Skip to main content

Theory and Modern Applications

The non-uniqueness of solution for initial value problem of impulsive differential equations involving higher order Katugampola fractional derivative

Abstract

In this paper we consider the initial value problem for some impulsive differential equations with higher order Katugampola fractional derivative (fractional order \(q \in (1,2]\)). The systems of impulsive higher order fractional differential equations can involve one or two kinds of impulses, and by analyzing the error between the approximate solution and exact solution it is found that these impulsive systems are equivalent to some integral equations with one or two undetermined constants correspondingly, which uncover the non-uniqueness of solution to these impulsive systems. Some numerical examples are offered to explain the obtained results.

1 Introduction

Fractional calculus serves as an important tool to characterize hereditary properties in many fields of science and engineering (such as chaotic behavior, epidemiology, thermal science, hydrology, and biology [117]). Since fractional calculus was put forward in the seventeenth century, there have appeared several definitions of fractional derivatives: Riemann–Liouville, Caputo, Hadamard, Grunwald–Letnikov etc. [18, 19]. To unify these fractional derivatives, some generalized fractional operators (such as Erdélyi–Kober fractional operator [18], Hilfer fractional operator [20, 21], Katugampola fractional operator [22, 23], and Atangana–Baleanu fractional operator [24] etc.) were presented, and some properties of these generalized fractional operators and differential equations involving these generalized fractional derivatives were widely studied [2533]. The potential application in quantum mechanics was considered for some properties of the Katugampola fractional derivative in [34], and the existence and uniqueness of solutions was studied for fractional Langevin equation with the nonlocal Katugampola fractional integral conditions in [35].

Furthermore, impulsive differential equations are used in description of some processes with impulsive effects [36], and the subject of impulsive fractional differential equations (IFrDE) has been getting an enormous amount of attention recently [3745]. In addition, IFrDE was considered from the short memory case that addressed the nonlocality and the impulsive conditions in [46]. For the studies of IFrDE, most of them considered impulsive differential equations involving the Caputo type fractional derivative, and a few of them were concerned with impulsive non-Caputo type fractional differential equations. Therefore, we consider the equivalent integral equation for the initial value problem (IVP) of impulsive differential equations involving higher order non-Caputo type fractional derivative (in the sense of Katugampola):

$$ \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)), \\ \quad t \in (t_{0},T],t \ne t_{k}(k = 1,2, \ldots ,m) \text{ and } t \ne \bar{t}_{l}(l = 1,2, \ldots ,n), \\ {}_{t_{0}}^{K}\mathcal{D}_{t_{k} +}^{q - 1,\rho } x(t_{k} + ) - {}_{t_{0}}^{K}\mathcal{D}_{t_{k} -}^{q - 1,\rho } x(t_{k} - ) = J_{k}(x(t_{k} - )),\quad k = 1,2, \ldots ,m, \\ {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} +}^{2 - q,\rho } x(\bar{t}_{l} + ) - {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} -}^{2 - q,\rho } x( \bar{t}_{l} - ) = \bar{J}_{l}(x(\bar{t}_{l} - )),\quad l = 1,2, \ldots ,n, \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}, \end{cases} $$
(1.1)

where \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } \) (here \(q \in (1,2]\) and \(\rho > 0\)) denotes the left-sided Katugampola fractional derivative of order q. \({}_{t_{0}}^{K}\mathcal{D}_{t_{k} +} ^{q - 1,\rho } x(t_{k} + ) = \lim_{\varepsilon \to 0 +} {}_{t_{0}} ^{K}\mathcal{D}_{t_{k} + \varepsilon }^{q - 1,\rho } x(t_{k} + \varepsilon )\) and \({}_{t_{0}}^{K}\mathcal{D}_{t_{k} -} ^{q - 1,\rho } x(t_{k} - ) = \lim_{\varepsilon \to 0 -} {}_{t_{0}} ^{K}\mathcal{D}_{t_{k} + \varepsilon }^{q - 1,\rho } x(t_{k} + \varepsilon )\) represent the right and left limits of \({}_{t_{0}} ^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t)\) at \(t = t_{k}\), respectively. \({}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} +}^{2 - q, \rho } x(\bar{t}_{l} + )\) and \({}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} -}^{2 - q,\rho } x(\bar{t}_{l} - )\) denote the right and left limits of \({}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } x(t)\) at \(t = \bar{t} _{l}\), respectively. Two kinds of impulsive points satisfy \(0 \le t _{0} < t_{1} < \cdots < t_{m} < t_{m + 1} = T\) and \(t_{0} < \bar{t} _{1} < \cdots < \bar{t}_{n} < \bar{t}_{n + 1} = T\), respectively. Moreover, for these impulsive points, two assumptions are given as follows:

  1. (H1)

    Let \(\{ t_{0},t_{1},t_{2}, \ldots ,t_{m}, \bar{t}_{1},\bar{t}_{2}, \ldots ,\bar{t}_{n},T \} = \{ t_{0},t'_{1},t'_{2}, \ldots ,t'_{M},T\}\) satisfy

    $$0 \le t_{0} < t'_{1} < t'_{2} < \cdots < t'_{M} < t'_{M + 1} = T. $$
  2. (H2)

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

In particular, letting \(J_{k}(x(t_{k} - )) = 0\) (for all \(k \in \{ 1,2, \ldots ,m \} \)) and \(\bar{J}_{l}(x(\bar{t}_{l} - )) = 0\) (for all \(l \in \{ 1,2, \ldots ,n \} \)) in (1.1) respectively, we obtain two simple impulsive systems:

$$ \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T] \text{ and } t \ne t_{k} \ (k = 1,2, \ldots ,m), \\ {}_{t_{0}}^{K}\mathcal{D}_{t_{k} +}^{q - 1,\rho } x(t_{k} + ) - {}_{t_{0}}^{K}\mathcal{D}_{t_{k} -}^{q - 1,\rho } x(t_{k} - ) = J_{k}(x(t_{k} - )),\quad k = 1,2, \ldots ,m, \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2} \end{cases} $$
(1.2)

and

$$ \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T] \text{ and } t \ne \bar{t}_{l}\ (l = 1,2, \ldots ,n), \\ {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} +}^{2 - q,\rho } x(\bar{t}_{l} + ) - {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} -}^{2 - q,\rho } x( \bar{t}_{l} - ) = \bar{J}_{l}(x(\bar{t}_{l} - )),\quad l = 1,2, \ldots ,n, \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}. \end{cases} $$
(1.3)

Moreover, letting \(\{ t_{1},t_{2}, \ldots ,t_{m}\} = \{ \bar{t}_{1}, \bar{t}_{2}, \ldots ,\bar{t}_{n}\}\) in (1.1), we get the impulsive system

$$ \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T] \text{ and } t \ne t_{k}\ (k = 1,2, \ldots ,m), \\ {}_{t_{0}}^{K}\mathcal{D}_{t_{k} +}^{q - 1,\rho } x(t_{k} + ) - {}_{t_{0}}^{K}\mathcal{D}_{t_{k} -}^{q - 1,\rho } x(t_{k} - ) = J_{k}(x(t_{k} - )),\quad k = 1,2, \ldots ,m, \\ {}_{t_{0}}^{K}\mathcal{I}_{t_{k} +}^{2 - q,\rho } x(t_{k} + ) - {} _{t_{0}}^{K}\mathcal{I}_{t_{k} -}^{2 - q,\rho } x(t_{k} - ) = \bar{J} _{k}(x(t_{k} - )),\quad k = 1,2, \ldots ,m, \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}. \end{cases} $$
(1.4)

Next we introduce some basic definitions and conclusions regarding the Katugampola fractional derivative in Sect. 2 and give some properties of IFrDEs (1.1)–(1.3) in Sect. 3. Then, we seek the equivalent integral equations of IFrDEs (1.1)–(1.4) in Sect. 4. Finally, we use some numerical examples to expound the obtained results in Sect. 5.

2 Preliminaries

Let [\(a,b\)] (\(- \infty \le a < b < \infty \)) be a finite interval on the real axis R and \(C[a,b]\) be the set of continuous functions on[\(a,b\)]. Define the function space

Xcp(a,b)={x:[a,b]C:xXcp<}(cR,1p)
(2.1)

endowed with the norm \(\Vert x \Vert _{X_{c}^{p}} = ( \int _{a}^{b} \vert t^{c}x(t) \vert ^{p}\,\frac{dt}{t} )^{1/p}\) (\(1 \le p < \infty \)) and \(\Vert x \Vert _{X_{c}^{\infty }} = \operatorname{ess}\sup_{t \in [a,b]} [ t^{c} \vert x(t) \vert ]\).

Definition 2.1

([22])

The left-sided Katugampola fractional integrals of order αC (\(\Re (\alpha ) > 0\)) of function \(x \in X_{c}^{p}(a,b)\) are defined by

$$ \bigl( {}_{a}^{K}\mathcal{I}_{t}^{\alpha ,\rho } x \bigr) (t) = \frac{1}{ \varGamma (\alpha )} \int _{a}^{t} \biggl( \frac{t^{\rho } - s^{\rho }}{ \rho } \biggr)^{\alpha - 1}\frac{x(s)\,ds}{s^{1 - \rho }}\quad (t > a \ge 0). $$
(2.2)

Definition 2.2

([23])

The left-sided Katugampola fractional derivatives of order αC (\(\Re (\alpha ) > 0\)) are defined by

$$\begin{aligned}& \bigl( {}_{a}^{K}\mathcal{D}_{t}^{\alpha ,\rho } x \bigr) (t) = \gamma ^{n} \bigl( {}_{a}^{K} \mathcal{I}_{t}^{n - \alpha ,\rho } x \bigr) (t) \\& \hphantom{\bigl( {}_{a}^{K}\mathcal{D}_{t}^{\alpha ,\rho } x \bigr) (t) }= \frac{\gamma ^{n}}{\varGamma (n - \alpha )} \int _{a}^{t} \biggl( \frac{t ^{\rho } - s^{\rho }}{\rho } \biggr)^{n - \alpha - 1}\frac{x(s)\,ds}{s ^{1 - \rho }} \\& \quad\biggl(\rho > 0,t > a \ge 0,\gamma = t^{1 - \rho } \,\frac{d}{dt} \biggr). \end{aligned}$$
(2.3)

Remark 2.3

From the L’Hospital rule, we have \(\lim_{\rho \to 0 +} ( \frac{t^{\rho } - \tau ^{\rho }}{\rho } )^{q - 1} = ( \ln \frac{t}{\tau } )^{q - 1}\). The Katugampola fractional operators with \(\rho \to 0 +\) and \(\rho = 1\) are the Hadamard fractional operator and the RiemannLiouville fractional operator, respectively.

For \(n - 1 < \alpha \le n\) (nN), a weighted space of continuous functions is defined by

$$\begin{aligned}& C_{n - \alpha ,\rho } [a,b] = \bigl\{ x(t): \bigl(t^{\rho } - a^{\rho } \bigr)^{n - \alpha } x(t) \in C[a,b], \Vert x \Vert _{C_{n - \alpha , \rho }} = \bigl\Vert \bigl(t^{\rho } - a^{\rho } \bigr)^{n - \alpha } x(t) \bigr\Vert _{C} \bigr\} \\& \quad (\rho \ne 0) \end{aligned}$$
(2.4)

and

$$\begin{aligned} C_{n - \alpha ,0}[a,b] =& \bigl\{ x(t):(\ln t - \ln a)^{n - \alpha } x(t) \in C[a,b], \\ &{}\Vert x \Vert _{C_{n - \alpha ,0}} = \bigl\Vert (\ln t - \ln a)^{n - \alpha } x(t) \bigr\Vert _{C} \bigr\} . \end{aligned}$$
(2.5)

Moreover, let

$$ C_{n - \alpha ,\rho }^{\alpha } [a,b] = \bigl\{ x(t) \in C_{n - \alpha ,\rho } [a,b]:{}_{a}^{K}\mathcal{D}_{t}^{\alpha , \rho } x(t) \in C_{n - \alpha ,\rho } [a,b] \bigr\} $$
(2.6)

and

$$ C_{2 - \alpha ,\rho }^{2}[a,T] = \biggl\{ x(t) \in C[a,b]:\gamma ^{2}x(t) \in C_{2 - \alpha ,\rho } [a,b],\gamma = t ^{1 - \rho } \,\frac{d}{dt} \biggr\} . $$
(2.7)

Lemma 2.4

Let\(q \in (1,2]\)and\(a,\rho > 0\), and letf:[a,T]×RRbe a function such that\(f( \cdot ,x( \cdot )) \in C_{2 - q,\rho } [a,T]\)for any\(x( \cdot ) \in C_{2 - q,\rho } [a,T]\).

If\(x( \cdot ) \in C_{2 - q,\rho }^{q}[a,T]\), then\(x(t)\)is a solution of the fractional differential equation

$$ \textstyle\begin{cases} {}_{a}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (a,T], \\ {}_{a}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to a +} = x_{1}, \qquad {}_{a}^{K}\mathcal{I}_{t}^{2 - q,\rho } x(t) |_{t \to a +} = x_{2} \end{cases} $$
(2.8)

if, and only if, \(x(t)\)satisfies the integral equation

$$\begin{aligned}& x(t) = \frac{x_{2}}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - a^{ \rho }}{\rho } \biggr]^{q - 2} + \frac{x_{1}}{\varGamma (q)} \biggl[ \frac{t ^{\rho } - a^{\rho }}{\rho } \biggr]^{q - 1} \\& \hphantom{x(t) =}{} + \frac{1}{\varGamma (q)} \int _{a}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr] ^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \\& \quad \textit{for } t \in (a,T] \textit{ and } f = f \bigl(\tau ,x(\tau ) \bigr). \end{aligned}$$
(2.9)

Proof

First, we prove the necessity. Let \(x(t) \in C_{2 - q, \rho }^{q}[a,T]\) be a solution of (2.8). By the hypotheses \(x(t) \in C_{2 - q,\rho }^{q}[a,T]\) and \({}_{a}^{K}\mathcal{D}_{t} ^{q,\rho } x(t) = \gamma ^{2} ( {}_{a}^{K}\mathcal{I}_{t}^{2 - q, \rho } x )(t)\), we have \({}_{a}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) \in C[a,T]\) and \({}_{a}^{K}\mathcal{I}_{t}^{2 - q,\rho } x \in C_{2 - q,\rho }^{2}[a,T]\). Therefore, by (2.8), we get

$${}_{a}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = \biggl( t^{1 - \rho } \,\frac{d}{dt} \biggr)^{2} \bigl( {}_{a}^{K}\mathcal{I}_{t}^{2 - q,\rho } x(t) \bigr) = f \bigl(t,x(t) \bigr). $$

Therefore

$$ {}_{a}^{K}\mathcal{I}_{t}^{2 - q,\rho } x(t) = x_{2} + \frac{t^{ \rho } - a^{\rho }}{\rho } x_{1} + \int _{a}^{t} \frac{t^{\rho } - \tau ^{\rho }}{\rho } \frac{f\,d\tau }{\tau ^{1 - \rho }}. $$
(2.10)

Applying the operator \({}_{a}^{K}\mathcal{I}_{t}^{q,\rho } \) to two sides of (2.10), we have

$$\begin{aligned} {}_{a}^{K}\mathcal{I}_{t}^{2,\rho } x(t) =& \frac{x_{2}}{\varGamma (q + 1)} \biggl[ \frac{t^{\rho } - a^{\rho }}{\rho } \biggr]^{q} + \frac{x _{1}}{\varGamma (q + 2)} \biggl[ \frac{t^{\rho } - a^{\rho }}{\rho } \biggr] ^{q + 1} \\ &{}+ \frac{1}{\varGamma (q + 2)} \int _{a}^{t} \biggl[ \frac{t^{ \rho } - \tau ^{\rho }}{\rho } \biggr]^{q + 1}\frac{f\,d\tau }{ \tau ^{1 - \rho }}. \end{aligned}$$
(2.11)

Using the operator \(\gamma ^{2}\) to two sides of (2.11), we obtain

$$\begin{aligned}& \begin{aligned} x(t) ={}& \frac{x_{2}}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - a^{ \rho }}{\rho } \biggr]^{q - 2} + \frac{x_{1}}{\varGamma (q)} \biggl[ \frac{t ^{\rho } - a^{\rho }}{\rho } \biggr]^{q - 1} \\ &{}+ \frac{1}{\varGamma (q)} \int _{a}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr] ^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \end{aligned} \\& \quad \text{for } t \in (a,T]. \end{aligned}$$

Now we prove the sufficiency. Let \(x(t) \in C_{2 - q,\rho }^{q}[a,T]\) satisfy Eq. (2.9), which can be written as (2.9). Moreover, by the hypotheses of Lemma 2.4, for any \(x( \cdot ) \in C_{2 - q, \rho } [a,T]\), we have \(f( \cdot ,x( \cdot )) \in C_{2 - q,\rho } [a,T]\). Applying the operators \({}_{a}^{K}\mathcal{D}_{t}^{q, \rho } \), \({}_{a}^{K}\mathcal{D}_{t}^{q - 1,\rho } \), and \({}_{a}^{K}\mathcal{I}_{t}^{2 - q,\rho } \) to both sides of (2.9), respectively, we obtain

$$\begin{aligned}& \begin{aligned} {}_{a}^{K} \mathcal{D}_{t}^{q,\rho } x(t) ={}& {}_{a}^{K} \mathcal{D}_{t}^{q,\rho } \biggl\{ \frac{x_{2}}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - a^{\rho }}{\rho } \biggr]^{q - 2} + \frac{x_{1}}{ \varGamma (q)} \biggl[ \frac{t^{\rho } - a^{\rho }}{\rho } \biggr]^{q - 1}\\ &{} + \frac{1}{\varGamma (q)} \int _{a}^{t} \biggl[ \frac{t^{\rho } - \tau ^{ \rho }}{\rho } \biggr]^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \biggr\} \\ ={}& f \bigl(t,x(t) \bigr)\quad \text{for } t \in (a,T], \end{aligned} \\& {}_{a}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) = x_{1} + \int _{a}^{t} \frac{f\,d\tau }{\tau ^{1 - \rho }} \quad \text{for } t \in (a,T], \end{aligned}$$

and

$${}_{a}^{K}\mathcal{I}_{t}^{2 - q,\rho } x(t) = x_{2} + \frac{t^{ \rho } - a^{\rho }}{\rho } x_{1} + \int _{a}^{t} \frac{t^{\rho } - \tau ^{\rho }}{\rho } \frac{f\,d\tau }{\tau ^{1 - \rho }} \quad \text{for } t \in (a,T]. $$

By the hypothesis \(f( \cdot ,x( \cdot )) \in C_{2 - q,\rho } [a,T]\), we have \((\tau ^{\rho } - a^{\rho } )^{2 - q}f(\tau ,x(\tau )) \in C[a,T]\). Therefore \(\vert (\tau ^{\rho } - a^{\rho } )^{2 - q}f \vert \le L\) (here L is a positive constant) and

$$\begin{aligned} \biggl\vert \int _{a}^{t} \frac{f\,d\tau }{\tau ^{1 - \rho }} \biggr\vert \le& \int _{a}^{t} \bigl\vert \bigl(\tau ^{\rho } - a^{\rho } \bigr)^{q - 2} \bigl[ \bigl( \tau ^{\rho } - a^{\rho } \bigr)^{2 - q}f \bigr] \bigr\vert \,\frac{d\tau ^{ \rho }}{\rho } \\ \le& \frac{L(t^{\rho } - a^{\rho } )^{q - 1}}{(q - 1) \rho }, \end{aligned}$$

and

$$\begin{aligned} \biggl\vert \int _{a}^{t} \frac{t^{\rho } - \tau ^{\rho }}{\rho } \frac{f\,d \tau }{\tau ^{1 - \rho }} \biggr\vert \le& \int _{a}^{t} \biggl\vert \frac{t ^{\rho } - \tau ^{\rho }}{\rho } \biggl[ \frac{\tau ^{\rho } - a^{ \rho }}{\rho } \biggr]^{q - 2} \bigl[ \bigl(\tau ^{\rho } - a^{\rho } \bigr)^{2 - q}f \bigr] \biggr\vert \,\frac{d\tau ^{\rho }}{\rho ^{3 - q}} \\ \le& \frac{LB(2,q - 1)}{\rho ^{2 - q}} \biggl[ \frac{t^{\rho } - a^{ \rho }}{\rho } \biggr]^{q}. \end{aligned}$$

Thus \({}_{a}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) \vert _{t \to a +} = x_{1}\) and \({}_{a}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) \vert _{t \to a +} = x_{2}\). The proof is completed. □

3 Some properties of (1.1)–(1.3)

In this section, we give some properties of three impulsive systems (1.1)–(1.3):

$$\begin{aligned}& \begin{aligned} \mbox{(i)}\quad& \lim_{\substack{J_{k}(x(t_{k} - )) \to 0 \text{ for all } k \in \{ 1,2, \ldots ,m\} \\ \bar{J}_{l}(x(\bar{t}_{l} - )) \to 0 \text{ for all } l \in \{ 1,2, \ldots ,n\}}} \bigl\{ \mbox{system (1.1)} \bigr\} \\ &\quad = \lim_{J_{k}(x(t_{k} - )) \to 0 \text{ for all } k \in \{ 1,2, \ldots ,m\}} \bigl\{ \mbox{system (1.2)} \bigr\} \\ &\quad = \lim_{\bar{J}_{l}(x(\bar{t}_{l} - )) \to 0 \text{ for all } l \in \{ 1,2, \ldots ,n\}} \bigl\{ \mbox{system (1.3)} \bigr\} \\ &\quad = \textstyle\begin{cases} {}_{a}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T], \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}. \end{cases}\displaystyle \\ &\quad \Leftrightarrow\quad x(t) = \frac{x_{2}}{\varGamma (q - 1)} \biggl[ \frac{t ^{\rho } - (t_{0})^{\rho }}{\rho } \biggr]^{q - 2} + \frac{x_{1}}{ \varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{0})^{\rho }}{\rho } \biggr] ^{q - 1} \\ &\hphantom{\quad \Leftrightarrow\quad x(t) =}{}+ \frac{1}{\varGamma (q)} \int _{t_{0}}^{t} \biggl[ \frac{t^{ \rho } - \tau ^{\rho }}{\rho } \biggr]^{q - 1}\frac{f\,d\tau }{ \tau ^{1 - \rho }}, \\ &\quad \text{for } t \in (t_{0},T] \text{ and } f = f \bigl(\tau ,x(\tau ) \bigr). \end{aligned} \\ & (\mbox{ii})\quad \lim_{\bar{J}_{l}(x(\bar{t}_{l} - )) \to 0 \text{ for all } l \in \{ 1,2, \ldots ,n\}} \bigl\{ \mbox{system (1.1)} \bigr\} = \bigl\{ \mbox{system (1.2)} \bigr\} . \\ & (\mbox{iii}) \lim_{J_{k}(x(t_{k} - )) \to 0 \text{ for all } k \in \{ 1,2, \ldots ,m\}} \bigl\{ \mbox{system (1.1)} \bigr\} = \bigl\{ \mbox{system (1.3)} \bigr\} . \\ & \begin{aligned} (\mbox{iv})\quad & \lim_{\substack{t_{k} \to t_{p} \text{ for all } k \in \{ 1,2, \ldots ,m\} \text{ and }\forall p \in \{ 1,2, \ldots ,m\}, \\ \bar{t}_{l} \to \bar{t}_{r} \text{ for all } l \in \{ 1,2, \ldots ,n\} \text{ and }\forall r \in \{ 1,2, \ldots ,n\}}} \bigl\{ \mbox{system (1.1)} \bigr\} \\ &\quad = \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T],t \ne t_{p}\text{ and } t \ne \bar{t}_{r}, \\ {}_{t_{0}}^{K}\mathcal{D}_{t_{p} +}^{q - 1,\rho } x(t_{p} + ) - {}_{t_{0}}^{K}\mathcal{D}_{t_{p} -}^{q - 1,\rho } x(t_{p} - ) = \sum_{k = 1}^{m} J_{k}(x(t_{p} - )), \\ {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{r} +}^{2 - q,\rho } x(\bar{t}_{r} + ) - {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{r} -}^{2 - q,\rho } x( \bar{t}_{r} - ) = \sum_{l = 1}^{n} \bar{J}_{l}(x(\bar{t}_{r} - )), \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}. \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned} (\mbox{v})\quad & \lim_{t_{k} \to t_{p} \text{ for all } k \in \{ 1,2, \ldots ,m \} \text{ and } \forall p \in \{ 1,2, \ldots ,m\}} \bigl\{ \mbox{system (1.2)} \bigr\} \\ &\quad = \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T] \text{ and } t \ne t_{p}, \\ {}_{t_{0}}^{K}\mathcal{D}_{t_{p} +}^{q - 1,\rho } x(t_{p} + ) - {}_{t_{0}}^{K}\mathcal{D}_{t_{p} -}^{q - 1,\rho } x(t_{p} - ) = \sum_{k = 1}^{m} J_{k}(x(t_{p} - )), \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}. \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned} (\mbox{vi})& \lim_{\bar{t}_{l} \to \bar{t}_{r} \text{ for all } l \in \{ 1,2, \ldots ,n\} \text{ and }\forall r \in \{ 1,2, \ldots ,n\}} \bigl\{ \mbox{system (1.3)} \bigr\} \\ &\quad = \textstyle\begin{cases} {}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t)),\quad t \in (t_{0},T], t \ne \bar{t}_{r}, \\ {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{r} +}^{2 - q,\rho } x(\bar{t}_{r} + ) - {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{r} -}^{2 - q,\rho } x( \bar{t}_{r} - ) = \sum_{l = 1}^{n} \bar{J}_{l}(x(\bar{t}_{r} - )), \\ {}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \to t_{0} +} = x_{1},\qquad {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q, \rho } x(t) |_{t \to t_{0} +} = x_{2}. \end{cases}\displaystyle \end{aligned} \end{aligned}$$

4 The equivalent integral equations of (1.1)–(1.4)

For simplicity, let \(f = f(\tau ,x(\tau ))\) and

$$\begin{aligned}& y(\ell ,t) = \frac{x_{2} + x_{1}\frac{\ell ^{\rho } - (t_{0})^{\rho }}{ \rho } + \int _{t_{0}}^{\ell } \frac{\ell ^{\rho } - \tau ^{\rho }}{ \rho } \frac{f\,d\tau }{\tau ^{1 - \rho }}}{\varGamma (q - 1)} \biggl[ \frac{t ^{\rho } - \ell ^{\rho }}{\rho } \biggr]^{q - 2} + \frac{x_{1} + \int _{t_{0}}^{\ell } \frac{f\,d\tau }{\tau ^{1 - \rho }}}{\varGamma (q)} \biggl[ \frac{t^{\rho } - \ell ^{\rho }}{\rho } \biggr]^{q - 1} \\& \hphantom{ y(\ell ,t) =}{}+ \frac{1}{\varGamma (q)} \int _{\ell }^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }}, \\& \quad \mbox{here } \ell \in \{ t_{0},t_{1},t_{2}, \ldots ,t_{m}, \bar{t}_{1},\bar{t}_{2}, \ldots , \bar{t}_{n},T \} . \end{aligned}$$
(4.1)

For \(1 < q \le 2\), define some function spaces:

Cˆ2q,ρ[t0,T]={x:(t0,T]R:[tρ(ti)ρ]2qx(t)C[ti,ti+1],i=0,1,,M}(ρ0),Cˆ2q,ρ[t0,T]={x:(t0,T]R:[lntln(ti)]2qx(t)C[ti,ti+1],i=0,1,,M}(ρ=0),Cˆ2q,ρq[t0,T]={x(t)Cˆ2q,ρ[t0,T]:t0KDtq,ρx(t)Cˆ2q,ρ[t0,T]},IC([t0,T],R)={x(t)Cˆ2q,ρ[t0,T]:Dtkq1,ρt0Kx(tk)=limttkt0KDtq1,ρx(t)=t0KDtkq1,ρx(tk)<and t0KDtk+q1,ρx(tk+)=limttk+t0KDtq1,ρx(t)<(here k=1,2,,m),and t0KIt¯l2q,ρx(t¯l)=limtt¯lt0KIt2q,ρx(t)=t0KIt¯l2q,ρx(t¯l)< andIt¯l+2q,ρt0Kx(t¯l+)=limtt¯l+t0KIt2q,ρx(t)<(here l=1,2,,n)};C˜2q,ρ[t0,T]={x:(t0,T]R:[tρ(ti)ρ]2qx(t)C[ti,ti+1],i=0,1,,m}(ρ0),C˜2q,ρ[t0,T]={x:(t0,T]R:[lntln(ti)]2qx(t)C[ti,ti+1],i=0,1,,m}(ρ=0),C˜2q,ρq[t0,T]={x(t)C˜2q,ρ[t0,T]:t0KDtq,ρx(t)C˜2q,ρ[t0,T]},IC1([t0,T],R)={x(t)C˜2q,ρ[t0,T]:t0KDtk+q1,ρx(tk+)=limttk+t0KDtq1,ρx(t)<,Dtkq1,ρt0Kx(tk)=limttkt0KDtq1,ρx(t)=t0KDtkq1,ρx(tk)<,and t0KItk+2q,ρx(tk+)=t0KItk2q,ρx(tk), here k=1,2,,m};C¯2q,ρ[t0,T]={x:(t0,T]R:[tρ(t¯j)ρ]2qx(t)C[t¯j,t¯j+1],j=0,1,,n}(ρ0),C¯2q,ρ[t0,T]={x:(t0,T]R:[lntln(t¯j)]2qx(t)C[t¯j,t¯j+1],j=0,1,,n}(ρ=0),C¯2q,ρq[t0,T]={x(t)C¯2q,ρ[t0,T]:t0KDtq,ρx(t)C¯2q,ρ[t0,T]},IC2([t0,T],R)={x(t)C¯2q,ρ[t0,T]:t0KIt¯l+2q,ρx(t¯l+)=limtt¯l+t0KIt2q,ρx(t)<,It¯l2q,ρt0Kx(t¯l)=limtt¯lt0KIt2q,ρx(t)=t0KIt¯l2q,ρx(t¯l)<and t0KDt¯l+q1,ρx(t¯l+)=t0KDt¯lq1,ρx(t¯l), here l=1,2,,n}.

Next, we seek the equivalent integral equation of (1.2). Considering \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } x(t) = f(t,x(t))\) on each piecewise interval (\(t_{k},t_{k + 1}\)] (\(k = 1,2, \ldots ,m\)) by Lemma 2.4, we find a piecewise function

$$ \tilde{x}(t) = \textstyle\begin{cases} y(t_{0},t) \quad \text{for } t \in (t_{0},t_{1}], \\ \frac{{}_{t_{0}}^{K}\mathcal{I}_{t_{k} +}^{2 - q,\rho } x(t_{k} + )}{ \varGamma (q - 1)} [ \frac{t^{\rho } - (t_{k})^{\rho }}{\rho } ] ^{q - 2} + \frac{{}_{t_{0}}^{K}\mathcal{D}_{t_{k} +}^{q - 1, \rho } x(t_{k} + )}{\varGamma (q)} [ \frac{t^{\rho } - (t_{k})^{ \rho }}{\rho } ]^{q - 1} \\ \quad {}+ \frac{1}{\varGamma (q)}\int _{t_{k}}^{t} [ \frac{t^{\rho } - \tau ^{\rho }}{\rho } ]^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \quad \text{for } t \in (t_{k},t_{k + 1}],k = 1,2, \ldots ,m, \end{cases} $$
(4.2)

with \({}_{t_{0}}^{K}\mathcal{D}_{t_{k} +}^{q - 1,\rho } x(t _{k} + ) = {}_{t_{0}}^{K}\mathcal{D}_{t_{k} -}^{q - 1,\rho } x(t_{k} - ) + J_{k}(x(t_{k} - ))\) and \({}_{t_{0}}^{K}\mathcal{I}_{t _{k} +}^{2 - q,\rho } x(t_{k} + ) = {}_{t_{0}}^{K}\mathcal{I}_{t_{k} -} ^{2 - q,\rho } x(t_{k} - )\).

Because (4.2) does not satisfy property (i), \(\tilde{x}(t)\) is only considered as an approximate solution of (1.2). And let

$$ e_{k}(t) = x(t) - \tilde{x}(t),\quad \text{for } t \in (t_{k},t_{k + 1}]\ (k=1,2, \ldots ,m), $$
(4.3)

where \(x(t)\) represents the exact solution of (1.2).

Lemma 4.1

Let\(q \in (1,2]\)and\(t_{0},\rho > 0\), and letf:[t0,T]×RRbe a function such that\(f( \cdot ,x( \cdot )) \in \tilde{C}_{2 - q,\rho } [t_{0},T]\)for any\(x( \cdot ) \in \tilde{C}_{2 - q,\rho } [t_{0},T]\).

Ifx()IC1([t0,T],R), then\(x(t)\)is a solution of (1.2) if, and only if, \(x(t)\)satisfies

$$ x(t) = \textstyle\begin{cases} y(t_{0},t)\quad \textit{for } t \in (t_{0},t_{1}], \\ y(t_{0},t) + \sum_{i = 1}^{k} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} [ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } ]^{q - 1} + \xi \sum_{i = 1}^{k} J_{i}(x(t_{i} - )) [ y(t_{i},t) - y(t_{0},t) ] \\ \quad \textit{for } t \in (t_{k},t_{k + 1}], k = 1,2, \ldots ,m, \end{cases} $$
(4.4)

whereξis an arbitrary constant.

Proof

First, we prove the necessity by applying mathematical induction. By Lemma 2.4, the solution of (1.2) as \(t \in (t _{0},t_{1}]\) satisfies

$$ x(t) = y(t_{0},t)\quad \text{for } t \in (t_{0},t_{1}]. $$
(4.5)

Using two operators \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1, \rho } \) and \({}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } \) to two sides of (4.5), respectively, we have

$$\begin{aligned} {}_{t_{0}}^{K}\mathcal{D}_{t_{1} +}^{q - 1,\rho } x(t_{1} + ) =& {}_{t_{0}}^{K} \mathcal{D}_{t_{1} -}^{q - 1,\rho } x(t_{1} - ) + J_{1} \bigl(x(t_{1} - ) \bigr) \\ =& x_{1} + \int _{t_{0}}^{t_{1}} \frac{f\,d\tau }{\tau ^{1 - \rho }} + J _{1} \bigl(x(t_{1} - ) \bigr) \end{aligned}$$
(4.6)

and

$$\begin{aligned} {}_{t_{0}}^{K}\mathcal{I}_{t_{1} +}^{2 - q,\rho } x(t_{1} + ) =& {} _{t_{0}}^{K} \mathcal{I}_{t_{1} -}^{2 - q,\rho } x(t_{1} - ) \\ =& x_{2} + x_{1}\frac{(t_{1})^{\rho } - (t_{0})^{\rho }}{\rho } + \int _{t_{0}}^{t_{1}} \frac{(t_{1})^{\rho } - \tau ^{\rho }}{\rho } \frac{f\,d \tau }{\tau ^{1 - \rho }}. \end{aligned}$$
(4.7)

Substituting (4.6)–(4.7) into (4.2), the approximate solution of (1.2) as \(t \in (t_{1},t_{2}]\) is given as

$$ \tilde{x}(t) = y(t_{1},t) + \frac{J_{1}(x(t_{1} - ))}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{1})^{\rho }}{\rho } \biggr]^{q - 1}\quad \text{for } t \in (t_{1},t_{2}]. $$
(4.8)

By (4.5) the exact solution of (1.2) as \(t \in (t_{1},t_{2}]\) satisfies

$$ \lim_{J_{1}(x(t_{1} - )) \to 0}x(t) = y(t_{0},t)\quad \text{for } t \in (t _{1},t_{2}]. $$
(4.9)

By (4.3) and (4.8)–(4.9), we get

$$ \lim_{J_{1}(x(t_{1} - )) \to 0}e_{1}(t) = y(t_{0},t) - y(t_{1},t)\quad \text{for } t \in (t_{1},t_{2}]. $$
(4.10)

From (4.10), let \(e_{1}(t) = \kappa (J_{1}(x(t_{1}^{ -} ))) \lim_{J_{1}(x(t_{1}^{ -} )) \to 0}e_{1}(t)\), where \(\kappa ( \cdot )\) is an undetermined function to satisfy \(\kappa (0) = 1\), and

$$ e_{1}(t) = \kappa \bigl( J_{1} \bigl(x(t_{1} - ) \bigr) \bigr) \lim_{J_{1}(x(t_{1} - )) \to 0}e_{1}(t) = - \kappa \bigl( J_{1} \bigl(x(t _{1} - ) \bigr) \bigr) \bigl[ y(t_{1},t) - y(t_{0},t) \bigr]. $$
(4.11)

Plugging (4.8) and (4.11) into (4.3), we obtain

$$\begin{aligned}& x(t) = y(t_{0},t) + \frac{J_{1}(x(t_{1} - ))}{\varGamma (q)} \biggl[ \frac{t ^{\rho } - (t_{1})^{\rho }}{\rho } \biggr]^{q - 1} + \bigl[ 1 - \kappa \bigl(J_{1} \bigl(x(t_{1} - ) \bigr) \bigr) \bigr] \bigl[ y(t_{1},t) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(4.12)

Because \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q,0 +} \) (\({}_{t_{0}} ^{K}\mathcal{D}_{t}^{q,\rho } \) with \(\rho \to 0 +\)) is the Hadamard fractional derivative, we get \(1 - \kappa ( J_{1}(x(t _{1}^{ -} )) ) = \xi J_{1}(x(t_{1}^{ -} ))\) (here ξ is an arbitrary constant) by applying Lemma 3.3 in [44] to (1.2) and (4.12) with \(\rho \to 0 +\). Thus (4.12) is rewritten as

$$\begin{aligned}& x(t) = y(t_{0},t) + \frac{J_{1}(x(t_{1} - ))}{\varGamma (q)} \biggl[ \frac{t ^{\rho } - (t_{1})^{\rho }}{\rho } \biggr]^{q - 1} + \xi J_{1} \bigl(x(t _{1} - ) \bigr) \bigl[ y(t_{1},t) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in (t_{1},t_{2}]. \end{aligned}$$
(4.13)

Therefore the solution of (1.2) satisfies (4.4) as \(t \in (t_{1},t_{2}]\).

For \(t \in (t_{k},t_{k + 1}]\), suppose that the solution of (1.2) satisfies

$$\begin{aligned}& x(t) = y(t_{0},t) + \sum_{i = 1}^{k} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr]^{q - 1} + \xi \sum_{i = 1}^{k} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in (t_{k},t_{k + 1}] \end{aligned}$$
(4.14)

to prove that the solution of (1.2) satisfies (4.4) as \(t \in (t_{k + 1},t_{k + 2}]\).

Using operators \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } \) and \({}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } \) to two sides of (4.14) respectively, we obtain

$$\begin{aligned} {}_{t_{0}}^{K}\mathcal{D}_{t_{k + 1} +}^{q - 1,\rho } x(t _{k + 1} + ) =& {}_{t_{0}}^{K} \mathcal{D}_{t_{k + 1} -}^{q - 1, \rho } x(t_{k + 1} - ) + J_{k + 1} \bigl(x(t_{k + 1} - ) \bigr) \\ =& x_{1} + \int _{t_{0}}^{t_{k + 1}} \frac{f\,d\tau }{\tau ^{1 - \rho }} + \sum _{i = 1}^{k + 1} J_{i} \bigl(x(t_{i} - ) \bigr) \end{aligned}$$
(4.15)

and

$$\begin{aligned} {}_{t_{0}}^{K}\mathcal{I}_{t_{k + 1} +}^{2 - q,\rho } x(t_{k + 1} + ) =& {}_{t_{0}}^{K} \mathcal{I}_{t_{k + 1} -}^{2 - q,\rho } x(t_{k + 1} - ) \\ =& x_{2} + x_{1}\frac{(t_{k + 1})^{\rho } - (t_{0})^{\rho }}{\rho } + \int _{t_{0}}^{t_{k + 1}} \frac{(t_{k + 1})^{\rho } - \tau ^{\rho }}{ \rho } \frac{f\,d\tau }{\tau ^{1 - \rho }} \\ &{}+ \sum_{i = 1}^{k} J_{i} \bigl(x(t_{i} - ) \bigr)\frac{(t_{k + 1})^{\rho } - (t _{i})^{\rho }}{\rho }. \end{aligned}$$
(4.16)

Plugging (4.15) and (4.16) into (4.2), the approximate solution of (1.2) as \(t \in (t_{k + 1},t_{k + 2}]\) is given by

$$\begin{aligned} \tilde{x}(t) =& y(t_{k + 1},t) + \frac{\sum_{i = 1}^{k} J_{i}(x(t_{i} - ))\frac{(t_{k + 1})^{\rho } - (t_{i})^{\rho }}{\rho }}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (t_{k + 1})^{\rho }}{\rho } \biggr]^{q - 2} \\ &{}+ \frac{\sum_{i = 1}^{k + 1} J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl[ \frac{t ^{\rho } - (t_{k + 1})^{\rho }}{\rho } \biggr]^{q - 1} \quad \text{for } t \in (t_{k + 1},t_{k + 2}]. \end{aligned}$$
(4.17)

On the other hand, by (4.14) the exact solution of (1.2) as \(t \in (t_{k + 1},t_{k + 2}]\) satisfies

$$ \lim_{J_{i}(x(t_{i} - )) \to 0 \text{ for all } i \in \{ 1,2, \ldots ,k + 1\}} x(t) = y(t_{0},t) \quad \text{for } t \in (t_{k + 1},t_{k + 2}] $$
(4.18)

and

$$\begin{aligned} \lim_{\substack{J_{p}(x(t_{p} - )) \to 0 \text{ here}\\ p \in \{ 1,2, \ldots ,k + 1\}}} x(t) =& y(t_{0},t) + \sum _{\substack{1 \le i \le k + 1\\ \text{and } i \ne p}} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr]^{q - 1} \\ &{}+ \xi \sum_{\substack{1 \le i \le k + 1 \\ \text{and } i \ne p}} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t _{0},t) \bigr] \quad \text{for } t \in (t_{k + 1},t_{k + 2}]. \end{aligned}$$
(4.19)

By (4.3) and (4.17)–(4.19), we have

$$ \lim_{\substack{J_{i}(x(t_{i} - )) \to 0 \text{ for} \\ \mathrm{all} i \in \{ 1,2, \ldots ,k + 1\}}} e_{k + 1}(t) = \lim _{\substack{J_{i}(x(t_{i} - )) \to 0 \text{ for}\\ \text{all } i \in \{ 1,2, \ldots ,k + 1\}}} \bigl\{ x(t) - \tilde{x}(t) \bigr\} = - \bigl[ y(t_{k + 1},t) - y(t_{0},t) \bigr] $$
(4.20)

and

$$\begin{aligned} \lim_{\substack{J_{p}(x(t_{p} - )) \to 0 \text{ here}\\ p \in \{ 1,2, \ldots ,k + 1\}}} e_{k + 1}(t) =& \lim _{\substack{J_{p}(x(t_{p} - )) \to 0 \text{ here} \\ p \in \{ 1,2, \ldots ,k + 1\}}} \bigl\{ x(t) - \tilde{x}(t) \bigr\} \\ =& - \bigl[ y(t_{k + 1},t) - y(t_{0},t) \bigr] + \xi \sum _{\substack{1 \le i \le k + 1 \\ \text{and }i \ne p}} J_{i} \bigl(x(t _{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] \\ &{}+ \sum_{\substack{ 1 \le i \le k + 1 \\ \text{and } i \ne p}} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl\{ \biggl[ \frac{t ^{\rho } - (t_{i})^{\rho }}{\rho } \biggr]^{q - 1} - \biggl[ \frac{t ^{\rho } - (t_{k + 1})^{\rho }}{\rho } \biggr]^{q - 1} \biggr\} \\ &{}- \sum_{\substack{1 \le i \le k + 1 \\ \text{and } i \ne p}} \frac{J_{i}(x(t_{i} - ))\frac{(t_{k + 1})^{\rho } - (t_{i})^{ \rho }}{\rho }}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (t_{k + 1})^{ \rho }}{\rho } \biggr]^{q - 2}. \end{aligned}$$
(4.21)

By (4.20) and (4.21), we obtain

$$\begin{aligned} e_{k + 1}(t) =& - \bigl[ y(t_{k + 1},t) - y(t_{0},t) \bigr] + \xi \sum_{i = 1}^{k + 1} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t _{0},t) \bigr] \\ &{}+ \sum_{i = 1}^{k + 1} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl\{ \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr]^{q - 1} - \biggl[ \frac{t^{\rho } - (t_{k + 1})^{\rho }}{\rho } \biggr]^{q - 1} \biggr\} \\ &{}- \sum_{i = 1}^{k + 1} \frac{J_{i}(x(t_{i} - ))\frac{(t_{k + 1})^{ \rho } - (t_{i})^{\rho }}{\rho }}{\varGamma (q - 1)} \biggl[ \frac{t^{ \rho } - (t_{k + 1})^{\rho }}{\rho } \biggr]^{q - 2}. \end{aligned}$$
(4.22)

Thus, substituting (4.17) and (4.22) into (4.3), we get

$$\begin{aligned}& x(t) = y(t_{0},t) + \sum_{i = 1}^{k + 1} \frac{J_{i}(x(t_{i} - ))}{ \varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr] ^{q - 1} + \xi \sum_{i = 1}^{k + 1} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in (t_{k + 1},t_{k + 2}]. \end{aligned}$$

Therefore the solution of (1.2) satisfies (4.4) as \(t \in (t_{k + 1},t _{k + 2}]\). Hence the necessity is proved.

Now we prove the sufficiency. Applying the operators \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q,\rho } \), \({}_{t_{0}}^{K}\mathcal{D}_{t} ^{q - 1,\rho } \), and \({}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } \) to two sides of (4.4) as \(t \in (t_{k},t_{k + 1}]\), respectively, we have

$$\begin{aligned}& {}_{t_{0}}^{K} \mathcal{D}_{t}^{q,\rho } x(t) | _{t \in (t_{k},t_{k + 1}]} \\& \quad = \Biggl\{ f \bigl(t,x(t) \bigr) |_{t \ge t_{0}}+ \xi \sum _{i = 1}^{k} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ f \bigl(t,x(t) \bigr) |_{t \ge t_{i}} - f \bigl(t,x(t) \bigr) |_{t \ge t_{0}} \bigr] \Biggr\} _{t \in (t_{k},t_{k + 1}]} \\& \quad = f \bigl(t,x(t) \bigr) |_{t \in (t_{k},t_{k + 1}]}, \\& {}_{t_{0}}^{K} \mathcal{D}_{t}^{q - 1,\rho } x(t) | _{t \in (t_{k},t_{k + 1}]} \\& \quad = \Biggl\{ x_{1} + \int _{t_{0}}^{t} \frac{f\,d \tau }{\tau ^{1 - \rho }} + \sum _{i = 1}^{k} J_{i} \bigl(x(t_{i} - ) \bigr) \\& \qquad {} + \sum_{i = 1}^{k} \xi J_{i} \bigl(x(t_{i} - ) \bigr) \biggl[ x_{1} + \int _{t_{0}}^{t_{i}} \frac{f\,d\tau }{\tau ^{1 - \rho }} + \int _{t_{i}} ^{t} \frac{f\,d\tau }{\tau ^{1 - \rho }} - x_{1} - \int _{t_{0}}^{t} \frac{f\,d \tau }{\tau ^{1 - \rho }} \biggr] \Biggr\} _{t \in (t_{k},t_{k + 1}]} \\& \quad = \Biggl\{ x_{1} + \int _{t_{0}}^{t} \frac{f\,d\tau }{\tau ^{1 - \rho }} + \sum _{i = 1}^{k} J_{i} \bigl(x(t_{i} - ) \bigr) \Biggr\} _{t \in (t_{k},t_{k + 1}]}, \end{aligned}$$

and

$$\begin{aligned}& \begin{gathered} {}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } x(t) |_{t \in (t_{k},t_{k + 1}]} \\ \quad = \Biggl\{ x_{2} + x_{1} \frac{t^{\rho } - (t _{0})^{\rho }}{\rho } + \int _{t_{0}}^{t} \frac{t^{\rho } - \tau ^{ \rho }}{\rho } \frac{f\,d\tau }{\tau ^{1 - \rho }} + \sum_{i = 1}^{k} J _{i} \bigl(x(t_{i} - ) \bigr)\frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \Biggr\} _{t \in (t_{k},t_{k + 1}]}. \end{gathered} \end{aligned}$$

Thus \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } x(t) \vert _{t \to t_{0} +} = x_{1}\), \({}_{t_{0}}^{K}\mathcal{I} _{t}^{2 - q,\rho } x(t) \vert _{t \to t_{0} +} = x_{2}\), \({}_{t_{0}} ^{K}\mathcal{D}_{t_{k} +}^{q - 1,\rho } x(t_{k} + ) - {}_{t _{0}}^{K}\mathcal{D}_{t_{k} -}^{q - 1,\rho } x(t_{k} - ) = J _{k}(x(t_{k} - ))\), and \({}_{t_{0}}^{K}\mathcal{I}_{t_{k} +}^{2 - q, \rho } x(t_{k} + ) = {}_{t_{0}}^{K}\mathcal{I}_{t_{k} -}^{2 - q, \rho } x(t_{k} - )\), and (4.4) satisfies the condition of fractional derivative in (1.2).

Letting \(J_{k}(x(t_{k} - )) = 0\) for all \(k \in \{ 1,2, \ldots ,m \} \) in (4.3), we obtain

$$\begin{aligned}& \lim_{J_{k}(x(t_{k} - )) \to 0 \text{ for all } k \in \{ 1,2,\ldots ,m\}} \bigl\{ \text{Eq.~(4.3)} \bigr\} \mbox{ is equivalent to}\\& \quad \lim_{J_{k}(x(t_{k} - )) \to 0 \text{ for all } k \in \{ 1,2,\ldots ,m\}} \bigl\{ \text{system (1.2)} \bigr\} . \end{aligned}$$

Moreover, it is obvious that (4.4) satisfies condition (v). Therefore (4.4) satisfies all the conditions of (1.2). Hence, this proof is completed. □

Remark 4.2

Similar to (4.2), an approximate solution of (1.3) is presented by

$$ \tilde{\tilde{x}}(t) = \textstyle\begin{cases} y(t_{0},t)\quad \text{for } t \in (t_{0},\bar{t}_{1}], \\ \frac{{}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} +}^{2 - q,\rho } x( \bar{t}_{l} + )}{\varGamma (q - 1)} [ \frac{t^{\rho } - (\bar{t} _{l})^{\rho }}{\rho } ]^{q - 2} + \frac{{}_{t_{0}}^{K}\mathcal{D}_{\bar{t}_{l} +}^{q - 1,\rho } x(\bar{t}_{l} + )}{\varGamma (q)} [ \frac{t^{\rho } - (\bar{t}_{l})^{\rho }}{\rho } ]^{q - 1} \\ \quad {}+ \frac{1}{\varGamma (q)}\int _{\bar{t}_{l}}^{t} [ \frac{t^{\rho } - \tau ^{\rho }}{\rho } ]^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \quad \text{for } t \in (\bar{t}_{l},\bar{t}_{l + 1}],l = 1,2, \ldots ,n, \end{cases} $$
(4.23)

with \({}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} +}^{2 - q,\rho } x( \bar{t}_{l} + ) = {}_{t_{0}}^{K}\mathcal{I}_{\bar{t}_{l} -}^{2 - q, \rho } x(\bar{t}_{l} - ) + \bar{J}_{l}(x(\bar{t}_{l} - ))\) and \({}_{t_{0}}^{K}\mathcal{D}_{\bar{t}_{l} +}^{q - 1,\rho } x( \bar{t}_{l} + ) = {}_{t_{0}}^{K}\mathcal{D}_{\bar{t}_{l} -} ^{q - 1,\rho } x(\bar{t}_{l} - )\).

Furthermore, using the thought of Lemma 4.1, we arrive at the following conclusion.

Lemma 4.3

Let\(q \in (1,2]\)and\(t_{0},\rho > 0\), and letf:[t0,T]×RRbe a function such that\(f( \cdot ,x( \cdot )) \in \bar{C}_{2 - q,\rho } [t_{0},T]\)for any\(x( \cdot ) \in \bar{C}_{2 - q,\rho } [t_{0},T]\).

Ifx()IC2([t0,T],R), then\(x(t)\)is a solution of (1.3) if, and only if, \(x(t)\)satisfies the following integral equation:

$$ x(t) = \textstyle\begin{cases} y(t_{0},t)\quad \textit{for } t \in (t_{0},\bar{t}_{1}], \\ y(t_{0},t) + \sum_{j = 1}^{l} \frac{\bar{J}_{j}(x(\bar{t}_{j} - ))}{ \varGamma (q - 1)} [ \frac{t^{\rho } - (\bar{t}_{j})^{\rho }}{ \rho } ]^{q - 2} + \eta \sum_{j = 1}^{l} \bar{J}_{j}(x(\bar{t} _{j} - )) [ y(\bar{t}_{j},t) - y(t_{0},t) ] \\ \quad \textit{for } t \in (\bar{t}_{l},\bar{t}_{l + 1}],l = 1,2, \ldots ,n, \end{cases} $$
(4.24)

whereηis an arbitrary constant.

The following theorem yields the equivalence between Cauchy problem (1.1) and the Volterra integral equation of the second kind:

$$ x(t) = \textstyle\begin{cases} y(t_{0},t)\quad \text{for } t \in (t_{0},t'_{1}], \\ y(t_{0},t) + \sum_{i = 1}^{k_{1}} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} [ \frac{t^{\rho } - (t_{i})^{ \rho }}{\rho } ]^{q - 1} + \sum_{j = 1}^{k_{2}} \frac{\bar{J} _{j}(x(\bar{t}_{j} - ))}{\varGamma (q - 1)} [ \frac{t^{\rho } - ( \bar{t}_{j})^{\rho }}{\rho } ]^{q - 2} \\ \qquad {}+ \xi \sum_{i = 1}^{k_{1}} J_{i}(x(t_{i} - )) [ y(t_{i},t) - y(t _{0},t) ]\\ \qquad {} + \eta \sum_{j = 1}^{k_{2}} \bar{J}_{j}(x(\bar{t} _{j} - )) [ y(\bar{t}_{j},t) - y(t_{0},t) ] \\ \quad \text{for } t \in (t'_{k},t'_{k + 1}], \end{cases} $$
(4.25)

where ξ and η are two arbitrary constants.

Theorem 4.4

Let\(q \in (1,2]\)and\(t_{0},\rho > 0\), and letf:[a,T]×RRbe a function such that\(f( \cdot ,x( \cdot )) \in \hat{C}_{2 - q,\rho } [t_{0},T]\)for any\(x( \cdot ) \in \hat{C}_{2 - q,\rho } [a,T]\).

Ifx()IC([t0,T],R), then\(x(t)\)is a solution of (1.1) if, and only if, \(x(t)\)satisfies (4.25).

Proof

First, we prove the necessity that the solution of (1.1) satisfies (4.25) by the mathematical induction. For \(t \in (t_{0},t'_{1}]\), by Lemma 2.4, the solution of system (1.1) satisfies (4.25) and

$$ x(t) = y(t_{0},t)\quad \text{for } t \in \bigl(t_{0},t'_{1}\bigr]. $$
(4.26)

For \(t \in (t'_{1},t'_{2}]\), there appear 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 \(t'_{1} = t_{1} < \bar{t}_{1}\) and \(t'_{1} = \bar{t}_{1} < t_{1}\), the solution of (1.1) satisfies (4.25) as \(t \in (t'_{1},t'_{2}]\) by Lemmas 4.1 and 4.3, respectively. Hence, we need only prove that the solution of (1.1) satisfies (4.25) as \(t \in (t'_{1},t'_{2}]\) with \(t'_{1} = t_{1} = \bar{t}_{1}\). Applying \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1, \rho } \) and \({}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } \) to two sides of (4.26), we have

$$ {}_{t_{0}}^{K}\mathcal{D}_{t'_{1} +}^{q - 1,\rho } x \bigl(t'_{1} + \bigr) = {}_{t_{0}}^{K} \mathcal{D}_{t'_{1} -}^{q - 1,\rho } x \bigl(t'_{1} - \bigr) + J_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) = x_{1} + \int _{t_{0}}^{t'_{1}} \frac{f\,d \tau }{\tau ^{1 - \rho }} + J_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) $$
(4.27)

and

$$\begin{aligned} {}_{t_{0}}^{K}\mathcal{I}_{t'_{1} +}^{2 - q,\rho } x \bigl(t'_{1} - \bigr) =& {}_{t_{0}}^{K} \mathcal{I}_{t'_{1} -}^{2 - q,\rho } x \bigl(t'_{1} - \bigr) + \bar{J}_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) \\ =& x_{2} + x_{1}\frac{(t'_{1})^{\rho } - (t_{0})^{\rho }}{\rho } + \int _{t_{0}}^{t'_{1}} \frac{(t'_{1})^{\rho } - \tau ^{\rho }}{\rho } \frac{f\,d \tau }{\tau ^{1 - \rho }} + \bar{J}_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr). \end{aligned}$$
(4.28)

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

$$\begin{aligned} \hat{x}(t) =& \frac{{}_{t_{0}}^{K}\mathcal{I}_{t'_{1} +}^{2 - q,\rho } x(t'_{1} + )}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (t_{1})^{ \rho }}{\rho } \biggr]^{q - 2} + \frac{{}_{t_{0}}^{K}\mathcal{D}_{t'_{1} +}^{q - 1,\rho } x(t'_{1} + )}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t'_{1})^{\rho }}{\rho } \biggr]^{q - 1} \\ &{}+ \frac{1}{\varGamma (q)} \int _{t'_{1}}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \quad \text{for } t \in \bigl(t'_{1},t'_{2}\bigr] \\ =& y \bigl(t'_{1},t \bigr) + \frac{ \bar{J}_{1}(x(t'_{1} - ))}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (t_{1})^{\rho }}{\rho } \biggr]^{q - 2} + \frac{J _{1}(x(t'_{1} - ))}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t'_{1})^{ \rho }}{\rho } \biggr]^{q - 1} \\ &\text{for } t \in \bigl(t'_{1},t'_{2}\bigr], \end{aligned}$$
(4.29)

with the error \(\hat{e}_{1}(t) = x(t) - \hat{x}(t)\) for \(t \in (t'_{1},t'_{2}]\), where \(x(t)\) is the exact solution of (1.1). Moreover, by Lemmas 4.1 and 4.3, the exact solution \(x(t)\) of (1.1) as \(t \in (t'_{1},t'_{2}]\) satisfies three conditions:

$$\begin{aligned}& \lim_{J_{1}(x(t'_{1} - )) \to 0,\bar{J}_{1}(x(t'_{1} - )) \to 0}x(t) = y(t_{0},t) \quad \text{for } t \in (t'_{1},t'_{2}], \end{aligned}$$
(4.30)
$$\begin{aligned}& \lim_{\bar{J}_{1}(x(t'_{1} - )) \to 0}x(t) = y(t_{0},t) + \frac{J_{1}(x(t'_{1} - ))}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t'_{1})^{ \rho }}{\rho } \biggr]^{q - 1} + \xi J_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) \bigl[ y \bigl(t'_{1},t \bigr) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in \bigl(t'_{1},t'_{2}\bigr], \end{aligned}$$
(4.31)
$$\begin{aligned}& \lim_{J_{1}(x(t'_{1} - )) \to 0}x(t) = y(t_{0},t) + \frac{\bar{J}_{1}(x(t'_{1} - ))}{\varGamma (q - 1)} \biggl[ \frac{t^{ \rho } - (t'_{1})^{\rho }}{\rho } \biggr]^{q - 2} + \eta \bar{J}_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) \bigl[ y \bigl(t'_{1},t \bigr) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in \bigl(t'_{1},t'_{2}\bigr]. \end{aligned}$$
(4.32)

By (4.29)–(4.32), we get

$$\begin{aligned}& \lim_{\substack{J_{1}(x(t'_{1} - )) \to 0, \\ \bar{J}_{1}(x(t'_{1} - )) \to 0}}\hat{e}_{1}(t) = y(t_{0},t) - y \bigl(t'_{1},t \bigr), \end{aligned}$$
(4.33)
$$\begin{aligned}& \lim_{\bar{J}_{1}(x(t'_{1} - )) \to 0}\hat{e}_{1}(t) = \bigl[ \xi J _{1} \bigl(x \bigl(t'_{1}{ -} \bigr) \bigr) - 1 \bigr] \bigl[ y \bigl(t'_{1},t \bigr) - y(t_{0},t) \bigr], \end{aligned}$$
(4.34)
$$\begin{aligned}& \lim_{J_{1}(x(t'_{1} - )) \to 0}\hat{e}_{1}(t) = \bigl[ \eta \bar{J} _{1} \bigl(x \bigl(t'_{1}{ -} \bigr) \bigr) - 1 \bigr] \bigl[ y \bigl(t'_{1},t \bigr) - y(t_{0},t) \bigr]. \end{aligned}$$
(4.35)

By (4.33)–(4.35), we obtain

$$ \hat{e}_{1}(t) = \bigl[ \xi J_{1} \bigl(x \bigl(t'_{1}{ -} \bigr) \bigr) + \eta \bar{J}_{1} \bigl(x \bigl(t'_{1} { -} \bigr) \bigr) - 1 \bigr] \bigl[ y \bigl(t'_{1},t \bigr) - y(t_{0},t) \bigr]\quad \text{for } t \in \bigl(t'_{1},t'_{2}\bigr]. $$
(4.36)

By (4.29) and (4.36), we have

$$ \begin{aligned}[b] x(t) ={}& \hat{x}(t) + \hat{e}_{1}(t) \\ ={}& y(t_{0},t) + \frac{J_{1}(x(t'_{1} - ))}{\varGamma (q)} \biggl[ \frac{t ^{\rho } - (t'_{1})^{\rho }}{\rho } \biggr]^{q - 1} + \frac{\bar{J} _{1}(x(t'_{1} - ))}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (t_{1})^{ \rho }}{\rho } \biggr]^{q - 2} \\ &{}+ \bigl[ \xi J_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) + \eta \bar{J}_{1} \bigl(x \bigl(t'_{1} - \bigr) \bigr) \bigr] \bigl[ y \bigl(t'_{1},t \bigr) - y(t_{0},t) \bigr] \quad \text{for } t \in \bigl(t'_{1},t'_{2}\bigr]. \end{aligned} $$
(4.37)

Therefore the solution of (1.1) satisfies (4.25) as \(t \in (t'_{1},t'_{2}]\).

Next, for \(t \in (t'_{k},t'_{k + 1}]\) (\(k \in \{ 1,2, \ldots ,M\}\)), suppose that the solution of (1.1) satisfies

$$ \begin{aligned}[b] &x(t) = y(t_{0},t) + \sum _{i = 1}^{k_{1}} \frac{J_{i}(x(t_{i} - ))}{ \varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr] ^{q - 1} + \sum _{j = 1}^{k_{2}} \frac{\bar{J}_{j}(x(\bar{t}_{j} - ))}{ \varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (\bar{t}_{j})^{\rho }}{ \rho } \biggr]^{q - 2} \\ &\hphantom{x(t) =}{}+ \xi \sum_{i = 1}^{k_{1}} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t _{0},t) \bigr] + \eta \sum _{j = 1}^{k_{2}} \bar{J}_{j} \bigl(x( \bar{t} _{j} - ) \bigr) \bigl[ y(\bar{t}_{j},t) - y(t_{0},t) \bigr] \\ &\quad \text{for } t \in \bigl(t'_{k},t'_{k + 1}\bigr]. \end{aligned} $$
(4.38)

Using \({}_{t_{0}}^{K}\mathcal{D}_{t}^{q - 1,\rho } \) and \({}_{t_{0}}^{K}\mathcal{I}_{t}^{2 - q,\rho } \) to two sides of (4.38) respectively, we get

$$ \begin{aligned}[b] {}_{t_{0}}^{K} \mathcal{D}_{t'_{k + 1} +}^{q - 1,\rho } x \bigl(t'_{k + 1} + \bigr) &= {}_{t_{0}}^{K}\mathcal{D}_{t'_{k + 1} -}^{q - 1, \rho } x \bigl(t'_{k + 1} - \bigr) + \sum _{i = k_{1} + 1}^{(k + 1)_{1}} J_{i} \bigl(x(t _{i} - ) \bigr) \\ &= x_{1} + \int _{t_{0}}^{t'_{k + 1}} \frac{f\,d\tau }{\tau ^{1 - \rho }} + \sum _{i = 1}^{(k + 1)_{1}} J_{i} \bigl(x(t_{i} - ) \bigr) \end{aligned} $$
(4.39)

and

$$ \begin{aligned}[b] {}_{t_{0}}^{K} \mathcal{I}_{t'_{k + 1} +}^{2 - q,\rho } x \bigl(t'_{k + 1} + \bigr) ={}& {}_{t_{0}}^{K}\mathcal{I}_{t'_{k + 1}{ -}}^{2 - q,\rho } x \bigl(t'_{k + 1} - \bigr) + \sum _{j = k_{2} + 1}^{(k + 1)_{2}} \bar{J}_{j} \bigl(x( \bar{t} _{j} - ) \bigr) \\ ={}& x_{2} + x_{1}\frac{(t'_{k + 1})^{\rho } - (t_{0})^{\rho }}{\rho } + \int _{t_{0}}^{t'_{k + 1}} \frac{(t'_{k + 1})^{\rho } - \tau ^{\rho }}{ \rho } \frac{f\,d\tau }{\tau ^{1 - \rho }} \\ &{}+ \sum_{j = 1}^{(k + 1)_{2}} \bar{J}_{j} \bigl(x(\bar{t}_{j} - ) \bigr) + \sum _{i = 1}^{k_{1}} J_{i} \bigl(x(t_{i} - ) \bigr)\frac{(t'_{k + 1})^{\rho } - (t _{i})^{\rho }}{\rho }. \end{aligned} $$
(4.40)

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

$$\begin{aligned} \tilde{x}(t) =& \frac{{}_{t_{0}}^{K}\mathcal{I}_{t'_{k + 1} +}^{2 - q, \rho } x(t'_{k + 1} + )}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr]^{q - 2} + \frac{{}_{t_{0}}^{K}\mathcal{D}_{t'_{k + 1} +}^{q - 1,\rho } x(t'_{k + 1} + )}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr]^{q - 1} \\ &{}+ \frac{1}{\varGamma (q)} \int _{t'_{k + 1}}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{q - 1}\frac{f\,d\tau }{\tau ^{1 - \rho }} \quad \text{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr] \\ =& y \bigl(t'_{k + 1},t \bigr) + \frac{\sum_{j = 1}^{(k + 1)_{2}} \bar{J}_{j}(x( \bar{t}_{j} - )) + \sum_{i = 1}^{k_{1}} J_{i}(x(t_{i} - ))\frac{(t'_{k + 1})^{\rho } - (t_{i})^{\rho }}{\rho }}{\varGamma (q - 1)} \biggl[ \frac{t ^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr]^{q - 2} \\ &{}+ \frac{\sum_{i = 1}^{(k + 1)_{1}} J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr]^{q - 1} \quad \text{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr], \end{aligned}$$
(4.41)

with \(\hat{e}_{k + 1}(t) = x(t) - \hat{x}(t)\) for \(t \in (t'_{k + 1},t'_{k + 2}]\), where \(x(t)\) is the exact solution of (1.1). By (4.38), the exact solution of (1.1) satisfies

$$\begin{aligned}& \lim_{\substack{J_{i}(x(t_{i} - )) \to 0,\bar{J}_{j}(x(\bar{t}_{j} - )) \to 0 \\ \text{for all } i \text{ and } j}}x(t) = y(t_{0},t) \quad \text{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr], \end{aligned}$$
(4.42)
$$\begin{aligned}& \lim_{\substack{ J_{i}(x(t_{i} - )) \to 0 \text{ for all } i \in \{ l_{1} + 1,l_{1} + 2, \ldots ,(l + 1)_{1} \} , \\ \bar{J}_{j}(x(\bar{t}_{j} - )) \to 0 \text{ for all } j \in \{ l_{2} + 1,l_{2} + 2, \ldots ,(l + 1)_{2} \} }}x(t) \\& \quad = y(t_{0},t) + \sum _{\substack{1 \le i \le (k + 1)_{1} \text{ and}\\ i \notin \{ l_{1} + 1,l_{1} + 2, \ldots ,(l + 1)_{1} \} }} \frac{J_{i}(x(t_{i} - ))}{ \varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr] ^{q - 1} \\& \qquad {}+ y(t_{0},t) + \sum_{\substack{1 \le j \le (k + 1)_{2} \text{ and}\\ j \notin \{ l_{2} + 1,l_{2} + 2, \ldots ,(l + 1)_{2} \} }} \frac{\bar{J}_{j}(x(\bar{t}_{j} - ))}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (\bar{t}_{j})^{\rho }}{\rho } \biggr]^{q - 2} \\& \qquad {}+ \xi \sum_{\substack{1 \le i \le (k + 1)_{1} \text{ and}\\ i \notin \{ l_{1} + 1,l_{1} + 2, \ldots ,(l + 1)_{1} \} }} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] \\& \qquad {}+ \eta \sum_{\substack{1 \le j \le (k + 1)_{2} \text{ and}\\ j \notin \{ l_{2} + 1,l_{2} + 2, \ldots ,(l + 1)_{2} \} }} \bar{J}_{j} \bigl(x(\bar{t}_{j} - ) \bigr) \bigl[ y( \bar{t}_{j},t) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr],l = 1,2, \ldots ,k + 1. \end{aligned}$$
(4.43)

By (4.41)–(4.43), we obtain

$$\begin{aligned}& \lim_{\substack{J_{i}(x(t_{i} - )) \to 0,\bar{J}_{j}(x(\bar{t}_{j} - )) \to 0 \\ \text{for all } i \text{ and } j}}\hat{e}_{k + 1}(t) = - \bigl[ y \bigl(t'_{k + 1},t \bigr) - y(t_{0},t) \bigr], \end{aligned}$$
(4.44)
$$\begin{aligned}& \lim_{\substack{J_{i}(x(t_{i} - )) \to 0 \text{ for all } i \in \{ l_{1} + 1,l_{1} + 2, \ldots ,(l + 1)_{1} \} , \\ \bar{J}_{j}(x(\bar{t}_{j} - )) \to 0 \text{ for all } j \in \{ l_{2} + 1,l_{2} + 2, \ldots ,(l + 1)_{2} \} }}\hat{e}_{k + 1}(t) \\& \quad = - \bigl[ y \bigl(t'_{k + 1},t \bigr) - y(t_{0},t) \bigr] + \xi \sum_{\substack{ 1 \le i \le (k + 1)_{1} \text{ and}\\ i \notin \{ l_{1} + 1, \ldots ,(l + 1)_{1} \} }} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] \\& \qquad {}+ \eta \sum_{\substack{ 1 \le j \le (k + 1)_{2} \text{ and}\\ j \notin \{ l_{2} + 1, \ldots ,(l + 1)_{2} \} }} \bar{J}_{j} \bigl(x(\bar{t}_{j} - ) \bigr) \bigl[ y(\bar{t}_{j},t) - y(t_{0},t) \bigr] \\& \qquad {}+ \sum_{\substack{ 1 \le i \le (k + 1)_{1} \text{ and}\\ i \notin \{ l_{1} + 1,l_{1} + 2, \ldots ,(l + 1)_{1} \} }} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl\{ \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr]^{q - 1} - \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr]^{q - 1} \biggr\} \\& \qquad {}+ \sum_{\substack{ 1 \le j \le (k + 1)_{2} \text{ and}\\ j \notin \{ l_{2} + 1,l_{2} + 2, \ldots ,(l + 1)_{2} \} }} \frac{\bar{J}_{j}(x(\bar{t}_{j} - ))}{\varGamma (q - 1)} \biggl\{ \biggl[ \frac{t^{\rho } - (\bar{t}_{j})^{\rho }}{\rho } \biggr] ^{q - 2} - \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr] ^{q - 2} \biggr\} \\& \qquad {}- \sum_{\substack{ 1 \le i \le k_{1} \text{ and}\\ i \notin \{ l_{1} + 1,l_{1} + 2, \ldots ,(l + 1)_{1} \} }} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q - 1)} \frac{(t'_{k + 1})^{\rho } - (t_{i})^{\rho }}{\rho } \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{ \rho }}{\rho } \biggr]^{q - 2} \\& \quad \text{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr],l = 1,2, \ldots ,k + 1. \end{aligned}$$
(4.45)

By (4.44) and (4.45), we have

$$\begin{aligned} \hat{e}_{k + 1}(t) =& - \bigl[ y \bigl(t'_{k + 1},t \bigr) - y(t_{0},t) \bigr] - \sum_{i = 1}^{k_{1}} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q - 1)}\frac{(t'_{k + 1})^{\rho } - (t_{i})^{\rho }}{\rho } \biggl[ \frac{t ^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr]^{q - 2} \\ &{}+ \xi \sum_{i = 1}^{(k + 1)_{1}} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] \\ &{}+ \eta \sum _{j = 1}^{(k + 1)_{2}} \bar{J}_{j} \bigl(x( \bar{t}_{j} - ) \bigr) \bigl[ y(\bar{t}_{j},t) - y(t_{0},t) \bigr] \\ &{}+ \sum_{i = 1}^{(k + 1)_{1}} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} \biggl\{ \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr] ^{q - 1} - \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{\rho }}{\rho } \biggr] ^{q - 1} \biggr\} \\ &{}+ \sum_{j = 1}^{(k + 1)_{2}} \frac{\bar{J}_{j}(x(\bar{t}_{j} - ))}{ \varGamma (q - 1)} \biggl\{ \biggl[ \frac{t^{\rho } - (\bar{t}_{j})^{ \rho }}{\rho } \biggr]^{q - 2} - \biggl[ \frac{t^{\rho } - (t'_{k + 1})^{ \rho }}{\rho } \biggr]^{q - 2} \biggr\} . \end{aligned}$$
(4.46)

By (4.41) and (4.46), we get

$$\begin{aligned}& x(t) = \hat{x}(t) + \hat{e}_{k + 1}(t) \\& \hphantom{x(t) }= y(t_{0},t) + \sum_{i = 1}^{(k + 1)_{1}} \frac{J_{i}(x(t_{i} - ))}{ \varGamma (q)} \biggl[ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } \biggr] ^{q - 1} + \sum_{j = 1}^{(k + 1)_{2}} \frac{\bar{J}_{j}(x(\bar{t}_{j} - ))}{\varGamma (q - 1)} \biggl[ \frac{t^{\rho } - (\bar{t}_{j})^{\rho }}{ \rho } \biggr]^{q - 2} \\& \hphantom{x(t) =}{}+ \xi \sum_{i = 1}^{(k + 1)_{1}} J_{i} \bigl(x(t_{i} - ) \bigr) \bigl[ y(t_{i},t) - y(t_{0},t) \bigr] + \eta \sum _{j = 1}^{(k + 1)_{2}} \bar{J}_{j} \bigl(x( \bar{t}_{j} - ) \bigr) \bigl[ y(\bar{t}_{j},t) - y(t_{0},t) \bigr] \\& \quad \text{for } t \in \bigl(t'_{k + 1},t'_{k + 2}\bigr]. \end{aligned}$$
(4.47)

Thus the solution of (1.1) satisfies (4.25) as \(t \in (t'_{k + 1},t'_{k + 2}]\), and the necessity is proved.

Now we verify the sufficiency that (4.25) satisfies all the conditions of system (1.1). It is easy to find that (4.25) satisfies conditions (i)–(iv) by Lemmas 4.1 and 4.3, and it is similar with the proof of Lemma 4.1 to verify that (4.25) satisfies the condition of generalized fractional derivative, impulsive conditions, and initial conditions in (1.1). The proof is completed. □

Corollary 4.5

Let\(q \in (1,2]\)and\(t_{0},\rho > 0\), and letf:[t0,T]×RRbe a function such that\(f( \cdot ,x( \cdot )) \in \hat{C}_{2 - q,\rho } [t_{0},T]\)for any\(x( \cdot ) \in \hat{C}_{2 - q,\rho } [t_{0},T]\).

Ifx()IC([t0,T],R), then\(x(t)\)is a solution of (1.4) if, and only if, \(x(t)\)satisfies the following integral equation:

$$ x(t) = \textstyle\begin{cases} y(t_{0},t)\quad \textit{for } t \in (t_{0},t_{1}], \\ y(t_{0},t) + \sum_{i = 1}^{k} \frac{J_{i}(x(t_{i} - ))}{\varGamma (q)} [ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } ]^{q - 1} + \sum_{i = 1}^{k} \frac{\bar{J}_{i}(x(t_{i} - ))}{\varGamma (q - 1)} [ \frac{t^{\rho } - (t_{i})^{\rho }}{\rho } ]^{q - 2} \\ \qquad {}+ \sum_{i = 1}^{k} [ \xi J_{i}(x(t_{i} - )) + \eta \bar{J}_{i}(x(t _{i} - )) ] [ y(t_{i},t) - y(t_{0},t) ] \\ \quad \textit{for } t \in (t_{k},t_{k + 1}],k = 1,2, \ldots ,m, \end{cases} $$
(4.48)

whereξandηare two arbitrary constants.

5 Examples

In this section, we consider the following IVP of three IFrDEs:

$$\begin{aligned}& \textstyle\begin{cases} {}_{1}^{K}\mathcal{D}_{t}^{\frac{3}{2},\rho } x(t) = x(t),\quad t \in (1,5],t \ne 3 \\ {}_{1}^{K}\mathcal{D}_{3^{ +}}^{\frac{1}{2},\rho } x(3^{ +} ) - {}_{1}^{K}\mathcal{D}_{3^{ -}}^{\frac{1}{2},\rho } x(3^{ -} ) = 1, \\ {}_{1}^{K}\mathcal{D}_{t}^{\frac{1}{2},\rho } x(t) | _{t \to 1 +} = 1,\qquad {}_{1}^{K}\mathcal{I}_{t}^{\frac{1}{2}, \rho } x(t) |_{t \to 1 +} = 0, \end{cases}\displaystyle \end{aligned}$$
(5.1)
$$\begin{aligned}& \textstyle\begin{cases} {}_{1}^{K}\mathcal{D}_{t}^{\frac{3}{2},\rho } x(t) = x(t),\quad t \in (1,5],t \ne 3 \\ {}_{1}^{K}\mathcal{I}_{3^{ +}}^{\frac{1}{2},\rho } x(3^{ +} ) - {} _{1}^{K}\mathcal{I}_{3^{ -}}^{\frac{1}{2},\rho } x(3^{ -} ) = 1, \\ {}_{1}^{K}\mathcal{D}_{t}^{\frac{1}{2},\rho } x(t) | _{t \to 1 +} = 1,\qquad {}_{1}^{K}\mathcal{I}_{t}^{\frac{1}{2}, \rho } x(t) |_{t \to 1 +} = 0, \end{cases}\displaystyle \end{aligned}$$
(5.2)
$$\begin{aligned}& \textstyle\begin{cases} {}_{1}^{K}\mathcal{D}_{t}^{\frac{3}{2},\rho } x(t) = x(t),\quad t \in (1,5],t \ne 3 \\ {}_{1}^{K}\mathcal{D}_{3^{ +}}^{\frac{1}{2},\rho } x(3^{ +} ) - {}_{1}^{K}\mathcal{D}_{3^{ -}}^{\frac{1}{2},\rho } x(3^{ -} ) = 1, \\ {}_{1}^{K}\mathcal{I}_{3^{ +}}^{\frac{1}{2},\rho } x(3^{ +} ) - {} _{1}^{K}\mathcal{I}_{3^{ -}}^{\frac{1}{2},\rho } x(3^{ -} ) = 1, \\ {}_{1}^{K}\mathcal{D}_{t}^{\frac{1}{2},\rho } x(t) | _{t \to 1 +} = 1,\qquad {}_{1}^{K}\mathcal{I}_{t}^{\frac{1}{2}, \rho } x(t) |_{t \to 1 +} = 0. \end{cases}\displaystyle \end{aligned}$$
(5.3)

By Lemma 4.1, Lemma 4.3, and Corollary 4.5, the equivalent integral equations of three systems (5.1)–(5.3) as \(t \in (1,3]\) are identical as follows:

$$ x(t) = \frac{2}{\sqrt{\pi }} \biggl[ \frac{t^{\rho } - 1}{\rho } \biggr] ^{\frac{1}{2}} + \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t ^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} \quad \text{for } t \in (1,3], $$
(5.4)

and the equivalent integral equations of three systems (5.1)–(5.3) as \(t \in (3,5]\) are respectively given by

$$\begin{aligned}& x(t) = \frac{2}{\sqrt{\pi }} \biggl[ \frac{t^{\rho } - 1}{\rho } \biggr] ^{\frac{1}{2}} + \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t ^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} + \frac{2}{\sqrt{\pi }} \biggl[ \frac{t ^{\rho } - 3^{\rho }}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{}+ \xi \biggl\{ \frac{\frac{3^{\rho } - 1}{\rho } + \int _{1}^{3} \frac{3^{ \rho } - \tau ^{\rho }}{\rho } \frac{x(\tau )\,d\tau }{\tau ^{1 - \rho }}}{\sqrt{ \pi }} \biggl[ \frac{t^{\rho } - 3^{\rho }}{\rho } \biggr]^{ - \frac{1}{2}} + \frac{1 + \int _{1}^{3} \frac{x(\tau )\,d\tau }{ \tau ^{1 - \rho }}}{\sqrt{\pi } /2} \biggl[ \frac{t^{\rho } - 3^{ \rho }}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{}+ \frac{2}{\sqrt{\pi }} \int _{3}^{t} \biggl[ \frac{t^{ \rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} - \frac{2}{\sqrt{\pi }} \biggl[ \frac{t ^{\rho } - 1}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{}- \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr] ^{\frac{1}{2}}\frac{x(\tau )\,d\tau }{\tau ^{1 - \rho }} \biggr\} \\& \quad \text{for } t \in (3,5], \end{aligned}$$
(5.5)
$$\begin{aligned}& x(t) = \frac{2}{\sqrt{\pi }} \biggl[ \frac{t^{\rho } - 1}{\rho } \biggr] ^{\frac{1}{2}} + \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t ^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} + \frac{1}{\sqrt{\pi }} \biggl[ \frac{t ^{\rho } - 3^{\rho }}{\rho } \biggr]^{ - \frac{1}{2}} \\& \hphantom{x(t) =}{}+ \eta \biggl\{ \frac{\frac{3^{\rho } - 1}{\rho } + \int _{1}^{3} \frac{3^{ \rho } - \tau ^{\rho }}{\rho } \frac{x(\tau )\,d\tau }{\tau ^{1 - \rho }}}{\sqrt{ \pi }} \biggl[ \frac{t^{\rho } - 3^{\rho }}{\rho } \biggr]^{ - \frac{1}{2}} + \frac{1 + \int _{1}^{3} \frac{x(\tau )\,d\tau }{ \tau ^{1 - \rho }}}{\sqrt{\pi } /2} \biggl[ \frac{t^{\rho } - 3^{ \rho }}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{} + \frac{2}{\sqrt{\pi }} \int _{3}^{t} \biggl[ \frac{t^{ \rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} - \frac{2}{\sqrt{\pi }} \biggl[ \frac{t ^{\rho } - 1}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{} - \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr] ^{\frac{1}{2}}\frac{x(\tau )\,d\tau }{\tau ^{1 - \rho }} \biggr\} \\& \quad \text{for } t \in (3,5], \end{aligned}$$
(5.6)
$$\begin{aligned}& x(t) = \frac{2}{\sqrt{\pi }} \biggl[ \frac{t^{\rho } - 1}{\rho } \biggr] ^{\frac{1}{2}} + \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t ^{\rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} \\& \hphantom{x(t) =}{} + \frac{2}{\sqrt{\pi }} \biggl[ \frac{t ^{\rho } - 3^{\rho }}{\rho } \biggr]^{\frac{1}{2}} + \frac{1}{\sqrt{ \pi }} \biggl[ \frac{t^{\rho } - 3^{\rho }}{\rho } \biggr]^{ - \frac{1}{2}} \\& \hphantom{x(t) =}{}+ [ \xi + \eta ] \biggl\{ \frac{\frac{3^{\rho } - 1}{ \rho } + \int _{1}^{3} \frac{3^{\rho } - \tau ^{\rho }}{\rho } \frac{x( \tau )\,d\tau }{\tau ^{1 - \rho }}}{\sqrt{\pi }} \biggl[ \frac{t^{ \rho } - 3^{\rho }}{\rho } \biggr]^{ - \frac{1}{2}} + \frac{1 + \int _{1}^{3} \frac{x(\tau )\,d\tau }{\tau ^{1 - \rho }}}{\sqrt{\pi } /2} \biggl[ \frac{t^{\rho } - 3^{\rho }}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{} + \frac{2}{\sqrt{\pi }} \int _{3}^{t} \biggl[ \frac{t^{ \rho } - \tau ^{\rho }}{\rho } \biggr]^{\frac{1}{2}}\frac{x(\tau )\,d \tau }{\tau ^{1 - \rho }} - \frac{2}{\sqrt{\pi }} \biggl[ \frac{t ^{\rho } - 1}{\rho } \biggr]^{\frac{1}{2}} \\& \hphantom{x(t) =}{}- \frac{2}{\sqrt{\pi }} \int _{1}^{t} \biggl[ \frac{t^{\rho } - \tau ^{\rho }}{\rho } \biggr] ^{\frac{1}{2}}\frac{x(\tau )\,d\tau }{\tau ^{1 - \rho }} \biggr\} \\& \quad \text{for } t \in (3,5], \end{aligned}$$
(5.7)

where ξ and η in (5.5)–(5.7) are two arbitrary constants.

Next we realize numerical simulation of (5.4) and (5.5)–(5.7) by using the Euler method with variable step size to give some solution trajectories of three systems (5.1)–(5.3) with given ρ, respectively.

Figures 14 denote the solution trajectories of (5.1) with \(\rho = 0.1, 0.5, 1, 2\), respectively. Moreover, in these figures three curves ‘\(\mbox{xi} = 0, 1, -1\)’, which are drawn by numerical simulation of (5.4)–(5.5) with \(\xi = 0, 1, - 1\), respectively, represent three solutions of (5.1) with the corresponding ρ.

Figure 1
figure 1

The solution trajectory of system (5.1) with ρ= 0.1

Figure 2
figure 2

The solution trajectory of system (5.1) with ρ= 0.5

Figure 3
figure 3

The solution trajectory of system (5.1) with ρ= 1

Figure 4
figure 4

The solution trajectory of system (5.1) with ρ= 2

Figures 58 denote the solution trajectories of (5.2) with \(\rho = 0.1, 0.5, 1, 2\), respectively. Moreover, in these figures three curves ‘\(\mbox{eta} = 0, 1, - 1\)’, which are drawn by numerical simulation of (5.4) and (5.6) with \(\eta = 0, 1, -1\), respectively, represent three solutions of (5.2) with the corresponding ρ.

Figure 5
figure 5

The solution trajectory of system (5.2) with \(\rho = 0.1\)

Figure 6
figure 6

The solution trajectory of system (5.2) with \(\rho = 0.5\)

Figure 7
figure 7

The solution trajectory of system (5.2) with \(\rho = 1\)

Figure 8
figure 8

The solution trajectory of system (5.2) with \(\rho = 2\)

Figures 912 denote the solution trajectories of (5.3) with \(\rho = 0.1, 0.5, 1, 2\), respectively. Moreover, in these figures five curves ‘\(\mbox{xi}+\mbox{eta} = 2, 1, 0, - 1, - 2\)’, which are drawn by numerical simulation of (5.4) and (5.7) with \(\xi +\eta = 2, 1, 0, - 1, - 2\), respectively, represent five solutions of (5.3) with the corresponding ρ.

Figure 9
figure 9

The solution trajectory of system (5.3) with \(\rho = 0.1\)

Figure 10
figure 10

The solution trajectory of system (5.3) with \(\rho = 0.5\)

Figure 11
figure 11

The solution trajectory of system (5.3) with \(\rho = 1\)

Figure 12
figure 12

The solution trajectory of system (5.3) with \(\rho = 2\)

6 Conclusion

The systems of impulsive high order fractional differential equations can involve one or two kinds of impulses. As a result, their equivalent integral equations include one or two arbitrary constants which uncover the non-uniqueness of solution for the systems of impulsive high order fractional differential equations.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Wu, G.C., Deng, Z.G., Baleanu, D., Zeng, D.Q.: New variable-order fractional chaotic systems for fast image encryption. Chaos 29, 083103 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wu, G.C., Baleanu, D., Xie, H.P., Zeng, S.D.: Discrete fractional diffusion equation of chaotic order. Int. J. Bifurc. Chaos 26, 1650013 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Shiri, B., Baleanu, D.: System of fractional differential algebraic equations with applications. Chaos Solitons Fractals 120, 203–212 (2019)

    Article  MathSciNet  Google Scholar 

  5. Alkahtani, B.S.T.: Chua’s circuit model with Atangana-Baleanu derivative with fractional order. Chaos Solitons Fractals 89, 547–551 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Koca, I.: Analysis of rubella disease model with non-local and non-singular fractional derivatives. Int. J. Optim. Control 8, 17–25 (2018)

    MathSciNet  Google Scholar 

  7. Gómez-Aguilar, J.: Irving-Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel. Chaos Solitons Fractals 95, 179–186 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tateishi, A.A., Ribeiro, H.V., Lenzi, E.K.: The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. 5, 1–9 (2017)

    Article  Google Scholar 

  9. Morales-Delgado, V., Gómez-Aguilar, J., Taneco-Hernandez, M.: Analytical solutions for the motion of a charged particle in electric and magnetic fields via non-singular fractional derivatives. Eur. Phys. J. Plus 132, 527 (2017)

    Article  Google Scholar 

  10. Dadkhah, E., Shiri, B., Ghaffarzadeh, H., Baleanu, D.: Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J. Appl. Math. Comput. (2019). https://doi.org/10.1007/s12190-019-01307-5

    Article  Google Scholar 

  11. Alijani, Z., Baleanu, D., Shiri, B., Wu, G.C.: Spline collocation methods for systems of fuzzy fractional differential equations. Chaos Solitons Fractals (2019). https://doi.org/10.1016/j.chaos.2019.109510

    Article  Google Scholar 

  12. Gómez-Aguilar, J.F., Atangana, A.: Fractional derivatives with the power-law and the Mittag-Leffler kernel applied to the nonlinear Baggs-Freedman model. Fractal Fract. 2, 1–14 (2018)

    Article  Google Scholar 

  13. Coronel-Escamilla, A., Gómez-Aguilar, J., Torres, L., Escobar-Jiménez, R.: A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A, Stat. Mech. Appl. 491, 406–424 (2018)

    Article  MathSciNet  Google Scholar 

  14. Zuniga-Aguilar, C., Gómez-Aguilar, J., Escobar-Jiménez, R., Romero-Ugalde, H.: Robust control for fractional variable-order chaotic systems with non-singular kernel. Eur. Phys. J. Plus 133, 1–13 (2018)

    Article  MATH  Google Scholar 

  15. Shiri, B., Baleanu, D.: Numerical solution of some fractional dynamical systems in medicine involving non-singular kernel with vector order. Results Nonlinear Anal. 2, 160–168 (2019)

    Google Scholar 

  16. Voyiadjis, G.Z., Sumelka, W.: Brain modelling in the framework of anisotropic hyperelasticity with time fractional damage evolution governed by the Caputo–Almeida fractional derivative. J. Mech. Behav. Biomed. Mater. 89, 209–216 (2019)

    Article  Google Scholar 

  17. Wu, G.C., Baleanu, D., Zeng, S.D., Deng, Z.G.: Discrete fractional diffusion equation. Nonlinear Dyn. 80, 281–286 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  19. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012)

    Book  MATH  Google Scholar 

  20. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  21. Hilfer, R., Luchko, Y., Tomovski, U.: Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12, 299–318 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6, 1–15 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Abdon, A., Dumitru, B.: New fractional derivatives with nonlocal and non-singular kernel; theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016)

    Article  Google Scholar 

  25. Gou, H., Li, B.: Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions. Chaos Solitons Fractals 112, 168–179 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, X.: Non-uniqueness of solution for initial value problem of impulsive Caputo–Katugampola fractional differential differential equations. Int. J. Dyn. Syst. Differ. Equ. (in press). https://www.inderscience.com/info/ingeneral/forthcoming.php?jcode=ijdsde

  27. Harrat, A., Nieto, J.J., Debbouche, A.: Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential. J. Comput. Appl. Math. 344, 725–737 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Debbouche, A., Antonov, V.: Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces. Chaos Solitons Fractals 102, 140–148 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jarad, F., Abdeljawad, T., Baleanu, D.: On the generalized fractional derivatives and their Caputo modification. J. Nonlinear Sci. Appl. 10, 2607–2619 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zeng, S., Baleanu, D., Bai, Y., Wu, G.: Fractional differential equations of Caputo–Katugampola type and numerical solutions. Appl. Math. Comput. 315, 549–554 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Almeida, R., Malinowska, A.B., Odzijewicz, T.: Fractional differential equations with dependence on the Caputo–Katugampola derivative. J. Comput. Nonlinear Dyn. 11, 061017 (2016)

    Article  Google Scholar 

  32. Baleanu, D., Shiri, B., Srivastava, H.M., Qurashi, Al.M.: A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel. Adv. Differ. Equ. 2018, 353 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Baleanu, D., Shiri, B.: Collocation methods for fractional differential equations involving non-singular kernel. Chaos Solitons Fractals 116, 136–145 (2018)

    Article  MathSciNet  Google Scholar 

  34. Anderson, D.R., Ulness, D.J.: Properties of the Katugampola fractional derivative with potential application in quantum mechanics. J. Math. Phys. 56, 063502 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thaiprayoon, C., Ntouyas, S.K., Tariboon, J.: On the nonlocal Katugampola fractional integral conditions for fractional Langevin equation. Adv. Differ. Equ. 2015, 374 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  37. Agarwal, R., Hristova, S., O’Regan, D.: A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19, 290–318 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, J.R., Feckan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, 806–831 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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)

    MathSciNet  MATH  Google Scholar 

  40. Zhang, X.: On the concept of general solutions for impulsive differential equations of fractional order \(q \in (1,2)\). Appl. Math. Comput. 268, 103–120 (2015)

    MathSciNet  Google Scholar 

  41. Stamova, I., Stamov, G.: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simul. 19(3), 702–709 (2014)

    Article  MathSciNet  Google Scholar 

  42. Wang, G., Ahmad, B., Zhang, L., Nieto, J.J.: Comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 19, 401–403 (2014)

    Article  MathSciNet  Google Scholar 

  43. Feckan, M., Zhou, Y., Wang, J.R.: Response to “Comments on the concept of existence of solution for impulsive fractional differential equations [Commun Nonlinear Sci Numer Simul 2014; 19:401-3.]”. Commun. Nonlinear Sci. Numer. Simul. 19, 4213–4215 (2014)

    Article  MathSciNet  Google Scholar 

  44. Zhang, X., Shu, T., Cao, H., Liu, Z., Ding, W.: The general solution for impulsive differential equations with Hadamard fractional derivative of order \(q \in (1,2)\). Adv. Differ. Equ. 2016, 14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Fan, Z.: A short note on the solvability of impulsive fractional differential equations with Caputo derivatives. Appl. Math. Lett. 38, 14–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wu, G.C., Zeng, D.Q., Baleanu, D.: Fractional impulsive differential equations: exact solutions, integral equations and short memory case. Fract. Calc. Appl. Anal. 22, 180–192 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is 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.

Availability of data and materials

Not applicable.

Funding

The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant No. 21576033, 21636004).

Author information

Authors and Affiliations

Authors

Contributions

The author wrote the first version of the manuscript and approved the final manuscript by himself.

Corresponding author

Correspondence to Xian-Min Zhang.

Ethics declarations

Competing interests

The author declares that he has no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, XM. The non-uniqueness of solution for initial value problem of impulsive differential equations involving higher order Katugampola fractional derivative. Adv Differ Equ 2020, 85 (2020). https://doi.org/10.1186/s13662-020-2536-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-020-2536-z

MSC

Keywords