Abstract

In this manuscript, a new family of contractions called Jaggi-type hybrid -contraction is introduced and some fixed point results in generalized metric space that are not deducible from their akin in metric space are obtained. The preeminence of this class of contractions is that its contractive inequality can be extended in a variety of manners, depending on the given parameters. Consequently, several corollaries that reduce our result to other well-known results in the literature are highlighted and analyzed. Substantial examples are constructed to validate the assumptions of our obtained theorems and to show their distinction from corresponding results. Additionally, one of our obtained corollaries is applied to set up unprecedented existence conditions for the solution of a family of integral equations.

1. Introduction

The prominent Banach contraction in metric space has laid a solid foundation for fixed point theory in metric space. The applications of fixed point range across inequalities, approximation theory, optimization, and so on. Researchers in this area have introduced several new concepts in metric space and obtained a great deal of fixed point results for linear and nonlinear contractions. Recently, Karapınar and Fulga [1] introduced a new notion of hybrid contraction which is a unification of some existing linear and nonlinear contractions in metric space.

On the other hand, Mustafa [2] pioneered an extension of a metric space by the name, generalized metric space (or more precisely, -metric space), and proved some fixed point results for Banach-type contraction mappings. This new generalization was brought to spotlight by Mustafa and Sims [3]. Subsequently, Mustafa et al. [4] obtained some engrossing fixed point results for Lipschitzian-type mappings on -metric space. However, Jleli and Samet [5] as well as Samet et al. [6] noted that most of the fixed point results in -metric space are direct consequences of existence results in corresponding metric space. Jleli and Samet [5] further observed that if a -metric is consolidated into a quasimetric, then the resultant fixed point results become the known fixed point results in the setting of quasimetric space. Motivated by the latter observation, many investigators (see for instance, [7, 8]) have established techniques of obtaining fixed point results in -metric space that are not deducible from their ditto ones in metric space or quasimetric space.

Following the existing literature, we realize that hybrid fixed point results in -metric space are not adequately investigated. Hence, motivated by the ideas in [1, 7, 8], we introduce a new concept of Jaggi-type hybrid -contraction in -metric space and prove some related fixed point theorems. An example is constructed to demonstrate that our result is valid, an improvement of existing results and the main ideas obtained herein do not reduce to any existence result in metric space. Some corollaries are presented to show that the concept proposed herein is a generalization and improvement of well-known fixed point results in -metric space. Finally, one of our obtained corollaries is applied to establish novel existence conditions for solution of a class of integral equations.

2. Preliminaries

In this section, we will present some fundamental notations and results that will be deployed subsequently.

Throughout, every set is considered nonempty, is the set of natural numbers, and represents the set of real numbers and the set of nonnegative real numbers.

Definition 1 (see [3]). Let be a nonempty set and let be a function satisfying
(G1) if
(G2) for all with
(G3) , for all with
(G4) (symmetry in all variables)
(G5) , for all (rectangle inequality)

Then, the function is called a generalized metric or, more precisely, a -metric on , and the pair is called a -metric space.

Example 2 (see [4]). Let be a usual metric space; then, and are -metric spaces, where

Definition 3 (see [4]). Let be a -metric space and let be a sequence of points of . Then, is said to be -convergent to if ; that is, for any , there exists such that , . We refer to as the limit of the sequence .

Proposition 4 (see [4]). Let be a -metric space. Then, the following are equivalent: (i) is -convergent to (ii), as (iii), as (iv), as

Definition 5 (see [4]). Let be a -metric space. A sequence is called -Cauchy if for any , we can find such that , , that is, , as .

Proposition 6 (see [4]). If is a -metric space, the following statements are equivalent: (i)The sequence is -Cauchy(ii)For every , there exists such that ,

Definition 7 (see [4]). Let and be -metric spaces and be a function. Then, is -continuous at a point if and only if for any , there exists such that ; and implies . A function is -continuous on if and only if it is -continuous at all .

Proposition 8 (see [4]). Let and be -metric spaces. Then, a function is said to be -continuous at a point if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

Definition 9 (see [4]). A -metric space is called symmetric -metric space if

Proposition 10 (see [4]). Let be a -metric space. Then, the function is jointly continuous in all variables.

Proposition 11 (see [4]). Every -metric space defines a metric space by Note that for a symmetric -metric space , On the other hand, if is not symmetric, then by the -metric properties, and that in general, these inequalities are sharp.

Definition 12 (see [4]). A -metric space is referred to as -complete (or complete -metric) if every -Cauchy sequence in is -convergent in .

Proposition 13 (see [4]). A -metric space is -complete if and only if is a complete metric space.
Mustafa [2] proved the following result in the framework of -metric space.

Theorem 14 (see [2]). Let be a complete -metric space, and let be a mapping satisfying the following condition: for all where ; then, has a unique fixed point (say , i.e., ), and is -continuous at .

Definition 15 (see [9]). Let be the set of all functions satisfying (i) is monotone increasing, that is, implies (ii)the series is convergent for all Then, is called a -comparison function.

Remark 16. If , then for any , , and is continuous at .

Karapınar and Fulga [1] gave the following definition of Jaggi-type hybrid contraction in metric space.

Definition 17 (see [1]). Let be a complete metric space. A self-mapping is called a Jaggi-type hybrid contraction; if there exists such that for all distinct , where with and .

3. Main Results

We begin this section by defining the notion of Jaggi-type hybrid -contraction in -metric space.

Definition 18. Let be a -metric space. A self-mapping is called a Jaggi-type hybrid -contraction, if there exists such that for all , where with and .

We now present the following results.

Theorem 19. Let be a complete -metric space and let be a continuous Jaggi-type hybrid -contraction on . Then, has a fixed point in (say ), and for any , the sequence converges to .

Proof. Let be an arbitrary point and define a sequence in by . If there exists some such that , then is a fixed point of , and so the proof is complete. Assume now that for any . Since is a Jaggi-type hybrid -contraction, then we have from (9) that

We then consider the given cases of (10).

Case 1. For , we have

Since is nondecreasing, if we assume that

then (11) becomes which is a contradiction. Therefore, for every , we have so that (11) becomes

Continuing inductively, we have

Now, since for all with , then for any with and by rectangle inequality, we have

Since is a -comparison function, then the series is convergent, and so denoting by , we have

Hence, as , we see that

Thus, is a -Cauchy sequence in and so by the completeness of , there exists such that is -convergent to , that is,

We will now show that is a fixed point of . By the assumption that is continuous, we have

so we get , that is, is a fixed point of .

Case 2. For , we have

Now, if , then (11) becomes which is a contradiction. Therefore,

Hence, by (11) we have

By similar argument as the case of , we can show that there exists a -Cauchy sequence in and a point in such that . Similarly, under the assumption that is continuous and by the uniqueness of limit, we have that , that is, is a fixed point of .

In the next result, we examine the existence of unique fixed point of under the restriction of continuity of some iterates of .

Theorem 20. Let be a complete -metric space and let be a Jaggi-type hybrid -contraction. If for some integer , we have that is continuous, then has a unique fixed point in .

Proof. In Theorem 19, we have established that there exists a -Cauchy sequence in with such that for some in . Let be a subsequence of where for all , fixed. Notice that is an identity self-mapping on so that . Hence, by the continuity of , we have that is, is a fixed point of .

To see that is a fixed point of , assume contrary that . Then in that case, for any . Hence, by (9), we have

Considering Case 1, we obtain

so that (29) becomes

Since , then for every , we have

This clearly implies that for every ,

In particular, letting and , the above inequality becomes

which is a contradiction. Hence, .

For Case 2, we have

so that (29) becomes implying that

By similar argument as in Case 1, we obtain a contradiction. Hence, .

Example 21. Let and let be a self-mapping on defined by for all . Define by Then, is a complete -metric space and is continuous for all . Define by for all .
To see that is a Jaggi-type hybrid -contraction, notice that for all . Hence, inequality (9) holds for all .

Now, for , if , then for all . If , then letting and , we obtain

Also, for , we have

If , then letting , , and , we obtain

Also, for , we take . Then,

Hence, inequality (9) is satisfied for all . Therefore, is a Jaggi-type hybrid -contraction. Consequently, all the assumptions of Theorem 19 are satisfied, and is the fixed point of .

We now demonstrate that our result is independent of the result of Karapınar and Fulga [1]. Let be defined by

Consider and take for Case 1, , , , and . Then, inequality (9) becomes while inequality (7) due to Karapınar and Fulga [1] yields

Also, Karapınar and Fulga [1] declared in Definition (17) that and are distinct, since is undefined for Case 1 if . However, our result is valid for all .

The above comparison is illustrated graphically for all , using the following Figures 1 and 2.

Therefore, Jaggi-type hybrid -contraction is not Jaggi-type hybrid contraction defined by Karapınar and Fulga [1], and so Theorem 1 due to Karapınar and Fulga [1] is not applicable to this example.

Corollary 22 (see Theorem 14). Let be a complete -metric space, and let be a mapping satisfying the following condition: for all where ; then, has a unique fixed point (say ) and is -continuous at .

Proof. Consider Definition (18) and let , , , , and for all and . Clearly, and is a Jaggi-type hybrid -contraction. Hence, (9) coincides with (6) of Theorem 14 due to Mustafa [2]. Therefore, it is easy to see that we can find a unique point in such that and is -continuous at .

Corollary 23 (see [10], Theorem 3.1). Let be a complete -metric space. Suppose the mapping satisfies for all . Then, has a unique fixed point (say ) and is -continuous at .

Proof. Consider Definition 18 and let , , and . Then, for all . Hence, inequality (9) becomes for all and . This coincides with Theorem 3.1 due to Shatanawi [10] and so the proof follows in a similar manner.

By specializing the parameters and , as well as letting for all and for , the following result is also a direct consequence of Theorem 19.

Corollary 24. Let be a complete -metric space. If there exists such that for all , the mapping satisfies then has a fixed point in .

4. Applications to Solution of Integral Equation

In this section, Corollary 24 is applied to examine the existence criteria for a solution to a class of integral equations. Ideas in this section are motivated by [7, 11, 12].

Consider the integral equation

Let be the set of all continuous real-valued functions. Define by

Then, is a complete -metric space.

Define a function as follows:

Then, a point is said to be a fixed point of if and only if is a solution to (52).

Now, we study existence conditions of the integral equation (52) under the following hypotheses.

Theorem 25. Assume that the following conditions are satisfied:
(C1) and are continuous
(C2) For all , , we have
(C3) for some

Then, the integral equation (52) has a solution in .

Proof. Observe that for any , using (55) and the above hypotheses, we obtain

Using this in (54), we have

Hence, all the required hypotheses of Corollary 24 are satisfied, implying that there exists a solution in of the integral equation (52).

Conversely, if is a solution of (52), then is also a solution of (55) so that , that is, is a fixed point of .

Remark 26. (i)We can deduce a number of corollaries by particularizing some of the parameters in Definition 18(ii)None of the results presented in this work can be expressed in the form or . Hence, they cannot be obtained from their corresponding versions in metric space

5. Conclusion

A generalization of metric space was introduced by Mustafa and Sims [3], namely, -metric space and several fixed point results were studied in that space. However, Jleli and Samet [5] as well as Samet et al. [6] established that most fixed point theorems obtained in -metric space are direct consequences of their analogues in metric space. Contrary to the above observation, a new family of contractions called Jaggi-type hybrid -contraction is introduced in this manuscript and some fixed point theorems that cannot be deduced from their corresponding ones in metric space are proved. The main distinction of this class of contractions is that its contractive inequality is expressible in a number of ways with respect to multiple parameters. Consequently, some corollaries including recently announced results in the literature are highlighted and analyzed. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. Furthermore, one of our obtained corollaries is applied to set up novel existence conditions for solution of a class of integral equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

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

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant no. UJ-21-DR-92. The authors, therefore, acknowledge with thanks the university technical and financial support.