key: cord-0745754-tcvvj9yi
authors: Abdo, Mohammed S.; Abdeljawad, Thabet; Shah, Kamal; Jarad, Fahd
title: Study of impulsive problems under Mittag-Leffler power law
date: 2020-10-02
journal: Heliyon
DOI: 10.1016/j.heliyon.2020.e05109
sha: 2fa836576c653c0156a6cacb7c50015ef3ca1f40
doc_id: 745754
cord_uid: tcvvj9yi

This article is fundamentally concerned with deriving the solution formula, existence, and uniqueness of solutions of two types of Cauchy problems for impulsive fractional differential equations involving Atangana-Baleanu-Caputo (ABC) fractional derivative which possesses nonsingular Mittag-Leffler kernel. Our investigation is based on nonlinear functional analysis and some fixed point techniques. Besides, some examples are given delineated to illustrate the effectiveness of our outcome.

Fractional calculus has risen as a significant area of examination and the fertile area for research in light of the application of its tools in science and engineering majors. Fractional-order operators lead us to the emergence of more useful and concrete mathematical models as opposed to ordinary integer-order. It has been because of the non-local nature of the fractional operators, which describes the memory and it enables us to earn a closer look at the dynamics behavior and hereditary properties of the related phenomena. For the recent development of the theme, see the monographs [1, 2, 3] and the references referred to in that.

Different kinds of nonlocal fractional derivatives were suggested in the current literature to handle the reduction of classical derivatives operators. For example, the idea of fractional derivative in a Riemann-Liouville sense was introduced based on power-law. A new fractional derivative has proposed by Caputo-Fabrizio [4] relying on the exponential kernel. However, this operator it some troubles with regard to the locality of the kernel. Just recently, to get rid of Caputo-Fabrizio's trouble, Atangana and Baleanu (AB) in [5] have introduced a new modified version of a fractional derivative with the help of Mittag-Leffler function (MLF) as a nonsingular and nonlocal kernel. Due to generalized MLF is utilized as the nonlocal kernel and does not guarantee singularity, the ABC derivative provides an excellent memory description [6, 7, 8, 9] . 

where 0 <∝≤ 1, ∝ [ ] denotes the Atangana-Baleanu-Caputo (ABC) fractional derivative of order ∝, ∶ Ω × ℝ → ℝ is a given continuous function. Further (0, (0)) = 0 and also it vanishes at impulsive points , = 1, ... ,  ∶ ℝ → ℝ, = 1, ... , 0 ∈ ℝ, satisfy 0 = 0 < 1 < ... < < +1 = , Δ | = = ( + ) − ( − ) = ( + ) − ( ), ( + ) = lim ℎ→0 + ( + ℎ), ( − ) = lim ℎ→0 − ( + ℎ) represent the right and left limits of ( ) at = , = 1, .., , and ∶ (Ω, ℝ) → ℝ is given function. Also, [ ] = if ∈ ( , +1 ], = 0, 1, ... and 0 = 0.

We mention here that on the light of Theorem 3.11 in [34] , we must always have the necessary condition (0, (0) = 0 to confirm the initial data for the solution. To our knowledge, there are no much studies on Cauchy problems for impulsive FDEs in the literature, especially those involving an ABC fractional operator. For instance, we mention [35, 36, 37] and the references therein.

The major aim of the article is to obtain the formula of solutions for two types of impulsive FDEs with ABC fractional operators. Moreover, we proved existence and uniqueness theorems by means of some fixed point theorems of Banach, Schaefer, and Kransosekliskii for proposed problems in the frame of ABC derivatives. We realized that the condition (0, (0) = ( , ( ) = 0 ( = 1, ... ) is necessary to guarantee a unique solution.

This paper is coordinated as follows. Section 1 deals with the introduction which contains a survey of the literature. Section 2 consists of some foundation preliminaries related the fractional calculus and nonlinear analysis. The formula of the solution for the proposed problems is presented in Section 3. The existence and uniqueness results on a Cauchy problem and nonlocal Cauchy problem are obtained in Sections 4 and 5. In Section 6, two examples are specified to the validation of our results.

Consider the following space Definition 1. [5, 38] Let ∝∈ [0, 1] and ∈ 1 ( , ) ( < ). Then the left AB-Caputo and AB-Riemann-Liouville fractional derivatives of order ∝ for a function are described by

respectively, where  (∝) is the normalization function satisfies the result  (0) =  (1) = 1, and ∝ is called the MLF defined by

Definition 2. [5, 38] Let ∝∈ (0, 1] and ∈ 1 ( , ). Then the left AB fractional integral of order ∝ for a function is specified by

Definition 3. [5] The Laplace transform of ABC fractional derivative of ( ) is specified by

where  is the Laplace transform starting from defined by 

Lemma 2. [34, 39] For ∝∈ (0, 1], the solution of the following problem

is given by

Theorem 1. [40] Let ℵ be a Banach space and  be a non-empty closed subset of ℵ. If Π ∶  ⟶  is a contraction, then there exists a unique fixed point of Π.

be a bounded set. Then Π has at least one fixed point in ℵ.

Let ℍ be a non-empty, closed, convex subset of a Banach space ℵ and let 1 , 2 be two operators such that (i) 1 + 2 ∈ ℍ, ∀ , ∈ ℍ; (ii) 1 is compact and continuous; (iii) 2 is a contraction mapping. Then there exists ∈ ℍ such that 1 + 2 = .

Definition 5. A function ∈ (Ω, ℝ) is a solution of (1) if satisfies the equation The following lemma is a direct consequence of Theorem 3.1 in [34] . 

if and only if is a solution of the impulsive ABC-fractional FDE

where [ ] = if ∈ ( , +1 ], = 0, 1, ... and 0 = 0.

Proof. The proof is derived by using Lemma 1 repeatedly. Assume satisfies (6) Using Lemma 1, we get

This means that

After the impulse ( ( − 1 ) = ( + 1 ) −  1 ( − 1 )), we get

If ∈ ( 1 , 2 ], then vanishes at 1 imply

This means that

If ∈ ( 2 , 3 ], then vanishes at 2 imply

This means that

After the impulse ( ( − 3 ) = ( + 3 ) −  3 ( − 3 )), we get

Assume that and hence the solution becomes

Thus (5) is satisfied. Conversely, assume that satisfies the equation (1). If ∈ [0, 1 ] and (0) = 0, then (0) = 0 . Using the concept that ∝ is the left inverse of ∝ and using Lemma 1, we find that As well, we can simply infer that 

then the impulsive ABC-fractional FDE (1) has a unique solution on Ω, where

Proof. Thanks to Lemma 3, we define the mapping  ∶ (Ω, ℝ) → (Ω, ℝ) by

Now, we are required to prove that  has a fixed point. First we show that  ⊂  . For ∈  , we have

Thus,  maps  into itself. Next, we show that  is contraction on (Ω, ℝ). Let , * ∈ (Ω, ℝ) and ∈ Ω. Then we obtain

The inequality (9) shows that  is contraction on (Ω, ℝ). It follows from Theorem 1 that the impulsive ABC-fractional FDE (1) has a unique solution. □ Now, we prove the existence result of (1) by applying Theorem 2. Since and ( = 1, .., ) are continuous, it can be to checked that  is continuous.

Next, let  1 = { ∈ (Ω, ℝ) ∶ ‖ ‖  ≤ 1 } be a ball set with

where 0 ∶= sup ∈Ω | ( )| and is given by (10) . Then for ∈  1 and ∈ Ω, we have

( 2 − 1 ) ∝ .

This confirms that ( 1 ) is relatively compact for ∈ Ω. By Arzela-Ascoli's theorem, the operator  is compact on  1 .

Step 2: We prove that the set  = { ∈ (Ω, ℝ) ∶ =  for some ∈ (0, 1)} is bounded. Let ∈ . Then =  for some ∈ (0, 1). Hence, for ∈ Ω we obtain

Thus for every ∈ Ω, we have

This confirms that  is bounded. So, the deduction of Theorem 2 implies that  has a fixed point which is a solution of the impulsive ABC-fractional FDE (1). □

Here we treatise the existence of solution for the impulsive ABCfractional FDE (2) . Suppose satisfies the following condition: 

where is given by (10) , then the impulsive ABC-fractional FDE (2) has a unique solution on Ω.

Proof. Define the nonlinear mapping  * ∶ (Ω, ℝ) → (Ω, ℝ) as follows

Then,  * has a fixed point if and only if the impulsive ABC-fractional FDE (2) possesses a solution. Choosing * ≥

. From the hypothesis (H 5 ), it is easy to find that  *  * ⊂  * , where  * = { ∈ (Ω, ℝ) ∶ ‖ ‖  ≤ * }. We need to prove that  * is contraction. Let , * ∈ (Ω, ℝ) and ∈ Ω. Then we have

The inequality (12) shows that  * is contraction on (Ω, ℝ). Then the impulsive ABC-fractional FDE (2) has a unique solution by the application of Theorem 1. □

The following theorem based on Theorem 3.

then the nonlocal impulsive ABC-FDE (2) has a solution on Ω.

Proof. Consider the operator  * ∶ (Ω, ℝ) → (Ω, ℝ) defined by (13) , and we define the operators  * 1 and  * 1 on  as 

Therefore, the hypotheses (H 1 ) and (H 2 ) hold with = 1 18 and = 1 10 . We shall examine that the condition (9) holds too with = 1, = 

Theory and Applications of Fractional Differential Equations

Fractional Differential Equations

Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering

A new definition of fractional derivative without singular kernel

New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model

Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties

Fractional derivatives with no-index law property: application to chaos and statistics

Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena

Numerical computation of a fractional derivative with non-local and non-singular kernel

Solving space-fractional Cauchy problem by modified finite-difference discretization scheme

Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlev equations in Hilbert space

Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system

Mathematical analysis of a fractional differential model of HBV infection with antibody immune response

A fractional epidemiological model for computer viruses pertaining to a new fractional derivative

Ulam-Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions

On impulsive nonlocal integro-initial value problems involving multi-order Caputo-type generalized fractional derivatives and generalized fractional integrals

Existence results for fractional impulsive integrodifferential equations in Banach spaces

On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces

Impulsive fractional differential equations in Banach spaces

Existence and uniqueness of solutions to impulsive fractional differential equations, Electron

A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions

Existence of solutions of abstract fractional impulsive semilinear evolution equations

A survey on impulsive fractional differential equations

Impulsive fractional functional differential equations

Nonlinear impulsive fractional differential equations in Banach spaces

Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order

A singular ABC-fractional differential equation with p-Laplacian operator

New exploration on existence of fractional neutral integro-differential equations in the concept of Atangana-Baleanu derivative

Existence and multiplicity for some boundary value problems involving Caputo and Atangana-Baleanu fractional derivatives: a variational approach

Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative

On nonlinear hybrid fractional differential equations with Atangana-Baleanu-Caputo derivative

On comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative

Existence theory and numerical analysis of three species prey-predator model under Mittag-Leffler power law

On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative

Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations

Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations

Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system

Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels

Basic Theory of Fractional Differential Equations

Thus,  * 1 1 +  * 2 2 ∈  0 . Next, for any ∈ Ω and 1 , 2 ∈ (Ω, ℝ) we haveIt follows from (14) that  * 1 is a contraction mapping. Now, let us show that  * 2 is continuous and compact. Continuity of implies that  * 2 is continuous. Also,  * 2 is uniformly bounded on  0 because, for ∈  0 and ∈ Ω, we getNow, the operator  * 2 is compactness on  0 , since  * 2 ⊂ . Thus all assumptions in Theorem (3) are satisfied. Therefore, the nonlocal impulsive ABC-FDE (2) has a solution on Ω. □

Example 1. For ∝∈ (0, 1], we consider the following impulsive FDE with ABC fractional derivative1+ , for ∈ Ω, ∈ ℝ + , and ( ) = 10+ , for ∈ ℝ + .Clearly, (0, (0)) = ( 1 2 , ( 1 2 )) = 0. Let ∈ Ω and , ∈ ℝ + . Thenwhere. Clearly, (0, (0)) = ( 1 3 , ( 1 3 )) = 0. Let ∈ Ω and , ∈ ℝ + . Then Next, for all ∈ Ω, and ∈ ℝ + ,Thus, all the assumptions in Theorem 7 are satisfied, our result can be applied to the nonlocal impulsive FDE (16).

The theory of fractional calculus with nonsingular kernels is new and there is a need to study qualitative properties of differential equations involving such operators. In this paper, we have studied two types of Cauchy problems for impulsive FDEs rely on ABC fractional derivative which contains Mittag-Leffler Power Law. Also, we have derived the formula for the solution and examine the existence and uniqueness of the results of the problems (1) and (2) . The acquired results are extended to the Caputo impulsive fractional differential equation. The arguments are upon some fixed point theorems of Banach, Schaefer, and Krasnoselskii. The obtained results play a significant role in developing the theory of fractional analytical. In our fractional impulsive system, we consider the case when the starting point of the Caputo fractional derivative changes with respect to the impulse points. In fact, if the starting point of the used Caputo fractional derivative is fixed then the solution representation of the impulsive problem will be more complicated. This case could be of interest in future works.

Author contribution statement M.S. Abdo, T. Abdeljawad, K. Shah, F. Jarad: Conceived and designed the analysis; Analyzed and interpreted the data; Contributed analysis tools or data; Wrote the paper.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

The authors declare no conflict of interest.

No additional information is available for this paper.