given by Theorem 2.3 is more general than Theorem 3.6 of O'Regan and Petruşel [29].

**Example 2.2**

*Let X =*[0, +∞)

*endowed with the partial metric p*(

*x, y*)

*=*max{

*x, y*}

*for all x*,

*y*∈

*X. We endow X with the usual order*≤.

*Consider the mappings F*,

*g*:

*X*→

*X and φ*: [0, +∞) → [0, +∞)

*defined by*

*Let y*≤

*x*.

*We have*

*Then*, (22) *is satisfied. It is easy to show that all the other hypotheses of Theorem* 2.3 *are also satisfied. Since F and g commute, we deduce that F and g have a common fixed point z* = 0, *that is*, 0 = *F*(0) = *g*(0).

*On the other hand, if we endow X with the standard metric d*(

*x*,

*y*)

*=*|

*x - y*|

*for all x*,

*y*∈

*X, we have*

*for x ≠ y and for any φ* : [0, +∞) → [0, +∞) *satisfying φ*(*t*) *< t for t >* 0. *Therefore, Theorem 3.6 of O'Regan and Petruşel* [29] *is not applicable*.

*Note that F also does not satisfy the contractive conditions in the rest theorems of O'Regan and Petruşel* [29].

## Declarations

### Acknowledgements

This work was supported by the Ministry of Sciences and technology of Republic Serbia (PROJECT 174025).

## Authors’ Affiliations

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