The general solution of impulsive systems with Riemann-Liouville fractional derivatives

Abstract In this paper, we study a kind of fractional differential system with impulsive effect and find the formula of general solution for the impulsive fractional-order system by analysis of the limit case (as impulse tends to zero). The obtained result shows that the deviation caused by impulses for fractional-order system is undetermined. An example is also provided to illustrate the result.

Motivated by the above-mentioned works, we will study the following impulsive Cauchy problem with Riemann-Liouville fractional derivative: (2) Therefore, it means that there exists a hidden condition lim 1 !0; 2 !0; ; m !0 fthe solution of impulsive system (1)g D fthe solution of system (2)g Therefore, the definition of solution for impulsive system (1) is provided as follows: Therefore, we will consider impulsive system (1) and seek some solutions of impulsive system (1) according to Definition 1.1.
The rest of this paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we give the formula of general solution for impulsive differential equations with Riemann-Liouville fractional derivatives. In Section 4, an example is provided to expound the main result.

Preliminaries
Firstly, we recall some concepts of fractional calculus [2] and a property for nonlinear fractional differential equations.
Definition 2.1. The left-sided Riemann-Liouville fractional integral of order˛2 C (<.˛/ > 0) of function x.t/ is defined by where is the gamma function.

Main results
Define a piecewise function .t s/ q 1 f .s; u.s//ds provided that the integral in .6/ exists.
The proof is now completed.

Example
For system (1) it is difficult to get the analytical solution when f is a nonlinear function in .1/. So, a linear example is given to illustrate the obtained result.
So, Eq. .26/ is the general solution of system .25/.