Abstract

This paper is aimed at proving the existence and uniqueness of a common fixed point for a pair of -interpolative Hardy-Rogers-Suzuki-type contractions in the context of quasipartial -metric space. Thus, several results in literature such as Hardy and Rogers, Suzuki, and others have been generalized in this work. We also offer a demonstrative example and an application of fractional differential equations to validate the findings.

1. Introduction and Preliminaries

Fixed-point theory is one of the fascinating research areas in pure mathematics, which has many applications in both pure and applied mathematics. Picard presented an iterative procedure for the solution of a functional equation first time in his research paper. This notion was later developed into an abstract framework by the Polish mathematician Stephan Banach [1] who presented a powerful tool known as the Banach contraction principle to determine the fixed point of mapping in complete metric space. It states as follows:

Theorem 1 (see [1]). Let be a complete metric space and let be a contraction; that is, there exists a number such that for all Then, has a unique fixed point in
By altering the contraction conditions, maps, and other conditions, several researchers have generalized the Banach contraction principle.

The Banach contraction principle needs continuity of the map involved in the contraction condition. In 1968, Kannan [2] relaxed the continuity condition and introduced a new fixed-point theorem with a new contraction condition as follows:

Theorem 2. Let be a complete metric space. A mapping is said to be a Kannan contraction if there exists such that for all Then, possesses a unique fixed point.

In 2018, Karapinar first established the interpolative Kannan-type contraction in his paper [3] as follows:

Definition 3. Let be a metric space. We say that the self-mapping is an interpolative Kannan-type contraction, if there exists a constant and such that for all with

Karapinar et al. [4] proved some results in the setting of -interpolative contractions. Again in 2021, Khan et al. [5] proved some fixed-point results on the interpolative -type -contraction. For more results on interpolative-type contractions, one can see [68] and the references therein.

Following the results due to Karapinar et al. [9], Gaba and Karapinar [10] introduced a new approach to the interpolative contraction as follows:

Definition 4 (see [10]). Let be a metric space and be a self-map. We shall call a relaxed -interpolative Kannan contraction, if there exists such that

Gaba and Karapinar [10] introduced the following definition of optimal interpolative triplet as follows:

Definition 5 (see [10]). Let be a metric space and be a relaxed -interpolative Kannan contraction. The triplet will be called an “optimal interpolative triplet” if for any the inequality (4) fails for at least one of the triplet

In view of the above definitions, Gaba and Karapinar [10] proved the following theorem:

Theorem 6 (see [10]). Let be a complete metric space, and be a -interpolative Kannan contraction with so that Then, has a fixed point in

In 1973, Hardy and Rogers [11] introduced a natural modification of the Banach contraction principle.

Theorem 7. Let be a complete metric space. The mapping is called an interpolative Hardy-Rogers type of contraction if there exist positive real numbers , with such that for each . Then, a mapping has a unique fixed point in .

Later in 2018, Karapinar et al. [12] introduced the following notion of interpolative Hardy-Rogers-type contraction.

Theorem 8 (see [12]). Let be a complete metric space. The mapping is called an interpolative Hardy-Rogers type of contraction if there exist and positive reals , with such that for each . Then, a mapping has a unique fixed point in .

Several other versions of this type of results were proven by researchers. Some of them can be seen in [9, 1315].

In 2008, Suzuki [16] introduced a generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. The generalized result is as follows:

Theorem 9 (see [16]). Let be a complete metric space and let be a mapping such that for all , where is a nonincreasing function defined by Then, there exists a unique fixed-point . A mapping satisfying (7) is called as the Suzuki contraction.

Example 10 (see [16]). Let with a metric be defined by Define a mapping Then, the map satisfies all the hypotheses of Theorem 9, and is the unique fixed point of . However, for and , Thus, does not satisfy the assumptions in Theorem 9 for any .

In 2021, Yeşilkaya [17] generalized the Banach contraction principle to -interpolative Kannan contraction as follows:

Definition 11 (see [17]). Let be a metric space. The mapping is called an interpolative Hardy-Rogers contraction of the Suzuki type. If there exist , , and positive reals with such that where is the set of all nondecreasing self-mappings on such that for all .

Similar results can be seen in [6, 7] and the references therein.

In 2012, Wardowski [18] generalized the Banach contraction principle into -contraction mapping principle as follows:

Definition 12 (see [18]). Let be a metric space. A mapping is called an -contraction if there exist and such that holds for any with , where is the set of all functions satisfying the following conditions:
() is strictly increasing: ,
() For each sequence in , ,
() There exists such that .

We denote by the set of all functions satisfying the conditions and

Example 13 (see [18]). The following are the elements of (1)(2)(3)(4)

In 2013, Salimi et al. [19] and Hussain et al. [20] modified the notions of -contractive and -admissible mappings and established certain fixed-point theorem as given below:

Definition 14 (see [19]). Let be a self-mapping on and be two functions. We say that is an -admissible mapping with respect to if

Remark 15. It should be noted that Definition 14 reduces to -admissible mapping definition due to Samet et al. [21] if we assume that . Furthermore, if we suppose that , we may argue that is an admissible -sub admissible mapping.

Note that a self-map can be -orbital admissible as stated in the definition below:

Definition 16 (see [11]). Let be a self-map defined on , and be a function. is said to be an -orbital admissible if for all , we have

Gopal et al. [22] established the idea of -type -contractions and -type -weak contractions by combining the concepts of -admissible mappings with -contractions and -weak contractions:

Definition 17 (see [22]). Let be a metric space and be a mapping. Suppose be a function. The function is said to be an -type -contraction if there exists such that for all

In 2019, Dey et al. [23] introduced the notion of generalized --contraction mapping as follows:

Theorem 18 (see [23]). Let be a metric space and be a mapping. Let be a function and The function is said to be a modified generalized -contraction mapping if there exists such that for all where

Later, Wangwe and Kumar [24] proved results for -type contractions. One can see more results in [2528] and the references therein.

contraction mapping of Hardy-Rogers type was introduced by Cosentino and Vetro [29] as follows:

Definition 19 (see [29]). Let be a metric space. A self-mapping on is called an -contraction of Hardy-Rogers type if there exists and such that for all with where Moreover, is said to be a -contraction of Suzuki-Hardy-Rogers type [30] if contraction Condition (18) holds for all with and

Many researchers generalized the concept of metric space. The concept of -metric space was first introduced by Bakhtin in 1989. By adding a variable to the definition of metric space, the triangle inequality in this concept was relaxed as follows:

Definition 20 (see [31]). A b-metric on a nonempty set is a function such that for all and for some real number , it satisfies the following: (i)if , then ,(ii)(iii),then, a pair is called -metric space.

In 2021, Pauline and Kumar [32] presented an extension of the fixed-point theorem for T-Hardy-Rodgers contraction mappings in -metric space. Czerwick [33] proved the existence of fixed point in -metric space as follows:

Theorem 21 (see [33]). Let be a topological space and let be a complete b-metric space. Let be continuous and satisfy for each for all where . Then for each , there exists a unique fixed-point of , i.e., and the function is continuous on

In 1994, Matthews [34] introduced partial metric space as a result of the failure of metric functions in computer science as follows:

Definition 22 (see [34]). Let . A partial metric is a function satisfying (i),(ii)If , then ,(iii) for all .Then, a pair is called partial metric space. It is clear that if , then ; however, if then may not be zero.

Remark 23 (see [34]). As partial metrics have a wider range of topological features and may easily support partial ordering, partial metrics are more versatile than metric spaces.

Künzi et al. [35] proposed the idea of partial quasimetric by eliminating symmetry condition from the notion of partial metric space.

Definition 24 (see [35]). A quasipartial metric on a nonempty set is a function such that (1) whenever ,(2) whenever ,(3) whenever ,(4) if and only if whenever .A pair is called a quasipartial metric space.

In 2015, Gupta and Gautam [36] introduced the notion of quasipartial -metric space as follows:

Definition 25 (see [36]). A quasipartial b-metric on a nonempty set is a function such that for some real number , it satisfies the following: (i)if , then (indistancy implies equality),(ii) (small self-distances),(iii) (small self-distances)(iv) (triangularity), for all .Then, the pair is quasipartial b-metric on space .

Example 26 (see [36]). Let be the set of all real numbers. Define by

Then, it is a quasipartial -metric on

Gautam et al. [37, 38] extended several results in quasipartial -metric spaces.

In this article, we establish the existence and uniqueness of fixed-point theorems for - interpolative Hardy-Rogers-Suzuki-type contraction in a compact quasipartial -metric spaces with an application to fractional differential equations. An example is given to use the results that have been proven. The outcomes of this study will generalize several results obtained in [11, 12, 1618, 25, 39, 40] and the references therein.

2. Main Results

To establish our first main results, we will begin by generalizing Definition 11 and extend it to a compact quasipartial -metric space.

Definition 27. Let be a compact quasipartial -metric space. A map is called -interpolative Hardy-Rogers contraction of Suzuki type, if there exist where is the set of all nondecreasing self-mappings on such that for all and , with

We now present our main theorem as follows:

Theorem 28. Let be a compact quasipartial -metric space and be -interpolative Hardy-Rogers contraction of Suzuki type. If is -orbital admissible mappings such that for some Then, a mapping has a fixed point in if at least one of the following properties holds (i) is -regular(ii) is a continuous map(iii) is continuous, where

Proof. Let satisfies We construct a sequence as shown below Assume that for some so that is a fixed point of . Thus on contrary, we can suppose that for each . As is -orbital admissible implies that Similarly, continuing this process, we get a sequence, By substituting and in Definition 27, we obtain Thus, using for , we have Assuming that, for all then Thus, which is a contradiction. Hence, we get Then, the positive sequence is a nonincreasing sequence with positive terms, so we attain that there exists such that Accordingly, we get Furthermore, using Equation (32), or equivalent Hence, by repeating this condition, we can write Now, we claim that is a Cauchy sequence in Then, we shall use the triangle inequality with Equation (41) for and find that Letting in Equation (42), we find that is a Cauchy sequence in . Regarding that is complete, there exists such that We will show that the point is a fixed point of . If Equation (32) holds, that is, is -regular, then verify Equation (32), that and we get We assert that or Assuming on the contrary that and Using triangle inequality for , we obtain which is a contradiction. Therefore, either or holds. In case that inequality (46) holds, we get If Equation (47) holds, we have Therefore, letting in Equations (54) and (55), we get , that is, In case that assumption (47) is true, that is the mapping is continuous, and we want to show that also Assuming on the contrary that Since, by Equation (47), we get which is a contradiction. Consequently, that is, is a fixed point of .

The following corollary is obtained by substituting in Theorem 28.

Corollary 29. Let be a complete and compact metric space and be self-mapping on , such that implies for each where and positive real , with . Then, has a fixed point in .

Proof. In Theorem 28, it is sufficient to get for proof.

Further, taking , with in Corollary 29, we obtain the following Corollary.

Corollary 30. Let be a compact quasipartial -metric space and be a self-mapping on space such that implies that for each , where positive reals , with . Then, has a fixed point in .

Remark 31. If we replace the quasipartial -metric space by the metric space in Theorem 28, then we get the result due to Yeşilkaya [17] as a corollary.

Kumar [27] discussed the concept of orbital continuity. Using this concept, we formulate the following example which validates the result proved in Theorem 28.

Example 32. Let and Here, is a complete and compact quasipartial -metric space defined by and further, let The mapping is not continuous but since we have is continuous mapping. Let a function defined as and we choose , and . Then, we have to check if Theorem 28 holds. We have to consider the following cases: (i)For , we haveimplies (ii)For and , we haveimplies For all other cases, Theorem 28 holds, since As a result, the assumptions of Theorem 28 are satisfied, also the mappings has a fixed point

3. An Application to Fractional Differential Equations

Several authors gave solutions of fractional differential equations using fixed-point theorems. Some of them are worth noting in this direction [4145]. In this section, Theorem 28 is used to establish the existence and uniqueness of the solution of the fractional order differential equation. Here, we consider the following initial valued problem (IVP) of the form where is the standard Riemann-Liouville fractional derivative and is called a phase, space, or state space. Consider a quasipartial -metric on given by

then, it is obvious that is a compact quasipartial -metric space. If and , then for every is a -valued continuous function on . The space is complete by a solution of problems (77) and (78); we mean a space Therefore, a function is called a solution of Equations (77) and (78) if it satisfies the equation on and condition on

Lemma 33 (see [41]). Let and be continuous and Then, is a solution of the fractional integral equation if and only if is a solution of the initial value problem for the fractional differential equation

Theorem 34. Let . Assume that there exists such that for and . If where and then, there exists a unique solution for (IVP) (77) and (78) on the interval

Proof. We first transform the given initial value problem into a fixed point problem. For this, we consider an operator defined by Let be a function defined by Then, For each with , we denote by the function defined by If for every and the function satisfies Set Now, let be Hardy-Rogers-Suzuki operator be defined by The operator has a fixed-point equivalent to ; hence, we have to prove that has a fixed point. Indeed, if we consider that then for all , we have Therefore, Suppose and with such that implies that Thus, we deduce that the operator satisfy all the hypothesis of Theorem 28. Therefore, has a unique fixed point.

Data Availability

There is no data required in this research.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.