Open Access

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

  • Bessem Samet1Email author,
  • Miloje Rajović2,
  • Rade Lazović3 and
  • Rade Stojiljković4
Fixed Point Theory and Applications20112011:71

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

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

partial metricordered setcommon fixed pointcoincidence pointpartial compatible

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 [145]). 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, 3638, 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+ 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 : + × ++ be defined by p(x, y) = max{x, y} for any x, y +. Then, (+ , p) is a partial metric space where the self-distance for any point x + is its value itself.

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

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

Each partial metric p on X generates a T0 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
B p ( x , ε ) = { y X | p ( x , y ) < p ( x , x ) + ε } for all x X and ε > 0 .
If p is a partial metric on X, then the function p s : X × X+ defined by
p s ( x , y ) = 2 p ( 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 X if and only if p(x, x) = limn→+∞p(x, x n ),

(ii) {x n } is a Cauchy sequence if there exists (and is finite) limn,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) = limn,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, limn→+∞p s (x n , x) = 0 if and only if
p ( x , x ) = lim n + p ( x n , x ) = lim n , m + p ( x n , x m ) .

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 :
p ( f x , f y ) c p ( x , y ) .

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 : XX be a map such that
p ( T x , T y ) φ max p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 [ p ( x , T y ) + p ( y , T x ) ]

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

(i) φ is continuous and non-decreasing,

(ii) n 1 φ n ( t ) 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 [13, 57, 11, 12, 14, 15, 17, 19, 2427, 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 : XX is a continuous and non-decreasing mapping (with respect to ) such that
p ( F x , F y ) φ max p ( x , y ) , p ( x , F x ) , p ( y , F y ) , 1 2 [ p ( x , F y ) + p ( y , F x ) ]

for all x, y X with y x, where φ : [0, +∞) → [0, +∞) satisfies conditions (i)-(ii) in Theorem 1.2. If there exists x0 X such that x0 Fx0, 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 : XX is a non-decreasing mapping such that
p ( F x , F y ) φ max p ( x , y ) , p ( x , F x ) , p ( y , F y ) , 1 2 [ p ( x , F y ) + p ( y , F x ) ]
for all x, y X with y x (y x and yx), where φ : [0, +∞) → [0, +∞) satisfies conditions (i)-(ii) in Theorem 1.2. Suppose also that the condition
i f { x n } X i s a i n c r e a s i n g s e q u e n c e w i t h x n x X , t h e n x n x f o r a l l n

holds. If there exists x0 X such that x0 Fx0, 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 : XX is a continuous and non-decreasing mapping such that
p ( F x , F y ) φ max p ( x , y ) , 1 2 [ p ( x , F x ) + p ( y , F y ) ] , 1 2 [ p ( x , F y ) + p ( y , F x ) ]

for all x, y X with y x, where φ : [0, +∞) → [0, +∞) satisfies conditions (i)-(ii) in Theorem 1.2. If there exists x0 X such that x0 Fx0, 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 n = 1 φ n ( t ) < 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 n = 1 φ n ( t ) < . 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 : XX be a given mapping. We say that F is continuous at x0 X, if for every ε > 0, there exists δ > 0 such that F(B p (x0, δ)) B p (Fx0, ε).

The following result is easy to check.

Lemma 2.1 Let (X, p) be a partial metric space, F : XX be a given mapping. Suppose that F is continuous at x0 X. Then, for all sequence {x n } X, we have
x n x 0 F x n F x 0 .
Definition 2.2 (Ćirić et al. [11]) Let (X, ) be a partially ordered set and F, g : XX are mappings of X into itself. One says F is g-non-decreasing if for x, y X, we have
g x g y F x F y .

We introduce the following definition.

Definition 2.3 Let (X, p) be a partial metric space and F, g: XX 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) limn→+∞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 : XX 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
p ( F x , F y ) φ max p ( g x , g y ) , p ( g x , F x ) , p ( g y , F y ) , 1 2 [ p ( g x , F y ) + p ( g y , F x ) ]
(1)

for all x, y X for which gy gx, where a function φ ϕ. If there exists x0 X with gx0 Fx0, 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 x0 X such that gx0 Fx0. Since FX gX, we can choose x1 X so that gx1 = Fx0. Again, from FX gX, there exists x2 X such that gx2 = Fx1. Continuing this process, we can choose a sequence {x n } X such that
g x n + 1 = F x n , n 0 .
Since gx0 Fx0 and Fx0 = gx1, then gx0 gx1. Since F is a g-non-decreasing mapping, we have Fx0 Fx1, that is, gx1 gx2. Again, using that F is a g-non-decreasing mapping, we have Fx1 Fx2, that is, gx2 gx3. Continuing this process, we get
g x 1 g x 2 g x 3 g x n g x n + 1
(2)
Suppose that there exists n N such that p(Fx n , Fxn+1) = 0. This implies that Fx n = Fxn+1, that is, gxn+1= Fxn+1. Then, xn+1is a coincidence point of F and g, and so we have finished the proof. Thus, we can assume that
p ( F x n , F x n + 1 ) > 0 , n .
(3)
We will show that
p ( F x n , F x n + 1 ) φ ( p ( F x n - 1 , F x n ) ) for all n 1 .
(4)
Using (2) and applying the considered contraction (1) with x = x n and y = xn+1, we get
p ( F x n , F x n + 1 ) φ max p ( g x n , g x n + 1 ) , p ( F x n , g x n ) , p ( F x n + 1 , g x n + 1 ) , 1 2 [ p ( g x n , F x n + 1 ) + p ( F x n , g x n + 1 ) ] = φ max p ( F x n - 1 , F x n ) , p ( F x n + 1 , F x n ) , 1 2 [ p ( F x n - 1 , F x n + 1 ) + p ( F x n , F x n ) ] φ max p ( F x n - 1 , F x n ) , p ( F x n + 1 , F x n ) , 1 2 [ p ( F x n - 1 , F x n ) + p ( F x n , F x n + 1 ) ] .
Hence, as
p ( F x n , F x n ) + p ( F x n - 1 , F x n + 1 ) p ( F x n - 1 , F x n ) + p ( F x n , F x n + 1 )
and φ is non-decreasing, we have
p ( F x n , F x n + 1 ) φ max p ( F x n - 1 , F x n ) , p ( F x n + 1 , F x n ) .
(5)
If we suppose that max p ( F x n - 1 , F x n ) , p ( F x n + 1 , F x n ) = p ( F x n + 1 , F x n ) , then from (5),
p ( F x n , F x n + 1 ) φ ( p ( F x n + 1 , F x n ) ) .
Using (3) and the fact that φ(t) < t for all t > 0, we have
p ( F x n , F x n + 1 ) φ ( p ( F x n + 1 , F x n ) ) < p ( F x n + 1 , F x n ) ,
a contradiction. Therefore,
max p ( F x n - 1 , F x n ) , p ( F x n + 1 , F x n ) = p ( F x n - 1 , F x n ) ,
and so from (5),
p ( F x n , F x n + 1 ) φ ( p ( F x n - 1 , F x n ) ) .

Thus, we proved (4).

Since φ is non-decreasing, repeating the inequality (4) n times, we get
p ( F x n , F x n + 1 ) φ n ( p ( F x 0 , F x 1 ) ) , n .
(6)
Letting n → +∞ in the inequality (6) and using the fact that φ n (t) → 0 as n → +∞ for all t > 0, we obtain
lim n + p ( F x n , F x n + 1 ) = 0 .
(7)
On the other hand, we have
p s ( F x n , F x n + 1 ) = 2 p ( F x n , F x n + 1 ) - p ( F x n , F x n ) - p ( F x n + 1 , F x n + 1 ) 2 p ( F x n , F x n + 1 ) .
Letting n → +∞ in this inequality, by (7), we get
lim n + p s ( F x n , F x n + 1 ) = 0 .
(8)
Now, we shall prove that {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
n ( k ) > m ( k ) > k and p s ( F x m ( k ) , F x n ( k ) ) ε .
(9)
Since p s (x, y) ≤ 2p(x, y) for all x, y X, from (9), for all positive integer k, we have
n ( k ) > m ( k ) > k and p ( F x m ( k ) , F x n ( k ) ) ε 2 .
Without loss of generality, we can suppose that also
n ( k ) > m ( k ) > k , p ( F x m ( k ) , F x n ( k ) ) ε 2 , p ( F x m ( k ) , F x n ( k ) - 1 ) < ε 2 .
(10)
From (10) and the triangular inequality (that holds for a partial metric), we have
ε 2 p ( F x m ( k ) , F x n ( k ) ) p ( F x m ( k ) , F x n ( k ) - 1 ) + p ( F x n ( k ) - 1 , F x n ( k ) ) - p ( F x n ( k ) - 1 , F x n ( k ) - 1 ) < ε 2 + p ( F x n ( k ) - 1 , F x n ( k ) ) .
Letting k → +∞ and using (7), we get
lim k + p ( F x m ( k ) , F x n ( k ) ) = ε 2 .
(11)
Again, using the triangular inequality, we obtain
ε 2 p ( F x m ( k ) , F x n ( k ) ) p ( F x m ( k ) , F x m ( k ) - 1 ) + p ( F x m ( k ) - 1 , F x n ( k ) ) p ( F x m ( k ) , F x m ( k ) - 1 ) + p ( F x n ( k ) , F x m ( k ) ) + p ( F x m ( k ) - 1 , F x m ( k ) ) .
Letting k → +∞ in this inequality, and using (11) and (7), we get
ε 2 lim k + p ( F x n ( k ) , F x m ( k ) - 1 ) ε 2 .
Hence,
lim k + p ( F x n ( k ) , F x m ( k ) - 1 ) = ε 2 .
(12)
On the other hand, we have
p ( F x n ( k ) , F x m ( k ) ) p ( F x n ( k ) , F x n ( k ) + 1 ) + p ( F x n ( k ) + 1 , F x m ( k ) ) .
(13)
From (1) with x = x n and y = xn+1, we get
p ( F x n ( k ) + 1 , F x m ( k ) ) φ ( m a x { p ( F x n ( k ) , F x m ( k ) 1 ) , p ( F x n ( k ) + 1 , F x n ( k ) ) , p ( F x m ( k ) , F x m ( k ) 1 ) , 1 2 [ p ( F x n ( k ) , F x m ( k ) ) + p ( F x n ( k ) + 1 , F x m ( k ) 1 ) ] } ) φ ( m a x { p ( F x n ( k ) , F x m ( k ) 1 ) , p ( F x n ( k ) + 1 , F x n ( k ) ) , p ( F x m ( k ) , F x m ( k ) 1 ) , 1 2 [ p ( F x n ( k ) , F x m ( k ) ) + p ( F x n ( k ) + 1 , F x n ( k ) ) + p ( F x n ( k ) , F x m ( k ) 1 ) ] } ) : = φ ( ξ ( k ) ) .
Therefore, from (13) and since φ is a non-decreasing function, we get
p ( F x n ( k ) , F x m ( k ) ) p ( F x n ( k ) , F n ( k ) + 1 ) + φ ( ξ ( k ) ) .
Letting k → +∞ in the above inequality, using (7), (11), (12) and the continuity of φ, we have
ε 2 φ ε 2 < ε 2 ,
a contradiction. Thus, our supposition that {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
lim m , n + p s ( F x n , F x m ) = 0 .
(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 X, that is,
lim n + p s ( F x n , x ) = lim n + p s ( g x n + 1 , x ) = 0 .
From the property (b) in Lemma 1.1, we have
p ( x , x ) = lim n + p ( F x n , x ) = lim n + p ( g x n + 1 , x ) = lim m , n + p ( F x n , F x m ) .
(15)
On the other hand, from property (p2) of a partial metric, we have
p ( F x n , F x n ) p ( F x n , F x n + 1 ) for all n .
Letting n → +∞ in the above inequality and using (7), we obtain
lim n + p ( F x n , F x n ) = 0 .
Therefore, from the definition of p s and using (14), we get limm,n→+∞p(Fx n , Fx m ) = 0. Thus, from (15), we have
p ( x , x ) = lim n + p ( F x n , x ) = lim m , n + p ( F x n , F x m ) = 0 .
(16)
Now, since F is continuous, from (16) and using Lemma 2.1, we get
lim n + p ( F ( F x n ) , F x ) = p ( F x , F x ) .
(17)
Using the triangular inequality, we obtain
p ( F x , g x ) p ( F x , F ( F x n ) ) + p ( F ( g x n + 1 ) , g ( F x n + 1 ) ) + p ( g ( F x n + 1 ) , g x ) .
(18)
Letting 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 ( F x , g x ) p ( F x , F x ) + p ( g x , g x ) = p ( F x , F x ) .
(19)
Now, suppose that p(Fx, gx) > 0. Then, from (1) with x = y, we get
p ( F x , F x ) φ ( max { p ( g x , g x ) , p ( F x , g x ) } ) = φ ( p ( F x , g x ) ) < p ( F x , g x ) .
Therefore, from (19), we have
p ( F x , g x ) < p ( F x , g x ) ,

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 : XX is a continuous and non-decreasing mapping (with respect to ) such that
p ( F x , F y ) φ max p ( x , y ) , p ( x , F x ) , p ( y , F y ) , 1 2 [ p ( x , F y ) + p ( y , F x ) ]
(20)

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 x0 X such that x0 Fx0, 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: XX 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 : XX 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 limn→+∞p(x n , x) = p(x, x) for some x X, that is, limn→+∞max{x n , x} = x. Using Lemma 2.1, we have to prove that limn→+∞p(Fx n , Fx) = p(Fx, Fx), that is, limn→+∞max{Fx n , Fx} = Fx.

Since F is a non-decreasing mapping, we have
max { F x n , F x } = F ( max { x n , x } ) .
(21)
Now, using that F is continuous with respect to the standard metric, we have
lim n + max { x n , x } = x lim n + F ( max { x n , x } ) = F x .
Therefore, from (21), it follows that
lim n + max { F x n , F x } = F x .

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 + is defined by p(x, y) = max{x, y}. Let us define a partial order on X as follows:
x y x = y o r ( x , y [ 0 , 1 ) w i t h x y ) .
Define F : XX by
F ( x ) = x 1 + x i f x [ 0 , 1 ) , x 2 i f x 1 ,
and let φ : [0, +∞) → [0, +∞) be defined by
φ ( t ) = t 1 + t i f t ( 0 , 1 ] , t 2 i f t > 1 .

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
p ( F x , F y ) = max x 1 + x , y 1 + y = x 1 + x = φ ( max { x , y } ) = φ ( p ( x , y ) ) .
Thus,
p ( F x , F y ) φ max p ( x , y ) , p ( F x , x ) , p ( F y , y ) , 1 2 [ p ( x , F y ) + p ( F x , y ) ] .

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

Also, for x0 = 0, we have x0 Fx0.

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 n = 1 φ n ( t ) < for each t> 0 of Theorems in [3, 4]. Indeed, let t0 (0, 1] be arbitrary. Then, it is easy to show by induction that φ n (t0) = t0/(1 + nt0). Thus,
n = 1 φ n ( t 0 ) = n = 1 t 0 1 + n t 0 = + .
Note that F does not satisfy the contractive condition (20) in Theorem 2.2 with a function
φ ( t ) = t 2 1 + t .
This function is given by Altun and Erduran in their illustrative example in [3]. It is easy to show that for y x,
p ( F x , F y ) = max x 1 + x , y 1 + y = x 1 + x > x 2 1 + x = φ max p ( x , y ) , p ( x , F x ) , p ( y , F y ) , 1 2 [ p ( x , F y ) + p ( y , F x ) ] φ max p ( x , y ) , p ( x , F x ) , p ( y , F y ) , 1 2 [ p ( x , F y ) + p ( y , F 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 : XX be two self-mappings of X such that FX gX, F is a g-non-decreasing mapping and,
p ( F x , F y ) φ max p ( g x , g y ) , p ( g x , F x ) , p ( g y , F y ) , 1 2 [ p ( g x , F y ) + p ( g y , F x ) ]
(22)
for all x, y X for which gx gy, whereφ ϕ. Also suppose
i f { g x n } X i s a i n c r e a s i n g s e q u e n c e w i t h g x n g z g X , t h e n g x n g z , g z g ( g z ) f o r a l l n
(23)

holds. Also suppose gX is closed. If there exists x0 X with gx0 Fx0, 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
M [ F , g ] ( x , y ) : = max p ( g x , g y ) , p ( g x , F x ) , p ( g y , F y ) , 1 2 [ p ( g x , F y ) + p ( g y , F x ) ]

for all x, y X.

As in the proof of Theorem 2.1, we can construct a sequence {x n } in X by gxn+1= Fx n for all n ≥ 0. Also, we can assume that Fx n Fxn+1for all n ≥ 0; otherwise, we are finished. Therefore, we have
g x 1 g x 2 g x n g x n + 1
(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 X such that
p ( y , y ) = lim n + p ( F x n , y ) = lim m , n + p ( F x n , F x m ) = 0 .
(25)
Since {Fx n } gX and gX is closed, there exists x X such that y = gx. From (24) and hypothesis (23), we have
g x n g x for all n , g x g ( g x ) .
(26)
Now, we will show that x is a coincidence point of F and g. Using the triangular inequality, we have
p ( g x , F x ) p ( g x , g x n + 1 ) + p ( F x n , F x ) .
From (26), using the considered contraction, we have
p ( F x , F x n ) φ ( M [ F , g ] ( x , x n ) ) .
Thus,
p ( g x , F x ) p ( g x , F x n ) + φ ( M [ F , g ] ( x , x n ) ) .
(27)
Now, we have
M [ F , g ] ( x , x n ) = max p ( g x , F x n - 1 ) , p ( F x , g x ) , p ( F x n , F x n - 1 ) , 1 2 [ p ( g x , F x n ) + p ( F x , F x n - 1 ) ] max p ( g x , F x n - 1 ) , p ( F x , g x ) , p ( F x n , F x n - 1 ) , 1 2 [ p ( g x , F x n ) + p ( F x , g x ) + p ( g x , F x n - 1 ) ] .
Since φ is a non-decreasing function, using (25), the above inequality and n → +∞ in (27), we get
p ( g x , F x ) φ ( p ( g x , F x ) ) .

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.

Suppose now that F and g commute at x. Set w = Fx = gx. Then,
F w = F ( g x ) = g ( F x ) = g w .
(28)
From the hypothesis (23), we have 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 ( F w , F x ) φ ( M [ F , g ] ( w , x ) ) ,
(29)
where
M [ F , g ] ( w , x ) = max p ( g w , g x ) , p ( F w , g w ) , p ( F x , g x ) , 1 2 [ p ( g w , F x ) + p ( F w , g x ) ] = max p ( F w , F x ) , p ( F w , F w ) , p ( F x , F x ) , 1 2 [ p ( F w , F x ) + p ( F w , F x ) ] = max { p ( F w , F x ) , p ( F w , F w ) } = p ( F w , F x ) .
Suppose that p(Fw, Fx) > 0, From (29), we get
p ( F w , F x ) φ ( M [ F , g ] ( w , x ) ) = φ ( p ( F w , F x ) ) < p ( F w , F x ) ,

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
i f { g x n } X i s a i n c r e a s i n g s e q u e n c e w i t h g x n g z g X , t h e n g x n g z , g z g ( g z ) f o r a l l n

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 : XX is a non-decreasing mapping such that
p ( F x , F y ) φ max p ( x , y ) , p ( x , F x ) , p ( y , F y ) , 1 2 [ p ( x , F y ) + p ( y , F x ) ] ,
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
i f { x n } X i s a i n c r e a s i n g s e q u e n c e w i t h x n x X , t h e n x n x f o r a l l n
(30)

holds. If there exists x0 X such that x0 Fx0, 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 : XX and φ : [0, +∞) → [0, +∞) defined by
F ( x ) = 2 x , g ( x ) = 4 x , φ ( t ) = ( 3 4 ) t .
Let yx. We have
p ( F ( x ) , F ( y ) ) = F ( x ) = 2 x < 3 1 4 4 x = 3 4 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 X, we have
d ( F ( x ) , F ( y ) ) = | F ( x ) - F ( y ) | = 2 | x - y | > φ ( | x - y | )

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

(1)
Université de Tunis, Ecole Supérieure des Sciences et Techniques de Tunis, Bab Menara, Tunisia
(2)
Faculty of Mechanical Engineering, Kraljevo, Serbia
(3)
Faculty of Organizational Science, Belgrade, Serbia
(4)
High School for Preschool Pedagogues Gnjilane, Bujanovac, Serbia

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