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Non-instantaneous impulses in Caputo fractional differential equations

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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 3

Abstract

Recent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.

MSC 2010: Primary 34A34; Secondary 34A08; 34A37
Key Words and Phrases: non-instantaneous impulses; Caputo fractional differential equation

1 Introduction

Fractional calculus is the theory of integrals and derivatives of arbitrary non-integer order, which unifies and generalizes the concepts of ordinary differentiation and integration. For more details on geometric and physical interpretations of fractional derivatives and for a general historical perspective, we refer the reader to the monographs [18, 31, 33], to the survey papers [37, 38], and the cited references therein.

Impulsive differential equations arise from real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are natural in biology, physics, engineering, etc.

In the literature there are two popular types of impulses:

  1. instantaneous impulses - the duration of these changes is relatively short compared to the overall duration of the whole process. For ordinary differential equations with impulses we refer the reader to the monographs [26], [34] and the cited references therein. There are also many recent contributions on fractional order differential equations with instantaneous impulses (see, for example, [4], [5], [9], [16], [17], [20], [ 24], [32], [40], [43]);

  2. non-instantaneous impulses - an impulsive action, starting abruptly at a fixed point and its action continues on a finite time interval. This kind of impulse is observed in lasers, and in the intravenous introduction of drugs in the bloodstream. Hernandez and O’Regan ([22]) introduced this new class of abstract differential equations where the impulses are not instantaneous, and they investigated the existence of mild and classical solutions. For recent works, we refer the reader to [6], [7], [8], [10], [12], [19], [23], [27], [29], [30], [35], [44].

The main goal of the survey is to present basic points in introducing non-instantaneous impulses in Caputo type fractional differential equations. In the literature there are two approaches in the interpretation of solutions. Both approaches are compared and their advantages/disadvantages are illustrated with examples. The existence of non-instantaneous impulsive fractional differential equations and the corresponding sufficient conditions are discussed using both approaches.

2 Preliminary notes on fractional derivatives and equations

Fractional calculus generalizes the derivatives and integrals of a function of a non-integer order [18, 31, 33].

In many applications in science and engineering, the fractional order q is often less than 1, so we restrictq ∈ (0,1) everywhere in the paper.

  1. The Riemann–Liouville (RL) fractional derivative of order q ∈ (0,1) of m(t) is given by ([18], [31], [33])

    t0Dtqm(t)=1Γ1−qddt∫t0tt−s−qm(s)ds,t≥t0,

    where Γ (.) denotes the Gamma function.

  2. The Caputo fractional derivative of order q ∈ (0,1) is defined by ([18], [31])

    t0cDqm(t)=1Γ1−q∫t0tt−s−qm′(s)ds,t≥t0.(2.1)

The Caputo and the Riemann-Liouville formulations coincide when the initial conditions are zero. Note, that the RL derivative is meaningful under weaker smoothness requirements, but the derivative in the Caputo sense allows an easier interpretation of conventional initial conditions.

3 Ordinary differential equations versus Caputo fractional differential equations

We compare some properties of the ordinary differential equations (ODE) and Caputo-type fractional differential equations (FrDE).

I. Ordinary differential equations. Consider the ODE

x′(t)=f(t,x)fort≥τ,(3.1)

with the initial condition

x(τ)=x~0,(3.2)

where x0~ ∈ ℝn.

Denote the solution of the IVP for ODE (3.1), (3.2) by x(t;τ, x0~).

Now consider the same ODE (3.1) with different initial time τ1 > τ, i.e. consider (3.1) with the following initial condition

x(τ1)=u~0,(3.3)

where u~0 ∈ ℝn.

Remark 3.1

For the IVP for ODE (3.1), (3.3) note that the right side part f(t,x) has to be defined only for t ≥ τ1.

We can look at IVP for ODE (3.1), (3.3) in two different ways:

(A1 for ODE). From the general solution x(t;τ,c) of (3.1) with initial condition x(τ) = c (c is an arbitrary constant) we choose the one x(t;τ,c1) with x(τ1;τ,c1) = u~0. We call it a solution of the IVP (3.1), (3.3) for t ≥ τ1 and denote it by x(t;τ1, u~0). Then using x(t1;τ,c1)=c1+∫ττ1f(s,x(s;τ,c1))ds we obtain that the solution x(t) = x(t;τ1, u~0) of (3.1), (3.3) will satisfy

x(t)=c1+∫τtf(s,x(s;τ,c1))ds=u~0−∫ττ1f(s,x(s;τ,c1))ds+∫τtf(s,x(s;τ,c1))ds,t≥τ1.(3.4)

(A2 for ODE). Consider (3.1), (3.3) as a new IVP and its solution, defined for t ≥ τ1, and we call this a solution of the IVP (3.1), (3.3). Then the solution will satisfy the following integral equation

x(t)=u~0+∫τ1tf(s,x(s))ds,t≥τ1.(3.5)

Remark 3.2

In the general case for ODE’s both points of view do not differ since ∫τtf(s,x(s))ds=∫ττ1f(s,x(s))ds+∫τ1tf(s,x(s))ds, i.e. x(t;τ,x~0)≡x(t;τ1,u~0) for t≥τ1 with x(τ1;τ,x~0)=u~0.

II. Caputo-type fractional differential equations. Consider the fractional differential equation (FrDE) with Caputo fractional derivatives

τcDqx(t)=f(t,x)fort≥τ(3.6)

with initial condition

x(τ)=x~0,(3.7)

where x0~ ∈ ℝn.

Denote the solution of the IVP for FrDE (3.6), (3.7) by x(t;τ, x0~).

The solution x(t) = x(t;τ, x0~) of IVP for FrDE (3.6), (3.7) satisfies the fractional Volterra integral equation

x(t)=x~0+1Γ(q)∫τt(t−s)q−1f(s,x(s))ds,t≥τ.(3.8)

Now change the initial time to τ1 > τ and consider FrDE (3.6) with the initial condition (3.3). Then as in the ordinary case (see (A1 for ODE) and (A2 for ODE)), there are two approaches to define the solution of the new IVP for the Caputo-type FrDE:

(A1 for FrDE). From the set of all solutions x(t;τ,c) of FrDE (3.6) with initial condition x(τ) = c, (c is an arbitrary constant), we choose the one x(t;τ,c1) with x(τ1;τ,c1) = u~0. We call it a solution of the IVP (3.6), (3.3) for t ≥ τ1 and denote it by x(t;τ1, u~0). Therefore, x(t;τ1, u0~) ≡ x(t;τ,c1) for t ≥ τ1. Then using Eq. (3.8) with t = τ1, x0~ = c1, the solution x(t) = x(t;τ1, u~0) of IVP for FrDE (3.6), (3.3) will satisfy the following integral equation

x(t)=c1+1Γ(q)∫τt(t−s)q−1f(s,x(s;τ,c1))ds=u~0−1Γ(q)∫ττ1(τ1−s)q−1f(s,x(s;τ,c1))ds+1Γ(q)∫τt(t−s)q−1f(s,x(s;τ,c1))ds,t≥τ1.(3.9)

(compare Eq. (3.4) in the ordinary case q = 1 with Eq. (3.9) in the fractional case).

K. Diethelm pointed out that the problem consisting of Eqs. (3.6) and (3.3) is more closely related to a boundary value problem than to an initial value problem. This is a contrast to the situation observed for first-order differential equations (see Section 6 in [18]).

Remark 3.3

Using (A1 for FrDE) we keep one of the basic properties of ODEs, namely, x(t;τ1, x(τ1;τ,c)) = x(t;τ,c) for t ≥ τ1.

(A2 for FrDE). Set up a new initial value problem for t ≥ τ1 whose solution will satisfy the following fractional integral equation

x(t)=u~0+1Γ(q)∫τ1t(t−s)q−1f(s,x(s))ds,t≥τ1.(3.10)

(Compare Eq. (3.5) in the ordinary case q = 1 with Eq. (3.10) in the fractional case).

The fractional integral equation (3.10) is equivalent to the following Caputo fractional differential equation

τ1cDqx(t)=f(t,x)fort≥τ1(3.11)

with initial condition (3.3).

Remark 3.4

In the general case both points of view (A1 for FrDE) and (A2 for FrDE) differ, since

∫τt(t−s)q−1f(s,x(s))ds≠∫ττ1(τ1−s)q−1f(s,x(s))ds+∫τ1t(t−s)q−1f(s,x(s))ds,

(compare with Remark 3.2).

Remark 3.5

Using (A2 for FrDE) we lose one of the basic properties of ODE’s, namely, x(t;τ1, x(τ1;τ,c)) ≠ x(t;τ,c) for t > τ1.

Remark 3.6

In (A2 for FrDE) the right side part f(t,x) of the IVP (3.6), (3.3) has to be defined only for t ≥ τ1.

Example 1

Consider FrDE(3.6) with n = 1, τ = 0, τ1 = 1, u~0 = 0.

Case 1. Let

f(t,x)≡h(t)=0t∈[0,1],1−tt≥1.

Case 1.1. (Approach (A1 for FrDE)). According to formula (3.9) we get

x(t)=0+1Γ(q)∫0t(t−s)q−1h(s)ds−1Γ(q)∫01(1−s)q−1h(s)ds=1Γ(q)∫1t(t−s)q−1(1−s)ds,t≥1.(3.12)

Case 1.2. (Approach (A2 for FrDE)). According to (3.10) we get

x(t)=0+1Γ(q)∫1t(t−s)q−1(1−s)ds,t≥1.(3.13)

In this particular case both solutions coincide.

Case 2. Let f(t,x) = 1 − t, t ≥ 0.

Case 2.1. (Approach (A1 for FrDE)). According to formula (3.9) we get

x(t)=−1Γ(q)∫01(1−s)q−1(1−s)ds+1Γ(q)∫0t(t−s)q−1(1−s)ds,t≥1.(3.14)

Case 2.2. (Approach (A2 for FrDE)). According to (3.10) the solution is given by (3.13).

In this particular case Eq. (3.14) differs from Eq. (3.13).

Therefore, the definition of the function f(t,x) to the left of the initial point has no influence in (A2 for FrDE) (similar to the ODE situation) but it has a huge influence in (A1 for FrDE). □

Remark 3.7

Note that (A1 for FrDE) is similar in some sense to a boundary value problem, whereas (A2 for FrDE) is close to the idea of initial value problems defined and studied in the classical books [18], [31] (the initial time coincides with the lower limit of the Caputo fractional derivative).

Example 2

Consider FrDE (3.6) with n = 1, q = 0.8, f(t,x) ≡ 1, τ = 0 and τ1 > 0.

The solution of IVP for FrDE (3.6), (3.7) is x(t;0,x0~)=x0~+1.25t0.8Γ(0.8).

Using (A1 for FrDE) we get the solution of IVP for FrDE (3.6), (3.3), namely, x(t;τ1,u0~)=u0~+1.25t0.8−τ10.8Γ(0.8).

Using (A2 for FrDE) the solution of IVP for FrDE (3.6), (3.3) (or the equivalent (3.11), (3.3)) is x(t;τ1,u0~)=u0~+1.25(t−τ1)0.8Γ(0.8).

In this particular case both solutions differ. □

Now consider a case when f(t,x) depends implicitly on x.

Example 3

Consider the FrDE (3.6) with n = 1, f(t,x) = x, τ = 0, τ1 > 0.

The solution of IVP for FrDE (3.6), (3.7) is x(t;0,x~0)=x~0Eq(tq).

Using (A1 for FrDE) and the general solution of IVP for FrDE (3.6), (3.7) we get x(τ1;0,c)=cEq(τ1q)=u~0, or c=u~0Eq(τ1q). Then from (3.9) we get the solution of IVP for FrDE (3.6), (3.3)x(t;τ1,u~0)=u~0Eq(tq)Eq(τ1q).

Using (A2 for FrDE) the solution of IVP for FrDE (3.6), (3.3) (or the equivalent (3.11), (3.3)) is x(t;τ1,u~0)=u~0Eq((t−τ1)q).

Solutions obtained by both approaches differ but in the ordinary case, (q = 1) the Mittag-Leffler function E1(t) = et and both solutions coincide. □

Remark 3.8

Both approaches described above usually differ and give different solutions in the general case.

4 Non-instantaneous impulses in Caputo fractional differential equations

Now we set up the IVP for Caputo fractional differential equations with non-instantaneous impulses.

In this paper we will assume two increasing sequences of points {ti}i=1∞ and {si}i=0∞ are given such that 0 < s0 < ti < si < ti + 1, i = 1,2,…, and t0 ∈ ℝ+. Without loss of generality we assume 0 ≤ t0 < s0.

Consider the initial value problem (IVP) for the nonlinear noninstantaneous impulsive fractional differential differential equation (NIFrDE)

t0cDqx(t)=f(t,x)fort∈(tk,sk],k=0,1,…,x(t)=ϕk(t,x(t),x(sk−1−0))fort∈(sk−1,tk],k=1,2,…,x(t0)=x0,(4.1)

where x0∈Rn,f:∪k=0∞[tk,sk]×Rn→Rn,ϕk:[sk−1,tk]×Rn×Rn→Rn, (k = 1,2,3,…).

Definition 1

For NIFrDE (4.1) the intervals (sk − 1,tk], k = 1,2,…, are called intervals of non-instantaneous impulses, and ϕk(t,x,y), k = 1,2,…, are called non-instantaneous impulsive functions.

Remark 4.1

If tk = sk − 1, k = 1,2,… then the IVP for NIFrDE (4.1) reduces to an IVP for impulsive fractional differential equations.

We give a brief description of the solution of IVP for NIFrDE (4.1). Based on the description of the solution of FrDE given in Section 3 we set up two approaches to the solutions of non-instantaneous fractional impulsive differential equations.

The definition of the solution x(t;t0,x0) for t > t0 depends on your point of view:

(A1 for NIFrDE). Let f(t,x) be defined for t ≥ t0, x ∈ ℝn. Following approach (A1 for FrDE) and Eq. (3.9) with τ = t0, τ1 = tk,k = 1,2,… and u~0 = x(tk − 0;t0,x0) = ϕk(tk, x(tk − 0;t0,x0), x(sk − 1 − 0;t0,x0)) given in Section 3, we get the solution of the IVP for NIFrDE (4.1) by the equalities (integral and algebraic)

x(t;t0,x0)=x0+1Γ(q)∫t0t(t−s)q−1f(s,x(s;t0,x0))ds,t∈(t0,s0]ϕk(t,x(t;t0,x0),x(sk−1−0;t0,x0)),t∈(sk−1,tk],k=1,2,3,…,ϕk(tk,x(tk;t0,x0),x(sk−1−0;t0,x0))−1Γ(q)∫t0tk(tk−s)q−1f(s,x(s;t0,x0))ds+1Γ(q)∫t0t(t−s)q−1f(s,x(s;t0,x0))ds,t∈(tk,sk],k=1,2,….(4.2)

Remark 4.2

In the special case ϕk(t, x(t), x(sk−1 − 0)) = gk(t, x(t)) the reduced formula of (4.2) is given in Section 9, [39], and Eq. (8), [44].

Remark 4.3

The approach (A1 for NIFrDE) is applied in [19], [41] for studying periodic solutions, and in [20], [39], [44] for studying existence.

(A2 for NIFrDE). Let f(t, x) be defined only for t∈∪k=0∞[tk,sk] and x ∈ ℝn, i.e. it is defined only on the intervals without non-instantaneous impulses. Then using the approach (A2 for FrDE) and Eq. (3.10) with τ = t0, τ1 = tk, k = 1, 2, … and ũ0 = x(tk − 0;t0, x0) = ϕk(tk, x(tk − 0;t0, x0), x(sk−1 − 0;t0, x0)) given in Section 3 we get the solution of the IVP for NIFrDE (4.1) by the equalities (integral and algebraic)

x(t;t0,x0)=x0+1Γ(q)∫t0t(t−s)q−1f(s,x(s;t0,x0))ds,for t∈[t0,s0],ϕk(t,x(t;t0,x0),x(sk−1−0;t0,x0))for t∈(sk−1,tk],k=1,2,…,ϕk(tk,x(tk;t0,x0),x(sk−1−0;t0,x0))+1Γ(q)∫tkt(t−s)q−1f(s,x(s;t0,x0))dsfor t∈[tk,sk],k=1,2,….(4.3)

Following the approach (A2 for FrDE) the solution of the IVP for NIFrDE (4.1) is given by

x(t;t0,x0)=Xk(t)for t∈(tk,sk],k=0,1,2,…,ϕk(t,x(t;t0,x0),Xk−1(sk−1−0)),for t∈(sk−1,tk],k=1,2,…,(4.4)

where

  1. the function X0(t) is the solution of IVP for FrDE (3.6), (3.7) with τ = t0 and x~0 = x0;

  2. the function Xk(t) is the solution of IVP for FrDE (3.6), (3.7) with τ = tk, and x~0 = ϕk(tk, x(tk; t0, x0), Xk−1(sk−1 − 0)), k = 1, 2, ….

Remark 4.4

The approach (A2 for NIFrDE) is applied in [1] for partial fractional differential equations, in [2], [3] for partial fractional inclusions, in [11] for stability, in [14] for abstract fractional differential equations, in [15], [25] for fractional integro-differential equations, in [21] for fractional functional differential equations of order α ∈ (1, 2), in [27] for boundary value problems, and in [45] for existence.

Now we discuss the statement of the problem (4.1) and the type of impulsive functions. In some papers (see, for example, [25], [39], [43], [44]) the special case ϕk(t, x(t), x(sk−1 − 0)) = gk(t, x(t)) is studied.

Example 4

Consider the IVP for NIFrDE (4.1) with n = 1, t0 = 0, q = 0.8, fk(t, x) = 1 for t ∈ [tk, sk], k = 0, 1, 2, ….

We will discuss several cases w.r.t. the type of impulsive function ϕk(t, x, y).

Case 1. Let ϕk(t, x, y) = gk(t) for t ∈ [sk−1, tk] (k = 1, 2, 3, …).

(A1 for NIFrDE). According to Eq. (4.2) and Lemma 2.7, [44] the solution is

x(t;0,x0)=x0+1.25t0.8Γ(0.8),t∈(0,s0],gk(t),t∈(sk−1,tk],k=1,2,3,…,gk(tk)+1.25t0.8−tk0.8Γ(0.8),t∈(tk,sk],k=1,2,….

The solution depends on the initial value x0 only on the interval (0, s0]. Therefore, the solutions x(t; 0, x0) and x(t;0, x~0 ) with different initial values x0 ≠ x~0 will differ only on the first interval (0, s0] and x(t;0, x0) ≠ x(t; 0, x~0 ) for all t > s0.

(A2 for NIFrDE). According to Eq. (4.3) the solution is given by

x(t;0,x0)=x0+1.25t0.8Γ(0.8),t∈(0,s0],gk(t),t∈(sk−1,tk],k=1,2,3,…,gk(tk)+1.25(t−tk)0.8Γ(0.8),t∈(tk,sk],k=1,2,….

Applying (A2 for NIFrDE) similarly to (A1 for NIFrDE) we obtain that the solutions x(t; 0, x0) and x(t; 0, x~0 ) with x0 ≠ x~0 coincide for all t > s0.

Case 2. Let ϕk(t, x, y) = gk(t, y) for t ∈ [sk−1, tk], (k = 1, 2, 3, …), i.e. the impulsive conditions are x(t) = gk(t, x(sk−1 − 0)), t ∈ (sk−1, tk] and the impulsive functions depend on the value of the solution before the jump.

Then applying (A1 for NIFrDE) the solution is

x(t;0,x0)=x0+1.25t0.8Γ(0.8)t∈(0,s0]g1(t,x0+1.25s00.8Γ(0.8))t∈(s0,t1]g1(t1,x0+1.25s00.8Γ(0.8))+1.25t0.8−t10.8Γ(0.8)t∈(t1,s1]g2(t,g1(t1,x0+1.25s00.8Γ(0.8))+1.25s10.8−t10.8Γ(0.8)))t∈(s1,t2]………

Now the solution depends on the initial value x0 for all t ≥ 0. The same happens with the application of (A2 for NIFrDE).

Case 3. Let ϕk(t, x, y) = akx + bk for t ∈ [sk−1, tk], (k = 1, 2, 3, …) where ak, bk are constants.

If ak = 1,bk = 0 then any function x(t) will satisfy the impulsive condition x(t) = x(t) for t ∈ [sk−1, tk], (k = 1, 2, 3, …) and obviously the IVP for NIFrDE (4.1) will have an infinite number of solutions.

If ak = 1, bk ≠ 0 then no function x(t) will satisfy the impulsive condition x(t) = x(t) + b for t ∈ [sk−1, tk], (k = 1, 2, 3, …) and obviously the IVP for NIFrDE (4.1) will have no solution.

If ak ≠ 1, bk = 0 then the only function x(t) that satisfies the impulsive condition x(t) = ax(t) for t ∈ [sk−1, tk], (k = 1, 2, 3, …) is the zero function, and therefore any solution of IVP for NIFrDE (4.1) will be zero on (sk−1, tk], (k = 1, 2, 3, …)

If ak ≠ 1, bk ≠ 0 then there will be a unique function x(t) that satisfies the impulsive condition x(t) = ax(t) + b for t ∈ [sk−1, tk], (k = 1, 2, 3, …) and we can talk about uniqueness of the solution IVP for NIFrDE (4.1).

Case 4. Let ϕk(t, x, y) = arctan(x) + cos(x) + y for t ∈ [sk−1, tk] (k = 1, 2, 3, …). Then the algebraic equation x = arctan(x) + cos(x) + y could have more than one solution (for example if y = 1, then there are 5 constant solutions), i.e. we do not have uniqueness. □

Remark 4.5

In the general case the impulsive functions in (4.1) have to depend on the value of the solution before the impulse, i.e. the impulsive condition has to be given by the function ϕk(t, x(t), x(sk−1 − 0)) for t ∈ (sk−1, tk], k = 1, 2, ….

Remark 4.6

To discuss the existence and uniqueness of the solution of NIFrDE (4.1) we need the equation corresponding to the impulsive condition x = ϕk(t, x, y), k = 1, 2, … to have a unique solution xk(t, y) for all k = 1, 2, ….

Example 5

Consider the IVP for NIFrDE (4.1) with n = 1, t0 = 0 and f(t, x) = Ax, t ≥ 0, i.e. consider

t0cDqx(t)=Ax for t∈(tk,sk],k=0,1,2,…,[6pt]x(t)=ϕk(t,x(t),x(sk−1−0)) fort∈(sk−1,tk],k=1,2,…,x(0)=x0,(4.5)

where x0 ∈ ℝ and A is a constant.

(A1 for NIFrDE). According to Eq. (4.2) the solution x(t; 0, x0) of (4.5) is given by

x(t;0,x0)=x0Eq(Atq)for t∈[0,s0],ϕk(t,x(t),x(sk−1−0))for t∈(sk−1,tk],k=1,2,…,ϕk(tk,x(tk),x(sk−1−0))−A1Γ(q)∫0tk(tk−s)q−1x(s)ds+A1Γ(q)∫0t(t−s)q−1x(s)dsfor t∈(tk,sk],k=1,2,….(4.6)

(A2 for NIFrDE). Applying formulas (4.4) and (4.3), we obtain the solution x(t;0, x0) of (4.5):

x(t;0,x0)=x0Eq(Atq) for t∈[0,s0],ϕk(t,x(t),x(sk−1−0))for t∈(sk−1,tk],k=1,2,…,ϕk(tk,x(tk),x(sk−1−0))Eq(A(t−tk)q)×(∏i=0k−1Eq(A(si−ti)q))for t∈[tk,sk],k=1,2,….(4.7)

Comparing (4.6) and (4.7) we can see the approach (A2 for NIFrDE) gives the solution in a closed form.

Case 1. Let ϕk(t, x, y) = ak x for t ∈ [sk−1, tk] (k = 1, 2, 3, …), where ak ≠ 1 are constants. Since the solution of the equation x = akx is the zero solution, then the non-instantaneous impulsive condition of (4.5) is reduced to x(t) = 0 for t ∈ (sk−1, tk], k = 1, 2, ….

According to (A1 for NIFrDE) and Eq. (4.6) the solution of (4.5) is

x(t;0,x0)=x0Eq(Atq) for t∈[0,s0],0 for t∈(sk−1,tk],k=1,2,…,−A1Γ(q)∫0tk(tk−s)q−1x(s)ds+A1Γ(q)∫0t(t−s)q−1x(s)ds for t∈(tk,sk],k=1,2,….(4.8)

According to (A2 for NIFrDE) and Eq. (4.7) the solution of (4.5) is

x(t;0,x0)=x0Eq(Atq)for t∈[0,s0],0for t>s0.(4.9)

Consider the ordinary case (q = 1) of (4.5), i.e. the non-instantaneous impulsive differential equation x’ = Ax for t ∈ (tk, sk], k = 0, 1, 2, … and x(t) = 0 for t ∈ (sk−1tk], k = 1, 2, …. Then the solution of the corresponding IVP for non-instantaneous impulsive differential equation is

x(t;0,x0)=x0eAtfor t∈[0,s0],0for t>s0.(4.10)

Eq. (4.9) is similar to Eq. (4.10), which shows the approach (A2 for NIFrDE) seems to be a natural generalization of the ordinary case.

Case 2. Let ϕk(t, x, y) = aky, ak = const, k = 1, 2, 3, ….

Applying (A1 for NIFrDE) and Eq. (4.6), we obtain the solution of (4.5)

x(t;0,x0)=x0Eq(Atq)for t∈[0,s0],akx(sk−1−0) for t∈(sk−1,tk],k=1,2,…,akx(sk−1−0)−A1Γ(q)∫t0tk(tk−s)q−1x(s)ds+A1Γ(q)∫t0t(t−s)q−1x(s)dsfor t∈(tk,sk],k=1,2,….(4.11)

Applying (A2 for NIFrDE) and Eq.(4.7), we get

x(t;0,x0)=x0Eq(Atq)for t∈[0,s0],x0∏i=0k−1(ai+1Eq(A(si−ti)q)) for t∈(sk−1,tk],k=1,2,…,x0Eq(A(t−tk)q)∏i=0k−1(ai+1Eq(A(si−ti)q))for t∈[tk,sk],k=1,2,….(4.12)

The approach (A2 for NIFrDE) gives the explicit form for the solution.

Case 3. Let A = 0 and ϕk(t, x, y) = ak(t)y, ak:[tk, sk] → ℝ, k = 1, 2, ….

Applying (A1 for NIFrDE) and Eq. (4.11) we obtain

x(t;0,x0)=x0for t∈[0,s0],x0ak(t)∏i=1k−1ai(ti) for t∈(sk−1,tk],k=1,2,…,x0∏i=1kak(ti)for t∈(tk,sk],k=1,2,….(4.13)

Applying (A1 for NIFrDE) and Eq. (4.11) we obtain (4.13), and therefore the formulas for the solutions, obtained by both approaches, coincide. □

Example 6

Consider the IVP for the scalar NIFrDE (4.1) with f(t, x) =1t−0.5(tk+sk−1) for t ≥ t0. The function f is not defined on the whole interval [sk−1, tk], k = 1, 2, ….

Applying (A1 for NIFrDE) the integral ∫t0t(t−s)q−1f(t,s)ds is not convergent for all t > s0, so the formula (4.2) is not applicable and this approach does not give a solution.

The application of (A2 for NIFrDE) and formula (4.3) causes no problem since we use the integral ∫tkt(t−s)q−11s−0.5(tk+sk−1)ds for t ∈ (tk, sk] which is convergent. □

Remark 4.7

The approach (A1 for NIFrDE) and the application of formula (4.2) for the solution of NIFrDE (4.1) require the function f(t, x) to be defined on the whole interval [t0, ∞) although this function is not used on ⋃k=1∞(sk−1,tk]. In [41] the conditions on the function f (such as the Lipschitz condition) are set up only on the intervals with no impulses [tk, sk], k = 1, 2, … and this causes conflicts in the proofs (see Theorem 3.1-3.4 in [41]).

The approach (A2 for NIFrDE) requires the function f(t, x) to be defined only on the interval [tk, sk], k = 1, 2, … on which this function is applied.

5 Instantaneous impulses in Caputo fractional differential equations

Consider the special case when sk−1 = tk, k = 1, 2, …. Then any interval of non-instantaneous impulses is reduced to a point and any impulsive function ϕk is reduced to ϕk(tk, x(tk), x(tk − 0)) for k = 1, 2, 3, …. Assume the equation x = ϕk(tk, x, y), k = 1, 2, … has a unique solution w.r.t. x : x = Bk(y) (see Remark 14). Then the impulsive condition could be presented as x(tk + 0) = Bk(x(tk − 0)) for k = 1, 2, 3, …. Then problem (4.1) will be reduced to an IVP of Caputo-type impulsive fractional differential equation (IFrDE)

t0cDqx(t)=f(t,x)for t≠tk,k=1,…,x(tk+0)=x(tk−0)+Ik(x(tk−0)) for k=1,2,…,x(t0)=x0,(5.1)

where x0 ∈ ℝn, f : [t0, ∞) × ℝn → ℝn, Ik : ℝn → ℝn, (k = 1, 2, 3, …) is defined by Ik(y) = Bk(y) − y, k = 1, 2, …..

Then both approaches (A1 for NIFrDE) and (A2 for NIFrDE) given in Section 4 reduce to IFrDE (5.1):

(A1 for IFrDE). Eq. (4.2) is reduced and the formula for the solution of (5.1) is given by

x(t;t0,x0)=x(tk−0;t0,x0)+Ik(x(tk−0;t0,x0))−1Γ(q)∫t0tk(tk−s)q−1f(s,x(s;t0,x0))ds+1Γ(q)∫t0t(t−s)q−1f(s,x(s;t0,x0))ds,t∈(tk,tk+1],k=0,1,2,…,(5.2)

where I0(x) ≢ 0 and x(t0 − 0; t0, x0) = x0.

Use induction and obtain that

x(t;t0,x0)=x0+1Γ(q)∫t0t(t−s)q−1f(s,x(s;t0,x0))ds+∑j=1kIj(x(tj−0;t0,x0)),t∈(tk,tk+1],k=0,1,2,….(5.3)

In the special case t0 = 0, Ik(x) = x + yk, yk = const and f(t, x) = h(t) an implicit formula for the solution is (see Lemma 3.2, [20]):

x(t;0,x0)=x0+∑i=1kyi+1Γ(q)∫0t(t−s)q−1h(s)ds for t∈(tk,tk+1],k=1,2,….

(A2 for IFrDE). Eq. (4.3) is reduced and the formula for the solution of (5.1) is given by

x(t;t0,x0)=x(tk−0;t0,x0)+Ik(x(tk−0;t0,x0))+1Γ(q)∫tkt(t−s)q−1f(s,x(s;t0,x0))ds,t∈(tk,tk+1],k=0,1,2,….(5.4)

where I0(x) ≡ 0 and x(t0 − 0; t0, x0) = x0.

Use induction in (5.4) and obtain

x(t;t0,x0)=x0+1Γ(q)∑j=1k∫tj−1tj(tj−s)q−1f(s,x(s;t0,x0))ds+1Γ(q)∫tkt(t−s)q−1fk(s,x(s;t0,x0))ds+∑j=1kIj(x(tj−0;t0,x0)),t∈(tk,tk+1],k=0,1,2,….(5.5)

Remark 5.1

Note that if the impulsive function is of the type gk(t, x(t)) as in some published papers, the non-instantaneous impulsive condition in (4.1) reduces to x(tk + 0) = gk(tk, x(tk + 0)), k = 1, 2, … which does not give the amount of the jump at the impulsive point tk for the unknown function.

Example 7

Consider the fractional comparison principle for FrDE (3.6) (see, for example, [28], [36]):

If x, y ∈ C(ℝ+) and0CDqx(t)≤0CDqy(t),thenx(0) ≤ y(0) impliesx(t) ≤ y(t), t ≥ 0.

Now we will discuss the application of the comparison principle to the IFrDE (5.1).

Applying the approach (A1 for IFrDE) the solution of (5.1) will be (see, for example, p. 5, [36]) x(t;t0,x0)=uk(t;tk,xk+)fort∈(tk,tk+1],k=1,2,…,whereuk(t;tk,xk+) is the solution of the FrDE without impulses (3.3) with initial condition (3.3) with τ=t0,τ1=tk,(~u0)=xk+=x(tk;t0,x0)+Ik(x(tk;t0,x0)). Therefore, the application of the above given fractional comparison principle on the interval [tk, tk+1] is not allowed since t0CDqx(t)≠tkCDqx(t) (this was used in [36]).

Applying (A2 for NIFrDE), the solution of (5.1) will be x(t; t0, x0) = uk(t;tk,xk+)fort∈(tk,tk+1],k=1,2,…,whereuk(t;tk,xk+) is the solution of the FrDE without impulses (3.6) with initial condition (3.3) with τ=tk,τ1=tk,u~0=xk+=x(tk;t0,x0)+Ik(x(tk;t0,x0)). Therefore, the application of the above given fractional comparison principle on the interval [tk, tk+1] is allowed since the lower limit of the fractional derivative and the initial time of the problem coincide.

6 Existence results

In this section we consider IVP for NFrDE (4.1) when n = 1, i.e. the scalar case on the finite interval J = [t0, T], T < ∞ with sm = T. We study existence for IVP for NIFrDE (4.1) using both approaches.

Introduce the following classes of functions

PC([t0,T])={u:[t0,T]→R:u∈C([t0,T]/{sk}k=0m,R):u(sk)=u(sk−0)=limt↑sku(t)<∞,u(sk+0)=limt↓sku(t)<∞,k=1,2,…,m},NPC1([t0,T])={u:[t0,T]→R:u∈C([t0,T]/{sk}k=0m,R),u∈C1(∪k=0m[tk,sk],R):u(sk)=u(sk−0)=limt↑sku(t)<∞,u′(sk)=limt↑sku′(t)<∞,u(sk+0)=limt↓sku(t)<∞,k=1,2,…,m}.

(A1 for NIFrDE). In [44] the NIFrDE (4.1) is studied when f(t, x) is continuously defined on the whole interval J and the impulsive functions do not depend on the value of the solution before the impulse, i.e. ϕk(t, x, y) = gk(t,x), k = 1, 2, … (see Examples 4, 5 and our comment in Remark 13).

Let Ψ > 0, φ ∈ PC(J, ℝ) and consider the fractional noninstantaneous differential inequalities

|t0cDqy(t)−f(t,y)|≤φ(t)fort∈(tk,sk],k=0,1,…,m|y(t)−ϕk(t,y(t))|≤Ψfort∈(sk−1,tk],k=1,2,…,m.(6.1)

Theorem 6.1

(by A1 for NIFrDE, Theorem 4.2, [44]), Let the following conditions be satisfied:

  1. The function f ∈ C(J × ℝ, ℝ) and there exists a positive constantLfsuch that |f(t, x) − f(t, y)| ≤ Lf|x − y| for eacht ∈ J, x, y ∈ ℝ.

  2. The functionsϕk(t, x) ∈ C([sk−1, tk] × ℝ, ℝ) and there exist constantsLksuch that |ϕk(t, x1) − ϕk(t, x2)| ≤ Lk| x1 − x2| for eacht ∈ [sk−1, tk], x1, x2 ∈ ℝ.

  3. The function y(t) satisfies the fractional non-instantaneous differential inequalities(6.1)with Ψ > 0 is a constant, φ ∈ C(J, ℝ) is a nondecreasing function in⋃i=0m[ti,si]such that there exists a constant Cφwith

    (∫t0t(φ(s))1pds)p≤Cφφ(t)fort∈J.

Then there exists a unique solutionx~(t)of the IVP for NIFrDE (4.1) withx0 = y(t0) such thatit satisfies the integral-algebraic equations(4.2)and

|y(t)−x~(t)|≤2CφΓ(q)(1−pq−p)1−pTq−p+11−M(φ(t)+Ψ)(6.2)

for allt ∈ J provided that 0 < p < q < 1, where the constant M = max{M1, M2} < 1, with

M1=max{Lk+LfCφΓ(q)(1−pq−p)1−p(skq−p+tkq−p),k=0,1,2…,m}<1(6.3)

and

M2=max{Lk+LfΓ(q+1)(tkq+skq)|,k=1,2…,m}<1.(6.4)

Remark 6.1

The function φ in condition 3 of Theorem 6.1 and the fractional non-instantaneous differential inequalities (6.1) is used only on the intervals [tk, sk], k = 0, 1, 2, …, m. However because of the application of (A1 for NIFrDE) this function has to be defined on the whole interval J = [t0, T].

Remark 6.2

Note that the condition M1, M2 < 1 concerning constants M1, M2 given in (6.3) and (6.4) requires conditions on the impulsive points tk, sk and on the Lipschitz constants. Also this condition does not allow the result to be generalized to the infinite interval [t0, ∞).

Remark 6.3

Note that in [44] the definition of Ulam–Hyers–Rassias stability w.r.t. (φ,Ψ) of NIFrDE (4.1) is given. However since the stability property is usually only meaningful for an infinite interval and Theorem 6.1 is true on a finite interval we will skip comments on this type of stability.

(A2 for NIFrDE). In our study we will use the result for FrDE (3.6).

Lemma 6.1

(Theorem 3.1, [42]), Let the following conditions be satisfied:

  1. The function f ∈ C(I, ℝ), I = [τ, T] and there exists a positive constant L such that |f(t, x) − f(t, y)| ≤ L|x − y|, t ∈ I, x, y ∈ ℝ.

  2. The function y ∈ C1(I, ℝ) satisfies the fractional differential equation

    |τcDqy(t)−f(t,y(t)|≤ϖ(t),t∈I,

where the functionϖ ∈ C(I, ℝ) is such that

1Γ(q)∫τt(t−s)q−1ϖ(s)ds≤Kϖ(t),t∈I,

with 0 < KL < 1.

Then there exists a unique functionx(t) ∈ C(I, ℝ) such that

x(t)=y(τ)+1Γ(q)∫τt(t−s)q−1f(s,x(s)ds,t∈I(6.5)

and

|y(t)−x(t)|≤K1−KLϖ(t),t∈I.(6.6)

Now we give sufficient conditions for existence of the NIFrDE (4.1) by an application of the approach (A2 for NIFrDE) for the solution.

Let Ψk > 0, φk ∈ C([tk, sk], ℝ), k = 0, 1, …, m and consider the fractional non-instantaneous differential inequalities

|tkcDqy(t)−f(t,y)|≤φk(t)fort∈(tk,sk],k=0,1,…,m|y(t)−ϕk(t,y(t),y(sk−1−0))|≤Ψkfort∈(sk−1,tk],k=1,2,…,m.(6.7)

Remark 6.4

Note if y(t) is a solution of the fractional non-instantaneous differential inequalities (6.7) then this solution satisfies the integral-algebraic inequalities

|y(t)−y(tk)+1Γ(q)∫tkt(t−s)q−1f(s,y(s))ds|≤1Γ(q)∫tkt(t−s)q−1φk(s)ds,t∈(tk,sk],k=0,1,2,…,m,|y(t)−ϕk(t,y(t),y(sk−1−0))|≤Ψk,t∈(sk−1,tk],k=1,2,…,m.

Theorem 6.2

(by A2 for NIFrDE) Let the following conditions be satisfied:

  1. The functionf∈C(∪k=0m[tk,sk]×R,R)and there exist positive constantsLk = Lk(f), k = 0, 1, 2, …, m, such that |f(t, x) − f(t, y)| ≤ Lk|x − y| for eacht ∈ [tk, sk], x, y ∈ ℝ, k = 0, 1, …, m.

  2. The functionsϕk(t, x, y) ∈ C([sk − 1, tk] × ℝ × ℝ, ℝ), k = 1, 2, …, mare such that for anyt ∈ [sk − 1, tk] and y ∈ ℝ there exists a unique solution x = γk(t, y) of the algebraic equation x = ϕk(t, x, y) w.r.t. x, and there exist constants lk = lk(ϕk) ∈ (0,1), k = 1, 2, …, m such that |ϕk(t, x1, y1) − ϕk(t, x2, y2)| ≤ lk(|x1 − x2| + |y1 − y2|) for eacht ∈ [sk − 1, tk], x1, x2, y1, y2 ∈ ℝ, k = 1,…, m.

  3. The functionsφk ∈ C([tk, sk], ℝ), k = 0, 1, … are nondecreasing functions and there exist constants Ck = Ck(φk) > 0, LkCk < 1, k = 0, 1, …, msuch that

    1Γ(q)∫tkt(t−s)q−1φk(s)ds≤Ckφk(t),t∈[tk,sk].(6.8)

Then for each solution y(t) ∈ NPC1([t0, T], ℝ) of the fractional differential inequality (6.7)there exists a solutionx(t) such that x ∈ NPC1([t0, T], ℝ) of the IVP for NIFrDE (4.1) with x0 = y(t0) and it satisfies the integral-algebraic equations (4.3) and

|y(t)−x(t)|≤C01−C0L0φ0(t)=F0(s),t∈(t0,s0],Ck1−CkLkφk(t)+11−lk11−CkLk(Ψk+lkFk−1(sk−1))=Fk(s),t∈(tk,sk],1,2,…,m11−lk(Ψk+lkFk−1(sk−1)),t∈(sk−1,tk],k=1,2,…,m.(6.9)

Proof

We will use induction.

Let t ∈ [t0, s0]. According to Lemma 6.1 with τ = t0, T = s0, L = L0, K = C0 and ϖ(t) = φ0(t) there exists a solution x0(t) ∈ C([t0, s0], ℝ) satisfying the integral equality

x0(t)=x0+1Γ(q)∫t0t(t−s)q−1f(s,x0(s))ds(6.10)

and the inequality

|y(t)−x0(t)|≤C01−C0L0φ0(t)=F0(t),t∈[t0,s0].(6.11)

Let t ∈ (s0, t1]. Denote the solution of the algebraic equation x = ϕ0(t,x,x0(s0−0))byx~0(t) and

|y(t)−x~0(t)|≤|y(t)−ϕ1(t,y(t),y(s0−0))|+|ϕ1(t,y(t),y(s0−0))−ϕ1(t,x~0(t),x0(s0−0))|≤Ψ1+l1(|y(t)−x~0(t)|+|y(s0−0)−x0(s0−0)|)≤Ψ1+l1|y(t)−x~0(t)|+l1F0(s0)(6.12)

or

|y(t)−x~0(t)|≤11−l1(Ψ1+l1F0(s0)),t∈(s0,t1].(6.13)

Let t ∈ (t1, s1]. Define the function

y~(t)=y(t)−y(t1)+ϕ1(t1,x~0(t1),x0(s0)).

Then

|t1cDqy~(t)−f1(t,y~(t)|≤|t1cDqy(t)−f1(t,y(t)|+|f1(t,y~(t)−f1(t,y(t)|≤φ1(t)+L1|y~(t)−y(t)|.(6.14)

From Remark 6.4, condition 2 and inequalities (6.11), (6.13) we obtain

|y~(t)−y(t)|≤|y(t1)−ϕ1(t1,y(t1),y(s0−0))|+|ϕ1(t1,y(t1),y(s0−0))−ϕ1(t1,x~0(t1),x0(s0))|≤Ψ1+l1|y(t1)−x~0(t1)|+l1|y(s0−0)−x0(s0)|≤Ψ1+l11−l1Ψ1+l1l1(1−l1)F0(s0)+l1F0(s0)=11−l1Ψ1+l11−l1F0(s0).(6.15)

From (6.14), (6.15) we get

|t1cDqy~(t)−f1(t,y~(t)|≤φ1(t)+L11−l1Ψ1+L1l11−l1F0(s0).(6.16)

According to Lemma 6.1 with τ=t1,T=s1,L=L1,K=C1,y(t)=y~(t) and ϖ(t)=φ1(t)+L11−l1Ψ1+L1l11−l1F0(s0) there exists a solution x1(t) ∈ C([t1,s1],ℝ) satisfying the integral equation

x1(t)=ϕ1(t1,x~0(t1),x0(s0))+1Γ(q)∫t1t(t−s)q−1f1(s,x1(s))ds,t∈(t1,s1](6.17)

and

|y~(t)−x1(t)|≤C11−C1L1(φ1(t)+L11−l1Ψ1+L1l11−l1F0(s0))t∈(t1,s1].(6.18)

Using inequalities (6.15) and (6.18), we get

|y(t)−x1(t)|≤|y~(t)−x1(t)|+|y~(t)−y(t)|≤C11−C1L1(φ1(t)+L11−l1Ψ1+L1l11−l1F0(s0))+11−l1Ψ1+l11−l1F0(s0)≤C11−C1L1φ1(t)+11−l111−C1L1Ψ1+l11−l111−C1L1F0(s0),(6.19)

i.e.

|y(t)−x1(t)|≤C11−C1L1φ1(t)+11−l111−C1L1(Ψ1+l1F0(s0))=F1(s),t∈(t1,s1].(6.20)

Following this inductive process we construct the function

x(t)=xk(t),t∈(tk,sk],k=0,1,2,…,mx~k−1(t),t∈(sk−1,tk],k=1,2,…,m(6.21)

which is a solution of IVP for the NIFrDE (4.1) with x0 = y(t0) and satisfies (6.9).

Remark 6.5

Note in (6.9) for the solution in Theorem 6.2 the points tk,sk are not included (compare with (6.2) in Theorem 6.1). This allows the result of Theorem 6.2 to be generalized to the infinite interval [t0, ∞) for appropriate values of the constants Lk,lk, Ck (for example, lk: ∏i=1∞(1−li)<K1<∞,∏i=1∞li1−li<K2<∞,Ck:∏i=0∞(1−CiLi)<K3<∞ and ∏i=0∞Ci1−CiLi<K4<∞).

Example 8

Let 0 = t0 < s0 = 1 < t1 = 2 < s1 = 4 < t2 = 5 < s2 = 7 < t3 = 9 < s3 = 10. Consider the IVP for NIFrDE (4.1) with n = 1 and q = 0.1, i.e.

0cD0.1x(t)=0.2xtan⁡(t) for t∈∪k=03(tk,sk],x(t)=1k+1(x(t)+x(sk−1−0)) for t∈(sk−1,tk],k=1,2,3,x(0)=1.(6.22)

The function f(t, x) = 0.2x tan(t) is not defined and continuous on the whole interval [0, 10]. Therefore the conditions of Theorem 1 are not satisfied for (6.22) and approach (A1 for NIFrDE) and Theorem 6.1 does not guarantee the existence.

The function f∈C(∪k=03[tk,sk]×R,R) and there exist positive constants L0 = 0.312, L1 = 0.44, L2 = 0.68, L3 = 0.13, i.e. condition 1 of Theorem 6.2 is satisfied. Let ϕk(t, x, y) = 1k+1 (x + y), k = 1, 2, 3. Then condition 2 of Theorem 6.2 is satisfied with lk = 1k+1 , k = 1, 2, 3.

Consider the function y(t) ≡ 1, t ∈ [0, 10] which satisfies the inequalities (6.7) with φk(t) ≡ Lk, t ∈ (tk, sk], k = 0, 1, 2, 3 and Ψk=|k−1k+1|,k=1,2,3. Then, LkΓ(q)∫tkt(t−s)q−1ds=(t−tk)qqΓ(q)Lk≤Ckφk(t),t∈[tk,sk] with Ck = (sk−tk)qqΓ(q),k = 0, 1, 2, 3, i.e. C0 = 1.052, C1 = 1.13, C2 = 1.13, C3 = 1.13. Condition 3 of Theorem 6.2 is satisfied.

According to Theorem 6.2 there exists a solution x(t) of the IVP for NIFrDE (6.22) for which the inequality (6.9) holds. In this case the solution is defined by approach (A2 for NIFrDE). □

7 Conclusions

Initial value problems for Caputo fractional differential equations with noninstantaneous impulses are discussed. We emphasize some basic points:

  1. the impulsive functions have to depend on not only the unknown function of the current argument but also on the value of the unknown function before the impulse, i.e. ϕk(t, x(t), x(sk−1−0)) (see Example 4 and Remark 4.5);

  2. the application of approach A1 requires the definition of the right side part of the Caputo fractional differential equation to be defined on the whole interval of consideration including the intervals of impulses. This is not the same as in approach A2 (see Example 6 and Remark 4.7);

  3. approach A1 does not allow the application of the step by step method w.r.t. to the intervals of impulses to be used directly. This is not the same as in approach A2 (see Example 7);

  4. approach A1 can not be applied to Caputo fractional differential equations with switching right sides parts, i.e. when f is defined in different ways on each interval without impulses. This is not the same as in approach A2 (see Example 1);

  5. in the application of approach A2 the basic property for ODE x(t; τ1, x(τ1; τ, c)) = x(t; τ, c), t ≥ τ1 is lost. This is not the same as in approach A1 (see Remark 3.4);

  6. impulsive fractional differential equations are a special case of fractional differential equations with noninstantaneous impulses (see Section 5);

  7. both approaches are used to study the existence of noninstantaneous impulsive fractional differential equations and the advantages/disadvantages in the corresponding conditions are discussed (see Section 6).

Acknowledgements

Research was partially supported by Fund FP17-FMI-008, University of Plovdiv Paisii Hilendarski.

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Received: 2016-10-27
Published Online: 2017-6-22
Published in Print: 2017-6-27

© 2017 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 20–3–2017)
  4. Survey Paper
  5. A survey of useful inequalities in fractional calculus
  6. Survey Paper
  7. Non-instantaneous impulses in Caputo fractional differential equations
  8. Research Paper
  9. Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models
  10. Research Paper
  11. Solvability of initial value problems with fractional order differential equations in banach spaces by α-dense curves
  12. Research Paper
  13. Periodic problem for two-term fractional differential equations
  14. Research Paper
  15. Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions
  16. Research Paper
  17. Identification problem for degenerate evolution equations of fractional order
  18. Research Paper
  19. Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations
  20. Research Paper
  21. Impact of fractional order methods on optimized tilt control for rail vehicles
  22. Historical Survey
  23. Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet union
  24. Short Paper
  25. Regular fractional differential equations in the Sobolev space
  26. Short Paper
  27. Monotonicity and convexity results for a function through its Caputo fractional derivative
https://doi.org/10.1515/fca-2017-0032
Keywords for this article
non-instantaneous impulses; Caputo fractional differential equation

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 20–3–2017)
  4. Survey Paper
  5. A survey of useful inequalities in fractional calculus
  6. Survey Paper
  7. Non-instantaneous impulses in Caputo fractional differential equations
  8. Research Paper
  9. Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models
  10. Research Paper
  11. Solvability of initial value problems with fractional order differential equations in banach spaces by α-dense curves
  12. Research Paper
  13. Periodic problem for two-term fractional differential equations
  14. Research Paper
  15. Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions
  16. Research Paper
  17. Identification problem for degenerate evolution equations of fractional order
  18. Research Paper
  19. Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations
  20. Research Paper
  21. Impact of fractional order methods on optimized tilt control for rail vehicles
  22. Historical Survey
  23. Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet union
  24. Short Paper
  25. Regular fractional differential equations in the Sobolev space
  26. Short Paper
  27. Monotonicity and convexity results for a function through its Caputo fractional derivative
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