On the fractional differential equations with not instantaneous impulses

1 Introduction

Fractional differential equations has gained much attention in literature because of its applications for description of hereditary properties in many fields, and some progresses were gotten in computation methods, controllability, existence etc. for fractional differential equations [1–6]. Moreover, impulsive fractional (partial) differential equations were widely studied [7–29] due to importance in description of some processes in which sudden, discontinuous jumps occur, and general solution has been discovered for several impulsive fractional order systems in [30–35].

However, Hernandez and O’Regan in [36] pointed out that the instantaneous impulses (considered in almost all papers about impulsive differential equations) cannot characterize some processes such as evolution processes in pharmacotherapy, and presented a kind of impulsive differential equations with non-instantaneous impulses. Moreover, the existence of solution is considered for some fractional order systems with non-instantaneous impulses in [37, 38].

Based on the above-works, we will study the following fractional order system with non- instantaneous impulses.

here CD0+q is the Caputo fractional derivative of order q. f : [0, T] ×R →R and g_k : (t_k, s_k] ×R →R are some appropriate functions, and g_k denote non-instantaneous impulses, and ǵ_k(sk,x(sk))exist (here k = 1,2,..., N),and0 = t₀ = s₀ < t₁ ≤ s₁ ≤ t₂ ≤ . . . ≤ t_N ≤ s_N ≤ t_N+1 = T.

Next, let us introduce the concept of the fractional derivative and some conclusions in Section 2, and provide main result in section 3, and give an example to show the usefulness of the obtained result.

2 Preliminaries

3 Main result

For convenience, letf = f(τ, x(τ)) in this section. Consider condition (1.1a) in system (1.1) by using two different approches:

Next, substituting (i) into system (1.1), we get

That is,

In fact, x͂(t) satisfies conditions (1.1a)–(1.1c) in system (1.1). But, we will show that x͂(t) isn’t a solution of system (1.1). For system (1.1), we have

And system (3.4) is equivalent to

Moreover, letting gk(t,x(t))=x0+x¯0t+1Γ(q)∫0t(t−τ)q−1fdτ (for all k∈ {1,2,..., N}) in (3.3), we get

Therefore, if x͂(t) is a solution of system (1.1), then (3.6) is equivalent to (3.5). Thus,

Eq. (3.7) is an unfit equation, which means that x͂(t) isn’t a solution of system (1.1). Therefore, we will regard x͂(t) as an approximate solution to seek the exact solution of system (1.1).

Substituting (ii) into system (1.1), we obtain

By initial conditions x(s_k) = g_k(s_k, x(s_k)) and x^′(s_k) = g^′_k(s_k, x(s_k)) (here k = 1, 2, ..., N), we obtain B₀ = 0, C₀ = 0,

and

Substituting (3.9)-(3.10) into (3.8), we get

In fact, Eq. (3.11) satisfies conditions (1.1a)–(1.1c) and

Therefore, Eq. (3.11) satisfies all conditions of system (1.1), and it is a solution of system (1.1).