Solving a Fractional Differential Equation via the Bipolar Parametric Metric Space

Abstract

In this paper, we propose the notion of the bipolar parametric metric space and prove fixed point theorems. The proved results generalize and extend some of the well-known results in the literature. An example and application to support our result is presented.

1. Introduction

Fixed point theory plays a vital role in applications of many fields of mathematics. Discovering FPs (fixed points) of generalized contraction maps has become an exciting field of study in the FP theory. Many researchers have recently released articles on FP theorems and applications in a variety of ways. One of the most recent topics in the FP theory is the presence of FPs in contraction maps in BPMSs (bipolar metric spaces), which can be thought of as generalizations of the Banach contraction principle. In 2016, Mutlu and Gurdal [1] have developed the concepts of BPMS, and they investigated certain basic FP and coupled FP results for covariant and contravariant maps under contractive conditions; see [1, 2]. In BPMSs, a lot of significant work has been done (see [3–9]). In 2021, Gaba et al. [10] proved FP theorems on BPMS. Mani et al. [11] developed the concept and proved coupled fixed point theorems in algebra-valued bipolar metric spaces (see [12–14]).

The notion of the parametric metric space was introduced in 2014. Rao et al. [15] presented parametric S-metric spaces and proved common FP theorems. In 2016, Krishnakumar and Nagaral [16] extended the Banach fixed point theorem to continuous mappings on complete parametric b-metric spaces. Tas and Ozgur [17] introduced parametric -metric spaces, obtained some FP results, and proved a fixed-circle theorem on a parametric -metric space as an application. Younis and Bahuguna [18] initiated the concept of controlled graphical metric type spaces, with integrate-controlled metric type spaces, extended b-metric type spaces, and graphical type spaces; also, finding a nonlinear model of a rocket’s ascending motion as an application. In 2023, Younis et al. [19] developed FP theorems in graphical spaces to show a solution to fourth-order two-point boundary value problem expressing elastic beam deformations. Smarandache et al. [20] demonstrated the quadruple neutrosophic theory and its applications. Ahmad et al. [21] demonstrated FP solutions in graphical bipolar b-metric spaces, applying covariant and contravariant mapping contractions. In this paper, we present the notion of BPPMS (bipolar parametric metric space) and prove FP theorems on BPPMS.

2. Preliminaries

In this section, we present some basic definitions. Mutlu and Grdal [1] proposed bipolar metric spaces and proved fixed point theorems.

Definition 1 (see [1]). Let and be nonempty sets and be a function s.t. (such that)(a)If , then , for all .(b)If , then , for all (c), for all (d), for all and .The triplet is called a BPMS.

Now, we introduce the notion of BPPMSs.

Definition 2. Let and be nonempty sets and be a function s.t.(a)If for all , then , for all .(b)If , then , for all and (c), for all and (d), for all , , and .The triplet is called a BPPMS.

We introduce the notions of covariant mapping, contravariant mapping, convergent sequence, Cauchy sequence, and continuous and contraction mapping as follows.

Definition 3.  (A1) Let be a BPPMS. Then, the points of the sets , , and are named as left, right, and central points, respectively, and any sequence, that is consisted of only left (or right, or central) point is called a left (or right, or central) sequence on . (A2) Let and be BPPMSs and be a function. If and , then is called a covariant map, or a map from to , and this is written as . If is a map, then is called a contravariant map from and this is denoted as .

Definition 4. Let be a BPPMS. A left sequence converges to a right point if and only if for every , there exists an s.t. for all and . Similarly, a right seqence converges to a left point if and only if, for every we can find an , satisfying whenever .

Definition 5. Let be a BPPMS.(i)A sequence on the set is called a bisequence on .(ii)If both and are convergent, then the bisequence is called convergent. If and both converge to a same point , then this bisequence is called biconvergent.(iii)A bisequence on is called Cauchy bisequence, if for each , we can find a number , satisfying for all positive integers .

Definition 6. Let and be BPPMSs.(i)A map is said to be continuous at a point , if for every , we can find a satisfying whenever , , and , . It is continuous at a point if for every , we can find a satisfying whenever , , and , . If is continuous at each point and , then it is called continuous.(ii)A contravariant map is continuous iff it is continuous as a covariant map .This definition implies that a contravariant map or a covariant from to is continuous, if and only if on implies on .

Definition 7. Let and be BPPMSs and . A covariant map s.t.or a contravariant map s.t.is called Lipschitz continuous. If  = 1, then this covariant or contravariant map is said to be nonexpansive, and if it is fulfilled for a , it is called a contraction.

3. Main Results

In this section, we prove FP theorems on BPPMS.

Theorem 8. Let be a complete BPPMS and given a covariant contraction . Then, the function has a UFP (unique fixed point).

Proof. Let and . For each , define and . Then, is a bisequence on . Say and . Then, for all ,and also,and similarly, . Let . Since , we can find an satisfying . Then,and is a Cauchy bisequence. Since is complete, converges and thus biconverges to a point andguarantees that has a unique limit. Since is continuous, , so . Hence, is a FP of . If is any FP of , then implies that and we havewhere , which implies , and so .

Example 1. Let and be equipped with for all , , and . Then, is a complete BPPMS. Define given by. Let and , thenTherefore, all the conditions of Theorem 8 are satisfied and has a UFP .

Example 2. Let , and the map defined byfor all , and . Then, is a complete BPPMS. Define given byfor all . Now,for all and . All the axioms of Theorem 8 are verified with and has a unique fixed point where is the null matrix.

Theorem 9. Let be a complete BPPMS and given a contravariant contraction . Then, the function has a UFP.

Proof. Let . For each , define and . Then, is a bisequence on . SayThen, for all ,Now, since , for any , we can find an integer satisfyingHence,and is a Cauchy bisequence. Since is a complete BPPMS, converges, and as a convergent Cauchy bisequence, in particular, it biconverges. Let , where . Since the contravariant map is continuous,which derives thatand combining this with gives . Let be a FP of , then implies so thatwhich gives . Hence, .

Example 3. Let and be equipped with for all , , and . Then, is a complete BPPMS. Define given by. Let and , then we can easily getTherefore, conditions of Theorem 9 are satisfied and has a UFP .

Finally, we express a theorem based of Kannan’s FP result [22].

Theorem 10. Let , where is a complete BPPMS and let satisfieswhich holds for all , , and . Then, the function has a UFP.

Proof. Let , for each non-negative integer , we define and . Then,for all integers . Then,andso thatIf we say , then we have since . Now,and for all ,if , andif . Since . Therefore, is a Cauchy bisequence. Since is complete. Then, and . SinceOn the other hand,which in turn implies that . Hence, . If is any FP of , then implies that is in . Then,Consequently, .
We conclude by establishing a theorem based on the Reich-type FP theorem [23].

Theorem 11. Let be a complete BPPMS. Consider the mapping s.t.for all and , where s.t. . Then, the function has a UFP.

Proof. Let . Define and for all . Then, we havefor all integers . Now,soIf we say and , then we have . Now,For all natural numbers , we haveFor all natural numbers , we haveTherefore, is a Cauchy bisequence. Since is complete BPPMS, , where . SinceOn the other hand,Therefore, . Hence, . If is any FP of , then , implies that . Then,Consequently, .

Example 4. Let and be equipped with for all , , and . Then, is a complete BPPMS. Define given by. Let and , thenAll the axioms of Theorem 11 are verified with , , and has a unique fixed point .

4. Application to Fractional Differential Equations

We recall many important definitions from the fractional calculus theory. For a function , the Reimann–Liouville fractional derivative of the order is given byprovided that the right hand side is pointwise defined on , where is the integer part of the number is the Euler gamma function.

Consider the following fractional differential equation:where is a continuous function and represents the Caputo fractional derivative of order and it is defined by

Let . Let . Define is given byfor all and . Obviously, axioms (a), (b), and (c) are satisfied. Now, we prove the axiom (d). For this,Taking the supremum on both sides, we getfor all , , and . Then, is a complete BPPMS.

Theorem 12. Assume the nonlinear fractional differential equation (46). Suppose that the following conditions are satisfied:(i)There exists , , and s.t. (ii)Then, equation (46) has a unique solution in .

Proof. The given equation (46) is equivalent to the succeeding integral equationwhereDefine the covariant mapping defined byNow,Taking the supremum on both sides, we getHence, all the hypothesis of Theorem 8 are satisfied and consequently, equation (46) has a unique solution.

5. Conclusion

The idea of BPPMS was introduced in this article and FP theorems were demonstrated. An illustrative example is provided that show the validity of the hypothesis and the degree of usefulness of our findings.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally in writing this paper. Furthermore, this manuscript were read and approved by all the authors.