Skip to content

Advertisement

  • Research
  • Open Access

The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect

Advances in Difference Equations20152015:215

https://doi.org/10.1186/s13662-015-0552-1

  • Received: 7 March 2015
  • Accepted: 23 June 2015
  • Published:

Abstract

In this paper, the formula of general solution for nonlinear systems with Caputo-Hadamard fractional derivatives and impulsive effect is found by analysis of the limit case (as impulse tends to zero), and it shows that the deviation caused by impulse for the fractional-order nonlinear systems is undetermined. Next, an example is given to illustrate the result.

Keywords

  • fractional differential equations
  • Caputo-Hadamard fractional derivative
  • impulse
  • general solution

MSC

  • 34A08
  • 34A37

1 Introduction

Fractional calculus was utilized as a powerful tool to reveal the hidden aspects of the dynamics of complex or hypercomplex systems [13]. And the subject of fractional differential equations is gaining much attention [411].

The Hadamard approach to fractional integral was based on the generalization of the nth integral [12],
$$ \bigl( \mathcal {J}_{a^{ +}}^{n}f \bigr) (x) = \int _{a}^{x} \frac{ds_{1}}{s_{1}}\int_{a}^{s_{1}} \frac{ds_{2}}{s_{2}} \cdots \int_{a}^{s_{n - 1}} f(s_{n})\frac{ds_{n}}{s_{n}} = \frac{1}{(n - 1)!}\int_{a}^{x} \biggl( \ln \frac{x}{s} \biggr)^{q - 1}f(s)\frac{ds}{s}, $$
and the works in [1315] were important to develop the fractional calculus within the frame of the Hadamard fractional derivative. Recently, Klimek investigated existence and uniqueness of the solution of sequential fractional differential equations with Hadamard derivative by using the contraction principle in [16]. Ahmad and Ntouyas studied two-dimensional fractional differential systems with Hadamard derivative in [17]. Thiramanus et al. studied existence and uniqueness of solutions for a kind of Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions in [18].

Next, Jarad et al. suggested a Caputo-type modification of the Hadamard fractional derivative in [12] (by the Caputo-Hadamard fractional derivative, we refer to this modified fractional derivative) and presented the fundamental theorem of fractional calculus in the Caputo-Hadamard setting in [12, 19].

Furthermore, impulsive effects exist widely in many processes in which their states can be described by impulsive differential equations. There have appeared a number of papers to research the subject of impulsive differential equations with Caputo fractional derivative [2025], and impulsive fractional partial differential equations were considered in [2631].

Recently, we found the formula of general solution for impulsive systems with Caputo fractional derivatives of order \(q \in (0,1)\) in [32]. Motivated by the above-mentioned works, we will consider the following system with Caputo-Hadamard fractional derivative and impulsive effect:
$$ \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta u |_{t = t_{k}} = u(t_{k}^{ +} ) - u(t_{k}^{ -} ) = \Delta_{k}( u(t_{k}^{ -} ) ) \in \mathbb{C},\quad k = 1,2,\ldots,m, \\ u(a) = u_{a},\quad u_{a} \in \mathbb{C}, \end{array}\displaystyle \right . $$
(1.1)
where \(q \in \mathbb{C}\) and \(\Re (q) \in (0,1)\), \({}_{C - H}D_{a^{ +}}^{q}\) denotes the left-sided Caputo-Hadamard fractional derivative of order q with the low limit a (>0), \(a = t_{0} < t_{1} <\cdots < t_{m} < t_{m + 1} = T\), \(f:(a,T] \times \mathbb{C} \to \mathbb{C}\) is an appropriate continuous function. Here \(u(t_{k}^{ +} ) = \lim_{\varepsilon \to 0^{ +}} u(t_{k} + \varepsilon )\) and \(u(t_{k}^{ -} ) = \lim_{\varepsilon \to 0^{ -}} u(t_{k} + \varepsilon )\) represent the right and left limits of \(u(t)\) at \(t = t_{k}\), respectively.

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 Caputo-Hadamard fractional derivatives. In Section 4, an example is provided to expound the main result in this paper.

2 Preliminaries

In this section, we shall introduce some basic definitions, notations and lemmas which are used throughout this paper.

Definition 2.1

([2], p.110)

Let \(0 \leq a \leq b \leq \infty\) be finite or infinite interval of the half-axis \(\mathbb{R}^{ +}\). The left-sided Hadamard fractional integral of order \(\alpha \in \mathbb{C}\) of function \(\varphi (x)\) is defined by
$$\bigl( {}_{H}\mathcal {J}_{a^{ +}}^{\alpha} \varphi \bigr) (x) = \frac{1}{\Gamma (\alpha )}\int_{a}^{x} \biggl(\ln \frac{x}{s}\biggr)^{\alpha - 1}\varphi (s)\frac{ds}{s} \quad (a < x < b), $$
where \(\Gamma(\cdot)\) is the gamma function.

Definition 2.2

([2], p.110)

The left-sided Hadamard fractional derivative of order \(\alpha \in \mathbb{C}\) with \(\Re (\alpha ) \ge 0\) on \((a, b)\) is defined by
$$\begin{aligned} &\bigl( {}_{H}D_{a^{+}}^{\alpha} \varphi \bigr) (x) = \delta^{n} \bigl( {}_{H}\mathcal {J}_{a^{ +}}^{n - \alpha} \varphi \bigr) (x) = \biggl( x\frac{d}{dx} \biggr)^{n} \frac{1}{\Gamma (n - \alpha )}\int_{a}^{x} \biggl(\ln \frac{x}{s}\biggr)^{n - \alpha - 1}\varphi (s)\frac{ds}{s}\\ &\quad (a < x < b), \end{aligned}$$
where \(n = [\Re (\alpha )] + 1\) and differential operator \(\delta = x\frac{d}{dx}\), \(\delta^{0}y(x) = y(x)\).

Lemma 2.3

([2], pp.114-116)

Let \(\alpha,\beta \in \mathbb{C}\) such that \(\Re (\alpha ) > \Re (\beta ) > 0\). For \(0 < a < b < \infty\), if \(\varphi \in L^{p}(a,b) \) (\(1 \leq p < \infty\)), then \({}_{H}D_{a^{ +}}^{\beta} {}_{H}\mathcal {J}_{a^{ +}}^{\alpha} \varphi = {}_{H}\mathcal {J}_{a^{ +}}^{\alpha - \beta} \varphi\) and \({}_{H}\mathcal {J}_{a^{ +}}^{\alpha} {}_{H}\mathcal {J}_{a^{ +}}^{\beta} \varphi = {}_{H}\mathcal {J}_{a^{ +}}^{\alpha + \beta} \varphi\).

In [12], the left-sided Caputo-Hadamard fractional derivative is suggested and defined by
$${}_{C - H}D_{a^{ +}}^{\alpha} \varphi (x) = {}_{H}D_{a^{ +}}^{\alpha} \Biggl[ \varphi (x) - \sum _{k = 0}^{n - 1} \frac{\delta^{k}\varphi (a)}{k!}\biggl(\ln \frac{t}{a}\biggr)^{k} \Biggr](x), $$
here \(\Re (\alpha ) \ge 0\), \(n = [\Re (\alpha )] + 1\), \(0 < a < b < \infty\), differential operator \(\delta = x\frac{d}{dx}\), \(\delta^{0}y(x) = y(x)\) and
$$\varphi (x) \in AC_{\delta}^{n}[a,b] = \biggl\{ \varphi:[a,b] \to \mathbb{C}:\delta^{(n - 1)}\varphi (x) \in AC[a,b],\delta = x \frac{d}{dx} \biggr\} . $$

For left-sided Caputo-Hadamard fractional derivatives, the following conclusions were given in [12].

Theorem 2.4

([12], p.4)

Let \(\Re (\alpha ) \ge 0\), \(n = [\Re (\alpha )] + 1\) and \(\varphi \in AC_{\delta}^{n}[a,b]\), \(0 < a < b < \infty\). Then \({}_{C - H}D_{a^{ +}}^{\alpha} \varphi (x)\) exist everywhere on \([a, b]\) and
  1. (a)
    if \(\alpha \notin \mathbb{N}_{0}\),
    $${}_{C - H}D_{a^{ +}}^{\alpha} \varphi (x) = \frac{1}{\Gamma (n - \alpha )}\int_{a}^{x} \biggl( \ln \frac{x}{s} \biggr)^{n - \alpha - 1}\delta^{n}\varphi (s) \frac{ds}{s} = {}_{H}\mathcal {J}_{a^{ +}}^{n - \alpha} \delta^{n}\varphi (x), $$
     
  2. (b)
    if \(\alpha = n \in \mathbb{N}_{0}\),
    $${}_{C - H}D_{a^{ +}}^{\alpha} \varphi (x) = \delta^{n}\varphi (x). $$
     

In particular, \({}_{C - H}D_{a^{ +}}^{0}\varphi (x) = \varphi (x)\).

Lemma 2.5

([12], p.5)

Let \(\Re (\alpha ) > 0\), \(n = [\Re (\alpha )] + 1\) and \(\varphi \in C[a,b]\). If \(\Re (\alpha ) \ne 0\) or \(\alpha \in \mathbb{N}\), then
$${}_{C - H}D_{a^{ +}}^{\alpha} \bigl( {}_{H} \mathcal {J}_{a^{ +}}^{\alpha} \varphi \bigr) (x) = \varphi (x). $$

Lemma 2.6

([12], p.6)

Let \(\varphi \in AC_{\delta}^{n}[a,b]\) or \(C_{\delta}^{n}[a,b]\) and \(\alpha \in \mathbb{C}\), then
$${}_{H}\mathcal {J}_{a^{ +}}^{\alpha} \bigl( {}_{C - H}D_{a^{ +}}^{\alpha} \varphi \bigr) (x) = \varphi (x) - \sum_{k = 0}^{n - 1} \frac{\delta^{k}\varphi (a)}{k!} \biggl( \ln \frac{x}{a} \biggr)^{k}. $$

3 Main result

For system (1.1), we have
$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, \ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\\ \quad t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta u |_{t = t_{k}} = u(t_{k}^{ +} ) - u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ) \in \mathbb{C},\quad k = 1,2,\ldots,m, \\ u(a^{ +} ) = u_{a},\quad u_{a} \in \mathbb{C} \end{array}\displaystyle \right . \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad t \in (a,T], \\ u(a^{ +} ) = u_{a},\quad u_{a} \in \mathbb{C}. \end{array}\displaystyle \right . \end{aligned}$$
(3.1)
That is,
$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, \ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \bigl\{ \mbox{the solution of system } (1.1) \bigr\} \\ &\quad= \bigl\{ \mbox{the solution of system } (3.1) \bigr\} . \end{aligned}$$
(3.2)
Thus, the definition of solution for system (1.1) is provided.

Definition 3.1

A function \(z(t):[a,T] \to \mathbb{C}\) is said to be a solution of the fractional Cauchy problem (1.1) if \(z(a) = u_{a}\), the equation condition \({}_{C - H}D_{a^{ +}}^{q}z(t) = f(t,z(t))\) for each \(t \in (a, T]\) is verified, the impulsive conditions \(\Delta z|_{t = t_{k}} = \Delta_{k} ( z(t_{k}^{ -} ) )\) (here \(k = 1, 2, \ldots, m\)) are satisfied, the restriction of \(z( \cdot )\) to the interval \((t_{k},t_{k + 1}]\) (here \(k = 0,1, 2, \ldots, m\)) is continuous, and condition (3.2) holds.

A piecewise function is defined by
$$\begin{aligned} &\tilde{u}(t) = u\bigl(t_{k}^{ +} \bigr) + \frac{1}{\Gamma (q)} \int_{t_{k}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}, \\ &\quad\mbox{for } t \in (t_{k},t_{k + 1}]\ (\mbox{where } k = 0,1,2,\ldots,m). \end{aligned}$$
By Theorem 2.4, we have
$$\begin{aligned} &\bigl[ {}_{C - H}D_{a^{ +}}^{q}\tilde{u}(t) \bigr]_{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (1 - q)}\int_{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{1 - q - 1} \biggl[ s\frac{d}{ds} \biggl( u\bigl(t_{k}^{ +} \bigr) \\ &\qquad{}+ \frac{1}{\Gamma (q)}\int _{t_{k}}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \biggr) \biggr] \frac{ds}{s} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{t_{k}}^{t} \biggl( \ln \frac{t}{s} \biggr)^{1 - q - 1} \biggl[ s\frac{d}{ds} \biggl( \int_{t_{k}}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \biggr) \biggr] \frac{ds}{s} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{t_{k}}^{t} \biggl( \ln \frac{t}{s} \biggr)^{1 - q - 1} \biggl( s\frac{ - 1}{q} \frac{d}{ds}\int_{t_{k}}^{s} f\bigl(\eta,u(\eta )\bigr)\,d\biggl(\ln \frac{s}{\eta} \biggr)^{q} \biggr) \frac{ds}{s} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{t_{k}}^{t} \biggl( \ln \frac{t}{s} \biggr)^{1 - q - 1} \biggl( s\frac{ - 1}{q} \frac{d}{ds} \biggl[ \biggl(\ln \frac{s}{\eta} \biggr)^{q}f\bigl(\eta,u(\eta )\bigr) \Big| _{t_{k}}^{s}\\ &\qquad{}- \int_{t_{k}}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q}f'\bigl(\eta,u(\eta )\bigr)\,d\eta \biggr] \biggr)\frac{ds}{s} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= f\bigl(t,u(t)\bigr) |_{t \in (t_{k},t_{k + 1}]}. \end{aligned}$$
It shows that the piecewise function \(\tilde{u}(t)\) satisfies the condition of fractional derivative in system (1.1). Thus, we assume that the piecewise function \(\tilde{u}(t)\) is an approximate solution of (1.1).

Theorem 3.2

Let \(0 < \Re (q) < 1\) and ħ be a constant. A function \(u(t):[a,T] \to \mathbb{C}\) is a general solution of system (1.1) if and only if \(u(t)\) satisfies the fraction integral equation
$$ u(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{a} + \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \quad \textit{for } t \in (a,t_{1}],\\ u_{a} + \sum_{i = 1}^{k} \Delta_{i} ( u(t_{i}^{ -} ) ) + \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \\ \quad{}+ \sum_{i = 1}^{k} \{ \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} [ \int_{a}^{t_{i}} (\ln \frac{t_{i}}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \\ \quad{}+ \int_{t_{i}}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} - \int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} ] \} \\ \quad\textit{for } t \in (t_{k},t_{k + 1}],k = 1,2,\ldots,m, \end{array}\displaystyle \right . $$
(3.3)
provided that the integral in (3.3) exists.

Proof

‘Necessity’, it will be verified that Eq. (3.3) satisfies the conditions of system (1.1).

Taking the Caputo-Hadamard fractional derivative to the both sides of Eq. (3.3), for \(t \in (a,t_{1}]\), we have
$$\begin{aligned} &{}_{C - H}D_{a^{ +}}^{q}u(t) |_{t \in (a,t_{1}]} \\ &\quad= {}_{C - H}D_{a^{ +}}^{q} \biggl( u_{a} + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr) \\ &\quad= {}_{C - H}D_{a^{ +}}^{q} \biggl( \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr) \\ &\quad= \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ s\frac{d}{ds} \biggl( \int _{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \biggr) \biggr] \frac{ds}{s} \\ &\quad=\frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ s\frac{ - 1}{q} \frac{d}{ds} \biggl( \int_{a}^{s} f\bigl( \eta,u(\eta )\bigr)\,d\biggl(\ln \frac{s}{\eta} \biggr)^{q} \biggr) \biggr]\frac{ds}{s} \\ &\quad= \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ s\frac{ - 1}{q} \frac{d}{ds} \biggl( \biggl(\ln \frac{s}{\eta} \biggr)^{q}f\bigl(\eta,u(\eta )\bigr) \Big| _{a}^{s}\\ &\qquad{}- \int_{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q}f'\bigl(\eta,u(\eta )\bigr)\,d\eta \biggr) \biggr]\frac{ds}{s} \\ &\quad= \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ s\frac{ - 1}{q} \frac{d}{ds} \biggl( - \biggl(\ln \frac{s}{a}\biggr)^{q}f \bigl(a,u(a)\bigr)\\ &\qquad{} - \int_{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q}f'\bigl(\eta,u(\eta )\bigr)\,d\eta \biggr) \biggr]\frac{ds}{s} \\ &\quad=\frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ \biggl(\ln \frac{s}{a} \biggr)^{q - 1}f\bigl(a,u(a)\bigr)\\ &\qquad{} + \int_{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f'\bigl(\eta,u( \eta )\bigr)\,d\eta \biggr]\frac{ds}{s} \\ &\quad= f\bigl(a,u(a)\bigr) + \frac{1}{\Gamma (1 - q)\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ \int _{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f'\bigl(\eta,u(\eta )\bigr)\,d\eta \biggr] \frac{ds}{s} \\ &\quad= f\bigl(t,u(t)\bigr). \end{aligned}$$
For \(t \in (t_{k},t_{k + 1}]\) (where \(k = 1, 2,\ldots, m\)), we get
$$\begin{aligned} &{}_{C - H}D_{a^{ +}}^{q} \Biggl( \sum _{i = 1}^{k} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int _{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \int _{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}- \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \Biggr) \bigg|_{t \in (t_{k},t_{k + 1}]} \\ &\quad= \sum_{i = 1}^{k} \biggl[ \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)}{}_{C - H}D_{a^{ +}}^{q} \biggl( \int _{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \int _{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}- \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr) \biggr]\Big|_{t \in (t_{k},t_{k + 1}]} \\ &\quad= \sum_{i = 1}^{k} \biggl[ \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (1 - q)\Gamma (q)}\int_{t_{0}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1} \biggl[ s\frac{d}{ds} \biggl( \int _{t_{i}}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \\ &\qquad{}- \int _{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \biggr) \biggr] \frac{ds}{s} \biggr] \Big|_{t \in (t_{k},t_{k + 1}]} \\ &\quad= \sum_{i = 1}^{k} \biggl[ \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (1 - q)\Gamma (q)} \biggl( \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1}s\frac{d}{ds} \biggl( \int _{t_{i}}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \biggr) \frac{ds}{s} \\ &\qquad{}- \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - q - 1}s\frac{d}{ds} \biggl( \int _{a}^{s} \biggl(\ln \frac{s}{\eta} \biggr)^{q - 1}f\bigl(\eta,u(\eta )\bigr)\frac{d\eta}{\eta} \biggr) \frac{ds}{s} \biggr) \biggr] \Big|_{t \in (t_{k},t_{k + 1}]} \\ &\quad= \sum_{i = 1}^{k} \bigl[ \hbar \Delta_{i} \bigl( u(t_{i}^{ -} ) \bigr) \bigl( f\bigl(t,u(t)\bigr) |_{t \ge t_{i}} - f \bigl(t,u(t)\bigr) \vert _{t \ge a} \bigr) \bigr]|_{t \in (t_{k},t_{k + 1}]} = 0. \end{aligned}$$
Thus,
$$\begin{aligned} &{}_{C - H}D_{a^{ +}}^{q}u(t) |_{t \in (t_{k},t_{k + 1}]} \\ &\quad= {}_{C - H}D_{a^{ +}}^{q}\Biggl\{ u_{a} + \sum _{i = 1}^{k} \Delta_{i} \bigl( u \bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}+ \sum_{i = 1}^{k} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}+ \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \Biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ {}_{C - H}D_{a^{ +}}^{q} \biggl( \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr) \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \bigl\{ f\bigl(t,u(t)\bigr) |_{t \ge a} \bigr\} _{t \in (t_{k},t_{k + 1}]} = f\bigl(t,u(t)\bigr)_{t \in (t_{k},t_{k + 1}]}. \end{aligned}$$
So, Eq. (3.3) satisfies the condition of fractional derivative in system (1.1).
Next, by (3.3), for each \(t_{k}\) (here \(k = 1, 2,\ldots, m\)), we have
$$\begin{aligned} u\bigl(t_{k}^{ +} \bigr) - u\bigl(t_{k}^{ -} \bigr) ={}& \lim_{t \to t_{k}^{ +}} u(t) - u(t_{k}) \\ ={}& u_{a} + \sum_{i = 1}^{k} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t_{k}} \biggl(\ln \frac{t_{k}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \sum_{i = 1}^{k} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \int_{t_{i}}^{t_{k}} \biggl(\ln \frac{t_{k}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{a}^{t_{k}} \biggl(\ln \frac{t_{k}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \\ &{}- u_{a} - \sum_{i = 1}^{k - 1} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t_{k}} \biggl(\ln \frac{t_{k}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \sum_{i = 1}^{k - 1} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \int_{t_{i}}^{t_{k}} \biggl(\ln \frac{t_{k}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{a}^{t_{k}} \biggl(\ln \frac{t_{k}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \\ ={}& \Delta_{k} \bigl( u\bigl(t_{k}^{ -} \bigr) \bigr). \end{aligned}$$
Therefore, Eq. (3.3) satisfies the impulsive condition of (1.1).

Finally, it can be easily verified that Eq. (3.3) satisfies condition (3.2).

‘Sufficiency’, we will prove that the solutions of system (1.1) satisfy Eq. (3.3) by using the inductive method. For \(t \in (a,t_{1}]\), we obtain the following equation by Lemma 2.6:
$$ u(t) = u_{a} + \frac{1}{\Gamma (q)}\int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (a,t_{1}]. $$
(3.4)
Using (3.4), we have
$$\begin{aligned} u\bigl(t_{1}^{ +} \bigr) &= u\bigl(t_{1}^{ -} \bigr) + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) \\ &= u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}. \end{aligned}$$
Then the approximate solution is given by
$$\begin{aligned} \tilde{u}(t) ={}& u\bigl(t_{1}^{ +} \bigr) + \frac{1}{\Gamma (q)}\int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}+ \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(3.5)
Let \(e_{1}(t) = u(t) - \tilde{u}(t)\), for \(t \in (t_{1},t_{2}]\). Due to
$$\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}u(t) = u_{a} + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} $$
for \(t \in (t_{1},t_{2}]\), we get
$$\begin{aligned} \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{1}(t) ={}& \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ ={}& \frac{1}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} - \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]. \end{aligned}$$
Then we assume
$$\begin{aligned} e_{1}(t) ={}& \sigma \bigl( \Delta_{1}\bigl(u \bigl(t_{1}^{ -} \bigr)\bigr) \bigr)\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{1}(t) \\ ={}& \frac{\sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} - \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr], \end{aligned}$$
where the function \(\sigma ( \cdot )\) is an undetermined function with \(\sigma (0) = 1\). Therefore,
$$\begin{aligned} u(t) ={}& \tilde{u}(t) + e_{1}(t) = u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}-\int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(3.6)
By (3.6), we get
$$\begin{aligned} u\bigl(t_{2}^{ +} \bigr) ={}& u\bigl(t_{2}^{ -} \bigr) + \Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) \\ ={}& u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]. \end{aligned}$$
Therefore, the approximate solution is provided by
$$\begin{aligned} \tilde{u}(t) ={}& u\bigl(t_{2}^{ +} \bigr) + \frac{1}{\Gamma (q)}\int_{t_{2}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)} \biggl[ \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}+ \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (t_{2},t_{3}]. \end{aligned}$$
(3.7)
Let \(e_{2}(t) = u(t) - \tilde{u}(t)\), for \(t \in (t_{2},t_{3}]\). For the exact solution \(u(t)\) of system (1.1), we have
$$\begin{aligned}& \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) = u_{a} + \frac{1}{\Gamma (q)}\int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{2},t_{3}], \\& \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}u(t) = u_{a} + \Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\& \hphantom{ \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}u(t) =}{}+ \frac{1 - \sigma ( \Delta_{2}(u(t_{2}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\& \hphantom{ \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}u(t) =}{}+ \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\& \hphantom{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}u(t) =} {}\mbox{for } t \in (t_{2},t_{3}], \\& \begin{aligned}[b] \lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) ={}& u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}+ \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}\mbox{for } t \in (t_{2},t_{3}]. \end{aligned} \end{aligned}$$
Thus,
$$\begin{aligned}& \begin{aligned}[b] \mathop{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, }}_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) &= \mathop{\lim_{ \Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, }}_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ &= \frac{1}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} - \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}-\int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr], \end{aligned} \end{aligned}$$
(3.8)
$$\begin{aligned}& \lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) = \lim _{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} = \frac{\sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) =}{}- \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) =}{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) =}{}- \int_{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr], \end{aligned}$$
(3.9)
$$\begin{aligned}& \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{2}(t) = \lim _{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\& \hphantom{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{2}(t)}= \frac{\sigma ( \Delta_{2}(u(t_{2}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\& \hphantom{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{2}(t) =}{}- \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]. \end{aligned}$$
(3.10)
By (3.8)-(3.10), we obtain
$$\begin{aligned} e_{2}(t) ={}& \frac{\sigma ( \Delta_{1}(u(t_{1}^{ -} )) ) + \sigma ( \Delta_{2}(u(t_{2}^{ -} )) ) - 1}{\Gamma (q)} \biggl[ \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} - \int_{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]. \end{aligned}$$
(3.11)
Therefore,
$$\begin{aligned} u(t) ={}& \tilde{u}(t) + e_{2}(t) \\ ={}& u_{a} + \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr) + \Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}+ \frac{1 - \sigma ( \Delta_{1}(u(t_{1}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}+ \frac{1 - \sigma ( \Delta_{2}(u(t_{2}^{ -} )) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]. \end{aligned}$$
(3.12)
Letting \(t_{2} \to t_{1}\), we have
$$\begin{aligned}& \lim_{t_{2} \to t_{1}} \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad t \in (a,t_{3}] \mbox{ and } t \ne t_{1} \mbox{ and } t \ne t_{2}, \\ \Delta u \vert _{t = t_{k}} = u(t_{k}^{ +} ) - u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ) \in \mathbb{C},\quad k = 1,2, \\ u(a^{ +} ) = u_{a},\quad u_{a} \in \mathbb{C} \end{array}\displaystyle \right . \end{aligned}$$
(3.13)
$$\begin{aligned}& \quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad t \in (a,t_{3}] \mbox{ and } t \ne t_{1}, \\ \Delta u \vert _{t = t_{1}} = u(t_{1}^{ +} ) - u(t_{1}^{ -} ) + u(t_{2}^{ +} ) - u(t_{2}^{ -} ) = \Delta_{1} ( u(t_{1}^{ -} ) ) + \Delta_{2} ( u(t_{2}^{ -} ) ), \\ u(a^{ +} ) = u_{a},\quad u_{a} \in \mathbb{C}. \end{array}\displaystyle \right . \end{aligned}$$
(3.14)
Using (3.6) and (3.12) to systems (3.14) and (3.13), respectively, we get
$$\begin{aligned} &1 - \sigma \bigl( \Delta_{1}\bigl(u\bigl(t_{1}^{ -} \bigr)\bigr) + \Delta_{2}\bigl(u\bigl(t_{2}^{ -} \bigr)\bigr) \bigr) = 1 - \sigma \bigl( \Delta_{1}\bigl(u \bigl(t_{1}^{ -} \bigr)\bigr) \bigr) + 1 - \sigma \bigl( \Delta_{2}\bigl(u\bigl(t_{2}^{ -} \bigr)\bigr) \bigr), \\ &\quad\forall \Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr),\Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) \in \mathbb{C}. \end{aligned}$$
Letting \(\rho (z) = 1 - \sigma (z)\) (\(\forall z \in \mathbb{C}\)), we have \(\rho (z + w) = \rho (z) + \rho (w)\) (\(\forall z,w \in \mathbb{C}\)). Therefore \(\rho (z) = \hbar z\), where ħ is a constant. So, we obtain the following two equations:
$$\begin{aligned} u(t) ={}& u_{a} + \Delta_{1} \bigl( u \bigl(t_{1}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \frac{\hbar \Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (t_{1},t_{2}] \end{aligned}$$
(3.15)
and
$$\begin{aligned} u(t) ={}& u_{a} + \Delta_{1} \bigl( u \bigl(t_{1}^{ -} \bigr) \bigr) + \Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \frac{\hbar \Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{1}} \biggl(\ln \frac{t_{1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}+ \frac{\hbar \Delta_{2} ( u(t_{2}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} + \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{} - \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (t_{2},t_{3}]. \end{aligned}$$
(3.16)
For \(t \in (t_{n},t_{n + 1}]\), suppose
$$\begin{aligned} u(t) ={}& u_{a} + \sum_{i = 1}^{n} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \sum_{i = 1}^{n} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{} - \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \quad\mbox{for } t \in (t_{n},t_{n + 1}]. \end{aligned}$$
(3.17)
By (3.17), we have
$$\begin{aligned} u\bigl(t_{n + 1}^{ +} \bigr) ={}& u\bigl(t_{n + 1}^{ -} \bigr) + \Delta_{n + 1} \bigl( u\bigl(t_{n + 1}^{ -} \bigr) \bigr) \\ ={}& u_{a} + \sum_{i = 1}^{n + 1} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \sum_{i = 1}^{n} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \int_{t_{i}}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{a}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr). \end{aligned}$$
So, the approximate solution is provided by
$$\begin{aligned} \tilde{u}(t) ={}& u\bigl(t_{n + 1}^{ +} \bigr) + \frac{1}{\Gamma (q)}\int_{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& u_{a} + \sum_{i = 1}^{n + 1} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)} \biggl[ \int_{a}^{t_{n + 1}} \biggl( \ln \frac{t_{n + 1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \int_{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] + \sum_{i = 1}^{n} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \int_{t_{i}}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{a}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \\ &{}\quad\mbox{for } t \in (t_{n + 1},t_{n + 2}]. \end{aligned}$$
(3.18)
Let \(e_{n + 1}(t) = u(t) - \tilde{u}(t)\), for \(t \in (t_{n + 1},t_{n + 2}]\). For the exact solution \(u(t)\) of system (1.1), we have
$$\begin{aligned}& \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, \ldots,\Delta_{n + 1} ( u(t_{n + 1}^{ -} ) ) \to 0}u(t) = u_{a} + \frac{1}{\Gamma (q)}\int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{n + 1},t_{n + 2}], \\& \begin{aligned}[b] \lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}u(t) ={}& u_{a} + \mathop{\sum_{1 \le i \le n + 1 }}_{ \mathrm{and}\ i \ne j} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \mathop{\sum_{1 \le i \le n + 1 }}_{ \mathrm{and}\ i \ne j} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\\ &{} + \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr), \\ &{}\mbox{here } 1 \le j \le n + 1. \end{aligned} \end{aligned}$$
Then
$$\begin{aligned}& \begin{aligned}[b] &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, \ldots, \Delta_{n + 1} ( u(t_{n + 1}^{ -} ) ) \to 0}e_{n + 1}(t)\\ &\quad=\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots, \Delta_{n + 1} ( u(t_{n + 1}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ &\quad= \frac{1}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} - \int_{a}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}- \int_{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr], \end{aligned} \end{aligned}$$
(3.19)
$$\begin{aligned}& \lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}e_{n + 1}(t) = \lim _{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\& \hphantom{\lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}e_{n + 1}(t)}= \frac{1 - \hbar \mathop{\sum_{1 \le i \le n + 1 }}\limits_{ \hphantom{ss}\mathrm{and}\ i \ne j} \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\& \hphantom{\lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}e_{n + 1}(t) =}{}- \int_{a}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\& \hphantom{\lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}e_{n + 1}(t) =}{}+ \mathop{\sum_{1 \le i \le n }}_{\mathrm{and}\ i \ne j} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\& \hphantom{\lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}e_{n + 1}(t) =}{}- \int_{t_{i}}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr), \\& \hphantom{\lim_{\Delta_{j} ( u(t_{j}^{ -} ) ) \to 0}e_{n + 1}(t) =}{}\mbox{here } 1 \le j \le n + 1. \end{aligned}$$
(3.20)
So, by (3.19) and (3.20), we obtain
$$\begin{aligned} e_{n + 1}(t) ={}& \frac{1 - \hbar \sum_{1 \le i \le n + 1} \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr) \frac{ds}{s} \\ &{}- \int_{a}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}+ \sum_{1 \le i \le n} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{t_{i}}^{t_{n + 1}} \biggl(\ln \frac{t_{n + 1}}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \int _{t_{n + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr). \end{aligned}$$
Thus,
$$\begin{aligned} u(t) ={}& \tilde{u}(t) + e_{n + 1}(t) \\ ={}& u_{a} + \sum_{i = 1}^{n + 1} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) + \frac{1}{\Gamma (q)}\int_{a}^{t} \biggl(\ln \frac{t}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \sum_{i = 1}^{n + 1} \biggl( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} \biggl[ \int_{a}^{t_{i}} \biggl(\ln \frac{t_{i}}{s}\biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \int_{t_{i}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \biggr) \quad\mbox{for } t \in (t_{n + 1},t_{n + 2}]. \end{aligned}$$
(3.21)
So, the solution of system (1.1) satisfies Eq. (3.3).

By the proof of Sufficiency and Necessity, it is shown that system (1.1) is equivalent to the integral equation (3.3). The proof is now completed. □

Remark 3.1

Due to uncertainty of the constant ħ, the deviation caused by impulse for the fractional order nonlinear systems is undetermined.

Next, we provide an analysis of the connection between impulsive Caputo-Hadamard fractional differential equations and impulsive first-order differential equations. Letting \(q \to 1^{ -}\) for system (1.1), we have
$$\begin{aligned} &\lim_{q \to 1^{ -}} \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta u \vert _{t = t_{k}} = u(t_{k}^{ +} ) - u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ) \in \mathbb{C},\quad k = 1,2,\ldots,m, \\ u(a) = u_{a},\quad u_{a} \in \mathbb{C} \end{array}\displaystyle \right . \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} t\frac{dx(t)}{dt} = f(t,x(t)),\quad t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta u |_{t = t_{k}} = u(t_{k}^{ +} ) - u(t_{k}^{ -} ) = \Delta_{k}( u(t_{k}^{ -} ) ) \in \mathbb{C},\quad k = 1,2,\ldots,m, \\ u(a) = u_{a},\quad u_{a} \in \mathbb{C}. \end{array}\displaystyle \right . \end{aligned}$$
(3.22)
On the other hand, by Theorem 3.2, the general solution of system (1.1) satisfies
$$\begin{aligned} \lim_{q \to 1^{ -}} u(t) &= \left \{ \textstyle\begin{array}{@{}l} \lim_{q \to 1^{ -}} [ u_{a} + \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} ] \quad\mbox{for } t \in (a,t_{1}], \\ \lim_{q \to 1^{ -}} \{ u_{a} + \sum_{i = 1}^{k} \Delta_{i} ( u(t_{i}^{ -} ) ) + \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \\ \quad{} + \sum_{i = 1}^{k} ( \frac{\hbar \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q)} [ \int_{a}^{t_{i}} (\ln \frac{t_{i}}{s})^{q - 1}f(s,u(s))\frac{ds}{s} + \int_{t_{i}}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \\ \quad{}- \int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} ] ) \} \quad\mbox{for } t \in (t_{k},t_{k + 1}], k = 1,2,\ldots,m \end{array}\displaystyle \right . \\ &= \left \{ \textstyle\begin{array}{@{}l} u_{a} + \int_{a}^{t} \frac{f(s,u(s))}{s}\,ds\quad \mbox{for } t \in (a,t_{1}], \\ u_{a} + \sum_{i = 1}^{k} \Delta_{i} ( u(t_{i}^{ -} ) ) + \int_{a}^{t} \frac{f(s,u(s))}{s}\,ds \\ \quad\mbox{for } t \in (t_{k},t_{k + 1}], k = 1,2,\ldots,m. \end{array}\displaystyle \right . \end{aligned}$$
(3.23)
Moreover, it is true that Eq. (3.23) is the solution of impulsive system (3.22) and indirectly verifies the formula of general solution for nonlinear systems with Caputo-Hadamard fractional derivatives and impulsive effect.

4 Example

In this section, an example is given to illustrate the usefulness of the result in this paper.

Example 1

Let us consider the general solution of the impulsive fractional system
$$ \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{1^{ +}}^{\frac{1}{2}}u(t) = \ln t,\quad t \in (1,3] \mbox{ and } t \ne 2, \\ \Delta u \vert = u(2^{ +} ) - u(2^{ -} ) = \Delta, \\ u(1^{ +} ) = u_{1}. \end{array}\displaystyle \right . $$
(4.1)
By Theorem 3.2, after some elementary computation, the general solution is obtained as follows:
$$ u(t) = \left \{ \textstyle\begin{array}{@{}l} u_{1} + \frac{1}{\Gamma (\frac{1}{2})}\int_{1}^{t} (\ln \frac{t}{s})^{\frac{1}{2} - 1}\ln s\frac{ds}{s}\quad \mbox{for } t \in (1,2], \\ u_{1} + \Delta + \frac{1}{\Gamma (\frac{1}{2})}\int_{1}^{t} (\ln \frac{t}{s})^{\frac{1}{2} - 1}\ln s\frac{ds}{s} + \frac{\hbar \Delta}{\Gamma (\frac{1}{2})} [ \int_{1}^{2} (\ln \frac{2}{s})^{\frac{1}{2} - 1}\ln s\frac{ds}{s} \\ \quad{}+ \int_{2}^{t} (\ln \frac{t}{s})^{\frac{1}{2} - 1}\ln s\frac{ds}{s} - \int_{1}^{t} (\ln \frac{t}{s})^{\frac{1}{2} - 1}\ln s\frac{ds}{s} ] \quad \mbox{for } t \in (2,3]. \end{array}\displaystyle \right . $$
(4.2)
That is,
$$ u(t) = \left \{ \textstyle\begin{array}{@{}l} u_{1} + \frac{4}{3}\frac{1}{\Gamma (\frac{1}{2})}(\ln t)^{\frac{3}{2}} \quad\mbox{for } t \in (1,2], \\ u_{1} + \Delta + \frac{4}{3}\frac{1}{\Gamma (\frac{1}{2})}(\ln t)^{\frac{3}{2}} \vert _{t \ge 1} + \frac{\hbar \Delta}{\Gamma (\frac{1}{2})} [ \frac{4}{3}(\ln 2)^{\frac{3}{2}} + ( \frac{4}{3}(\ln \frac{t}{2})^{\frac{3}{2}}\\ \quad{} + 2(\ln \frac{t}{2})^{\frac{1}{2}}\ln 2 ) \vert _{t \ge 2} - \frac{4}{3}(\ln t)^{\frac{3}{2}} \vert _{t \ge 1} ] \quad \mbox{for } t \in (2,3]. \end{array}\displaystyle \right . $$
(4.2a)

Next, it is verified that Eq. (4.2a) satisfies the condition of system (4.1).

Taking the Caputo-Hadamard fractional derivative to the both sides of Eq. (4.2a), after some elementary computation, we have

(i) for \(t \in (1,2]\),
$$\begin{aligned} {}_{C - H}D_{1^{ +}}^{\frac{1}{2}}u(t) |_{t \in (1,2]} ={}& \biggl\{ \frac{1}{\Gamma (1 - \frac{1}{2})}\int_{1}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - \frac{1}{2} - 1} \biggl( s\frac{d}{ds} \biggr) \biggl[ u_{1} + \frac{4}{3}\frac{1}{\Gamma (\frac{1}{2})} ( \ln s )^{\frac{3}{2}} \biggr] \frac{ds}{s} \biggr\} _{t \in (1,2]} \\ ={}& \biggl\{ \frac{1}{\Gamma (1 - \frac{1}{2})\Gamma (\frac{1}{2})}\int_{1}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - \frac{1}{2} - 1} \bigl[ 2 ( \ln s )^{\frac{1}{2}} \bigr] \frac{ds}{s} \biggr\} _{t \in (1,2]} \\ ={}& \ln t |_{t \in (1,2]}, \end{aligned}$$
(ii) for \(t \in (2,3]\),
$$\begin{aligned} &{}_{C - H}D_{1^{ +}}^{\frac{1}{2}}u(t) \vert _{t \in (2,3]} \\ &\quad= \frac{1}{\Gamma (1 - \frac{1}{2})}\int_{1}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - \frac{1}{2} - 1} \biggl( s\frac{d}{ds} \biggr) \biggl\{ u_{1} + \Delta + \frac{4}{3} \frac{1}{\Gamma (\frac{1}{2})}(\ln s)^{\frac{3}{2}} \Big| _{t \ge 1} \\ &\qquad{} + \frac{\hbar \Delta}{\Gamma (\frac{1}{2})} \biggl[ \frac{4}{3}(\ln 2)^{\frac{3}{2}} + \biggl( \frac{4}{3}\biggl(\ln \frac{s}{2}\biggr)^{\frac{3}{2}} + 2\biggl(\ln \frac{s}{2} \biggr)^{\frac{1}{2}}\ln 2 \biggr) \Big| _{t \ge 2} - \frac{4}{3}(\ln s)^{\frac{3}{2}} \Big| _{t \ge 1} \biggr] \biggr\} \frac{ds}{s} \Big| _{t \in (2,3]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (1 - \frac{1}{2})\Gamma (\frac{1}{2})}\int_{1}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - \frac{1}{2} - 1} \bigl[ 2(\ln s)^{\frac{1}{2}} - 2\hbar \Delta (\ln s)^{\frac{1}{2}} \bigr]\frac{ds}{s} \\ &\qquad{} + \frac{\hbar \Delta}{\Gamma (1 - \frac{1}{2})\Gamma (\frac{1}{2})}\int_{2}^{t} \biggl(\ln \frac{t}{s}\biggr)^{1 - \frac{1}{2} - 1} \biggl( 2\biggl(\ln \frac{s}{2}\biggr)^{\frac{1}{2}} + \ln 2\biggl(\ln \frac{s}{2} \biggr)^{ - \frac{1}{2}} \biggr)\frac{ds}{s} \biggr\} _{t \in (2,3]} \\ &\quad= \biggl\{ \ln t \vert _{t \ge 1} - \hbar \Delta \ln t \vert _{t \ge 1} + \hbar \Delta \biggl( \biggl(\ln \frac{t}{2}\biggr) + \ln 2 \biggr) \Big| _{t \ge 2} \biggr\} _{t \in (2,3]} \\ &\quad= \ln t \vert _{t \in (2,3]}. \end{aligned}$$
So, Eq. (4.2a) satisfies the Hadamard fractional derivative condition of system (4.1).
By Eq. (4.2a), we have
$$\begin{aligned} u\bigl(2^{ +} \bigr) - u\bigl(2^{ -} \bigr) ={}& \lim _{t \to 2^{ +}} \biggl\{ u_{1} + \Delta + \frac{4}{3}\frac{1}{\Gamma (\frac{1}{2})}(\ln t)^{\frac{3}{2}} \Big| _{t \ge 1} + \frac{\hbar \Delta}{\Gamma (\frac{1}{2})} \biggl[ \frac{4}{3}(\ln 2)^{\frac{3}{2}} + \biggl( \frac{4}{3}\biggl(\ln \frac{t}{2}\biggr)^{\frac{3}{2}} \\ &{}+ 2\biggl(\ln \frac{t}{2} \biggr)^{\frac{1}{2}}\ln 2 \biggr) \Big| _{t \ge 2} - \frac{4}{3}(\ln t)^{\frac{3}{2}} \Big| _{t \ge 1} \biggr] \biggr\} - u_{1} - \frac{4}{3}\frac{1}{\Gamma (\frac{1}{2})}(\ln 2)^{\frac{3}{2}} \\ ={}& \Delta. \end{aligned}$$

That is, Eq. (4.2a) satisfies the impulsive condition in system (4.1).

Finally, it is obvious that Eq. (4.2a) satisfies the following limit case:
$$\begin{aligned} &\lim_{\Delta \to 0} \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{1^{ +}}^{\frac{1}{2}}u(t) = \ln t, \quad t \in (1,3] \mbox{ and } t \ne 2, \\ \Delta u \vert _{t = 2} = u(2^{ +} ) - u(2^{ -} ) = \Delta \in \mathbb{R}, \\ u(1^{ +} ) = u_{1} \in \mathbb{R} \end{array}\displaystyle \right . \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{C - H}D_{1^{ +}}^{\frac{1}{2}}u(t) = \ln t,\quad t \in (1,3], \\ u(1^{ +} ) = u_{1} \in \mathbb{R}. \end{array}\displaystyle \right . \end{aligned}$$
(4.3)

So, Eq. (4.2a) is the general solution of system (4.1).

5 Conclusion

For the first-order impulsive differential equations, the solution is determined by initial value. However, Caputo-Hadamard fractional differential equations with impulsive effect have a general solution and need more conditions to decide the constant ħ than the first-order impulsive ones. Moreover, due to the non-uniqueness of solution for the impulsive fractional-order nonlinear system, it means that there appear new problems on impulsive control of the fractional-order nonlinear systems.

Declarations

Acknowledgements

The author is deeply grateful to the anonymous referees for their kind comments, correcting errors and improving written language, which has been very useful for improving the quality of this paper. The work described in this paper is financially supported by State Key Development Program for Basic Research of Health and Family Planning Commission of Jiangxi Province China (Grant No. 20143246) and Jiujiang University Research Foundation (Grant No. 8400183).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi, 332005, China

References

  1. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  2. Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  3. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012) MATHGoogle Scholar
  4. Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075-1081 (2007) MathSciNetView ArticleMATHGoogle Scholar
  5. Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340-1350 (2008) MathSciNetView ArticleMATHGoogle Scholar
  6. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973-1033 (2010) MathSciNetView ArticleMATHGoogle Scholar
  7. Odibat, ZM: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171-1183 (2010) MathSciNetView ArticleMATHGoogle Scholar
  8. Ahmad, B, Nieto, JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35(2), 295-304 (2010) MathSciNetMATHGoogle Scholar
  9. Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. TMA 72, 916-924 (2010) View ArticleMATHGoogle Scholar
  10. Mophou, GM, N’Guérékata, GM: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61-69 (2010) MathSciNetView ArticleGoogle Scholar
  11. Deng, W: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. TMA 72, 1768-1777 (2010) View ArticleMATHGoogle Scholar
  12. Jarad, F, Abdeljawad, T, Baleanu, D: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012) View ArticleGoogle Scholar
  13. Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191-1204 (2001) MathSciNetMATHGoogle Scholar
  14. Butzer, PL, Kilbas, AA, Trujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387-400 (2002) MathSciNetView ArticleMATHGoogle Scholar
  15. Butzer, PL, Kilbas, AA, Trujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1-15 (2002) MathSciNetView ArticleMATHGoogle Scholar
  16. Klimek, M: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16, 4689-4697 (2011) MathSciNetView ArticleMATHGoogle Scholar
  17. Ahmad, B, Ntouyas, SK: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17, 348-360 (2014) MathSciNetView ArticleMATHGoogle Scholar
  18. Thiramanus, P, Ntouyas, SK, Tariboon, J: Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions. Abstr. Appl. Anal. (2014). doi:10.1155/2014/902054 Google Scholar
  19. Gambo, YY, Jarad, F, Baleanu, D, Abdeljawad, T: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, 10 (2014) MathSciNetView ArticleGoogle Scholar
  20. Ahmad, B, Sivasundaram, S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3, 251-258 (2009) MathSciNetView ArticleMATHGoogle Scholar
  21. Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4, 134-141 (2010) MathSciNetView ArticleMATHGoogle Scholar
  22. Tian, Y, Bai, Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59, 2601-2609 (2010) MathSciNetView ArticleMATHGoogle Scholar
  23. Cao, J, Chen, H: Some results on impulsive boundary value problem for fractional differential inclusions. Electron. J. Qual. Theory Differ. Equ. 2010, 11 (2010) MathSciNetGoogle Scholar
  24. Wang, X: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383-2391 (2011) MathSciNetView ArticleMATHGoogle Scholar
  25. Stamova, I, Stamov, G: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simul. 19, 702-709 (2014) MathSciNetView ArticleGoogle Scholar
  26. Abbas, A, Benchohra, M: Impulsive hyperbolic functional differential equations of fractional order with state-dependent delay. Fract. Calc. Appl. Anal. 13, 225-242 (2010) MathSciNetMATHGoogle Scholar
  27. Abbas, A, Benchohra, M: Upper and lower solutions method for impulsive hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4, 406-413 (2010) MathSciNetView ArticleMATHGoogle Scholar
  28. Abbas, A, Agarwal, RP, Benchohra, M: Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay. Nonlinear Anal. Hybrid Syst. 4, 818-829 (2010) MathSciNetView ArticleMATHGoogle Scholar
  29. Abbas, A, Benchohra, M, Gorniewicz, L: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn. 72(1), 49-60 (2010) MathSciNetMATHGoogle Scholar
  30. Benchohra, M, Seba, D: Impulsive partial hyperbolic fractional order differential equations in Banach spaces. J. Fract. Calc. Appl. 1(4), 1-12 (2011) MathSciNetGoogle Scholar
  31. Guo, T, Zhang, K: Impulsive fractional partial differential equations. Appl. Math. Comput. 257, 581-590 (2015) MathSciNetView ArticleGoogle Scholar
  32. Zhang, X, Zhang, X, Zhang, M: On the concept of general solution for impulsive differential equations of fractional order \(\mathrm{q} \in (0, 1)\). Appl. Math. Comput. 247, 72-89 (2014) MathSciNetView ArticleMATHGoogle Scholar

Copyright

Advertisement