Open Access

Fixed point theorems for weakly contractive mappings in partially ordered metric-like spaces

Fixed Point Theory and Applications20132013:51

https://doi.org/10.1186/1687-1812-2013-51

Received: 25 November 2012

Accepted: 21 February 2013

Published: 12 March 2013

Abstract

In this article, we establish some fixed point theorems for weakly contractive mappings defined in ordered metric-like spaces. We provide an example and some applications in order to support the useability of our results. These results generalize some well-known results in the literature. We also derive some new fixed point results in ordered partial metric spaces.

MSC:54H25, 47H10.

Keywords

fixed pointweak contractionpartially ordered setpartial metric spacemetric-like space

1 Introduction and preliminaries

During the last decades many authors have worked on domain theory in order to equip semantics domain with a notion of distance. In 1994, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle can be generalized to the partial metric context for applications in program verification. Later on, many researchers studied fixed point theorems in partial metric spaces as well as ordered partial metric spaces.

A partial metric on a nonempty set X is a function p : X × X R + such that for all x , y , z X :
  1. (p1)

    x = y p ( x , x ) = p ( x , y ) = p ( y , y ) ;

     
  2. (p2)

    p ( x , x ) p ( x , y ) ;

     
  3. (p3)

    p ( x , y ) = p ( y , x ) ;

     
  4. (p4)

    p ( x , y ) p ( x , z ) + p ( z , y ) p ( z , z ) .

     
The pair ( X , p ) is then called a partial metric space. A sequence { x n } in a partial metric space ( X , p ) converges to a point x X if lim n p ( x n , x ) = p ( x , x ) . A sequence { x n } of elements of X is called p-Cauchy if the limit lim m , n p ( x m , x n ) exists and is finite. The partial metric space ( X , p ) is called complete if for each p-Cauchy sequence { x n } n = 0 , there is some x X such that
p ( x , x ) = lim n p ( x n , x ) = lim n , m p ( x n , x m ) .

A basic example of a partial metric space is the pair ( R + , p ) , where p ( x , y ) = max { x , y } for all x , y R + . For some other examples of partial metric spaces, see [216].

Another important development is reported in fixed point theory via ordered metric spaces. The existence of a fixed point in partially ordered sets has been considered recently in [1732]. Tarski’s theorem is used in [25] to show the existence of solutions for fuzzy equations and in [27] to prove existence theorems for fuzzy differential equations. In [26, 29] some applications to ordinary differential equations and to matrix equations are presented, respectively. In [1921, 30] some fixed point theorems were proved for a mixed monotone mapping in a metric space endowed with partial order and the authors applied their results to problems of existence and uniqueness of solutions for some boundary value problems.

Recently, Amini-Harandi [33] introduced the notion of a metric-like space which is a new generalization of a partial metric space. The purpose of this paper is to present some fixed point theorems involving weakly contractive mappings in the context of ordered metric-like spaces. The presented theorems extend some recent results in the literature.

Weakly contractive mappings and mappings satisfying other weak contractive inequalities have been discussed in several works, some of which are noted in [3440]. Alber and Guerre-Delabriere in [34] suggested a generalization of the Banach contraction mapping principle by introducing the concept of a weak contraction in Hilbert spaces. Rhoades [35] showed that the result which Alber et al. had proved in Hilbert spaces [34] was also valid in complete metric spaces.

Definition 1 A mapping σ : X × X R + , where X is a nonempty set, is said to be metric-like on X if for any x , y , z X , the following three conditions hold true:

(σ 1) σ ( x , y ) = 0 x = y ;

(σ 2) σ ( x , y ) = σ ( y , x ) ;

(σ 3) σ ( x , z ) σ ( x , y ) + σ ( y , z ) .

The pair ( X , σ ) is then called a metric-like space. Then a metric-like on X satisfies all of the conditions of a metric except that σ ( x , x ) may be positive for x X . Each metric-like σ on X generates a topology τ σ on X whose base is the family of open σ-balls
B σ ( x , ε ) = { y X : | σ ( x , y ) σ ( x , x ) | < ε } for all  x X  and  ε > 0 .

Then a sequence { x n } in the metric-like space ( X , σ ) converges to a point x X if and only if lim n σ ( x n , x ) = σ ( x , x ) .

Let ( X , σ ) and ( Y , τ ) be metric-like spaces, and let F : X Y be a continuous mapping. Then
lim n x n = x lim n F x n = F x .
A sequence { x n } n = 0 of elements of X is called σ-Cauchy if the limit lim m , n σ ( x m , x n ) exists and is finite. The metric-like space ( X , σ ) is called complete if for each σ-Cauchy sequence { x n } n = 0 , there is some x X such that
lim n σ ( x n , x ) = σ ( x , x ) = lim m , n σ ( x m , x n ) .

Every partial metric space is a metric-like space. Below we give another example of a metric-like space.

Example 1 Let X = { 0 , 1 } and
σ ( x , y ) = { 2 if  x = y = 0 ; 1 otherwise .

Then ( X , σ ) is a metric-like space, but since σ ( 0 , 0 ) σ ( 0 , 1 ) , then ( X , σ ) is not a partial metric space.

Remark 1 Let X = { 0 , 1 } , and σ ( x , y ) = 1 for each x , y X , and x n = 1 for each n N . Then it is easy to see that x n 0 and x n 1 , and so in metric-like spaces the limit of a convergent sequence is not necessarily unique.

2 Main results

Throughout the rest of this paper, we denote by ( X , , σ ) a complete partially ordered metric-like space, i.e., is a partial order on the set X and σ is a complete metric-like on X.

A mapping F : X X is said to be nondecreasing if x , y X , x y F x F y .

Theorem 1 Let ( X , , σ ) be a complete partially ordered metric-like space. Let F : X X be a continuous and nondecreasing mapping such that for all comparable x , y X
ψ ( σ ( F x , F y ) ) ψ ( M ( x , y ) ) ϕ ( M ( x , y ) ) ,
(2.1)
where M is given by
M ( x , y ) = max { σ ( x , y ) , σ ( x , F x ) , σ ( y , F y ) , σ ( x , x ) , σ ( y , y ) , [ σ ( x , F y ) + σ ( F x , y ) ] / 2 } ,
and
  1. (a)

    ψ : [ 0 , ) [ 0 , ) is a continuous monotone nondecreasing function with ψ ( t ) = 0 if and only if t = 0 ;

     
  2. (b)

    ϕ : [ 0 , ) [ 0 , ) is a lower semi-continuous function with ϕ ( t ) = 0 if and only if t = 0 .

     

If there exists x 0 X with x 0 F x 0 , then F has a fixed point.

Proof Since F is a nondecreasing function, we obtain by induction that
x 0 F x 0 F 2 x 0 F n x 0 F n + 1 x 0 .
Put x n + 1 = F x n . Then, for each integer n = 0 , 1 , 2 ,  , as the elements x n + 1 and x n are comparable, from (2.1) we get
ψ ( σ ( x n + 1 , x n ) ) = ψ ( σ ( F x n , F x n 1 ) ) ψ ( M ( x n , x n 1 ) ) ϕ ( M ( x n , x n 1 ) ) ,
(2.2)
which implies ψ ( σ ( x n + 1 , x n ) ) ψ ( M ( x n , x n 1 ) ) . Using the monotone property of the ψ-function, we get
σ ( x n + 1 , x n ) M ( x n , x n 1 ) .
(2.3)
Now, from the triangle inequality, for σ we have
M ( x n , x n 1 ) = max { σ ( x n , x n 1 ) , σ ( x n , F x n ) , σ ( x n 1 , F x n 1 ) , σ ( x n , x n ) , σ ( x n 1 , x n 1 ) , [ σ ( x n , F x n 1 ) + σ ( F x n , x n 1 ) ] / 2 } = max { σ ( x n , x n 1 ) , σ ( x n , x n + 1 ) , σ ( x n 1 , x n ) , σ ( x n , x n ) , σ ( x n 1 , x n 1 ) , [ σ ( x n , x n ) + σ ( x n + 1 , x n 1 ) ] / 2 } max { σ ( x n , x n 1 ) , σ ( x n + 1 , x n ) , [ σ ( x n , x n + 1 ) + σ ( x n , x n 1 ) ] / 2 } = max { σ ( x n , x n 1 ) , σ ( x n + 1 , x n ) } .
If σ ( x n + 1 , x n ) > σ ( x n , x n 1 ) , then M ( x n , x n 1 ) = σ ( x n + 1 , x n ) > 0 . By (2.2) it furthermore implies that
ψ ( σ ( x n + 1 , x n ) ) ψ ( σ ( x n + 1 , x n ) ) ϕ ( σ ( x n + 1 , x n ) ) ,
which is a contradiction. So, we have
σ ( x n + 1 , x n ) M ( x n , x n 1 ) σ ( x n , x n 1 ) .
(2.4)
Therefore, the sequence { σ ( x n + 1 , x n ) } is monotone nonincreasing and bounded. Thus, there exists r 0 such that
lim n σ ( x n + 1 , x n ) = lim n M ( x n , x n 1 ) = r .
(2.5)
We suppose that r > 0 . Then, letting n in the inequality (2.2), we get
ψ ( r ) ψ ( r ) ϕ ( r ) ,
which is a contradiction unless r = 0 . Hence,
lim n σ ( x n + 1 , x n ) = 0 .
(2.6)
Next we prove that { x n } is a σ-Cauchy sequence. Suppose that { x n } is not a σ-Cauchy sequence. Then, there exists ε > 0 for which we can find subsequences { x m k } and { x n k } of { x n } with n k > m k > k such that
σ ( x n k , x m k ) ε .
(2.7)
Further, corresponding to m k , we can choose n k in such a way that it is the smallest integer with n k > m k satisfying (2.7). Then
σ ( x n k 1 , x m k ) < ε .
(2.8)
Using (2.7), (2.8) and the triangle inequality, we have
ε σ ( x n k , x m k ) σ ( x n k , x n k 1 ) + σ ( x n k 1 , x m k ) < σ ( x n k , x n k 1 ) + ε .
Letting k and using (2.6), we obtain
lim k σ ( x n k , x m k ) = ε .
(2.9)
Again, the triangle inequality gives us
Then we have
| σ ( x n k 1 , x m k ) σ ( x n k , x m k ) | σ ( x n k , x n k 1 ) .
Letting k in the above inequality and using (2.6) and (2.9), we get
lim k σ ( x n k 1 , x m k ) = ε .
(2.10)
Similarly, we can show that
lim k σ ( x n k , x m k 1 ) = lim k σ ( x n k 1 , x m k 1 ) = lim k σ ( x n k , x m k + 1 ) = lim k σ ( x n k + 1 , x m k ) = ε .
(2.11)
As
M ( x n k 1 , x m k 1 ) = max { σ ( x n k 1 , x m k 1 ) , σ ( x n k 1 , x n k ) , σ ( x m k 1 , x m k ) , σ ( x n k 1 , x n k 1 ) , σ ( x m k 1 , x m k 1 ) , [ σ ( x n k 1 , x m k ) + σ ( x n k , x m k 1 ) ] / 2 }
using (2.6) and (2.9)-(2.11), we have
lim k M ( x n k 1 , x m k 1 ) = max { ε , 0 , 0 , 0 , 0 , ε } = ε .
(2.12)
As n k > m k and x n k 1 and x m k 1 are comparable, setting x = x n k 1 and y = x m k 1 in (2.1), we obtain
ψ ( σ ( x n k , x m k ) ) = ψ ( σ ( F x n k 1 , F x m k 1 ) ) ψ ( M ( x n k 1 , x m k 1 ) ) ϕ ( M ( x n k 1 , x m k 1 ) ) .
Letting k in the above inequality and using (2.9) and (2.12), we get
ψ ( ε ) ψ ( ε ) ϕ ( ε ) ,
which is a contradiction as ε > 0 . Hence { x n } is a σ-Cauchy sequence. By the completeness of X, there exists z X such that lim n x n = z , that is,
lim n σ ( x n , z ) = σ ( z , z ) = lim m , n σ ( x m , x n ) = 0 .
(2.13)
Moreover, the continuity of F implies that
lim n σ ( x n + 1 , z ) = lim n σ ( F x n , z ) = σ ( F z , z ) = 0

and this proves that z is a fixed point. □

Notice that the continuity of F in Theorem 1 is not necessary and can be dropped.

Theorem 2 Under the same hypotheses of Theorem  1 and without assuming the continuity of F, assume that whenever { x n } is a nondecreasing sequence in X such that x n x X implies x n x for all n N , then F has a fixed point in X.

Proof Following similar arguments to those given in Theorem 1, we construct a nondecreasing sequence { x n } in X such that x n z for some z X . Using the assumption of X, we have x n z for every n N . Now, we show that F z = z . By (2.1), we have
ψ ( σ ( F z , x n + 1 ) ) = ψ ( σ ( F z , F x n ) ) ψ ( M ( z , x n ) ) ϕ ( M ( z , x n ) ) ,
(2.14)
where
σ ( F z , z ) M ( z , x n ) = max { σ ( z , x n ) , σ ( F z , z ) , σ ( x n , x n + 1 ) , σ ( z , z ) , σ ( x n , x n ) , [ σ ( z , x n + 1 ) + σ ( F z , x n ) ] / 2 } max { σ ( z , x n ) , σ ( F z , z ) , σ ( x n , x n + 1 ) , σ ( z , z ) , σ ( x n , x n ) , [ σ ( z , x n + 1 ) + σ ( F z , z ) + σ ( z , x n ) ] / 2 } .
Taking limit as n , by (2.13), we obtain
lim n M ( z , x n ) = σ ( F z , z ) .
Therefore, letting n in (2.14), we get
ψ ( σ ( F z , z ) ) ψ ( σ ( F z , z ) ) ϕ ( σ ( F z , z ) ) ,

which is a contradiction unless σ ( F z , z ) = 0 . Thus F z = z . □

Next theorem gives a sufficient condition for the uniqueness of the fixed point.

Theorem 3 Let all the conditions of Theorem  1 (resp. Theorem  2) be fulfilled and let the following condition be satisfied: For arbitrary two points x , y X , there exists z X which is comparable with both x and y. Then the fixed point of F is unique.

Proof Suppose that there exist z , x X which are fixed points. We distinguish two cases.

Case 1. If x is comparable to z, then F n x = x is comparable to F n z = z for n = 0 , 1 , 2 , and
ψ ( σ ( z , x ) ) = ψ ( σ ( F n z , F n x ) ) ψ ( M ( F n 1 z , F n 1 x ) ) ϕ ( M ( F n 1 z , F n 1 x ) ) ψ ( M ( z , x ) ) ϕ ( M ( z , x ) ) ,
(2.15)
where
M ( z , x ) = max { σ ( z , x ) , σ ( z , F z ) , σ ( x , F x ) , σ ( z , z ) , σ ( x , x ) , [ σ ( z , F x ) + σ ( F z , x ) ] / 2 } = max { σ ( z , x ) , σ ( z , z ) , σ ( x , x ) , [ σ ( z , x ) + σ ( z , x ) ] / 2 } = σ ( z , x ) .
(2.16)
Using (2.15) and (2.16), we have
ψ ( σ ( z , x ) ) ψ ( σ ( z , x ) ) ϕ ( σ ( z , x ) ) ,

which is a contradiction unless σ ( z , x ) = 0 . This implies that z = x .

Case 2. If x is not comparable to z, then there exists y X comparable to x and z. The monotonicity of F implies that F n y is comparable to F n x = x and F n z = z , for n = 0 , 1 , 2 ,  . Moreover,
ψ ( σ ( z , F n y ) ) = ψ ( σ ( F n z , F n y ) ) ψ ( M ( F n 1 z , F n 1 y ) ) ϕ ( M ( F n 1 z , F n 1 y ) ) ,
(2.17)
where
M ( F n 1 z , F n 1 y ) = max { σ ( F n 1 z , F n 1 y ) , σ ( F n 1 z , F n z ) , σ ( F n 1 y , F n y ) , σ ( F n 1 z , F n 1 z ) , σ ( F n 1 y , F n 1 y ) , [ σ ( F n 1 z , F n y ) + σ ( F n z , F n 1 y ) ] / 2 } = max { σ ( z , F n 1 y ) , σ ( z , z ) , σ ( F n 1 y , F n y ) , σ ( F n 1 y , F n 1 y ) , [ σ ( z , F n y ) + σ ( z , F n 1 y ) ] / 2 } max { σ ( z , F n 1 y ) , σ ( z , F n y ) }
(2.18)
for n sufficiently large, because σ ( F n 1 y , F n 1 y ) 0 and σ ( F n 1 y , F n y ) 0 when n . Similarly as in the proof of Theorem 1, it can be shown that σ ( z , F n y ) M ( z , F n 1 y ) σ ( z , F n 1 y ) . It follows that the sequence { σ ( z , F n y ) } is nonnegative decreasing and, consequently, there exists α 0 such that
lim n σ ( z , F n y ) = lim n M ( z , F n 1 y ) = α .
We suppose that α > 0 . Then letting n in (2.17), we have
ψ ( α ) ψ ( α ) ϕ ( α )
which is a contradiction. Hence α = 0 . Similarly, it can be proved that
lim n σ ( x , F n y ) = 0 .

Now, passing to the limit in σ ( x , z ) σ ( x , F n y ) + σ ( F n y , z ) , it follows that σ ( x , z ) = 0 , so x = z , and the uniqueness of the fixed point is proved. □

Now, we present an example to support the useability of our results.

Example 2 Let X = { 0 , 1 , 2 } and a partial order be defined as x y whenever y x , and define σ : X × X R + as follows:

Then ( X , , σ ) is a complete partially ordered metric-like space.

Let F : X X be defined by
F 0 = 1 , F 1 = 2 and F 2 = 2 .

Define ψ , ϕ : [ 0 , ) [ 0 , ) by ψ ( t ) = t and ϕ ( t ) = t 2 . We next verify that the function F satisfies the inequality (2.1). For that, given x , y X with x y , so y x . Then we have the following cases.

Case 1. If x = 1 , y = 0 , then
σ ( F 1 , F 0 ) = σ ( 2 , 1 ) = 5
and
M ( 1 , 0 ) = max { σ ( 1 , 0 ) , σ ( 1 , F 1 ) , σ ( 0 , F 0 ) , σ ( 1 , 1 ) , σ ( 0 , 0 ) , [ σ ( 1 , F 0 ) + σ ( F 1 , 0 ) ] / 2 } = max { 9 , 5 , 3 , 1 , 3 + 4 2 } = 9 .

As ψ ( σ ( F 1 , F 0 ) ) = 5 < 9 9 2 = ψ ( M ( 1 , 0 ) ) ϕ ( M ( 1 , 0 ) ) , the inequality (2.1) is satisfied in this case.

Case 2. If x = 2 , y = 0 , then
σ ( F 2 , F 0 ) = σ ( 2 , 1 ) = 5
and
M ( 2 , 0 ) = max { σ ( 2 , 0 ) , σ ( 2 , F 2 ) , σ ( 0 , F 0 ) , σ ( 2 , 2 ) , σ ( 0 , 0 ) , [ σ ( 2 , F 0 ) + σ ( F 2 , 0 ) ] / 2 } = max { 4 , 0 , 9 , 1 , 5 + 4 2 } = 9 .

As ψ ( σ ( F 2 , F 0 ) ) = 5 < 9 9 2 = ψ ( M ( 2 , 0 ) ) ϕ ( M ( 2 , 0 ) ) , the inequality (2.1) is satisfied in this case.

Case 3. If x = 2 , y = 1 , then as σ ( F 2 , F 1 ) = 0 and M ( 2 , 1 ) = 5 , the inequality (2.1) is satisfied in this case.

Case 4. If x = 0 , y = 0 , then as σ ( F 0 , F 0 ) = 3 and M ( 0 , 0 ) = 9 , the inequality (2.1) is satisfied in this case.

Case 5. If x = 1 , y = 1 , then as σ ( F 1 , F 1 ) = 0 and M ( 1 , 1 ) = 5 , the inequality (2.1) is satisfied in this case.

Case 6. If x = 2 , y = 2 , then as σ ( F 2 , F 2 ) = 0 and M ( 2 , 2 ) = 0 , the inequality (2.1) is satisfied in this case.

So, F, ψ and ϕ satisfy all the hypotheses of Theorem 1. Therefore F has a unique fixed point. Here 2 is the unique fixed point of F.

If we take ψ ( t ) = t in Theorem 1, we have the following corollary.

Corollary 1 Let ( X , , σ ) be a complete partially ordered metric-like space. Let F : X X be a nondecreasing mapping such that for all comparable x , y X with
σ ( F x , F y ) M ( x , y ) ϕ ( M ( x , y ) ) ,
where M is given by
M ( x , y ) = max { σ ( x , y ) , σ ( x , F x ) , σ ( y , F y ) , σ ( x , x ) , σ ( y , y ) , [ σ ( x , F y ) + σ ( F x , y ) ] / 2 } ,
ϕ : [ 0 , ) [ 0 , ) is lower semi-continuous, and ϕ ( t ) = 0 if and only if t = 0 . If there exists x 0 X with x 0 F x 0 and one of the following two conditions is satisfied:
  1. (a)

    F is continuous in ( X , σ ) ;

     
  2. (b)

    { x n } is a nondecreasing sequence in X such that x n x X implies x n x for all n N .

     

Then F has a fixed point. Moreover, if the following condition is satisfied: For arbitrary two points x , y X , there exists z X which is comparable with both x and y, then the fixed point of F is unique.

If we take ϕ ( t ) = ( 1 k ) t for k [ 0 , 1 ) in Corollary 1, we have the following corollary.

Corollary 2 Let ( X , , σ ) be a complete partially ordered metric-like space. Let F : X X be a nondecreasing mapping such that for all comparable x , y X
σ ( F x , F y ) k M ( x , y ) ,
where M is given by
M ( x , y ) = max { σ ( x , y ) , σ ( x , F x ) , σ ( y , F y ) , σ ( x , x ) , σ ( y , y ) , [ σ ( x , F y ) + σ ( F x , y ) ] / 2 } ,
and k [ 0 , 1 ) . If there exists x 0 X with x 0 F x 0 and one of the following two conditions is satisfied:
  1. (a)

    F is continuous in ( X , σ ) ;

     
  2. (b)

    { x n } is a nondecreasing sequence in X such that x n x X implies x n x for all n N .

     

Then F has a fixed point. Moreover, if the following condition is satisfied: For arbitrary two points x , y X , there exists z X which is comparable with both x and y, then the fixed point of F is unique.

The following corollary improves Theorem 2.7 in [33].

Corollary 3 Let ( X , , σ ) be a complete partially ordered metric-like space. Let F : X X be a nondecreasing mapping such that for all comparable x , y X
σ ( F x , F y ) σ ( x , y ) ϕ ( σ ( x , y ) ) ,
where ϕ : [ 0 , ) [ 0 , ) is a lower semi-continuous, and ϕ ( t ) = 0 if and only if t = 0 . If there exists x 0 X with x 0 F x 0 and one of the following two conditions is satisfied:
  1. (a)

    F is continuous in ( X , σ ) ;

     
  2. (b)

    { x n } is a nondecreasing sequence in X such that x n x X implies x n x for all n N .

     

Then F has a fixed point. Moreover, if the following condition is satisfied: For arbitrary two points x , y X , there exists z X which is comparable with both x and y, then the fixed point of F is unique.

The following corollary improves Theorem 2.1 in [10].

Corollary 4 Let ( X , , p ) be a complete partially ordered partial metric space. Let F : X X be a nondecreasing mapping such that for all comparable x , y X
ψ ( p ( F x , F y ) ) ψ ( M ( x , y ) ) ϕ ( M ( x , y ) ) ,
where M is given by
M ( x , y ) = max { p ( x , y ) , p ( x , F x ) , p ( y , F y ) , p ( x , x ) , p ( y , y ) , [ p ( x , F y ) + p ( F x , y ) ] / 2 } ,
ψ , ϕ : [ 0 , ) [ 0 , ) , ψ is continuous monotone nondecreasing, ϕ is lower semi-continuous, and ψ ( t ) = ϕ ( t ) = 0 if and only if t = 0 . If there exists x 0 X with x 0 F x 0 and one of the following two conditions is satisfied:
  1. (a)

    F is continuous in ( X , p ) ;

     
  2. (b)

    { x n } is a nondecreasing sequence in X such that x n x X implies x n x for all n N .

     

Then F has a fixed point. Moreover, the set of fixed points of F is well ordered if and only if F has one and only one fixed point.

3 Applications

Denote by Λ the set of functions α : [ 0 , + ) [ 0 , + ) satisfying the following hypotheses:
  1. (h1)

    α is a Lebesgue-integrable mapping on each compact subset of [ 0 , + ) ;

     
  2. (h2)
    For every ε > 0 , we have
    0 ε α ( s ) d s > 0 .
     

We have the following results.

Corollary 5 Let ( X , , σ ) be a complete partially ordered metric-like space. Let F : X X be a continuous and nondecreasing mapping such that for all comparable x , y X
0 σ ( F x , F y ) α 1 ( s ) d s 0 M ( x , y ) α 1 ( s ) d s 0 M ( x , y ) α 2 ( s ) d s ,

where α 1 , α 2 Λ . If there exists x 0 X with x 0 F x 0 , then F has a fixed point.

Proof Follows from Theorem 1 by taking ψ ( t ) = 0 t α 1 ( s ) d s and ϕ ( t ) = 0 t α 2 ( s ) d s . □

Corollary 6 Under the same hypotheses of Corollary  5 and without assuming the continuity of F, assume that whenever { x n } is a nondecreasing sequence in X such that x n x X implies x n x for all n N , then F has a fixed point in X.

Proof Follows from Theorem 2 by taking ψ ( t ) = 0 t α 1 ( s ) d s and ϕ ( t ) = 0 t α 2 ( s ) d s . □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, University of Gazi
(2)
Department of Mathematics, Faculty of Science and Arts, University of Amasya

References

  1. Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology and Applications 1994, 183–197.Google Scholar
  2. O’Neill SJ: Partial metrics, valuations and domain theory. Ann. New York Acad. Sci. 806. Proc. 11th Summer Conference on General Topology and Applications 1996, 304–315.Google Scholar
  3. Valero O: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 2005, 6: 229–240.MATHMathSciNetView ArticleGoogle Scholar
  4. Oltra S, Valero O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 2004, 36: 17–26.MATHMathSciNetGoogle Scholar
  5. Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157: 2778–2785. 10.1016/j.topol.2010.08.017MATHMathSciNetView ArticleGoogle Scholar
  6. Romaguera S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 493298Google Scholar
  7. Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1: 1–8.MATHMathSciNetGoogle Scholar
  8. Abdeljawad T, Karapnar E, Taş K: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 2011, 24: 1900–1904. 10.1016/j.aml.2011.05.014MATHMathSciNetView ArticleGoogle Scholar
  9. Abdeljawad T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 2011, 54(11–12):2923–2927. 10.1016/j.mcm.2011.07.013MATHMathSciNetView ArticleGoogle Scholar
  10. Abbas M, Nazir T: Fixed point of generalized weakly contractive mappings in ordered partial metric spaces. Fixed Point Theory Appl. 2012, 2012: 1. 10.1186/1687-1812-2012-1View ArticleGoogle Scholar
  11. Karapınar E, Shatanawi W:On weakly ( C , ψ , φ ) -contractive mappings in ordered partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 495892Google Scholar
  12. Erhan İM, Karapinar E, Türkoğlu D: Different types Meir-Keeler contractions on partial metric spaces. J. Comput. Anal. Appl. 2012, 14(6):1000–1005.MATHMathSciNetGoogle Scholar
  13. Aydi H:Common fixed point results for mappings satisfying ( ψ , φ ) -weak contractions in ordered partial metric space. Int. J. Math. Stat. 2012, 12(2):38–52.MathSciNetGoogle Scholar
  14. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.MATHMathSciNetGoogle Scholar
  15. Romaguera S, Valero O: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 2009, 19(3):541–563. 10.1017/S0960129509007671MATHMathSciNetView ArticleGoogle Scholar
  16. Aydi H, Vetro C, Sintunavarat W, Kumam P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 124Google Scholar
  17. Nashine HK, Samet B:Fixed point results for mappings satisfying ( ψ , φ ) -weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 2011, 74(6):2201–2209. 10.1016/j.na.2010.11.024MATHMathSciNetView ArticleGoogle Scholar
  18. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151MATHMathSciNetView ArticleGoogle Scholar
  19. Burgic D, Kalabusic S, Kulenovic MRS: Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 762478Google Scholar
  20. Ciric L, Cakid N, Rajovic M, Uma JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294Google Scholar
  21. Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017MATHMathSciNetView ArticleGoogle Scholar
  22. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240MATHMathSciNetView ArticleGoogle Scholar
  23. Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003MATHMathSciNetView ArticleGoogle Scholar
  24. Lakshmikantham V, Ciric L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020MATHMathSciNetView ArticleGoogle Scholar
  25. Nieto JJ, Rodríguez-López R: Existence of extremal solutions for quadratic fuzzy equations. Fixed Point Theory Appl. 2005, 2005: 321–342.MATHView ArticleGoogle Scholar
  26. Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5MATHMathSciNetView ArticleGoogle Scholar
  27. Nieto JJ, Rodríguez-López R: Applications of contractive like mapping principles to fuzzy equations. Rev. Mat. Complut. 2006, 19: 361–383.MATHMathSciNetView ArticleGoogle Scholar
  28. O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026MATHMathSciNetView ArticleGoogle Scholar
  29. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4MATHMathSciNetView ArticleGoogle Scholar
  30. Wu Y: New fixed point theorems and applications of mixed monotone operator. J. Math. Anal. Appl. 2008, 341: 883–893. 10.1016/j.jmaa.2007.10.063MATHMathSciNetView ArticleGoogle Scholar
  31. Amini-Harandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023MATHMathSciNetView ArticleGoogle Scholar
  32. Shatanawi W, Al-Rawashdeh A:Common fixed points of almost generalized ( ψ , φ ) -contractive mappings in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 80Google Scholar
  33. Amini-Harandi A: Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory Appl. 2012., 2012: Article ID 204Google Scholar
  34. Alber YI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. Advances and Appl. 98. In New Results in Theory Operator Theory. Edited by: Gohberg I, Lyubich Y. Birkhäuser, Basel; 1997:7–22.Google Scholar
  35. Rhoades BH: Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1MATHMathSciNetView ArticleGoogle Scholar
  36. Dutta PN, Choudhury BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 406368Google Scholar
  37. Popescu O:Fixed points for ( ψ , ϕ ) -weak contractions. Appl. Math. Lett. 2011, 24(1):1–4. 10.1016/j.aml.2010.06.024MATHMathSciNetView ArticleGoogle Scholar
  38. Zhang Q, Song Y: Fixed point theory for generalized ϕ -weak contractions. Appl. Math. Lett. 2009, 22(1):75–78. 10.1016/j.aml.2008.02.007MATHMathSciNetView ArticleGoogle Scholar
  39. Doric D:Common fixed point for generalized ( ψ , φ ) -weak contractions. Appl. Math. Lett. 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001MATHMathSciNetView ArticleGoogle Scholar
  40. Rouhani BD, Moradi S: Common fixed point of multivalued generalized φ -weak contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 708984Google Scholar

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