Abstract

Motivated by some preliminary works about general solution of impulsive system with fractional derivative, the generalized impulsive differential equations with Caputo-Hadamard fractional derivative of   () are further studied by analyzing the limit case (as impulses approach zero) in this paper. The formulas of general solution are found for the impulsive systems.

1. Introduction

Hadamard fractional calculus is a key part of the theory of fractional calculus. The authors in [1–6] made an important development of the fractional calculus within the frame of Hadamard fractional derivative. For the general theory of Hadamard fractional calculus, one can see the monograph of Kilbas et al. [7].

Recently, Jarad et al. made a progress on Hadamard fractional derivative to present the definition of Caputo-Hadamard fractional derivative in [8] and developed the fundamental theorem of this fractional derivative in [8, 9].

Furthermore, impulsive differential equations are utilized as a valuable tool to describe the dynamics of processes in which sudden, discontinuous jumps occur, and impulsive differential equations with Caputo fractional derivative were widely researched in [10–26]. Next, the general solutions of several kinds of impulsive fractional differential equations have been found in [27–30], respectively.

Motivated by the above-mentioned works, we will further seek the general solution of generalized impulsive system with Caputo-Hadamard fractional derivative:Here and , denotes left-sided Caputo-Hadamard fractional derivative of order and , is an appropriate continuous function, , and . Here and represent the right and left limits of at , respectively, and and have similar meaning. Let us queue to such thatFor each , suppose (here ) and (here ), respectively.

In order to get the solution of (1), we will first consider the following system:where differential operator , .

Next, some definitions and conclusions are introduced in Section 2, and the formulas of general solution will be given for some impulsive differential equations with Caputo-Hadamard fractional derivative in Section 3.

2. Preliminaries

Definition 1 (see [7, p. 110]). Let be finite or infinite interval of the half-axis . The left-sided Hadamard fractional integral of order of function is defined bywhere is the Gamma function.

Definition 2 (see [7, p. 110]). The left-sided Hadamard fractional derivative of order () on (, ) is defined bywhere and differential operator and .

Lemma 3 (see [7, p. 114–116]). Let such that . For , if , then and .

The left-sided Caputo-Hadamard fractional derivative was defined in [8] by here , , , differential operator , , and

Theorem 4 (see [8, p. 4]). Let , , and , . Then, exist everywhere on and(a)if ,(b)if ,In particular,

Lemma 5 (see [8, p. 5]). Let , and . If or , then

Lemma 6 (see [8, p. 6]). Let or let and ; then,

Lemma 7 (see [29, p. 4]). Let and , and let be a constant. A function is general solution of system if and only if satisfies the integral equationprovided that the integral in (14) exists.

3. Main Results

Firstly, let us consider some limit cases in system (3a), (3b), (3c), and (3d):Thus, Thus, the definition of solution of system (3a), (3b), (3c), and (3d) is presented as follows.

Definition 8. A function is said to be a solution of (3a), (3b), (3c), and (3d) if and , the equation condition for each is verified, the impulsive conditions (here ) and (here ) are satisfied, the restriction of to the interval (here ) is continuous, and the conditions (i)–(iii) hold.

Next, define a function byBy Theorem 4, we haveThis means that satisfies (3a), and satisfies (3b)–(3d). However, does not satisfy the conditions (i)–(iii), and it is not a solution of system (3a), (3b), (3c), and (3d). Therefore, is considered as an approximate solution to seek the exact solutions of (3a), (3b), (3c), and (3d). Next, let us prove some useful conclusions.

Lemma 9. Let , , and is a constant. System (16) is equivalent to the integral equationprovided that the integral in (21) exists.