Katugampola-type fractional differential equations with delay and impulses

Because of its wide applicability in biology, medicine and in more and more fields, the theory of fractional differential equations(FDEs) has recently been attracting increasing interest, see for instance1‒8 and references therein. Impulsive differential equations have played an important role in modelling phenomena, especially in describing dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so forth, some authors have used impulsive differential systems to describe the model since the last century. For the basic theory on impulsive differential equations, the reader can refer to the books9‒12 and the papers.1,13‒16 In addition, some modelling is done via impulsive functional differential equations when these processes involve hereditary phenomena such as biological and social macrosystems. For fractional functional differential equations, the initial value problem, for a class of nonlinear fractional functional differential equations is discussed. For more details, see.17‒24 Motivated by the papers,25,26 the aim of this note is to discuss the existence and uniqueness of solutions of Katugampolatype FDEs with delay and impulses.


Introduction
Because of its wide applicability in biology, medicine and in more and more fields, the theory of fractional differential equations(FDEs) has recently been attracting increasing interest, see for instance [1][2][3][4][5][6][7][8] and references therein. Impulsive differential equations have played an important role in modelling phenomena, especially in describing dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so forth, some authors have used impulsive differential systems to describe the model since the last century. For the basic theory on impulsive differential equations, the reader can refer to the books 9-12 and the papers. 1,[13][14][15][16] In addition, some modelling is done via impulsive functional differential equations when these processes involve hereditary phenomena such as biological and social macrosystems. For fractional functional differential equations, the initial value problem, for a class of nonlinear fractional functional differential equations is discussed. For more details, see. [17][18][19][20][21][22][23][24] Motivated by the papers, 25,26 the aim of this note is to discuss the existence and uniqueness of solutions of Katugampolatype FDEs with delay and impulses.
Consider the Katugampola-type FDEs with delay and impulse of the form, C are given functions satisfying some assumptions that will be specified later. ( represent the right and left limits of ( ) t Z at = k t t respectively, and they satisfy that ( ) = ( ) The rest of this paper is organized as follows. In Section 2, we give some notations and recall some concepts and preliminary results. In Section 3, the existence and uniqueness of the problem(1) are obtained by successive approximation method. In Section 4, an example is given to demonstrate the effectiveness of the main results.

Preliminaries
In this section, we recollect several definitions of fractional derivatives and integals from the papers [27][28][29][30] where Γ is the gamma function.

Definition 4.2 For a function H given on the interval [ , ]
a b , the Caputo fractional order derivative of H , is defined by n ω + .

Definition 4.3 The generalized left-sided fractional integral
if the integral exists.

Definition 4.6
The generalized fractional derivative in Caputo sense, corresponding to the generalized fractional integral in Caputo sense (6), is defined for 0 < a t ≤ , by Remark 4.7 In Caputo sense, the Katugampola fractional derivative and the following holds Proof. Assume that Z satisfies (9). One can see, from Remark 2.7 It follows that, for 1 2 ( , ], t t t ∈ ( ) In consequence, we can see, by means of ( ) ( ) ( ) ( ) Repeating the above process, the solution ( ) t Z for 1 ( , ] k k t t t + ∈ can be written as Conversely, if Z is a solution of (10), one can obtain by a direct computation, that

Existence and uniqueness results
Initially, set Therefore, the problem (1) can be transformed into the following fixed point problem of the operator 0 : , Now, let us present our main results.

Proof
To Since 0 ( )=0, v t it is easy to see from(11) that ( ) s T ∈ Thus we have, which implies that Note that for any > >0, r n we have for sufficiently large numbers , , r n it follows from the above inequalities with 1 <1 N that 0. In what follows, we shall show that ( ) v t is a solution of the equation (1). Observe that there exists a sufficiently large number 0 >0 n such that for all 0 > , n n we have In consequence, we can see that for a sufficiently large number On the other hand, we get that 3 L By a direct computation, we obtain that

Conclusion
In this note, the existences of solutions of a Katugampolatype fractional impulsive differential equation with delay were investigated. The successive approximation method was employed to show the existence of solutions. The example reflects the applicability of the proposed method.