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Boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses

Abstract

This paper considers the boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses. Some sufficient conditions of the existence and uniqueness are obtained by means of the Banach contraction principle and Leray–Schauder alternative. We also give some interesting examples to illustrate the effectiveness of our main results.

Introduction

In 1695, L’Hôpital asked what was \(\frac{d^{n}y}{dx^{n}}\) if \(n=\frac{1}{2}\) in his letter to Leibniz. This year is generally regarded as the birthday of fractional calculus. Hereafter, Leibniz, J. Bernoulli, Euler, Lagrange, Laplace, Lacroix, Fourier, Abel, Cantor, De Morgen, Ya Sonin, Riemann, Liouville, Caputo, et al. have made important contributions to the definition of fractional calculus. In 1830s, Riemann and Liouville defined the integral and derivative which is now called Riemann–Liouville (R-L) fractional calculus by the Cauchy integral formula. Subsequently, many famous and important fractional integrals and derivatives have been proposed, for example, Grünwald–Letnikov fractional derivative, Caputo fractional derivative, Weyl fractional calculus, Hadamard fractional calculus, and so on. As for the history of fractional calculus, the readers can refer to the literature [1, 2].

The fractional-order calculus as a good tool is used to establish the mathematical model describing many actual phenomena and processes. For example, the fractional differential equations can describe the diffusion processes (see [3, 4]), the mechanical properties of materials (see [5,6,7,8]), the signal processing (see [9]), the image processing (see [10]), the behavior of viscoelastic and visco-plastic materials under external influences (see [11, 12]), the pharmacokinetics (see [13,14,15]), the bioengineering (see [16, 17]), the control theory (see [18, 19]), and so on. In addition there are some applications of fractional calculus within various fields of mathematics itself, e.g., in the analytical investigation of various types of special functions (see [20]). Therefore, the fractional differential equation has been widely focused and studied in depth. There have been some monographs and textbooks for the readers to learn use fractional calculus theories and methods (see [2, 21,22,23,24,25]). In the last few decades, there have been many papers dealing with fractional differential equation involving Riemann–Liouville and Caputo fractional derivatives (see [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]). In fact, the Hadamard fractional derivative is one of the most famous fractional calculi which was put forward by Hadamard in 1892. This type of fractional derivative differs from other types of derivatives. Its main feature is that the integral kernel contains a logarithmic function of arbitrary exponent in definition. Recently, there have been several papers dealing with Hadamard fractional differential equation (see [44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67]). However, these papers rarely considered the Hadamard fractional differential coupled equations. Therefore, it is interesting and challenging to study the Hadamard nonlinear fractional differential coupled system with impulses. So, in this paper we mainly study the following impulsive fractional differential coupled system with Hadamard fractional calculus:

$$\begin{aligned} \textstyle\begin{cases} _{\mathrm{RLH}}D_{t_{k}}^{\alpha }[u(t)-{}_{H}J_{t_{k}}^{\alpha }e(t,u(t),v(t))] =g(t,u(t),v(t)), \quad t\in J=[a,T], t\neq t_{k}, \\ _{\mathrm{RLH}}D_{t_{k}}^{\beta }[v(t)-{}_{H}J_{t_{k}}^{\beta }f(t,u(t),v(t))] =h(t,u(t),v(t)), \quad t\in J=[a,T], t\neq t_{k}, \\ {}_{\mathrm{RLH}}D_{t_{k}}^{\alpha -1}u(t_{k}^{+}) -{}_{\mathrm{RLH}}D_{t_{k}}^{\alpha -1}u(t _{k}^{-})=I_{k}(u(t_{k})), \quad k=1,2,\ldots ,m, \\ {}_{\mathrm{RLH}}D_{t_{k}}^{\beta -1}v(t_{k}^{+}) -{}_{\mathrm{RLH}}D_{t_{k}}^{\beta -1}v(t _{k}^{-})=J_{k}(v(t_{k})), \quad k=1,2,\ldots ,m, \\ c\cdot {}_{\mathrm{RLH}}D_{a}^{\alpha -1}u(a)=u(T), \qquad d\cdot {}_{\mathrm{RLH}}D_{a}^{\beta -1}v(a)=v(T), \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(a>0\), \(1<\alpha ,\beta <2\), \(c,d\in \mathbb{R}\), \(I_{k},J_{k} \in C(\mathbb{R},\mathbb{R})\). \({}_{\mathrm{RLH}}D_{t_{k}}^{\alpha }\), \(_{\mathrm{RLH}}D_{t_{k}}^{\beta }\) denote the left-sided Riemann–Liouville type Hadamard fractional derivatives of order α and β. \({}_{H}J_{t_{k}}^{\alpha }\), \({}_{H}J_{t_{k}}^{\beta }\) denote the left-sided Hadamard fractional integrals of order α and β. \(e, f, g, h: J\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) are some given continuous functions and impulsive points, \(\{t_{k}\}_{k=1} ^{m}\) satisfies \(a=t_{0}< t_{1}< t_{2}<\cdots <t_{m}<t_{m+1}=T\). \({}_{\mathrm{RLH}}D_{t_{k}}^{\alpha -1}u(t_{k}^{+})\), \({}_{\mathrm{RLH}}D_{t_{k}}^{\alpha -1}u(t _{k}^{-})\), \({}_{\mathrm{RLH}}D_{t_{k}}^{\beta -1}v(t_{k}^{+})\), and \({}_{\mathrm{RLH}}D _{t_{k}}^{\beta -1}v(t_{k}^{-})\) represent the right and left limits and satisfy the left continuity at \(t=t_{k}\), \(k=1,2,\ldots ,m\).

In addition, some other inspiration for studying system (1.1) comes from the literature [48, 49]. In [48], the authors considered the existence and finite-time stability results of Hadamard type impulsive fractional differential equations as follows:

$$\begin{aligned} \textstyle\begin{cases} _{H}D_{1}^{\alpha }u(t)=f (t,u(t),\max_{\xi \in [\beta t,t]}u( \xi ) ), \quad \alpha \in (0,1), t\in [1,e]\backslash \varTheta , \beta \in (0,1), \\ {}_{H}J_{1}^{1-\alpha }u(t_{i}^{+}) -{}_{H}J_{1}^{1-\alpha }u(t_{i} ^{-}) =a_{i}u(t_{i}^{-})+b_{i}, \quad a_{i}, b_{i}>0, i=1,2,\ldots ,m, \\ _{H}J_{1}^{1-\alpha }u(1)=u_{0}, \quad u_{0}>0, \end{cases}\displaystyle \end{aligned}$$

with initial condition \(u(t)=\phi (t)\), \(t\in [\beta ,1]\), where \({}_{H}D_{1}^{\alpha }\) denotes the left-sided Riemann–Liouville type Hadamard fractional derivative of order α, \({}_{H}J_{1}^{1- \alpha }\) denotes the left-sided Hadamard fractional integral of order \(1-\alpha \), and \(\varTheta =\{t_{1},t_{2},\ldots ,t_{m}\}\) satisfying \(1=t_{0}< t_{1}<\cdots <t_{m}\) < \(t_{m+1}=e\). \(f: J\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R}\) is a Carathéodory function, and \(u(t_{i}^{+})=\lim_{\epsilon \rightarrow 0^{+}}u(t_{i}+\epsilon )\), \(u(t_{i}^{-})=\lim_{\epsilon \rightarrow 0^{-}}u(t_{i}+\epsilon )\).

In [49], the author discussed the existence and uniqueness results of solutions for the Hadamard and Riemann–Liouville fractional neutral functional integro-differential equations with finite delay described by

$$\begin{aligned} \textstyle\begin{cases} _{H}D^{\alpha } [u(t)-\sum_{i=1}^{m}I^{\beta _{i}}h_{i}(t,u _{t}) ] =f(t,u_{t}), \quad t\in J=[1,T], \\ u(t)=\varphi (t), \quad t\in [1-r,1], r>0, \end{cases}\displaystyle \end{aligned}$$

where \({}_{H}D^{\alpha }\) denotes the left-sided Riemann–Liouville type Hadamard fractional derivative of order α, \(0<\alpha \leq 1\), \(I^{\beta _{i}}\) is the Riemann–Liouville fractional integral of order \(\beta _{i}>0\), \(i=1,2,\ldots ,m\), \(f, h_{i}: J\times C([-r,0], \mathbb{R})\)\(\mathbb{R}\) are given continuous functions, \(\varphi \in C([1-r,1],\mathbb{R})\) with \(\varphi (1)=0\). For any function u defined on \([1-r, T]\) and any \(t\in J\), \(u_{t}(\theta )=u(t+ \theta )\), \(\theta \in [-r,0]\) denotes the element of \(C([-r,0], \mathbb{R})\). The author derived the existence of solutions by the Leray–Schauder alternative and established the uniqueness of solutions by the Banach contraction principle.

The rest of this paper is organized as follows. In Sect. 2, we recall some useful preliminaries. In Sect. 3, we shall prove the existence and uniqueness of solutions for system (1.1). In Sect. 4, some examples are also provided to illustrate the effectiveness of our main results. Finally, the conclusion is given to simply recall our studied contents and obtained results in Sect. 5.

Preliminaries

In this section, we introduce some notations and definitions of Hadamard fractional calculus and present preliminary results needed in our proofs later.

Definition 2.1

([22])

For \(a\geq 0\), the left-sided Hadamard fractional integral of order \(\alpha >0\) for a function \(u: (a,\infty )\rightarrow \mathbb{R}\) is defined as

$$\begin{aligned} {}_{H}J_{a}^{\alpha }u(t)=\frac{1}{\varGamma (\alpha )} \int _{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\alpha -1}u(s)\,\frac{ds}{s}, \end{aligned}$$

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

Definition 2.2

([22])

For \(a\geq 0\), the left-sided Riemann–Liouville type Hadamard fractional derivative of order \(\alpha >0\) for a function \(u: (a, \infty )\rightarrow \mathbb{R}\) is defined by

$$\begin{aligned} {}_{\mathrm{RLH}}D_{a}^{\alpha }u(t)=\frac{1}{\varGamma (n-\alpha )} \biggl(t \,\frac{d}{dt} \biggr)^{n} \int _{a}^{t} \biggl(\ln \frac{t}{s} \biggr)^{n- \alpha +1}u(s)\,\frac{ds}{s}, \quad n-1< \alpha < n,n=[\alpha ]+1, \end{aligned}$$

where \([\alpha ]\) denotes the integer part of the real number \(\alpha >0\), and \(\varGamma (\cdot )\) is the gamma function.

Lemma 2.1

([22])

For \(a>0\), assume that \(u\in C(a,T)\cap L^{1}(a,T)\) with a left-sided Riemann–Liouville type Hadamard fractional derivative of order \(\alpha >0\). Then

$$\begin{aligned} _{H}J_{a}^{\alpha }{}_{\mathrm{RLH}}D_{a}^{\alpha }u(t)=u(t) +c_{1} \biggl( \ln \frac{t}{a} \biggr)^{\alpha -1}+c_{2} \biggl(\ln \frac{t}{a} \biggr)^{ \alpha -2} +\cdots +c_{n} \biggl(\ln \frac{t}{a} \biggr)^{\alpha -n} \end{aligned}$$

for some \(c_{i}\in \mathbb{R}\), \(i=1,2,\ldots ,n-1\), \(n=[\alpha ]+1\).

Lemma 2.2

([44])

Let \(\alpha >0\), \(\beta >0\), and \(0< a<\infty \). Then the following properties hold:

$$\begin{aligned} &{}_{\mathrm{RLH}}D_{a}^{\alpha } \biggl(\ln \frac{t}{a} \biggr)^{\beta -1}(x) =\frac{ \varGamma (\beta )}{\varGamma (\beta -\alpha )} \biggl(\ln \frac{x}{a} \biggr)^{ \beta -\alpha -1}, \\ &{}_{H}J_{a}^{\alpha } \biggl(\ln \frac{t}{a} \biggr)^{\beta -1}(x) =\frac{ \varGamma (\beta )}{\varGamma (\beta +\alpha )} \biggl(\ln \frac{x}{a} \biggr)^{ \beta +\alpha -1}, \\ &{}_{\mathrm{RLH}}D_{a}^{\alpha }{}_{H}J_{a}^{\alpha }u(t) =u(t). \end{aligned}$$

Lemma 2.3

([68])

If E is a real Banach space and \(F: E \rightarrow E\) is a contraction mapping, then F has a unique fixed point in E.

Lemma 2.4

(Leray–Schauder alternative theorem [61])

Let U be a normed linear space and \(F: U \rightarrow U\) be a completely continuous operator (i.e., a map that restricted to any bounded set in U is compact). Let

$$\begin{aligned} \varepsilon (F)=\bigl\{ x\in U: x=kF(x), 0< k< 1\bigr\} , \end{aligned}$$

then either the set \(\varepsilon (F)\) is unbounded, or F has at least one fixed point.

For the convenient statements, we introduce the notation as follows: \(t_{0}=a\), \(t_{m+1}=T\), \(I_{0}(t)=J_{0}(t)\equiv 0\). Let \(C[a,T]\) be the Banach space of all continuous functions from \([a,T] \rightarrow \mathbb{R}\) with the norm \(\|\omega \|=\sup_{t\in [a,T]}|\omega (t)|\). For \(1<\gamma <2\), we define

$$\begin{aligned} \mathrm{PC}_{\gamma }[a,T]={} & \bigl\{ \omega :\omega (t)\in C(t_{k},t_{k+1}], {}_{\mathrm{RLH}}D^{\gamma -1}_{t_{k}} \omega \bigl(t_{k}^{-}\bigr), {}_{\mathrm{RLH}}D^{\gamma -1} _{t_{k}}\omega \bigl(t_{k}^{+}\bigr) \mbox{ all exist} \quad \\ &\mbox{and satify } {}_{\mathrm{RLH}}D^{\gamma -1}_{t_{k}}\omega \bigl(t_{k}^{-}\bigr)={}_{\mathrm{RLH}}D^{\gamma -1}_{t_{k}} \omega (t_{k}), k=0,1,2,\ldots ,m \bigr\} . \end{aligned}$$

Obviously, \(\mathrm{PC}_{\gamma }[a,T]\) is a Banach space equipped with the norm \(\|\omega \|_{\mathrm{PC}_{\gamma }}=\|\omega (t)\|_{C}\). The space \(X=\mathrm{PC}_{ \alpha }[a,T]\times \mathrm{PC}_{\beta }[a,T]\) equipped with the norm \(\|(u,v)\|=\max \{\|u\|_{\mathrm{PC}_{\alpha }},\|v\|_{\mathrm{PC}_{\beta }}\}\) is also a Banach space.

Definition 2.3

A pair of functions \((u(t),v(t))\in X=\mathrm{PC}_{\alpha }[a,T]\times \mathrm{PC}_{ \beta }[a,T]\) is called to be a solution of (1.1) if \((u(t),v(t))\) satisfy all the equations and boundary value conditions of system (1.1).

Lemma 2.5

Assume that the functions \(e, g\in C[a,T])\) and \(I_{k}\in C( \mathbb{R},\mathbb{R})\). If \(\delta \triangleq c\varGamma (\alpha )- (\ln \frac{T}{t_{m}} )^{\alpha -1}\neq 0\), then, for given \(v(t)\in \mathrm{PC}_{\beta }[a,T]\), a function \(u(t)\in \mathrm{PC}_{\alpha }[a,T]\) is a solution of the impulsive Hadamard fractional differential equation

$$\begin{aligned} \textstyle\begin{cases} _{\mathrm{RLH}}D_{t_{k}}^{\alpha }[u(t)-{}_{H}J_{t_{k}}^{\alpha }e(t,u(t),v(t))] =g(t,u(t),v(t)), \quad 1< \alpha < 2, \\ {}_{\mathrm{RLH}}D_{t_{k}}^{\alpha -1}u(t_{k}^{+}) -{}_{\mathrm{RLH}}D_{t_{k}}^{\alpha -1}u(t _{k}^{-})=I_{k}(u(t_{k})), \quad k=1,2,\ldots ,m, \\ c\cdot {}_{\mathrm{RLH}}D_{a}^{\alpha -1}u(a)=u(T), \end{cases}\displaystyle \end{aligned}$$
(2.1)

if and only if \(u(t)\in C[a,T]\cap \mathrm{PC}_{\alpha }[a,T]\) is a solution of the integral equation

$$\begin{aligned} \begin{aligned}[b] u(t)= &{}_{H}J_{t_{k}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{k}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c^{*} \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} +\frac{ \varLambda }{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl[I_{i}\bigl(u(t_{i})\bigr) \\ &{}+{}_{H}J_{t_{i-1}}^{1}g\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr)+{}_{H}J_{t_{i-1}} ^{1}e\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1}, \quad t\in (t_{k},t_{k+1}], \end{aligned} \end{aligned}$$
(2.2)

where \(k=0,1,2,\ldots ,m\), \(\varLambda = \bigl\{\scriptsize{\begin{array}{ll} 0,& t\in [a,t_{1}], \\ 1,& t\in (t_{1},T], \end{array}} \) and

$$\begin{aligned} c^{*}={} &\frac{1}{\delta } \Biggl({}_{H}J_{t_{m}}^{\alpha }g \bigl(T,u(T),v(T)\bigr)+ {}_{H}J_{t_{m}}^{\alpha }e \bigl(T,u(T),v(T)\bigr) +\frac{ (\ln \frac{T}{t _{m}} )^{\alpha -1}}{\varGamma (\alpha )} \sum_{i=1}^{m} \bigl[I_{i}\bigl(u(t_{i})\bigr) \\ &{}+{}_{H}J_{t_{i-1}}^{1}g\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr)+{}_{H}J_{t_{i-1}} ^{1}e\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \Biggr). \end{aligned}$$

Proof

When \(t\in [a,t_{1}]=[t_{0},t_{1}]\), applying the Hadamard fractional integral operator on both sides of the first equation in (2.1), that is,

$$\begin{aligned} {}_{H}J_{t_{0}}^{\alpha }{}_{\mathrm{RLH}}D_{t_{0}}^{\alpha } \bigl[u(t)-{}_{H}J_{t _{0}}^{\alpha }e\bigl(t,u(t),v(t)\bigr) \bigr] ={}_{H}J_{t_{0}}^{\alpha }g\bigl(t,u(t),v(t)\bigr), \end{aligned}$$

we have

$$\begin{aligned} u(t)={}_{H}J_{t_{0}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{0}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c_{1} \biggl(\ln \frac{t}{t_{0}} \biggr)^{\alpha -1}+d_{1} \biggl(\ln \frac{t}{t_{0}} \biggr)^{\alpha -2}, \end{aligned}$$
(2.3)

where \(c_{1}\) and \(d_{1}\) are some constants. In the light of the existence of \(u(a)\), we have \(d_{1}=0\).

In view of Lemmas 2.12.2, we obtain

$$\begin{aligned} {}_{\mathrm{RLH}}D_{t_{0}}^{\alpha -1}u(t)={} &{}_{\mathrm{RLH}}D_{t_{0}}^{\alpha -1}{}_{H}J_{t_{0}}^{\alpha }g \bigl(t,u(t),v(t)\bigr) +{}_{\mathrm{RLH}}D_{t_{0}}^{\alpha -1} {}_{H}J_{t_{0}}^{\alpha }e\bigl(t,u(t),v(t)\bigr) \\ &{}+c_{1}{}_{H}D_{t_{0}}^{\alpha -1} \biggl(\ln \frac{t}{t_{0}} \biggr) ^{\alpha -1} \\ = {}&{}_{H}J_{t_{0}}^{1}g\bigl(t,u(t),v(t) \bigr)+{}_{H}J_{t_{0}}^{1}e\bigl(t,u(t),v(t)\bigr) +c_{1}\varGamma (\alpha ). \end{aligned}$$
(2.4)

(2.4) gives that

$$\begin{aligned} _{\mathrm{RLH}}D_{t_{0}}^{\alpha -1}u(t_{0})=c_{1} \varGamma (\alpha ). \end{aligned}$$
(2.5)

According to (2.3) and (2.4), we get

$$\begin{aligned} _{\mathrm{RLH}}D_{t_{0}}^{\alpha -1}u \bigl(t_{1}^{-}\bigr) ={}_{H}J_{t_{0}}^{1}g \bigl(t_{1},u(t _{1}),v(t_{1}) \bigr)+{}_{H}J_{t_{0}}^{1}e\bigl(t_{1},u(t_{1}),v(t_{1}) \bigr) +c_{1} \varGamma (\alpha ) \end{aligned}$$
(2.6)

and

$$\begin{aligned} u(t)={}_{H}J_{t_{0}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{0}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c_{1} \biggl(\ln \frac{t}{t_{0}} \biggr)^{\alpha -1}, \quad t\in [t_{0},t_{1}]. \end{aligned}$$
(2.7)

When \(t\in (t_{1},t_{2}]\), there are similar to have

$$\begin{aligned} u(t)={}_{H}J_{t_{1}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{1}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c_{2} \biggl(\ln \frac{t}{t_{1}} \biggr)^{\alpha -1}+d_{2} \biggl(\ln \frac{t}{t_{1}} \biggr)^{\alpha -2}, \end{aligned}$$
(2.8)

where \(c_{2}\) and \(d_{2}\) are some constants. In the light of the existence of \({}_{\mathrm{RLH}}D_{t_{1}}^{\alpha -1}u(t_{1}^{+})\), we have \(d_{2}=0\), and

$$\begin{aligned} _{\mathrm{RLH}}D_{t_{1}}^{\alpha -1}u \bigl(t_{1}^{+}\bigr)=c_{2}\varGamma (\alpha ). \end{aligned}$$
(2.9)

It follows from (2.6), (2.9), and the second equation of (2.1) that

$$\begin{aligned} c_{2}-c_{1}=\frac{1}{\varGamma (\alpha )} \bigl[I_{1}\bigl(u(t_{1})\bigr)+{}_{H}J _{t_{0}}^{1}g\bigl(t_{1},u(t_{1}),v(t_{1}) \bigr)+{}_{H}J_{t_{0}}^{1}e\bigl(t_{1},u(t _{1}),v(t_{1})\bigr) \bigr] \end{aligned}$$
(2.10)

and

$$\begin{aligned} u(t)={}_{H}J_{t_{1}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{1}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c_{2} \biggl(\ln \frac{t}{t_{1}} \biggr)^{\alpha -1}, \quad t\in (t_{1},t_{2}]. \end{aligned}$$
(2.11)

Repeating the above calculation process, for \(t\in (t_{k},t_{k+1}]\), \(k=1,2,\ldots ,m\), we obtain

$$\begin{aligned} c_{k+1}-c_{k}=\frac{1}{\varGamma (\alpha )} \bigl[I_{k}\bigl(u(t_{k})\bigr)+{}_{H}J _{t_{k-1}}^{1}g\bigl(t_{k},u(t_{k}),v(t_{k}) \bigr)+{}_{H}J_{t_{k-1}}^{1}e\bigl(t_{k},u(t _{k}),v(t_{k})\bigr) \bigr] \end{aligned}$$
(2.12)

and

$$\begin{aligned} u(t)={}_{H}J_{t_{k}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{k}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c_{k+1} \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1}, \quad t\in (t_{k},t_{k+1}]. \end{aligned}$$
(2.13)

From (2.12) and (2.13), we have

$$\begin{aligned} c_{m+1}-c_{1}= {}&\frac{1}{\varGamma (\alpha )}\sum _{k=1}^{m} \bigl[I_{k}\bigl(u(t _{k})\bigr)+{}_{H}J_{t_{k-1}}^{1}g \bigl(t_{k},u(t_{k}),v(t_{k}) \bigr) \\ &{}+{}_{H}J_{t_{k-1}} ^{1}e\bigl(t_{k},u(t_{k}),v(t_{k}) \bigr) \bigr] \end{aligned}$$
(2.14)

and

$$\begin{aligned} u(T)=u(t_{m+1})={}&{}_{H}J_{t_{m}}^{\alpha }g \bigl(T,u(T),v(T)\bigr)+{}_{H}J_{t _{m}}^{\alpha }e \bigl(T,u(T),v(T)\bigr) \\ &{}+c_{m+1} \biggl(\ln \frac{T}{t_{m}} \biggr) ^{\alpha -1}. \end{aligned}$$
(2.15)

In the light of (2.5), (2.14), (2.15), and the third equation of (2.1), we have

$$\begin{aligned} c_{1}={} &\frac{1}{\delta } \Biggl({}_{H}J_{t_{m}}^{\alpha }g\bigl(T,u(T),v(T)\bigr)+ {}_{H}J_{t_{m}}^{\alpha }e\bigl(T,u(T),v(T)\bigr) + \frac{ (\ln \frac{T}{t _{m}} )^{\alpha -1}}{\varGamma (\alpha )} \sum_{i=1}^{m} \bigl[I_{i}\bigl(u(t_{i})\bigr) \\ &{}+{}_{H}J_{t_{i-1}}^{1}g\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr)+{}_{H}J_{t_{i-1}} ^{1}e\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \Biggr). \end{aligned}$$
(2.16)

Thus, for \(k=1,2,\ldots ,m\), we have

$$\begin{aligned} u(t)= &{}_{H}J_{t_{k}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{k}}^{\alpha }e \bigl(t,u(t),v(t)\bigr) +c_{1} \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1}\\ &{} +\frac{1}{ \varGamma (\alpha )}\sum_{i=1}^{k} \bigl[I_{i}\bigl(u(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}g\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \\ &{}+{}_{H}J_{t_{i-1}} ^{1}e\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1}, \quad t\in (t_{k},t_{k+1}]. \end{aligned}$$
(2.17)

Conversely, if \(u(t)\) satisfies (2.2), it is easy to verify \(u(t)\) satisfying (2.1). The proof is complete. □

Similarly, we obtain the following lemma.

Lemma 2.6

Assume that the functions \(f, h\in C[a,T]\) and \(J_{k}\in C(\mathbb{R}, \mathbb{R})\). If \(\rho \triangleq d\varGamma (\beta )- (\ln \frac{T}{t _{m}} )^{\beta -1}\neq 0\), then, for given \(u(t)\in \mathrm{PC}_{\alpha }[a,T]\), a function \(v(t)\in \mathrm{PC}_{\beta }[a,T]\) is a solution of the impulsive Hadamard fractional differential equation

$$\begin{aligned} \textstyle\begin{cases} _{\mathrm{RLH}}D_{t_{k}}^{\beta }[v(t)-{}_{H}J_{t_{k}}^{\beta }f(t,u(t),v(t))] =h(t,u(t),v(t)), \quad 1< \beta < 2, \\ {}_{\mathrm{RLH}}D_{t_{k}}^{\beta -1}v(t_{k}^{+}) -{}_{\mathrm{RLH}}D_{t_{k}}^{\beta -1}v(t _{k}^{-})=J_{k}(v(t_{k})), \quad k=1,2,\ldots ,m, \\ d\cdot {}_{\mathrm{RLH}}D_{a}^{\beta -1}v(a)=v(T) \end{cases}\displaystyle \end{aligned}$$
(2.18)

if and only if \(v(t)\in C[a,T]\cap \mathrm{PC}_{\beta }[a,T]\) is a solution of the integral equation

$$\begin{aligned} v(t)={} &{}_{H}J_{t_{k}}^{\beta }h \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{k}}^{\beta }f \bigl(t,u(t),v(t)\bigr) +d^{*} \biggl(\ln \frac{t}{t_{k}} \biggr)^{\beta -1}\\ &{} +\frac{\varLambda }{ \varGamma (\beta )}\sum_{i=1}^{k} \bigl[J_{i}\bigl(v(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}h\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \\ &{}+{}_{H}J_{t_{i-1}} ^{1}f\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\beta -1}, \quad t\in (t_{k},t_{k+1}], \end{aligned}$$
(2.19)

where \(k=0,1,2,\ldots ,m\), \(\varLambda = \bigl\{\scriptsize{\begin{array}{ll} 0, & t\in [a,t_{1}], \\ 1, & t\in (t_{1},T], \end{array}} \) and

$$\begin{aligned} d^{*}={} &\frac{1}{\rho } \Biggl({}_{H}J_{t_{m}}^{\beta }h \bigl(T,u(T),v(T)\bigr)+{}_{H}J_{t_{m}}^{\beta }f \bigl(T,u(T),v(T)\bigr) +\frac{ (\ln \frac{T}{t_{m}} )^{\beta -1}}{\varGamma (\beta )} \sum_{i=1}^{m} \bigl[J_{i}\bigl(v(t_{i})\bigr) \\ &{}+{}_{H}J_{t_{i-1}}^{1}h\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr)+{}_{H}J_{t_{i-1}} ^{1}f\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \Biggr). \end{aligned}$$

Main results

In this section, we shall employ Lemmas 2.3 and 2.4 to prove the existence of solutions to system (1.1). In the light of Lemmas 2.5 and 2.6, we define the operator \(S: X=\mathrm{PC}_{\alpha } \times \mathrm{PC} _{\beta } \rightarrow X\) by

$$\begin{aligned} S(u,v) (t)= \bigl(S_{1}(u,v) (t), S_{2}(u,v) (t) \bigr)^{T}, \quad \forall (u,v)\in X, t\in [a,T], \end{aligned}$$
(3.1)

where

$$\begin{aligned} S_{1}(u,v) (t)={}&{}_{H}J_{t_{k}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t _{k}}^{\alpha }e \bigl(t,u(t),v(t)\bigr)\\ &{} +c^{*} \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} +\frac{\varLambda }{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl[I _{i}\bigl(u(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}g\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr)\\ &{}+{}_{H}J_{t_{i-1}} ^{1}e\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1}, \quad t\in (t_{k},t_{k+1}], 0\leq k\leq m, \end{aligned}$$

and

$$\begin{aligned} S_{2}(x,y) (t)={}&{}_{H}J_{t_{k}}^{\beta }h \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t _{k}}^{\beta }f \bigl(t,u(t),v(t)\bigr) \\ &{}+d^{*} \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\beta -1} +\frac{\varLambda }{\varGamma (\beta )}\sum_{i=1}^{k} \bigl[J _{i}\bigl(v(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}h\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr)\\ &{}+{}_{H}J_{t_{i-1}} ^{1}f\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\beta -1}, \quad t\in (t_{k},t_{k+1}], 0\leq k\leq m. \end{aligned}$$

Solving system (1.1) is equivalent to finding the fixed point of the operator S defined by (3.1). Now we present and prove our main results.

Theorem 3.1

If the following conditions \((H_{1})\)\((H_{3})\) hold, then the Hadamard impulsive fractional differential coupled system (1.1) has a pair of unique solutions \((u^{*}(t),v^{*}(t))\) \(\in \mathrm{PC}_{\alpha } \times \mathrm{PC}_{\beta }\).

\((H_{1})\) :

Let \(e, f, g, h\in C[a,T]\), \(I_{k}\), \(J_{k}\in C( \mathbb{R},\mathbb{R})\), \(k=1,2,\ldots ,m\). For \(u_{i}, v_{i}\in \mathbb{R}\) (\(i=1,2\)), there exist some positive constants \(M_{i}\), \(\overline{M}_{i}\), \(N_{i}\), \(\overline{N}_{i}(i=1,2)\), \(P_{k}\), and \(Q_{k}\) (\(k=1,2,\ldots ,m\)) such that

$$\begin{aligned} & \bigl\vert g(t,u_{1},v_{1})-e(t,u_{2},v_{2}) \bigr\vert \leq M_{1} \vert u_{1}-u_{2} \vert +M_{2} \vert v _{1}-v_{2} \vert , \\ & \bigl\vert e(t,u_{1},v_{1})-g(t,u_{2},v_{2}) \bigr\vert \leq N_{1} \vert u_{1}-u_{2} \vert +N_{2} \vert v _{1}-v_{2} \vert , \\ & \bigl\vert f(t,u_{1},v_{1})-f(t,u_{2},v_{2}) \bigr\vert \leq \overline{M}_{1} \vert u_{1}-u _{2} \vert +\overline{M}_{2} \vert v_{1}-v_{2} \vert , \\ & \bigl\vert h(t,u_{1},v_{1})-h(t,u_{2},v_{2}) \bigr\vert \leq \overline{N}_{1} \vert u_{1}-u _{2} \vert +\overline{N}_{2} \vert v_{1}-v_{2} \vert , \\ & \bigl\vert I_{k}(u_{1})-I_{k}(v_{1}) \bigr\vert \leq P_{k} \vert u_{1}-v_{1} \vert , \\ & \bigl\vert J_{k}(u_{1})-J_{k}(v_{1}) \bigr\vert \leq Q_{k} \vert u_{1}-v_{1} \vert , \quad k=1,2\ldots ,m. \end{aligned}$$
\((H_{2})\) :

\(\delta \triangleq c\varGamma (\alpha )- (\ln \frac{T}{t _{m}} )^{\alpha -1}>0\), \(\rho \triangleq d\varGamma (\beta )- (\ln \frac{T}{t _{m}} )^{\beta -1}>0\).

\((H_{3})\) :
$$\begin{aligned} \kappa \triangleq{} &\frac{M_{1}+M_{2}+N_{1}+N_{2}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha }\\ &{}\times \biggl[1+\frac{1}{\delta } \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} +\frac{1}{\delta \varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1}+\frac{1}{\varGamma (\alpha )} \biggr] \\ &{}+\frac{\sum_{i=1}^{m}P_{i}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr) ^{\alpha -1} \biggr]< 1, \\ \varrho \triangleq{} & \frac{\overline{M}_{1}+\overline{M}_{2}+ \overline{N}_{1}+\overline{N}_{2}}{\varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{\beta } \\ &{}\times\biggl[1+\frac{1}{\rho } \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} +\frac{1}{\rho \varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{\beta -1}+\frac{1}{\varGamma (\beta )} \biggr] \\ &{}+\frac{\sum_{i=1}^{m}Q_{i}}{\varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr) ^{\beta -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} \biggr]< 1. \end{aligned}$$

Proof

Now, we apply the Banach contraction principle to prove that \(S: X\rightarrow X\) defined by (3.1) has a unique fixed point. We shall show that S is a contraction. In fact, from (3.1) and conditions \((H_{1})\)\((H_{2})\), for \(t\in J=[a,T]\), \((u_{1},v_{1}), (u _{2},v_{2})\in X\), we have

$$\begin{aligned} & \bigl\vert S_{1}(u_{1},v_{1}) (t)-S_{1}(u_{2},v_{2}) (t) \bigr\vert \\ &\quad = \Biggl\vert {}_{H}J_{t_{k}}^{\alpha } \bigl[g\bigl(t,u_{1}(t),v_{1}(t)\bigr)-g \bigl(t,u_{2}(t),v _{2}(t)\bigr)\bigr] +{}_{H}J_{t_{k}}^{\alpha } \bigl[e\bigl(t,u_{1}(t),v_{1}(t)\bigr)-e \bigl(t,u_{2}(t),v _{2}(t)\bigr)\bigr] \\ &\qquad {}+\frac{1}{\delta } \Biggl[{}_{H}J_{t_{m}}^{\alpha } \bigl[g\bigl(T,u_{1}(T),v _{1}(T)\bigr)-g \bigl(T,u_{2}(T),v_{2}(T)\bigr)\bigr]+{}_{H}J_{t_{m}}^{\alpha } \bigl[e\bigl(T,u_{1}(T),v _{1}(T)\bigr) \\ &\qquad {}-e\bigl(T,u_{2}(T),v_{2}(T)\bigr) \bigr] +\frac{ (\ln \frac{T}{t_{m}} )^{ \alpha -1}}{\varGamma (\alpha )}\sum_{i=1}^{m} \bigl( \bigl[I_{i}\bigl(u_{1}(t _{i}) \bigr)-I_{i}\bigl(u_{2}(t_{i})\bigr) \bigr]\\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl[g\bigl(t_{i},u _{1}(t_{i}),v_{1}(t_{i})\bigr) -g\bigl(t_{i},u_{2}(t_{i}),v_{2}(t_{i}) \bigr) \bigr]\\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl[e \bigl(t_{i},u_{1}(t_{i}),v_{1}(t_{i}) \bigr)-e\bigl(t_{i},u_{2}(t_{i}),v_{2}(t _{i})\bigr) \bigr] \bigr) \Biggr] \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} \\ &\qquad {}+\frac{\varLambda }{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl( \bigl[I_{i}\bigl(u _{1}(t_{i}) \bigr)-I_{i}\bigl(u_{2}(t_{i})\bigr) \bigr] +{}_{H}J_{t_{i-1}}^{1} \bigl[g\bigl(t _{i},u_{1}(t_{i}),v_{2}(t_{i}) \bigr)-g\bigl(t_{i},u_{1}(t_{i}),v_{2}(t_{i}) \bigr) \bigr] \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl[e \bigl(t_{i},u_{1}(t_{i}),v_{1}(t_{i}) \bigr)-e\bigl(t _{i},u_{2}(t_{i}),v_{2}(t_{i}) \bigr) \bigr] \bigr) \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} \Biggr\vert \\ &\quad \leq {}_{H}J_{t_{k}}^{\alpha } \bigl\vert g\bigl(t,u_{1}(t),v_{1}(t)\bigr)-g \bigl(t,u_{2}(t),v _{2}(t)\bigr) \bigr\vert +{}_{H}J_{t_{k}}^{\alpha } \bigl\vert e \bigl(t,u_{1}(t),v_{1}(t)\bigr)-e\bigl(t,u_{2}(t),v _{2}(t)\bigr) \bigr\vert \\ &\qquad {}+\frac{1}{\delta } \Biggl[{}_{H}J_{t_{m}}^{\alpha } \bigl\vert g\bigl(T,u_{1}(T),v _{1}(T)\bigr)-g \bigl(T,u_{2}(T),v_{2}(T)\bigr) \bigr\vert +{}_{H}J_{t_{m}}^{\alpha } \bigl\vert e \bigl(T,u_{1}(T),v _{1}(T)\bigr) \\ &\qquad {}-e\bigl(T,u_{2}(T),v_{2}(T)\bigr) \bigr\vert +\frac{ (\ln \frac{T}{t_{m}} )^{ \alpha -1}}{\varGamma (\alpha )}\sum_{i=1}^{m} \bigl( \bigl\vert I_{i}\bigl(u_{1}(t _{i}) \bigr)\\ &\qquad {}-I_{i}\bigl(u_{2}(t_{i})\bigr) \bigr\vert +{}_{H}J_{t_{i-1}}^{1} \bigl\vert g \bigl(t_{i},u _{1}(t_{i}),v_{1}(t_{i}) \bigr) -g\bigl(t_{i},u_{2}(t_{i}),v_{2}(t_{i}) \bigr) \bigr\vert \\ &\qquad {} +{}_{H}J_{t_{i-1}}^{1} \bigl\vert e\bigl(t_{i},u_{1}(t_{i}),v_{1}(t_{i}) \bigr)-e\bigl(t_{i},u_{2}(t_{i}),v_{2}(t _{i})\bigr) \bigr\vert \bigr) \Biggr] \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} \\ &\qquad {}+\frac{1}{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl( \bigl\vert I_{i}\bigl(u_{1}(t _{i}) \bigr)-I_{i}\bigl(u_{2}(t_{i})\bigr) \bigr\vert +{}_{H}J_{t_{i-1}}^{1} \bigl\vert g \bigl(t_{i},u _{1}(t_{i}),v_{2}(t_{i}) \bigr)-g\bigl(t_{i},u_{1}(t_{i}),v_{2}(t_{i}) \bigr) \bigr\vert \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl\vert e\bigl(t_{i},u_{1}(t_{i}),v_{1}(t_{i}) \bigr)-e\bigl(t _{i},u_{2}(t_{i}),v_{2}(t_{i}) \bigr) \bigr\vert \bigr) \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} \vert \\ &\quad \leq {}_{H}J_{t_{k}}^{\alpha } \bigl[M_{1} \bigl\vert u_{1}(t)-u_{2}(t) \bigr\vert +M_{2} \bigl\vert v _{1}(t)-v_{2}(t) \bigr\vert \bigr] \\ &\qquad {}+{}_{H}J_{t_{k}}^{\alpha } \bigl[N_{1} \bigl\vert u_{1}(t)-u_{2}(t) \bigr\vert +N _{2} \bigl\vert v_{1}(t)-v_{2}(t) \bigr\vert \bigr] \\ &\qquad {}+\frac{1}{\delta } \Biggl[{}_{H}J_{t_{m}}^{\alpha } \bigl[M_{1} \bigl\vert u_{1}(T)-u _{2}(T) \bigr\vert +M_{2} \bigl\vert v_{1}(T)-v_{2}(T) \bigr\vert \bigr]+{}_{H}J_{t_{m}}^{\alpha } \bigl[N_{1} \bigl\vert u _{1}(T)-u_{2}(T) \bigr\vert \\ &\qquad {}+N_{2} \bigl\vert v_{1}(T)-v_{2}(T) \bigr\vert \bigr] +\frac{ (\ln \frac{T}{t_{m}} )^{ \alpha -1}}{\varGamma (\alpha )}\sum_{i=1}^{m} \bigl(P_{i} \bigl\vert u_{1}(t_{i})-u _{2}(t_{i}) \bigr\vert \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl[M_{1} \bigl\vert u_{1}(t_{i})-u_{2}(t_{i}) \bigr\vert +M_{2} \bigl\vert v_{1}(t_{i})-v_{2}(t_{i}) \bigr\vert \bigr] \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl[N_{1} \bigl\vert u _{1}(t_{i})-u_{2}(t_{i}) \bigr\vert +N_{2} \bigl\vert v_{1}(t_{i})-v_{2}(t_{i}) \bigr\vert \bigr] \bigr) \Biggr] \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} \\ &\qquad {}+\frac{1}{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl(P_{i} \bigl\vert u_{1}(t_{i})-u _{2}(t_{i}) \bigr\vert +{}_{H}J_{t_{i-1}}^{1} \bigl[M_{1} \bigl\vert u_{1}(t_{i})-u_{2}(t_{i}) \bigr\vert +M _{2} \bigl\vert v_{1}(t_{i})-v_{2}(t_{i}) \bigr\vert \bigr] \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1}[N_{1} \bigl\vert u_{1}(t_{i})-u_{2}(t_{i}) \bigr\vert +N_{2}\bigl[ \bigl\vert v _{1}(t_{i})-v_{2}(t_{i}) \bigr\vert \bigr] \bigr) \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} \\ &\quad \leq {}_{H}J_{t_{k}}^{\alpha } \bigl[M_{1} \Vert u_{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }}+M _{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr]\\ &\qquad {} +{}_{H}J_{t_{k}}^{\alpha } \bigl[N_{1} \Vert u _{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }}+N_{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr] \\ &\qquad {}+\frac{1}{\delta } \Biggl[{}_{H}J_{t_{m}}^{\alpha } \bigl[M_{1} \Vert u_{1}-u _{2} \Vert _{\mathrm{PC}_{\alpha }}+M_{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr]+{}_{H}J_{t _{m}}^{\alpha } \bigl[N_{1} \Vert u_{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }} \\ &\qquad {}+N_{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr] +\frac{ (\ln \frac{T}{t _{m}} )^{\alpha -1}}{\varGamma (\alpha )}\sum _{i=1}^{m} \bigl(P_{i} \Vert u _{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }} +{}_{H}J_{t_{i-1}}^{1} \bigl[M_{1} \Vert u_{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }} \\ &\qquad {}+M_{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr]+{}_{H}J_{t_{i-1}}^{1} \bigl[N_{1} \Vert u _{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }}+N_{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr] \bigr) \Biggr] \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} \\ &\qquad {}+\frac{1}{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl(P_{i} \Vert u_{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }} +{}_{H}J_{t_{i-1}}^{1} \bigl[M_{1} \Vert u_{1}-u_{2} \Vert _{\mathrm{PC}_{ \alpha }}+M_{2} \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr] \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1}[N_{1} \Vert u_{1}-u_{2} \Vert _{\mathrm{PC}_{\alpha }}+N_{2} \bigl[ \Vert v_{1}-v_{2} \Vert _{\mathrm{PC}_{\beta }}\bigr] \bigr) \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} \\ &\quad \leq (M_{1}+M_{2}+N_{1}+N_{2}) \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \frac{1}{ \varGamma (\alpha )} \int _{t_{k}}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\alpha -1}\,\frac{ds}{s} \\ &\qquad {}+\frac{1}{\delta } \Biggl[(M_{1}+M_{2}+N_{1}+N_{2}) \bigl\Vert (u_{1}-u_{2},v _{1}-v_{2}) \bigr\Vert \frac{1}{\varGamma (\alpha )} \int _{t_{m}}^{T} \biggl(\ln \frac{T}{s} \biggr)^{\alpha -1}\,\frac{ds}{s} \\ & \qquad {}+\frac{ (\ln \frac{T}{t_{m}} )^{\alpha -1}}{\varGamma (\alpha )}\sum_{i=1}^{m} \biggl(P_{i} \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \\ & \qquad {}+(M_{1}+M _{2}+N_{1}+N_{2}) \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \\ &\qquad {}\times \frac{1}{\varGamma (\alpha )} \int _{t_{i-1}}^{t_{i}} \,\frac{ds}{s} \biggr) \Biggr] \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} +\frac{1}{\varGamma (\alpha )}\sum _{i=1}^{k} \biggl(P_{i} \bigl\Vert (u_{1}-u_{2},v _{1}-v_{2}) \bigr\Vert \\ &\qquad {}+(M_{1}+M_{2}+N_{1}+N_{2}) \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \frac{1}{ \varGamma (\alpha )} \int _{t_{i-1}}^{t_{i}}\,\frac{ds}{s} \biggr) \biggl(\ln \frac{t}{t _{k}} \biggr)^{\alpha -1} \\ &\quad \leq \frac{M_{1}+M_{2}+N_{1}+N_{2}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha } \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \\ &\qquad {}+\frac{1}{\delta } \Biggl[\frac{M_{1}+M_{2}+N_{1}+N_{2}}{\varGamma ( \alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha } \bigl\Vert (u_{1}-u_{2},v_{1}-v _{2}) \bigr\Vert \\ & \qquad {}+\frac{ (\ln \frac{T}{a} )^{\alpha -1}}{\varGamma (\alpha )} \Biggl(\sum _{i=1}^{m}P_{i}+ \frac{M_{1}+M_{2}+N_{1}+N_{2}}{\varGamma ( \alpha )}\ln \frac{T}{a} \Biggr) \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \Biggr] \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} \\ &\qquad {}+\frac{1}{\varGamma (\alpha )} \Biggl(\sum_{i=1}^{k}P_{i} +\frac{M_{1}+M _{2}+N_{1}+N_{2}}{\varGamma (\alpha )}\ln \frac{T}{a} \Biggr) \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \\ &\quad \leq \biggl\{ \frac{M_{1}+M_{2}+N_{1}+N_{2}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha } \biggl[1+\frac{1}{\delta } \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} +\frac{1}{\delta \varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1}+\frac{1}{\varGamma (\alpha )} \biggr] \\ &{}\qquad +\frac{\sum_{i=1}^{m}P_{i}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr) ^{\alpha -1} \biggr] \biggr\} \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert \\ &\quad =\kappa \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert . \end{aligned}$$
(3.2)

Similarly, we derive

$$\begin{aligned} & \bigl\vert S_{2}(u_{1},v_{1}) (t)-S_{2}(u_{2},v_{2}) (t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{ \overline{M}_{1}+\overline{M}_{2}+\overline{N}_{1}+\overline{N}_{2}}{ \varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{\beta } \biggl[1+\frac{1}{ \rho } \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} +\frac{1}{\rho \varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{ \beta -1}+\frac{1}{\varGamma (\beta )} \biggr] \\ &\qquad {}+\frac{\sum_{i=1}^{m}Q_{i}}{ \varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} \biggr] \biggr\} \bigl\Vert (u_{1}-u_{2},v _{1}-v_{2}) \bigr\Vert \\ &\quad =\varrho \bigl\Vert (u_{1}-u_{2},v_{1}-v_{2}) \bigr\Vert . \end{aligned}$$
(3.3)

According to \((H_{3})\), (3.2), and (3.3), we get

$$\begin{aligned} \bigl\Vert S(u_{1},v_{1})-S(u_{2},v_{2}) \bigr\Vert &=\max \bigl\{ \bigl\Vert S_{1}(u_{1},v_{1})-S_{1}(u _{2},v_{2}) \bigr\Vert _{\mathrm{PC}_{\alpha }}, \bigl\Vert S_{2}(u_{1},v_{1})-S_{2}(u_{2},v_{2}) \bigr\Vert _{\mathrm{PC}_{\beta }}\bigr\} \\ & \leq \max \{\kappa ,\varrho \} \bigl\Vert (u_{1},v_{1}) \bigr\Vert < \bigl\Vert (u_{1},v_{1}) \bigr\Vert . \end{aligned}$$
(3.4)

Therefore, (3.4) means that \(S: X\rightarrow X\) defined by (3.1) is a contraction. According to Lemma 2.3, S has a unique fixed point \((u^{*}(t),v^{*}(t))\in X\), which is a pair of unique solutions of system (1.1). The proof of Theorem 3.1 is completed. □

Theorem 3.2

Let \(e, f, g, h\in C[a,T])\), \(I_{k}\), \(J_{k}\in C(\mathbb{R}, \mathbb{R})\), \(k=1,2,\ldots ,m\). Assume that there exist some positive constants \(L_{1}\), \(L_{2}\), \(\overline{L}_{1}\), \(\overline{L}_{2}\), \(O_{k}\), and \(\overline{O}_{k}\) (\(k=1,2,\ldots ,m\)) such that

\((H_{4})\) :

\(|e(t,u,v)|\leq L_{1}\), \(|g(t,u,v)|\leq L_{2}\), \(|f(t,u,v)|\leq \overline{L}_{1}\), \(|h(t,u,v)|\leq \overline{L}_{2}\), \(|I_{k}(u)|\leq O_{k}\) and \(|J_{k}(v)|\leq \overline{O}_{k}\) (\(k=1,2, \ldots ,m\)) for all \(t\in (a,T]\), \(u,v\in \mathbb{R}\).

If \((H_{2})\) and \((H_{4})\) hold, then the Hadamard impulsive fractional differential coupled system (1.1) has at least a pair of solutions \((u^{*}(t),v^{*}(t))\).

Proof

Define the operator \(S: X\rightarrow X\) as (3.1). In order to apply the Leray–Schauder alternative theorem, we need first to prove that S is completely continuous. Indeed, in view of the continuities of e, g, f, h, \(I_{k}\), and \(J_{k}\), it is easy to know that T is continuous.

Now we show that the operator S is uniformly bounded. Let \(r>0\), \(B_{r}=\{(u,v)\in X, \|(u,v)\|\leq r\}\) be any bounded subset of X. For all \((u,v)\in B_{r}\), \(t\in [a,T]\), it follows from \((H_{4})\) that

$$\begin{aligned} &\bigl\vert S_{1}(u,v) (t) \bigr\vert \\ &\quad = \Biggl\vert {}_{H}J_{t_{k}}^{\alpha }g \bigl(t,u(t),v(t)\bigr)+{}_{H}J_{t_{k}}^{ \alpha }e \bigl(t,u(t),v(t)\bigr) +\frac{1}{\delta } \Biggl({}_{H}J_{t_{m}}^{ \alpha }g \bigl(T,u(T),v(T)\bigr) \\ &\qquad {}+{}_{H}J_{t_{m}}^{\alpha }e\bigl(T,u(T),v(T) \bigr) +\frac{ (\ln \frac{T}{t _{m}} )^{\alpha -1}}{\varGamma (\alpha )} \sum_{i=1}^{m} \bigl[I_{i}\bigl(u(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}g \bigl(t_{i},u(t _{i}),v(t_{i})\bigr) \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1}e \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr] \Biggr) \biggl(\ln \frac{t}{t _{k}} \biggr)^{\alpha -1} +\frac{\varLambda }{\varGamma (\alpha )} \sum_{i=1} ^{k} \bigl[I_{i} \bigl(u(t_{i})\bigr) \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1}g \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) +{}_{H}J_{t_{i-1}} ^{1}e\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} \Biggr\vert \\ &\quad \leq {}_{H}J_{t_{k}}^{\alpha } \bigl\vert g\bigl(t,u(t),v(t)\bigr) \bigr\vert +{}_{H}J_{t_{k}}^{\alpha } \bigl\vert e\bigl(t,u(t),v(t)\bigr) \bigr\vert +\frac{1}{\delta } \Biggl({}_{H}J_{t_{m}}^{\alpha } \bigl\vert g \bigl(T,u(T),v(T)\bigr) \bigr\vert \\ &\qquad {}+{}_{H}J_{t_{m}}^{\alpha } \bigl\vert e \bigl(T,u(T),v(T)\bigr) \bigr\vert +\frac{ (\ln \frac{T}{t _{m}} )^{\alpha -1}}{\varGamma (\alpha )}\sum _{i=1}^{m} \bigl[ \bigl\vert I_{i} \bigl(u(t _{i})\bigr) \bigr\vert +{}_{H}J_{t_{i-1}}^{1} \bigl\vert g\bigl(t_{i},u(t_{i}),v(t_{i}) \bigr) \bigr\vert \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl\vert e \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr\vert \bigr] \Biggr) \biggl(\ln \frac{t}{t _{k}} \biggr)^{\alpha -1} + \frac{1}{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl[ \bigl\vert I_{i}\bigl(u(t_{i})\bigr) \bigr\vert \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1} \bigl\vert g \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr\vert +{}_{H}J_{t_{i-1}} ^{1} \bigl\vert e \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr\vert \bigr] \biggl(\ln \frac{t}{t_{k}} \biggr) ^{\alpha -1} \\ &\quad \leq \frac{L_{1}+L_{2}}{\varGamma (\alpha )} \int _{t_{k}}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\alpha -1}\,\frac{ds}{s} +\frac{1}{\delta } \Biggl( \frac{L_{1}+L_{2}}{\varGamma (\alpha )} \int _{t_{m}}^{T} \biggl( \ln \frac{T}{s} \biggr)^{\alpha -1}\,\frac{ds}{s} +\frac{ (\ln \frac{T}{t _{m}} )^{\alpha -1}}{\varGamma (\alpha )}\sum _{i=1}^{m} \biggl[O_{i} \\ &\qquad {}+\frac{L_{1}+L_{2}}{\varGamma (\alpha )} \int _{t_{i-1}}^{t_{i}} \,\frac{ds}{s} \biggr] \Biggr) \biggl(\ln \frac{t}{t_{k}} \biggr)^{\alpha -1} +\frac{1}{\varGamma (\alpha )}\sum _{i=1}^{k} \biggl[O_{i} + \frac{L_{1}+L _{2}}{\varGamma (\alpha )} \int _{t_{i-1}}^{t_{i}}\,\frac{ds}{s} \biggr] \biggl(\ln \frac{t}{t _{k}} \biggr)^{\alpha -1} \\ &\quad \leq \frac{L_{1}+L_{2}}{\varGamma (\alpha )} \int _{a}^{T} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1}\,\frac{ds}{s} +\frac{1}{\delta } \Biggl( \frac{L _{1}+L_{2}}{\varGamma (\alpha )} \int _{a}^{T} \biggl(\ln \frac{T}{a} \biggr)^{ \alpha -1}\,\frac{ds}{s}\\ &\qquad {} +\frac{ (\ln \frac{T}{a} )^{\alpha -1}}{ \varGamma (\alpha )} \Biggl[\sum _{i=1}^{m}O_{i} +\frac{L_{1}+L_{2}}{\varGamma (\alpha )} \int _{a}^{T}\,\frac{ds}{s} \Biggr] \Biggr) \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1}\\ &\qquad {} +\frac{1}{ \varGamma (\alpha )} \Biggl[ \sum_{i=1}^{m}O_{i} + \frac{L_{1}+L_{2}}{\varGamma (\alpha )} \int _{a}^{T}\,\frac{ds}{s} \Biggr] \biggl(\ln \frac{T}{a} \biggr) ^{\alpha -1} \\ &\quad = \frac{L_{1}+L_{2}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{ \alpha } \biggl[1+\frac{1}{\delta } \biggl(\ln \frac{T}{a} \biggr)^{ \alpha -1} +\frac{1}{\delta \varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr) ^{\alpha -1}+\frac{1}{\varGamma (\alpha )} \biggr] \\ &\qquad {}+\frac{\sum_{i=1}^{m}P_{i}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr) ^{\alpha -1} \biggr]\triangleq A. \end{aligned}$$
(3.5)

Similarly, we have

$$\begin{aligned} \bigl\vert S_{2}(u,v) (t) \bigr\vert \leq{} &\frac{\overline{L}_{1}+\overline{L}_{2}}{\varGamma (\beta )} \biggl( \ln \frac{T}{a} \biggr)^{\beta } \biggl[1+\frac{1}{\rho } \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} + \frac{1}{\rho \varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{\beta -1}+ \frac{1}{\varGamma (\beta )} \biggr] \\ &{}+\frac{\sum_{i=1}^{m}\overline{O}_{i}}{\varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr) ^{\beta -1} \biggr]\triangleq B. \end{aligned}$$
(3.6)

From (3.5) and (3.6), we know that S is uniformly bounded.

Next, we show that the operator S is equicontinuous. In fact, for any \(\tau _{1}, \tau _{2}\in [a,T]\), \(\tau _{1}<\tau _{2}\), \(\tau _{2}-\tau _{1}\) is small enough such that \(\tau _{1}, \tau _{2}\in [t_{k},t_{k+1}]\), \(k=0,1,2, \ldots , m\), we have

$$\begin{aligned} &\bigl\vert S_{1}(u,v) (\tau _{2})-S_{1}(u,v) (\tau _{1}) \bigr\vert \\ &\quad = \Biggl\vert {}_{H}J_{t_{k}} ^{\alpha }g\bigl(\tau _{2},u(\tau _{2}),v(\tau _{2})\bigr)-{}_{H}J_{t_{k}}^{\alpha }g\bigl( \tau _{1},u(\tau _{1}),v(\tau _{1})\bigr) \\ &\qquad {}+{}_{H}J_{t_{k}}^{\alpha }e\bigl(\tau _{2},u(\tau _{2}),v(\tau _{2}) \bigr)-{}_{H}J_{t_{k}}^{\alpha }e\bigl(\tau _{1},u(\tau _{1}),v(\tau _{1})\bigr) \\ &\qquad {}+c^{*} \biggl[ \biggl(\ln \frac{\tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] +\frac{ \varLambda }{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl[I_{i}\bigl(u(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}g \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1}e \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr] \biggl[ \biggl(\ln \frac{ \tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] \Biggr\vert \\ &\quad =\Biggl\vert \frac{1}{\varGamma (\alpha )} \int _{t_{k}}^{\tau _{2}} \biggl(\ln \frac{ \tau _{2}}{s} \biggr)^{\alpha -1}g\bigl(s,u(s),v(s)\bigr)\,\frac{ds}{s} - \frac{1}{ \varGamma (\alpha )} \int _{t_{k}}^{\tau _{1}} \biggl(\ln \frac{\tau _{1}}{s} \biggr) ^{\alpha -1}g\bigl(s,u(s),v(s)\bigr)\,\frac{ds}{s} \\ &\qquad {}+\frac{1}{\varGamma (\alpha )} \int _{t_{k}}^{\tau _{2}} \biggl(\ln \frac{ \tau _{2}}{s} \biggr)^{\alpha -1}e\bigl(s,u(s),v(s)\bigr)\,\frac{ds}{s} - \frac{1}{ \varGamma (\alpha )} \int _{t_{k}}^{\tau _{1}} \biggl(\ln \frac{\tau _{1}}{s} \biggr) ^{\alpha -1}e\bigl(s,u(s),v(s)\bigr)\,\frac{ds}{s} \\ &\qquad {}+c^{*} \biggl[ \biggl(\ln \frac{\tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] +\frac{ \varLambda }{\varGamma (\alpha )}\sum_{i=1}^{k} \bigl[I_{i}\bigl(u(t_{i})\bigr) +{}_{H}J_{t_{i-1}}^{1}g \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \\ &\qquad {}+{}_{H}J_{t_{i-1}}^{1}e \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr] \biggl[ \biggl(\ln \frac{ \tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] \Biggr\vert \\ &\quad \leq\frac{1}{\varGamma (\alpha )} \biggl\{ \int _{t_{k}}^{\tau _{1}} \biggl[ \biggl(\ln \frac{\tau _{2}}{s} \biggr)^{\alpha -1} - \biggl(\ln \frac{ \tau _{1}}{s} \biggr)^{\alpha -1} \biggr] \bigl\vert g\bigl(s,u(s),v(s)\bigr) \bigr\vert \,\frac{ds}{s} + \int _{\tau _{1}}^{\tau _{2}} \biggl(\ln \frac{\tau _{2}}{s} \biggr)^{ \alpha -1} \\ &\qquad {}\times \bigl\vert g\bigl(s,u(s),v(s)\bigr) \bigr\vert \,\frac{ds}{s} + \int _{t_{k}}^{\tau _{1}} \biggl[ \biggl(\ln \frac{\tau _{2}}{s} \biggr)^{\alpha -1} - \biggl(\ln \frac{ \tau _{1}}{s} \biggr)^{\alpha -1} \biggr] \bigl\vert e\bigl(s,u(s),v(s)\bigr) \bigr\vert \,\frac{ds}{s} \\ &\qquad {}+ \int _{\tau _{1}}^{\tau _{2}} \biggl(\ln \frac{\tau _{2}}{s} \biggr) ^{\alpha -1} \bigl\vert e\bigl(s,u(s),v(s)\bigr) \bigr\vert \,\frac{ds}{s} \biggr\} + \bigl\vert c^{*} \bigr\vert \biggl[ \biggl(\ln \frac{ \tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] \\ &\qquad {}+\frac{1}{\varGamma (\alpha )}\sum_{i=1}^{m} \bigl[ \bigl\vert I_{i}\bigl(u(t_{i})\bigr) \bigr\vert +{}_{H}J_{t_{i-1}}^{1} \bigl\vert g \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr\vert +{}_{H}J_{t_{i-1}} ^{1} \bigl\vert e \bigl(t_{i},u(t_{i}),v(t_{i})\bigr) \bigr\vert \bigr] \\ &\qquad {}\times \biggl[ \biggl(\ln \frac{\tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] \\ &\quad \leq\frac{1}{\varGamma (\alpha )} \biggl\{ L_{1} \int _{t_{k}}^{\tau _{1}} \biggl[ \biggl(\ln \frac{\tau _{2}}{s} \biggr)^{\alpha -1} - \biggl(\ln \frac{ \tau _{1}}{s} \biggr)^{\alpha -1} \biggr]\,\frac{ds}{s} +L_{1} \int _{\tau _{1}}^{\tau _{2}} \biggl(\ln \frac{\tau _{2}}{s} \biggr)^{\alpha -1} \,\frac{ds}{s} \\ &\qquad {}+L_{2} \int _{t_{k}}^{\tau _{1}} \biggl[ \biggl(\ln \frac{\tau _{2}}{s} \biggr) ^{\alpha -1} - \biggl(\ln \frac{\tau _{1}}{s} \biggr)^{\alpha -1} \biggr] \,\frac{ds}{s} +L_{2} \int _{\tau _{1}}^{\tau _{2}} \biggl(\ln \frac{\tau _{2}}{s} \biggr)^{\alpha -1}\,\frac{ds}{s} \biggr\} \\ &\qquad {}+ \bigl\vert c^{*} \bigr\vert \biggl[ \biggl(\ln \frac{\tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{\tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr] +\frac{1}{ \varGamma (\alpha )} \Biggl[\sum _{i=1}^{m}O_{i} +\frac{L_{1}}{\varGamma ( \alpha )} \int _{a}^{t_{m}}\,\frac{ds}{s} \\ &\qquad {}+\frac{L_{2}}{\varGamma (\alpha )} \int _{a}^{t_{m}}\,\frac{ds}{s} \Biggr] \biggl[ \biggl(\ln \frac{\tau _{2}}{t_{k}} \biggr)^{\alpha -1}- \biggl(\ln \frac{ \tau _{1}}{t_{k}} \biggr)^{\alpha -1} \biggr]\rightarrow 0, \quad \mbox{as } \tau _{1} \rightarrow \tau _{2}. \end{aligned}$$
(3.7)

We similarly get

$$\begin{aligned} \bigl\vert S_{2}(u,v) (\tau _{2})-S_{2}(u,v) (\tau _{1}) \bigr\vert \rightarrow 0, \quad \mbox{as } \tau _{1}\rightarrow \tau _{2}. \end{aligned}$$
(3.8)

(3.7) and (3.8) mean that S is equicontinuous. By the Ascoli–Arzelá theorem, we know that S is completely continuous.

Finally, we prove that the set \(\varepsilon (S)=\{(u,v)\in X |(u,v)= \lambda S(u,v), 0<\lambda <1\}\) is bounded. Let \((u,v)\in \varepsilon (S)\), then \((u,v)=\lambda S(u,v)\), for any \(t\in [a,T]\), we have

$$\begin{aligned} u(t)=\lambda S_{1}(u,v) (t), \qquad v(t)=\lambda S_{2}(u,v) (t). \end{aligned}$$

For \(t\in (t_{k},t_{k+1}]\), \(k=0,1,2,\ldots ,m\), it follows from (3.5) and (3.6) that

$$\begin{aligned} \bigl\vert u(t) \bigr\vert = \bigl\vert \lambda S_{1}(u,v) (t) \bigr\vert =\lambda \bigl\vert S_{1}(u,v) (t) \bigr\vert \leq \lambda A \end{aligned}$$
(3.9)

and

$$\begin{aligned} \bigl\vert v(t) \bigr\vert = \bigl\vert \lambda S_{2}(u,v) (t) \bigr\vert =\lambda \bigl\vert S_{2}(u,v) (t) \bigr\vert \leq \lambda B. \end{aligned}$$
(3.10)

(3.9) and (3.10) implicate that \(\varepsilon (S)\) is bounded for any \(t\in [a,T]\). In view of Lemma 2.4, the operator S defined by (3.1) has at least one fixed point. Hence, the Hadamard impulsive fractional differential coupled system (1.1) has at least a pair of solutions \((u^{*}(t), v^{*}(t))\). The proof is completed. □

Illustrative examples

Consider the nonlinear Hadamard fractional integro-differential coupled system with impulses as follows:

$$\begin{aligned} \textstyle\begin{cases} {}_{\mathrm{RLH}}D_{t_{k}}^{\frac{3}{2}}[u(t)-{}_{H}J_{t_{k}}^{\frac{3}{2}}e(t,u(t),v(t))] =g(t,u(t),v(t)), \quad t\in J=[1,e], t\neq t_{1}=\frac{4}{3}, \\ {}_{\mathrm{RLH}}D_{t_{k}}^{\frac{5}{4}}[v(t)-{}_{H}J_{t_{k}}^{\frac{5}{4}}f(t,u(t),v(t))] =h(t,u(t),v(t)), \quad t\in J=[1,e], t\neq t_{1}=\frac{4}{3}, \\ {}_{\mathrm{RLH}}D_{t_{1}}^{\frac{1}{2}}u(t_{1}^{+}) -{}_{\mathrm{RLH}}D_{t_{1}}^{ \frac{1}{2}}u(t_{1}^{-})=I_{1}(u(t_{1})), \\ _{\mathrm{RLH}}D_{t_{1}}^{\frac{1}{4}}v(t_{1}^{+}) -{}_{\mathrm{RLH}}D_{t_{1}}^{ \frac{1}{4}}v(t_{1}^{-})=J_{1}(v(t_{1})), \\ 3{}_{H}D_{1}^{\frac{1}{2}}u(1)=u(e), \qquad 4{}_{H}D_{1}^{\frac{1}{4}}v(1)=v(e). \end{cases}\displaystyle \end{aligned}$$
(4.1)

Case 1

Take \(g(t,u,v)=\frac{(u^{2}+v^{3})\cos 2t}{180}\), \(e(t,u,v)=\frac{(u+v)\sin t}{50}\), \(f(t,u,v)=\frac{e^{-t}(\sqrt[3]{u}+ \sqrt[5]{v})}{150}\), \(h(t,u,v)=\frac{\sin u\cos v\arcsin t}{20\pi }\), \(I_{1}(u)=\frac{u^{2}}{10}\), \(J_{1}(v)=\frac{u^{4}}{20}\). Obviously, \(e,g,f,h\in C[1,e]\), \(I_{1},J_{1}\in C(\mathbb{R},\mathbb{R})\). By the simple calculation, we have

$$\begin{aligned} & \bigl\vert g(t,u_{1},v_{1})-g(t,u_{2},v_{2}) \bigr\vert \leq \frac{1}{90} \vert u_{1}-u_{2} \vert + \frac{1}{60} \vert v_{1}-v_{2} \vert , \\ & \bigl\vert e(t,u_{1},v_{1})-e(t,u_{2},v_{2}) \bigr\vert \leq \frac{1}{50} \vert u_{1}-u_{2} \vert + \frac{1}{50} \vert v_{1}-v_{2} \vert , \\ & \bigl\vert f(t,u_{1},v_{1})-f(t,u_{2},v_{2}) \bigr\vert \leq \frac{1}{50} \vert u_{1}-u_{2} \vert + \frac{1}{30} \vert v_{1}-v_{2} \vert , \\ & \bigl\vert h(t,u_{1},v_{1})-h(t,u_{2},v_{2}) \bigr\vert \leq \frac{1}{10} \vert u_{1}-u_{2} \vert + \frac{1}{10} \vert v_{1}-v_{2} \vert , \\ & \bigl\vert I_{1}(u_{1})-e(u_{2}) \bigr\vert \leq \frac{1}{5} \vert u_{1}-u_{2} \vert , \qquad \bigl\vert J_{1}(v_{1})-J_{1}(v_{2}) \bigr\vert \leq \frac{1}{5} \vert v_{1}-v_{2} \vert , \end{aligned}$$

that is, \(M_{1}=\frac{1}{90}\), \(M_{2}=\frac{1}{60}\), \(N_{1}=N_{2}= \frac{1}{50}\), \(\overline{M}_{1}=\frac{1}{50}\), \(\overline{M}_{2}= \frac{1}{30}\), \(\overline{N}_{1}=\overline{N}_{1}=\frac{1}{10}\), \(P_{1}=Q_{1}=\frac{1}{5}\). Thus, we obtain

$$\begin{aligned} &\delta =c\varGamma (\alpha )- \biggl(\ln \frac{T}{t_{1}} \biggr)^{\alpha -1} \approx 1.8147>0,\qquad \rho =d\varGamma (\beta )- \biggl(\ln \frac{T}{t _{1}} \biggr)^{\beta -1}\approx 2.7069>0, \\ &\begin{aligned} \kappa ={}& \frac{M_{1}+M_{2}+N_{1}+N_{2}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha } \biggl[1+\frac{1}{\delta } \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} +\frac{1}{\delta \varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1}+\frac{1}{\varGamma (\alpha )} \biggr] \\ &{}+\frac{\sum_{i=1}^{m}P_{i}}{\varGamma (\alpha )} \biggl(\ln \frac{T}{a} \biggr)^{\alpha -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr) ^{\alpha -1} \biggr]\approx 0.7038< 1, \end{aligned} \\ &\begin{aligned} \varrho = {}& \frac{\overline{M}_{1}+\overline{M}_{2}+\overline{N}_{1}+ \overline{N}_{2}}{\varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr)^{ \beta } \biggl[1+\frac{1}{\rho } \biggl(\ln \frac{T}{a} \biggr)^{ \beta -1} +\frac{1}{\rho \varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr) ^{\beta -1}+\frac{1}{\varGamma (\beta )} \biggr] \\ &{}+\frac{\sum_{i=1}^{m}Q_{i}}{\varGamma (\beta )} \biggl(\ln \frac{T}{a} \biggr) ^{\beta -1} \biggl[1+ \biggl(\ln \frac{T}{a} \biggr)^{\beta -1} \biggr] \approx 0.7379< 1. \end{aligned} \end{aligned}$$

Therefore, conditions \((H_{1})\)\((H_{3})\) of Theorem 3.1 hold. Then (4.1) has a pair of unique solutions \((u^{*}(t),v^{*}(t)) \in \mathrm{PC}_{\frac{3}{2}}[1,e]\times \mathrm{PC}_{\frac{5}{4}}[1,e]\).

Case 2

Take \(e(t,u,v)=g(t,u,v)=f(t,u,v)=h(t,u,v)=\sin \sqrt[2]{3}t+e^{-(u+v)^{2}}+\arctan (tuv)\), \(I_{1}(u)=\arccos u^{2}\), \(J_{1}(v)=\frac{1}{1+v^{2}}\). Obviously, \(e,g,f,h\in C[1,e]\), \(I_{1},J_{1}\in C(\mathbb{R},\mathbb{R})\). \(|e(t,u,v)|=|g(t,u,v)|=|f(t,u,v)|=|h(t,u,v)|=| \sin \sqrt{3}t+e^{-(u+v)^{2}}+\arctan (tuv)|\leq \frac{\pi }{2}+1+\frac{ \pi }{2}=\pi +1\), \(|I_{1}(u)|=|\arccos u^{2}|\leq \pi \), \(|J_{1}(v)|=|\frac{1}{1+v ^{2}}|\leq 1\). Thus, conditions \((H_{1})\) and \((H_{4})\) hold. According to Theorem 3.2, we know that (4.1) has at least a pair of solutions \((u^{*}(t),v^{*}(t))\).

Conclusions

In describing some phenomena and processes of many fields such as physics, chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, capacitor theory, electrical circuits, biology, control theory, fitting of experimental data, and so on, the fractional differential equation is better and more accurate than the integer-order differential equations. Therefore, the study of fractional differential equations has attracted the eyes of many scholars. Good papers involving the dynamics of the fractional differential equation are emerging in large numbers. However, it was noticed that most of these works are based on Riemann–Liouville and Caputo fractional derivatives. In fact, another kind of fractional derivatives was introduced by Hadamard in 1892. It differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition contains a logarithmic function of arbitrary exponent. Relatively speaking, this fractional differential equation with Hadamard derivatives is still studied less than that of Riemann–Liouville and Caputo. So it is worth studying the Hadamard fractional differential equations. In this paper, we consider the boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses. By means of the Banach contraction principle and Leray–Schauder alternative theorem, some new sufficient criteria are established to guarantee the existence and uniqueness of solutions.

References

  1. 1.

    Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4, 75–89 (1977)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    Google Scholar 

  3. 3.

    Oldham, K., Spanier, J.: The Fractional Calculus. Academic, New York (1974)

    Google Scholar 

  4. 4.

    Olmstead, W., Handelsman, R.: Diffusion in a semi-infinite region with nonlinear surface dissipation. SIAM Rev. 18, 275–291 (1976)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent – II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967) reprinted in Fract. Calc. Appl. Anal. 11, 4–14 (2008)

    Article  Google Scholar 

  6. 6.

    Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971) reprinted in Fract. Calc. Appl. Anal. 10, 310–323 (2007)

    Article  Google Scholar 

  7. 7.

    Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971)

    Article  Google Scholar 

  8. 8.

    Marks, I.I., Hall, M.: Differintegral interpolation from a bandlimited signal’s samples. IEEE Trans. Acoust. Speech Signal Process. 29, 872–877 (1981)

    Article  Google Scholar 

  9. 9.

    Torvik, P., Bagley, R.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)

    Article  Google Scholar 

  10. 10.

    Bai, J., Feng, X.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492–2502 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chern, J.: Finite element modeling of viscoelastic materials on the theory of fractional calculus. Ph.D. thesis, Pennsylvania State University (1993)

  12. 12.

    Freed, A., Diethelm, K., Luchko, Y.: Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus (first annual report). Technical Memorandum 2002-211914, NASA Glenn Research Center, Cleveland (2002)

  13. 13.

    Dokoumetzidis, A., Magin, R., Macheras, P.: A commentary on fractionalization of multicompartmental models. J. Pharmacokinet. Pharmacodyn. 37, 203–207 (2010)

    Article  Google Scholar 

  14. 14.

    Popović, J., Atanacković, M., Pilipović, A., Rapaić, M., Pilipović, S., Atanacković, T.: A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac. J. Pharmacokinet. Pharmacodyn. 37, 119–134 (2010)

    Article  Google Scholar 

  15. 15.

    Verotta, D.: Fractional compartmental models and multi-term Mittag-Leffler response functions. J. Pharmacokinet. Pharmacodyn. 37, 209–215 (2010)

    Article  Google Scholar 

  16. 16.

    Freed, A., Diethelm, K.: Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional-derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 5, 203–215 (2006)

    Article  Google Scholar 

  17. 17.

    Magin, R.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)

    Google Scholar 

  18. 18.

    Caponetto, R., Dongola, G., Fortuna, L., Petráš, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific, River Edge (2010)

    Google Scholar 

  19. 19.

    Podlubny, I.: Fractional-order systems and fractional-order controllers. Technical report UEF-03-94, Institute for Experimental Physics, Slovak Acad. Sci. (1994)

  20. 20.

    Kiryakova, V.: The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl. 59, 1128–1141 (2010)

    MathSciNet  Article  Google Scholar 

  21. 21.

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

    Google Scholar 

  22. 22.

    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential. Equations: North-Holland Mathematics Studies., vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  23. 23.

    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)

    Google Scholar 

  24. 24.

    Tarasov, V.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2010)

    Google Scholar 

  25. 25.

    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Google Scholar 

  26. 26.

    Goodrich, C.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Lakshmikantham, V., Leela, S.: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal. 71, 2886–2889 (2009)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Zhang, X., Liu, L., Wu, Y.: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55(3), 1263–1274 (2012)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Tariboon, J., Ntouyas, S., Sudsutad, W.: Fractional integral problems for fractional differential equations via Caputo derivative. Adv. Differ. Equ. 2014, 181 (2014)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Zhao, K., Gong, P.: Existence of positive solutions for a class of higher-order Caputo fractional differential equation. Qual. Theory Dyn. Syst. 14(1), 157–171 (2015)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Zhu, C., Zhang, X., Wu, Z.: Solvability for a coupled system of fractional differential equations with integral boundary conditions. Taiwan. J. Math. 17(6), 2039–2054 (2013)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Zhao, K.: Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays. Dyn. Syst. 30(2), 208–223 (2015)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Yang, W.: Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 63(1), 288–297 (2012)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Zhao, K.: Impulsive boundary value problems for two classes of fractional differential equation with two different Caputo fractional derivatives. Mediterr. J. Math. 13, 1033–1050 (2016)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Henderson, J., Luca, R.: Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Appl. Math. Comput. 249, 182–197 (2014)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Liu, B., Li, J., Liu, L., Wang, Y.: Existence and uniqueness of nontrivial solutions to a system of fractional differential equations with Riemann–Stieltjes integral conditions. Adv. Differ. Equ. 2018, 306 (2018)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Agarwal, P., Chand, M., Baleanu, D., O’Regan, D., Jain, S.: On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function. Adv. Differ. Equ. 2018, 249 (2018)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Lu, Z., Zhu, Y.: Comparison principles for fractional differential equations with the Caputo derivatives. Adv. Differ. Equ. 2018, 237 (2018)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Xu, M., Han, Z.: Positive solutions for integral boundary value problem of two-term fractional differential equations. Bound. Value Probl. 2018, 100 (2018)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Zhang, W., Liu, W.: Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance. Bound. Value Probl. 2018, 36 (2018)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Gaafar, F.: The existence of solutions for a nonlinear first-order differential equation involving the Riemann–Liouville fractional-order and nonlocal condition. Mediterr. J. Math. 15, 191 (2018)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Zhai, C., Li, P.: Nonnegative solutions of initial value problems for Langevin equations involving two fractional orders. Mediterr. J. Math. 15, 164 (2018)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Yukunthorn, W., Suantai, S., Ntouyas, S., Tariboon, J.: Boundary value problems for impulsive multi-order Hadamard fractional differential equations. Bound. Value Probl. 2015, 148 (2015)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Benchohra, M., Bouriah, S., Graef, J.: Boundary value problems for nonlinear implicit Caputo–Hadamard-type fractional differential equations with impulses. Mediterr. J. Math. 14, 206 (2017)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Ntouyas, S., Tariboon, J., Sudsutad, W.: Boundary value problems for Riemann–Liouville fractional differential inclusions with nonlocal Hadamard fractional integral conditions. Mediterr. J. Math. 13, 939–954 (2016)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Ahmad, B., Nitouyas, S.: Boundary value problems of Hadamard-type fractional differential equations and inclusions with nonlocal conditions. Vietnam J. Math. 45, 409–423 (2017)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Zhang, Y., Wang, J.: Existence and finite-time stability results for impulsive fractional differential equations with maxima. J. Appl. Math. Comput. 51, 67–79 (2016)

    MathSciNet  Article  Google Scholar 

  49. 49.

    Mohamed, I.: On the Hadamard and Riemann–Liouville fractional neutral functional integrodifferential equations with finite delay. J. Pseudo-Differ. Oper. Appl. (2018). https://doi.org/10.1007/s11868-018-0244-1

    Article  Google Scholar 

  50. 50.

    Wang, H., Liu, Y., Zhu, H.: Existence and stability for Hadamard p-type fractional functional differential equations. J. Appl. Math. Comput. 55, 549–562 (2017)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Kiataramkul, C., Ntouyas, S., Tariboon, J., Kijjathanakon, A.: Generalized Sturm–Liouville and Langevin equations via Hadamard fractional derivatives with anti-periodic boundary conditions. Bound. Value Probl. 2016, 217 (2016)

    MathSciNet  Article  Google Scholar 

  52. 52.

    Ahmad, B., Ntouyas, S., Alsaedi, A.: New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions. Bound. Value Probl. 2013, 275 (2013)

    MathSciNet  Article  Google Scholar 

  53. 53.

    Alsaedi, A., Ntouyas, S., Ahmad, B., Hobiny, A.: Nonlinear Hadamard fractional differential equations with Hadamard type nonlocal non-conserved conditions. Adv. Differ. Equ. 2015, 285 (2015)

    MathSciNet  Article  Google Scholar 

  54. 54.

    Tariboon, J., Ntouyas, S., Sudsutad, W.: Nonlocal Hadamard fractional integral conditions for nonlinear Riemann–Liouville fractional differential equations. Bound. Value Probl. 2014, 253 (2014)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Ahmad, B., Ntouyas, S.: On Hadamard fractional integro-differential boundary value problems. J. Appl. Comput. 47, 119–131 (2015)

    MathSciNet  Article  Google Scholar 

  56. 56.

    Wang, J., Zhang, Y.: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 39, 85–90 (2015)

    MathSciNet  Article  Google Scholar 

  57. 57.

    Huang, H., Liu, W.: Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter. Adv. Differ. Equ. 2018, 96 (2018)

    MathSciNet  Article  Google Scholar 

  58. 58.

    Thiramanus, P., Ntouyas, S., Tariboon, J.: Positive solutions for Hadamard fractional differential equations on infinite domain. Adv. Differ. Equ. 2016, 83 (2016)

    MathSciNet  Article  Google Scholar 

  59. 59.

    Yang, W.: Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions. J. Appl. Math. Comput. 49, 357–381 (2015)

    MathSciNet  Article  Google Scholar 

  60. 60.

    Sudsutad, W., Ntouyas, S., Tariboon, J.: Systems of fractional Langevin equations of Riemann–Liouville and Hadamard types. Adv. Differ. Equ. 2015, 235 (2015)

    MathSciNet  Article  Google Scholar 

  61. 61.

    Zhang, X., Shu, T., Cao, H., 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  Google Scholar 

  62. 62.

    Zhang, X.: The general solution of differential equations with Caputo–Hadamard fractional derivatives and impulsive effect. Adv. Differ. Equ. 2015, 215 (2015)

    MathSciNet  Article  Google Scholar 

  63. 63.

    Zhang, W., Liu, W.: Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval. Bound. Value Probl. 2018, 134 (2018)

    MathSciNet  Article  Google Scholar 

  64. 64.

    Butzer, P., Kilbas, A., Trujillo, J.: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387–400 (2002)

    MathSciNet  Article  Google Scholar 

  65. 65.

    Butzer, P., Kilbas, A., Trujillo, J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)

    MathSciNet  Article  Google Scholar 

  66. 66.

    Wang, J., Fečkan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19(4), 806–831 (2016)

    MathSciNet  Article  Google Scholar 

  67. 67.

    Yang, D., Wang, J., O’Regan, D.: On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses. C. R. Math. 356(2), 150–171 (2018)

    MathSciNet  Article  Google Scholar 

  68. 68.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cone. Academic Press, Orlando (1988)

    Google Scholar 

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Acknowledgements

The authors thank the referees for a number of suggestions which have improved many aspects of this article.

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This work was supported by the National Natural Sciences Foundation of Peoples Republic of China under Grant (Nos. 11161025, 11661047).

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Zhao, K., Suo, L. & Liao, Y. Boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses. Bound Value Probl 2019, 105 (2019). https://doi.org/10.1186/s13661-019-1219-8

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MSC

  • 34B10
  • 34B15
  • 34B37

Keywords

  • Boundary value problem
  • Hadamard fractional differential coupled system
  • Impulses
  • Leray–Schauder alternative theorem
  • Existence and uniqueness