New generalized conformable fractional impulsive delay differential equations with some illustrative examples

Hua Wang1, Tahir Ullah Khan2,3, Muhammad Adil Khan2,∗ and Sajid Iqbal4 1 School of Mathematics and Statistics Changsha University of Science and Technology, Changsha 410114, China 2 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan 3 Higher Education Department, Directorate General of Commerce Education and Management Sciencs KP, Peshawar, Pakistan 4 Department of Mathematics, Riphah International University, Faisalabad Campus, Satyana Road, Faisalabad, Pakistan


Introduction
Fractional Differential Equations (FDE) are of immense significance as they are great contributors to research fields of applied sciences [1]. They have gained substantial popularity and importance due to their attractive applications in extensive areas of science and engineering [2,3]. In addition to this, Impulsive FDE (I-FDE) have also played an influential role in describing phenomena, particularly in modeling dynamics of populations subject to abrupt changes [4,5]. They provide a realistic framework of modeling systems in fields like control theory, population dynamics, biology, physics, and medicine [6,7]. Similarly, Delay Differential Equations (DDE) are significant because they have the ability to describe processes with retarded time. The importance of DDE in various sciences like biology, physics, economics, medical science, and social sciences has been acknowledged [8,9]. When the above-mentioned classes of equations come to a single platform, and when they are studied combined, they are then called Impulsive Delay FDE (ID-FDE). Such type of equations have been getting worthwhile attention from researchers in the present age. For the theory of ID-FDE and recent development on this topic, one can see [10][11][12][13][14][15][16][17][18][19][20][21][22] and the references therein. Recently, Khan et al. have defined fractional integral and derivative operators [23]. Unlike other fractional operators, they satisfy properties like continuity, boundedness, linearity and unify some previously-presented operators into a single form. They are defined as under: Definition 1 ( [23]). Let φ be a function that is conformable integrable on the interval [p, q] ⊆ [0, ∞). The left-sided and right-sided Generalized Conformable Fractional (GCF) integral operators σ θ K ν p + and σ θ K ν q − of order ν > 0 with θ ∈ (0, 1], σ ∈ R, and θ + σ 0 are defined by: and respectively, and σ θ K 0 The integral q p d θ w, in the Definition 1, represents the conformable integration defined as [24]: q p φ(w)d θ w := q p φ(w)w θ−1 dw. (1. 3) The associated left-and right-sided GCF derivative operators are defined as follows [23]: Definition 2. Let φ be a function that is conformable integrable on the interval [p, q] ⊆ [0, ∞) such that θ ∈ (0, 1], σ ∈ R and θ + σ 0. The left-and right-sided GCF derivative operators σ θ T ν p + and σ θ T ν q − of order ν ∈ (0, 1) are defined, respectively, by: Here T θ represents the θth-order conformable derivative with respect to τ, and it is defined as in the following definition.

Definition 3 ( [24]
). Consider the real-valued function φ defined on the interval [0, ∞). The θth-order conformable derivative T θ of the function φ, where θ ∈ (0, 1], is defined as: The relation between ordinary derivative φ (w) and the conformable derivative T θ φ(w), is given as follows [24]: The conformable operators have gained a considerable attention of many researchers in a very short span of time. Due to their classical properties, they have been used in various fields, for example one can see [25][26][27][28][29][30][31][32][33][34][35][36] and the references therein. The following Remark 1 highlights that how the GCF operators unify various early-defined operators. Here, the left-sided operators are only taking into account. A similar methodology can be carried out also for the right-sided operator.
The inverse property of the newly introduced GCF derivative operators is given below, which will be used in the proofs of our results.

Theorem 1 ( [23]
). Let σ ∈ R, θ ∈ (0, 1] such that σ + θ 0 and ν ∈ (0, 1). For any continuous function φ : [p, q] ⊆ [0, ∞) → R, in the domain of σ θ K ν p + and σ θ K ν q − we have: One of the fundamental theorems in mathematical analysis, in the theory of double integrals, is Fubini's theorem. This theorem allows the order of integration to be changed in certain iterated integrals. This is stated as follows [38]: The scope and novelty of the present paper is that it addresses a new class of generalized fractional impulsive delay differential equations. This class is defined using newly introduced GCF operators which are the generalizations of fractional operators of the types Katugampola, Riemann-Lioville, Hadamard, Riemann-Lioville's type, conformable and ordinary or classical operators [23]. That is, while considering the generalized problem containing GCF operators, we work with various (above-mentioned) operators at the same time. Therefore, the paper combines various previously defined operators (or work) into a single form and is expected to provide a unique platform for the researchers working with different operators in this field. Moreover, since researchers commonly face the problem of choosing a convenient approach or a suitable operator to solve a problem, thus this kind of study, in which one can work with several operators at a time, is helpful in this regard.

Main results
In the present work, we start by stating the GCFDE with delay and impulse terms as under: ∆φ(τ k ) = I k (φ(τ k )) , k = 1, 2, 3...m; is the GCF derivative of order ν ∈ (0, 1), θ ∈ (0, 1] and σ ∈ R where σ + θ 0, ω is a non-negative real number and 0 = τ 0 < τ 1 < τ 2 ... < τ m+1 = T. Also f : denotes the left and right hand limits of the function φ at the point τ k respectively such that φ( A motivation to study the system in Eq (2.1), as compare to other systems in the literature, is that it contains fractional operators having such properties which are not satisfied by those obtained earlier [24]. These operators are simple, friendly (while dealing with them) and have properties analogous to ordinary derivative and integral operators [23]. Moreover, there are some classes of differential equations which cannot be solved easily using previous definitions of fractional derivatives. An example of such type of equation has been given by Khalil et. al., in the paper [24]. Therefore, to establish results related to the solution of such type of problems, we have chosen a generalized problem in the form of Eq (2.1), which covers equations of the type given in [24] as well others in the literature.
To establish our main results, first we need to prove the following lemma.
Proof. First using definition of the integral operator σ θ K ν p + (Eq (1.1)), then of the derivative operator σ θ T ν p + (Eq (1.4)), and then definition of conformable derivative and integral operators (Eq (1.7) and Eq (1.3)) in sequence, we have: where To find the value of I(w), Using integration by parts formula, we get: This means that: Thanks to Lebnitz rule of differentiating integral: We have from Eq (2.6): Putting value of I(w) in Eq (2.4), we get: Switching the order of integration (using Fubini's Theorem) and changing variables to u by defining w σ+θ = s σ+θ + (r σ+θ − s σ+θ )u, we have: where in the last step fundamental theorem of calculus has been used and in the second last step definition of Euler Beta function B and its relation with Gamma function have been used as under: The proof of the Eq (2.3) is same to the procedure developed for the proof of Eq (2.2). It can easily be obtained by first applying the definition of σ θ K ν q − and then of σ θ T ν q − . The rest of the process is same as above. This completes our proof.
Next we will show that z(τ) is a solution of the Eq (2.1). Keeping the Eq (2.29) and Eq (2.30) in mind, we proceed: (2.38) Since z n (τ) → z(τ), as n → ∞. By definition of convergence, for any > 0, there exists a sufficiently large number p 0 > 0, such that for n > p 0 , we have (2.39) Therefore, using Eq (2.38) we get: (2.40) And also (2.42) In consequence, we can see that for a sufficiently large number n > p 0 : (2.43) using Eq. (2.40), Eq (2.41) and Eq (2.42) we get that |z(τ) − Θz(τ)| → 0. This shows that z(τ) is the solution of Eq (2.1). Now we show that the solution is unique. On contrary suppose that there exists two solutions z 1 and z 2 of Eq (2.1). Then (2.45) Using the condition 2 in the theorem hypothesis, the uniqueness of the solution of Eq (2.1) follows immediately, which completes the proof.

Illustrative examples
To illustrate the obtained results, some examples are presented in this section. Example 1. A particular GCF differential equation with delay and impulse is considered as follows: where ω is a non-negative constant.
To check whether a unique solution of the problem in Eq (3.1) exists or not, we have to verify all the three conditions of the Theorem 3. We consider: where h(τ) = 1 (10+τ) 2 , which shows that the condition 1 of Theorem 3 is satisfied. Also we have: which shows that the condition 2 of Theorem 3 is also satisfied. Finally: So the condition 3 of Theorem 3 is also satisfied. Now using the Theorem 3, it is concluded that the solution of the Eq (3.1) exists and it is unique.
Also we have: Thus the condition 2 of Theorem 3 also holds true. Finally: where τ ∈ [0, 1]. So the condition 3 of Theorem 3 is also satisfied. Thus using Theorem 3, it is established that solution of the Eq (3.6) exists and it will be unique.

Conclusions
A new generalized class of ID-FDE has been constructed successfully. A sufficient criterion for the existence and uniqueness of the solution of this type of systems have been developed. The results have been supported by the successive approximation method. All the results have been given in terms of newly introduced GCF operators. To illustrate the obtained results, some particular examples have been presented. The present attempt also allows direct applications of the obtained results to FDE of the types Katugampola, Riemann-Liovilles, Hadamard, New Riemann-Lioville's, conformable and ordinary differential equations, which can be considered as special cases of our established results.
Since there exist many fractional derivative and integral operators, which have been defined with the passage of time. Each operator satisfies some useful properties and also has some flaws. In most of the cases there arises a confusion regarding selection of a suitable fractional operator for solving a given mathematical problem. In this context, there is a need for such operators that combine most of the previously defined operators into a single form. In this regard, GCF operators nicely fulfill this criterion using which one can work with multiple number of operators at the same time.