Some new common fixed point results through generalized altering distances on partial metric spaces

We establish common fixed point results for two pairs of weakly compatible mappings on a partial metric space, satisfying a weak contractive condition involving generalized control functions. The presented theorems extend and unify various known fixed point results. Examples are given to show that our results are proper extensions of the known ones.

MSC: 47H10, 54H25, 54H10.

In [1], Matthews introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification.

Subsequently, several authors (see, e.g., Altun and Erduran [2], Oltra et al.[3], Romaguera and Schellekens [4], Romaguera and Valero [5], Rus [6], Djukić et al.[7], Nashine et al.[8], Di Bari and Vetro [9], Paesano and Vetro [10], Shatanawi et al.[11], Shatanawi and Nashine [12], Aydi et al.[13]) derived fixed point theorems in partial metric spaces.

Altering distance functions (also called control functions) were introduced by Khan et al.[14]. Subsequently, they were used by many authors to obtain fixed point results, including those in partial metric spaces (e.g., Abdeljawad [15], Abdeljawad et al.[16,17], Altun et al.[18], Ćirić et al.[19], Karapinar and Yüksel [20]). Generalized altering distance functions with several variables were used on metric spaces by Berinde [21], Choudhury [22] and Rao et al.[23].

In this paper, an attempt has been made to derive some common fixed point theorems for two pairs of weakly compatible mappings on partial metric spaces, satisfying a weak contractive condition involving generalized control functions. The presented theorems extend and unify various known fixed point results. Examples are given to show that our results are proper extensions of the known ones.

The following definitions and details about partial metrics can be seen, e.g., in [1,24-28].

Definition 1 A partial metric on a nonempty set X is a function such that for all :

(p₁) ,

(p₂) ,

(p₃) ,

(p₄) .

The pair is called a partial metric space.

It is clear that, if , then from (p₁) and (p₂), it follows that . But may not be 0.

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 . A sequence in converges to a point , with respect to , if . This will be denoted as , or . If is a partial metric space, and is a mapping, continuous at (in ) then, for each sequence in X, we have

Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function need not be continuous in the sense that and implies .

Definition 2 Let be a partial metric space. Then:

1 A sequence in is called a Cauchy sequence if exists (and is finite).

2 The space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .

It is easy to see that every closed subset of a complete partial metric space is complete.

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

is a metric on X. Furthermore, if and only if

Lemma 1Letbe a partial metric space.

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

(b) The spaceis complete if and only if the metric spaceis complete.

Definition 3 ([22,23])

A function is said to be a generalized altering distance function if:

1 ψ is continuous;

2 ψ is increasing in each of its variables;

3 if and only if .

The set of generalized altering distance functions with n variables will be denoted by . If , we will write (obviously, this function belongs to ).

Simple examples of generalized altering distance functions with, say, four variables are:

Recall also the following notions. Let X be a nonempty set and be given self-maps on X. If for some , then x is called a coincidence point of and , and w is called a point of coincidence of and . The pair is said to be weakly compatible if , whenever for some t in X.

3.1 Some auxiliary results

Assertions similar to the following lemma (see, e.g., [29]) were used (and proved) in the course of proofs of several fixed point results in various papers.

Lemma 2Letbe a metric space and letbe a sequence inXsuch thatis nonincreasing and

Ifis not a Cauchy sequence, then there existand two sequencesandof positive integers such thatand the following four sequences tend toεwhen:

As a corollary (putting for a partial metric p), we obtain

Lemma 3Letbe a partial metric space and letbe a sequence inXsuch thatis nonincreasing and

Ifis not a Cauchy sequence in, then there existand two sequencesandof positive integers such thatand the following four sequences tend toεwhen:

3.2 Main results

Theorem 1Letbe a complete partial metric space. Letbe given mappings satisfying for every pair:

whereandare generalized altering distance functions, and. Suppose that

(i) and;

(ii) one of the rangesIX, JX, TXandSXis a closed subset of.

Then

(a) IandShave a coincidence point,

(b) JandThave a coincidence point.

Moreover, if the pairsandare weakly compatible, thenI, J, TandShave a unique common fixed point.

Proof Let be an arbitrary point in X. Since and , we can define sequences and in X by

Without loss of the generality, we may assume that

If not, then and hence , for some n. Taking and , from (3.4) and the considered contraction condition (3.3), we have

since

Suppose that . Using (3.6) together with and the properties of the generalized altering distance functions , we get

which is a contradiction. It follows that and hence . Following similar arguments, we obtain . Thus becomes an eventually constant sequence and is a point of coincidence of I and S, while is a point of coincidence of J and T.

Assume further that (3.5) holds. We claim that

Suppose that, for some ,

Using this together with the properties of generalized altering distance functions , , we get from (3.6) that

This implies that

which yields that . Hence, we obtain a contradiction with (3.5). We deduce that

By a similar reasoning, we obtain that

Combining (3.8) and (3.9), we obtain

Then, is a nonincreasing sequence of positive real numbers. This implies that there exists such that

By (3.6), we have

Letting in (3.11) and using continuity of and , we obtain

which implies that , and thus . Hence, (3.7) is proved.

Next, we claim that is a Cauchy sequence in the space (and also in the metric space by Lemma 1). For this it is sufficient to show that is a Cauchy sequence. Suppose that this is not the case. Then, using Lemma 3 we get that there exist and two sequences and of positive integers such that and sequences (3.2) tend to ε when . Applying condition (3.3) to elements and , we get that

Passing to the limit as in the last inequality (and using the continuity of the functions , ), we obtain

which implies that , that is a contradiction since . We deduce that is a Cauchy sequence.

Finally, we prove the existence of a common fixed point of the four mappings I, J, S and T.

Since is complete, then from Lemma 1, is a complete metric space. Therefore, the sequence -converges to some that is, . From (2.2), we have

Moreover, since is a Cauchy sequence in the metric space , then . On the other hand, by (p₂) and (3.7), we have , and hence

Thus from the definition of and (3.13), we have . Therefore, from (3.12), we have

This implies that

Thus we have

and

Now we can suppose, without loss of generality, that IX is a closed subset of the partial metric space . From (3.15), there exists such that . We claim that . Suppose, to the contrary, that . By (p₄) we get

It follows by (3.15) that

Then, since is increasing and continuous, we get that

Now, from (3.3)

Passing to the upper limit as in (3.17), we obtain using (3.14) and the continuity of , that

Therefore, from (3.16) we have

which is a contradiction. Thus we deduce that

We get that , so u is a coincidence point of I and S.

From and (3.18), we have . Hence we deduce that there exists such that . We claim that . Suppose, to the contrary, that . From (3.3), we have

which is a contradiction. Then, we deduce that

We get that , so v is a coincidence point of J and S.

Since the pair is weakly compatible, from (3.18), we have . We claim that . Suppose, to the contrary, that . Then we have

Again from (3.15) we get that

Then, since is increasing and continuous, we get

Now, from (3.3)

Passing to the upper limit as , we obtain (since )

Therefore, from (3.20) we have

a contradiction. This implies that

Hence, we have

Since the pair is weakly compatible, from (3.19), we have . We claim that . Suppose, to the contrary, that , then by (3.3), we have

Therefore, . Hence, we have and

Now, combining (3.21) and (3.22), we deduce

so z is a common fixed point of the four mappings I, J, S and T.

We claim that there is a unique common fixed point of S, T, I and J. Assume on contrary that, and with . By supposition, we can replace x by u and y by v in (3.3) to obtain

a contradiction. Hence , that is, . We conclude that S, T, I and J have only one common fixed point in X. The proof is complete. □

It is easy to state the corollary of Theorem 1 involving a contraction of integral type.

Corollary 1LetT, S, IandJas well as, satisfy the conditions of Theorem 1, except that condition (3.3) is replaced by the following: there exists a positive Lebesgue integrable functionuonsuch thatfor eachand that

for all. Then, S, T, IandJhave a unique common fixed point.

If in Theorem 1 is the identity mapping on X, then we have the following consequence:

Theorem 2Letbe a complete partial metric space. Letbe given mappings satisfying for every pair

whereandare altering distance functions, and. ThenTandShave a unique common fixed point.

Remark 1 Several corollaries of Theorems 1 and 2 could be derived for particular choices of and . We state some of them.

Putting and for , [15], Theorem 9] is obtained.

It is clear from the proof of Theorem 1 that condition (3.3), resp. (3.23), can be replaced by

resp.

where . Hence, Theorem 1 can be considered an extension of [23], Theorem 2.1] to the frame of partial metric spaces (since semi-compatible mappings are weakly compatible).

Putting and , with , in Theorem 2 (with condition (3.25)), we obtain [20], Theorem 8]. The same substitution in Theorem 1 (with (3.24)) gives an improvement of [20], Theorem 12] (since only weak compatibility and not commutativity of the respective mappings is assumed).

Putting and for in Theorem 2 (with condition (3.25)), [16], Theorem 5] is obtained.

Of course, several known results from the frame of standard metric spaces (see, e.g., [30] and [31]) are also special cases of these theorems. For example, the following corollary can be obtained as a consequence of Theorem 2, which is a generalization and extension of [31], Corollary 3.2].

Corollary 2Letbe a complete partial metric space. Letbe given mappings satisfying for every pair

wheremandnare positive integers, andare altering distance functions, and. ThenTandShave a unique common fixed point.

Remark 2 However, it is not possible to use in Theorems 1 and 2, as the following example, adapted from [23], Example 2.3], shows.

Example 1 Let and be given by for , , , and for . Then is a (complete) partial metric space. Consider the mappings defined by

and the functions given as and . It is easy to check that

holds for all . However, these mappings have no common fixed points; hence, condition (3.23) (or (3.25)) of Theorem 2 cannot be replaced by the respective condition with 5 variables. At the same time, condition (3.25) is not satisfied since, for , , and

hence,

whatever is chosen.

This example also shows (as in [23], Remark 2.4]) the importance of the second generalized altering distance function in Theorems 1 and 2.

The next example shows that Theorems 1 and 2 are proper extensions of the respective results in standard metric spaces.

Example 2 Let be endowed with the partial metric . Consider the mappings defined by

and the functions , given by

Take arbitrary elements, say , from X. Then

On the other hand,

and

Hence, condition (3.25) is satisfied, as well as other conditions of Theorem 2. Mappings have a common fixed point .

On the other hand, consider the same problem in the standard metric and take and . Then

and

and hence

Thus, condition (3.25) for does not hold and the existence of a common fixed point of these mappings cannot be derived from [23], Theorem 2.1].

The authors declare that they have no competing interests.

All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

The first and second author would like to acknowledge the financial support received from University Kebangsaan Malaysia under the research grant OUP-UKM-FST-2012. The fourth and fifth author are thankful to the Ministry of Science and Technological Development of Serbia.

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