Cyclic -Multiplicative -Hardy–Rogers-Type Local Contraction and Related Results in -Multiplicative and -Metric Spaces

Abstract

The aim of this paper is to define cyclic -multiplicative Hardy–Rogers-type local contraction in the context of generalized spaces named as -multiplicative spaces to extend various results of the literature including the main results of Yamaod et al. In this way, we apply a new generalized contractive condition only on a closed set instead of a whole set and by using -multiplicative space instead of multiplicative metric space. We apply our results to obtain new results in -metric spaces. Examples are given to show the usability of our results, when others cannot.

1. Introduction and Preliminaries

Bakhtin [1] was the first who gave the idea of -metric. After that, Czerwik [2] gave an axiom and formally defined a -metric space. For further results on the -metric space, see [3, 4]. Ozaksar and Cevical [5] investigated the multiplicative metric space and proved its topological properties. Mongkolkeha and Sintunavarat [6] described the concept of multiplicative proximal contraction mapping and proved some results for such mappings. Recently, Abbas et al. [7] proved some common fixed point results of quasi-weak commutative mappings on a closed ball in the setting of multiplicative metric spaces. For further results on the multiplicative metric space, see [8–12]. In 2017, Ali et al. [13] introduced the notion of the -multiplicative space and proved some fixed point results. As an application, they established an existence theorem for the solution of a system of Fredholm multiplicative integral equations. For further results on the -multiplicative space, see [14]. Shoaib et al. [4] discussed some results for the mappings satisfying contraction condition on a closed ball in a -metric space. For further results on a closed ball, see [15–25]. In this paper, we generalized the results in [12] by using cyclic -multiplicative -Hardy–Rogers-type local contraction on a closed ball in a -multiplicative space. Moreover, we show that our results can be applied on those mappings where the other results cannot be applied. The following definitions and results will be used to understand this paper.

Definition 1 (see [13]). Let be a nonempty set, and let be a given real number. A mapping is called -multiplicative with coefficient , if the following conditions hold:(i) for all with and if and only if (ii) for all (iii) for all The pair is called a -multiplicative space. If , , then and are called the closed ball and the open ball in , respectively. Note that if , then we obtain empty open balls because .

Example 1. Let . Define a mapping :where is any fixed real number. Then, for each , is -multiplicative on with . Note that is not a multiplicative metric on . Considering , , and , then is a closed ball in .

Definition 2 (see [13]). Let be a -multiplicative space.(i)A sequence is called -multiplicative Cauchy iff(ii)A sequence is -multiplicative convergent iff there exists such that(iii)A -multiplicative space is said to be -multiplicative complete if every -multiplicative Cauchy sequence in is -multiplicative convergent to some .

Example 2. The space defined in Example 1 is a -multiplicative complete space. Taking , then , as . Hence, is a -multiplicative Cauchy sequence. Now, , as implies that is -multiplicative convergent to .

Definition 3 (see [3]). Let and be a real number. A mapping is said to be -metric with coefficient “,” if for all , the following assertions hold:(i)(ii)(iii)The pair is said to be a -metric space. If , , then is called a closed ball in .

Remark 1 (see [13]). Every -metric space generates a -multiplicative space defined as

Remark 2. Let be a -multiplicative space generated by a -metric space , and . If and are closed balls in and , respectively, then .

2. Results on b-Multiplicative Spaces

Definition 4. Let , , and . We say that is a locally cyclic -admissible mapping on set , if

Definition 5. Let be a -multiplicative space with and be a closed set. A self-mapping is said to be cyclic -multiplicative -Hardy–Rogers-type local contraction on , if the following conditions hold:(1) and (2) is a locally cyclic -admissible mapping on (3) impliesfor , where and .

Example 3. Let and be a mapping defined asDefine a complete -multiplicative metric with asConsidering and , then . is defined bynow, and . Clearly, is a locally cyclic -admissible mapping on . Note that, if we take , then but () so is a not-cyclic -admissible mapping on , and the results in [12] cannot be applied. Taking , , and , now, and . For each with , we haveThus, all conditions of Definition 5 hold. Therefore, is a cyclic -multiplicative -Hardy–Rogers-type local contraction on . Note thatfor all , so again the results in [12] cannot be applied because cannot satisfy any definition in [12].

Definition 6. If we take in Definition 5, then we say it is cyclic -multiplicative -Banach-type local contraction on . If we take in Definition 5, then we say it is cyclic -multiplicative -Kannan-type local contraction on . If we take in Definition 5, then we say it is -multiplicative cyclic -Chatterjea-type local contraction on . If we take in Definition 5, then we say it is cyclic multiplicative -Hardy–Rogers-type local contraction on . If we exclude the role of functions and in Definition 5, that is, if we exclude conditions (1) and (2) and the restriction from Definition 5, then we say it is -multiplicative Hardy–Rogers-type local contraction on .

Example 4. If we take in Example 3, then we obtain an example of cyclic -multiplicative -Banach-type local contraction on . If we take in Example 3, then we obtain an example of cyclic -multiplicative -Kannan-type local contraction on . If we take in Example 3, then we obtain an example of cyclic -multiplicative -Chatterjea-type local contraction on . If we take in Example 3, then we obtain an example of cyclic multiplicative -Hardy–Rogers-type local contraction on . If we exclude the role of functions and in Example 3, then we obtain an example of -multiplicative Hardy–Rogers-type local contraction on .

Definition 7. Let be a -multiplicative complete space with coefficient and and be two mappings. We say that is cyclic regular on a closed ball with respect to , if one of the following conditions holds:(a)If contains a sequence such that for all and as , then (b) is continuous on

Example 5. The mapping in Example 3 is cyclic regular on a closed ball with respect to because for any sequence in such that for all and as , then . Also, is continuous on .

Theorem 1. Let be a -multiplicative complete space with coefficient and be a cyclic -multiplicative -Hardy–Roger-type local contraction mapping on . Suppose thatwhere . Then, there exists a -multiplicative convergent sequence in . Also, if is cyclic regular on with respect to , then there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of , then the fixed point of will be unique.

Proof. Consider that and and . As is a cyclic admissible mapping on , we haveNote thatBy assumption, we haveSince , so . Now, and implies . Hence, . AsAssume that for some . By a similar method, as above for and , we getThis implies thatBy using (6), we haveNow, using inequality (19), we getBy using triangle inequality and inequality (20), we getBy using inequality (12), we haveThis implies that . By induction on , we conclude that for all . By a similar method, for all , we getThis implies thatNow, inequality (20) implies thatNow, we prove that is a -multiplicative Cauchy sequence in . Let , so ; . By using the triangle inequality, we haveBy using inequality (25), we getTaking limit as , we get . Hence, the sequence is a -multiplicative Cauchy sequence. By the completeness of , it follows that . Suppose that is continuous. Thus, we get . Now, we assume that condition (a) of Definition 7 holds. As and , so . Then, we haveLetting , we getHence, , that is, . This proves that is a fixed point of . Eventually we prove that is the unique fixed point of . Suppose that is another fixed point of . By the hypothesis, we find that and . Thus,This proves that and then . Thus, is the unique fixed point of .

Example 6. In Example 3, we have proved that is a cyclic -multiplicative -Hardy–Rogers-type local contraction on . It has been proved in Example 5 that the mapping in Example 3 is cyclic regular on a closed ball with respect to . Now,Now,Hence, all the conditions of Theorem 1 are satisfied, and zero is the unique fixed point of the mapping . Note that the results in [12] cannot ensure the existence of a fixed point of mapping because cannot satisfy the contractive condition of any theorem in [12].
The following results for various other contractions on b-multiplicative spaces can be proved by following the proof of Theorem 1.

Theorem 2. Let be a -multiplicative complete space with coefficient and be a cyclic -multiplicative -Banach-type local contraction mapping on . Suppose that

Then, there exists a -multiplicative convergent sequence in . Also, if is cyclic regular on , then there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of then the fixed point of will be unique.

Theorem 3. Let be a -multiplicative complete space with coefficient and be a cyclic -multiplicative -Kannan-type local contraction mapping on . Suppose thatwhere . Then, there exists a -multiplicative convergent sequence in . Also, if is cyclic regular on , then there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of , then the fixed point of will be unique.

Theorem 4. Let be a -multiplicative complete space with coefficient and be a cyclic -multiplicative -Chatterjea-type local contraction mapping on . Suppose thatwhere . Then, there exists a -multiplicative convergent sequence in . Also, if is cyclic regular on , then there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of , then the fixed point of will be unique.

Theorem 5. Let be a -multiplicative complete space with coefficient and be a -multiplicative Hardy–Roger-type local contraction mapping on . Suppose that

Then, there exists a unique fixed point in .

Theorem 6. Let be a -multiplicative complete space with coefficient and be a cyclic -multiplicative -Hardy–Roger-type local contraction mapping on . Then, there exists a -multiplicative convergent sequence in . Also, if is cyclic regular on , then there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of , then the fixed point of will be unique.

The following result is a multiplicative metric version of Theorem 1.

Theorem 7. Let be a complete multiplicative space and be a multiplicative -Hardy–Roger-type local contraction mapping on . Suppose thatwhere . Then, there exists a multiplicative convergent sequence in . Also, if one of the following conditions holds:(a)If contains a sequence such that for all and as , then (b) is continuous on Then, there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of , then the fixed point of will be unique.

As an application, we give an existence theorem for the Fredholm multiplicative integral equation of the following type:where is an integrable function.

Let , and , be the space of all positive, continuous real-valued functions, endowed with the -multiplicative:

Clearly, the set is a closed ball in .

Theorem 8. Let , , , , , and :where is an integrable function. Assume that the following conditions hold:(1) and ;(2) is a cyclic -admissible mapping on ;(3)For each and for , belongs to closed set , such that ; then, this implies(4)The constant is such that and Also, if one of the following conditions holds:(5) is continuous on or(6)If is a sequence in such that as and for all , then .

Then, the integral equation (38) has a solution. Moreover, if and for all in the set of fixed points of , then equation (38) has a unique solution.

Proof. Let . Now, we haveThus, we get , . As , so . Also, hypothesis (4) impliesTherefore, by Theorem 2, there exists a unique fixed point of the operator . Hence, the integral equation (38) has a unique solution.

3. Results on b-Metric Spaces

Definition 8. Let be a -metric space and be a closed set. A self-mapping is said to be cyclic --Hardy–Rogers-type local contraction on , if the following conditions hold:(1) and (2) is a locally cyclic -admissible mapping on (3) impliesfor , where and .

Theorem 9. Let be a complete -metric space with coefficient and be a cyclic --Hardy–Roger-type local contraction mapping on . Suppose thatwhere . Then, there exists a convergent sequence in . Also, if one of the following conditions holds:(a)If contains a sequence such that for all and as , then (b) is continuous on .

Then, there exists a fixed point of in . Moreover, if and , for all in the set of fixed points of , then the fixed point of will be unique.

Proof. Defining . Then, by Remark 1 is a -multiplicative space. By taking exponential on both sides of inequality (46), we havewhere . Now, by taking exponential on both sides of inequality (45) and by using Remark 2, we havefor all belong to the closed set . Now, by using Remarks 1 and 2, we havefor all belong to the closed set . Now, by Theorem 1, has a unique fixed point in or .

Example 7. Let endowed with the -metric for all and be defined byand be given byLet and , then is closed. Now, and . Also, is a locally -admissible mapping on . If belong to such that , then and . Taking , , and , we haveSo, is cyclic --Hardy–Rogers-type local contraction on . Also,Now, if is a sequence in such that and as . Then, . Hence, and . So, all hypotheses of Theorem 9 are satisfied, and therefore has a unique fixed point.
Note that the results of Alizadeh et al. [26] and other results for -admissible mapping cannot be applied. Since , but . Also, , but . Therefore, is not a cyclic -admissible mapping.
Now, we give an example of a mapping which is a cyclic -admissible, but none of the previously defined contractions in other papers holds. Therefore, other results for -admissible mapping fail to ensure the existence of a fixed point. However, has a fixed point, and our result is valid for such mappings.

Example 8. Let endowed with the -metric for all and be defined byand be given byBy taking , , , , and , all hypotheses of Theorem 9 are satisfied, and therefore has a unique fixed point. Note that, is a cyclic -admissible mapping on , but all other results for -admissible mapping cannot be applied. For example, defining by and . Let . Then, and . If and , thenThat is, Theorem 2.4 of [26] cannot be applied here.

Definition 9. In Definition 8, if(1)If , then we say it is cyclic --Banach-type local contraction on (2)If , then we say it is cyclic --Kannan-type local contraction on (3)If , then we say it is cyclic --Chatterjea-type local contraction on (4)If , then we say it is cyclic -Hardy–Rogers-type local contraction on (5)We exclude the role of functions and ; that is, if we exclude conditions (1) and (2) and the restriction from Definition 8, then we say it is -Hardy–Rogers-type local contraction on

Example 9. If we consider endowed with the -metric for all and define as in Example 7, then by taking and , we can get cyclic --Banach-type local contraction on . Similarly, if we consider and in Example 7, we can get cyclic --Kannan-type local contraction on . Also, if we take, and , then we can get cyclic --Chatterjea-type local contraction on . If we exclude the role of functions and in Example 7, then we can get an example of -Hardy–Rogers-type local contraction on .

Remark 3. By using Definition 9, we can make five new theorems in b-metric spaces.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.