Results on Implicit Fractional Pantograph Equations with Mittag-Leffler Kernel and Nonlocal Condition

Almalahi, Mohammed A.; Panchal, Satish K.; Jarad, Fahd

doi:https://doi.org/10.1155/2022/9693005

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9693005 | https://doi.org/10.1155/2022/9693005

Results on Implicit Fractional Pantograph Equations with Mittag-Leffler Kernel and Nonlocal Condition

Mohammed A. Almalahi,¹Satish K. Panchal,²and Fahd Jarad^3,4

Academic Editor: Phang Chang

Received12 Feb 2022

Revised11 Mar 2022

Accepted21 Mar 2022

Published17 May 2022

Abstract

In this study, the main focus is on an investigation of the sufficient conditions of existence and uniqueness of solution for two-classess of nonlinear implicit fractional pantograph equations with nonlocal conditions via Atangana–Baleanu–Riemann–Liouville (ABR) and Atangana–Baleanu–Caputo (ABC) fractional derivative with order . We introduce the properties of solutions as well as stability results for the proposed problem without using the semigroup property. In the beginning, we convert the given problems into equivalent fractional integral equations. Then, by employing some fixed-point theorems such as Krasnoselskii and Banach’s techniques, we examine the existence and uniqueness of solutions to proposed problems. Moreover, by using techniques of nonlinear functional analysis, we analyze Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability results. As an application, we provide some examples to illustrate the validity of our results.

1. Introduction and Motivation

Fractional calculus and its applications have increased in popularity because of its utility in modeling a wide range of intricate processes in science and engineering [1–5]. In order to meet the need to model more real-world problems, new approaches and techniques have been created in various fields of science and engineering to characterize the dynamics of real-world events. Until 2015, all fractional derivatives had single kernels. So, simulating physical events based on these singularities is difficult. In 2015, Caputo and Fabrizio (C-F) studied a novel type of fractional derivative (FD) in the exponential kernel [6]. In [7], Atangana and Baleanu (AB) investigated a novel form of FD using Mittag-Leffler kernels. In [8], Abdeljawad expanded the Atangana and Baleanu FD to higher arbitrary orders and established the integral operators associated with them. In [9, 10], Abdeljawad and Baleanu discussed the discrete forms of the new operators. For some theoretical work on Atangana–Baleanu FD, we refer the reader to a series of papers [11–14]. Traditional fractional operators cannot adequately describe some models of dissipative events, which is why fractional derivatives with nonsingular kernels are useful. For further details on the modeling and applications of the AB fractional operator (see [15–17]). The ABC fractional derivative is often used to simulate physical dynamical systems because it accurately represents the processes of heterogeneity and diffusion at various scales (see [18–21]). For the existence and uniqueness, as well as stability results regarding ABC and ABR operators, we refer the readers to a series of papers [22–25]. The challenge arises from the fact that the semigroup property in the ABC fractional derivative is not satisfied. In this paper, we introduce some properties of solutions to the implicit pantograph fractional differential equation without using the semigroup property.

The topic of stability arose from Ulam’s question regarding the stability of group homomorphisms in 1940 (see [26]). In the next year, Hyers [27] offered a positive interpretation of the Ulam issue in Banach spaces, which was the first significant development and step toward additional answers in this area. Since then, some researchers have published different generalizations of the Ulam result and Hyers theory. In 1978, Rassias [28] presented a generalized Hyers concept of mappings over Banach spaces. The Rassias result grabbed the attention of a large number of mathematicians from across the world, who began investigating the problems of functional equation stability. In stochastic analysis, financial mathematics, and actuarial science, these stability results are often employed. Calculating the Lyapunov stability for various nonlinear fractional differential equations is difficult and time-consuming, as everyone knows, and constructing the correct Lyapunov function is also a difficulty. Stability means that the solution of the differential equation will not leave the -ball. But asymptotic stability means that the solution does not leave the -ball and goes to the origin. Asymptotic stability implies stability, but the converse is not true in general (see [29]). For nonlinear fractional differential equations that deal with the nonlocal conditions, Ulam–Hyers’s stability is ideal. Not only Ulam-Hyers’s stability but also the existence and uniqueness of fractional differential equation solutions have attracted a large number of scholars.

The pantograph is a vital component of electric trains that collects electric current from overload lines. The pantograph equations have been modeled by Ockendon and Tayler [30]. Many researchers who are convinced of the relevance of these equations have extended them into numerous types and shown the solvability of such problems both theoretically and quantitatively (for additional details, see [31–35] and the references therein). Many researchers have investigated the existence and UH stability results of fractional pantograph differential equations using various forms of FD. For example, Almalahi et al. [36] studied the existence and uniqueness results of the following Hilfer–Katugampola boundary value problems.where are the Hilfer–Katugampola (FD) of order and , respectively, and type , , are the generalized fractional integral of order , respectively, , and are prefixed points.

Ahmed et al. [37] studied some properties of the solutions of the boundary impulsive fractional pantograph differential equation. In [38], the authors considered the pantograph problem as follows:the existence and uniqueness results were investigated using Banach’s contraction principle and Krasnoselskii fixed point theorem, and the Ulam–Hyers stabilities were addressed using Gronwall’s inequality in the context of ABC. Almalahi et al. [39] via Banach’s contraction principle and Krasnoselskii fixed point theorem studied the existence, uniqueness, and UH stability results of the following problems:where and are the ABR and ABC fractional derivatives of order and , respectively, is the AB-integral operator such that , , and is a continuous function.

Motivated by the argumentations above and due to the fact that the nonlocal condition is a suitable tool to describe memory phenomena like nonlocal elasticity, propagation in complex media, polymers, biological, porous media, viscoelasticity, electromagnetics, electrochemistry, etc. We intend to analyze and investigate the sufficient conditions of solution for the following two-class of nonlinear implicit fractional pantograph equations with ABR and ABC fractional derivatives in order with nonlocal conditions as follows:where are respectively the ABR and ABC-FD of order , and are prefixed points such that , , and is continuous function satisfies some condition described later.

It is notable that nonlocal Cauchy type problems may be employed to explain differential rules in the growth of a system. These equations are frequently used to explain non-negative values such as a species’ concentration or the distribution of mass or temperature. Before studying any model of real-world phenomena, the first question to address is whether the problem genuinely exists or not. The fixed-point theory provides the answer to this question.

The contribution of the current works is as follows:(i)In this paper, we will study two types of fractional problems involving new higher-order fractional operators via ABC and ABR operators, which have recently been expanded by Abdeljawad.(ii)To our knowledge, this is the first study that deals with high-order ABC and ABR fractional derivatives. As a result, our findings will be a valuable addition to the current literature on these fascinating operators.(iii)We use a novel method to establish the existence and uniqueness of solutions for problems (4) and (5), as well as different types of stability results, without relying on the semigroup property and with a minimal number of hypotheses.(iv)If , then problems (4) and (5), respectively, reduces to the following implicit fractional differential equations:

The rest of this paper is organized as follows: in Section 2, we review several notations, definitions, and lemmas that are necessary for our analysis. In Section 3, we examine the existence and uniqueness results for problems (4) and (5) with ABC and ABR derivatives with the nonlocal condition. In section 4, we address the stability results of problems (4) and (5). We present two examples to demonstrate the validity of our results in section 5. In the concluding part, we will provide some last observations regarding our findings.

2. Preliminaries and Auxiliary Results

Let , , and be the space of continuous functions with the norm . Then is a Banach space.

Definition 1 (see [7]). Let . Then, the following expressions,are called and fractional derivatives of order for a function , respectively. is the normalization function that satisfies and , and is the Mittag-Leffler function defined byThe fractional integral is given by

Definition 2 (see [8] Definition 3.1). Let us assume that and . We set . Then, and the following expressionsare called the left-sided and fractional derivatives of order for a function . The correspondent (FI) is given by

Lemma 1 (see [8] Proposition 3.1). If is a function defined on and , then, for some , we have(i)(ii)(iii)

Theorem 1 (see [40]). Let be a closed subset from a Banach space , and let be a strict contraction such that for some for all . Then has a fixed point in .

Theorem 2 (see [41]). Let be a Banach space, let a set be a nonempty, closed, convex, and bounded set. If there are two operators such that (i) , for all , (ii) is compact and continuous, and (iii) is a contraction mapping, then there exists a function such that .

Lemma 2 (see [8] example 3.3). Let and . Then, the solution to the following linear problemis given bywhere

3. Equivalent Integral Equations

In this section, we will derive the formula of the equivalent integral equations for problems (4) and (5).

3.1. Equivalent Integral Equations for the Problem (4)

Lemma 3. Let and . A function is a solution to the following ABR-problemthen, satisfies the following fractional integral equation:

Proof. By (see [8] Theorem 4.2), the solution of is given aswhere is an arbitrary constant andNow, we replace with into (17) and multiply by , we getMaking use of the condition , we haveSubstituting in (17), we get (16). Conversely, let us assume that satisfies (16). Then, by applying the operator on both sides of (16) and using Lemmas 1, we obtainNext, we replace by in (16) and multiply by , we getThus, the nonlocal condition is satisfied.

Theorem 3. Let be a continuous function such that and . A function is a solution to the problem (4) if and only if satisfies the following fractional integral equation:where

Proof. According to Lemma 3, the solution to problem (4) is given byBy definition in the case , we haveBy (26), we can rewrite (25) as follows:By (24), we get

3.2. Equivalent Integral Equations for the Problem (5)

Theorem 4. Let be a continuous function such that and . A function is a solution to the problem (5) if and only if satisfies the following fractional integral equation:where

Proof. Let us assume that is a solution of the first equation of (5). Then, by Lemma 2, we getBy conditions and by the same technique of Theorem 3, we can easily get (29).

4. Main Results

4.1. Existence and Uniqueness of Solutions for Problem (4)

In this subsection, we will discuss the existence and uniqueness results for the ABR problem (4). For simplicity, we set

Theorem 5. Suppose that is a continuous function such that and . Moreover, we assume that there is a constant number such thatThen the ABR problem (4) has a unique solution provided that .

Proof. On the light of Theorem 3, we define the operator Let us consider a closed ball defined aswith radius , whereNow, we show that . For all and , we haveBy , we haveHenceThus, . Now, we will prove that is a contraction map. Let and . ThenFrom our assumption, we obtainHenceSince , we deduce that is a contraction. Hence, Theorem 1 implies that has a unique fixed point. Consequently, the ABR problem (4) has a unique solution.

Theorem 6. Let us assume that the hypothesis in Theorem 5 is satisfied. Then, the ABR problem (4) has at least one solution.

Proof. Let us consider the operator , which is defined in Theorem 5 such that , whereLet be a closed ball defined aswith radius , whereIn order to apply Krasnoselskii fixed point theorem, we split the proof into the following steps:

Step 1. We show that for all . First, for the operator . For and , we haveBy (38), we haveNext, for the operator , we haveBy inequalities (47) and (48), we haveThus .

Step 2. is a contraction map. Due to the operator being a contraction map, we conclude that is a contraction too.

Step 3. is continuous and compact. Since is continuous, is continuous too. Also, by (48), is uniformly bounded on . Now, we show that is equicontinuous. For this purpose, let . Then, we haveThusIn view of the previous steps with the theorem of Arzela–Ascoli, we deduce that is relatively compact. Consequently, is completely continuous. Hence, Theorem 2 shows that ABR problem (4) has at least one solution.

4.2. Existence of Unique Solutions for Problem (5)

Theorem 7. Suppose that is a continuous function such that and . Moreover, we assume that there is a constant number such thatThen the ABC problem (5) has a unique solution, provided thatwhere

Proof. In view of Theorem 4, we define the operator byLet us consider a closed ball aswith radius , whereNow, we show that . For all and , we haveBy , we haveHenceThus, . Now, we prove that is a contraction. Let and . ThenFrom (41), we obtainDue to condition (53), we conclude that is a contraction. Hence, via Theorem 1, we conclude that has a unique fixed point. Consequently, the ABC problem (5) has a unique solution.

4.3. Ulam–Hyers Stability for the Problem (4)

The UH and GUH stabilities for problem (4) are discussed in this subsection. For , the following inequality is taken into account:

Definition 3 (see [42]). The problem (4) is UH stable if there exists a real number such that, for each and each solution of inequality (63), there is a unique solution of (4) withFurthermore, the problem (4) is GUH stable if we can identify with such that

Remark 1. Let be the solution to inequality (63) if and only if we have a function that depends on such that(i) for all ,(ii), .

Lemma 4. If is a solution to inequality (63), then satisfies the following inequality:where

Proof. In view of Remark 1, we haveThen, by Lemma 3, we getwhich implies

Theorem 8. Suppose that is a continuous function such that and . Moreover, we assume that there is a constant number such thatIfthen, the ABR problem (4) is UH stable.

Proof. Let and satisfies the inequality (63), and let be a unique solution to the following problem:Then, by Lemma 3, we getwhereSince and . Then and hence by Lemma 4, we haveThuswhereNow, by choosing such that , then the ABR problem (4) has GUH stability.

4.4. Ulam–Hyers Stability for the Problem (5)

The UH and generalized UH stabilities for problem (5) are discussed in this subsection.

Lemma 5. If is a solution of the inequalitythen satisfies the following inequality:where

Proof. By the same technique of Lemma 4, one can prove it. So, we omit the proof here.

Theorem 9. Suppose that is a continuous function such that and . Moreover, we assume that there is a constant number such thatIfthen the ABC problem (5) is UH stable.

Proof. Let and satisfies inequality (79), and let be the unique solution to the following problem:Then, by Theorem 4, we getwhereSince and . Then and hence by Lemma 5, we haveThuswhereNow, by choosing such that , then the ABC problem (5) has GUH stability.

4.5. Examples

Example 1. Consider the following ABR fractional problem:Here andLet . ThenTherefore, hypothesis holds with . Also , and . Then all conditions in Theorem 5 are satisfied and hence the ABR-problem (4) has a unique solution. For every and each satisfiesThere exists a solution to the ABC problem (5) withwhereTherefore, all conditions in Theorem 8 are satisfied and hence the ABR problem (4) is UH stable.

Example 2. Consider the following ABC fractional problemHere . Let . ThenTherefore, the hypothesis holds with . Also and . Then all conditions in Theorem 7 are satisfied and hence the ABC problem (5) has a unique solution.

5. Conclusion remarks

The theory of fractional operators in the Atangana–Baleanu framework has recently sparked interest, prompting some scholars to investigate and create certain qualitative features of solutions to FDEs employing such operators. We developed and investigated adequate guarantee conditions for the existence and uniqueness of solutions for two classes of nonlinear implicit fractional pantograph equations with the interval ABC and ABR fractional derivatives, subjected to nonlocal condition.

The reduction of ABC-type pantograph FDEs to FIEs, as well as various Banach and Krasnoselskii’s fixed point theorems, are the foundations of our technique. In addition, we used Gronwall’s inequality in the context of the AB fractional integral operator to derive suitable conclusions for various forms of UH stability. The results are supported by relevant instances. The problems under consideration are also true in some particular circumstances, i.e., they may be reduced to problems containing the Caputo–Fabrizio fractional derivative operator. Furthermore, the examination of the generated findings was kept to a bare minimum.

Data Availability

The data available upon requested.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is conducted during our work at Hajjah University (Yemen). The authors would like to thank the reviewers and editor for useful discussions and helpful comments that improved the manuscript.

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