Fractional calculus was utilized as a powerful tool to reveal the hidden aspects of the dynamics of the complex or hypercomplex systems [1-3], and the subject of fractional differential equations is gaining much attention. For details see [4-14] and the references therein.
Impulsive effects exist widely in many processes in which their states can be described by impulsive differential equations. Moreover, in case of impulsive differential equations with Caputo fractional derivative there have been numerous works about the subject [15-23], and impulsive fractional partial differential equations are widely considered in [24-29].
Motivated by the above-mentioned works, we will study the following impulsive Cauchy problem with Riemann-Liouville fractional derivative: (1) where q ∊ ℂ and ℜ(q) ∊ (0, 1), denotes left-sided Riemann-Liouville fractional derivative of order q and denotes left-sided Riemann-Liouville fractional integral of order 1 − q. f : J × ℂ → ℂ is an appropriate continuous function, and a = t0 < t1 < ... <tm < tm+1 = T. Here and represent the right and left limits of respectively.
For impulsive system (1) we have (2)
Therefore, it means that there exists a hidden condition (3)
Therefore, the definition of solution for impulsive system (1) is provided as follows:
A function z(t) : [a,T] → ℂ is said to be a solution of the fractional Cauchy problem (1) if the equation condition for each t ∊ (a, T] is verified, the impulsive conditions (here i = 1,2,..., m)are satisfied, the restriction of z(t) to the interval (tk, tk+1] (here k = 0, 1, 2,...,m) is continuous, and the condition (3) holds.
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.
Firstly, we recall some concepts of fractional calculus  and a property for nonlinear fractional differential equations.
The left-sided Riemann-Liouville fractional integral of order α ∊ ℂ (ℜ(α) > 0) of function x(t) is defined by where Γ is the gamma function.
The left-sided Riemann-Liouville fractional derivative of order q ∊ ℂ (ℜ(q) ≥ 0) of function x(t) is defined by
By Lemma 2.2 in , the initial value problem (4) is equivalent to the following nonlinear Volterra integral equation of the second kind, (5)
3 Main results
Define a piecewise function with By Definition 2.2, ..., we have So, ũ(t) satisfies the condition of fractional derivative of (1), and it doesn’t satisfy the condition (3). Thus, we assume that ũ(t) is an approximate solution to seek the exact solution of impulsive system (1).
Next, Taking Riemann-Liouville fractional derivative to Eq. (6) for each t ∊ (tk, tk+1] (where k = 0, 1, 2, ...,m), we have
By (7), we have and the approximate solution ũ(t) (for t ∊ (t1, t2]) is given by (8) with e1(t) = u(t) – ũ(t) for t ∊ (t1, t2]. By we get
Then, we assume where function σ(.) is an undetermined function with σ(0) = 1. Thus, (9)
Moreover, letting t2 → t1, we have (16) (17)
Using (9) and (15), we have 1 – σ(Δ1 + Δ2) = 1 – σ(Δ1) + 1 – σ(Δ2). Letting ρ(z) = 1 – σ(z), we get ρ(z + w) = ρ(z) + ρ(w) for ∀z, w ∊ ℂ. So, ρ(z)ξz, here ξ is a constant. Thus, (18) and (19) Next, for t ∊ (tn, tn+1], suppose (20) Using (20), we have Thus, the approximate solution ũ(t) for t ∊ (tn+1, tn+2] is given by (21) with en+1(t) = u(t) – ũ(t) for t ∊ (tn+1, tn+2]. By (20), the exact solution u(t) of (1) satisfies Therefore, (22) (23)
Let us consider the general solution of the impulsive fractional system (25)
By Theorem 3.1, the general solution of impulsive system (25) is obtained as follows: (26) Next, it is verified that Eq. (26) satisfies the condition of system (25). Taking Riemann-Liouville fractional derivative to the both sides of Eq. (26), we have
(i) for t ∊ (1,2]
(ii)for t ∊ (2, 3]
The authors are deeply grateful to the anonymous referees for their kind comments, correcting errors and improving written language, which have been very useful for improving the quality of this paper.
The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant No. 21576033, 21636004, 61563023, 61261046) and State Key Development Program for Basic Research of Health and Family planning Commission of Jiangxi Province China (Grant No. 20143246), the Natural Science Foundation of Jiangxi Province (Grant No. 20151BAB207013) and the Jiujiang University Research Foundation (Grant No. 8400183).
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Published Online: 2016-12-30
Published in Print: 2016-01-01