#### Abstract

We establish some common fixed point results for a new class of pair of contraction mappings having functions as contractive parameters, and satisfying minimal noncommutative operators property.

#### 1. Introduction and Preliminaries

The metric fixed point theory has a vast literature; the Banach-Caccioppoli contraction principle is one of the most outstanding results in this theory. Since its appearance, several generalizations of this result have appeared in the literature. In 1976, Jungck [1] generalized this principle by considering two commutating mappings and proved a common fixed point theorem for these mappings. Afterwards, the commutative property of the mappings assumed by Jungck has been relaxed by introducing “weak” alternative notions as *weak commutativity*, (*non*)*compatibility*, *R-weak commutativity,* and *weak compatibility,* among others, which allowed extending several well-known common fixed point theorems for Lipschitz type of mapping pairs.

Here, we are going to establish existence and uniqueness results of common fixed point for a pair of contractive type of mappings whose contractive parameters are nonconstants and its contractive inequality is controlled by a positive function satisfying a stability condition at 0 (see (11)). As a particular case, our results are valid if we control the mentioned inequality by using the well-known *altering distance functions* [2]. To attain our goals, we will assume that the mappings under consideration are weakly compatible, which is a minimal noncommuting notion for contractive type of mapping pairs. Also, alternatively, we are going to assume that the pair of mappings satisfy some strong conditions like E. A. property and property.

In order to establish our results the following notions will be needed. A pair of self-mappings on a metric space is said to be *compatible* [3] if and only if , whenever is such that
for some . A pair of mappings will be said to be *noncompatible* if there exits at least one sequence such that for some , but is either nonzero or nonexistent. A pair of self-mappings is said to satisfy the *property* (E.A.) [4] if there exists a sequence such that
for some .

A pair of self-mappings is said to satisfy the *common limit in the range of**property* (in short ) [5] if there exists a sequence such that
for some . It may be observed that the property avoids the requirement of the condition of closeness of the ranges of the involved mappings.

If is a self-map of a metric space then the set is called the *orbit* of at and is called *orbitally continuous* if implies [6].

If and are self-maps of a metric space and if is a sequence in such that , then the set is called the *orbit at* and and are called *orbitally continuous* if implies and .

A point is called a *coincidence point* (CP) of and if . The set of coincidence points of and will be denoted by . If , then is called a *point of coincidence* (POC) of and .

A pair of mappings is said to be *nontrivially weakly compatible* [7], whenever implies . Finally, a pair of mappings is said to be *occasionally weakly compatible* (OWC) [8] if there exists some such that .

*Remark 1. *We would like to show that weak compatibility is a necessary, hence minimal, condition for the existence of common fixed points of contractive type mapping pairs. Suppose and are a contractive type pair of self-mappings of a metric space having a common fixed point, say ; then and . If possible, suppose that and are not weakly compatible. Then there exists a point in such that while ; we thus have and with . This is not possible in view of contractive conditions. For example, if and satisfy the contractive condition
then we get
a contradiction.

This shows that weak compatibility is a necessary, hence minimal, condition for the existence of common fixed points of contractive type mapping pairs.

The following result due to Babu and Sailaja in [9] will be useful in the sequel.

Lemma 2. *Let be a metric space. Let be a sequence in such that
**
If is not a Cauchy sequence in , then there exist an and sequences of integers positive and with
**
such that
**
and*(i)*,*(ii)*,*(iii)*.*

#### 2. The Class of Pairs of Mappings with Nonconstant Contractive Parameters

In order to introduce the class of mappings which will be the focus of study of this paper, as in [10], we are going to use the functions which satisfy that , for all , and

Now, we introduce the following class of pair of contraction type of mappings.

*Definition 3. *Let be a metric space and let be mappings. The pair is called a-contraction pair if for all
where are functions satisfying the conditions (9) and is a continuous function satisfying that

Proposition 4. *Let and be two self-maps on a metric space . Let one assume that the pair is a -contraction pair. If and have a POC in then it is unique.*

*Proof. *Let be a POC of the pair . Then there exits such that . Suppose that, for some , with . Then, by (10) we have
It follows that
Thus, we get
which is a contradiction. Therefore, .

Proposition 5. *Let be a metric space and let be mappings with . If the pair is a -contraction pair, then for any , the sequence defined by
**
satisfies *(1);(2)*is a Cauchy sequence in*.

*Proof. *To prove (1), let be an arbitrary point. Since , then there exists such that . By continuing this process inductively we obtain a sequence in such that
Now, we have
It follows that
Therefore, we obtain
from which, together with (9), we conclude that
Thus, is a nondecreasing sequence, bounded below by zero, and so converges to . Now, if then by taking from both sides of the above inequality we have a contradiction. Thus, . Now, from condition (11) we conclude that .

To prove (2), we are going to suppose that is not a Cauchy sequence. Then, from Lemma 2 there exist and sequences and with such that

In this way we have
which is a contradiction; hence, is a Cauchy sequence in .

#### 3. On the Existence and Uniqueness of Common Fixed Points

In this section we prove our main results concerning the existence and uniqueness of common fixed points for a -contraction pair of mappings without continuity requirement.

Theorem 6. *Let and be self-maps on a complete metric space such that*(i)*;*(ii)*the pair is a -contraction pair.**Then,*(1)*the pair has a unique POC;*(2)*if and are orbitally continuous and if the pair is compatible, then and have a unique common fixed point.*

*Proof. *Let , , be the Cauchy sequence defined in Proposition 5 which, as was proved, satisfies that . Since is complete, there exists a point in such that

Compatibility and orbital continuity of and imply ; hence is a POC of and . From Proposition 4 we conclude that is the unique POC.

On the other hand, since the pair is compatible and compatible mappings commute at their coincidence point, . Using (ii), we get
that is, . Hence, , and is a common fixed point of the pair . The uniqueness of the common fixed point follows easily.

Theorem 7. *Let and be self-mappings on a metric space such that*(i)*;*(ii)* is a complete subspace of ;*(iii)*the pair is a -contraction pair.**Then,*(1)*the pair has a unique POC;*(2)*if the pair is nontrivially weakly compatible, then and have a unique common fixed point.*

*Proof. *Let , , be the Cauchy sequence defined in Proposition 5 which, as was proved, satisfies that . Since is a complete subspace of , then there exists such that
and thus we can find such that . Now, we are going to assume that . Then,
Letting , we obtain
which is a contradiction; therefore ; hence is a POC of and . The theorem now follows easily using arguments similar to those which have been used in the corresponding part of Theorem 6.

Theorem 7 is also true if we replace conditions (i) and (ii) (of Theorem 7) by a single condition. In the next theorem denotes the closure of the range of the mapping .

Theorem 8. *Let and be self-mappings on a metric space such that*(i)*;*(ii)*the pair is a -contraction pair.**Then,*(1)*the pair has a unique POC;*(2)*if the pair is nontrivially weakly compatible, then and have a unique common fixed point.*

*Remark 9. *Notice that in Theorem 7 we cannot replace nontrivially weakly compatible mappings by OWC. In fact, under the -contraction pair of Theorem 6, assumption of OWC and the existence of a unique common fixed point are equivalent conditions. To see this, first suppose that and satisfy the -contraction condition of Theorem 6 above. If and have a common fixed point, say , then , and and are, therefore, OWC mappings. On the other hand, if and are OWC mappings such that and for some then, using -contraction condition, we get
That is, . Since -contraction condition excludes the existence of two coincidence points for and , we get . This means that is a common fixed point of and . Therefore, one should be really careful before using OWC under any contractive conditions (see, also, [11]).

Theorem 10. *Let be a metric space and nontrivially weakly compatible mappings satisfying the property (E.A.). Let one suppose that the pair is a -contraction pair. If is closed, then and have a unique common fixed point.*

*Proof. *Since the pair satisfies the property (E.A.), then there exists a sequence such that
for some . Since is closed, then and for some . As in the proof of Theorem 7, we can prove that and that is the unique POC of and . The rest of the proof of the theorem follows from Theorem 6.

*Remark 11. *Since two noncompatible self-mappings on a metric space satisfy the property (E.A.), then the conclusion of Theorem 10 remains valid if we consider and noncompatible self-mappings.

In the next theorem we drop closeness of the range of mapping and replace property (E.A.) with property.

Theorem 12. *Let be a metric space and satisfying the property. Let us suppose that the pair is a -contraction pair. If the pair is nontrivially weakly compatible, then and have a unique common fixed point.*

*Proof. *Since the pair satisfies the property, then there exists a sequence such that
for some .

The rest of the proof of the theorem follows easily.

#### 4. Controlling by Altering Distance Functions

In 1976, Delbosco [12] initiated the study of fixed point for contractive conditions using altering distance functions; however, his study was limited to some power functions only. Subsequently, his result was extended by Skof [13] and Khan et. al [2] in 1977 and 1984, respectively. Since then, it has been used to solve several problems in the metric fixed point theory (see, e.g., [14–21]).

*Definition 13. *A function is called an altering distance function if the following properties are satisfied: if and only if ; is monotonically nondecreasing; is continuous.

By we are going to denote the set of all the altering distance functions.

Since every nondecreasing map satisfies (11) (but the converse is not true), then all the previous results are valid, in particular, if we replace functions satisfying the condition (11) for altering distance functions.

On the other hand, in 2002, Branciari [22] extended the Banach-Caccioppoli theorem by using some Lebesgue integrable functions. Since then, several well-known fixed point criteria for contractive type of mappings have been generalized in this way. See, for example, [5, 10, 16, 17, 23–27] and a lot of references therein. In 2009, Jachymski [28] showed that most contractive conditions of integral type given by many authors are mere consequences of classical known ones (see [28] and references therein).

By is denoted the set of all mappings satisfying the following conditions: is a Lebesgue integrable mapping which is summable on each compact subset of ; is nonnegative;for each , .

A relation between these two classes of functions and is given in the following result ([29, 30]).

Lemma 14. *For each , the function defined by , is such that .*

In this way, additionally to the class of -contraction pairs, we can consider a type of pair of mappings satisfying the following inequality contraction of integral type: for all , where , , and are functions satisfying (9). Since this class can be rewritten as for all , where is the function defined in Lemma 14, then we have that all the conclusions given for -contraction pairs are automatically valid for pair of mappings satisfying the inequality contraction (31).

#### 5. Conclusions and Examples

Notice that due to the minor restrictions on the functions involved in the definition of the class of -contraction pairs and the minimal commutative requirements of the mappings, our results extend several common fixed point theorems for classes of well-known contractive type of mappings, including various classes of contractive mappings with inequalities controlled by altering distance functions as well as contractive mappings of the integral type. Even more, the mappings and considered here are not necessarily continuous, so in this way our results are more general compared with other results in this line of research.

Next, we are going to show some examples in support of our results.

*Example 15. *Let be equipped with the Euclidean metric. We consider the following mappings: defined by and for all . Let defined by and for all and given by the formula , .

Notice that and the functions satisfy the conditions (9); also note that and is a complete subspace of . Moreover, it is not difficult to show that the pair is a -contraction pair. Besides, and , which mean that is nontrivially weakly compatible. Then, Theorem 7 guarantees that is the unique common fixed point of and .

*Example 16. *As in the example before, with the usual metric. We define the self-maps on by
and for all . Let defined as follows:
Let defined by , . Then, the pair is a -contraction pair satisfying the hypotheses of Theorem 6; thus is the unique POC and moreover the unique common fixed point of and .

*Example 17. *Let with the usual metric on . In this case we consider the mappings defined by

Let given by the formula , . given by , , . Notice that and . Moreover, is the unique POC of and ; thus the pair is nontrivially weakly compatible. On the other hand, by considering the sequence , in , it is clear that the pair satisfies the property (E.A.). Finally, it is easy to show that in fact the mappings and satisfy all the hypotheses of Theorem 10, so is the unique common fixed point of and .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are thankful to the referees for the very constructive comments and suggestions that led to an improvement of the paper. E. M. Rojas is sponsored by Pontificia Universidad Javeriana under Grant no. 000000000005781.