Common Fixed Point Results for Generalized Contractive Mappings with Applications

Abstract

In this paper, we introduce a new class of admissible mappings and prove some common fixed point theorems involving this new class of mappings which satisfy generalized contractive conditions in the framework of metric spaces. We also provide two examples to show the applicability and validity of our results. Meanwhile, we present an application to the existence of solutions to an integral equation by means of one of our results.

1. Introduction

The Banach contraction principle [1] is one of the essential pillars of the theory of metric fixed points. Many authors have obtained generalizations, extensions, and applications of their findings by investigating the Banach contraction principle in many directions. One of the most popular and interesting topics among them is the study of new classes of spaces and their fundamental properties.

Czerwik [2] introduced the concept of metric space and proved some fixed point theorems of contractive mappings in metric space. Subsequently, some authors have studied on the fixed point theorems of a various new type of contractive conditions in metric space. Aydi et al. in [3] proved common fixed point results for mappings satisfying a weak contraction in metric spaces. Following the results of Berinde [4], Pacurar [5] obtained the existence and uniqueness of fixed point of contractions, and Zada et al. [6] got fixed point results satisfying contractive conditions of rational type. In 2019, Hussain et al. studied the existence and uniqueness of a periodic common fixed point for pairs of mappings via rational type contraction in [7]. After that, authors obtained fixed point theorems for cyclic contractions and cyclic rational type contractions and discussed the existence of a unique solution to nonlinear fractional differential equations in [8, 9], respectively. Also using rational type contractive conditions, Hussain et al. [10] got the existence and uniqueness of common tupled fixed point for a pair of mappings. Using a contractive condition defined by means of a comparison function, [11] established results regarding the common fixed points of two mappings. In 2014, Abbas et al. obtained the results on common fixed points of four mappings in metric space in [12].

To generalize the concept of metric spaces, Hussain and Shah in [13] introduced the notion of a cone metric space, which means that it is a generalization of metric spaces and cone metric spaces; they considered topological properties of cone metric spaces and obtained some results on KKM mappings in the setting of cone metric spaces. In [14], some fixed point results for weakly contractive mappings in ordered partial metric space were obtained. Recently, Samet et al. [15] introduced the concept of admissible and contractive mappings and presented fixed point theorems for them. In [16], Jamal et al. used weak contraction to generalize coincidence point results which are established in the context of partially ordered metric spaces. In [17, 18], Zoto et al. studied generalized contractive mappings and contractions in metriclike space. Recently, in [16], Jamal et al. used weak contraction to generalize coincidence point results which are established in the context of partially ordered metric spaces. Abu-Donia et al. [19] proved the uniqueness and existence of the fixed points for five mappings from a complete intuitionistic fuzzy metric space into itself under weak compatible of type and asymptotically regular. For recent development on fixed point theory, we refer to [20–26].

Motivated and inspired by Theorems 27 and 29 in [17], Theorem 3.13 in [18], and Theorem 2.1 in [20], in this paper, our purpose is to introduce the concept of admissible mappings and obtain a few common fixed point results involving generalized contractive conditions in the framework of metric space. Furthermore, we provide examples that elaborated the useability of our results. Meanwhile, we present an application to the existence of solutions to an integral equation by means of one of our results.

2. Preliminaries

First of all, we introduce some definitions as follows:

Definition 1 (see [2]). Let be a nonempty set and be a given real number. A mapping is said to be a metric if and only if, for all the following conditions are satisfied: (i) if and only if (ii)(iii)Generally, we call a metric space with parameter .

Remark 2. We should note that a metric space with is a metric space. We can find several examples of metric spaces which are not metric spaces. (see [24]).

Example 3 (see [20]). Let be a metric space, and , where is a real number
Then, is a metric space with .

Definition 4 (see [12]). Let be a metric space with parameter . Then, a sequence in is said to be: (i)convergent if and only if there exists such that as (ii)a Cauchy sequence if and only if when In general, a metric space is called complete if and only if each Cauchy sequence in this space is convergent.

Definition 5 (see [21]). Let and be two self-mappings on a nonempty set . If , for some then is said to be the coincidence point of and , where is called the point of coincidence of and . Let denote the set of all coincidence points of and .

Definition 6 (see [21]). Let and be two self-mappings defined on a nonempty set . Then, and are said to be weakly compatible if they commute at every coincidence point, that is, for every .

We need the following lemma to obtain our main results:

Lemma 7 (see [20]). Let be a metric space with parameter . Assume that and are convergent to and , respectively. Then, we have In particular, if , then we have . Moreover, for each , we have

3. Main Results

In this section, we will show the existence and uniqueness of common fixed point for generalized contractive mappings in complete metric space. Meanwhile, we give two examples to support our results.

Definition 8. Let be a metric space with parameter , and let and be given mappings and be an arbitrary constant. The mapping is said to be admissible if, for all implies .

Remark 9. (i)Note that, for , the definition reduces to an -admissible mapping in a metric space(ii)For , the definition reduces to the definition of an admissible mapping in a metric spaceLet be a complete metric space with parameter and be a function. Then,
If is a sequence in such that as , then there exists a subsequence of with for all
For all we have the condition of or .

We know that contraction-type mappings are extended in several directions. Since Samet introduced the concept of admissible mappings and contractive mapping, some papers have been published to study a series of generalizations. Afterwards, these classes of mappings are used under generalized weakly contractive conditions.

We shall consider the contractive conditions in this section are constructed via auxiliary functions defined with the families , respectively:

Now, we introduce the notion of rational contraction in the setting of metric spaces.

Definition 10. Let be a metric space with parameter , and let be two self-mappings. Assume that and is a constant. A mapping is called a generalized contractive mapping, if there exist such that for all with , where

Example 11. Let and Define mappings by

Define mappings by and with .

It is clear that . For such that , we can know that and this implies that . By definitions, we obtain and . That is, is a admissible mapping. For all , we have

According to the above inequalities, we get that

It follows that is a generalized contractive mapping.

Theorem 12. Let be a complete metric space with parameter and let be given self-mappings on such that . Also, is a closed subset of , and is a given mapping. If the following conditions are satisfied: (i) is a admissible mapping(ii) is a generalized contractive mapping(iii)there is with (iv)properties and are satisfied(v) has a transitive property type , that is, for Then, and have a unique point of coincidence in . Moreover, and have a unique common fixed point provided that and are weakly compatible.

Proof. According to condition (3), there exists an such that . Define the sequences and in by for all . If for some , then we have and it is easy to see that and have a point of coincidence. Without loss of generality, assume that for all . By the condition (1), we get Therefore, by induction, we obtain for all . It follows from (4) that where If we assume that, for some , then from inequalities (13) and (14), we have Using (12), (16), and (17), one can obtain that which gives and then , a contradiction. It follows that , that is, is a nonincreasing sequence and so there exists such that By virtue of (13) and (14) again, we have It follows that Now suppose that , then taking the limit as in above inequality, we have , which gives a contradiction. Hence, Next, we aim to prove that is a Cauchy sequence. Suppose on the contrary that, , then there exists for which one can find sequences and of satisfying is the smallest index for which , In view of the triangle inequality, we have Taking the upper limit as in the above inequality and using (22), we have Also, From (24) and (27), we obtain Using (28) and (29), we have Similarly, so there is In view of the definition of , one can deduce that which yields that Similarly, we obtain So there is that is, Using the transitive property type of , we get Applying (4) with and ,we get By (35) and (38), we have which implies that a contradiction to (38). Therefore, is a Cauchy sequence in . The completeness of ensures that there exists a such that Since is closed, we have . It follows that one can choose a such that , and we can write (43) as The property yields that there exists a subsequence of so that for all . If , applying contractive condition (4) with and , we have where It is easy to show that Taking the upper limit as in (45), we have which implies that That is, . Therefore, is a point of coincidence for and . By using contractive condition (4) and the property , one can conclude that the point of coincidence is unique. Assume on the contrary that, there exist and . According to the property of , without loss of generality, we assume that Applying (4) with and , we obtain that that is, . By the weak compatibility of and , it is easy to show that is a unique common fixed point. This completes the proof.

Remark 13. It is obvious that the mappings defined in Example 11 satisfy the conditions of Theorem 12, so and have a unique common fixed point

In Theorem 12, put , one can get the following result.

Corollary 14. Let be a complete metric space with parameter and let be given self-mappings on with . Also, is a closed subset of , and is a given mapping. If the following conditions are satisfied: (i) is a admissible mapping(ii)there is function such thatwhere are same as Theorem 12, (iii)there exists with (iv)properties and are satisfied(v) has a transitive property type , that is, for Then, and have a unique point of coincidence in . Moreover, and have a unique common fixed point provided that and are weakly compatible.

Theorem 15. Let be a complete metric space with parameter , and let be given self-mappings on with . Also, is a closed subset of , and is a given mapping. Suppose that the following conditions are satisfied: (i) is a admissible mapping(ii)there are functions and such that for all where is same as Theorem 12 and (iii)there exists with (iv)properties and are satisfied(v) has a transitive property type , that is, for then and have a unique point of coincidence in . Moreover, and have a unique common fixed point provided that and are weakly compatible.

Proof. It is the same as the proof of Theorem 12, we also define the sequences and in by for and suppose that for each , so one can get that It follows from (54) that where If we assume that, for some then according to inequalities (59) and (60), we obtain In view of (58), we get which implies that , a contradiction to . It follows that . Hence, is a nonincreasing sequence. Consequently, the limit of the sequence is a nonnegative number, say . That is, .
By (59) and (60), we have So, If , then letting in above inequality, we obtain that , which implies that , i.e., Now, we prove that is a Cauchy sequence. If not, as the proof of Theorem 12, there exists for which one can find sequences and of so that is the smallest index for which , and the following inequalities hold: In view of the definitions of and , we have Letting and using (67)–(70), one can obtain Similarly, we get that That is, Using the transitive property type of , we have Taking and in (54), one can deduce that Therefore, It follows that , and which gives a contradiction to (74). Hence, From the completeness of and the closure of , there exists such that It follows that one can choose a such that , and write the above equality as In view of the property , one can get a subsequence of with for all . If , taking and in contractive condition (54), we have where Consequently, Taking the upper limit as in (81), we get It follows that . That is, is a point of coincidence for and . Using the same technique in the proof of Theorem 12, one can complete the proof.

Example 16. Let and . Define mappings by Define mappings by and with .
It is clear that and is closed. For such that , we can know that and which implies that . It follows that and , that is, is a admissible mapping.
For , we have Obviously, we conclude It follows that all conditions of Theorem 15 are satisfied. It is obvious that 0 is the unique common fixed point of and .

Remark 17. Taking in Theorem 2.1 of [12], Roshan et al. give that the existence of common fixed point for mappings such that where is a constant. Suppose all hypotheses in Example 16 are true. For , it is easy to calculate that which implies that Theorem 2.1 of [12] cannot be applied to testify the existence of common fixed points of the mappings and in .

If and in Theorem 15, we get the following result immediately:

Corollary 18. Let be a complete metric space with parameter and let be given self-mappings on such that . Also, is a closed subset of , and is a given mapping. If the following conditions are satisfied: (i) is a admissible mapping,(ii)for where are same as Theorem 15, (iii)there exists with (iv)properties and are satisfied(v) has a transitive property type , that is, for Then, and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Let , , and ( is a constant), we obtain that

Theorem 19. Let be a complete metric space with parameter and be a given self-mapping on . Let be a given mapping. If the following conditions are satisfied: (i) is a admissible mapping(ii)for where (iii)there exists with (iv)properties and are satisfied when (v) has a transitive property type , that is, for Then, has a unique fixed point.

Proof. The proof of Theorem 19 is similar to that of Theorem 15, we omit it.

Remark 20. Since a metric space is a metric space when , so our results can be viewed as the generalization and the extension of comparable results.

4. Application

In this section, we will use Theorem 19 to show that there is a solution to the integral equation:

Let be the set of real continuous functions defined on . The standard metric given by

Now for we define

It is obvious that is a complete metric space with .

Consider the mapping defined by and let be a given function.

Theorem 21. Consider equation (96) and suppose that (i) is continuous(ii)there exists such that for all (iii)for all and , implies (iv)properties and are satisfied when (v)there exists a continuous function such that(vi)there exists a constant such that for Then, the integral equation (96) has a unique solution

Proof. Define by It is easy to prove that is -admissible. For , by virtue of assumptions (1)–(6), we have which implies that Therefore, all the conditions of Theorem 19 hold. As a result, the mapping has a unique fixed point , which is a solution of the integral equation (96).

5. Conclusions

In this manuscript, we introduced a new class of admissible mappings and obtained common fixed point theorems for generalized contractive mappings in the framework of metric space. Further, we provided examples that elaborated the useability of our results. As an application of our result, we obtained a solution to an integral equation. The obtained results will be helpful for the variational iteration method, so we are going to study this topic in future.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This work was financially supported by the Science and Research Project Foundation of Liaoning Province Education Department (No:LQN201902, LJC202003).