Open Access

Fixed point results for cyclic ( ψ , ϕ , A , B ) -contraction in partial metric spaces

Fixed Point Theory and Applications20122012:165

https://doi.org/10.1186/1687-1812-2012-165

Received: 2 July 2012

Accepted: 31 August 2012

Published: 28 September 2012

Abstract

Very recently, Agarwal et al. (Fixed Point Theory Appl. 2012:40, 2012) initiated the study of fixed point theorems for mappings satisfying cyclical generalized contractive conditions in complete partial metric spaces. In the present paper, we study some fixed point theorems for a mapping satisfying a cyclical generalized contractive condition based on a pair of altering distance functions in complete partial metric spaces. Also, we introduce an example and an application to support the usability of our paper.

MSC:54H25, 47H10.

Keywords

partial metric spaces fixed point altering distance function cyclic map

1 Introduction

The existence and uniqueness of fixed and common fixed point theorems of operators has been a subject of great interest since Banach [1] proved the Banach contraction principle in 1922. Many authors generalized the Banach contraction principle in various spaces such as quasi-metric spaces, generalized metric spaces, cone metric spaces and fuzzy metric spaces. Matthews [2] introduced the notion of partial metric spaces in such a way that each object does not necessarily have to have a zero distance from itself and proved a modified version of the Banach contraction principle. Afterwards, many authors proved many existing fixed point theorems in partial metric spaces (see [321] for examples).

We recall below the definition of partial metric space and some of its properties.

Definition 1 [2]

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

( p 1 ) x = y p ( x , x ) = p ( x , y ) = p ( y , y ) ,

( p 2 ) p ( x , x ) p ( x , y ) ,

( p 3 ) p ( x , y ) = p ( y , x ) ,

( p 4 ) p ( x , y ) p ( x , z ) + p ( z , y ) p ( z , z ) .

A partial metric space is a pair ( X , p ) such that X is a nonempty set and p is a partial metric on X. It is clear that, if p ( x , y ) = 0 , then from ( p 1 ) and ( p 2 ), x = y . But if x = y , p ( x , y ) may not be 0. The function p ( x , y ) = max { x , y } for all x , y R + defines a partial metric on  R + .

Each partial metric 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 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 d p : X × X R + given by
d p ( x , y ) = 2 p ( x , y ) p ( x , x ) p ( y , y )

is a metric on X.

Definition 2 Let ( X , p ) be a partial metric space. Then
  1. (1)

    A sequence { x n } in a partial metric space ( X , p ) converges to a point x X if and only if p ( x , x ) = lim n p ( x , x n ) .

     
  2. (2)

    A sequence { x n } in a partial metric space ( X , p ) is called a Cauchy sequence iff lim n , m p ( x n , x m ) exists (and is finite).

     
  3. (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 ) .

     
  4. (4)

    A subset A of a partial metric space ( X , p ) is closed if whenever { x n } is a sequence in A such that { x n } converges to some x X , then x A .

     

Remark 1 The limit in a partial metric space is not unique.

Lemma 1 ([2, 17])

Let ( X , p ) be a partial metric space.
  1. (a)

    { x n } is a Cauchy sequence in ( X , p ) if and only if it is a Cauchy sequence in the metric space ( X , d p ) .

     
  2. (b)
    A partial metric space ( X , p ) is complete if and only if the metric space ( X , d p ) is complete. Furthermore, lim n d p ( 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 ) .
     

Now, we define the cyclic map.

Definition 3 Let A and B be nonempty subsets of a metric space ( X , d ) and T : A B A B . Then T is called a cyclic map if T ( A ) B and T ( B ) A .

In 2003, Kirk et al. [22] gave the following fixed point theorem for a cyclic map.

Theorem 1 [22]

Let A and B be nonempty closed subsets of a complete metric space ( X , d ) . Suppose that T : A B A B is a cyclic map such that
d ( T x , T y ) k d ( x , y ) x A , y B .

If k [ 0 , 1 ) , then T has a unique fixed point in A B .

Karapınar and Erhan [23] introduced the following types of cyclic contractions:

Definition 4 [23]

Let A and B be nonempty closed subsets of a metric space ( X , d ) . A cyclic map T : A B A B is said to be a Kannan type cyclic contraction if there exists k ( 0 , 1 2 ) such that
d ( T x , T y ) k ( d ( T x , x ) + d ( T y , y ) ) x A , y B .

Definition 5 [23]

Let A and B be nonempty closed subsets of a metric space ( X , d ) . A cyclic map T : A B A B is said to be a Reich type cyclic contraction if there exists k ( 0 , 1 3 ) such that
d ( T x , T y ) k ( d ( x , y ) + d ( T x , x ) + d ( T y , y ) ) x A , y B .

Definition 6 [23]

Let A and B be nonempty closed subsets of a metric space ( X , d ) . A cyclic map T : A B A B is said to be a Ćirić type cyclic contraction if there exists k ( 0 , 1 3 ) such that
d ( T x , T y ) k max { d ( x , y ) , d ( T x , x ) , d ( T y , y ) } x A , y B .

Moreover, Karapınar and Erhan [23] obtained the following results:

Theorem 2 [23]

Let A and B be nonempty closed subsets of a complete metric space ( X , d ) , and let T : A B A B be a Kannan type cyclic contraction. Then T has a unique fixed point in A B .

Theorem 3 [23]

Let A and B be nonempty closed subsets of a complete metric space ( X , d ) , and let T : A B A B be a Reich type cyclic contraction. Then T has a unique fixed point in A B .

Theorem 4 [23]

Let A and B be nonempty closed subsets of a complete metric space ( X , d ) , and let T : A B A B be a Ćirić type cyclic contraction. Then T has a unique fixed point in A B .

For more results on cyclic contraction mappings, see [24, 25].

Very recently, Agarwal et al. [26] initiated the study of fixed point theorems for mappings satisfying cyclical generalized contractive conditions in complete partial metric spaces.

Khan et al. [27] introduced the notion of altering distance function as follows.

Definition 7 (Altering distance function [27])

The function ϕ : [ 0 , + ) [ 0 , + ) is called an altering distance function if the following properties are satisfied:
  1. (1)

    ϕ is continuous and nondecreasing.

     
  2. (2)

    ϕ ( t ) = 0 if and only if t = 0 .

     

For some work on altering distance function, we refer the reader to [2833].

The purpose of this paper is to study some fixed point theorems for a mapping satisfying a cyclical generalized contractive condition based on a pair of altering distance functions in partial metric spaces.

2 Main result

We start with the following definition.

Definition 8 Let ( X , p ) be a partial metric space and A, B be nonempty closed subsets of X. A mapping T : X X is called a cyclic ( ψ , ϕ , A , B ) -contraction if
  1. (1)

    ψ and ϕ are altering distance functions;

     
  2. (2)
    A B has a cyclic representation w.r.t. T; that is, T ( A ) B and T ( B ) A ; and(3)
    ψ ( p ( T x , T y ) ) ψ ( max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 ( p ( x , T y ) + p ( T x , y ) ) } ) ϕ ( max { p ( x , y ) , p ( y , T y ) } )
    (2.1)
     

for all x A and y B .

From now on, by ψ and ϕ we mean altering distance functions unless otherwise stated.

In the rest of this paper, N stands for the set of nonnegative integer numbers.

Theorem 5 Let A and B be nonempty closed subsets of a complete partial metric space ( X , p ) . If T : X X is a cyclic ( ψ , ϕ , A , B ) -contraction, then T has a unique fixed point u A B .

Proof Let x 0 A . Since T A B , we choose x 1 B such that T x 0 = x 1 . Also, since T B A , we choose x 2 A such that T x 1 = x 2 . Continuing this process, we can construct sequences { x n } in X such that x 2 n A , x 2 n + 1 B , x 2 n + 1 = T x 2 n and x 2 n + 2 = T x 2 n + 1 . If x 2 n 0 + 1 = x 2 n 0 + 2 for some n N , then x 2 n 0 + 1 = T x 2 n 0 + 1 . Thus, x 2 n 0 + 1 is a fixed point of T in A B . Thus, we may assume that x 2 n + 1 x 2 n + 2 for all n N .

Given n N . If n is even, then n = 2 t for some t N . By (2.1), we have
ψ ( p ( x n + 1 , x n + 2 ) ) = ψ ( p ( x 2 t + 1 , x 2 t + 2 ) ) = ψ ( p ( T x 2 t , T x 2 t + 1 ) ) ψ ( max { p ( x 2 t , x 2 t + 1 ) , p ( T x 2 t , x 2 t ) , p ( T x 2 t + 1 , x 2 t + 1 ) , 1 2 ( p ( x 2 t , T x 2 t + 1 ) + p ( T x 2 t , x 2 t + 1 ) ) } ) ϕ ( max { p ( x 2 t , x 2 t + 1 ) , p ( T x 2 t + 1 , x 2 t + 1 ) } ) = ψ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) , 1 2 ( p ( x 2 t , x 2 t + 2 ) + p ( x 2 t + 1 , x 2 t + 1 ) ) } ) ϕ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) .
By ( p 4 ), we have
ψ ( p ( x n + 1 , x n + 2 ) ) = ψ ( p ( x 2 t + 1 , x 2 t + 2 ) ) ψ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) , 1 2 ( p ( x 2 t , x 2 t + 1 ) + p ( x 2 t + 1 , x 2 t + 2 ) ) } ) ϕ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) ψ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) ϕ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) ψ ( max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) .
If
max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } = p ( x 2 t + 2 , x 2 t + 1 ) ,
then
ψ ( p ( x 2 t + 1 , x 2 t + 2 ) ) ψ ( p ( x 2 t + 2 , x 2 t + 1 ) ) ϕ ( p ( x 2 t + 2 , x 2 t + 1 ) ) .
Therefore, ϕ ( p ( x 2 t + 1 , x 2 t + 2 ) ) = 0 , and hence p ( x 2 t + 1 , x 2 t + 2 ) = 0 . By ( p 1 ) and ( p 2 ), we have x 2 t + 1 = x 2 t + 2 , which is a contradiction. Therefore,
max { p ( x 2 t , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } = p ( x 2 t , x 2 t + 1 ) .
Hence,
p ( x n + 1 , x n + 2 ) = p ( x 2 t + 2 , x 2 t + 1 ) p ( x 2 t , x 2 t + 1 ) = p ( x n , x n + 1 )
(2.2)
and
ψ ( p ( x n + 1 , x n + 2 ) ) ψ ( p ( x n , x n + 1 ) ) ϕ ( p ( x n , x n + 1 ) ) .
(2.3)
If n is odd, then n = 2 t + 1 for some t N . By (2.1), we have
ψ ( p ( x n + 1 , x n + 2 ) ) = ψ ( p ( x 2 t + 2 , x 2 t + 3 ) ) = ψ ( p ( x 2 t + 3 , x 2 t + 2 ) ) = ψ ( p ( T x 2 t + 2 , T x 2 t + 1 ) ) ψ ( max { p ( x 2 t + 2 , x 2 t + 1 ) , p ( T x 2 t + 2 , x 2 t + 2 ) , p ( T x 2 t + 1 , x 2 t + 1 ) , 1 2 ( p ( x 2 t + 2 , T x 2 t + 1 ) + p ( T x 2 t + 2 , x 2 t + 1 ) ) } ) ϕ ( max { p ( x 2 t + 2 , x 2 t + 1 ) , p ( T x 2 t + 1 , x 2 t + 1 ) } ) = ψ ( max { p ( x 2 t + 3 , x 2 t + 2 ) , p ( x 2 t + 2 , x 2 t + 1 ) , 1 2 ( p ( x 2 t + 2 , x 2 t + 2 ) + p ( x 2 t + 3 , x 2 t + 1 ) ) } ) ϕ ( max { p ( x 2 t + 2 , x 2 t + 1 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) .
By ( p 4 ), we have
ψ ( p ( x n + 1 , x n + 2 ) ) = ψ ( p ( x 2 t + 3 , x 2 t + 2 ) ) ψ ( max { p ( x 2 t + 3 , x 2 t + 2 ) , p ( x 2 t + 2 , x 2 t + 1 ) , 1 2 ( p ( x 2 t + 3 , x 2 t + 2 ) + p ( x 2 t + 2 , x 2 t + 1 ) ) } ) ϕ ( max { p ( x 2 t + 2 , x 2 t + 1 ) , p ( T x 2 t + 1 , x 2 t + 1 ) } ) ψ ( max { p ( x 2 t + 3 , x 2 t + 2 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) ϕ ( max { p ( x 2 t + 2 , x 2 t + 1 ) , p ( T x 2 t + 1 , x 2 t + 1 ) } ) ψ ( max { p ( x 2 t + 3 , x 2 t + 2 ) , p ( x 2 t + 2 , x 2 t + 1 ) } ) .
If
max { p ( x 2 t + 3 , x 2 t + 2 ) , p ( x 2 t + 2 , x 2 t + 1 ) } = p ( x 2 t + 3 , x 2 t + 2 ) ,
then
ϕ ( p ( x 2 t + 3 , x 2 t + 2 ) ) ψ ( p ( x 2 t + 3 , x 2 t + 2 ) ) ϕ ( p ( x 2 t + 2 , x 2 t + 1 ) ) .
Therefore, ϕ ( p ( x 2 t + 2 , x 2 t + 1 ) ) = 0 , and hence p ( x 2 t + 3 , x 2 t + 2 ) = 0 . By ( p 1 ) and ( p 2 ), we have x 2 t + 2 = x 2 t + 1 , which is a contradiction. Therefore,
max { p ( x 2 t + 3 , x 2 t + 2 ) , p ( x 2 t + 2 , x 2 t + 1 ) } = p ( x 2 t + 2 , x 2 t + 1 ) .
Hence,
(2.4)
(2.5)
From (2.2) and (2.4), we have { p ( x n + 1 , x n ) : n N } is a nonincreasing sequence and hence there exists r 0 such that
lim n + p ( x n , x n + 1 ) = r .
Also, from (2.3) and (2.5), we have
ψ ( p ( x n + 2 , x n + 1 ) ) ψ ( p ( x n , x n + 1 ) ) ϕ ( p ( x n , x n + 1 ) ) n N .
(2.6)
Letting n + in (2.6) and using the fact that ψ and ϕ are continuous, we get that
ψ ( r ) ψ ( r ) ϕ ( r ) .
Therefore, ϕ ( r ) = 0 and hence r = 0 . Thus
lim n + p ( x n , x n + 1 ) = 0 .
(2.7)
By ( p 2 ), we get that
lim n + p ( x n , x n ) = 0 .
(2.8)
Since d p ( x , y ) 2 p ( x , y ) for all x , y X , we get that
lim n + d p ( x n , x n + 1 ) = 0 .
(2.9)
Next, we show that { x n } is a Cauchy sequence in the metric space ( A B , d p ) . It is sufficient to show that { x 2 n } is a Cauchy sequence in ( A B , d p ) . Suppose the contrary; that is, { x 2 n } is not a Cauchy sequence in ( A B , d p ) . Then there exists ϵ > 0 for which we can find two subsequences { x 2 m ( i ) } and { x 2 n ( i ) } of { x 2 n } such that n ( i ) is the smallest index for which
n ( i ) > m ( i ) > i , d p ( x 2 m ( i ) , x 2 n ( i ) ) ϵ .
(2.10)
This means that
d p ( x 2 m ( i ) , x 2 n ( i ) 2 ) < ϵ .
(2.11)
From (2.10), (2.11) and the triangular inequality, we get that
ϵ d p ( x 2 m ( i ) , x 2 n ( i ) ) d p ( x 2 m ( i ) , x 2 n ( i ) 2 ) + d p ( x 2 n ( i ) 2 , x 2 n ( i ) 1 ) + d p ( x 2 n ( i ) 1 , x 2 n ( i ) ) < ϵ + d p ( x 2 n ( i ) 2 , x 2 n ( i ) 1 ) + d p ( x 2 n ( i ) 1 , x 2 n ( i ) ) .
On letting i + in the above inequalities and using (2.9), we have
lim i + d p ( x 2 m ( i ) , x 2 n ( i ) ) = ϵ .
(2.12)
Again, from (2.10) and the triangular inequality, we get that
ϵ d p ( x 2 m ( i ) , x 2 n ( i ) ) d p ( x 2 n ( i ) , x 2 n ( i ) 1 ) + d p ( x 2 n ( i ) 1 , x 2 m ( i ) ) d p ( x 2 n ( i ) , x 2 n ( i ) 1 ) + d p ( x 2 n ( i ) 1 , x 2 m ( i ) + 1 ) + d p ( x 2 m ( i ) + 1 , x 2 m ( i ) ) d p ( x 2 n ( i ) , x 2 n ( i ) 1 ) + d p ( x 2 n ( i ) 1 , x 2 m ( i ) ) + 2 d p ( x 2 m ( i ) + 1 , x 2 m ( i ) ) 2 d p ( x 2 n ( i ) , x 2 n ( i ) 1 ) + d p ( x 2 n ( i ) , x 2 m ( i ) ) + 2 d p ( x 2 m ( i ) + 1 , x 2 m ( i ) ) .
Letting i + in the above inequalities and using (2.9) and (2.12), we get that
lim i + d p ( x 2 m ( i ) , x 2 n ( i ) ) = lim i + d p ( x 2 m ( i ) + 1 , x 2 n ( i ) 1 ) = lim i + d p ( x 2 m ( i ) + 1 , x 2 n ( i ) ) = lim i + d p ( x 2 m ( i ) , x 2 n ( i ) 1 ) = ϵ .
Since
d p ( x , y ) = 2 p ( x , y ) p ( x , x ) p ( y , y )
for all x , y X , then
lim i + p ( x 2 m ( i ) , x 2 n ( i ) ) = lim i + p ( x 2 m ( i ) + 1 , x 2 n ( i ) 1 ) = lim i + p ( x 2 m ( i ) + 1 , x 2 n ( i ) ) = lim i + p ( x 2 m ( i ) , x 2 n ( i ) 1 ) = ϵ 2 .
By (2.1), we have
ψ ( p ( x 2 m ( i ) + 1 , x 2 n ( i ) ) ) = ψ ( p ( T x 2 m ( i ) , T x 2 n ( i ) 1 ) ) ψ ( max { p ( x 2 m ( i ) , x 2 n ( i ) 1 ) , p ( x 2 m ( i ) , T x 2 m ( i ) ) , p ( x 2 n ( i ) 1 , T x 2 n ( i ) 1 ) , 1 2 ( p ( x 2 m ( i ) , T x 2 n ( i ) 1 ) + p ( x 2 n ( i ) 1 , T x 2 m ( i ) ) ) } ) ϕ ( max { p ( x 2 m ( i ) , x 2 n ( i ) 1 ) , p ( x 2 n ( i ) 1 , T x 2 n ( i ) 1 ) } ) = ψ ( max { p ( x 2 m ( i ) , x 2 n ( i ) 1 ) , p ( x 2 m ( i ) , x 2 m ( i ) + 1 ) , p ( x 2 n ( i ) 1 , x 2 n ( i ) ) , 1 2 ( p ( x 2 m ( i ) , x 2 n ( i ) ) + p ( x 2 n ( i ) 1 , x 2 m ( i ) + 1 ) ) } ) ϕ ( max { p ( x 2 m ( i ) , x 2 n ( i ) 1 ) , p ( x 2 n ( i ) 1 , x 2 n ( i ) ) } ) .
Letting i + and using the continuity of ϕ and ψ, we get that
ψ ( ϵ 2 ) ψ ( ϵ 2 ) ϕ ( ϵ 2 ) .
Therefore, we get that ϕ ( ϵ 2 ) = 0 . Hence, ϵ = 0 is a contradiction. Thus { x n } is a Cauchy sequence in ( A B , d p ) . Since ( X , p ) is complete and A B is a closed subspace of ( X , p ) , then we have ( A B , p ) is complete. From Lemma 1, the sequence { x n } converges in the metric space ( A B , d p ) , say lim n d p ( x n , u ) = 0 . Again from Lemma 1, we have
p ( u , u ) = lim n p ( x n , u ) = lim n , m p ( x n , x m ) .
(2.13)
Moreover, since { x n } is a Cauchy sequence in the metric space ( A B , d p ) , we have
lim n , m d p ( x n , x m ) = 0 .
(2.14)
From the definition of d p we have
d p ( x n , x m ) = 2 p ( x n , x m ) p ( x n , x n ) p ( x m , x m ) .
Letting n , m + in the above equality and using (2.8) and (2.14), we get
lim n , m p ( x n , x m ) = 0 .
Thus by (2.13), we have
lim n + p ( x n , u ) = p ( u , u ) = 0 .
(2.15)
Since p ( x 2 n , u ) 0 = p ( u , u ) , { x 2 n } is a sequence in A, and A is closed in ( X , p ) , we have u A . Similarly, we have u B , that is u A B . Again, from the definition of p, we have
p ( x n , T u ) p ( x n , u ) + p ( u , T u ) p ( u , u ) p ( x n , u ) + p ( u , x n ) + p ( x n , T u ) p ( x n , x n ) p ( u , u ) .
Letting n + in the above inequalities and using (2.9) and (2.15), we get that
lim n + p ( x n , T u ) = p ( u , T u ) .

Now, we claim that T u = u .

Since x 2 n A and u B , by (2.1) we have
ψ ( p ( x 2 n + 1 , T u ) ) = ψ ( p ( T x 2 n , T u ) ) ψ ( max { p ( x 2 n , u ) , p ( T x 2 n , x 2 n ) , p ( T u , u ) , 1 2 ( p ( x 2 n , T u ) + p ( u , T x 2 n ) ) } ) ϕ ( max { p ( x 2 n , u ) , p ( T u , u ) } ) = ψ ( max { p ( x 2 n , u ) , p ( x 2 n , x 2 n + 1 ) , p ( T u , u ) , 1 2 ( p ( x 2 n , T u ) + p ( u , x 2 n + 1 ) ) } ) ϕ ( max { p ( x 2 n , u ) , p ( u , T u ) } ) .
Letting n + , we get that
ψ ( p ( u , T u ) ) ψ ( p ( u , T u ) ) ϕ ( p ( u , T u ) ) .

Therefore, ϕ ( p ( u , T u ) ) = 0 . Since ϕ is an altering distance function, p ( u , T u ) = 0 , that is, u = T u .

Therefore, u is a fixed point of T. To prove the uniqueness of the fixed point, we let v be any other fixed point of T in A B . It is an easy matter to prove that p ( v , v ) = 0 . Now, we prove that u = v . Since u A B A and v A B B , we have
ψ ( p ( u , v ) ) = ψ ( p ( T u , T v ) ) ψ ( max { p ( u , v ) , p ( u , u ) , p ( v , v ) } ) ϕ ( max { p ( u , v ) , p ( v , v ) } ) = ψ ( p ( u , v ) ) ϕ ( p ( u , v ) ) .

Thus ϕ ( p ( u , v ) ) = 0 and hence p ( u , v ) = 0 . Therefore, u = v . □

Taking ψ = I [ 0 , + ) (the identity function) in Theorem 5, we have the following result.

Corollary 1 Let A and B be nonempty closed subsets of a complete partial metric space ( X , p ) . Let T : X X be a mapping such that A B has a cyclic representation w.r.t. T. Suppose there exists an altering distance function ϕ such that
p ( T x , T y ) max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 ( p ( x , T y ) + p ( T x , y ) ) } ϕ ( max { p ( x , y ) , p ( y , T y ) } )

for all x A and y B . Then T has a unique fixed point u A B .

Corollary 2 Let A and B be nonempty closed subsets of a complete partial metric space ( X , p ) . Let T : X X be a mapping such that A B has a cyclic representation w.r.t. T. Suppose there exists an altering distance function ϕ such that
p ( T x , T y ) max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) } ϕ ( max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) } )

for all x A and y B . Then T has a unique fixed point u A B .

Now, we introduce an example to support the usability of our results.

Example 1 Let X = [ 0 , 1 ] . Define the partial metric p on X by
p ( x , y ) = { 0 , if  x = y ; max { x , y } , if  x y .
Also, define the mapping T : X X by T ( x ) = x 2 1 + x and the functions ψ , ϕ : [ 0 , + ) [ 0 , + ) by ψ ( t ) = 2 t and ϕ ( t ) = t 1 + 2 t . Take A = [ 0 , 1 2 ] and B = [ 0 , 1 ] . Then
  1. (1)

    ( X , p ) is a complete partial metric space.

     
  2. (2)

    A B has a cyclic representation w.r.t. T.

     
  3. (3)
    For all x A and y B , we have
    ψ ( p ( T x , T y ) ) ψ ( max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 ( p ( x , T y ) + p ( T x , y ) ) } ) ϕ ( max { p ( x , y ) , p ( y , T y ) } ) .
     
Proof Note that T A = [ 0 , 1 6 ] B and T B = [ 0 , 1 2 ] A . Thus A B has a cyclic representation of T. To prove (3), given x A and y B , without loss of generality, we may assume that x y . So,
ψ ( p ( T x , T y ) ) = ψ ( p ( x 2 1 + x , y 2 1 + y ) ) = ψ ( y 2 1 + y ) = 2 y 2 1 + y , ψ ( max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 ( P ( x , T y ) + p ( T x , y ) ) } ) = ψ ( max { y , p ( x , x 2 1 + x ) , p ( y , y 2 1 + y ) , 1 2 ( p ( x , y 2 1 + y ) + p ( x 2 1 + x , y ) ) } ) ψ ( y ) = 2 y ,
and
ϕ ( max { p ( x , y ) , p ( y , T y ) } ) = ϕ ( max { y , p ( y , y 2 1 + y ) } ) = ϕ ( y ) = y 1 + 2 y .
Since
2 y 2 1 + y 2 y y 1 + 2 y ,
we have
ψ ( p ( T x , T y ) ) ψ ( max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 ( p ( x , T y ) + p ( T x , y ) ) } ) ϕ ( max { p ( x , y ) , p ( y , T y ) } ) .

 □

Note that Example 1 satisfies all the hypotheses of Theorem 5.

3 Application

Denote by Λ the set of functions μ : [ 0 , + ) [ 0 , + ) satisfying the following hypotheses:

(h1) μ is a Lebesgue-integrable mapping on each compact of [ 0 , + ) .

(h2) For every ϵ > 0 , we have
0 ϵ μ ( t ) d t > 0 .
Theorem 6 Let A and B be nonempty closed subsets of a complete partial metric space ( X , p ) . Let T : X X be a mapping such that A B has a cyclic representation w.r.t. T. Suppose that for x A and y B , we have
0 p ( T x , T y ) μ 1 ( t ) d t 0 max { p ( x , y ) , p ( x , T x ) , p ( y , T y ) , 1 2 ( p ( x , T y ) + p ( T x , y ) ) } μ 1 ( t ) d t 0 max { p ( x , y ) , p ( y , T y ) } μ 2 ( t ) d t ,

where μ 1 , μ 2 Λ . Then T has a unique fixed point u A B .

Proof Follows from Theorem 5 by defining ψ , ϕ : [ 0 , + ) [ 0 , + ) via ψ ( t ) = 0 t μ 1 ( s ) d s and ϕ ( t ) = 0 t μ 2 ( s ) d s and noting that ψ, ϕ are altering distance functions. □

Remark 2 Theorem 2.1 of [23] is a special case of Corollary 2.

Remark 3 Theorem 2.3 of [23] is a special case of Corollary 2.

Remark 4 Theorem 2.4 of [23] is a special case of Corollary 2.

Remark 5 Theorem 1.1 of [22] is a special case of Corollary 2.

Declarations

Acknowledgements

The authors thank the Editor and the referees for their useful comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Hashemite University
(2)
School of Mathematics and Computer Applications, Thapar University

References

  1. Banach S: Sur les operations dans les ensembles et leur application aux equation sitegrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
  2. Matthews SG: Partial metric topology. Ann. N.Y. Acad. Sci. 1994, 728: 183–197. Proc. 8th Summer Conference on General Topology and Applications 10.1111/j.1749-6632.1994.tb44144.xMathSciNetView ArticleGoogle Scholar
  3. Abdeljawad T, Karapinar 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.2011.05.014MathSciNetView ArticleGoogle Scholar
  4. Abdeljawad T, Karapinar E, Taş K: A generalized contraction principle with control functions on partial metric spaces. Comput. Math. Appl. 2012, 6: 716–719.View ArticleGoogle Scholar
  5. Abdeljawad T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 2011, 54: 2923–2927. 10.1016/j.mcm.2011.07.013MathSciNetView ArticleGoogle Scholar
  6. Altun I, Erduran A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 508730Google Scholar
  7. Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157: 2778–2785. 10.1016/j.topol.2010.08.017MathSciNetView ArticleGoogle Scholar
  8. Aydi H: Some fixed point results in ordered partial metric spaces. J. Nonlinear Sci. Appl 2011, 4: 1–12.MathSciNetGoogle Scholar
  9. Aydi H: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 647091Google Scholar
  10. Aydi H: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. J. Nonlinear Anal. Optim. 2011, 2: 33–48.MathSciNetGoogle Scholar
  11. Aydi H, Karapinar E, Shatanawi W:Coupled fixed point results for ( ψ , φ ) -weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 2011, 62: 4449–4460. 10.1016/j.camwa.2011.10.021MathSciNetView ArticleGoogle Scholar
  12. Golubović Z, Kadelburg Z, Radenović S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 192581Google Scholar
  13. Heckmann R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 1999, 7: 71–83. 10.1023/A:1008684018933MathSciNetView ArticleGoogle Scholar
  14. Karapinar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. (2011, in press)Google Scholar
  15. Karapinar E, Erhan IM: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 2011. doi:10.1016/j.aml.2011.05.013Google Scholar
  16. Nashine, HK, Kadelburg, Z, Radenović, S: Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces. Math. Comput. Model. (2012, in press)Google Scholar
  17. Oltra S, Valero O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 2004, 36: 17–26.MathSciNetGoogle Scholar
  18. Romaguera S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 493298Google Scholar
  19. Samet B, Rajović M, Lazović R, Stoiljković R: Common fixed point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 71Google Scholar
  20. Shatanawi W, Nashine HK: A generalization of Banach’s contraction principle for nonlinear contraction in a partial metric space. J. Nonlinear Sci. Appl 2012, 5: 37–43.MathSciNetGoogle Scholar
  21. Valero O: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 2005, 6: 229–240.MathSciNetView ArticleGoogle Scholar
  22. Kirk WA, Srinavasan PS, Veeramani P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4: 79–89.MathSciNetGoogle Scholar
  23. Karapinar E, Erhan IM: Best proximity point on different type contractions. Appl. Math. Inf. Sci. 2011, 5: 342–353.MathSciNetGoogle Scholar
  24. Karapinar E, Erhan IM, Ulus AY: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 2012, 6: 239–244.MathSciNetGoogle Scholar
  25. Karapinar E, Erhan IM: Cyclic contractions and fixed point theorems. Filomat 2012, 26: 777–782.MathSciNetView ArticleGoogle Scholar
  26. Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40Google Scholar
  27. Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 1–9. 10.1017/S0004972700001659MathSciNetView ArticleGoogle Scholar
  28. Cho YJ, Rhoades BE, Saadati R, Samet B, Shatanawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8. doi:10.1186/1687–1812–2012–8Google Scholar
  29. Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for ( ψ , ϕ ) -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012, 63: 298–309. 10.1016/j.camwa.2011.11.022MathSciNetView ArticleGoogle Scholar
  30. Lakzian H, Samet B:Fixed points for ( ψ , ϕ ) -weakly contractive mappings in generalized metric spaces. Appl. Math. Lett. 2012, 25: 902–906. 10.1016/j.aml.2011.10.047MathSciNetView ArticleGoogle Scholar
  31. Shatanawi, W, Al-Rawashdeh, A: Common fixed points of almost generalized ( ψ , ϕ ) -contractive mappings in ordered metric spaces. Fixed Point Theory Appl. (accepted)Google Scholar
  32. Shatanawi W, Mustafa Z, Tahat N: Some coincidence point theorems for nonlinear contraction in ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 68Google Scholar
  33. Shatanawi W, Samet B:On ( ψ , ϕ ) -weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. 2011, 62: 3204–3214. 10.1016/j.camwa.2011.08.033MathSciNetView ArticleGoogle Scholar

Copyright

© Shatanawi and Manro; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.