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

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.

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 [3-21] 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 such that for all :

() ,

() ,

() ,

() .

A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X. It is clear that, if , then from () and (), . But if , may not be 0. The function for all defines a partial metric on .

Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and .

If p is a partial metric on X, then the function given by

is a metric on X.

Definition 2 Let be a partial metric space. Then

(1) A sequence in a partial metric space converges to a point if and only if .

(2) A sequence in a partial metric space is called a Cauchy sequence iff exists (and is finite).

(3) A partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .

(4) A subset A of a partial metric space is closed if whenever is a sequence in A such that converges to some , then .

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

Lemma 1 ([2,17])

Letbe a partial metric space.

(a) is a Cauchy sequence inif and only if it is a Cauchy sequence in the metric space.

(b) A partial metric spaceis complete if and only if the metric spaceis complete. Furthermore, if and only if

Now, we define the cyclic map.

Definition 3 Let A and B be nonempty subsets of a metric space and . Then T is called a cyclic map if and .

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

Theorem 1[22]

LetAandBbe nonempty closed subsets of a complete metric space. Suppose thatis a cyclic map such that

If, thenThas a unique fixed point in.

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 . A cyclic map is said to be a Kannan type cyclic contraction if there exists such that

Definition 5[23]

Let A and B be nonempty closed subsets of a metric space . A cyclic map is said to be a Reich type cyclic contraction if there exists such that

Definition 6[23]

Let A and B be nonempty closed subsets of a metric space . A cyclic map is said to be a Ćirić type cyclic contraction if there exists such that

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

Theorem 2[23]

LetAandBbe nonempty closed subsets of a complete metric space, and letbe a Kannan type cyclic contraction. ThenThas a unique fixed point in.

Theorem 3[23]

LetAandBbe nonempty closed subsets of a complete metric space, and letbe a Reich type cyclic contraction. ThenThas a unique fixed point in.

Theorem 4[23]

LetAandBbe nonempty closed subsets of a complete metric space, and letbe a Ćirić type cyclic contraction. ThenThas a unique fixed point in.

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 is called an altering distance function if the following properties are satisfied:

(1) ϕ is continuous and nondecreasing.

(2) if and only if .

For some work on altering distance function, we refer the reader to [28-33].

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.

We start with the following definition.

Definition 8 Let be a partial metric space and A, B be nonempty closed subsets of X. A mapping is called a cyclic -contraction if

(1) ψ and ϕ are altering distance functions;

(2) has a cyclic representation w.r.t. T; that is, and ; and

(3)

for all and .

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 5LetAandBbe nonempty closed subsets of a complete partial metric space. Ifis a cyclic-contraction, thenThas a unique fixed point.

Proof Let . Since , we choose such that . Also, since , we choose such that . Continuing this process, we can construct sequences in X such that , , and . If for some , then . Thus, is a fixed point of T in . Thus, we may assume that for all .

Given . If n is even, then for some . By (2.1), we have

By (), we have

If

then

Therefore, , and hence . By () and (), we have , which is a contradiction. Therefore,

Hence,

and

If n is odd, then for some . By (2.1), we have

By (), we have

If

then

Therefore, , and hence . By () and (), we have , which is a contradiction. Therefore,

Hence,

From (2.2) and (2.4), we have is a nonincreasing sequence and hence there exists such that

Also, from (2.3) and (2.5), we have

Letting in (2.6) and using the fact that ψ and ϕ are continuous, we get that

Therefore, and hence . Thus

By (), we get that

Since for all , we get that

Next, we show that is a Cauchy sequence in the metric space . It is sufficient to show that is a Cauchy sequence in . Suppose the contrary; that is, is not a Cauchy sequence in . Then there exists for which we can find two subsequences and of such that is the smallest index for which

This means that

From (2.10), (2.11) and the triangular inequality, we get that

On letting in the above inequalities and using (2.9), we have

Again, from (2.10) and the triangular inequality, we get that

Letting in the above inequalities and using (2.9) and (2.12), we get that

Since

for all , then

By (2.1), we have

Letting and using the continuity of ϕ and ψ, we get that

Therefore, we get that . Hence, is a contradiction. Thus is a Cauchy sequence in . Since is complete and is a closed subspace of , then we have is complete. From Lemma 1, the sequence converges in the metric space , say . Again from Lemma 1, we have

Moreover, since is a Cauchy sequence in the metric space , we have

From the definition of we have

Letting in the above equality and using (2.8) and (2.14), we get

Thus by (2.13), we have

Since , is a sequence in A, and A is closed in , we have . Similarly, we have , that is . Again, from the definition of p, we have

Letting in the above inequalities and using (2.9) and (2.15), we get that

Now, we claim that .

Since and , by (2.1) we have

Letting , we get that

Therefore, . Since ϕ is an altering distance function, , that is, .

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 . It is an easy matter to prove that . Now, we prove that . Since and , we have

Thus and hence . Therefore, . □

Taking (the identity function) in Theorem 5, we have the following result.

Corollary 1LetAandBbe nonempty closed subsets of a complete partial metric space. Letbe a mapping such thathas a cyclic representation w.r.t. T. Suppose there exists an altering distance functionϕsuch that

for alland. ThenThas a unique fixed point.

Corollary 2LetAandBbe nonempty closed subsets of a complete partial metric space. Letbe a mapping such thathas a cyclic representation w.r.t. T. Suppose there exists an altering distance functionϕsuch that

for alland. ThenThas a unique fixed point.

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

Example 1 Let . Define the partial metric p on X by

Also, define the mapping by and the functions by and . Take and . Then

(1) is a complete partial metric space.

(2) has a cyclic representation w.r.t. T.

(3) For all and , we have

Proof Note that and . Thus has a cyclic representation of T. To prove (3), given and , without loss of generality, we may assume that . So,

and

Since

we have

 □

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

Denote by Λ the set of functions satisfying the following hypotheses:

(h1) μ is a Lebesgue-integrable mapping on each compact of .

(h2) For every , we have

Theorem 6LetAandBbe nonempty closed subsets of a complete partial metric space. Letbe a mapping such thathas a cyclic representation w.r.t. T. Suppose that forand, we have

where. ThenThas a unique fixed point.

Proof Follows from Theorem 5 by defining via and 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.

The authors declare that they have no competing interests.

All authors read and approved the manuscript.

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

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