New Fixed Point Theorems for Admissible Hybrid Maps

Noorwali, Maha; Yeşilkaya, Seher Sultan

doi:https://doi.org/10.1155/2022/5800790

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Fixed-Point Techniques and Applications to Real World Problems

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Volume 2022 | Article ID 5800790 | https://doi.org/10.1155/2022/5800790

New Fixed Point Theorems for Admissible Hybrid Maps

Maha Noorwali¹and Seher Sultan Yeşilkaya²

Academic Editor: Santosh Kumar

Received30 Oct 2021

Accepted18 Mar 2022

Published04 Apr 2022

Abstract

The aim of this work is to investigate the concept of a new hybrid Suzuki contractive by using the Rus-Reich-Ćirić-type interpolative mappings in -metric spaces. We seek the presence and uniqueness of a fixed point of such new contraction type mappings and prove some related results. We further give an application of Ulam-Hyers-type stability to show the well-posedness of our results.

1. Introduction and Preliminaries

Fixed point hypothesis has been a considerable area of research for mathematics and other sciences for the last century. It is the basis of functional analysis in mathematics, which is one of the critical topics of mathematics. The first concept of fixed point theory is knowing to appear in the work of Liouville in 1837 and Picard in 1890. But the main fixed point theorem was introduced by Banach [1]. The theorem is named after Banach. There are many generalizations of Banach theorem in the literature. In 1968, one of the most famous generalizations due to know, Kannan [2] introduced a new and useful contraction using Banach’s theorem. Suzuki [3] introduced important extensions of Banach’s main theorem, which we refer to [4–6]. In one of these studies [7], the researchers investigated a new extensive result by using simulation function. On the other hand, in [8], by using other auxiliary functions, called the Wardowski functions, they observed a contraction that combines both linear and nonlinear contractions. We also mention that in [9], the author obtained a fixed point theorem without the Picard operator. For more interesting results, see, e.g., [10–19]. In addition, Banach’s fixed point theorem is a significant mean in the theory of metric spaces. The metric concept has been generalized from different angles. One of the significant generalizes is defined -metric which was defined as follows.

Definition 1 (see [20, 21]). Let be a (nonempty) set and a real number. A function is a -metric space on if following conditions are satisfied: (i), if (ii)(iii), for every In this case, the pair is called a -metric space.

We recollect some basic notions that are used in our study.

A map is defined as a comparison function if it is increasing and , for any . We state by the class of all the comparison functions see, e.g., [22–24]. Defined by .

Lemma 2 (see [22, 23]). For a comparison function, satisfying the below statements take (1)every iterate of is a comparison function(2) is continuous(3), for any

Lemma 3 (see [25]). If is a -comparison function, then, (1)the series converges for any (2)the function defined by is increasing and continuous at We state that any -comparison function is a comparison function because of Lemma 2.3, and thus, in Lemma 2.2 any -comparison function satisfies .

Karapinar [26] introduced interpolation Kannan-type contraction generalized from the famous Kannan fixed point theorem by using interpolative operator. In the following, the common fixed point theorem for this contraction was obtained [27]. In [28], the authors extended the results in [26] by introducing the interpolative Reich-Rus-Ćirić contractive in a general framework, in the setting of partial metric space. In addition, the interpolative Hardy-Rogers-type contractive was defined and discussed in [28]. The contraction, defined in [29], was generalized in [30] by involving the admissibility into the contraction inequality. Furthermore, in [31], hybrid contractions were considered. Indeed, the notion of hybrid contraction here refers to combination of interpolative (nonlinear) contraction and linear contraction. For more interesting papers, see [32–34].

In 2019, inspired by interpolative contraction, researchers [35] obtained and published a hybrid contractive that integrates Reich-Rus-Ćirić-type contractive and interpolative-type mappings. In particular, this approach was applied for Pata-Suzuki-type contraction in [36]. On the other hand, by using hybrid contraction, a solution for a Volterra fractional integral equation was proposed in [37]. Furthermore, the hybrid contractions were discussed in a distinct abstract space, namely, Branciari-type distance space, in [38]. Another advance was recorded in [39] where the authors investigated the Ulam-type stability of this consideration. In addition, new hybrid contractions were developed in -metric spaces [40]. As a result, as can be seen in the literature review, many papers were published on the subject of interpolative contraction and hybrid contraction inspired by it. The contractions are a current study topic for fixed point theory. Therefore, the results of the study contribute to the existing literature.

Now we give the idea of -admissibility defined by Samet et al. [41] and Karapnar and Samet [42].

Definition 4. A mapping is defined -admissible if for each we have where is a given function.

The mapping of -orbital admissibility was presented by Popescu [43] as a modification of -admissibility as follows:

Definition 5. Let be a mapping and . A map is defined -orbital admissible if for every , we get

The following condition has often been considered on account of refraining from the continuity of the concerned contractive mappings.

Definition 6. A space is defined -regular, if is a sequence in such that for all and as ; then, there exists a subsequence of such that for all .

The framework of this study is organized into four sections. After the first introduction section, in Section 2, we introduced the definitions, theorems, and some results on the Ćirić-Rus-Reich-Suzuki-type hybrid. In Section 3, we give an application Ulam-Hyers-type stability to show the well-posedness for our fixed point theorem. Finally, in the last section, the conclusions are drawn.

2. Main Results

We begin with the definition of the Ćirić-Rus-Reich-Suzuki-type hybrid contraction:

Definition 7. Let be a -metric space and be a function. A map is a Ćirić-Rus-Reich-Suzuki-type hybrid contraction (CRRS-type hybrid contraction) if there exist such that for each , where and , such that ,

Theorem 8. Let be a complete -metric space and -orbital admissible map also for some . Given that be a CRRS-type hybrid contraction satisfying one of the following conditions:
(h₁) is -regular
(h₂) is continuous
(h₃) is continuous and , where
Thereupon, admits a fixed point in .

Proof. We install an iterative sequence of points such that for and with . If for some integers , then . Thus, suppose that , as is -orbital admissible, then implies that . Continuing this process, we get Condition 1: , by taking choosing and in (3) we get where Whereupon, we deduce that If we have given that , then, accompanying that is nondecreasing with (9), we get which is a contradiction. Thus, we obtain As a result, from (9), we will turn into and by similarly this process, we obtain that for any .
We argue that is a fundamental sequence in . Then, let such that and using the triangle inequality with (13), we take By using Lemma 3, the series is convergent where , the above inequality finds and , we obtain Thus, is a fundamental sequence. Accompanying this together with the fact that the space is complete, it will imply that there exists such that We argue that is a fixed point of .
If the suppose takes, we get , and we assert that for every . Since, if we have given that then, by using conditions of -metric spaces , since the sequence is decreasing, we write that a contradiction. Therefore, for all , either or provides. In the condition that (21) takes, then by (3), we get If the second condition, (22) holds, we obtain Thereupon, taking in (23) and (24), which is contraction. Therefore, we get that that is
If the presume is correct, and the map is continuous, we get In case that last supposition, holds, from above, we write , we want to show that . Let us pretend otherwise, that is, from using (3) we obtain that a contradiction. Eventually, .
Condition 2: if , in the equation taking and in (3) we write From (30), we find and from , we attain that for every . Using (30), we take and as in condition 1, we can prove that Since the equal methods as in the case of , we clearly prove that is a fundamental sequence in a complete -metric space. Additionally, for so, also we assert that . In the meanwhile, is -regular; thus, as confirm (5), and for each we obtain . Moreover, as in the proof of condition 1, we know that either or holds, for each . If (34) is taken, we conclude that Let us assume that inequality (35) is satisfied, then Then, getting to the limit, we conclude that and Now, the continuity of implies (from condition 1). Therefore, supposition lead to . We will prove that . Let’s presume otherwise, that is, using (3) we find that a contradiction. Consequently, . Thus, the proof of the Theorem is completed.

Theorem 9. Adding for any and if supplying to all the hypothesis of Theorem 8, we prove the uniqueness of fixed point.

Proof. Supposing that different is fixed point of , that is with In the case that , then, from (3) we have Thus, which is contradiction. In the case that , then, from (4) we get that a contradiction. Eventually, , so is a unique fixed point of .

Example 1. Let , for every with and also, the function with . Define a mapping as also, , we get that is continuous but is not continuous, where .
We choose and , then we obtain the following conditions: (a): For we get , then, (3) holds(b): If and Other conditions are confirmed, from Consequently, the assumptions of Theorem 8, being supplied, has a fixed point ().

Corollary 10. Let be a complete -metric space and let a continuous map satisfying the following inequality: where is defined by (4), and for all , where and with In the case of or functions continuity, admits a fixed point in .

Proof. It is sufficient to get for in Theorem 8.

Corollary 11. Let be a complete -metric space and let a continuous map satisfying the following inequality where is defined by (4), and for each where and with . In the event of or functions continuity, admits a fixed point in .

Proof. It is adequate get for any in Corollary 10.

Corollary 12. Let be a complete -metric space and a continuous map. If there exist such that for each in the case of or functions continuity, admits a fixed point in .

Proof. If , using Corollary 11, getting .

Corollary 13. Let be a complete -metric space and a continuous map. If there exist such that for each in the case of or functions continuity, admits a fixed point in .

Proof. By using Corollary 11, letting and .

Corollary 14. Let be a complete -metric space and a continuous map. If there exist such that for each in the case of or functions continuity, admits a fixed point in .

Proof. By using Corollary 11, taking and .

3. An Application: Ulam-Hyers-Type Stability

The stability of the solution is a considerable important subject of nonlinear analysis. Recently, Ulam stability [44, 45] results in fixed point theory have been investigated heavily. In what follows. we investigate the Ulam stability of our main theorem.

Consider the following function:

Assume that is a map on a -metric spaces . The fixed point problem of is to notice an such that

Equality (58) is also known as fixed point implication. The fixed point implication is called to be general Ulam-Hyers stable if and only if there exists a function so that for all also for every which satisfies the following inequality, there exists providing the equation (58) such that

Moreover, if there exists a such that for all then the fixed point equation (58) is said to be Ulam-Hyers stable. On the -metric spaces , fixed point problem (58) and are defined to be well-known if the following suppositions are satisfy:

(l₁) has a unique fixed point

(l₂) for every sequence such that

Theorem 15. Let be a complete -metric space. If we joint the condition and where and where or or also suppositions of Theorem 9, thus the following conditions hold: (a)the fixed point problem (58) is Ulam-Hyers stable, if for any satisfying the condition (59)(b)the fixed point problem (58) is well-known, if for any in such that and

Proof. (a)Taking into account Theorem 9, we consider that there is a unique in such that Assume that is a solution of (59), that is for Clearly, holds (59), then we get that and using triangular inequality satisfiesSince is CRRS-type hybrid contraction, we obtain Thus, we get then, where for any and such that . (b)The Picard iterations is -stable, that is, let such that and From the triangular inequality, we can writeThus, is a CRRS contraction, we have Then, we calculate process then, Taking in the above inequality and using we obtain the fixed point equation (58) is well posed.

4. Conclusion

In this study, we present new hybrid fixed point theorems in -metric spaces. We obtain the extended results of the interpolative Reich-Rus-Ćirić fixed point theorem by using -orbital admissible and Suzuki-type mapping. We also offer an example to show the availability of introduced results. Further, we obtain Ulam-Hyers-type stability of the fixed point theorem which is the application of our study.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

The authors contributed equally to this manuscript. All authors have read and agreed to the published version of the manuscript.

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Copyright © 2022 Maha Noorwali and Seher Sultan Yeşilkaya. 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.

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