RESEARC H Open Access
Common fixed-point results for nonlinear
contractions in ordered partial metric spaces
Bessem Samet
1*
, Miloje Rajović
2
, Rade Lazović
3
and Rade Stojiljković
4
* Correspondence: bessem.
[email protected]
1
Université de Tunis, Ecole
Supérieure des Sciences et
Techniques de Tunis, 5, Avenue
Taha Hussein-Tunis, B.P.:56, 1008
Bab Menara, Tunisia
Full list of author information is
available at the end of the article
Abstract
In this paper, a new class of a pair of generalize d 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.

Example 1.1 Let a function p : ℝ
+
× ℝ
+
® ℝ
+
be defined by p(x, y)=max{x, y} for
any x, y Î ℝ
+
. Then,(ℝ
+
, p) is a partial metric space where the self-distance for any
point x Î ℝ
+
is its value itself.
Example 1.2 Cons ider a fu nct ion p : ℝ
-
× ℝ
-
® ℝ
+
defined by p(x, y)=- min(x, y)
for any x, y Î 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 Î ℝ

)
=2p
(
x, y
)
− p
(
x, x
)
− p
(
y, y
)
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 Î 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

p
s
(x
n
, x)=0if and only if
p(x, x)= lim
n→+∞
p(x
n
, x)= lim
n
,
m→+∞
p(x
n
, x
m
)
.
Matt hews [22] obtained the following Banach fixed-point theorem on complete par-
tial metric spaces.
Theorem 1.1 (Matthews [22]) Let f be a mapping of a complet e partial metric space
(X, p) into itself such that there is a constant c Î [0,1) satisfying for all x, y Î X :
p
(
fx, fy
)
≤ cp
(
x, y

Then, T has a unique fixed point.
On the other hand, existence of fixed points in partially ordered sets has been con-
sidered 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. J achymski [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 fro m 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  :[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 sup-
pose 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
p(Fx, Fy) ≤ ϕ

max

p(x, y), p(x, Fx), p(y, Fy),
1
2
[p(x, Fy)+p(y, Fx)]


n
≺ xforall
n
holds. If there exists x
0
Î X such that x
0
≼ Fx
0
, then there exists x Î X such that Fx =
x. Moreover, p(x, x)=0.
Theorem 1.5 (A ltun and Erduran [3]) Let ( X, ≼) be a partially ordered set and s up-
pose 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
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p(Fx, Fy) ≤ ϕ

max

p(x, y),
1
2
[p(x, Fx)+p(y, Fy)],
1
2
[p(x, Fy)+p(y, Fx)]

for all x, y Î Xwithy≼ x, where  : [0, +∞) ® [0, +∞) satisfies c onditions (i)-(ii) in

n
=1
ϕ
n
(t ) < ∞
.
Presented theorems gen-
eralize 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
Î 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. Sup-
pose that F is continuous at x
0
Î X. Then, for all sequence {x
n

} isasequenceinXsuchthatFx
n
® t
and gx
n
® t for some t Î X.
It is cl ear that Definition 2.3 extends a nd generalizes the notion of compatibility
introduced by Jungck [21].
Define by j the set of functions  :[0,+∞) ® [0, +∞) satisf ying the following
conditions:
(c1)  is continuous and non-decreasing,
(c2) (t)<t for each t >0.
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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 p artial metric space. Let F, g : X ® Xbe
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
p(Fx, Fy) ≤ ϕ

max

p(gx, gy), p(gx, Fx), p(gy, Fy),
1
2
[p(gx, Fy)+p(gy, Fx)]

(1)

n
} ⊂ X such that
g
x
n+1
= Fx
n
, ∀n ≥ 0
.
Since 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
,thatis,gx
1
≼ gx
2

·
(2)
Suppose th at t here exists n Î N such that p(Fx
n
, Fx
n+1
) = 0. Thi s implies that Fx
n
=
Fx
n+1
,thatis,gx
n+1
= Fx
n+1
.Then,x
n+1
is a coincidence point of F and g,andsowe
have finished the proof. Thus, we can assume that
p
(
Fx
n
, Fx
n+1
)
> 0, ∀n ∈ N
.
(3)
We will show that

ϕ

max

p(gx
n
, gx
n+1
), p(Fx
n
, gx
n
), p(Fx
n+1
, gx
n+1
),
1
2
[p(gx
n
, Fx
n+1
)+p(Fx
n
, gx
n+1
)]



n−1
, Fx
n
), p(Fx
n+1
, Fx
n
),
1
2
[p(Fx
n−1
, Fx
n
)+p(Fx
n
, Fx
n+1
)]

.
Hence, as
p
(
Fx
n
, Fx
n
)
+ p

n−1
, Fx
n
), p(Fx
n+1
, Fx
n
)

.
(5)
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If we suppose that
max

p(Fx
n−1
, Fx
n
), p(Fx
n+1
, Fx
n
)

= p(Fx
n+1
, Fx

(
Fx
n+1
, Fx
n
))
< p
(
Fx
n+1
, Fx
n
),
a contradiction. Therefore,
max

p(Fx
n−1
, Fx
n
), p(Fx
n+1
, Fx
n
)

= p(Fx
n−1
, Fx
n

(
p
(
Fx
0
, Fx
1
))
, ∀n ∈ N
.
(6)
Letting n ® +∞ in the inequality (6) and using the fact that 
n
(t) ® 0asn ® +∞
for all t > 0, we obtain
lim
n
→+
∞
p(Fx
n
, Fx
n+1
)=0
.
(7)
On the other hand, we have
p
s
(Fx

(Fx
n
, Fx
n+1
)=0
.
(8)
Now, we shall prove that {Fx
n
} is a Cauchy sequence in the metric space (X, p
s
). Sup-
pose, 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 inte-
gers {m(k)} and {n(k)} such that
n(k) > m(k) > k and p
s
(Fx
m
(
k
)
, Fx
n
(
k

.
(10)
From (10) and the triangular inequality (that holds for a partial metric), we have
ε
2
≤ p(Fx
m(k)
, Fx
n(k)
)
≤ p(Fx
m(k)
, Fx
n(k)−1
)+p(Fx
n(k)−1
, Fx
n(k)
) −p(Fx
n(k)−1
, Fx
n(k)−1
)
<
ε
2
+ p(Fx
n(k)−1
, Fx
n(k)

, Fx
n(k)
)
≤ p(Fx
m
(
k
)
, Fx
m
(
k
)
−1
)+p(Fx
n
(
k
)
, Fx
m
(
k
)
)+p(Fx
m
(
k
)
−1

ε
2
.
(12)
On the other hand, we have
p(Fx
n
(
k
)
, Fx
m
(
k
)
) ≤ p(Fx
n
(
k
)
, Fx
n
(
k
)
+1
)+p(Fx
n
(
k

m(k)−1
), p(Fx
n(k)+1
, Fx
n(k)
), p(Fx
m(k)
, Fx
m(k)−1
),
1
2
[p(Fx
n(k)
, Fx
m(k)
)+p(Fx
n(k)+1
, Fx
m(k)−1
)]

≤ ϕ

max

p(Fx
n(k)
, Fx
m(k)−1

k
))
.
Therefore, from (13) and since  is a non-decreasing function, we get
p(Fx
n
(
k
)
, Fx
m
(
k
)
) ≤ p(Fx
n
(
k
)
, F
n
(
k
)
+1
)+ϕ(ξ (k))
.
Lettin g k ® + ∞ in the above inequality, using (7), (11), (12) and the continuity of ,
we have
ε

(14)
Now, since (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 Î X, that is,
lim
n
→+∞
p
s
(Fx
n
, x) = lim
n
→+∞
p
s
(gx
n+1
, x)=0
.
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From the property (b) in Lemma 1.1, we have
p(x, x) = lim
n→+∞
p(Fx

)
for all n ∈ N
.
Letting n ® +∞ in the above inequality and using (7), we obtain
lim
n
→+
∞
p(Fx
n
, Fx
n
)=0
.
Therefore, from the definition of p
s
and using (14), we get lim
m,n®+∞
p(Fx
n
, Fx
m
)=
0. Thus, from (15), we have
p(x, x) = lim
n→+∞
p(Fx
n
, x) = lim
m

))
+ p
(
F
(
gx
n+1
)
, g
(
Fx
n+1
))
+ p
(
g
(
Fx
n+1
)
, gx
).
(18)
Letting n ® +∞ in the above inequality, using (17), (15), (16), the partial compatibil-
ity of {F, g}, the continuity of g and Lemma 2.1, we have
p
(
Fx, gx
)
≤ p

(
p
(
Fx, gx
))
< p
(
Fx, gx
).
Therefore, from (19), we have
p
(
Fx, gx
)
< p
(
Fx, gx
),
a contradiction. Thu s, we have p(Fx, gx) = 0, which implies that Fx = gx, that is, x is
a coincidence point of F and g. Moreover, fr om (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 ® Xis
a continuous and non-decreasing mapping (with respect to ≼) such that
p(Fx, Fy) ≤ ϕ

max

p(x, y), p(x, Fx), p(y, Fy),

n®+∞
p(x
n
, x)=p(x, x) for some x Î
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.
Since F is a non-decreasing mapping, we have
max{Fx
n
, Fx} = F
(
max{x
n
, x}
).
(21)
Now, using that F is continuous with respect to the standard metric, we have
lim

with x ≤ y
).
Define F : X ® Xby
F( x )=
⎧
⎪
⎨
⎪
⎩
x
1+x
if x ∈ [0, 1)
,
√
x
2
if x ≥ 1,
and let  : [0, +∞) ® [0, +∞) be defined by
ϕ(t)=
⎧
⎨
⎩
t
1+t
if t ∈ (0, 1]
,
t
2
if t > 1.
Clearly the function  Îj, that is,  is continuous non-decreasing and (t)<tfor


max

p(x, y), p(Fx, x), p(Fy, y),
1
2
[p(x, Fy)+p(Fx, y)]

.
Therefore, the contractive condition (20) is satisfied for all x, y Î 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,  does not satisfy the condition

∞
n
=1
ϕ
n
(t ) <
∞
for

.
Note that F does not sat isfy the contractive condition (20) in Theorem 2.2 wit h a
function
ϕ(t)=
t
2
1+
t
.
This function is given by Altun and Erduran in their illustrativ e example i n [3]. It is
easy to show that for y ≼ x,
p(Fx, Fy)=max

x
1+x
,
y
1+y

=
x
1+x
>
x
2
1+x
= ϕ

max



if {gx
n
}⊂X is a increasing sequence
with gx
n
→ gz ∈ gX, then gx
n
≺ gz, gz  g(gz) for all
n
(23)
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holds. Also suppose gX is closed. If there exists x
0
Î Xwithgx
0
≼ Fx
0
, then F and g
have a coincidence point x Î 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
M[F, g](x, y):=max

p(gx, gy), p(gx, Fx), p(gy, Fy),
1
2
[p(gx, Fy)+p(gy, Fx)]

≺··
·
(24)
Again, as in the proof of Theorem 2.1, we can show that {Fx
n
} is a Cauchy sequence
in the complete metric space (X, p
s
), and therefore, there exists y Î X such that
p(y, y) = lim
n→+∞
p(Fx
n
, y) = lim
m
,
n→+∞
p(Fx
n
, Fx
m
)=0
.
(25)
Since {Fx
n
} ⊂ gX and gX is closed, there exists x Î X such that y = gx. From (24) and
hypothesis (23), we have
g
x

(
M[F, g]
(
x, x
n
)).
Thus,
p
(
gx, Fx
)
≤ p
(
gx, Fx
n
)
+ ϕ
(
M[F, g]
(
x, x
n
)).
(27)
Now, we have
M
[
F, g
](
x, x

2
[p(gx, Fx
n
)+p(Fx, gx)+p(gx, Fx
n−1
)]

.
Since  is a non-decreasing function, using (25), the above inequality and n ® +∞ in
(27), we get
p
(
gx, Fx
)
≤ ϕ
(
p
(
gx, Fx
)).
If p(gx, Fx) > 0, we obtain p(gx, Fx) ≤ (p(gx, Fx)) <p(gx, Fx): a contradiction. We
deduce that p(gx, Fx) = 0, which implies that gx = Fx,thatis,x is a coincidence point
of F and g.
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Suppose now that F and g commute at x. Set w = Fx = gx. Then,
Fw = F
(
gx


=max

p(Fw, Fx), p(Fw, Fw), p(Fx, Fx),
1
2
[p(Fw, Fx)+p(Fw, Fx)]

=max{p(Fw, Fx), p(Fw, Fw)}
= p
(
Fw, Fx
)
.
Suppose that p(Fw, Fx) > 0, From (29), we get
p
(
Fw, Fx
)
≤ ϕ
(
M[F, g]
(
w, x
))
= ϕ
(
p
(
Fw, Fx

2
[p(x, Fy)+p(y, Fx)]

,
for all x, y Î X with y ≺ x, whe re  : [0, +∞) ® [0, +∞) is continuous non-decreasing
and (t) < t for all t >0.Suppose also that the condition

if {x
n
}⊂X is a increasing sequence
with x
n
→ x ∈ X , then x
n
≺ xforall
n
(30)
holds. If there exists x
0
Î X such that x
0
≼ Fx
0
, then there exists x Î 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, +∞) end owed with the partial metric p (x, y) = max{x, y} for
all x, y Î X. We endow X with the usual order ≤ . Consider the m appings F, g : X ® X
Samet et al. Fixed Point Theory and Applications 2011, 2011:71

p(g(x), g(y)) = ϕ(p(g(x), g(y))
.
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 Î X, we have
d
(
F
(
x
)
, F
(
y
))
= |F
(
x
)
− F
(
y
)
| =2|x −y| >ϕ
(
|x −y|
)
for x ≠ yandforany :[0,+∞) ® [0, +∞) satisfying (t) <tfort>0. Therefore,

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Cite this article as: Samet et al.: Common fixed-point results for nonlinear contractions in ordered partial metric