CUBO, A Mathematical Journal Vol. 24, no. 03, pp. 439–455, December 2022 DOI: 10.56754/0719-0646.2403.0439 A class of nonlocal impulsive differential equations with conformable fractional derivative Mohamed Bouaouid 1, B Ahmed Kajouni 1 Khalid Hilal 1 Said Melliani 1 1 Sultan Moulay Slimane University, Faculty of Sciences and Technics, Department of Mathematics, BP 523, 23000, Béni Mellal, Morocco. bouaouidfst@gmail.com B kajjouni@gmail.com Khalid.hilal.usms@gmail.com said.melliani@gmail.com ABSTRACT In this paper, we deal with the Duhamel formula, existence, uniqueness, and stability of mild solutions of a class of non- local impulsive differential equations with conformable frac- tional derivative. The main results are based on the semi- group theory combined with some fixed point theorems. We also give an example to illustrate the applicability of our abstract results RESUMEN En este art́ıculo, tratamos la fórmula de Duhamel, la exis- tencia, unicidad y estabilidad de soluciones mild de una clase de ecuaciones diferenciales no locales impulsivas con derivadas fraccionarias conformables. Los resultados prin- cipales se basan en teoŕıa de semigrupos, combinada con algunos teoremas de punto fijo. También entregamos un ejemplo para ilustrar la aplicabilidad de nuestros resultados abstractos. Keywords and Phrases: Functional-differential equations with fractional derivatives; Groups and semigroups of linear operators; Nonlocal conditions; Impulsive conditions; Conformable fractional derivatives. 2020 AMS Mathematics Subject Classification: 34K37, 47D03. Accepted: 30 September, 2022 Received: 10 March, 2022 c©2022 M. Bouaouid et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0439 https://orcid.org/0000-0003-0474-0121 https://orcid.org/0000-0001-8484-6107 https://orcid.org/0000-0002-0806-2623 https://orcid.org/0000-0002-5150-1185 mailto:bouaouidfst@gmail.com mailto:kajjouni@gmail.com mailto:Khalid.hilal.usms@gmail.com mailto:said.melliani@gmail.com 440 M. Bouaouid, A. Kajouni, K. Hilal & S. Melliani CUBO 24, 3 (2022) 1 Introduction Fractional calculus has attracted the attention of many researchers, due to its wide range of applications in modeling of various natural phenomena in different fields of sciences and engineering including: physics, engineering, biology, finance, chemistry [3, 26, 31, 35, 38, 41, 43, 44, 45, 46, 47, 48]. For better understanding these phenomena, several definitions of fractional derivatives have been introduced such as Riemann-Liouville and Caputo definitions, for more details we refer to the books [31, 41]. Unfortunately, these definitions are very complicated to handle in real models. However, in [30] a new definition of fractional derivative named conformable fractional derivative was initiated. This novel fractional derivative is very easy and satisfies all the properties of the classical derivative. The advantage of the conformable fractional derivative is very remarkable compared to other fractional derivatives in many comparisons. Indeed, for example, in the work [15] the authors gave the solution of conformable-fractional telegraph equations in terms of the classical exponential function, however for the Caputo-fractional telegraph equations considered in the very good papers [19, 20, 36], the fundamental solution cannot be given in terms of the exponential function as in the conformable-fractional case, and therefore the authors have been introduced the so-called Mittag-Leffler function. Another comparison, we notice that the constants of increases of the norms of the control bounded operators W and W −1 in the application of the work [27] are given directly in a simple way in terms of the exponential function, contrary, for the Caputo fractional derivative in the application of the nice work [51] these constants are given in terms of the so-called Mittag-Leffler function. For more details and conclusions concerning the uses and applications of conformable fractional calculus, we refer to the works [2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, 17, 22, 23, 24, 25, 28, 29, 42, 49]. On the other hand, impulsive differential equations are crucial in description of dynamical processes with short-time perturbations [6, 32, 50]. Actually, the Cauchy problem of impulsive differential equations attracts the attention of many authors [1, 9, 33, 34, 37]. For example, Liang et al. [33] have proved the existence and uniqueness of mild solutions for the Cauchy problem        ẋ(t) = Ax(t) + f(t, x(t)), t ∈ [0, τ], t 6= t1, t2, . . . , tn, x(0) = x0 + g(x), x(t+i ) = x(t − i ) + hi(x(ti)), i = 1, 2, . . . , n, . (1.1) by using the following classical Duhamel formula: x(t) = T (t)[x0 + g(x)] + ∑ 0 0 and dαx(0) dtα = lim t−→0+ dαx(t) dtα , provided that the limits exist. The fractional integral Iα(.) associated with the conformable fractional derivative is defined by Iα(x)(t) = ∫ t 0 sα−1x(s)ds. Theorem 2.2 ([30]). If x(.) is a continuous function in the domain of Iα(.), then we have dα(Iα(x)(t)) dtα = x(t). Definition 2.3 ([41]). The Laplace transform of a function x(.) is defined by L(x(t))(λ) := ∫ +∞ 0 e −λt x(t)dt, λ > 0. It is remarkable that the above transform is not adequate to solve conformable fractional differential equations. For this reason, we consider the following definition, which appeared in [2]. Definition 2.4 ([2]). The fractional Laplace transform of order α of a function x(.) is defined by Lα(x(t))(λ) := ∫ +∞ 0 t α−1 e −λ t α α x(t)dt, λ > 0. The following proposition gives us the actions of the fractional integral and the fractional Laplace transform on the conformable fractional derivative, respectively. Proposition 2.5 ([2]). If x(.) is a differentiable function, then we have the following results I α ( dαx(.) dtα ) (t) = x(t) − x(0), Lα ( dαx(t) dtα ) (λ) = λLα(x(t))(λ) − x(0). We end this preliminaries by the following remark. CUBO 24, 3 (2022) A class of nonlocal impulsive differential equations... 443 Remark 2.6 ([14]). For two arbitrary functions x(.) and y(.), we have Lα ( x ( tα α )) (λ) = L(x(t))(λ), Lα ( ∫ t 0 sα−1x ( tα − sα α ) y(s)ds ) (λ) = L(x(t))(λ)Lα(y(t))(λ). 3 Main results We first prove the conformable fractional Duhamel formula (1.6). To do so, for t ∈ [0, t1], we apply the fractional Laplace transform in equation (1.5), we obtain Lα(x(t))(λ) = (λ − A)−1[x0 + g(x)] + (λ − A)−1Lα(f(t, x(t)))(λ). According to the inverse fractional Laplace transform and Remark (2.6), we get x(t) = T ( tα α ) [x0 + g(x)] + ∫ t 0 s α−1 T ( tα − sα α ) f(s, x(s))ds, where (T (t))t≥0 is the semigroup generated by the linear part A on the Banach space X, that is, (T (t))t≥0 is one parameter family of bounded linear operators on X satisfying the following properties (1) T (0) = I, (2) T (s + t) = T (s)T (t) for all t, s ∈ R+, (3) lim t↓0 ‖ T (t)x − x ‖= 0 for each fixed x ∈ X, (4) lim t↓0 T (t)x − x t = Ax, for x ∈ X, provided that the limit exists. As in [37], we assume that the solution of equation (1.5) is such that at the point of discontinuity tk, we have x(t − k ) = x(tk). Hence, one has x(t−1 ) = T ( tα1 α ) [x0 + g(x)] + ∫ t1 0 sα−1T ( tα1 − sα α ) f(s, x(s))ds. For t ∈ (t1, t2], using the fractional Laplace transform in equation (1.5), we obtain x(t) = T ( tα − tα1 α ) x(t+1 ) + ∫ t t1 sα−1T ( tα − sα α ) f(s, x(s))ds = T ( tα − tα1 α ) [x(t−1 ) + h1(x(t1))] + ∫ t t1 sα−1T ( tα − sα α ) f(s, x(s))ds. Replacing x(t−1 ) by its expression in the above equation, we get x(t) = T ( tα − tα1 α )[ T ( tα1 α ) (x0 + g(x)) + ∫ t1 0 sα−1T ( tα1 − sα α ) f(s, x(s))ds + h1(x(t1)) ] + ∫ t t1 s α−1 T ( tα − sα α ) f(s, x(s))ds. 444 M. Bouaouid, A. Kajouni, K. Hilal & S. Melliani CUBO 24, 3 (2022) By using a computation, the above equation becomes x(t) = T ( tα α ) [x0 + g(x)] + T ( tα − tα1 α ) [h1(x(t1))] + ∫ t 0 s α−1 T ( tα − sα α ) f(s, x(s))ds. In particular, for t = t−2 , one has x(t−2 ) = T ( tα2 α ) [x0 + g(x)] + T ( tα2 − tα1 α ) [h1(x(t1))] + ∫ t2 0 sα−1T ( tα2 − sα α ) f(s, x(s))ds. As the same, for t ∈ (t2, t3], we obtain x(t) = T ( tα − tα2 α ) x(t+2 ) + ∫ t t2 sα−1T ( tα − sα α ) f(s, x(s))ds = T ( tα − tα1 α ) [x(t−2 ) + h2(x(t2))] + ∫ t t2 sα−1T ( tα − sα α ) f(s, x(s))ds. Hence, replacing x(t−2 ) by its expression, we have x(t) = T ( tα − tα2 α )[ T ( tα2 α ) [x0 + g(x)] + T ( tα2 − tα1 α ) [h1(x(t1))] + ∫ t2 0 sα−1T ( tα2 − sα α ) f(s, x(s))ds + h2(x(t2)) ] + ∫ t t2 s α−1 T ( tα − sα α ) f(s, x(s))ds. Using a computation, we get x(t) = T ( tα α ) [x0 + g(x)] + T ( tα − tα1 α ) [h1(x(t1))] + T ( tα − tα2 α ) [h2(x(t2))] + ∫ t t2 s α−1 T ( tα − sα α ) f(s, x(s))ds. Repeating the same process, we obtain the following conformable fractional Duhamel formula x(t) = T ( tα α ) [x0 + g(x)] + ∑ 0 0 there exists a function µr ∈ L∞([0, τ], R+) such that sup ‖x‖≤r ‖ f(t, x) ‖≤ µr(t), for all t ∈ [0, τ]. CUBO 24, 3 (2022) A class of nonlocal impulsive differential equations... 445 (H2) The function f(., x) : [0, τ] −→ X is continuous, for all x ∈ X. (H3) There exists a constant L1 > 0 such that ‖ g(y) − g(x) ‖≤ L1 | y − x |c, for all x, y ∈ C. (H4) There exist constants Ci > 0 such that ‖ hi(y(ti))−hi(x(ti)) ‖≤ Ci | y −x |c, for all x, y ∈ C. Theorem 3.2. If (T (t))t>0 is compact and (H1) − (H4) are satisfied, then the conformable frac- tional Cauchy problem (1.5) has at least one mild solution, provided that ( L1 + n ∑ i=1 Ci ) sup t∈[0,τ] ∣ ∣ ∣ ∣ T ( tα α ) ∣ ∣ ∣ ∣ < 1. Proof. Let Br = {x ∈ C, | x |c≤ r}, where r ≥ sup t∈[0,τ] ∣ ∣ ∣ ∣ T ( tα α ) ∣ ∣ ∣ ∣ [ ‖ x0 ‖ + ‖ g(0) ‖ + n ∑ i=1 ‖ hi(0) ‖ + τα α | µr |L∞([0,τ],R+) ] 1 − ( L1 + n ∑ i=1 Ci ) sup t∈[0,τ] ∣ ∣ ∣ ∣ T ( tα α ) ∣ ∣ ∣ ∣ . In order to use the Krasnoselskii fixed-point theorem, we consider the following operators Γ1 and Γ2 defined by Γ1(x)(t) = T ( tα α ) [x0 + g(x)] + ∑ 00 assures that lim t2−→t1 ∣ ∣ ∣ ∣ T ( tα2 − tα1 α ) − I ∣ ∣ ∣ ∣ = 0. Hence, combining this fact with the above inequality, we conclude that Γ2(x), x ∈ Br are equicontinuous on [0, τ]. Claim 2: We prove that the set {Γ2(x)(t), x ∈ Br} is relatively compact in X. For some fixed t ∈]0, τ] let ε ∈]0, t[, x ∈ Br and define the operator Γε2 as follows Γε2(x)(t) = T ( εα α ) ∫ (tα−εα) 1 α 0 sα−1T ( tα − sα − εα α ) f(s, x(s))ds. Since (T (t))t>0 is compact, then the set {Γε2(x)(t), x ∈ Br} is relatively compact in X. By using a computation combined with assumption (H1), we get ‖ Γε2(x)(t) − Γ2(x)(t) ‖≤| µr |L∞([0,τ],R+) sup t∈[0,τ] ∣ ∣ ∣ ∣ T ( tα α ) ∣ ∣ ∣ ∣ εα α . Therefore, we deduce that the {Γ2(x)(t), x ∈ Br} is relatively compact in X. For t = 0 the set {Γ2(x)(0), x ∈ Br} is compact. Thus, the set {Γ2(x)(t), x ∈ Br} is relatively compact in X for all t ∈ [0, τ]. By using the Arzelà-Ascoli theorem, we conclude that the operator Γ2 is compact. In conclusion, by the above steps combined with the Krasnoselskii fixed-point theorem, we conclude that Γ1 + Γ2 has at least one fixed point in C, which is a mild solution of conformable fractional Cauchy problem (1.5). To obtain the uniqueness of the mild solution, we need the following assumption: (H5) There exists a constant L2 > 0 such that ‖ f(t, y) − f(t, x) ‖≤ L2 ‖ y − x ‖, for all x, y ∈ X and t ∈ [0, τ]. Theorem 3.3. Assume that (H2) − (H5) hold, then the conformable fractional Cauchy problem (1.5) has an unique mild solution, provided that ( L1 + n ∑ i=1 Ci + τα α L2 ) sup t∈[0,τ] ∣ ∣ ∣ ∣ T ( tα α ) ∣ ∣ ∣ ∣ < 1. Proof. Define the operator Γ : C −→ C by: Γ(x)(t) = T ( tα α ) [x0 + g(x)] + ∑ 0