Theory and Modern Applications

The general solution for impulsive differential equations with Hadamard fractional derivative of order $$q \in(1, 2)$$

Abstract

In this paper, we find formulas of general solution for a kind of impulsive differential equations with Hadamard fractional derivative of order $$q \in(1, 2)$$ by analysis of the limit case (as the impulse tends to zero) and provide an example to illustrate the importance of our results.

1 Introduction

Fractional differential equations are an excellent tool in the modeling of many phenomena in various fields of science and engineering [13], and the subject of fractional differential equations is gaining much attention (see [411] and the references therein).

Recently, Hadamard fractional derivative was studied in [1215], 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 [2026]); impulsive fractional partial differential equations are also considered (see [2732]).

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).

References

1. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)

2. Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

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)

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)

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)

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)

7. Odibat, ZM: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171-1183 (2010)

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)

9. Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. TMA 72, 916-924 (2010)

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)

11. Deng, W: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. TMA 72, 1768-1777 (2010)

12. Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191-1204 (2001)

13. 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)

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)

15. 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

16. Klimek, M: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16, 4689-4697 (2011)

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)

19. Gambo, YY, Jarad, F, Baleanu, D, Abdeljawad, T: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, 10 (2014)

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)

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)

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)

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)

24. Wang, X: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383-2391 (2011)

25. Zhang, X, Zhang, X, Zhang, M: On the concept of general solution for impulsive differential equations of fractional order $$q \in (0, 1)$$. Appl. Math. Comput. 247, 72-89 (2014)

26. Stamova, I, Stamov, G: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simul. 19, 702-709 (2014)

27. Abbas, S, Benchohra, M: Impulsive hyperbolic functional differential equations of fractional order with state-dependent delay. Fract. Calc. Appl. Anal. 13, 225-242 (2010)

28. Abbas, S, Benchohra, M: Upper and lower solutions method for impulsive hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4, 406-413 (2010)

29. Abbas, S, 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)

30. Abbas, S, 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)

31. Benchohra, M, Seba, D: Impulsive partial hyperbolic fractional order differential equations in Banach spaces. J. Fract. Calc. Appl. 1(4), 1-12 (2011)

32. Guo, T, Zhang, K: Impulsive fractional partial differential equations. Appl. Math. Comput. 257, 581-590 (2015)

Acknowledgements

The authors are deeply grateful to the anonymous referees for their kind comments, correcting errors, and improving written language, which have been very useful for improving the quality of this paper. The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant Nos. 21576033, 61261046), State Key Development Program for Basic Research of Health and Family Planning Commission of Jiangxi Province China (Grant No. 20143246), the Natural Science Foundation of Jiangxi Province (Grant No. 20151BAB207013), and the Research Foundation of Education Bureau of Jiangxi Province, China (Grant No. GJJ14738).

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Zhang, X., Shu, T., Cao, H. et al. The general solution for impulsive differential equations with Hadamard fractional derivative of order $$q \in(1, 2)$$ . Adv Differ Equ 2016, 14 (2016). https://doi.org/10.1186/s13662-016-0744-3

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• DOI: https://doi.org/10.1186/s13662-016-0744-3