Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 8101802, 20 pages
http://dx.doi.org/10.1155/2016/8101802

Research Article

The General Solution of Impulsive Systems with Caputo-Hadamard Fractional Derivative of Order

Xianmin Zhang,¹ Xianzhen Zhang,² Zuohua Liu,³ Hui Peng,¹ Tong Shu,¹ and Shiyong Yang¹

¹School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
²School of Chemical and Environmental Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
³School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China

Received 21 December 2015; Accepted 19 January 2016

Academic Editor: José A. T. Machado

Copyright © 2016 Xianmin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 that

For 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 by

where is the Gamma function.

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

where 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 equation

provided 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 by

By Theorem 4, we have

This 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 equation

provided that the integral in (21) exists.

Proof.    
Necessity. Letting in (16), we have

That is,

In fact, we can verify that (21) satisfies the condition (23).
Next, taking fractional derivative to (21) for (here ), we get

So, (21) satisfies the condition of fractional derivative in system (16).
Finally, using (21) for each (here ), we have

It means that (21) satisfies the impulsive condition of (16). Hence, (21) satisfies all conditions of system (16).
Sufficiency (by mathematical induction). By Lemma 6, the solution of (16) satisfies

Using (26), we obtain

Thus, the approximate solution is given by

Let , for . By (26), the exact solution of system (16) satisfies

Then,

This shows that is connected with and . Thus, we assume

where function is an undetermined function with . So,

Letting , we get

Using (33), we obtain

Therefore, the approximate solution is provided by

Let for . Moreover, by (33), the exact solution of system (16) satisfies

Then,

By (37), we get

Then,

Consider the following limit case

Using (33) and (39) for (41) and (40), respectively, we have

Therefore,   ; here is a constant. Thus,

Next, suppose

Using (44), we obtain

Therefore, the approximate solution is presented by

Let for . In addition, by (44), the exact solution of system (16) satisfies

Then,

By (48), we obtain

Thus, we have

Then, the solution of system (16) satisfies (21).
By the proof of Sufficiency and Necessity, system (16) is equivalent to (21). The proof is completed.

Lemma 10. Let , , and is a constant. System (17) is equivalent to the integral equation

provided that the integral in (51) exists.

Remark 11. For (17), we have

Then,

In fact, we can verify that (51) satisfies the condition (53). Moreover, the approximate solution of system (17) is defined by

here and , .
Due to similarity with Lemma 9, the proof is omitted.

Corollary 12. Let , , and is a constant. A function is general solution of the system (16); then,

provided that the integral in (55) exists.

Corollary 13. Let , , and is a constant. A function is general solution of the system (17); then,

provided that the integral in (56) exists.

Remark 14. By Corollaries 12 and 13, it is shown that two kinds of impulses and have similar effect on of system (15).

Lemma 15. Let , , and and are two constants. A function is general solution of the system (3a), (3b), (3c), and (3d); then,

provided that the integral in (57) exists.

Proof. According to Corollaries 12 and 13, the solutions of system (3a), (3b), (3c), and (3d) satisfy

By the definition of Caputo-Hadamard fractional derivative, system (3a), (3b), (3c), and (3d) satisfies

Moreover, it is reasonable that impulses () are considered as special impulses () in system (59) by Remark 14. Therefore, using Lemma 7 for system (59) (as , here ), we have

where () and () are undetermined constants. Letting (for all ) and (for all ), respectively, we get (for all ) and (for all ) by Corollaries 12 and 13. Thus,

This proof is completed.

Theorem 16. Let , , and and are two constants. System (3a), (3b), (3c), and (3d) is equivalent to the integral equation

provided that the integral in (62) exists.

Proof.   
Necessity. We can verify that (62) satisfies conditions (i)–(iii) by Lemmas 9, 10, and 6.
Next, taking the fractional derivative to (62) for (here ), we get

So, (62) satisfies (3a).
Finally, it is straightforward to verify that (62) satisfies (3b) and (3c). So, (62) satisfies all conditions of system (3a), (3b), (3c), and (3d).
Sufficiency. According to Lemmas 9 and 10, the solutions of system (3a), (3b), (3c), and (3d) satisfy

Next, by Lemma 15, the solutions of system (3a), (3b), (3c), and (3d) satisfy

Using (65), we have

Letting (for all ) and (for all ) in (66), respectively, by Lemmas 9 and 10, we obtain

Thus,

So, the solutions of system (3a), (3b), (3c), and (3d) satisfy (62). This proof is completed.

Corollary 17. Let , , and and are two constants. System (1) is equivalent to the integral equation

provided that the integral in (69) exists.

Remark 18. Substituting and into (62), (69) can be obtained. Next, let us analyze the limited case of system (1):

On the other hand, by (69), we have

It can be verified that (71) is the solution of (70), which indirectly supports our conclusion.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant no. 21576033), the Natural Science Foundation of Jiangxi Province (Grant no. 20151BAB207013), the Research Foundation of Education Bureau of Jiangxi Province, China (Grant no. GJJ14738), and Jiujiang University Research Foundation (Grant no. 8400183).

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