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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 213876, 19 pages
http://dx.doi.org/10.1155/2012/213876
Research Article

Some Generalizations of Jungck's Fixed Point Theorem

1Departamento de Matemáticas, Universidad de Los Andes, Mérida 5101, Venezuela
2Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia

Received 28 March 2012; Revised 25 July 2012; Accepted 24 September 2012

Academic Editor: A. Zayed

Copyright © 2012 J. R. Morales and E. M. Rojas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are going to generalize the Jungck's fixed point theorem for commuting mappings by mean of the concepts of altering distance functions and compatible pair of mappings, as well as, by using contractive inequalities of integral type and contractive inequalities depending on another function.

1. Introduction and Preliminary Facts

In 1922, Banach introduced his famous result in the metric fixed point theory, the Banach contraction Principle (BCP), as follows.

Theorem 1.1 (see [1]). Let be a complete metric space and let be a self-map that satisfies the following condition: there is such that for all . Then, has a unique fixed point such that for each . One says that a mapping belongs to the class if it satisfies the condition .

Since then, several generalizations of the BCP have been appeared, some of them can be found, for instance, in [27] and into the references therein. In this paper we will focus our attention on an extension of the BCP introduced in 1976 by Jungck. More precisely, we are going to improve and generalize the following extension of the BCP.

Theorem 1.2 (see [8]). Let be two self-maps on a complete metric space, such that is a commuting pair, is continuous, , there is such that
for all .
Then, and have a unique common fixed point .

The class of all pair of mappings satisfying condition will be denoted by .

Notice that by taking , for all , in we obtain the condition . That is, the Theorem 1.1 is obtained from Theorem 1.2.

In order to attain our goals, we are going to use the notions of altering distance functions [9] and compatible pair of mappings [10], also we will use some contractive inequalities of integral type [11] and contractive inequalities depending on another function [12].

We would like to start recalling that in 1984 Khan et al. [9] introduced the concept of altering distance function as follows. A function is called an altering distance function if the following conditions are satisfied: , is monotonically nondecreasing, is continuous. We will denote by the set of all altering distance functions. Using this notion the same authors introduce the following class of mappings and studied the conditions for the existence of fixed points.

Definition 1.3 (see [9]). Let be a metric space, and be a self-map. The map is called a -Banach contraction, , if for each there exists such that The class of all mappings satisfying condition will be called the class. Notice that letting in the inequality we obtain , so the class generalizes the ones.

In 2002, Branciari [12] introduced a new generalization of Banach contraction mappings, which satisfies a general contractive condition of integral type. For that generalization first let us consider the set where satisfies the following conditions: is nonnegative, is Lebesgue integrable mapping which is summable on each compact subset of , for each

Definition 1.4 (see [12]). Let be a metric space. A mapping is said to be a -Banach contraction, , if there is such that for all , one has where .

The class of all the mappings satisfying the definition above will be denoted by .

Notice that the Definition 1.4 is a generalization of the BCP. In fact, letting for each in , we have Thus, a Banach contraction mapping also satisfies the condition . The converse is not true as we will see in [12, Example 3.6].

Theorem 1.5 (see [12]). Let be a complete metric space and let be a Then has a unique fixed point such that for each .

General contractive inequalities of integral type are becoming popular for extend classes of contractive mappings with fixed points. See for example, [1320] and references therein.

In 2009, Beiranvand et al. [11] introduced a new class of contraction mappings, by generalizing the contraction condition in terms of another function.

Definition 1.6 (see [11]). Let be a metric space and be two mappings. The mapping is said to be a -Banach contraction, (or belongs to the class). If there is such that for all .

By taking , for all , we get that the and are equivalent conditions.

Contraction conditions depending on another function have been used in order to generalize other well-known contraction type maps as the Contractive, Chatterjea, Ćirić, Hardy-Rogers, Kannan, Reich and Rhoades mappings [11, 2125].

Definition 1.7 (see [11]). Let be a metric space. A mapping is said sequentially convergent if one has, for every sequence , if is convergent, then also is convergent. is said subsequentially convergent if one has, for every sequence , if is convergent, then has a convergent subsequence.

The conditions for the existence of a unique fixed point for mappings in the class are given in the following result.

Theorem 1.8 (see [11]). Let be a complete metric space and be a one to one, continuous, and subsequentially convergent mapping. Then, every continuous mapping , has a unique fixed point . Moreover, if is sequentially convergent, then for each .

2. A Version of the Jungck's Fixed Point Theorem Using Altering Distance Functions

In this section we are going to generalize the Jungck’s fixed point Theorem 1.2 by using the altering distance function and the class. More precisely, we will introduce the class of -contraction mappings which generalize the class, and therefore the class of mappings.

Definition 2.1. Let be a metric space, be self-mappings and . The pair is called a -Jungck contraction, ( Contraction) if for each there exists such that

It is clear that letting in we obtain .

Example 2.2. Let endowed with the Euclidean metric. We define , , with and . Then, From here we get that the pair belongs to the class of mappings . On the other hand, Hence, the pair does not belong to the class .

In order to prove the fixed point theorem for pair of mappings belonging to the class we will need the following notions.

Definition 2.3 (see [26]). Let be a metric space and let be two mappings. Suppose that and for every one considers the sequence defined by, , for all , one says that is a sequence of initial point .

On the other hand, recall that a pair of mappings is said to be compatible [10] if and only if , whenever is such that for some .

Theorem 2.4. Let be a complete metric space and let be self-mappings such that(a) is a continuous function,(b) ,(c) is compatible,(d)the pair belongs to the class of mappings . Then, and have a unique common fixed point .

Proof. Let be an arbitrary point, we will prove that the -sequence of initial point is a Cauchy sequence in . For each from inequality contraction we have since , it follows that From conditions and of the altering distance function we obtain Now, we want to prove that is a Cauchy sequence in . Suppose that is not a Cauchy sequence in . So, there exists for which we can find subsequences and of with such that and for this we can choose in such a way that it is the smallest integer with satisfying (2.7). Then Thus, we have From (2.6), (2.7), and (2.9), it follows that In a similar way we obtain Now, using the definition of mappings for and we have Letting and using (2.11) we have which is a contraction if . This shows that is a Cauchy sequence in and hence it is convergent in the complete metric space . So there is such that Since is a continuous function which proves that the pair is compatible. Thus Now letting we have then, it follows that which implies that , so . Again, letting we have that , so it follows that . Then .
Therefore, is a common fixed point of and . Now, we are going to suppose that is another common fixed point of and . Then thus we conclude that , therefore , or equivalently, .

Notice that letting , in the definition of the class, we obtain a natural generalization of Jungck’s fixed point Theorem 1.2 to compatible pair of mappings, by replacing condition for the following:

is compatible.

Also, notice that the fixed point criterion for the class of mappings introduced by Khan et al. in [9] is obtained now as a consequence of Theorem 2.4 by letting .

3. The Jungck's Fixed Point Theorem with Contractive Inequality Depending on Another Function

In this section our main goal is to generalize the Theorem 1.2 by considering its contractive condition depending on another function. Now, using the ideas of Beiranvand et al. given in [11], we introduce the following class of pair of mappings.

Definition 3.1. Let be a metric space and let be mappings. The pair is said to be a contraction, if there is such that for all .

Notice that taking in the inequality above, we get that and are equivalent conditions. The next example shows that the class is more general than the ones.

Example 3.2. Let endowed with the Euclidean metric. We define , , and for all . Then, Thus, the pair does not belong to . However, Hence, the pair belongs to the class .

Theorem 3.3. Let be a complete metric space and let be mappings such that(a) is one to one, continuous, and sequentially convergent,(b) is continuous,(c) ,(d) and are commuting mappings,(e)the pair is a member of the class . Then, and have a unique common fixed point .

Proof. Let be an arbitrary point. We will prove that the sequence of initial point is a Cauchy sequence in . For each from the condition we get consequently, by repeating the argument above we can conclude that Taking limits in the last inequality we have that From (3.3), we have that is a Cauchy sequence in . Since is a complete metric space, then is convergent in , and due to the fact that is sequentially convergent, then is convergent in . So, there exists such that Since and are continuous maps, then is a continuous mapping, and using the contractive inequality we conclude then that is continuous. Thus Using that and are commuting mappings we obtain Therefore
Now Then, it follows that so, . Since is one to one, we obtain that , which implies that is a fixed point of and also that is a fixed point of . Hence, is the common fixed point of and .
Now, we will prove that is a common fixed point of and . Using the condition, we have taking the limit as we get That is, Therefore, , then . Since is one to one, we conclude that . Finally, we suppose that and . Then which implies that , or equivalently, . Therefore, the common fixed point of and is unique.

Example 3.4. As in Example 3.2, let be endowed with the euclidean metric and we define , , and , for all . Then,(a) is one to one, continuous and sequentially convergent,(b) is continuous,(c) , (d) and are commuting mappings,(e)In the Example 3.2, we proved that the pair does not belong to , thus we cannot apply the Theorem 1.2.(f)In the Example 3.2 we have proved that the pair is a , so we apply the Theorem 3.3,(g)It is clear that is the unique common fixed point of and .

Notice that if in the Theorem 3.3 we take , for all , then we obtain Theorem 1.2.

4. Contractive Conditions of Integral Type for the Jungck's Fixed Point Theorem

This section is devoted to generalize the Theorem 1.2 using the ideas of Branciari [12]. More precisely, for a pair of mappings we introduce contractive conditions of integral type depending on another function. Afterwards, we will give (common) fixed point results for this new class of mappings.

Definition 4.1. Let be a metric space and let be mappings. The pair is called if there is such that for all , we have where .

The class of pair of mappings satisfying the definition above will be denoted by .

Example 4.2. Let us consider and as in Example 3.2 and consider the function defined by . So, the pair belongs to the class if we have that Or, equivalently which is hold if and only if On the other hand, from Example 3.2 we have that the inequality above is valid if we take . Therefore, the pair belongs to with contractive constant .

Theorem 4.3. Let be a complete metric space and let be mappings such that(a) is one to one, continuous, and sequentially convergent,(b) and are continuous mappings,(c) , (d) and are commuting mappings,(e)The pair is a . Then, and have a unique common fixed point.

Proof. Let be an arbitrary point. We must prove that the sequence of initial point is a Cauchy sequence in . For each from the condition we have Consequently since , it follows from (4.5) that using that for each , we conclude that Now, we can prove that is a Cauchy sequence in . We are going to follow the proof of the Theorem 2.1 of [27].
In first place, we prove that for all , . Suppose that it is not true. Then, there exist and subsequences and such that for each positive integer , is minimal in the sense that Now, From (4.7) and (4.8), letting , we have letting , we have the contradiction Thus Now, we prove that is a Cauchy sequence in . From (4.7)–(4.12), which is a contradiction. Therefore is a Cauchy sequence in , and since is a complete metric space, then is a convergent sequence in . Now, from the fact that is sequentially convergent, we get that is a convergent sequence in . Thus, there exists satisfying using that , and are continuous mappings, we conclude that and are continuous, hence from (4.14) we get On the other hand, by the commutative property of and , it follows that therefore, and because is injective, we have It follows for the commuting of the pair that Now, suppose that . Using the condition we have That is, or equivalently which is a contradiction. Therefore, Since for each , then which implies that again, since is one to one, we have that . Thus, is a fixed point of . Now, due to the equalities and because is one to one, we get that therefore, is a fixed point of , hence is a common fixed point of and .
Now we are going to prove that is a common fixed point of and . Using the inequality condition , we have Taking the limit as we obtain thus we have since , then we get that which is a contradiction, therefore From here we conclude that therefore, , or equivalently, . Now, we will prove the uniqueness of the common fixed point. Let us suppose that with . Then then which implies that thus, . That is, and since is one to one, then . Therefore, we have proved that is the unique common fixed point of and .

5. Further Generalizations

Using contractive conditions of integral type and altering distance functions we can introduce new classes of mapping that generalize the class.

Definition 5.1. Let be a metric space, and be self-mappings. The pair is called a -Jungck contraction, if for each there exists such that where .

The class of pairs of mappings fulfilling inequality above will be denoted by .

Similarly, we introduce the following class of mappings.

Definition 5.2. Let be a metric space, and be self-mappings. The pair is called a -Jungck contraction, if for each there exists such that where .

By we mean the class of mappings given by definition above.

The existence and uniqueness of the common fixed point of pair of mappings in these new classes of self-maps is a consequence of the Theorem 2.4.

Proposition 5.3. Let be a complete metric space and be self-mappings satisfying the following conditions: (a) is a continuous function,(b) ,(c)The pair is compatible,(d)The pair belongs to Then, and have a unique common fixed point .

Proof. Let be a function defined by where . It is clear that . Moreover, it follows that for all and . Hence by the Theorem 2.4, the mappings and have a unique common fixed point .

In similar form we can prove the following result.

Proposition 5.4. Let be a complete metric space and be self-mappings satisfying the conditions:(a) is a continuous mapping,(b) ,(c)The pair is compatible,(d)The pair belongs to . Then, and have a unique common fixed point .

Notice that taking in or we obtain the results given by Kumar et al. in [28].

Acknowledgments

The authors sincerely thank the referees for their valuable comments and suggestions which improved the presentation of the paper. E. M. Rojas is sponsored by the Pontificia Universidad Javeriana under Grant no. 004805.

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