On Some Interpolative Contractions of Suzuki Type Mappings

Abstract

The goal of this study is to propose a new interpolative contraction mapping by using an interpolative approach in the setting of complete metric spaces. We present some fixed point theorems for interpolative contraction operators using -admissible maps which satisfy Suzuki type mappings. In addition, some results are given. These results generalize several new results present in the literature. Moreover, examples are provided to show the suitability of our given results.

1. Introduction

In 1922, Banach [1] proved his famous remarkable fixed-point theorem; the result is known as the Banach contraction principle, which states that “Let be a complete metric space and be a contraction, then has a unique fixed point.” The Banach contraction principle is one of the essential and most valuable theorems of analysis and is accepted as the main results of metric fixed-point theory. In the last century, the fixed point and its applications have been the subject of research by many authors in the literature, since it provides useful tools to solve many complex problems that have applications in different sciences like computer science, engineering, data science, physics, economics, game theory, and biosciences [2–7]. Due to several applications of “fixed point theory,” researchers were motivated to further generalize it in different directions, by generalizing the contractive conditions underlying the space concept of completeness.

The background literature on the famous Banach contraction principle has been extended in various comprehensive directions by many researchers. One of the exciting generalizations was given by Kannan [8], which characterize the completeness of underlying metric spaces. Kannan introduced the following theorem.

Theorem 1. [8] Let be a complete metric space. A mapping is said to be a Kannan contraction if there exists such thatfor all . Then, posses a unique fixed point.

The Kannan theorem has been generalized in different aspects by many authors; one of the crucial generalizations was given by Karapinar in [9]. Karapinar [9] introduced the notion of an interpolative Kannan contraction mapping and proved the following:

A mapping on a complete metric space such thatwhere and , for each . Then, has a unique fixed point in . Subsequently, Karapinar et al. [10] introduced the following notion of interpolative -Reich-Rus contractions.

Theorem 2 (see [10]). Letbe a partial metric space. The mappingis called an interpolative Ciric-Reich-Rus contraction if there existand positive reals, with, such thatfor each . Then, the mapping has a fixed point in .

Afterward, this concept has been extended in different aspects, for example, [11–15].

Let be the set of all nondecreasing self-mappings on such that . Notice that for , we have and for all (see [16, 17]).

The concept of -orbital admissible mappings was introduced by Popescu as a clarification of the concept of -admissible mappings of Samet et al. [18].

Definition 3 (see [19]). Letbe a self-map defined onandbe a function.is said to be an-orbital admissible if for all, we have

In our appointed theorems, if the continuity of the involved contractive mappings is removed, to handle this defect, it is necessary that be -regular.

A space is defined as -regular, if is a sequence in such that for each and as , then for all .

Some curious results in this sense are found in the works in [20–24].

Another most interesting Banach contraction principle generalization was given by Suzuki [25, 26]. He introduced a weaker notion of contraction and discussed the existence of some new fixed point theorems. Besides the famous theorem, Suzuki generalized also the results of Nemytzki [27] and Edelstein [28] for compact metric spaces. One of the recently popular topics in fixed point theory is addressing the existence of fixed points of Suzuki type mappings. As with many generalizations of the famous Banach theorems, Suzuki type generalization can be said to have many applications, such as in computer science [29], game theory [30], and biosciences [31] and in other areas of mathematical sciences such as in dynamic programming, integral equations, data dependence, and homotopy [32, 33]. Subsequently, Popescu [34] has modified the nonexpansiveness situation with the weaker -condition presented by Suzuki. Accordingly, the existence of fixed points of maps satisfying the -condition has been extensively studied (see [35–38]). Karapınar et al. [39] introduced the definition of a nonexpansive mapping satisfying the -condition:

Definition 4. A mappingon a metric spacesatisfies the-condition iffor each

2. Main Results

We start the section with the following essential definition:

Definition 5. Letbe a metric space. A mappingis called an--interpolative Kannan contraction of Suzuki type if there exist, , and a real number, such thatfor each .

Theorem 6. Letbe a complete metric space andbe an--interpolative Kannan contraction of the Suzuki type. Suppose thatis an-orbital admissible mapping andfor some. Then,has a fixed point inprovided that at least one of the following conditions holds:(a) is -regular(b) is continuous(c) is continuous and where

Proof. Let such that and be the sequence constructed by for each positive integer Assuming that for some , , we get , so is a fixed point of Then, for each positive integer As is -orbital admissible, implies that . Similarly, continuing this process, we haveThereupon, choosing and in (6), we getwhence it follows thator equivalentThus, on the one hand, it follows that the sequence is a nonincreasing sequence with positive terms, so there exists such that . On the other hand, combining (8) and (10) and keeping in mind that the function is nondecreasing, we obtainNow, applying the triangle inequality and using (11), for all , we getwhere But, , the series is convergent, so there exists a positive real number such that . Consequently, letting in the above inequality, we getTherefore, is a Cauchy sequence, and taking into account the completeness of the space , it follows that there exists such thatand we claim that this is a fixed point of .
In case that the assumption (a) holds, we have , and we claim thatorfor every . Supposingon the account of the triangle inequality, we havewhich is a contradiction. Thereupon, for every , eitherorholds. In the case that (19) holds, we obtainIf the second condition, (20), holds, we haveTherefore, letting in (21) and (22), we get that that is,
In the case that the assumption (b) is true, that is, the mapping is continuous,If the last assumption, (c), holds, as above, we have and we want to show that also . Supposing on the contrary, that , sinceby (6), we getwhich is a contradiction. Consequently, , that is, is a fixed point of the mapping .☐

Example 7. Letandbe the usual distance on. Consider the mappingbe defined asLet also , where

We remark that the space is not regular since, for example, considering the sequence , with we have as , , but . On the other hand, the mapping is not continuous, but since , we have that is a continuous mapping. Let the function defined as and we choose . Thus, we have to check that (6) holds. We have to consider the following cases:(1)For , respectively, , we have , so (6) holds(2)For and (3)All other cases are not interesting because

Consequently, the assumptions of Theorem 6 being satisfied, it follows that the mapping has a fixed point, which is .

Corollary 8. Letbe a complete metric space andbe a self-mapping on, such that,for each , where and . Then, possesses a fixed point in .

Proof. Theorem 6 is sufficient to get for the proof.☐

Moreover, taking , with in Corollary (8), we obtain the following consequence.

Corollary 9. Letbe a complete metric space andbe a self-mapping on, such thatfor each , where . Then, the mapping possesses a fixed point in .

Definition 10. Letbe a metric space. The mappingis called an--interpolative Ćirić-Reich-Rus contraction of Suzuki type if there exist, , and positive reals, with, such thatfor each .

Theorem 11. Letbe a complete metric space and the mappingbe an--interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Suppose thatis-orbital admissible andfor some. Ifis-regular or either(1) is continuous or(2) is continuous and for any , then the mapping has a fixed point in

Proof. Let satisfy and be the sequence defined by for each positive integer If for some , we get , that means is a fixed point of Then, we can assume that for each positive integer Moreover, due to the assumption that is -orbital admissible, as in the previous proof, we haveBy letting and in (31), we obtainthen, using for every .or equivalentSo,for every . Therefore, the positive sequence is decreasing. Eventually, by (33), we haveand by repeating this process, we find thatWe assert that is a fundamental sequence in . Thus, using the triangle inequality with (38), we can writeTaking in (39), we deduce that is a fundamental sequence in , and using the completeness , there exists such thatWe claim that the point is a fixed point of In the case of the space being -regular and verifies (32), that is, for every we get . On the other hand, we know (see the proof of Theorem 6) that eitherorholds, for every . If (41) is holds, we obtainLetting in the above inequality, we get that that is, If the second condition (42) is true, we get that is a fixed point by a similar argument.
Furthermore, if the -regular of is removed and, instead, is continuous, we get that has a fixed point in , becauseFinally, if the mapping is such that is continuous, we easily obtain . Supposing that , since for any and , we haveThat is a contradiction. Thereupon, .☐

Example 12. Let, , , and, where, , and. Consider the function,let , , and the real constants . Taking into account the definition of the function , the only interesting situations are for , respectively, . For the first case, we haveFor the second case,

Definition 13. Letbe a metric space. The mappingis called an-interpolative Ćirić-Reich-Rus contraction of Suzuki type if there existand the constants, with, such thatfor each .

Theorem 14. Letbe a complete metric space and the mappingbe an-interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Then, the mappinghas a fixed point in.

Proof. Put in Theorem 11.☐

Definition 15. Letbe a metric space. A mappingis called an interpolative Ćirić-Reich-Rus contraction of the Suzuki type if there existand positive reals, with, such thatfor each .

Theorem 16. Letbe a complete metric space andbe an interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Therefore,has a fixed point in.

Proof. Put , for all , in Theorem 14.☐

3. Conclusions

In this manuscript, we introduce new concepts on completeness of --interpolative Kannan contraction of Suzuki type and --interpolative Ćirić-Reich-Rus contraction of Suzuki type mappings in metric space. We prove the existence of some fixed point theorems for mappings these concepts. Further, we obtain some fixed point results and give examples to show that the new results are applicable. Interpolation contraction, which is generalized from the Kannan type contraction, is a new and interesting contraction in fixed point theory, and different interpolation contractions of Suzuki type studies can be obtained by combining it with a Suzuki type contraction in the future. Additionally, these proposed contractions can be generalized in other well-known spaces and can give new fixed point results.

Data Availability

No data were used to support this study.

Conflicts of Interest

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

Authors’ Contributions

Both authors contributed equally to this work. Both authors have read and approved the final manuscript.