1 Introduction

Fractional differential equations are an excellent tool in the modeling of many phenomena in various fields of science and engineering [1–3], and the subject of fractional differential equations is gaining much attention (see [4–11] and the references therein).

Recently, Hadamard fractional derivative was studied in [12–15], and Klimek [16] studied the existence and uniqueness of the solution of a sequential fractional differential equation with Hadamard derivative by using the contraction principle and a new equivalent norm and metric. Ahmad and Ntouyas [17] studied two-dimensional fractional differential systems with Hadamard derivative. Next, Jarad et al. [18, 19] presented a Caputo-type modification about Hadamard fractional derivative and developed the fundamental theorem of fractional calculus in the Caputo-Hadamard setting.

Furthermore, impulsive effects exist widely in many processes in which their states can be described by impulsive differential equations, and the subject of impulsive Caputo fractional differential equations is widely studied (see [20–26]); impulsive fractional partial differential equations are also considered (see [27–32]).

Motivated by the above-mentioned works, we consider the following impulsive system with Hadamard fractional derivative:

$$ \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m)\\ \quad\mbox{and } t \ne \bar{t}_{l} \ (l = 1,2,\ldots,n), \\ \Delta ({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = t_{k}} = {}_{H}\mathcal {J}_{a^{ +}}^{2 - q}u(t_{k}^{ +} ) - {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ),\quad k = 1,2,\ldots,m, \\ \Delta ( {}_{H}D_{a^{ +}}^{q - 1}u ) \vert _{t = \bar{t}_{l}} = {}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ +} ) - {}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ -} ) = \bar{\Delta}_{l} ( u(\bar{t}_{l}^{ -} ) ),\quad l = 1,2,\ldots,n, \\ {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2},\qquad{}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1}, \end{array}\displaystyle \right . $$
(1.1)

where \(a > 0\), \({}_{H}D_{a^{ +}}^{q}\) denotes left-sided Hadamard fractional derivative of order q, \(f:J \times \mathbb{R} \to \mathbb{R}\) is an appropriate continuous function, \(a = t_{0} < t_{1} <\cdots < t_{m} < t_{m + 1} = T\) and \(a = \bar{t}_{0} < \bar{t}_{1} <\cdots < \bar{t}_{n} < \bar{t}_{n + 1} = T\), \({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}\) denotes the left-sided Hadamard fractional integral of order \(2- q\), and \({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ +} ) = \lim_{\varepsilon \to 0^{ +}} {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k} + \varepsilon )\) and \({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ -} ) = \lim_{\varepsilon \to 0^{ -}} {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k} + \varepsilon )\) represent the right and left limits of \({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t)\) at \(t = t_{k}\), respectively. The derivatives \({}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ +} )\) and \({}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ -} )\) have a similar meaning for \({}_{H}D_{a^{ +}}^{q - 1}u(t)\). Moreover, \(a,t_{1},t_{2}, \ldots,t_{m},\bar{t}_{1},\bar{t}_{2}, \ldots,\bar{t}_{n},T\) is queued to \(a = t'_{0} < t'_{1} <\cdots < t'_{\Omega} < t'_{\Omega + 1} = T\) so that

$$\operatorname{set} \{ t_{1},t_{2}, \ldots,t_{m}, \bar{t}_{1},\bar{t}_{2}, \ldots,\bar{t}_{n} \} = \operatorname{set} \bigl\{ t'_{1},t'_{2}, \ldots,t'_{\Omega} \bigr\} . $$

For each interval \([a,t'_{k}]\) (here \(k = 1,2,\ldots,\Omega\)), suppose that \([a,t_{k_{0}}] \subseteq [a,t'_{k}] \subset [a,t_{k_{0} + 1}]\) (here \(k_{0} \in \{ 1,2,\ldots,m\}\)) and \([a,\bar{t}_{k_{1}}] \subseteq [a,t'_{k}] \subset [a,\bar{t}_{k_{1} + 1}]\) (here \(k_{1} \in \{ 1,2,\ldots,n\}\)), respectively.

Next, we simplify system (1.1) to obtain the following system:

$$ \left \{ \textstyle\begin{array}{l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = t_{k}} = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ +} ) - {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ),\quad k = 1,2,\ldots,m, \\ \Delta ( {}_{H}D_{a^{ +}}^{q - 1}u ) \vert _{t = t_{k}} = {}_{H}D_{a^{ +}}^{q - 1}u(t_{k}^{ +} ) - {}_{H}D_{a^{ +}}^{q - 1}u(t_{k}^{ -} ) = \bar{\Delta}_{k} ( u(t_{k}^{ -} ) ),\quad k = 1,2,\ldots,m, \\ {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2},\qquad {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1}. \end{array}\displaystyle \right . $$
(1.2)

Let \(a = t_{0} < t_{1} <\cdots < t_{m} < t_{m + 1} = T\), \(J_{0} = [a,t_{1}]\), and \(J_{k} = (t_{k},t_{k + 1}] \) (\(k = 1,2,\ldots,m\)).

The rest of this paper is organized as follows. In Section 2, some definitions and conclusions are presented. In Section 3, we give formulas of a general solution for a kind of impulsive differential equations with Hadamard fractional derivative of order \(q \in(1,2)\). In Section 4, an example is provided to expound our results.

2 Preliminaries

In this section, we introduce some basic definitions, notation, and lemmas used in this paper.

Definition 2.1

([2], p.110)

The left-sided Hadamard fractional integral of order \(q \in \mathbb{R}^{ +}\) of a function \(x(t)\) is defined by

$$\bigl( {}_{H}\mathcal {J}_{a^{ +}}^{q}x \bigr) (t) = \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{q - 1}x(s)\frac{ds}{s}\quad (0 < a < t \leq T), $$

where \(\Gamma(\cdot)\) is the gamma function.

Definition 2.2

([2], p.111)

The left-sided Hadamard fractional derivative of order \(q \in [n-1, n]\), \(n \in \mathbb{Z}^{ +}\) of a function \(x(t)\) is defined by

$$\bigl( {}_{H}D_{a^{ +}}^{q}x \bigr) (t) = \frac{1}{\Gamma (n - q)} \biggl( t\frac{d}{dt} \biggr)^{n} \int_{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{n - q - 1}x(s)\frac{ds}{s}\quad (0 < a < x \leq T). $$

Lemma 2.3

([2], Theorem 3.28)

Let \(q > 0\), \(n = -[- q]\), and \(0 \leq \gamma < 1\). Let G be an open set in \(\mathbb{R}\), and \(f:(a,b] \times G \to \mathbb{R}\) be a function such that \(f(t,x) \in C_{\gamma\cdot\ln} [a,b]\) for any \(y \in G\). A function \(x \in C_{n - q\cdot\ln} [a,b]\) is a solution of the fractional integral equation

$$x(t) = \sum_{i = 1}^{n} \frac{b_{i}}{\Gamma (q - i + 1)} \biggl( \ln \frac{t}{a} \biggr)^{q - i} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{q - 1}f \bigl(s,x(s) \bigr)\frac{ds}{s}\quad (0 < a < t), $$

if and only if x is a solution of the fractional Cauchy problem

$$ \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}x(t) = f(t,x(t)), \quad q \in (n - 1,n],t \in (a,b], \\ {}_{H}D_{a^{ +}}^{q - i}x(a^{ +} ) = b_{i} \in \mathbb{R},\quad i = 1,2,\ldots,n;n = - [ - q]. \end{array}\displaystyle \right . $$
(2.1)

Lemma 2.4

([2], Properties 2.26, 2.28, 2.37)

For \(q > 0\), \(p > 0\), and \(0 < a < b < \infty\), if \(f \in C_{\gamma\cdot\ln} [a,b] \) (\(0 \leq \gamma < 1\)), then \({}_{H}\mathcal{J}_{a^{ +}}^{q}{}_{H}\mathcal {J}_{a^{ +}}^{p}f = {}_{H}\mathcal{J}_{a^{ +}}^{q + p}f\) and \({}_{H}D_{a^{ +}}^{q}{}_{H}\mathcal{J}_{a^{ +}}^{q}f = f\).

3 Main results

For system (1.1) we have

$$\begin{aligned} &\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0,\ldots,\bar{\Delta}_{n} ( u(\bar{t}_{n}^{ -} ) ) \to 0} \bigl\{ \mbox{system } (1.1) \bigr\} \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)), \quad q \in (1,2),t \in (a,T] \mbox{ and } t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = t_{k}} = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ +} ) - {}_{H}\mathcal {J}_{a^{ +}}^{2 - q}u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ),\\ \quad k = 1,2,\ldots,m, \\ {}_{H}\mathcal {J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2},\qquad{}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1}. \end{array}\displaystyle \right . \end{aligned}$$
(3.1)
$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \bigl\{ \mbox{system } (1.1) \bigr\} \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T] \mbox{ and } t \ne \bar{t}_{l}\ (l = 1,2,\ldots,n), \\ \Delta ( {}_{H}D_{a^{ +}}^{q - 1}u ) \vert _{t = \bar{t}_{l}} = {}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ +} ) - {}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ -} ) = \bar{\Delta}_{l} ( u(\bar{t}_{l}^{ -} ) ),\\ \quad l = 1,2,\ldots,n, \\ {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2},\qquad{}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1}. \end{array}\displaystyle \right . \end{aligned}$$
(3.2)
$$\begin{aligned} & \mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0,\ldots,\bar{\Delta}_{n} ( u(\bar{t}_{n}^{ -} ) ) \to 0, }}\limits_{ \Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \bigl\{ \mbox{system } (1.1) \bigr\} \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T], \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}, \end{array}\displaystyle \right . \end{aligned}$$
(3.3)

Therefore, the solution of system (1.1) satisfies the following three conditions:

$$\begin{aligned} (\mathrm{i})&\quad \lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, \ldots,\bar{\Delta}_{n} ( u(\bar{t}_{n}^{ -} ) ) \to 0} \bigl\{ \mbox{the solution of system } (1.1) \bigr\} \\ &\qquad= \bigl\{ \mbox{the solution of system } (3.1) \bigr\} , \\ (\mathrm{ii})&\quad \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, \ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \bigl\{ \mbox{the solution of system } (1.1) \bigr\} \\ &\qquad= \bigl\{ \mbox{the solution of system } (3.2) \bigr\} , \\ (\mathrm{iii})& \quad \mathop{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, \ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0}}\limits _{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, \ldots,\bar{\Delta}_{n} ( u(\bar{t}_{n}^{ -} ) ) \to 0} \bigl\{ \mbox{the solution of system } (1.1) \bigr\} \\ &\qquad= \bigl\{ \mbox{the solution of system } (3.3) \bigr\} \end{aligned}$$

Therefore, we give the definition of a solution of system (1.1).

Definition 3.1

A function \(z(t):[a,T] \to \mathbb{R}\) is said to be a solution of the fractional Cauchy problem (1.1) if \({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}z(a^{ +} ) = u_{2}\) and \({}_{H}D_{a^{ +}}^{q - 1}z(a^{ +} ) = u_{1}\), the equation condition \({}_{H}D_{a^{ +}}^{q}z(t) = f(t,z(t))\) for each \(t \in (a, T]\) is satisfied, the impulsive conditions \(\Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}z ) |_{t = t_{k}} = \Delta_{k} ( z(t_{k}^{ -} ) ) \) (\(k = 1, 2, \ldots, m\)) and \(\Delta ( {}_{H}D_{a^{ +}}^{q - 1}z ) \vert _{t = \bar{t}_{l}} = \bar{\Delta}_{l} ( z(\bar{t}_{l}^{ -} ) ) \) (\(l =1, 2, \ldots, n\)) are satisfied, the restriction of \(z( \cdot )\) to the interval \((t'_{k},t'_{k + 1}] \) (\(k = 0,1, 2, \ldots, \Omega\)) is continuous, and conditions (i)-(iii) hold.

Using the equality \(\ln \frac{t}{t_{k}} = \int_{t_{k}}^{t} \frac{ds}{s}\) (\(k = 0, 1, 2, \ldots, m\)), define the piecewise function

$$\begin{aligned} \tilde{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k}^{ +} \bigr) \bigr) \biggl( \int_{t_{k}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{k}^{ +} \bigr) \bigr) \biggl( \int_{t_{k}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{k}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{k},t_{k + 1}] \ (\mbox{where } k = 0,1,2,\ldots,m). \end{aligned}$$

By Definition 2.2 we have

$$\begin{aligned} &\bigl[ {}_{H}D_{a^{ +}}^{q}\tilde{u}(t) \bigr]_{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (2 - q)} \biggl( t\frac{d}{dt} \biggr)^{2} \int_{a}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{2 - q - 1} \biggl[ \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k}^{ +} \bigr) \bigr) \biggl(\ln \frac{\eta}{t_{k}} \biggr)^{q - 1} \\ &\qquad{} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q} u\bigl(t_{k}^{ +} \bigr) \bigr) \biggl(\ln \frac{\eta}{ t_{k}}\biggr)^{q - 2} \\ &\qquad{}+ \frac{1}{\Gamma (q)} \int_{t_{k}}^{\eta} \biggl(\ln \frac{\eta}{ s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \frac{d\eta}{\eta} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (2 - q)} \biggl( t\frac{d}{dt} \biggr)^{2} \int_{t_{k}}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{2 - q - 1} \biggl[ \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k}^{ +} \bigr) \bigr) \biggl(\ln \frac{\eta}{t_{k}} \biggr)^{q - 1} \\ &\qquad{} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{k}^{ +} \bigr) \bigr) \biggl(\ln \frac{\eta}{ t_{k}}\biggr)^{q - 2} \\ &\qquad{}+ \frac{1}{\Gamma (q)} \int_{t_{k}}^{\eta} \biggl(\ln \frac{\eta}{ s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \frac{d\eta}{\eta} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \frac{1}{\Gamma (2 - q)\Gamma (q)} \biggl( t\frac{d}{dt} \biggr)^{2} \int_{t_{k}}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{2 - q - 1} \biggl[ \int_{t_{k}}^{\eta} \biggl(\ln \frac{\eta}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \frac{d\eta}{\eta} \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= \biggl\{ \biggl( t\frac{d}{dt} \biggr)^{2} \biggl( \int_{t_{k}}^{t} \biggl(\ln \frac{t}{s}\biggr)f \bigl(s,u(s)\bigr)\frac{ds}{s} \biggr) \biggr\} _{t \in (t_{k},t_{k + 1}]} \\ &\quad= f\bigl(t,u(t)\bigr) \vert _{t \in (t_{k},t_{k + 1}]}. \end{aligned}$$

So, \(\tilde{u}(t)\) satisfies the condition of Hadamard fractional derivative and does not satisfy conditions (i)-(iii). Thus, we will assume that \(\tilde{u}(t)\) is an approximate solution to seek the exact solution of system (1.1). First, let us prove two useful conclusions.

Lemma 3.2

Let \(q \in(1, 2)\), and let be a constant. System (3.1) is equivalent to the fractional integral equation

(3.4)

provided that the integral in (3.4) exists.

Proof

Necessity. We will verify that Eq. (3.4) satisfies the conditions of system (3.1).

For system (3.1), we have

$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T] \mbox{ and}\\ \quad t \ne t_{k}\ (k = 1,2,\ldots,m), \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = t_{k}} = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ +} ) - {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ),\\ \quad k = 1,2,\ldots,m, \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad{}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2} \end{array}\displaystyle \right . \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T], \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1}, \qquad{}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}. \end{array}\displaystyle \right . \end{aligned}$$

Therefore,

$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots,\Delta_{m} ( u(t_{m}^{ -} ) ) \to 0} \bigl\{ \mbox{the solution of system } (3.1) \bigr\} \\ &\quad= \bigl\{ \mbox{the solution of system } (3.3) \bigr\} . \end{aligned}$$
(3.5)

Moreover, we easily verify that Eq. (3.4) satisfies condition (3.5).

Next, taking the Hadamard fractional derivative of Eq. (3.4) for each \(t \in (t_{k},t_{k + 1}] \) (\(k = 0,1, 2,\ldots, m\)), we have

So, Eq. (3.4) satisfies the Hadamard fractional derivative of system (3.1).

Finally, for each \(t_{k}\) (\(k = 1, 2,\ldots, m\)) in Eq. (3.4), we have

$$\begin{aligned} &{}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{k}^{ +} \bigr) - {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{k}^{ -} \bigr) \\ &\quad= \biggl\{ \frac{1}{\Gamma (2 - q)} \int_{a}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{2 - q - 1}u(\eta )\frac{d\eta}{\eta} \biggr\} _{t \to t_{k}^{ +}} - \biggl\{ \frac{1}{\Gamma (2 - q)} \int_{a}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{2 - q - 1}u(\eta )\frac{d\eta}{\eta} \biggr\} _{t = t_{k}^{ -}} \\ &\quad= \Delta_{k} \bigl( u\bigl(t_{k}^{ -} \bigr) \bigr). \end{aligned}$$

Therefore, Eq. (3.4) satisfies the impulsive conditions of (3.1). Thus, Eq. (3.4) satisfies all conditions of system (3.1).

Sufficiency. We will prove that the solutions of system (3.1) satisfy Eq. (3.4) by mathematical induction. By Definitions 2.1 and 2.2 the solution of system (3.1) satisfies

$$\begin{aligned} u(t) ={}& \frac{u_{1}}{\Gamma (q)}\biggl(\ln \frac{t}{a} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)}\biggl(\ln \frac{t}{a} \biggr)^{q - 2} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (a,t_{1}]. \end{aligned}$$
(3.6)

Using (3.6) and Definitions 2.1 and 2.2, we get \({}_{H}D_{a^{ +}}^{q - 1}u(t_{1}^{ +} ) = {}_{H}D_{a^{ +}}^{q - 1}u(t_{1}^{ -} ) = u_{1} + \int_{a}^{t_{1}} f(s, u(s))\frac{ds}{s}\) and \({}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{1}^{ +} ) = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{1}^{ -} ) + \Delta_{1} ( u(t_{1}^{ -} ) ) = u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s} + \Delta_{1} ( u(t_{1}^{ -} ) )\). Therefore, the approximate solution for \(t \in (t_{1},t_{2}]\) is provided by

$$\begin{aligned} \tilde{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{1}^{ +} \bigr) \bigr) \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{1}^{ +} \bigr) \bigr) \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& \frac{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s} + \Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(3.7)

Let \(e_{1}(t) = u(t) - \tilde{u}(t)\) for \(t \in (t_{1},t_{2}]\), where \(u(t)\) is the exact solution of system (3.1). Due to

$$\begin{aligned} \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}u(t) = {}&\frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad \mbox{for } t \in (t_{1},t_{2}], \end{aligned}$$

we obtain

$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{1}(t) \\ &\quad= \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ &\quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}- \frac{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{}- \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(3.8)

By (3.8) we assume that

$$\begin{aligned} e_{1}(t) ={}& \sigma \bigl(\Delta_{1} \bigl( u \bigl(t_{1}^{ -} \bigr) \bigr)\bigr)\lim_{\Delta_{1} \to 0}e_{1}(t) \\ ={}& \sigma \bigl(\Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr)\bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\biggr]\quad \mbox{for } t \in (t_{1},t_{2}], \end{aligned}$$

where the function \(\sigma ( \cdot )\) is an undetermined function with \(\sigma (0) = 1\). Then,

$$\begin{aligned} u(t) ={}& \tilde{u}(t) + e_{1}(t) \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}+ \frac{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \bigl( 1 - \sigma \bigl(\Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{} - \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\biggr] \quad\mbox{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(3.9)

Using (3.9), we obtain \({}_{H}D_{a^{ +}}^{q - 1}u(t_{2}^{ +} ) = {}_{H}D_{a^{ +}}^{q - 1}u(t_{2}^{ -} ) = u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}\) and

$$\begin{aligned} {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{2}^{ +} \bigr)&= {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{2}^{ -} \bigr) + \Delta_{2} \bigl( u \bigl(t_{2}^{ -} \bigr) \bigr) \\ &= u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)f \bigl(s,u(s) \bigr)\frac{ds}{s} + \Delta_{1} \bigl( u \bigl(t_{1}^{ -} \bigr) \bigr) + \Delta_{2} \bigl( u \bigl(t_{2}^{ -} \bigr) \bigr). \end{aligned}$$

So, the approximate solution for \(t \in (t_{2},t_{3}]\) is given by

$$\begin{aligned} \tilde{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{2}^{ +} \bigr) \bigr) \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{2}^{ +} \bigr) \bigr) \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& \frac{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s} + \Delta_{1} ( u(t_{1}^{ -} ) ) + \Delta_{2} ( u(t_{2}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{2},t_{3}]. \end{aligned}$$
(3.10)

Let \(e_{2}(t) = u(t) - \tilde{u}(t)\) for \(t \in (t_{2},t_{3}]\). By (3.6) and (3.9) the exact solution \(u(t)\) of system (3.1) satisfies

$$\begin{aligned}& \mathop{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, }}\limits _{ \Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) = \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \hphantom{\mathop{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, }}\limits_{ \Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t)=}{}+ \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) \\& \quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}+ \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{\Delta_{2} ( u(t_{2}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}- \bigl( 1 - \sigma \bigl(\Delta_{2} \bigl( u \bigl(t_{2}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl\{ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}+ \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{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\& \qquad{}- \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}- \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} \biggr\} \quad\mbox{for } t \in (t_{2},t_{3}], \\& \lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) = \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2}\\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) =}{} + \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{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) =}{}- \bigl( 1 - \sigma \bigl(\Delta_{1} \bigl( u \bigl(t_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl\{ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) =}{}+ \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{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) =}{}- \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2}\\& \hphantom{\lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}u(t) =}{} - \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} \biggr\} \quad\mbox{for } t \in (t_{2},t_{3}]. \end{aligned}$$

Therefore,

$$\begin{aligned} \mathop{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0, }}\limits _{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) ={}& \mathop{\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,}}\limits _{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}, \end{aligned}$$
(3.11)
$$\begin{aligned} \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0}e_{2}(t) ={}& \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ ={}& \sigma \bigl(\Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr)\bigr) \biggl\{ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{} - \frac{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{} - \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\biggr\} , \end{aligned}$$
(3.12)
$$\begin{aligned} \lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0}e_{2}(t) ={}& \lim_{\Delta_{2} ( u(t_{2}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \frac{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} - \frac{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &{}- \bigl( 1 - \sigma \bigl(\Delta_{1} \bigl( u \bigl(t_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl\{ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr\} . \end{aligned}$$
(3.13)

So, by (3.11)-(3.13) we obtain

$$\begin{aligned} e_{2}(t) ={}& \sigma \bigl(\Delta_{2} \bigl( u \bigl(t_{2}^{ -} \bigr) \bigr)\bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] + \frac{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \bigl( 1 - \sigma \bigl(\Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2}\\ &{} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1}\\ &{} - \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (t_{2},t_{3}]. \end{aligned}$$

Thus,

$$\begin{aligned} u(t) ={}& \tilde{u}(t) + e_{2}(t) \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{\Delta_{1} ( u(t_{1}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} + \frac{\Delta_{2} ( u(t_{2}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \bigl( 1 - \sigma \bigl(\Delta_{1} \bigl( u\bigl(t_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{t_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{1}}{a} + u_{2} + \int_{a}^{t_{1}} (\ln \frac{t_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \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} \biggr] \\ &{}- \bigl( 1 - \sigma \bigl(\Delta_{2} \bigl( u\bigl(t_{2}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{t_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t_{2}}{a} + u_{2} + \int_{a}^{t_{2}} (\ln \frac{t_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{} - \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}\biggr] \quad\mbox{for } t \in (t_{2},t_{3}]. \end{aligned}$$
(3.14)

Letting \(t_{2} \to t_{1}\), we have

$$\begin{aligned}& \lim_{t_{2} \to t_{1}} \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,t_{3}] \mbox{ and } t \ne t_{1} \mbox{ and } t \ne t_{2}, \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = t_{k}} = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ +} ) - {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(t_{k}^{ -} ) = \Delta_{k} ( u(t_{k}^{ -} ) ),\quad k = 1,2, \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}, \end{array}\displaystyle \right . \end{aligned}$$
(3.15)
$$\begin{aligned}& \quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (0,1),t \in (a,t_{3}] \mbox{ and } t \ne t_{1}, \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = t_{1}} = \Delta_{1} ( u(t_{1}^{ -} ) ) + \Delta_{2} ( u(t_{2}^{ -} ) ), \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1}, \qquad {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}. \end{array}\displaystyle \right . \end{aligned}$$
(3.16)

Using (3.9) and (3.14) in systems (3.16) and (3.15), respectively, we obtain

$$1 - \sigma (\Delta_{1} + \Delta_{2}) = 1 - \sigma ( \Delta_{1}) + 1 - \sigma (\Delta_{2}) \quad \forall \Delta_{1},\Delta_{2} \in \mathbb{R}. $$

Letting \(\rho (z) = 1 - \sigma (z)\) (\(\forall z \in \mathbb{R}\)), we have \(\rho (z + w) = \rho (z) + \rho (w) \) (\(\forall z,w \in \mathbb{R}\)). Therefore, , where is a constant. So, by (3.9) and (3.14) we get

(3.17)

and

(3.18)

For \(t \in (t_{k},t_{k + 1}]\), suppose that

(3.19)

Using (3.19), we obtain \({}_{H}D_{a^{ +}}^{q - 1}u(t_{k + 1}^{ +} ) = {}_{H}D_{a^{ +}}^{q - 1}u(t_{k + 1}^{ -} ) = u_{1} + \int_{a}^{t_{k + 1}} f(s,u(s))\frac{ds}{s}\) and

$$\begin{aligned} {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{k + 1}^{ +} \bigr) &= {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{k + 1}^{ -} \bigr) + \Delta_{k + 1} \bigl( u \bigl(t_{k + 1}^{ -} \bigr) \bigr) \\ &= u_{1}\ln \frac{t_{k + 1}}{a} + u_{2} + \int_{a}^{t_{k + 1}} \biggl(\ln \frac{t_{k + 1}}{s} \biggr)f \bigl(s,u(s) \bigr)\frac{ds}{s} + \sum_{i = 1}^{k + 1} \Delta_{i} \bigl( u \bigl(t_{i}^{ -} \bigr) \bigr). \end{aligned}$$

So, the approximate solution for \(t \in (t_{k + 1},t_{k + 2}]\) is provided by

$$\begin{aligned} \tilde{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k + 1}^{ +} \bigr) \bigr) \biggl( \int_{t_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{k + 1}^{ +} \bigr) \bigr) \biggl( \int_{t_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{k + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ =&{} \frac{u_{1} + \int_{a}^{k + 1} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{t_{k + 1}}{a} + u_{2} + \int_{a}^{t_{k + 1}} (\ln \frac{t_{k + 1}}{s})f(s,u(s))\frac{ds}{s} + \sum_{i = 1}^{k + 1} \Delta_{i} ( u(t_{i}^{ -} ) )}{\Gamma (q - 1)} \biggl( \int_{t_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t_{k + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (t_{k + 1},t_{k + 2}]. \end{aligned}$$
(3.20)

Let \(e_{k + 1}(t) = u(t) - \tilde{u}(t)\) for \(t \in (t_{k + 1},t_{k + 2}]\). By (3.19) the exact solution \(u(t)\) of system (3.1) satisfies

Therefore,

(3.21)
$$\begin{aligned} &\lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots,\Delta_{k + 1} ( u(t_{k + 1}^{ -} ) ) \to 0}e_{k + 1}(t) \\ &\quad= \lim_{\Delta_{1} ( u(t_{1}^{ -} ) ) \to 0,\ldots,\Delta_{k + 1} ( u(t_{k + 1}^{ -} ) ) \to 0} \bigl\{ u(t) - \tilde{u}(t) \bigr\} \\ &\quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{k + 1} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{}- \frac{u_{1}\ln \frac{t_{k + 1}}{a} + u_{2} + \int_{a}^{t_{k + 1}} (\ln \frac{t_{k + 1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{t_{k + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in (t_{k + 1},t_{k + 2}]. \end{aligned}$$
(3.22)

So, by (3.21) and (3.22) we obtain

Thus,

So, the solution of system (3.1) satisfies Eq. (3.4).

So, system (3.1) is equivalent to Eq. (3.4). The proof is now completed. □

Lemma 3.3

Let \(q \in(1, 2)\), and let ħ be a constant. System (3.2) is equivalent to the fractional integral equation

$$ u(t) = \left \{ \textstyle\begin{array}{@{}l} \frac{u_{1}}{\Gamma (q)} ( \int_{a}^{t} \frac{ds}{s} )^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} ( \int_{a}^{t} \frac{ds}{s} )^{q - 2} \\ \quad{}+ \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \quad\textit{for } t \in (a,\bar{t}_{1}], \\ \frac{u_{1}}{\Gamma (q)} ( \int_{a}^{t} \frac{ds}{s} )^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} ( \int_{a}^{t} \frac{ds}{s} )^{q - 2} + \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \\ \quad{}+ \sum_{j = 1}^{l} \frac{\bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) )}{\Gamma (q)} ( \int_{\bar{t}_{j}}^{t} \frac{ds}{s} )^{q - 1} - \sum_{j = 1}^{l} \hbar \bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) ) [ \frac{u_{1}}{\Gamma (q)} ( \int_{a}^{t} \frac{ds}{s} )^{q - 1} \\ \quad{}+ \frac{u_{2}}{\Gamma (q - 1)} ( \int_{a}^{t} \frac{ds}{s} )^{q - 2} + \frac{1}{\Gamma (q)}\int_{a}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} \\ \quad{} - \frac{u_{1} + \int_{a}^{\bar{t}_{j}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} ( \int_{\bar{t}_{j}}^{t} \frac{ds}{s} )^{q - 1} - \frac{u_{1}\ln \frac{\bar{t}_{j}}{a} + u_{2} + \int_{a}^{\bar{t}_{j}} (\ln \frac{\bar{t}_{j}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} ( \int_{\bar{t}_{j}}^{t} \frac{ds}{s} )^{q - 2} \\ \quad{}- \frac{1}{\Gamma (q)}\int_{\bar{t}_{j}}^{t} (\ln \frac{t}{s})^{q - 1}f(s,u(s))\frac{ds}{s} ] \quad\textit{for } t \in (\bar{t}_{l},\bar{t}_{l + 1}], 1 \le l \le n \end{array}\displaystyle \right . $$
(3.23)

provided that the integral in (3.23) exists.

Proof

Necessity. We will verify that Eq. (3.23) satisfies the conditions of system (3.2).

For system (3.2), there exists an implicit condition

$$\begin{aligned} &\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, \ldots,\bar{\Delta}_{n} ( u(\bar{t}_{n}^{ -} ) ) \to 0} \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T] \mbox{ and}\\ \quad t \ne \bar{t}_{l}\ (l = 1,2,\ldots,n), \\ \Delta ( {}_{H}D_{a^{ +}}^{q - 1}u ) \vert _{t = \bar{t}_{l}} = {}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ +} ) - {}_{H}D_{a^{ +}}^{q - 1}u(\bar{t}_{l}^{ -} ) = \bar{\Delta}_{l} ( u(\bar{t}_{l}^{ -} ) ),\\ \quad l = 1,2,\ldots,n, \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad{}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}, \end{array}\displaystyle \right . \\ &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,T], \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}, \end{array}\displaystyle \right . \end{aligned}$$

that is,

$$\begin{aligned} &\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, \ldots,\bar{\Delta}_{n} ( u(\bar{t}_{n}^{ -} ) ) \to 0} \bigl\{ \mbox{the solution of system } (3.2) \bigr\} \\ &\quad= \bigl\{ \mbox{the solution of system } (3.3) \bigr\} . \end{aligned}$$
(3.24)

Obviously, Eq. (3.23) satisfies condition (3.24).

Next, taking the Hadamard fractional derivative to Eq. (3.23) for each \(t \in (\bar{t}_{l},\bar{t}_{l + 1}] \) (\(l = 0,1, 2,\ldots, n\)), we have

$$\begin{aligned} {}_{H}D_{a^{ +}}^{q}u(t) ={}& {}_{H}D_{a^{ +}}^{q} \Biggl\{ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \sum _{j = 1}^{l} \frac{\bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{j}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \sum_{j = 1}^{l} \hbar \bar{ \Delta}_{j} \bigl( u\bigl(\bar{t}_{j}^{ -} \bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{j}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{j}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{j}}{a} + u_{2} + \int_{a}^{\bar{t}_{j}} (\ln \frac{\bar{t}_{j}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{j}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{j}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \Biggr\} \\ ={}& f\bigl(t,u(t)\bigr)_{t \ge a} \vert _{t \in (\bar{t}_{l},\bar{t}_{l + 1}]} \\ &{}- \Biggl\{ \sum _{j = 1}^{l} \hbar \bar{\Delta}_{j} \bigl( u\bigl(\bar{t}_{j}^{ -} \bigr) \bigr) \bigl[ f \bigl(t,u(t)\bigr)_{t \ge a} - f\bigl(t,u(t)\bigr)_{t \ge \bar{t}_{j}} \bigr] \Biggr\} _{t \in (\bar{t}_{l},\bar{t}_{l + 1}]} \\ =&{} f\bigl(t,u(t)\bigr) \vert _{t \in (\bar{t}_{l},\bar{t}_{l + 1}]}. \end{aligned}$$

So, Eq. (3.23) satisfies the Hadamard fractional derivative of system (3.2).

Finally, for each \(\bar{t}_{l} \) (\(l = 1, 2,\ldots, n\)) in Eq. (3.23), we have

$$\begin{aligned} &{}_{H}D_{a^{ +}}^{q - 1}u\bigl(\bar{t}_{l}^{ +} \bigr) - {}_{H}D_{a^{ +}}^{q - 1}u\bigl( \bar{t}_{l}^{ -} \bigr) \\ &\quad= \biggl\{ \frac{1}{\Gamma (2 - q)} \biggl( t\frac{d}{dt} \biggr) \int_{a}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{1 - q}u(\eta )\frac{d\eta}{\eta} \biggr\} _{t \to \bar{t}_{l}^{ +}} \\ &\qquad{}- \biggl\{ \frac{1}{\Gamma (2 - q)} \biggl( t\frac{d}{dt} \biggr) \int_{a}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{1 - q}u(\eta )\frac{d\eta}{\eta} \biggr\} _{t = \bar{t}_{l}^{ -}} \\ &\quad= \bar{\Delta}_{l} \bigl( u\bigl(\bar{t}_{l}^{ -} \bigr) \bigr). \end{aligned}$$

Therefore, Eq. (3.23) satisfies the impulsive conditions of (3.2). Thus, Eq. (3.23) satisfies all conditions of (3.2).

Sufficiency. We will prove that the solutions of system (3.2) satisfy Eq. (3.23) by mathematical induction. For \(t \in (a,t_{1}]\), by Definitions 2.1 and 2.2, the solution of system (3.2) satisfies

$$\begin{aligned} u(t) ={}& \frac{u_{1}}{\Gamma (q)}\biggl(\ln \frac{t}{a} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)}\biggl(\ln \frac{t}{a} \biggr)^{q - 2} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in (a,\bar{t}_{1}]. \end{aligned}$$
(3.25)

By (3.25) and Definitions 2.1 and 2.2 we have

$${}_{H}D_{a^{ +}}^{q - 1}u \bigl(\bar{t}_{1}^{ +} \bigr) = {}_{H}D_{a^{ +}}^{q - 1}u \bigl( \bar{t}_{1}^{ -} \bigr) = u_{1} + \int_{a}^{\bar{t}_{1}} f \bigl(s,u(s) \bigr)\frac{ds}{s} + \bar{\Delta}_{1} \bigl( u \bigl(\bar{t}_{1}^{ -} \bigr) \bigr) $$

and

$${}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl( \bar{t}_{1}^{ +} \bigr) = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(\bar{t}_{1}^{ -} \bigr) = u_{1} \biggl( \int_{a}^{\bar{t}_{1}} \frac{ds}{s} \biggr) + u_{2} + \int_{a}^{\bar{t}_{1}} \biggl(\ln \frac{\bar{t}_{1}}{s} \biggr)f \bigl(s,u(s) \bigr)\frac{ds}{s}. $$

Therefore, the approximate solution for \(t \in (\bar{t}_{1},\bar{t}_{2}]\) is provided by

$$\begin{aligned} \bar{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(\bar{t}_{1}^{ +} \bigr) \bigr) \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(\bar{t}_{1}^{ +} \bigr) \bigr) \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{} + \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& \frac{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s} + \bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in (\bar{t}_{1},\bar{t}_{2}]. \end{aligned}$$
(3.26)

Let \(\bar{e}_{1}(t) = u(t) - \bar{u}(t)\) for \(t \in (\bar{t}_{1},\bar{t}_{2}]\). Due to

$$\begin{aligned} \lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0}u(t) ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in ( \bar{t}_{1},\bar{t}_{2}], \end{aligned}$$

we have

$$\begin{aligned} &\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0}\bar{e}_{1}(t) \\ &\quad= \lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0} \bigl\{ u(t) - \bar{u}(t) \bigr\} \\ &\quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in ( \bar{t}_{1},\bar{t}_{2}]. \end{aligned}$$

Therefore, we assume that

$$\begin{aligned} \bar{e}_{1}(t) ={}& \delta \bigl(\bar{\Delta}_{1} \bigl( u \bigl(\bar{t}_{1}^{ -} \bigr) \bigr)\bigr)\lim _{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0}\bar{e}_{1}(t) \\ ={}& \delta \bigl(\bar{\Delta}_{1} \bigl( u\bigl( \bar{t}_{1}^{ -} \bigr) \bigr)\bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\biggr], \end{aligned}$$

where \(\delta ( \cdot )\) is an undetermined function with \(\delta (0) = 1\). So,

$$\begin{aligned} u(t) ={}& \bar{u}(t) + \bar{e}_{1}(t) \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{1} \bigl( u\bigl( \bar{t}_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (\bar{t}_{1},\bar{t}_{2}]. \end{aligned}$$
(3.27)

By (3.27) we obtain

$${}_{H}D_{a^{ +}}^{q - 1}u \bigl(\bar{t}_{2}^{ +} \bigr) = {}_{H}D_{a^{ +}}^{q - 1}u \bigl( \bar{t}_{2}^{ -} \bigr) + \bar{\Delta}_{2} \bigl( u \bigl(\bar{t}_{2}^{ -} \bigr) \bigr) = u_{1} + \int_{a}^{\bar{t}_{2}} f \bigl(s,u(s) \bigr)\frac{ds}{s} + \bar{\Delta}_{1} \bigl( u \bigl(\bar{t}_{1}^{ -} \bigr) \bigr) + \bar{\Delta}_{2} \bigl( u \bigl(\bar{t}_{2}^{ -} \bigr) \bigr) $$

and

$${}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u\bigl( \bar{t}_{2}^{ +} \bigr) = {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(\bar{t}_{2}^{ -} \bigr) = u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \bar{ \Delta}_{1} \bigl( u\bigl(\bar{t}_{1}^{ -} \bigr) \bigr)\ln \frac{\bar{t}_{2}}{\bar{t}_{1}} + \int_{a}^{\bar{t}_{2}} \ln \frac{\bar{t}_{2}}{s}f\bigl(s,u(s) \bigr)\frac{ds}{s}. $$

So, the approximate solution for \(t \in (\bar{t}_{2},\bar{t}_{3}]\) is given by

$$ \begin{aligned}[b] \bar{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(\bar{t}_{2}^{ +} \bigr) \bigr) \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(\bar{t}_{2}^{ +} \bigr) \bigr) \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\\ ={}& \frac{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s} + \bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) + \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )\ln \frac{\bar{t}_{2}}{\bar{t}_{1}} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2}\\ &{}+ \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in ( \bar{t}_{2},\bar{t}_{3}]. \end{aligned} $$
(3.28)

Let \(\bar{e}_{2}(t) = u(t) - \bar{u}(t)\) for \(t \in (\bar{t}_{2},\bar{t}_{3}]\). By (3.27) the exact solution \(u(t)\) of system (3.2) satisfies

$$\begin{aligned}& \mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}u(t) = \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2}\\& \hphantom{\mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}u(t) =}{} + \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 ( \bar{t}_{2},\bar{t}_{3}], \\& \lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0}u(t) \\& \quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}+ \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{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\& \qquad{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{2} \bigl( u\bigl( \bar{t}_{2}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}+ \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{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\& \qquad{}- \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (\bar{t}_{2},\bar{t}_{3}], \\& \lim_{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}u(t) \\& \quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}+ \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{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\& \qquad{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{1} \bigl( u\bigl( \bar{t}_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2}\\& \qquad{} + \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{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\& \qquad{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\& \qquad{}- \frac{1}{\Gamma (q)} \int_{\bar{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 (\bar{t}_{2},\bar{t}_{3}]. \end{aligned}$$

Therefore,

$$\begin{aligned} &\mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{ \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}\bar{e}_{2}(t) = \mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0} \bigl\{ u(t) - \bar{u}(t) \bigr\} \\ &\hphantom{\mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{ \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}\bar{e}_{2}(t)}= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\hphantom{\mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{ \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}\bar{e}_{2}(t)=}{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\hphantom{\mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{ \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}\bar{e}_{2}(t)=}{} - \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\hphantom{\mathop{\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0, }}\limits _{ \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}\bar{e}_{2}(t)=}{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in ( \bar{t}_{2},\bar{t}_{3}], \end{aligned}$$
(3.29)
$$\begin{aligned} &\lim_{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0}\bar{e}_{2}(t) \\ &\quad= \lim_{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ) \to 0} \bigl\{ u(t) - \bar{u}(t) \bigr\} \\ &\quad= \frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} - \frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} - \frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q - 1)}\ln \frac{\bar{t}_{2}}{\bar{t}_{1}} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}+ \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} + \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{}- \frac{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{}- \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{ \Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &\qquad{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{1} \bigl( u\bigl( \bar{t}_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}+ \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{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{\bar{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 (\bar{t}_{2},\bar{t}_{3}], \end{aligned}$$
(3.30)
$$\begin{aligned} &\lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0}\bar{e}_{2}(t) \\ &\quad= \lim_{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) \to 0} \bigl\{ u(t) - \bar{u}(t) \bigr\} \\ &\quad= \delta \bigl(\bar{\Delta}_{2} \bigl( u\bigl( \bar{t}_{2}^{ -} \bigr) \bigr)\bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{}- \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\biggr] \quad\mbox{for } t \in (\bar{t}_{2},\bar{t}_{3}]. \end{aligned}$$
(3.31)

So, by (3.29)-(3.31) we obtain

$$\begin{aligned} \bar{e}_{2}(t) = {}&\frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} - \frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} - \frac{\Delta_{1} ( u(\bar{t}_{1}^{ -} ) )\ln \frac{\bar{t}_{2}}{\bar{t}_{1}}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \delta \bigl(\bar{\Delta}_{2} \bigl( u\bigl( \bar{t}_{2}^{ -} \bigr) \bigr)\bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{1} \bigl( u\bigl( \bar{t}_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{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 (\bar{t}_{2},\bar{t}_{3}]. \end{aligned}$$
(3.32)

Thus,

$$\begin{aligned} u(t) ={}& \bar{u}(t) + \bar{e}_{2}(t) \\ ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} + \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{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{1} \bigl( u\bigl( \bar{t}_{1}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}- \bigl( 1 - \delta \bigl(\bar{\Delta}_{2} \bigl( u\bigl( \bar{t}_{2}^{ -} \bigr) \bigr)\bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{ \Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \quad\mbox{for } t \in (\bar{t}_{2},\bar{t}_{3}]. \end{aligned}$$
(3.33)

Letting \(\bar{t}_{2} \to \bar{t}_{1}\), we have

$$\begin{aligned} &\lim_{\bar{t}_{2} \to \bar{t}_{1}} \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (1,2),t \in (a,\bar{t}_{3}] \mbox{ and } t \ne \bar{t}_{1} \mbox{ and } t \ne \bar{t}_{2}, \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = \bar{t}_{l}} = {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(\bar{t}_{l}^{ +} ) - {}_{H}\mathcal{J}_{a^{ +}}^{1 - q}u(\bar{t}_{l}^{ -} ) = \bar{\Delta}_{l} ( u(\bar{t}_{l}^{ -} ) ), \quad l = 1,2, \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2} \end{array}\displaystyle \right . \end{aligned}$$
(3.34)
$$\begin{aligned} &\quad\to \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{a^{ +}}^{q}u(t) = f(t,u(t)),\quad q \in (0,1),t \in (a,\bar{t}_{3}] \mbox{ and } t \ne \bar{t}_{1}, \\ \Delta ( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u ) \vert _{t = \bar{t}_{1}} = \bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) ) + \bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) ), \\ {}_{H}D_{a^{ +}}^{q - 1}u(a^{ +} ) = u_{1},\qquad {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u(a^{ +} ) = u_{2}. \end{array}\displaystyle \right . \end{aligned}$$
(3.35)

Using (3.27) and (3.33) in systems (3.35) and (3.34), respectively, we have

$$1 - \delta (\bar{\Delta}_{1} + \bar{\Delta}_{2}) = 1 - \delta (\bar{\Delta}_{1}) + 1 - \delta (\bar{\Delta}_{2}) \quad \forall \bar{\Delta}_{1},\bar{\Delta}_{2} \in \mathbb{R}. $$

Letting \(\gamma (z) = 1 - \delta (z) \) (\(\forall z \in \mathbb{R}\)), we have \(\gamma (z + w) = \gamma (z) + \gamma (w)\) (\(\forall z,w \in \mathbb{R}\)). Therefore, \(\gamma (z) = \hbar z\), where ħ is a constant. So, by (3.27) and (3.33) we get

$$\begin{aligned} u(t) ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} + \frac{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \hbar \bar{\Delta}_{1} \bigl( u\bigl(\bar{t}_{1}^{ -} \bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{} - \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{ \Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]\quad \mbox{for } t \in (\bar{t}_{1},\bar{t}_{2}], \end{aligned}$$

and

$$\begin{aligned} u(t) ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} + \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{\bar{\Delta}_{1} ( u(\bar{t}_{1}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{\bar{\Delta}_{2} ( u(\bar{t}_{2}^{ -} ) )}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \hbar \bar{\Delta}_{1} \bigl( u\bigl(\bar{t}_{1}^{ -} \bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{\bar{t}_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{} - \frac{u_{1}\ln \frac{\bar{t}_{1}}{a} + u_{2} + \int_{a}^{\bar{t}_{1}} (\ln \frac{\bar{t}_{1}}{s})f(s,u(s))\frac{ds}{s}}{ \Gamma (q - 1)} \biggl( \int_{\bar{t}_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr] \\ &{}- \hbar \bar{\Delta}_{2} \bigl( u\bigl(\bar{t}_{2}^{ -} \bigr) \bigr) \biggl[ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \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{u_{1} + \int_{a}^{\bar{t}_{2}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{} - \frac{u_{1}\ln \frac{\bar{t}_{2}}{a} + u_{2} + \int_{a}^{\bar{t}_{2}} (\ln \frac{\bar{t}_{2}}{s})f(s,u(s))\frac{ds}{s}}{ \Gamma (q - 1)} \biggl( \int_{\bar{t}_{2}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{\bar{t}_{2}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr]\quad \mbox{for } t \in (\bar{t}_{2},\bar{t}_{3}]. \end{aligned}$$

Due to similarity of the proof with Lemma 3.2, the rest of proof is omitted.

So, system (3.2) is equivalent to Eq. (3.23). The proof is now completed. □

Theorem 3.4

Let \(q \in(1, 2)\), and let , ħ be two constants. System (1.1) is equivalent to the fractional integral equation

(3.36)

provided that the integral in (3.36) exists.

Proof

Necessity. We will verify that Eq. (3.36) satisfies the conditions of system (1.1).

First, we can easily verify that Eq. (3.36) satisfies the three implicit conditions (i)-(iii) by Lemmas 3.2 and 3.3.

Second, the proof that Eq. (3.36) satisfies the Hadamard fractional derivative and impulsive conditions of system (1.1) is similar to that of Lemma 3.2 and is omitted.

Sufficiency. We will prove that the solutions of system (3.1) satisfy Eq. (3.36) by mathematical induction. For \(t \in [a,t'_{1}]\), it is clear that the solution of system (1.1) satisfies Eq. (3.36) with

$$\begin{aligned} u(t) ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f \bigl(s,u(s) \bigr)\frac{ds}{s}\quad \mbox{for } t \in \bigl[a,t'_{1} \bigr]. \end{aligned}$$
(3.37)

For \(t \in (t'_{1},t'_{2}]\), there exist three cases \(t'_{1} = t_{1} < \bar{t}_{1}\), \(t'_{1} = \bar{t}_{1} < t_{1}\), and \(t'_{1} = t_{1} = \bar{t}_{1}\). For the two cases (\(t'_{1} = t_{1} < \bar{t}_{1}\) and \(t'_{1} = \bar{t}_{1} < t_{1}\)), we can verify that the solution of (1.1) satisfies Eq. (3.36) for \(t \in (t'_{1},t'_{2}]\) by Lemmas 3.2 and 3.3, respectively. Hence, we will consider the case \(t'_{1} = t_{1} = \bar{t}_{1}\). Using (3.37), we have

$$\begin{aligned}& {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{1}^{\prime +} \bigr) = {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{1}^{\prime -} \bigr) + \bar{\Delta}_{1} \bigl( u \bigl(t_{1}^{\prime -} \bigr) \bigr) = u_{1} + \int_{a}^{t'_{1}} f \bigl(s,u(s) \bigr)\frac{ds}{s} + \bar{\Delta}_{1} \bigl( u \bigl(t_{1}^{\prime -} \bigr) \bigr) \quad\mbox{and} \\& \begin{aligned}[b] {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{1}^{\prime +} \bigr) &= {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{1}^{\prime -} \bigr) + \Delta_{1} \bigl( u \bigl(t_{1}^{\prime -} \bigr) \bigr) \\ &= u_{1}\ln \frac{t'_{1}}{a} + u_{2} + \int_{a}^{t'_{1}} \ln \frac{t'_{1}}{s}f \bigl(s,u(s) \bigr)\frac{ds}{s} + \Delta_{1} \bigl( u \bigl(t_{1}^{\prime -} \bigr) \bigr). \end{aligned} \end{aligned}$$

Therefore, the approximate solution of system (1.1) for \(t \in (t'_{1},t'_{2}]\) is given by

$$\begin{aligned} \hat{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{1}^{\prime +} \bigr) \bigr) \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H} \mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{1}^{\prime +} \bigr) \bigr) \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t'_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ =&{} \frac{u_{1} + \int_{a}^{t'_{1}} f(s,u(s))\frac{ds}{s} + \bar{\Delta}_{1} ( u(t_{1}^{\prime -} ) )}{\Gamma (q)} \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{t'_{1}}{a} + u_{2} + \int_{a}^{t'_{1}} \ln \frac{t'_{1}}{s}f(s,u(s))\frac{ds}{s} + \Delta_{1} ( u(t_{1}^{\prime -} ) )}{\Gamma (q - 1)} \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t'_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in \bigl(t'_{1},t'_{2}\bigr]. \end{aligned}$$
(3.38)

Let \(\hat{e}_{1}(t) = u(t) - \hat{u}(t)\) for \(t \in (t'_{1},t'_{2}]\). By Lemmas 3.2 and 3.3 the exact solution \(u(t)\) of system (1.1) satisfies

Therefore,

$$\begin{aligned} \mathop{\lim_{\Delta_{1} ( u(t_{1}^{\prime -} ) ) \to 0, }}\limits _{ \bar{\Delta}_{1} ( u(t_{1}^{\prime -} ) ) \to 0}\hat{e}_{1}(t) ={}& \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t'_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t'_{1}}{a} + u_{2} + \int_{a}^{t'_{1}} \ln \frac{t'_{1}}{s}f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}- \frac{1}{\Gamma (q)} \int_{t'_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in \bigl(t'_{1},t'_{2}\bigr], \end{aligned}$$
(3.39)
$$\begin{aligned} \lim_{\Delta_{1} ( u(t_{1}^{\prime -} ) ) \to 0}\hat{e}_{1}(t) ={}& \bigl[ 1 - \hbar \bar{\Delta}_{1} \bigl( u \bigl(t_{1}^{\prime -} \bigr) \bigr) \bigr] \biggl\{ \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} - \frac{u_{1} + \int_{a}^{t'_{1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}- \frac{u_{1}\ln \frac{t'_{1}}{a} + u_{2} + \int_{a}^{t'_{1}} (\ln \frac{t'_{1}}{s})f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t'_{1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{} - \frac{1}{\Gamma (q)} \int_{t'_{1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \biggr\} \quad\mbox{for } t \in \bigl(t'_{1},t'_{2}\bigr], \end{aligned}$$
(3.40)
(3.41)

By (3.39)-(3.41) we get

Then,

(3.42)

Next, for \(t \in (t'_{k},t'_{k + 1}]\) (\(k =1, 2, \ldots, \Omega\)), suppose that

(3.43)

By (3.43) we have

$$\begin{aligned} {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k + 1}^{\prime +} \bigr) &= {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k + 1}^{\prime -} \bigr) + \sum_{k_{1} + 1}^{(k + 1)_{1}} \bar{ \Delta}_{j} \\ &= u_{1} + \int_{a}^{t'_{k + 1}} f \bigl(s,u(s) \bigr)\frac{ds}{s} + \sum_{j = 1}^{(k + 1)_{1}} \bar{ \Delta}_{j} \bigl( u \bigl(\bar{t}_{j}^{ -} \bigr) \bigr) \end{aligned}$$

and

$$\begin{aligned}[b] {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u \bigl(t_{k + 1}^{\prime +} \bigr) ={}& {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{k + 1}^{\prime -} \bigr) + \sum_{k_{0} + 1}^{(k + 1)_{0}} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) \\ ={}& u_{1} \int_{a}^{t'_{k + 1}} \frac{ds}{s} + u_{2} + \int_{a}^{t'_{k + 1}} \ln \frac{t'_{k + 1}}{s}f\bigl(s,u(s) \bigr)\frac{ds}{s} + \sum_{i = 1}^{(k + 1)_{0}} \Delta_{i} \bigl( u\bigl(t_{i}^{ -} \bigr) \bigr) \\ &{}+ \sum_{j = 1}^{k_{1}} \bar{\Delta}_{j} \bigl( u\bigl(\bar{t}_{j}^{ -} \bigr) \bigr)\ln \frac{t'_{k + 1}}{\bar{t}_{j}}. \end{aligned} $$

Therefore, the approximate solution of system (1.1) for \(t \in (t'_{k + 1},t'_{k + 2}]\) is given by

$$\begin{aligned} \hat{u}(t) ={}& \frac{1}{\Gamma (q)} \bigl( {}_{H}D_{a^{ +}}^{q - 1}u \bigl(t_{k + 1}^{\prime +} \bigr) \bigr) \biggl( \int_{t'_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{1}{\Gamma (q - 1)} \bigl( {}_{H}\mathcal{J}_{a^{ +}}^{2 - q}u\bigl(t_{k + 1}^{\prime +} \bigr) \bigr) \biggl( \int_{t'_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &{}+ \frac{1}{\Gamma (q)} \int_{t'_{k + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ ={}& \frac{u_{1} + \int_{a}^{t'_{k + 1}} f(s,u(s))\frac{ds}{s} + \sum_{j = 1}^{(k + 1)_{1}} \bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) )}{ \Gamma (q)} \biggl( \int_{t'_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &{}+ \frac{u_{1}\ln \frac{t'_{k + 1}}{a} + u_{2} + \int_{a}^{t'_{k + 1}} \ln \frac{t'_{k + 1}}{s}f(s,u(s))\frac{ds}{s} + \sum_{i = 1}^{(k + 1)_{0}} \Delta_{i} ( u(t_{i}^{ -} ) ) + \sum_{j = 1}^{k_{1}} \bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) )\ln \frac{t'_{k + 1}}{\bar{t}_{j}}}{\Gamma (q - 1)} \\ &{}\times\biggl( \int_{t'_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 2} + \frac{1}{\Gamma (q)} \int_{t'_{k + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s}\quad \mbox{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr]. \end{aligned}$$
(3.44)

Let \(\hat{e}_{k + 1}(t) = u(t) - \hat{u}(t)\) for \(t \in (t'_{k + 1},t'_{k + 2}]\). By (3.43) the exact solution of system (1.1) satisfies

Therefore,

$$\begin{aligned} &\mathop{\lim_{\Delta_{i} ( u(t_{i}^{ -} ) ) \to 0,\bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) ) \to 0}}\limits _{\mathrm{for\ all}\ i\ \mathrm{and}\ j}\hat{e}_{k + 1}(t) \\ &\quad= \mathop{\lim_{\Delta_{i} ( u(t_{i}^{ -} ) ) \to 0,\bar{\Delta}_{j} ( u(\bar{t}_{j}^{ -} ) ) \to 0 }}\limits _{\mathrm{for\ all}\ i\ \mathrm{and}\ j} \bigl\{ u(t) - \hat{u}(t) \bigr\} \\ &\quad= \frac{u_{1}}{\Gamma (q)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 1} + \frac{u_{2}}{\Gamma (q - 1)} \biggl( \int_{a}^{t} \frac{ds}{s} \biggr)^{q - 2} + \frac{1}{\Gamma (q)} \int_{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \\ &\qquad{}- \frac{u_{1} + \int_{a}^{t'_{k + 1}} f(s,u(s))\frac{ds}{s}}{\Gamma (q)} \biggl( \int_{t'_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 1} \\ &\qquad{} - \frac{u_{1}\int_{a}^{t'_{k + 1}} \frac{ds}{s} + u_{2} + \int_{a}^{t'_{k + 1}} \ln \frac{t'_{k + 1}}{s}f(s,u(s))\frac{ds}{s}}{\Gamma (q - 1)} \biggl( \int_{t'_{k + 1}}^{t} \frac{ds}{s} \biggr)^{q - 2} \\ &\qquad{}- \frac{1}{\Gamma (q)} \int_{t'_{k + 1}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{q - 1}f\bigl(s,u(s)\bigr)\frac{ds}{s} \quad\mbox{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr], \end{aligned}$$
(3.45)
(3.46)

By (3.45) and (3.46) we have

(3.47)

Thus,

So, system (1.1) is equivalent to the integral equation (3.36). The proof is now completed. □

Corollary 3.5

Let \(q \in (1, 2)\), and let , ħ be two constants. System (1.2) is equivalent to the fractional integral equation

(3.48)

provided that the integral in (3.48) exists.

4 Example

In this section, we give an example to illustrate the usefulness of our results.

Example 1

Let us consider the general solution of the impulsive fractional system

$$ \left \{ \textstyle\begin{array}{@{}l} {}_{H}D_{1^{ +}}^{\frac{3}{2}}u(t) = \ln t, \quad t \in (1,3] \mbox{ and } t \ne 2, \\ \Delta ( {}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u ) \vert _{t = 2} = {}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u(2^{ +} ) - {}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u(2^{ -} ) = \delta, \\ \Delta ( {}_{H}D_{1^{ +}}^{\frac{1}{2}}u ) \vert _{t = 2} = {}_{H}D_{1^{ +}}^{\frac{1}{2}}u(2^{ +} ) - {}_{H}D_{1^{ +}}^{\frac{1}{2}}u(2^{ -} ) = \bar{\delta}, \\ {}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u(1^{ +} ) = u_{2},\qquad{}_{H}D_{1^{ +}}^{\frac{1}{2}}u(1^{ +} ) = u_{1}. \end{array}\displaystyle \right . $$
(4.1)

By the Theorem 3.4 the general solution is obtained by

(4.2)

Here , ħ are two constants. Next, we will verify that Eq. (4.2) satisfies all conditions of system (4.1).

Taking the Hadamard fractional derivative of the both sides of Eq. (4.2), we have

(i) for \(t \in (1,2]\),

$$\begin{aligned} &{}_{H}D_{1^{ +}}^{\frac{3}{2}}u(t) \\ &\quad= \frac{1}{\Gamma (2 - \frac{3}{2})} \biggl( t\frac{d}{dt} \biggr)^{2} \int_{1}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{2 - \frac{3}{2} - 1} \biggl[ \frac{u_{1}}{\Gamma (\frac{3}{2})} \biggl( \int_{1}^{\eta} \frac{ds}{s} \biggr)^{\frac{3}{2} - 1} + \frac{u_{2}}{\Gamma (\frac{3}{2} - 1)} \biggl( \int_{1}^{\eta} \frac{ds}{s} \biggr)^{\frac{3}{2} - 2} \\ &\qquad{}+ \frac{1}{\Gamma (\frac{3}{2})} \int_{1}^{\eta} \biggl(\ln \frac{\eta}{s} \biggr)^{\frac{3}{2} - 1}\ln s\frac{ds}{s} \biggr]\frac{d\eta}{\eta} \\ &\quad= \frac{1}{\Gamma (\frac{1}{2})\Gamma (\frac{3}{2})} \biggl( t\frac{d}{dt} \biggr)^{2} \int_{1}^{t} \biggl(\ln \frac{t}{\eta} \biggr)^{\frac{1}{2} - 1} \biggl[ \int_{1}^{\eta} \biggl(\ln \frac{\eta}{s} \biggr)^{\frac{3}{2} - 1}\ln s\frac{ds}{s} \biggr]\frac{d\eta}{\eta} = \ln t, \end{aligned}$$

(ii) for \(t \in (2,3]\),

So, Eq. (4.2) satisfies the Hadamard fractional derivative condition of system (4.1).

By Definition 2.1 we obtain

$${}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{1}\ln t + u_{2} + \int_{1}^{t} \ln \frac{t}{s}\ln s \frac{ds}{s} &\mbox{for } t \in [1,2], \\ u_{1}\ln t + u_{2} + \int_{1}^{t} \ln \frac{t}{s}\ln s \frac{ds}{s} + \delta + \bar{\delta} (\ln t - \ln 2) &\mbox{for } t \in (2,3], \end{array}\displaystyle \right . $$

and

$${}_{H}D_{1^{ +}}^{\frac{1}{2}}u(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{1} + \int_{1}^{t} \ln s \frac{ds}{s} &\mbox{for } t \in [1,2], \\ u_{1} + \int_{1}^{t} \ln s \frac{ds}{s} + \bar{\delta} &\mbox{for } t \in (2,3]. \end{array}\displaystyle \right . $$

Therefore,

$${}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u \bigl(2^{ +} \bigr) - {}_{H}\mathcal{J}_{1^{ +}}^{\frac{1}{2}}u \bigl(2^{ -} \bigr) = \delta\quad\mbox{and}\quad {}_{H}D_{1^{ +}}^{\frac{1}{2}}u \bigl(2^{ +} \bigr) - {}_{H}D_{1^{+}}^{\frac{1}{2}}u \bigl(2^{ -} \bigr) = \bar{\delta}. $$

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

Finally, it is obvious that Eq. (4.2) satisfies the three implicit conditions (i)-(iii). So, Eq. (4.2) is the general solution of system (4.1).