Discussion on Geraghty Type Hybrid Contractions

Abstract

In this manuscript, we define the notion of Geraghty type hybrid contractions in the setting of -metric spaces. We prove the existence of a fixed point for such mappings whenever -metric space is complete. Our observed results not only unify several existing results but also extend some known results.

1. Introduction and Preliminaries

The distance notion is one of the ancient and most basic concepts in the history of mathematics. In modern mathematics history, this notion was formally formulated by Frechét as “-space.” Later, it was redefined as “metric space” by Hausdorff. After then, this concept has been extended and generalized in several ways. From all these generalizations of metric notions, the -metric is the most interesting.

In order to introduce the subject clearly, we first fix the basic concepts and notations. A function , defined from (where is a nonempty set) to nonnegative reals, is said to be a distance function, if it is symmetrical, that is , for every and if and only if .

Moreover, a distance function is a (standard) metric in case that

As we mentioned above, the distance notion, as well as the notion of the metric, has been extended and generalized in several directions. One of the outstanding generalizations of metric notion is named -metric. Indeed, the concept of -metric was considered by distinct authors, in various periods of the time, involving Bakhtin [1] and Czerwik [2]. Later, many researchers were interested in this topic, and thus, a series of interesting results were obtained, see, e.g., [3–19].

Definition 1. A distance function on is said to be a -metric over constant if the inequality (weighted triangle inequality) holds .

In what follows, we consider that denotes a -metric space.

An immediate observation is that the notion of -metric is more general than the concept of metric; for instance, when , we recover the notion of metric space. Moreover, we mention that a -metric is not necessarily continuous, see, e.g., [20, 21].

Example 2. The function on , where , is a -metric over , but not a metric.

Definition 3. On a -metric space , let be a sequence in . (a)The sequence is convergent in to , if for every , there exists such that for all . (We denote by as or .)(b)The sequence is Cauchy, if for every , there exists such that for all (c)If every Cauchy sequence in converges to a point , then the triplet is said to be complete

In short, denotes a complete -metric space over .

Recently, Mitrovic et al. [22] introduced the following type of contractions.

Definition 4 (see [22]). Let and be a self-mapping. We say that is a -weight type contraction, if there exists such that where , such that and for all , where .

In 1973, Geraghty [23] introduced a class of auxiliary functions to refine the Banach contraction principle. Let be the set defined as

Theorem 5 (see Geraghty [23]). On a complete metric space , a mapping admits a unique fixed point provided that there exists a function such that

2. Main Results

Let the set .

Definition 6. On , a mapping is called Geraghty type hybrid contraction, if there exists such that where , , with , and

Theorem 7. On , a Geraghty type hybrid contraction admits a unique fixed point if one of the following hypotheses is satisfied: (i) is continuous at (ii)(iii)Moreover, for any the sequence converges to .

Proof. We take an arbitrary point . Starting from this initial point, we shall construct a recursive sequence with the following formula: It is evident that if there exists such that , then becomes a fixed point of . Therefore, from now, we assume that

We shall discuss all possible situations.

Case 8. Suppose that From (7), we have that where

It yields which is equivalent to

Therefore, we have

It yields is nonincreasing sequence bounded by 0. Thus, the sequence converges to a nonnegative real number, say . We assert that is 0. On the one hand, by taking the of all sides of (13), we deduce that

Suppose on the contrary, that , we obtain

Thus, the limit . Consequently, .

We assert that the sequence is -Cauchy.

On contrary, supposing that the sequence is not -Cauchy, we can find and two sequences of positive integers and , such that

On the other hand, where

Taking of (21), we find

If we combine the observed inequalities above, in particular, (20) and (22), we have since . It implies that

Since , we conclude . Attendantly,

Under these observations, by employing the weighted triangle axiom together with (18), we get

Therefore, is a -Cauchy sequence in , so we can find a point such that

We assert now that this point, is a fixed point of . (i)Assuming that the mapping is continuous at , since , we haveand from (49), we get , i.e., .

For the other cases, we consider the inequality, for any . (ii)Suppose that . If , we haveand when , we get

Since , we get a contradiction, that is, . (iii)Suppose that Assuming that , we have

Taking , we have which is a contradiction, since . Therefore, .

Case 9. Here, (7) and (8) become for every , where and .

As in the proof of the case , we shall consider a recursive sequence , starting with an arbitrary point where . By using the same argument of this part of the proof, we presume that

Employing and in (34), we find that

It yields that for each . Attendantly, we deduce that the sequence of nonnegative numbers is a nonincreasing sequence. Ergo, there is a real number such that .

As in the previous case, we assert that . Supposing on the contrary, that , by taking in (37), we derive that

Since , we obtain

Thus, and consequently, .

We claim that is a -Cauchy sequence. On contrary, if we suppose that is not a -Cauchy sequence, that is, we can find and the sequences , of positive integers with such that

By the weighted triangle inequality, we have

Since where taking of (43), we find

If we combine the observed inequalities above, in particular, (42) and (45), we get that we get

Therefore, the sequence is -Cauchy in , so it is convergent at a point , that is

Now, we assert that is a fixed point of .

If the assumption (i) holds, since we get and from the inequality for any , we obtain , i.e., .

Suppose that or . Assuming that , we have

At the limit as , we have So, .

Example 10. We shall derive several distinct contractions from Definition 6. Some examples are given below. Let be a self-mapping on . (1)If , we obtain the following condition(2)If , we obtain the following condition(3)If with we obtainwhich means that is an interpolative Kannan type Geraghty-contraction; (4)If with , we havethat is is an interpolative Reich-Rus-iri type Geraghty-contraction.

Related to these examples, we can establish some consequences, by choosing proper values for in Theorem 15.

Corollary 11. Let and a self-mapping on . If there exists a function such that for all then admits a unique fixed point .

Corollary 12. Let and a self-mapping on . If there exists a function such that for all then admits a unique fixed point .

Corollary 13. Let be a complete -metric space, a self-mapping on and . If there exists a function such that for all , then admits a fixed point .

Corollary 14. Let be a complete -metric space, a self-mapping on and . If there exists a function such that for all , then admits a fixed point .

3. Immediate Consequences

By letting , we shall observe the Definition 4, [22].

Theorem 15 (see [22]). Let . A -weight type contraction mapping admits a fixed point if one of the following holds: (i) is continuous at such point (ii)(iii)Moreover, for any the sequence converges to

We list the following corollaries.

Corollary 16. On the complete -metric space let be a mapping. If there exists such that for all , and , then has a fixed point . the sequence converges to .

Proof. Put in Theorem 15, and

Corollary 17. On the complete -metric let be a mapping such that for all , where Then, has a fixed point .

Proof. Put in Theorem 15, and

Corollary 18. Let be a complete -metric space and be a mapping such that for every where The mapping has a fixed point provided that one of the following hold: (i) is continuous at (ii)Then, has a fixed point . Moreover for any , the sequence converges to .

Proof. Let and in Theorem 15.

Corollary 19. Let be a complete -metric space and be a mapping such that for all , where Assume that one of the following conditions hold: (i) is continuous at (ii)Then, has a fixed point and for any , the sequence converges to .

Proof. Take and in Theorem 15.

Corollary 20 (see [24]). Let , be a self-mapping on and . If there exists a function such that for all ; then, admits a unique fixed point .

Proof. Choose in Corollary 13.

Corollary 21 (see [25]). Let , be a self-mapping on and . If there exists such that for all ; then, admits a unique fixed point .

Proof. Choose in Corollary 14.

4. Conclusions

In this paper, we combine linear and nonlinear contractions to unify and extend the several existing results. This approach may bring new frames to the topic of metric fixed point theory. In particular, interpolative contraction may extend several results in the setting of Banach space.

We also mention that, for the case , we find a series of results known in the context of metric spaces, see, e.g., [25–36].

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

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no RG-1441-420.