Open Access

Solvability of a boundary value problem for singular multi-term fractional differential system with impulse effects

Boundary Value Problems20152015:244

https://doi.org/10.1186/s13661-015-0504-4

Received: 12 August 2015

Accepted: 4 December 2015

Published: 30 December 2015

Abstract

In this article, we first of all convert the boundary value problems for impulsive fractional differential equations to integral equations. Second, we construct a weighted function space and prove the completely continuous property of a nonlinear operator. Finally, we establish existence results for solutions of a boundary value problem for a nonlinear impulsive fractional differential system. Our analysis relies on the well-known Schauder fixed point theorem. An example is given to illustrate main results.

Keywords

impulsive multi-term fractional differential system boundary value problem Schauder’s fixed point theorem

MSC

92D25 34A37 34K15

1 Introduction

Fractional differential equations have many applications in modeling of physical and chemical processes [1, 2]. In its turn, mathematical aspects of fractional differential equations and methods of their solutions were discussed by many authors; see the text books [35]. Existence of solutions or positive solutions of boundary value problems (BVPs for short) of fractional differential equations with or without impulse effects have been studied by many authors; see [616] and [1719].

In recent years, some authors have studied the solvability or existence of positive solutions of BVPs of fractional differential systems [2028]. In order to show motivations of this paper, we address some of them.

In [29], Su investigated the existence of positive solutions (continuous on \([0,1]\)) of the following boundary value problem of nonlinear multi-term fractional differential system:
$$ \textstyle\begin{cases} D_{0+}^{\alpha} u+ f(t,v(t), D_{0^{+}}^{p}v(t))=0,\quad 0< t< 1,\\ D_{0+}^{\beta} v+ g(t,u(t), D_{0^{+}}^{q}u(t))=0,\quad 0< t< 1,\\ u(0)=0, \quad\quad u(1)=0,\\ v(0)=0,\quad\quad v(1)=0, \end{cases} $$
(1)
where \(\alpha,\beta\in(1,2)\), \(D_{0+}\) is the Riemann-Liouville fractional derivative, \(0< p<\beta-1\), \(0< q<\alpha-1\), \(\gamma\eta ^{\alpha-1}<1\), and \(\gamma\eta^{\beta-1}<1\), \(f,g:[0,1]\times \mathbb {R}^{+}\times \mathbb {R}\to \mathbb {R}\) are continuous functions.
In [30], authors studied the existence of multiple positive solutions (continuous on \([0,1]\)) of the following boundary value problem of N-dimension nonlinear fractional differential system:
$$ \textstyle\begin{cases} D_{0+}^{\alpha_{1}} u_{1}+ f_{1}(t,u_{2}(t), D_{0^{+}}^{\mu_{1}}u_{2}(t))=0, \quad 0< t< 1,\\ \ldots,\\ D_{0+}^{\alpha_{N-1}} u_{N-1}+ f_{N-1}(t,u_{N}(t), D_{0^{+}}^{\mu _{N}}u_{N}(t))=0,\quad 0< t< 1,\\ D_{0+}^{\alpha_{N}} u_{N}+ f_{N}(t,u_{1}(t), D_{0^{+}}^{\mu_{N}}u_{1}(t))=0, \quad 0< t< 1,\\ u_{1}(0)=\cdots=u_{N}(0)=0,\\ u_{1}(1)=\cdots=u_{N}(1)=0, \end{cases} $$
(2)
where \(\alpha_{i}\in(1,2)\), \(D_{0+}\) is the Riemann-Liouville fractional derivative, \(0<\mu_{i-1}<\alpha_{i}-1\) with \(\mu_{0}=\mu_{N}\), \(f_{i}:[0,1]\times \mathbb {R}^{+}\times \mathbb {R}\to \mathbb {R}\) (\(i=1,2,\ldots,N\)) are continuous functions.
In [31], the authors investigated the existence of positive solutions (continuous on \([0,1]\)) of the following boundary value problem of nonlinear multi-term fractional differential system:
$$ \textstyle\begin{cases} D_{0+}^{\alpha} u+ f(t,v(t), D_{0^{+}}^{p}v(t))=0,\quad 0< t< 1,\\ D_{0+}^{\beta} v+ g(t,u(t), D_{0^{+}}^{q}u(t))=0,\quad 0< t< 1,\\ u(0)=0,\quad\quad u(1)=\gamma u(\eta),\\ v(0)=0,\quad\quad v(1)=\gamma v(\eta), \end{cases} $$
(3)
where \(\alpha,\beta\in(1,2)\), \(D_{0+}\) is the Riemann-Liouville fractional derivative, \(0< p\le\beta-1\), \(0< q\le\alpha-1\), \(\gamma\eta ^{\alpha-1}<1\), and \(\gamma\eta^{\beta-1}<1\), \(f,g:[0,1]\times \mathbb {R}^{+}\times \mathbb {R}\to \mathbb {R}\) are continuous functions.
In [32], the authors studied the existence of solutions of the following four-point coupled boundary value problem for nonlinear fractional differential equation:
$$ \textstyle\begin{cases} D_{0+}^{\alpha} u= f(t,u(t),D_{0^{+}}^{\alpha-1}u(t),v(t),D_{0^{+}}^{\beta -1}v(t)),\quad 0< t< 1,\\ D_{0+}^{\beta} v=g(t,u(t),D_{0^{+}}^{\alpha-1}u(t),v(t),D_{0^{+}}^{\beta -1}v(t)),\quad 0< t< 1,\\ I_{0^{+}}^{2-\alpha}u(0)=0,\quad\quad u(1)=a v(\xi),\\ I_{0^{+}}^{2-\beta}v(0)=0,\quad\quad v(1)=b u(\eta), \end{cases} $$
(4)
where \(1 <\alpha,\beta<2\), \(D_{0^{+}}^{*}\), and \(I_{0^{+}}^{*}\) are the standard Riemann-Liouville differentiation and integration, \(f,g:[0,1]\times \mathbb {R}^{4}\to \mathbb {R}\) are continuous functions, \(a,b\in \mathbb {R}\), \(\xi ,\eta\in(0,1)\) with \(ab\xi^{\beta-1}\eta^{\alpha-1}=1\).
In [33], the following four-point boundary value problem of multi-term fractional differential system:
$$ \textstyle\begin{cases} D_{0+}^{\alpha} u= f(t,v(t),D_{0^{+}}^{m}v(t)),\quad 0< t< 1,\\ D_{0+}^{\beta} v=g(t,u(t),D_{0^{+}}^{n}u(t)),\quad 0< t< 1,\\ u(0)=\gamma u(\xi),\quad\quad u(1)=\delta u(\eta),\\ v(0)=\gamma v(\xi),\quad\quad v(1)=\delta v(\eta), \end{cases} $$
(5)
was studied, where \(1 <\alpha,\beta<2\), \(0< m\le\beta-1\), \(0< n\le\alpha -1\), \(\gamma>0\), \(\delta>0\), \(0<\xi<\eta<1\), \(D_{0^{+}}^{*}\) is the standard Riemann-Liouville differentiation, \(f,g:[0,1]\times \mathbb {R}^{4}\to \mathbb {R}\) are continuous functions.

In [34], the authors studied a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations involving Caputo type fractional derivatives. Sufficient conditions for the existence and uniqueness of positive solutions were established by using Banach’s fixed point theorem and Krasnoselskii’s fixed point theorem.

In [35], the authors studied the following 2m-point boundary value problem for a coupled system of impulsive fractional differential equations at resonance:
$$\textstyle\begin{cases} D_{0+}^{\alpha} u= f(t,v(t),D_{0^{+}}^{p}v(t)),\quad 0< t< 1,\\ D_{0+}^{\beta} v=g(t,u(t),D_{0^{+}}^{q}u(t)), \quad 0< t< 1,\\ \Delta u(t_{i})=A_{i}(v(t_{i}),D_{0^{+}}^{p} v(t_{i})),\quad\quad \Delta D_{0^{+}}^{q} u(t_{i})=B_{i}(v(t_{i}),D_{0^{+}}^{p} v(t_{i})),\quad i=1,2,\ldots,k,\\ \Delta v(t_{i})=C_{i}(u(t_{i}),D_{0^{+}}^{q} u(t_{i})),\quad\quad \Delta D_{0^{+}}^{p} v(t_{i})=D_{i}(u(t_{i}),D_{0^{+}}^{q} u(t_{i})),\quad i=1,2,\ldots,k,\\ D_{0^{+}}^{\alpha-1}u(0)=\sum _{i=1}^{m} a_{i}u(\xi_{i}),\quad\quad u(1)=\sum _{i=1}^{m}b_{i}\eta_{i}^{2-\alpha}u(\eta_{i}),\\ D_{0^{+}}^{\beta-1}v(0)=\sum _{i=1}^{m} c_{i}v(\zeta_{i}), \quad\quad v(1)=\sum _{i=1}^{m}d_{i}\theta_{i}^{2-\beta}v(\eta_{i}), \end{cases} $$
where \(\alpha,\beta\in(1,2)\), \(\alpha-q\ge1\), \(\beta-p\ge1\), \(0<\xi _{1}<\cdots<\xi_{m}<1\), \(0<\eta_{1}<\cdots<\eta_{m}<1\), \(0<\zeta_{1}<\cdots<\zeta _{m}<1\), and \(0<\theta_{1}<\cdots<\theta_{m}<1\), \(f,g:[0,1]\times \mathbb {R}^{2}\to \mathbb {R}\) satisfy Carathéodory conditions, \(A_{i},B_{i},C_{i},D_{i}:\mathbb {R}\times \mathbb {R}\to \mathbb {R}\), \(\Delta w(t_{i})=w(t_{i}^{+})-w(t_{i}^{-})\), \(\Delta D_{0^{+}}^{r}w(t_{i})=D_{0^{+}}^{r}w(t_{i}^{+})-D_{0^{+}}^{r}w(t_{i}^{-})\) with \(w\in\{u,v\} \) and \(r\in\{p,q\}\), \(w(t^{+}_{i})\) and \(w(t_{i}^{-})\) denote the right and left limits of \(w(t)\) at \(t = t_{i}\), respectively, and the fractional derivative is understood in the Riemann-Liouville sense. k, m, \(a_{i}\), \(b_{i}\), \(c_{i}\), \(d_{i}\) (\(i=1,2,\ldots,m\)) are fixed constants satisfying \(\sum _{i=1}^{m}a_{i}=\sum _{i=1}^{m}b_{i}=\sum _{i=1}^{m}c_{i}=\sum _{i=1}^{m}d_{i}=1\) and \(\sum _{i=1}^{m}b_{i}\eta_{i}=\sum _{i=1}^{m}d_{i}\theta_{i}\). This system happens to be at resonance in the sense that the associated linear homogeneous coupled system
$$\textstyle\begin{cases} D_{0+}^{\alpha} u=0,\quad\quad D_{0+}^{\beta} v=0,\quad 0< t< 1,\\ D_{0^{+}}^{\alpha-1}u(0)=\sum _{i=1}^{m} a_{i}D_{0^{+}}^{\alpha-1}u(\xi _{i}),\quad\quad u(1)=\sum _{i=1}^{m}b_{i}\eta_{i}^{2-\alpha}u(\eta_{i}),\\ D_{0^{+}}^{\beta-1}v(0)=\sum _{i=1}^{m} c_{i}D_{0^{+}}^{\beta-1}v(\zeta _{i}),\quad\quad v(1)=\sum _{i=1}^{m}d_{i}\theta_{i}^{2-\beta}v(\eta_{i}) \end{cases} $$
has \((u(t),v(t))=(h_{1}t^{\alpha-1}+h_{2}t^{\alpha-2},h_{3}t^{\beta -1}+h_{4}t^{\beta-2})\) (\(h_{i}\in R\), \(i=1,2,3,4\)) as a nontrivial solution.

We find in the papers mentioned that f in fractional differential equations is supposed to be continuous, the solutions obtained are also continuous on \([0,1]\). So it is interesting to study the solvability of boundary value problems of singular fractional differential equations and to obtain discontinuous solutions of this kind of problems.

In this paper, we study the existence of solutions of the following boundary value problems (BVP for short) for the multi-term impulsive fractional differential system:
$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=p(t)f (t,v(t),D_{0^{+}}^{\theta}v(t),D_{0^{+}}^{\beta-1}v(t) ),\quad t\in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\ D_{0^{+}}^{\beta}v(t)=q(t)g (t,u(t),D_{0^{+}}^{\delta}u(t),D_{0^{+}}^{\alpha-1}u(t) ),\quad t\in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\ D_{0^{+}}^{\alpha-1}u(0)=0, \quad\quad u(1)=0,\quad\quad D_{0^{+}}^{\beta-1}v(0)=0, \quad\quad v(1)=0,\\ \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha }u(t)=I_{1}(t_{i},v(t_{i}),D_{0^{+}}^{\delta}v(t_{i}),D_{0^{+}}^{\beta -1}v(t_{i})),\quad i\in \mathbb {N}_{1}^{m},\\ \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}u(t)-D_{0^{+}}^{\alpha -1}u(t_{i})=J_{1}(t_{i},v(t_{i}),D_{0^{+}}^{\delta}v(t_{i}),D_{0^{+}}^{\beta -1}v(t_{i})),\quad i\in \mathbb {N}_{1}^{m},\\ \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\beta }v(t)=I_{2}(t_{i},u(t_{i}),D_{0^{+}}^{\delta}u(t_{i}),D_{0^{+}}^{\alpha -1}u(t_{i})),\quad i\in \mathbb {N}_{1}^{m},\\ \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\beta-1}v(t)-D_{0^{+}}^{\beta -1}v(t_{i})=J_{2}(t_{i},u(t_{i}),D_{0^{+}}^{\delta}u(t_{i}),D_{0^{+}}^{\alpha -1}u(t_{i})),\quad i\in \mathbb {N}_{1}^{m}, \end{cases} $$
(6)
where
  1. (a)

    \(\alpha,\beta\in(1,2)\), \(0<\delta<\alpha-1\), \(0<\theta <\beta-1\), \(D_{a^{+}}^{b}\) is the Riemann-Liouville fractional derivative of order \(b>0\) with the start point \(a\in R\), see Definition 2.2,

     
  2. (b)

    m is a positive integer, \(0=t_{0}< t_{1}< t_{2}<\cdots <t_{m}<t_{m+1}=1\), \(\mathbb {N}_{0}^{m}=\{0,1,2,\ldots,m\}\), and \(\mathbb {N}_{1}^{m}=\{1,2,\ldots ,m\}\),

     
  3. (c)

    \(p,q\in C^{0}(0,1)\) there exists \(k_{i}>-1\), \(l_{i}\le0\) with \(2+k_{i}+l_{i}>0\), \(\alpha+l_{1}-\delta>0\), \(\beta+l_{2}-\theta>0\) such that \(|p(t)|\le t^{k_{1}}(1-t)^{l_{1}}\) and \(|q(t)|\le t^{k_{2}}(1-t)^{l_{2}}\) for all \(t\in(0,1)\),

     
  4. (d)

    \(f:(0,1)\times \mathbb {R}^{3}\to \mathbb {R}\) is a \((\theta,\beta;p)\)-Carathéodory function, \(g:(0,1)\times \mathbb {R}^{3}\to \mathbb {R}\) a \((\delta,\alpha ;q)\)-Carathéodory function, see Definition 2.3, \(I_{1},J_{1}:\{t_{i}\} \times \mathbb {R}^{3}\to \mathbb {R}\) are \((\theta,\beta)\)-Carathéodory functions, \(I_{2},J_{2}:\{t_{i}\}\times \mathbb {R}^{3}\to \mathbb {R}\) \((\delta,\alpha)\)-Carathéodory functions, see Definition 2.4.

     
A pair of functions \(u,v:(0,1]\to \mathbb {R}\) is called a solution of (6) if
$$\begin{aligned}& v|_{(t_{i},t_{i+1}]},D_{0^{+}}^{\theta}v|_{(t_{i},t_{i+1}]},D_{0^{+}}^{\beta -1}v|_{(t_{i},t_{i+1}]} \in C^{0}(t_{i},t_{i+1}],\quad i\in \mathbb {N}_{0}^{m}, \\& u|_{(t_{i},t_{i+1}]},D_{0^{+}}^{\delta}u|_{(t_{i},t_{i+1}]},D_{0^{+}}^{\alpha -1}u|_{(t_{i},t_{i+1}]} \in C^{0}(t_{i},t_{i+1}],\quad i\in \mathbb {N}_{0}^{m}, \\& \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\beta}v(t), \lim _{t\to t_{i}^{+}}(t-t_{i})^{2+\theta-\beta}D_{0^{+}}^{\theta}v(t), \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\beta-1}v(t)\hbox{ are finite}, \quad i \in \mathbb {N}_{0}^{m}, \\& \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha}u(t), \lim _{t\to t_{i}^{+}}(t-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}u(t), \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}u(t) \hbox{ are finite},\quad i \in \mathbb {N}_{0}^{m}, \end{aligned}$$
\(D_{0^{+}}^{\alpha}u\) and \(D_{0^{+}}^{\beta}v\) are an α-well integrable function and a β-well integrable function, respectively, and all equations in (6) are satisfied. Here the definitions of an α-well integrable function and a β-well integrable function may be found in Definition 2.5.

The purpose of this paper is to obtain the results on the existence of solutions of BVP (6) by using the Schauder fixed point theorem [16] under some suitable assumptions. The solutions obtained may be discontinuous on \([0,1]\).

The main features of our paper are as follows. First of all, compared with the well-known papers [2933], we construct a new Banach space and establish the existence results of solutions of (6). Second, boundary conditions are different from the known ones and impulse effects are imposed. Third, boundary conditions and impulse effects in (6) imply that solutions obtained in this paper are continuous on \((t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)) but they may be unbounded on \((0,1)\). Finally, both \(pf:(t,x,y)\mapsto p(t)f(t,x,y)\) and \(qg:(t,x,y)\mapsto q(t)g(t,x,y)\) may be singular at \(t=0,1\) while in the well-known papers mentioned nonlinearities are supposed to be continuous.

The remainder of the paper is organized as follows: In Section 2, we present some preliminary results. In Section 3, the existence results for solutions of BVP (6) are established. Finally, in Section 4, we present an example to illustrate the main theorems.

2 Preliminary results

In this section, we present some necessary definitions from the fractional calculus theory which can be found in the literature [15].

Definition 2.1

[1]

The Riemann-Liouville fractional integral of order \(\alpha>0\) of a function \(h:(a,+\infty)\mapsto R\) is given by
$$ I_{a^{+}}^{\alpha}h(t)=\frac{1}{\Gamma(\alpha)} \int_{a}^{t}(t-s)^{\alpha-1}h(s)\,ds, $$
(7)
provided that the right-hand side exists.

Definition 2.2

[1]

The Riemann-Liouville fractional derivative of order \(\alpha>0\) of a function \(h:(a,+\infty)\mapsto R\) is given by
$$ D_{a^{+}}^{\alpha}h(t)=\frac{1}{\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}} \int_{a}^{t}\frac{h(s)}{(t-s)^{\alpha-n+1}}\,ds, $$
(8)
where \(n-1\le\alpha< n\), provided that the right-hand side exists.

Lemma 2.1

(Schauder fixed point theorem)

Let Ω be a closed convex subset of the Banach space X. Suppose \(T:\Omega\mapsto \Omega\) and T is compact (i.e., bounded sets in Ω are mapped into relatively compact sets). Then T has a fixed point in Ω.

Remark 2.1

Let \(a< b\). From (2.106) in [5], we have \(D_{a^{+}}^{\gamma}I_{a^{+}}^{\gamma}h(t)=h(t)\), \(t\in[a,b]\), \(\gamma>0\) if f satisfies some suitable assumptions. For example we know \((\int _{a}^{t}h(s)\,ds )'=h(t)\) for all continuous function \(h:[a,b]\to \mathbb {R}\). From (2.108), in [5], for a function \(h:[a,b]\mapsto \mathbb {R}\), if \(D_{a^{+}}^{\alpha}h(t)\) (\(D_{a^{+}}^{\delta}h(t)\), \(D_{a^{+}}^{\alpha-1} h(t)\)) is integrable, then we have \(A,B,C,D\in \mathbb {R}\) such that
$$\begin{aligned}& I_{a^{+}}^{\alpha}D_{a^{+}}^{\alpha}h(t)=h(t)+A(t-a)^{\alpha-1}+B(t-a)^{\alpha -2},\quad t\in(a,b], \\& I_{a^{+}}^{\delta}D_{a^{+}}^{\delta}h(t)=h(t)+C(t-a)^{\delta-1},\quad t\in(a,b], \\& I_{a^{+}}^{\alpha-1} D_{a^{+}}^{\alpha-1} h(t)=h(t)+D(t-a)^{\alpha-2},\quad t\in(a,b]. \end{aligned}$$
These results are generalizations of \(\int h'(s)\,ds=h(t)+C\) and \([\int h(t)\,dt ]'=h(t)\) if h is an absolutely continuous function. If h is a piecewise continuous function, what are the results? We give the following lemma.

Lemma 2.2

Let \(\alpha\in(1,2)\). Suppose that \(h:(0,1]\mapsto \mathbb {R}\) satisfies \(h|_{(t_{i},t_{i+1}]}\), \(D_{0^{+}}^{\alpha-1}h\in C^{0}(t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)), \(\lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha}h(t)\), and \(\lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}h(t)\) exist for all \(i\in \mathbb {N}_{0}^{m}\). We can prove that there exist constants \(c_{i,j}\in \mathbb {R}\) (\(i\in N_{0}^{m}\), \(j=1,2\)) such that
$$ I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(t)=h(t)+ \sum _{j=0}^{i} \bigl(c_{1,j}(t-t_{j})^{\alpha-1}+c_{2,j}(t-t_{j})^{\alpha-2} \bigr),\quad t\in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, $$
(9)
and
$$ D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}h(t)=h(t),\quad t \in(0,1]. $$
(10)

Proof

Step 1. We first of all prove (10). Since \(\lim _{t\to0^{+}}t^{2-\alpha}h(t)=A_{0}\) exists, for any \(\epsilon>0\), we have \(A_{0}-\epsilon< t^{2-\alpha}h(t)< A_{0}+\epsilon\) for sufficiently small \(t\in(0,t_{1}]\). Then there exists \(A>0\) such that \(|h(t)|\le At^{\alpha-2}\) for all \(t\in(t_{0},t_{1}]\). We have for \(t\in(t_{0},t_{1}]\)
$$\begin{aligned} \biggl\vert \int_{0}^{t}(t-v)^{\alpha-1}h(v)\,dv\biggr\vert \le& \int_{0}^{t}(t-v)^{\alpha -1}\bigl|h(v)\bigr|\,dv \\ \le& A \int_{0}^{t}(t-v)^{\alpha-1}v^{\alpha-2}\,dv \quad\hbox{by }\frac{v}{t}=w \\ =&At^{2\alpha-2} \int_{0}^{1}(1-w)^{\alpha-1}w^{\alpha-2}\,dw. \end{aligned}$$
Then \(\int_{0}^{t}(t-v)^{\alpha-1}h(v)\,dv\) is convergent for all \(t\in (t_{0},t_{1}]\). So for \(t\in(0,t_{1}]\), we have
$$\begin{aligned} D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}h(t) =& \frac{1}{\Gamma(2-\alpha)\Gamma (\alpha)} \biggl[ \int_{0}^{t}(t-s)^{1-\alpha} \int_{0}^{s}(s-v)^{\alpha -1}h(v)\,dv\,ds \biggr]'' \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)} \biggl[ \int_{0}^{t} \int _{s}^{t}(t-s)^{1-\alpha}(s-v)^{\alpha-1}\,dsh(v)\,dv \biggr]'' \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)} \biggl[ \int_{0}^{t}(t-v) \int _{0}^{1}(1-w)^{1-\alpha}w^{\alpha-1}\,dwh(v)\,dv \biggr]'' \\ =& \biggl[ \int_{0}^{t}(t-v)h(v)\,dv \biggr]''=h(t). \end{aligned}$$
Then \(D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}h(t)=h(t)\), \(t\in(t_{0},t_{1}]\). For \(t\in(t_{i+1},t_{i+2}]\) (\(i\ge0\)), similar to above discussion, use that \(\lim _{t\to t_{j}^{+}}(t-t_{j})^{2-\alpha}h(t)\) exists, we know that \(\int_{t_{j}}^{t_{j+1}}(t-v)^{\alpha-1}h(v)\,dv\) and \(\int _{t_{i+1}}^{t}(t-v)^{\alpha-1}h(v)\,dv\) are convergent. So \(\int _{0}^{t}(t-v)^{\alpha-1}h(v)\,dv\) is convergent. Then similarly to the above discussion \(D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}h(t)=h(t)\). The proof of (10) is complete.
Step 2. We prove (9). Let \(D_{0^{+}}^{\alpha}h(t)=H(t)\). Since \(\lim _{t\to0^{+}}t^{2-\alpha}h(t)=A_{0}\) exists, for any \(\epsilon >0\), we have \(A_{0}-\epsilon< t^{2-\alpha}h(t)< A_{0}+\epsilon\) for sufficiently small \(t\in(0,t_{1}]\). Then for sufficiently small \(t\in (0,t_{1}]\), we have
$$\begin{aligned} (A_{0}-\epsilon)\mathbf{B}(2-\alpha,\alpha-1) \le&(A_{0}- \epsilon) \int _{0}^{t}(t-s)^{1-\alpha}s^{\alpha-2}\,ds< \int_{0}^{t}(t-s)^{1-\alpha}h(s)\,ds \\ < &(A_{0}+\epsilon) \int_{0}^{t}(t-s)^{1-\alpha}s^{\alpha-2}\,ds=(A_{0}+ \epsilon )\mathbf{B}(2-\alpha,\alpha-1). \end{aligned}$$
Then
$$\begin{aligned} (A_{0}-\epsilon)\mathbf{ B}(2-\alpha,\alpha-1) \le& \mathop{\underline{\lim}}_{t\to0^{+}} \int_{0}^{t}(t-s)^{1-\alpha}h(s)\,ds \\ \le& \mathop{\overline{\lim}} _{t\to0^{+}} \int_{0}^{t}(t-s)^{1-\alpha}h(s)\,ds\le (A_{0}+\epsilon)\mathbf{B}(2-\alpha,\alpha-1). \end{aligned}$$
Let \(\epsilon\to0\), we get \(\lim _{t\to0^{+}}\int _{0}^{t}(t-s)^{1-\alpha}h(s)\,ds=A_{0}\mathbf{ B}(2-\alpha,\alpha-1)\). Furthermore, since \(\lim _{t\to0^{+}}D_{0^{+}}^{\alpha-1}h(t)=B\) exists, \(\lim _{t\to0^{+}} (\int_{0}^{t}(t-s)^{1-\alpha}h(s)\,ds )'=\Gamma (2-\alpha)B\) exists by Definition 2.2. So
$$\begin{aligned} I_{0^{+}}^{\alpha}H(t) =&I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(t)\\ =&\frac{1}{\Gamma (2-\alpha)\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \biggl( \int _{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)''\,ds \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)} \biggl[ \int_{0}^{t}(t-s)^{\alpha -1}\,d \biggl( \int_{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)' \biggr] \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)}\biggl[ (t-s)^{\alpha-1} \biggl( \int _{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)'\bigg|_{0}^{t} \\ &{}+(\alpha-1) \int_{0}^{t}(t-s)^{\alpha-2} \biggl( \int_{0}^{s}(s-u)^{1-\alpha }h(u)\,du \biggr)'\,ds\biggr] \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)}\biggl[-t^{\alpha-1}\lim _{s\to0^{+}} \biggl( \int_{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)' \\ &{}+ \biggl( \int_{0}^{t}(t-s)^{\alpha-1} \biggl( \int_{0}^{s}(s-u)^{1-\alpha }h(u)\,du \biggr)'\,ds \biggr)'\biggr] \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)}\biggl[-t^{\alpha-1}\lim _{s\to0^{+}} \biggl( \int_{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)' \\ &{}+ \biggl[ \int_{0}^{t}(t-s)^{\alpha-1}\,d \biggl( \int_{0}^{s}(s-u)^{1-\alpha }h(u)\,du \biggr) \biggr]'\biggr] \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)}\biggl[-t^{\alpha-1}\lim _{s\to0^{+}} \biggl( \int_{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)' \\ &{}- \biggl(t^{\alpha-1}\lim _{s\to0^{+}} \int_{0}^{s}(s-u)^{1-\alpha }h(u)\,du \biggr)' \\ &{}+(\alpha-1) \biggl( \int_{0}^{t}(t-s)^{\alpha-2} \int_{0}^{s}(s-u)^{1-\alpha }h(u)\,du\,ds \biggr)'\biggr] \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)}\biggl[-t^{\alpha-1}\lim _{s\to0^{+}} \biggl( \int_{0}^{s}(s-u)^{1-\alpha}h(u)\,du \biggr)' \\ &{}-(\alpha-1)t^{\alpha-2}\lim _{s\to0^{+}} \int_{0}^{s}(s-u)^{1-\alpha }h(u)\,du \\ &{}+(\alpha-1) \biggl( \int_{0}^{t} \int_{s}^{t}(t-s)^{\alpha-2}(s-u)^{1-\alpha }\,dsh(u)\,du \biggr)'\biggr]\quad\hbox{by }\frac{s-u}{t-u} = w \\ =&\frac{1}{\Gamma(2-\alpha)\Gamma(\alpha)}\biggl[-t^{\alpha-1}\Gamma(2-\alpha )B -( \alpha-1)t^{\alpha-2}A_{0}\mathbf{B}(2-\alpha,\alpha-1) \\ &{}+(\alpha-1) \biggl( \int_{0}^{t} \int_{0}^{1}(1-w)^{\alpha-2}w^{1-\alpha }\,dwh(u)\,du \biggr)'\biggr] \\ =:&c_{10}t^{\alpha-1}+c_{20}t^{\alpha-2}+h(t). \end{aligned}$$
So
$$ I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(t)=I_{0^{+}}^{\alpha}H(t)=h(t)+c_{10}t^{\alpha-1}+c_{20}t^{\alpha-2},\quad t\in(0,t_{1}]. $$
(11)
It follows from (11) that (9) holds for \(i=0\). Now we suppose that (9) holds for \(i=0,1,2,\ldots,n< m\). We will prove that (9) holds for \(i=n+1\). Then by the method of mathematical induction, we know (9) holds for all \(i\in \mathbb {N}_{0}^{m}\).
In fact, suppose that
$$ I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(t)=h(t)+ \sum _{j=0}^{n} \bigl(c_{1,j}(t-t_{j})^{\alpha-1}+c_{2,j}(t-t_{j})^{\alpha-2} \bigr)+\Phi (t),\quad t\in(t_{n+1},t_{n+2}]. $$
(12)
Use the assumption, we get for \(t\in(t_{n+1},t_{n+2}]\)
$$\begin{aligned} H(t) =&D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}H(t)=D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(t)\\ =&\frac{1}{\Gamma(2-\alpha)} \biggl[ \int _{0}^{t}(t-s)^{1-\alpha}I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(s)\,ds \biggr]'' \\ =&\frac{1}{\Gamma(2-\alpha)} \Biggl[\sum _{j=0}^{n} \int _{t_{j}}^{t_{j+1}}(t-s)^{1-\alpha}I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(s)\,ds+ \int_{t_{n+1}}^{t}(t-s)^{1-\alpha}I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}h(s)\,ds \Biggr]'' \\ =&\frac{1}{\Gamma(2-\alpha)}\Biggl[\sum _{j=0}^{n} \int _{t_{j}}^{t_{j+1}}(t-s)^{1-\alpha} \Biggl( h(s)+ \sum _{\nu=0}^{j} \bigl(c_{1,\nu}(s-t_{\nu})^{\alpha-1}+c_{2,\nu }(s-t_{\nu})^{\alpha-2} \bigr) \Biggr)\,ds \\ &{}+ \int_{t_{n+1}}^{t}(t-s)^{1-\alpha} \Biggl(h(s)+\sum _{\nu=0}^{n} \bigl(c_{1,\nu}(s-t_{\nu})^{\alpha-1}+c_{2,\nu}(s-t_{j})^{\alpha-2} \bigr)+\Phi(s) \Biggr)\,ds\Biggr]'' \\ =&D_{t_{n+1}^{+}}^{\alpha}\Phi(t)+D_{0^{+}}^{\alpha}h(t) +\frac{1}{\Gamma(2-\alpha)}\Biggl[\sum _{j=0}^{n}\sum _{\nu =0}^{j}c_{1,\nu} \int_{t_{j}}^{t_{j+1}}(t-s)^{1-\alpha} (s-t_{\nu})^{\alpha-1}\,ds \\ &{}+\sum _{j=0}^{n}\sum _{\nu=0}^{j}c_{2,\nu} \int _{t_{j}}^{t_{j+1}}(t-s)^{1-\alpha} (s-t_{\nu})^{\alpha-2}\,ds \\ &{}+\sum _{\nu=0}^{n}c_{1,\nu} \int_{t_{n+1}}^{t}(t-s)^{1-\alpha}(s-t_{\nu})^{\alpha-1}\,ds+\sum _{\nu=0}^{n} c_{2,\nu} \int_{t_{n+1}}^{t}(t-s)^{1-\alpha}(s-t_{j})^{\alpha-2}\,ds \Biggr]'' \\ =&D_{t_{n+1}^{+}}^{\alpha}\Phi(t)+D_{0^{+}}^{\alpha}h(t) +\frac{1}{\Gamma(2-\alpha)}\Biggl[\sum _{j=0}^{n}\sum _{\nu =0}^{j}c_{1,\nu}(t-t_{\nu}) \int_{\frac{t_{j}-t_{\nu}}{t-t_{\nu}}}^{\frac {t_{j+1}-t_{\nu}}{t-t_{\nu}} }(1-w)^{1-\alpha} w^{\alpha-1}\,dw \\ &{}+\sum _{j=0}^{n}\sum _{\nu=0}^{j}c_{2,\nu} \int_{\frac {t_{j}-t_{\nu}}{t-t_{\nu}}}^{\frac{t_{j+1}-t_{\nu}}{t-t_{\nu}} }(1-w)^{1-\alpha} w^{\alpha-2}\,dw \\ &{}+\sum _{\nu=0}^{n}c_{1,\nu}(t-t_{\nu}) \int_{\frac{t_{n+1}-t_{\nu}}{t-t_{\nu}}}^{1}(1-w)^{1-\alpha}w^{\alpha-1}\,dw \\ &{}+ \sum _{\nu=0}^{n} c_{2,\nu} \int_{\frac{t_{n+1}-t_{\nu}}{t-t_{\nu}}}^{1}(1-w)^{1-\alpha}w^{\alpha-2}\,dw \Biggr]'' \\ =&D_{t_{n+1}^{+}}^{\alpha}\Phi(t)+D_{0^{+}}^{\alpha}h(t) +\frac{1}{\Gamma(2-\alpha)}\Biggl[\sum _{\nu=0}^{n}c_{1,\nu}(t-t_{\nu})\sum _{j=\nu}^{n} \int_{\frac{t_{j}-t_{\nu}}{t-t_{\nu}}}^{\frac {t_{j+1}-t_{\nu}}{t-t_{\nu}} }(1-w)^{1-\alpha} w^{\alpha-1}\,dw \\ &{}+\sum _{\nu=0}^{n}c_{2,\nu}\sum _{j=\nu}^{n} \int_{\frac {t_{j}-t_{\nu}}{t-t_{\nu}}}^{\frac{t_{j+1}-t_{\nu}}{t-t_{\nu}} }(1-w)^{1-\alpha} w^{\alpha-2}\,dw \\ &{}+\sum _{\nu=0}^{n}c_{1,\nu}(t-t_{\nu}) \int_{\frac{t_{n+1}-t_{\nu}}{t-t_{\nu}}}^{1}(1-w)^{1-\alpha}w^{\alpha-1}\,dw \\ &{}+ \sum _{\nu=0}^{n} c_{2,\nu} \int_{\frac{t_{n+1}-t_{\nu}}{t-t_{\nu}}}^{1}(1-w)^{1-\alpha}w^{\alpha -2}\,dw \Biggr]'' \\ =&D_{t_{n+1}^{+}}^{\alpha}\Phi(t)+H(t) +\frac{1}{\Gamma(2-\alpha)}\Biggl[\sum _{\nu=0}^{n}c_{1,\nu}(t-t_{\nu}) \int_{1}^{1}(1-w)^{1-\alpha} w^{\alpha-1}\,dw \\ &{}+\sum _{\nu=0}^{n}c_{2,\nu} \int_{0}^{1}(1-w)^{1-\alpha} w^{\alpha-2}\,dw\Biggr]'' \\ =&D_{t_{n+1}^{+}}^{\alpha}\Phi(t)+H(t). \end{aligned}$$
It follows that \(D_{t_{n+1}^{+}}^{\alpha}\Phi(t)=0\) on \((t_{n+1},t_{n+2}]\). Similarly to the proof of (11), there exist constants \(c_{1,n+1},c_{2,n+1}\in R\) such that \(\Phi (t)=c_{1,n+1}(t-t_{n+1})^{\alpha-1}+c_{2,n+1}(t-t_{n+1})^{\alpha-2}\). Substituting \(\Phi(t)\) into (12), we know that (9) holds for \(i=n+1\). This completes the proof of (9). □

Remark 2.2

Let \(\delta\in(0,1)\). Suppose that \(h:(0,1]\to \mathbb {R}\) satisfies \(h|_{(t_{i},t_{i+1}]}\in C^{0}(t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)) and \(\lim _{t\to t_{i}^{+}}(t-t_{i})^{1-\delta}h(t)\) exists for all \(i\in \mathbb {N}_{0}^{m}\). We can prove that there exist constants \(c_{i}\in \mathbb {R}\) (\(i\in \mathbb {N}_{0}^{m}\)) such that
$$\begin{aligned}& I_{0^{+}}^{\delta}D_{0^{+}}^{\delta}h(t)=h(t)+ \sum _{j=0}^{i}c_{j}(t-t_{j})^{\delta-1},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\& D_{0^{+}}^{\delta}I_{0^{+}}^{\delta}h(t)=h(t),\quad t \in(0,1]. \end{aligned}$$

Proof

The proof is similar to that of Lemma 2.2 and is omitted. □

Definition 2.3

Let \(b>0\), \(c,d\in R\) with \(c< d\) be fixed. A function \(h:(c,d)\to \mathbb {R}\) is called a b-well integrable function on \((c,d)\) if both \(\int_{0}^{t}(t-s)^{b-1}|h(s)|\,ds\) and \(\int _{0}^{t}(t-s)^{b-1}h(s)\,ds\) are convergent for all \(t\in[c,d]\).

Definition 2.4

Let \(b\in(1,2)\) and \(a\in(0,b-1)\) and \(\mu :(0,1)\to R\). \(h:(0,1)\times \mathbb {R}^{3}\to \mathbb {R}\) is called a \((a,b;\mu )\)-Carathéodory function if
  1. (i)

    \(t\mapsto\mu(t)h (t,\frac{x_{1}}{(t-t_{i})^{2-b}},\frac {x_{2}}{(t-t_{i})^{2+a-b}},x_{3} )\) is a b-well integrable function on \((0,1]\) for every \((x_{1},x_{2},x_{3})\in \mathbb {R}^{3}\),

     
  2. (ii)

    \((x_{1},x_{2},x_{3})\mapsto h (t,\frac {x_{1}}{(t-t_{i})^{2-b}},\frac{x_{2}}{(t-t_{i})^{2+a-b}},x_{3} )\) is continuous on \(\mathbb {R}^{3}\) for each \(t\in(t_{i},t_{i+1})\) (\(i\in \mathbb {N}_{0}^{m}\)),

     
  3. (iii)
    for each \(r>0\), there exists \(M_{r}>0\) such that \(|x_{i}|\le r\) (\(i=1,2,3\)) imply that
    $$\biggl\vert h \biggl(t,\frac{x_{1}}{(t-t_{i})^{2-b}},\frac {x_{2}}{(t-t_{i})^{2+a-b}},x_{3} \biggr)\biggr\vert \le M_{r},\quad t\in(t_{i},t_{i+1}),i \in \mathbb {N}_{0}^{m}. $$
     

Definition 2.5

Let \(b\in(1,2)\) and \(a\in(0,b-1)\). \(I:\{ t_{i}:i\in \mathbb {N}_{1}^{m}\}\times \mathbb {R}^{3}\to \mathbb {R}\) is a \((a,b)\)-Carathéodory function if
  1. (i)

    \((x_{1},x_{2},x_{3})\mapsto I (t_{i},\frac {x_{1}}{(t_{i}-t_{i-1})^{2-b}},\frac{x_{2}}{(t_{i}-t_{i-1})^{2+a-b}},x_{3} )\) is continuous on \(\mathbb {R}^{3}\) for each \(i\in N_{1}^{m}\),

     
  2. (ii)
    for each \(r>0\), there exists \(M_{I,r}>0\) such that \(|x_{i}|\le r\) (\(i=1,2,3\)) imply that
    $$\biggl\vert I \biggl(t_{i},\frac{x_{1}}{(t_{i}-t_{i-1})^{2-b}},\frac {x_{2}}{(t_{i}-t_{i-1})^{2+a-b}},x_{3} \biggr)\biggr\vert \le M_{I,r},\quad i\in \mathbb {N}_{1}^{m}. $$
    Choose
    $$\begin{aligned} X_{\delta,\alpha} =& \Bigl\{ u:(0,1]\to R , u, D^{\delta}u, D^{\alpha-1}u \in C^{0}(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m},\\ &\hbox{the following limits exist: } \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha}u(t), i\in \mathbb {N}_{0}^{m},\\ & \lim _{t\to t_{i}^{+}}(t-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}u(t),i\in \mathbb {N}_{0}^{m}, \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}u(t),i \in \mathbb {N}_{0}^{m} \Bigr\} . \end{aligned}$$
    For \(u\in X_{\delta,\alpha}\), define
    $$\|u\|=:\|u\|_{\delta,\alpha}=\max \left \{ \textstyle\begin{array}{l}\displaystyle\sup _{t\in(t_{i},t_{i+1}]}(t-t_{i})^{2-\alpha }\bigl|u(t)\bigr|\\ \displaystyle\sup _{t\in(t_{i},t_{i+1}]}(t-t_{i})^{2+\delta-\alpha}\bigl|D_{0^{+}}^{\delta}u(t)\bigr|\\ \displaystyle\sup _{t\in(t_{i},t_{i+1}]}\bigl|D_{0^{+}}^{\alpha-1}u(t)\bigr| \end{array}\displaystyle :i\in \mathbb {N}_{0}^{m} \right \}. $$
     

Lemma 2.3

\(X_{\delta,\alpha}\) is a Banach space with the norm defined.

Proof

In fact, it is easy to see that X is a normed linear space with the norm \(\|\cdot\|\). Let \(\{x_{u}\}\) be a Cauchy sequence in \(X_{\delta,\alpha}\). Then \(\|x_{u}-x_{v}\|\to0\), \(u,v \to+\infty\). It follows that
$$\begin{aligned}& \sup _{t\in(t_{i},t_{i+1}]}(t-t_{i})^{2-\alpha}\bigl|x_{u}(t)-x_{v}(t)\bigr| \to 0,\quad v,u\to+\infty,i\in \mathbb {N}_{0}^{m}, \\& \sup _{t\in(t_{i},t_{i+1}]}(t-t_{i})^{2+\delta-\alpha }\bigl|D_{0^{+}}^{\delta}x_{u}(t)-D_{0^{+}}^{\delta}x_{v}(t)\bigr| \to0,\quad v,u\to+\infty,i\in \mathbb {N}_{0}^{m}, \\& \sup _{t\in(t_{i},t_{i+1}]}\bigl|D_{0^{+}}^{\alpha -1}x_{u}(t)-D_{0^{+}}^{\alpha-1}x_{v}(t)\bigr| \to0,\quad v,u\to+\infty,i\in \mathbb {N}_{0}^{m}. \end{aligned}$$
Denote \(x_{u,i}=x_{u}|_{(t_{i},t_{i+1}]}\). Since
$$\lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha}x_{u}(t), \lim _{t\to t_{i}^{+}}(t-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}x(t), \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}x_{u}(t) $$
exist, we know that \(t\mapsto(t-t_{i})^{2-\alpha}x_{u,i}(t)\) are continuous on \([t_{i},t_{i+1}]\). Thus \(t\mapsto(t-t_{i})^{2-\alpha}x_{u,i}(t)\) are Cauchy sequences in \(C[t_{i},t_{i+1}]\). So \((t-t_{i})^{2-\alpha }x_{u,i}(t)\) uniformly converges to some \(x_{0,i}\) in \(C[t_{i},t_{i+1}]\) as \(u\to+\infty\). It follows that
$$\sup _{t\in[t_{i},t_{i+1}]}\bigl|(t-t_{i})^{2-\alpha }{x}_{u,i}(t)-x_{0,i}(t)\bigr| \to0,\quad u\to+\infty,i\in \mathbb {N}_{0}^{m}. $$
That is,
$$\sup _{t\in[t_{i},t_{i+1}]}(t-t_{i})^{2-\alpha }\bigl|{x}_{u,i}(t)-(t-t_{i})^{\alpha-1}x_{0,i}(t)\bigr| \to0,\quad u\to+\infty,i\in \mathbb {N}_{0}^{m}. $$
Let \(x_{0}(t)=(t-t_{i})^{\alpha-2}x_{0,i}(t)\) for \(t\in(t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)). It is easy to see that \(x_{0}\in C^{0}(t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)) and the limit \(\lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha}x_{0}(t)\) exists for all \(i\in \mathbb {N}_{0}^{m}\).
Similarly there exist \(y_{0,i},z_{0,i}:(0,1]\to \mathbb {R}\) such that
$$\begin{aligned}& \sup _{t\in[t_{i},t_{i+1}]}\bigl|(t-t_{i})^{2+\delta-\alpha }D_{0^{+}}^{\delta}{x}_{u,i}(t)-y_{0,i}(t)\bigr| \to0,\quad u\to+\infty,i\in \mathbb {N}_{0}^{m}, \\& \sup _{t\in[t_{i},t_{i+1}]}\bigl|D_{0^{+}}^{\alpha -1}x_{u,i}(t)-z_{0,i}(t)\bigr| \to0,\quad u\to+\infty,i\in \mathbb {N}_{0}^{m}. \end{aligned}$$
Let \(y_{0}(t)= (t-t_{i})^{\alpha-\delta-2}y_{0,i}(t)\) and \(z_{0}(t)=z_{0,i}(t)\) for \(t\in(t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)). Then \(y_{0},z_{0}\in C^{0}(t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)) and the limits \(\lim _{t\to t_{i}^{+}}(t-t_{i})^{2+\delta-\alpha}y_{0}(t)\) and \(\lim _{t\to t_{i}^{+}}z_{0}(t)\) exist for all \(i\in \mathbb {N}_{0}^{m}\).
Furthermore, using Lemma 2.2, for \(t\in(t_{i},t_{i+1}]\) there exists \(c_{uj}\in \mathbb {R}\) such that
$$\begin{aligned}& \Biggl\vert x_{u}(t)+\sum _{j=0}^{i}c_{uj}(t-t_{j})^{\delta -1}-I_{0^{+}}^{\delta}y_{0}(t)\Biggr\vert \\& \quad= \bigl|I_{0^{+}}^{\delta}D_{0^{+}}^{\delta}x_{u}(t)-I_{0^{+}}^{\delta}y_{0}(t)\bigr| \\& \quad\le \int_{0}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}\bigl\vert D_{0^{+}}^{\delta}x_{u}(s)- y_{0}(s)\bigr\vert \,ds \\& \quad= \sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}\frac{(t-s)^{\delta -1}}{\Gamma(\delta)}(s-t_{j})^{\alpha-\delta-2} \bigl\vert (s-t_{j})^{2+\delta -\alpha} D_{0^{+}}^{\delta}x_{u}(s)- (s-t_{j})^{2+\delta-\alpha}y_{0}(s)\bigr\vert \, ds \\& \quad\quad{}+ \int_{t_{i}}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}(s-t_{i})^{\alpha -\delta-2} \bigl\vert (s-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}x_{u}(s)- (s-t_{i})^{2+\delta-\alpha}y_{0}(s)\bigr\vert \, ds \\& \quad= \sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}\frac{(t-s)^{\delta -1}}{\Gamma(\delta)}(s-t_{j})^{\alpha-\delta-2} \bigl\vert (s-t_{j})^{2+\delta -\alpha} D_{0^{+}}^{\delta}x_{u}(s)- y_{0,j}(s)\bigr\vert \,ds \\& \quad\quad{}+ \int_{t_{i}}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}(s-t_{i})^{\alpha -\delta-2} \bigl\vert (s-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}x_{u}(s)- y_{0,i}(s)\bigr\vert \, ds \\& \quad\le \Biggl( \sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}\frac {(t-s)^{\delta-1}}{\Gamma(\delta)}(s-t_{j})^{\alpha-\delta-2}\,ds + \int_{t_{i}}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}(s-t_{i})^{\alpha -\delta-2}\,ds \Biggr) \\& \quad\quad{}\times \max \Bigl\{ \sup _{t\in(t_{i},t_{i+1}]}\bigl\vert (t-t_{i})^{2+\delta -\alpha}D_{0^{+}}^{\delta}x_{u}(t)- y_{0,i}(t)\bigr\vert :i\in N_{0} \Bigr\} \\& \quad= \Biggl( \sum _{j=0}^{i-1}(t-t_{j})^{\alpha-2} \int_{0}^{\frac {t_{j+1}-t_{j}}{t-t_{j}}}\frac{(1-w)^{\delta-1}}{\Gamma(\delta)}w^{\alpha -\delta-2}\,dw +(t-t_{i})^{\alpha-2} \int_{0}^{1}\frac{(1-w)^{\delta-1}}{\Gamma(\delta )}w^{\alpha-\delta-2}\,dw \Biggr) \\& \quad\quad{}\times \max \Bigl\{ \sup _{t\in(t_{i},t_{i+1}]}\bigl\vert (t-t_{i})^{2+\delta -\alpha}D_{0^{+}}^{\delta}x_{u}(t)- y_{0,i}(t)\bigr\vert :i\in N_{0} \Bigr\} \\& \quad\le\frac{\mathbf{B}(\delta,\alpha-\delta-1)}{\Gamma(\delta)}\sum _{j=0}^{i}(t-t_{j})^{\alpha-2} \max \Bigl\{ \sup _{t\in(t_{i},t_{i+1}]}\bigl\vert (t-t_{i})^{2+\delta -\alpha}D_{0^{+}}^{\delta}x_{u}(t)- y_{0,i}(t)\bigr\vert :i\in N_{0} \Bigr\} \\& \quad\to0\quad\hbox{as }u\to+\infty. \end{aligned}$$
Then \(\lim _{u\to+\infty} [x_{u}(t)+\sum _{j=0}^{i}c_{uj}(t-t_{j})^{\delta-1} ]=I_{0^{+}}^{\delta}y_{0}(t)\). So \((t-t_{i})^{\alpha-2}x_{0,i}(t)+\sum _{j=0}^{i}c_{0j}(t-t_{j})^{\delta -1}=I_{0^{+}}^{\delta}y_{0}(t)\) (\(i\in \mathbb {N}_{0}^{m}\)). It follows that \(x_{0}(t)+\sum _{j=0}^{i}c_{0j}(t-t_{j})^{\delta-1}=I_{0^{+}}^{\delta}y_{0}(t)\). Thus \(y_{0}(t)=D_{0^{+}}^{\delta}x_{0}(t)\) for \(t\in(t_{i},t_{i+1}]\).
We have similarly for \(t\in(t_{i},t_{i+1}]\)
$$\begin{aligned}& \Biggl\vert x_{u}(t)+\sum _{j=0}^{i}c_{uj}(t-t_{j})^{\alpha -2}-I_{0^{+}}^{\alpha-1}z_{0}(t) \Biggr\vert \\& \quad = \bigl|I_{0^{+}}^{\alpha-1} D_{0^{+}}^{\alpha-1} x_{u}(t)-I_{0^{+}}^{\alpha -1}z_{0}(t)\bigr| \\& \quad\le \int_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert D_{0^{+}}^{\alpha-1} x_{u}(s)- z_{0}(s)\bigr\vert \,ds \\& \quad= \sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}\frac{(t-s)^{\alpha -2}}{\Gamma(\alpha-1)}\bigl\vert D_{0^{+}}^{\alpha-1} x_{u,j}(s)- z_{0,j}(s)\bigr\vert \, ds \\& \quad\quad{}+ \int_{t_{i}}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert D_{0^{+}}^{\alpha-1} x_{u,i}(s)- z_{0,i}(s)\bigr\vert \,ds \\& \quad\le \int_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\,ds \max \Bigl\{ \sup _{t\in(t_{i},t_{i+1}]}\bigl\vert D_{0^{+}}^{\alpha-1}x_{u,i}(t)- z_{0,i}(t)\bigr\vert :i\in N_{0} \Bigr\} \\& \quad= \frac{t^{\alpha-1}}{\Gamma(\alpha)} \max \Bigl\{ \sup _{t\in (t_{i},t_{i+1}]}\bigl\vert D_{0^{+}}^{\alpha-1}x_{u,i}(t)- z_{0,i}(t)\bigr\vert :i\in N_{0} \Bigr\} \to0\quad\hbox{ as }u\to+\infty. \end{aligned}$$
Then \(\lim _{u\to+\infty} [x_{u}(t)+\sum _{j=0}^{i}c_{uj}(t-t_{j})^{\alpha-2} ]=I_{0^{+}}^{\alpha-1}z_{0}(t)\). So \((t-t_{i})^{\alpha-2}x_{0,i}(t)+\sum _{j=0}^{i}c_{0j}(t-t_{j})^{\alpha -2}=I_{0^{+}}^{\alpha-1}z_{0}(t)\) (\(i\in \mathbb {N}_{0}^{m}\)). It follows that \(x_{0}(t)+\sum _{j=0}^{i}c_{0j}(t-t_{j})^{\alpha-2}=I_{0^{+}}^{\alpha-1}z_{0}(t)\). Thus \(z_{0}(t)=D_{0^{+}}^{\alpha-1} x_{0}(t)\) for \(t\in(t_{i},t_{i+1}]\).

From the above discussion, we know that \(x_{u}\to x_{0}\) as \(u\to+\infty\) in \(X_{\delta,\alpha}\). It follows that \(X_{\delta,\alpha}\) is a Banach space. The proof is complete. □

Denote \(E=X_{\delta,\alpha}\times X_{\theta,\beta}\). Define \(\|(x,y)\| \max\{\|x\|_{\delta,\alpha},\|y\|_{\theta,\beta}\}\). Then E is a Banach space. For \(y\in X_{\theta,\beta}\) and \(x\in X_{\delta,\alpha}\), denote
$$\begin{aligned}& F_{y}(t)=p(t)f \bigl(t,y(t),D_{0^{+}}^{\theta}y(t),D_{0^{+}}^{\beta-1}y(t) \bigr), \\& G_{x}(t)=q(t)g\bigl(t,x(t),D_{0^{+}}^{\delta}x(t),D_{0^{+}}^{\alpha-1}x(t)\bigr), \\& I_{1y}(t_{i})=I_{1}\bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i})\bigr), \\& J_{1y}(t_{i})=J_{1}\bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i})\bigr), \\& I_{2x}(t_{i})=I_{2}\bigl(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha -1}x(t_{i})\bigr), \\& J_{2x}(t_{i})=J_{2}\bigl(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha-1}x(t_{i})\bigr). \end{aligned}$$

Lemma 2.4

Suppose that (a)-(d) hold and \((x,y)\in E\). Then \((u,v)\in E\) is a solution of
$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=p(t)f (t,y(t),D_{0^{+}}^{\theta}y(t),D_{0^{+}}^{\beta-1}y(t) ),\quad t\in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\ D_{0^{+}}^{\beta}v(t)=q(t)g (t,x(t),D_{0^{+}}^{\delta}x(t),D_{0^{+}}^{\alpha-1}x(t) ),\quad t\in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\ D_{0^{+}}^{\alpha-1}u(0)=0, \quad\quad u(1)=0, \quad\quad D_{0^{+}}^{\beta-1}v(0)=0, \quad\quad v(1)=0,\\ \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha }u(t)=I_{1}(t_{i},y(t_{i}),D_{0^{+}}^{\delta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i})),\quad i\in \mathbb {N}_{1}^{m},\\ \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}u(t)-D_{0^{+}}^{\alpha -1}u(t_{i})=J_{1}(t_{i},y(t_{i}),D_{0^{+}}^{\delta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i})),\quad i\in \mathbb {N}_{1}^{m},\\ \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\beta }v(t)=I_{1}(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha -1}x(t_{i})),\quad i\in \mathbb {N}_{1}^{m},\\ \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\beta-1}v(t)-D_{0^{+}}^{\beta -1}v(t_{i})=J_{2}(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha -1}x(t_{i})),\quad i\in \mathbb {N}_{1}^{m}, \end{cases} $$
(13)
if and only if
$$\begin{aligned} u(t) =& \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds \\ &{}- \Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds+ \sum _{j=1}^{m}\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha )}J_{1y}(t_{j})+ \sum _{j=1}^{m} (1-t_{j})^{\alpha-2}I_{1y}(t_{j}) \Biggr]t^{\alpha-2} \\ &{}+\sum _{j=1}^{i}I_{1y}(t_{j}) (t-t_{j})^{\alpha-2}+\sum _{j=1}^{i} \frac{J_{1y}(t_{j}) }{\Gamma(\alpha)}(t-t_{j})^{\alpha-1},\quad t\in(t_{i},t_{i+1}],i \in \mathbb {N}_{0}^{m}, \end{aligned}$$
(14)
and
$$\begin{aligned} v(t) =& \int_{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}G_{x}(s)\,ds \\ &{}- \Biggl[ \int_{0}^{1}\frac{(1-s)^{\beta-1}}{\Gamma(\beta)}G_{x}(s)\,ds+ \sum _{j=1}^{m}I_{2x}(t_{j}) (1-t_{j})^{\beta-2} +\sum _{j=1}^{m} \frac{J_{2x}(t_{j})}{\Gamma(\beta)}(1-t_{j})^{\beta -1} \Biggr]t^{\beta-2} \\ &{}+\sum _{j=1}^{i}I_{2x}(t_{j}) (t-t_{j})^{\beta-2}+\sum _{j=1}^{i} \frac{J_{2x}(t_{j}) }{\Gamma(\beta)}(t-t_{j})^{\beta-1},\quad t\in(t_{i},t_{i+1}],i \in \mathbb {N}_{0}^{m}. \end{aligned}$$
(15)

Proof

From \(x\in X\) and \(y\in Y\), we see that there exists a constant \(r>0\) such that \(\|x\|=r<+\infty\). Then there exist constants \(M_{r,f},M_{r,g},M_{1,r,I},M_{2,r,I},M_{1,r,J},M_{2,r,J}\ge0\) such that
$$\begin{aligned} \bigl|F_{y}(t)\bigr| =&\bigl|p(t)\bigr|\bigl\vert f \bigl(t,y(t),D_{0^{+}}^{\theta}y(t),D_{0^{+}}^{\beta -1}y(t) \bigr)\bigr\vert \\ =&\bigl|p(t)\bigr|\biggl\vert f \biggl(t,\frac{(t-t_{i})^{2-\beta}y(t)}{(t-t_{i})^{2-\beta }},\frac{(t-t_{i})^{2+\theta-\beta}D_{0^{+}}^{\theta}y(t)}{(t-t_{i})^{2+\theta -\beta}},D_{0^{+}}^{\beta-1}x(t) \biggr)\biggr\vert \\ \le& M_{r,f}t^{k_{1}}(1-t)^{l_{1}},\quad t\in(t_{i},t_{i+1}),i \in \mathbb {N}_{0}^{m}, \end{aligned}$$
(16)
and similarly
$$\begin{aligned}& \bigl|G_{x}(t)\bigr|\le M_{r,g}t^{k_{2}}(1-t)^{l_{2}},\quad t \in(t_{i},t_{i+1}),i\in \mathbb {N}_{0}^{m}, \\& \bigl|I_{1}\bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta-1}y(t_{i})\bigr)\bigr|\le M_{1,r,I},\quad i\in \mathbb {N}_{1}^{m}, \\& \bigl|I_{2}\bigl(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha-1}x(t_{i})\bigr)\bigr|\le M_{2,r,I},\quad i\in \mathbb {N}_{1}^{m}, \\& \bigl|J_{1}\bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta-1}y(t_{i})\bigr)\bigr|\le M_{1,r,J},\quad i\in \mathbb {N}_{1}^{m}, \\& \bigl|J_{2}\bigl(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha-1}x(t_{i})\bigr)\bigr|\le M_{2,r,J},\quad i\in \mathbb {N}_{1}^{m}. \end{aligned}$$
(17)
Hence
$$\begin{aligned} \biggl\vert \int _{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha )}F_{y}(s)\,ds \biggr\vert \le& \int _{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\bigl|F_{y}(s)\bigr|\,ds \\ \le& \int _{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha )}s^{k_{1}}(1-s)^{l_{1}} M_{r,f}\,ds \\ \le &M_{r,f} \int _{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha )}s^{k_{1}}(t-s)^{l_{1}}\,ds \\ =& M_{r,f}t^{\alpha+l_{1}+k_{1}} \int _{0}^{1}\frac{(1-w)^{\alpha +l_{1}-1}}{\Gamma(\alpha)}w^{k_{1}}\,dw \\ =& M_{r,f}t^{\alpha+l_{1}+k_{1}}\frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma (\alpha)}< \infty,\quad t\in(0,1]. \end{aligned}$$
This means \(F_{y}\) is an α-integral function. Similarly we get
$$\biggl\vert \int _{0}^{t}F_{y}(s)\,ds\biggr\vert \le M_{r,f}\mathbf{B}(l_{1}+1,k_{1}+1)< \infty,\quad t\in(0,1]. $$
From \(u\in X\) and (9) in Lemma 2.3, we know that there exist constants \(A_{i}\), \(B_{i}\) (\(i\in \mathbb {N}_{0}^{m}\)) such that
$$\begin{aligned}& u(t)= \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds +\sum _{j=0}^{i}\bigl[A_{j}(t-t_{j})^{\alpha-1}+B_{j}(t-t_{j})^{\alpha-2} \bigr], \\ & \quad t\in (t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}. \end{aligned}$$
(18)
So
$$\begin{aligned} D_{0^{+}}^{\delta}u(t) =& \int_{0}^{t}\frac{(t-s)^{\alpha-\delta-1}}{\Gamma(\alpha -\delta)}F_{y}(s)\,ds +\frac{\Gamma(\alpha)}{\Gamma(\alpha-\delta)}\sum _{j=0}^{i}A_{j}(t-t_{j})^{\alpha-\delta-1} \\ &{}+\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)}\sum _{j=0}^{i} B_{j}(t-t_{j})^{\alpha-\delta-2},\quad t\in(t_{i},t_{i+1}],i \in \mathbb {N}_{0}^{m}, \end{aligned}$$
(19)
and
$$ D_{0^{+}}^{\alpha-1}u(t)= \int_{0}^{t}F_{y}(s)\,ds +\Gamma(\alpha) \sum _{j=0}^{i}A_{j},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}. $$
(20)
From \(D_{0^{+}}^{\alpha-1}u(0)=0 \) and \(u(1)=0\) imply that \(A_{0}=0\) and
$$ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds +\sum _{j=0}^{m}\bigl[A_{j}(1-t_{j})^{\alpha-1}+B_{j}(1-t_{j})^{\alpha-2} \bigr]=0. $$
(21)
Now we get by using the impulse conditions
$$\begin{aligned}& B_{i}=I_{1}\bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta-1}y(t_{i})\bigr),\quad i \in \mathbb {N}_{1}^{m}, \\& \Gamma(\alpha)A_{i}=J_{1}\bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i})\bigr),\quad i \in \mathbb {N}_{1}^{m}. \end{aligned}$$
Together with (21), we obtain
$$B_{0}=- \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds- \sum _{j=1}^{m} \biggl[\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}J_{1y}(t_{j}) +(1-t_{j})^{\alpha-2}I_{1y}(t_{j}) \biggr]. $$
Substitute \(A_{i}\), \(B_{i}\) into (19), we get (14). Similarly we get (15).

Now we suppose that u satisfies (14) and v satisfies (15). We will prove that \(u\in X\) and \(v\in Y\), u, v is a solution of BVP (13).

It is easy to see that \(u\in X\), \(v\in Y\). Furthermore, by direct computation, we get
$$\begin{aligned}& D_{0^{+}}^{\alpha}u(t)=F_{y}(t),\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\& D_{0^{+}}^{\alpha-1}u(0)=0,\quad\quad u(1)=0, \\& D_{0^{+}}^{\beta}v(t)=G_{x}(t),\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\& D_{0^{+}}^{\beta-1}v(0)=0,\quad\quad v(1)=0, \\& \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\alpha}u(t)=I_{1y}(t_{i}),\quad i \in \mathbb {N}_{1}^{m}, \\& \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\alpha-1}u(t)-D_{0^{+}}^{\alpha -1}u(t_{i})=J_{1y}(t_{i}),\quad i \in \mathbb {N}_{1}^{m}, \\& \lim _{t\to t_{i}^{+}}(t-t_{i})^{2-\beta}v(t)=I_{2y}(t_{i}),\quad i \in \mathbb {N}_{1}^{m}, \\& \lim _{t\to t_{i}^{+}}D_{0^{+}}^{\beta-1}v(t)-D_{0^{+}}^{\beta -1}v(t_{i})=J_{2y}(t_{i}),\quad i \in \mathbb {N}_{1}^{m}. \end{aligned}$$
Then \((u,v)\) is a solution of BVP (13). The proof is completed. □
For \((x,y)\in E\), define \(T(x,y)\) by \(T(x,y)(t)=((T_{1}y)(t),(T_{2}x)(t))\) with
$$\begin{aligned} (T_{1}y) (t) =& \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds \\ &{}- \Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds+ \sum _{j=1}^{m}\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha )}J_{1y}(t_{j})+ \sum _{j=1}^{m} (1-t_{j})^{\alpha-2}I_{1y}(t_{j}) \Biggr]t^{\alpha-2} \\ &{}+\sum _{j=1}^{i}I_{1y}(t_{j}) (t-t_{j})^{\alpha-2}+\sum _{j=1}^{i} \frac{J_{1y}(t_{j}) }{\Gamma(\alpha)}(t-t_{j})^{\alpha-1},\quad t\in(t_{i},t_{i+1}],i \in \mathbb {N}_{0}^{m}, \end{aligned}$$
(22)
and
$$\begin{aligned} (T_{2}x) (t) =& \int_{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}G_{x}(s)\,ds \\ &{}- \Biggl[ \int_{0}^{1}\frac{(1-s)^{\beta-1}}{\Gamma(\beta)}G_{x}(s)\,ds+ \sum _{j=1}^{m}I_{2x}(t_{j}) (1-t_{j})^{\beta-2} +\sum _{j=1}^{m} \frac{J_{2x}(t_{j})}{\Gamma(\beta)}(1-t_{j})^{\beta -1} \Biggr]t^{\beta-2} \\ &{}+\sum _{j=1}^{i}I_{2x}(t_{j}) (t-t_{j})^{\beta-2}+\sum _{j=1}^{i} \frac{J_{2x}(t_{j}) }{\Gamma(\beta)}(t-t_{j})^{\beta-1},\quad t\in(t_{i},t_{i+1}],i \in \mathbb {N}_{0}^{m}. \end{aligned}$$
(23)

Lemma 2.5

Suppose that (a)-(d) hold. Then \(T:E\to E\) is well defined and is completely continuous, \((x,y)\) is a solution of BVP (6) if and only if \((x,y)=T(x,y)\).

Proof

By Lemma 2.4, we know that \(T_{1}y\in X\) and \(T_{2}x\in Y\). Then \(T:E\to E\) is well defined. It is easy to show from Lemma 2.4 that \((x,y)\) is a solution of BVP (6) if and only if \((x,y)=T(x,y)\).

Now, we prove that T is completely continuous. It suffices to prove that \(T_{1}:Y\to X\) and \(T_{2}:X\to Y\) are completely continuous. We divide the proof of completely continuous property of \(T_{1}\) into four steps. Similarly we can prove the completely continuous property of \(T_{2}\).

Step 1. Prove that \(T_{1}\) is continuous.

Let \(y_{n}\in Y\) (\(n=0,1,2,\ldots\)) with \(y_{n}\to y_{0}\) as \(n\to+\infty\). We will prove that \(T_{1}y_{n}\to T_{1}y_{0}\) as \(n\to+\infty\). It is easy to show that there exists \(r>0\) such that \(\|y_{n}\|\le r\), \(n=0,1,2,\ldots\) , and \(\|y_{n}-y_{0}\|\to0\) as \(n\to+\infty\). Then there exist constants \(M_{r,f}, M_{1,r,I}, M_{1,r,J}\ge0\) such that
$$\begin{aligned}& \bigl|F_{y_{n}}(t)\bigr|\le M_{r,f}t^{k_{1}}(1-t)^{l_{1}},\quad t \in(t_{i},t_{i+1}), i\in \mathbb {N}_{0}^{m}0,n=0,1,2, \ldots, \\& \bigl|I_{1}\bigl(t_{i},y_{n}(t_{i}),D_{0^{+}}^{\theta}y_{n}(t_{i}),D_{0^{+}}^{\beta -1}y_{n}(t_{i}) \bigr)\bigr|\le M_{1,r,I},\quad i\in \mathbb {N}_{1}^{m}, n=0,1,2,\ldots, \\& \bigl|J_{1}\bigl(t_{i},y_{n}(t_{i}),D_{0^{+}}^{\theta}y_{n}(t_{i}),D_{0^{+}}^{\beta -1}y_{n}(t_{i}) \bigr)\bigr|\le M_{1,r,J},\quad i\in \mathbb {N}_{1}^{m}, n=0,1,2,\ldots. \end{aligned}$$
(24)
Using the definition in (22), one sees for \(t\in(t_{i},t_{i+1}]\) that
$$\begin{aligned}& (t-t_{i})^{2-\alpha}\bigl|(T_{1}y_{n}) (t)-(T_{1}{y_{0}}) (t)\bigr| \\& \quad\le(t-t_{i})^{2-\alpha} \int _{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\bigl|F_{y_{n}}(s)-F_{y_{0}}(s)\bigr|\,ds \\& \quad\quad{}+(t-t_{i})^{2-\alpha}\Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha )}\bigl|F_{y_{n}}(s)-F_{y_{0}}(s)\bigr|\,ds +\sum _{j=1}^{m}(1-t_{j})^{\alpha-2}\bigl|I_{1y_{n}}(t_{j})-I_{1y_{0}}(t_{j})\bigr| \\& \quad\quad{}+\sum _{j=1}^{m}\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)} \bigl|J_{1y_{n}}(t_{j})-J_{1y_{0}}(t_{j})\bigr| \Biggr]t^{\alpha-2} +(t-t_{i})^{2-\alpha}\sum _{j=1}^{i}(t-t_{j})^{\alpha -2}\bigl|I_{1y_{n}}(t_{j})-I_{1y_{0}}(t_{j})\bigr| \\& \quad\quad{}+(t-t_{i})^{2-\alpha}\sum _{j=1}^{i} \frac {|J_{1y_{n}}(t_{j})-J_{1y_{0}}(t_{j})|}{\Gamma(\alpha)}(t-t_{j})^{\alpha-1} \\& \quad\le2M_{r,f}(t-t_{i})^{2-\alpha} \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma (\alpha)}s^{k_{1}}(1-s)^{l_{1}}\,ds \\& \quad\quad{}+2M_{r,f} \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}s^{k_{1}}(1-s)^{l_{1}}\,ds +2M_{1,r,I}\sum _{j=1}^{m}(1-t_{j})^{\alpha-2}+2M_{1,r,J} \sum _{j=1}^{m}\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)} \\& \quad\quad{}+2mM_{1,r,I}+2M_{1,r,J}\sum _{j=1}^{m} \frac{t_{j+1}-t_{j}}{\Gamma (\alpha)} \\& \quad\le2M_{r,f}(t-t_{i})^{2-\alpha}t^{\alpha+k_{1}+l_{1}} \int_{0}^{1}\frac {(1-w)^{\alpha+l_{1}-1}}{\Gamma(\alpha)}w^{k_{1}}\,dw \\& \quad\quad{}+2M_{r,f}\frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha)} +2M_{1,r,I}\sum _{j=1}^{m}(1-t_{j})^{\alpha-2}+2M_{1,r,J} \sum _{j=1}^{m}\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)} \\& \quad\quad{}+2mM_{1,r,I}+2M_{1,r,J}\sum _{j=1}^{m} \frac{t_{j+1}-t_{j}}{\Gamma (\alpha)} \\& \quad\le4M_{r,f}\frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha )}+2M_{1,r,I} \Biggl( \sum _{j=1}^{m}(1-t_{j})^{\alpha-2}+m \Biggr) \\& \quad\quad{}+2M_{1,r,J}\sum _{j=1}^{m} \biggl( \frac{(1-t_{j})^{\alpha-1}}{\Gamma (\alpha)}+ \frac{t_{j+1}-t_{j}}{\Gamma(\alpha)} \biggr). \end{aligned}$$
Similarly we can prove for \(t\in(t_{i},t_{i+1}]\) that
$$\begin{aligned}& (t-t_{i})^{2+\delta-\alpha}\bigl|D_{0^{+}}^{\delta}(T_{1}y_{n}) (t)-D_{0^{+}}^{\delta}(T_{1}y_{0}) (t)\bigr| \\& \quad\le2M_{r,f} \biggl(\frac{\mathbf{B}(\alpha+l_{1}-\delta,k_{1}+1)}{\Gamma(\alpha -\delta)}+ \frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \frac{\mathbf{B}(\alpha +l_{1},k_{1}+1)}{\Gamma(\alpha)} \biggr) \\& \quad\quad{}+ 2M_{1,r,J} \Biggl(\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \sum _{j=1}^{m} \frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}+\frac {\Gamma(\alpha)}{\Gamma(\alpha-\delta)}\sum _{j=1}^{m} \frac{t_{j+1}-t_{j} }{\Gamma(\alpha)} \Biggr) \\& \quad\quad{}+2M_{1,r,I} \Biggl(\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \sum _{j=1}^{m} (1-t_{j})^{\alpha-2}+\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta -1)}m \Biggr) \end{aligned}$$
and
$$\begin{aligned}& \bigl|D_{0^{+}}^{\alpha-1}(T_{1}y_{n}) (t)-D_{0^{+}}^{\alpha-1}(T_{1}{y_{0}}) (t)\bigr| \\& \quad\le \int_{0}^{t}\bigl|F_{y_{n}}(s)-F_{y_{0}}(s)\bigr|\,ds +\sum _{j=1}^{i}\bigl|J_{1y_{n}}(t_{j})-J_{1y_{0}}(t_{j})\bigr| \le2M_{r,f}\mathbf{B}(l_{1}+1,k_{1}+1)+2mM_{1,r,J}. \end{aligned}$$
By Lebesgue’s dominated convergence theorem, we can show that
$$\begin{aligned}& \lim _{n\to\infty}\sup _{t\in (t_{i},t_{i+1}]}(t-t_{i})^{2-\alpha}\bigl|(Tx_{n}) (t)-(T{x_{0}}) (t)\bigr|=0, \\ & \lim _{n\to\infty}\sup _{t\in (t_{i},t_{i+1}]}(t-t_{i})^{2+\delta-\alpha}\bigl|D_{0^{+}}^{\delta}(Tx_{n}) (t)-D_{0^{+}}^{\delta}(T{x_{0}}) (t)\bigr|=0, \\ & \lim _{n\to\infty}\sup _{t\in (t_{i},t_{i+1}]}\bigl|D_{0^{+}}^{\alpha-1}(Tx_{n}) (t)-D_{0^{+}}^{\alpha-1}(T{x_{0}}) (t)\bigr|=0. \end{aligned}$$
Hence \(\|Tx_{n}-Tx_{0}\|\to0\) as \(n\to\infty\). Then T is continuous.

Let \(\Omega_{1}\subseteq X\) and \(\Omega_{2}\subseteq Y\) be bounded sets of X and Y, respectively. Then there exists \(r>0\) such that \(\|x\|,\|y\|\le r\), \(x\in\Omega_{1}\), \(y\in\Omega_{2}\). So there exist constants \(M_{r,f} , M_{r,g} , M_{1,r,I} , M_{2,r,I} , M_{1,r,J} , M_{2,r,J}\ge0\) such that (16) and (17) hold for all \(x\in\Omega _{1}\), \(y\in\Omega_{2}\).

Step 2. Prove that \(\{T_{1}y:y\in\Omega\}\) is bounded.

We have similarly to Step 1 for \(t\in(t_{i},t_{i+1}]\)
$$\begin{aligned}& (t-t_{i})^{2-\alpha}\bigl|(Ty) (t)\bigr| \le2M_{r,f} \frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha )}+M_{1,r,I} \Biggl( \sum _{j=1}^{m}(1-t_{j})^{\alpha-2}+m \Biggr) \\& \hphantom{(t-t_{i})^{2-\alpha}\bigl|(Ty) (t)\bigr|\le}{}+M_{1,r,J}\sum _{j=1}^{m} \biggl( \frac{(1-t_{j})^{\alpha-1}}{\Gamma (\alpha)}+ \frac{t_{j+1}-t_{j}}{\Gamma(\alpha)} \biggr), \\& (t-t_{i})^{2+\delta-\alpha}\bigl|D_{0^{+}}^{\delta}(T_{1}y) (t)\bigr| \\& \quad \le M_{r,f} \biggl(\frac{\mathbf{B}(\alpha+l_{1}-\delta,k_{1}+1)}{\Gamma(\alpha -\delta)}+ \frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \frac{\mathbf{B}(\alpha +l_{1},k_{1}+1)}{\Gamma(\alpha)} \biggr) \\& \quad\quad{}+ M_{1,r,J} \Biggl(\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \sum _{j=1}^{m} \frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}+\frac {\Gamma(\alpha)}{\Gamma(\alpha-\delta)}\sum _{j=1}^{m} \frac{t_{j+1}-t_{j} }{\Gamma(\alpha)} \Biggr) \\& \quad\quad{}+M_{1,r,I} \Biggl(\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \sum _{j=1}^{m} (1-t_{j})^{\alpha-2}+\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta -1)}m \Biggr), \\& \bigl|D_{0^{+}}^{\alpha-1}(T_{1}y) (t)\bigr|\le M_{r,f}\mathbf{B} (l_{1}+1,k_{1}+1)+mM_{1,r,J}. \end{aligned}$$
It follows that there exists a constant \(M>0\) such that \(\|T_{1}y\|\le M_{1}\) for all \(y\in\Omega_{2}\). Similarly we see that there exists a constant \(M_{1}>0\) such that \(\|T_{2}x\|\le M_{2}\) for all \(x\in\Omega_{1}\). From the above discussion, we see that \(\{T(x,y):x\in\Omega_{1},y\in\Omega_{2}\}\) is bounded.
Step 3. Prove that
$$\begin{aligned}& \bigl\{ t\to(t-t_{i})^{2-\alpha}(T_{1}y) (t):y\in\Omega \bigr\} , \\& \bigl\{ t\to(t-t)^{2+\delta-\alpha}D_{0^{+}}^{\delta}(T_{1}y) (t):x\in\Omega\bigr\} , \\& \bigl\{ t\to D_{0^{+}}^{\alpha-1}(T_{1}y) (t):y\in\Omega \bigr\} \end{aligned}$$
are equi-continuous on each \((t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)).
Since \(T:X\rightarrow X\), we can take
$$\begin{aligned}& (t-t_{i})^{2-\alpha}(Tx) (t)|_{t=t_{i}}=\lim {t\rightarrow t_{i}^{+}} (t-t_{i})^{2-\alpha}(Tx) (t), \\& (t-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}(Tx) (t)|_{t=t_{i}}=\lim _{t\rightarrow t_{i}^{+}}(t-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}(Tx) (t), \\& D_{0^{+}}^{\alpha-1}(Tx) (t)|_{t=t_{i}}=\lim _{t\rightarrow t_{i}^{+}} D_{0^{+}}^{\alpha-1}(Tx) (t). \end{aligned}$$
Then
$$(t-t_{i})^{2-\alpha}(Tx) (t), (t-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}(Tx) (t), D_{0^{+}}^{\alpha-1}(Tx) (t) $$
are continuous on \([t_{i},t_{i+1}]\) for each \(x\in\Omega\). So
$$t^{2-\alpha}(Tx) (t), t^{2+\delta-\alpha}D_{0^{+}}^{\delta}(Tx) (t), D_{0^{+}}^{\alpha-1}(Tx) (t) $$
are uniformly continuous, for any \(\varepsilon>0\), there exists \(\delta _{0}>0\), when \(s_{1}, s_{2}\in[t_{i},t_{i+1}]\), \(|s_{1}-s_{2}|<\delta_{0}\), and \(x\in \Omega\), we can get
$$\begin{aligned}& \bigl|(s_{1}-t_{i})^{2-\alpha}(Tx) (s_{1})-(s_{2}-t_{i}) ^{2-\alpha}(Tx) (s_{2})\bigr|< \varepsilon, \\& \bigl|(s_{1}-t_{i})^{2+\delta-\alpha}D_{0^{+}}^{\delta}(Tx) (s_{1})-( s_{2}-t_{i})^{2-\alpha}(Tx) (s_{2})\bigr|< \varepsilon, \\& \bigl|D_{0^{+}}^{\alpha-1}(Tx) (s_{1})-D_{0^{+}}^{\alpha-1}(Tx) (s_{2})\bigr|< \varepsilon. \end{aligned}$$
This shows us that
$$\begin{aligned}& \bigl\{ t\to t^{2-\alpha}(Tx) (t):x\in\Omega\bigr\} , \\& \bigl\{ t\to t^{2+\delta-\alpha }D_{0^{+}}^{\delta}(Tx) (t):x\in\Omega\bigr\} ,\\& \bigl\{ t\to D_{0^{+}}^{\alpha-1}(Tx) (t):x\in\Omega\bigr\} \end{aligned}$$
are equi-continuous on any closed subinterval of \([t_{i},t_{i+1}]\) (\(i\in \mathbb {N}_{0}^{m}\)).

It follows from Steps 1-3 that T is completely continuous. The proof is ended. □

3 Main results

In this section, we prove the main theorem. Suppose that \(\sigma_{j},\tau _{j}\ge0\) (\(j=1,2,3\)) are constants. We need the following assumptions:
  1. (H1)
    there exist nonnegative constants \(A_{j}\), \(B_{j}\) (\(j=1,2,3\)) and two functions \(\phi_{0}\), \(\psi_{0}\) such that \(p\psi_{0}\) is an α-well integrable function and \(q\phi_{0}\) a β-well integrable function and
    $$\begin{aligned}& \biggl\vert f \biggl(t,\frac{y_{1}}{(t-t_{i})^{2-\beta}},\frac {y_{2}}{(t-t_{i})^{2-\beta+\theta}},y_{3} \biggr)-\psi_{0}(t)\biggr\vert \\& \quad\le \sum _{j=1}^{3}A_{j}|y_{j}|^{\sigma_{j}},\quad t\in(t_{i},t_{i+1}], y_{j}\in R\ (j=1,2,3),i\in \mathbb {N}_{0}^{m}0, \\& \biggl\vert g \biggl(t,\frac{x_{1}}{(t-t_{i})^{2-\alpha}},\frac {x_{2}}{(t-t_{i})^{2-\alpha+\delta}},x_{3} \biggr)-\phi_{0}(t)\biggr\vert \\& \quad\le \sum _{j=1}^{3}B_{j}|x_{j}|^{\tau_{j}},\quad t\in(t_{i},t_{i+1}], x_{j}\in R\ (j=1,2,3),i\in \mathbb {N}_{0}^{m}, \end{aligned}$$
     
  2. (H2)
    there exist constants \(I_{1i}\), \(J_{1i}\), \(I_{2i}\), \(J_{2i}\) (\(i\in N\)), \(C_{j},D_{j}\ge0\) (\(j=1,2,3\)) such that
    $$\begin{aligned}& \biggl\vert I_{1} \biggl(t_{i},\frac{y_{1}}{(t_{i}-t_{i-1})^{2-\beta }}, \frac{y_{2}}{(t_{i}-t_{i-1})^{2-\beta+\theta}},y_{3} \biggr)-I_{1i}\biggr\vert \le\sum _{j=1}^{3}C_{j}|y_{j}|^{\sigma_{j}},\quad i \in \mathbb {N}_{1}^{m}, \\& \biggl\vert J_{1} \biggl(t,\frac{y_{1}}{(t-t_{i})^{2-\beta}},\frac {y_{2}}{(t-t_{i})^{2-\beta+\theta}},y_{3} \biggr)-J_{1i}\biggr\vert \le\sum _{j=1}^{3}D_{j}|y_{j}|^{\sigma_{j}},\quad i \in \mathbb {N}_{1}^{m}, \\& \biggl\vert I_{2} \biggl(t_{i},\frac{x_{1}}{(t_{i}-t_{i-1})^{2-\alpha}}, \frac {x_{2}}{(t_{i}-t_{i-1})^{2-\alpha+\delta}},x_{3} \biggr)-I_{2i}\biggr\vert \le\sum _{j=1}^{3}E_{j}|x_{j}|^{\tau_{j}},\quad i \in \mathbb {N}_{1}^{m}, \\& \biggl\vert J_{2} \biggl(t,\frac{x_{1}}{(t-t_{i})^{2-\alpha}},\frac {x_{2}}{(t-t_{i})^{2-\alpha+\delta}},x_{3} \biggr)-J_{2i}\biggr\vert \le\sum _{j=1}^{3}F_{j}|x_{j}|^{\tau_{j}},\quad i \in \mathbb {N}_{1}^{m}. \end{aligned}$$
     
Denote
$$\begin{aligned} \Phi(t) =& \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}p(s) \psi_{0}(s)\,ds \\ &{}- \Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}p(s) \psi_{0}(s)\,ds+ \sum _{j=1}^{m} \frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}J_{1j}+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2}I_{1j} \Biggr]t^{\alpha-2} \\ &{}+\sum _{j=1}^{i}(t-t_{j})^{\alpha-2}I_{1j}+ \sum _{j=1}^{i}\frac{J_{1j} }{\Gamma(\alpha)}(t-t_{j})^{\alpha-1},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \end{aligned}$$
and
$$\begin{aligned} \Psi(t) =& \int_{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}q(s) \phi_{0}(s)\,ds \\ &{}- \Biggl[ \int_{0}^{1}\frac{(1-s)^{\beta-1}}{\Gamma(\beta)}q(s) \phi_{0}(s)\,ds+\sum _{j=1}^{m}I_{2j}(1-t_{j})^{\beta-2} +\sum _{j=1}^{m}\frac{J_{2j}}{\Gamma(\beta)}(1-t_{j})^{\beta-1} \Biggr]t^{\beta-2} \\ &{}+\sum _{j=1}^{i}I_{2j}(t-t_{j})^{\beta-2}+ \sum _{j=1}^{i}\frac{J_{2j} }{\Gamma(\beta)}(t-t_{j})^{\beta-1},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}. \end{aligned}$$
Denote
$$\begin{aligned}& P_{1}=\frac{2\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha)}+ \frac{\mathbf{B}(\alpha-\delta+l_{1},k_{1}+1)}{\Gamma(\alpha-\delta)}+\frac {\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \frac{\mathbf{B}(\alpha +l_{1},k_{1}+1)}{\Gamma(\alpha)}\\& \hphantom{P_{1}=}{}+\mathbf{B}(l_{1}+1,k_{1}+1), \\& P_{2}=m+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2}+\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \Biggl(m+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2} \Biggr), \\& P_{3}=\frac{2m}{\Gamma(\alpha)}+m \biggl(\frac{1}{\Gamma(\alpha-\delta )}+ \frac{\Gamma(\alpha-1)}{\Gamma(\alpha)\Gamma(\alpha-\delta-1)} \biggr)+m, \end{aligned}$$
and
$$\begin{aligned}& Q_{1}=\frac{2\mathbf{B}(\beta+l_{2},k_{2}+1)}{\Gamma(\beta)}+ \frac{\mathbf{B}(\beta-\theta+l_{2},k_{2}+1)}{\Gamma(\beta-\theta)}+\frac {\Gamma(\beta-1)}{\Gamma(\beta-\theta-1)} \frac{\mathbf{B}(\beta +l_{2},k_{2}+1)}{\Gamma(\beta)}\\& \hphantom{Q_{1}=}{}+\mathbf{B}(l_{2}+1,k_{2}+1), \\& Q_{2}=m+\sum _{j=1}^{m} (1-t_{j})^{\beta-2}+\frac{\Gamma(\beta-1)}{\Gamma(\beta-\theta-1)} \Biggl(m+\sum _{j=1}^{m} (1-t_{j})^{\beta-2} \Biggr), \\& Q_{3}=\frac{2m}{\Gamma(\beta)}+m \biggl(\frac{1}{\Gamma(\beta-\theta)}+ \frac {\Gamma(\beta-1)}{\Gamma(\beta)\Gamma(\beta-\theta-1)} \biggr)+m, \\& \overline{P_{j}}=P_{1}A_{j}+P_{2}C_{j}+P_{3}D_{j},\quad\quad \overline{Q_{j}}=Q_{1}B_{j}+Q_{2}E_{j}+Q_{3}F_{j}. \end{aligned}$$

Theorem 3.1

Let \(\sigma=\max\{\sigma_{i}\ (i=1,2,3)\}\) and \(\tau =\max\{\tau_{i}\ (i=1,2,3)\}\). Suppose that (a)-(d) and (H1)-(H2) hold. Then BVP (6) has at least one solution if
  1. (i)

    \(\sigma\tau\in[0,1)\) or

     
  2. (ii)
    \(\sigma\tau=1\) with
    $$\sum _{j=1}^{3}Q_{j}\|\Phi \|^{\tau_{j}-\tau}< \biggl(\frac{1}{\sum _{j=1}^{3}P_{j}\|\Psi\|^{\sigma_{j}-\sigma}} \biggr) ^{1/\sigma}\quad\textit{or}\quad\sum _{j=1}^{3}P_{j}\|\Psi\|^{\sigma_{j}-\sigma}< \biggl(\frac{1}{\sum _{j=1}^{3}Q_{j}\|\Phi\|^{\tau_{j}-\tau}} \biggr)^{1/\tau} $$
    or
     
  3. (iii)
    \(\sigma\tau>1\) with
    $$\begin{aligned}& \frac{\sigma\tau-1}{\|\Psi\|}\sum _{j=1}^{3}Q_{j} \|\Phi\|^{\tau_{j}-\tau } \Biggl[\|\Phi\|+ \biggl(\frac{\sigma\tau\|\Psi\|}{\sigma\tau-1} \biggr)^{\sigma}\sum _{j=1}^{3}P_{j} \|\Psi\|^{\sigma_{j}-\sigma} \Biggr]^{\tau}\le1,\quad\textit{or } \\& \frac{\sigma\tau-1}{\|\Phi\|}\sum _{j=1}^{3}P_{j} \|\Psi\|^{\sigma _{j}-\sigma} \Biggl[\|\Psi\|+ \biggl(\frac{\tau\sigma\|\Phi\|}{\sigma\tau-1 } \biggr)^{\sigma}\sum _{j=1}^{3}Q_{j} \|\Phi\|^{\tau_{j}-\tau} \Biggr]^{\sigma}\le1. \end{aligned}$$
     

Proof

It is easy to see that \((\Phi,\Psi)\in E\). For \(r_{1},r_{2}>0\), denote \(\Omega_{r_{1},r_{2}}=\{(x,y)\in E:\|x-\Phi\|\le r_{1},\|y-\Psi\|\le r_{2}\}\). One sees that \(\|x\|\le\|x-\Phi\|+\|\Phi\|\le r_{1}+\|\Phi\|\) and \(\| y\|\le r_{2}+\|\Psi\|\) for all \((x,y)\in\Omega_{r_{1},r_{2}}\).

Use (H1), for \((x,y)\in\Omega_{r_{1},r_{2}}\), we have
$$\begin{aligned}& \bigl|f\bigl(t,y(t),D_{0^{+}}^{\theta}y(t),D_{0^{+}}^{\beta-1}y(t) \bigr)-\psi_{0}(t)\bigr| \\& \quad=\biggl\vert f \biggl(t,\frac{(t-t_{i})^{2-\beta}y(t)}{(t-t_{i})t^{2-\beta}},\frac {(t-t_{i})^{2+\theta-\beta}D_{0^{+}}^{\theta}y(t)}{(t-t_{i})^{2+\delta-\alpha }},D_{0^{+}}^{\beta-1}y(t) \biggr)-\psi_{0}(t)\biggr\vert \\& \quad\le A_{1}\bigl\vert (t-t_{i})^{2-\beta}y(t)\bigr\vert ^{\sigma_{1}}+A_{2}\bigl\vert (t-t_{i})^{2+\theta-\beta}D_{0^{+}}^{\theta}y(t)\bigr\vert ^{\sigma_{2}}+A_{3}\bigl\vert D_{0^{+}}^{\beta-1}y(t) \bigr\vert ^{\sigma_{3}} \\& \quad\le\sum _{j=1}^{3}A_{j} \bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}},\quad t\in (t_{i},t_{i+1}),i\in N_{0}. \end{aligned}$$
It follows that
$$ \bigl|f\bigl(t,y(t),D_{0^{+}}^{\theta}y(t),D_{0^{+}}^{\beta-1}y(t) \bigr)-\psi_{0}(t)\bigr|\le\sum _{j=1}^{3}A_{j} \bigl[r_{2}+\|\Psi\| \bigr]^{\sigma_{j}}. $$
(25)
Similarly using (H1)-(H2), we get for \((x,y)\in\Omega_{r}\)
$$\begin{aligned}& \bigl|g\bigl(t,x(t),D_{0^{+}}^{\delta}x(t),D_{0^{+}}^{\alpha-1}x(t) \bigr)-\phi_{0}(t)\bigr|\le\sum _{j=1}^{3}B_{j} \bigl[r_{1}+\|\Phi\| \bigr]^{\tau_{j}}, \\& \bigl\vert I_{1} \bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i}) \bigr)-I_{1i}\bigr\vert \le\sum _{j=1}^{3}C_{j}\bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}},\quad i\in \mathbb {N}_{1}^{m}, \\& \bigl\vert J_{1} \bigl(t_{i},y(t_{i}),D_{0^{+}}^{\theta}y(t_{i}),D_{0^{+}}^{\beta -1}y(t_{i}) \bigr)-J_{1i}\bigr\vert \le\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}},\quad i\in \mathbb {N}_{1}^{m}, \\& \bigl\vert I_{2} \bigl(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha -1}x(t_{i}) \bigr)-I_{2i}\bigr\vert \le\sum _{j=1}^{3}E_{j}\bigl[r_{1}+ \|\Phi\| \bigr]^{\tau_{j}},\quad i\in \mathbb {N}_{1}^{m}, \\& \bigl\vert J_{2} \bigl(t_{i},x(t_{i}),D_{0^{+}}^{\delta}x(t_{i}),D_{0^{+}}^{\alpha -1}x(t_{i}) \bigr)-J_{2i}\bigr\vert \le\sum _{j=1}^{3}F_{j}\bigl[r_{1}+ \|\Phi\| \bigr]^{\tau_{j}},\quad i\in \mathbb {N}_{1}^{m}. \end{aligned}$$
(26)
Use (c) and (22), (25), (26), we get for \(t\in(t_{i},t_{i+1}]\)
$$\begin{aligned}& (t-t_{i})^{2-\alpha}\bigl|(T_{1}y) (t)-\Phi(t)\bigr| \\& \quad\le(t-t_{i})^{2-\alpha} \int_{0}^{t}\frac {(t-s)^{\alpha-1}}{\Gamma(\alpha)}\bigl|F_{y}(s)-p(s) \psi_{0}(s)\bigr|\,ds \\& \quad\quad{}+(t-t_{i})^{2-\alpha}\Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha )}\bigl|F_{y}(s)-p(s) \psi_{0}(s)\bigr|\,ds + \sum _{j=1}^{m} \frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha )}\bigl|J_{1y}(t_{j})-J_{1j}\bigr| \\& \quad\quad{}+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2}\bigl|I_{1y}(t_{j})-I_{1j}\bigr| \Biggr]t^{\alpha-2} \\& \quad\quad{}+(t-t_{i})^{2-\alpha}\sum _{j=1}^{i}\bigl|I_{1y}(t_{j})-I_{1j}\bigr|(t-t_{j})^{\alpha-2}+(t-t_{i})^{2-\alpha} \sum _{j=1}^{i}\frac{|J_{1y}(t_{j})-J_{1j}| }{\Gamma(\alpha)}(t-t_{j})^{\alpha-1} \\& \quad\le(t-t_{i})^{2-\alpha} \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha )}s^{k_{1}}(1-s)^{l_{1}}\,ds \sum _{j=1}^{3}A_{j} \bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}} \\& \quad\quad{}+(t-t_{i})^{2-\alpha}\Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha )}s^{k_{1}}(1-s)^{l_{1}}\,ds \sum _{j=1}^{3}A_{j} \bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}} \\& \quad\quad{} + \sum _{j=1}^{m} \frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}} \\& \quad\quad{}+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2} \sum _{j=1}^{3}C_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma _{j}}\Biggr]t^{\alpha-2} \\& \quad\quad{}+(t-t_{i})^{2-\alpha}\sum _{j=1}^{i}(t-t_{j})^{\alpha-2} \sum _{j=1}^{3}C_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}}\\& \quad\quad{}+(t-t_{i})^{2-\alpha}\sum _{j=1}^{i}\frac{(t-t_{j})^{\alpha-1} }{\Gamma(\alpha)}\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}} \\& \quad\le(t-t_{i})^{2-\alpha} \int_{0}^{t}\frac{(t-s)^{\alpha+l_{1}-1}}{\Gamma(\alpha )}s^{k_{1}}\,ds \sum _{j=1}^{3}A_{j} \bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}} \\& \quad\quad{}+ \int_{0}^{1}\frac{(1-s)^{\alpha+l_{1}-1}}{\Gamma(\alpha)}s^{k_{1}}\,ds \sum _{j=1}^{3}A_{j} \bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}}\\& \quad\quad{} + \sum _{j=1}^{m} \frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}} \\& \quad\quad{}+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2} \sum _{j=1}^{3}C_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}} +i\sum _{j=1}^{3}C_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}}\\& \quad\quad{}+\sum _{j=1}^{i} \frac{t_{i+1}-t_{j}}{\Gamma(\alpha)}\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \| \Psi\|\bigr]^{\sigma_{j}} \\& \quad\le\frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha)}\sum _{j=1}^{3}A_{j} \bigl[r_{2}+\|\Psi\| \bigr]^{\sigma_{j}} +\frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha)}\sum _{j=1}^{3}A_{j} \bigl[r_{2}+\|\Psi\| \bigr]^{\sigma_{j}} \\& \quad\quad{}+ \frac{m}{\Gamma(\alpha)}\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma _{j}} +\sum _{j=1}^{m} (1-t_{j})^{\alpha-2} \sum _{j=1}^{3}C_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}} \\& \quad\quad{}+m\sum _{j=1}^{3}C_{j}\bigl[r+ \|\Psi\|\bigr]^{\sigma_{j}}+\frac{m}{\Gamma(\alpha )}\sum _{j=1}^{3}D_{j}\bigl[r_{2}+ \|\Psi\|\bigr]^{\sigma_{j}} \\& \quad=\sum _{j=1}^{3} \Biggl[\frac{2\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma (\alpha)}A_{j}+ \Biggl(m+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2} \Biggr)C_{j}+\frac{2m}{\Gamma(\alpha)}D_{j} \Biggr] \bigl[r_{2}+\| \Psi\|\bigr]^{\sigma_{j}}. \end{aligned}$$
One has from (22)
$$\begin{aligned}& D_{0^{+}}^{\delta}(T_{1}y) (t)= \int_{0}^{t}\frac{(t-s)^{\alpha-\delta-1}}{\Gamma (\alpha-\delta)}F_{y}(s)\,ds +\frac{\Gamma(\alpha)}{\Gamma(\alpha-\delta)}\sum _{j=1}^{i} \frac {(t-t_{j})^{\alpha-\delta-1}}{\Gamma(\alpha)}J_{1y}(t_{j}) \\& \hphantom{D_{0^{+}}^{\delta}(T_{1}y) (t)=}{}-\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} t^{\alpha-\delta-2} \\& \hphantom{D_{0^{+}}^{\delta}(T_{1}y) (t)=}{}\times\Biggl[ \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}F_{y}(s)\,ds+ \sum _{j=1}^{m}\frac{(1-t_{j})^{\alpha-1}}{\Gamma(\alpha)}J_{1y}(t_{j}) +\sum _{j=1}^{m}(1-t_{j})^{\alpha-2}I_{1y}(t_{j}) \Biggr] \\& \hphantom{D_{0^{+}}^{\delta}(T_{1}y) (t)=}{}+\frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)}\sum _{j=1}^{i} (t-t_{j})^{\alpha-\delta-2}I_{1y}(t_{j}),\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}, \\& D_{0^{+}}^{\alpha-1}(T_{1}y) (t)= \int_{0}^{t}F_{y}(s)\,ds +\sum _{j=1}^{i}{J_{1y}(t_{j})},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}. \end{aligned}$$
Similarly we have
$$\begin{aligned}& (t-t_{i})^{2+\delta-\alpha}\bigl|D_{0^{+}}^{\delta}(T_{1}y) (t)-D_{0^{+}}^{\delta}\Phi (t)\bigr| \\& \quad\le\sum _{j=1}^{3}\Biggl[ \biggl( \frac{\mathbf{B}(\alpha-\delta +l_{1},k_{1}+1)}{\Gamma(\alpha-\delta)}+\frac{\Gamma(\alpha-1)}{\Gamma (\alpha-\delta-1)}\frac{\mathbf{B}(\alpha+l_{1},k_{1}+1)}{\Gamma(\alpha)} \biggr)A_{j}\\& \quad\quad{}+ \frac{\Gamma(\alpha-1)}{\Gamma(\alpha-\delta-1)} \Biggl(m+\sum _{j=1}^{m} (1-t_{j})^{\alpha-2} \Biggr)C_{j} \\& \quad\quad{}+m \biggl(\frac{1}{\Gamma(\alpha-\delta)}+\frac{\Gamma(\alpha-1)}{\Gamma (\alpha)\Gamma(\alpha-\delta-1)} \biggr)D_{j}\Biggr] \bigl[r_{2}+\|\Psi\|\bigr]^{\sigma_{j}} \end{aligned}$$
and
$$\bigl|D_{0^{+}}^{\alpha-1}(T_{1}y) (t)-D_{0^{+}}^{\alpha-1} \Phi(t)\bigr|\le\sum _{j=1}^{3} \bigl[\mathbf{B}(l_{1}+1,k_{1}+1)A_{j}+mD_{j} \bigr] \bigl[r_{2}+\|\Psi\|\bigr]^{\sigma_{j}}. $$
It follows that
$$\begin{aligned} \|T_{1}y-\Phi\| \le&\sum _{j=1}^{3}(P_{1}A_{j}+P_{2}C_{j}+P_{3}D_{j})\bigl[r_{2}+ \|\Psi\| \bigr]^{\sigma_{j}} \\ \le&\bigl[r_{2}+\|\Psi\|\bigr]^{\sigma}\sum _{j=1}^{3}(P_{1}A_{j}+P_{2}C_{j}+P_{3}D_{j}) \| \Psi\|^{\sigma_{j}-\sigma}. \end{aligned}$$
(27)
Similarly we can get
$$\begin{aligned} \|T_{2}x-\Psi\| \le&\sum _{j=1}^{3}(Q_{1}B_{j}+Q_{2}E_{j}+Q_{3}F_{j})\bigl[r_{1}+ \|\Phi\| \bigr]^{\tau_{j}} \\ \le&\bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3}(Q_{1}B_{j}+Q_{2}E_{j}+Q_{3}F_{j}) \|\Phi \|^{\tau_{j}-\tau}. \end{aligned}$$
(28)
From (27), (28), we will seek \(r_{1},r_{2}>0\) such that
$$ \begin{aligned} &\bigl[r_{2}+\|\Psi\|\bigr]^{\sigma}\sum _{j=1}^{3} \overline{P_{j}}\|\Psi\|^{\sigma _{j}-\sigma}\le r_{1},\\ & \bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3} \overline {Q_{j}}\|\Phi\|^{\tau_{j}-\tau}\le r_{2}. \end{aligned} $$
(29)
Then one has \(T\Omega_{r_{1},r_{2}}\subseteq\Omega_{r_{1},r_{2}}\). By Schauder’s fixed point theorem, T has at least one fixed point \((x,y)\in\Omega_{r_{1},r_{2}}\) which is a solution of BVP (6). It suffices to get positive solutions of the following inequality:
$$ \bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3}Q_{j} \|\Phi\|^{\tau_{j}-\tau}\le r_{2}\le \biggl(\frac{r_{1}}{\sum _{j=1}^{3}P_{j}\|\Psi\|^{\sigma_{j}-\sigma }} \biggr)^{1/\sigma}-\|\Psi\| $$
(30)
or
$$ \bigl[r_{2}+\|\Psi\|\bigr]^{\sigma}\sum _{j=1}^{3}P_{j} \|\Psi\|^{\sigma_{j}-\sigma }\le r_{1}\le \biggl(\frac{r_{2}}{\sum _{j=1}^{3}Q_{j}\|\Phi\|^{\tau_{j}-\tau }} \biggr)^{1/\tau}-\|\Phi\|. $$
(31)

Case 1. \(\sigma\tau<1\).

It is easy to see that
$$\bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3} \overline{Q_{j}}\|\Phi\|^{\tau _{j}-\tau}\le \biggl(\frac{r_{1}}{\sum _{j=1}^{3}\overline{P_{j}}\|\Psi\| ^{\sigma_{j}-\sigma}} \biggr)^{1/\sigma}-\|\Psi\| $$
has a positive solution \(r_{1}>0\) sufficiently large. Choose \(r_{2}\) satisfying
$$\bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3} \overline{Q_{j}}\|\Phi\|^{\tau _{j}-\tau}\le r_{2}\le \biggl( \frac{r_{1}}{\sum _{j=1}^{3}\overline{P_{j}}\| \Psi\|^{\sigma_{j}-\sigma}} \biggr)^{1/\sigma}-\|\Psi\|. $$
Then (29) has positive solutions \(r_{1}>0\) and \(r_{2}>0\). Then \(T(x,y)\in\Omega_{r_{1},r_{2}}\) for \((x,y)\in\Omega_{r_{1},r_{2}}\). By Schauder’s fixed point theorem, T has at least one fixed point \((x,y)\in\Omega_{r_{1},r_{2}}\). Then \((x,y)\) is a solution of BVP (6).

Case 2.1. \(\sigma\tau=1\) and \(\sum _{j=1}^{3}\overline{Q_{j}}\|\Phi\|^{\tau_{j}-\tau} ({\sum _{j=1}^{3}\overline{P_{j}}\|\Psi\|^{\sigma_{j}-\sigma}} )^{1/\sigma}<1 \).

Since
$$\lim _{r_{1}\to+\infty}\frac{[r_{1}+\|\Phi\|]^{\tau}\sum _{j=1}^{3}\overline{Q_{j}}\|\Phi\|^{\tau_{j}-\tau}}{ (\frac{r_{1}}{\sum _{j=1}^{3}P_{j}\|\Psi\|^{\sigma_{j}-\sigma }} )^{1/\sigma}-\|\Psi\|} =\frac{\sum _{j=1}^{3}Q_{j}\|\Phi\|^{\tau_{j}-\tau}}{ (\frac{1}{\sum _{j=1}^{3}\overline{P_{j}}\|\Psi\| ^{\sigma_{j}-\sigma}} )^{1/\sigma}}< 1, $$
we know that there exists \(r_{1}>0\) sufficiently large such that
$$ \bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3} \overline{Q_{j}}\|\Phi\|^{\tau _{j}-\tau}\le \biggl(\frac{r_{1}}{\sum _{j=1}^{3}\overline{P_{j}}\|\Psi\| ^{\sigma_{j}-\sigma}} \biggr)^{1/\sigma}-\|\Psi\|. $$
(32)
Choose \(r_{2}\) satisfying
$$ \bigl[r_{1}+\|\Phi\|\bigr]^{\tau}\sum _{j=1}^{3} \overline{Q_{j}}\|\Phi\|^{\tau _{j}-\tau}\le r_{2}\le \biggl( \frac{r_{1}}{\sum _{j=1}^{3}\overline{P_{j}}\| \Psi\|^{\sigma_{j}-\sigma}} \biggr)^{1/\sigma}-\|\Psi\|. $$
(33)
Then (29) has a positive solution \(r_{1}\), \(r_{2}\). Then \(T(x,y)\in\Omega_{r_{1},r_{2}}\) for \((x,y)\in\Omega_{r_{1},r_{2}}\). By Schauder’s fixed point theorem, T has at least one fixed point \((x,y)\in\Omega_{r_{1},r_{2}}\). Then \((x,y)\) is a solution of BVP (6).

Case 2.2. \(\sigma\tau=1\) and \(\sum _{j=1}^{3}\overline{P_{j}}\|\Psi\|^{\sigma_{j}-\sigma} ({\sum _{j=1}^{3}\overline{Q_{j}}\|\Phi\|^{\tau_{j}-\tau}} )^{1/\tau}<1 \).

Similarly to Case 2.1, use (31), we get solutions of BVP (6) by using the Schauder fixed point theorem.

Case 3. \(\sigma\tau>1\).

Choose
$$r_{1}= \biggl(\frac{\sigma\tau\|\Psi\|}{\sigma\tau-1} \biggr)^{\sigma}\sum _{j=1}^{3}\overline{P_{j}}\|\Psi \|^{\sigma_{j}-\sigma}. $$
Since
$$\frac{\sigma\tau-1}{\|\Psi\|}\sum _{j=1}^{3} \overline{Q_{j}}\|\Phi\| ^{\tau_{j}-\tau} \Biggl[\|\Phi\|+ \biggl( \frac{\sigma\tau\|\Psi\|}{\sigma\tau -1} \biggr)^{\sigma}\sum _{j=1}^{3} \overline{P_{j}}\|\Psi\|^{\sigma_{j}-\sigma} \Biggr]^{\tau}\le1, $$
we know that (32) has a positive solution \(r_{1}>0\). Choose \(r_{2}\) such that (33) holds. Then we have \(T(x,y)\in\Omega_{r_{1},r_{2}}\) for \((x,y)\in\Omega _{r_{1},r_{2}}\). By Schauder’s fixed point theorem, T has at least one fixed point \((x,y)\in\Omega_{r_{1},r_{2}}\). Then \((x,y)\) is a solution of BVP (6).
If
$$\frac{\sigma\tau-1}{\|\Phi\|}\sum _{j=1}^{3} \overline{P_{j}}\|\Psi\| ^{\sigma_{j}-\sigma} \Biggl[\|\Psi\|+ \biggl( \frac{\tau\sigma\|\Phi\|}{\sigma \tau-1 } \biggr)^{\sigma}\sum _{j=1}^{3} \overline{Q_{j}}\|\Phi\|^{\tau_{j}-\tau } \Biggr]^{\sigma}\le1, $$
we choose
$$r_{2}= \biggl(\frac{\tau\sigma \|\Phi\|}{\sigma\tau-1} \biggr)^{\tau}\sum _{j=1}^{3}\overline{Q_{j}}\| \Phi \|^{\tau_{j}-\tau}. $$
It is easy to verify that \(r_{2}\) satisfies
$$\bigl[r_{2}+\|\Psi\|\bigr]^{\sigma}\sum _{j=1}^{3} \overline{P_{j}}\|\Psi\|^{\sigma _{j}-\sigma}\le \biggl(\frac{r_{2}}{\sum _{j=1}^{3}\overline{Q_{j}}\|\Phi\| ^{\tau_{j}-\tau}} \biggr)^{1/\tau}-\|\Phi\|. $$
Choose \(r_{1}\) such that
$$\bigl[r_{2}+\|\Psi\|\bigr]^{\sigma}\sum _{j=1}^{3} \overline{P_{j}}\|\Psi\|^{\sigma _{j}-\sigma}\le r_{1}\le \biggl( \frac{r_{2}}{\sum _{j=1}^{3}\overline {Q_{j}}\|\Phi\|^{\tau_{j}-\tau}} \biggr)^{1/\tau}-\|\Phi\|. $$
Then we have \(T(x,y)\in\Omega_{r_{1},r_{2}}\) for \((x,y)\in\Omega _{r_{1},r_{2}}\). By Schauder’s fixed point theorem, T has at least one fixed point \((x,y)\in\Omega_{r_{1},r_{2}}\). Then \((x,y)\) is a solution of BVP (6).

The proof of Theorem 3.1 is completed. □

Theorem 3.2

Suppose that (a)-(d) hold and there exist constants \(M_{f},M_{g},M_{I1},M_{J1},M_{I2}, M_{J2}\ge0\) such that
$$\begin{aligned}& \biggl\vert f \biggl(t,\frac{y_{1}}{(t-t_{i})^{2-\beta}},\frac{y_{2}}{(t-t_{i})^{2-\beta +\theta}},y_{3} \biggr)\biggr\vert \le M_{f},\quad t\in(t_{i},t_{i+1}), y_{j}\in R\ (j=1,2,3),i\in \mathbb {N}_{0}^{m}, \\& \biggl\vert g \biggl(t,\frac{x_{1}}{(t-t_{i})^{2-\alpha}},\frac {x_{2}}{(t-t_{i})^{2-\alpha+\delta}},x_{3} \biggr)\biggr\vert \le M_{g},\quad t\in (t_{i},t_{i+1}), x_{j}\in R\ (j=1,2,3),i\in \mathbb {N}_{0}^{m}, \\& \biggl\vert I_{1} \biggl(t_{i},\frac{y_{1}}{(t_{i}-t_{i-1})^{2-\beta}}, \frac {y_{2}}{(t_{i}-t_{i-1})^{2-\beta+\theta}},y_{3} \biggr)\biggr\vert \le M_{I1},\quad i \in \mathbb {N}_{1}^{m}, \\& \biggl\vert J_{1} \biggl(t,\frac{y_{1}}{(t-t_{i})^{2-\beta}},\frac {y_{2}}{(t-t_{i})^{2-\beta+\theta}},y_{3} \biggr)\biggr\vert \le M_{J1},\quad i\in \mathbb {N}_{1}^{m}, \\& \biggl\vert I_{2} \biggl(t_{i},\frac{x_{1}}{(t_{i}-t_{i-1})^{2-\alpha}}, \frac {x_{2}}{(t_{i}-t_{i-1})^{2-\alpha+\delta}},x_{3} \biggr)\biggr\vert \le M_{I2},\quad i\in \mathbb {N}_{1}^{m}, \\& \biggl\vert J_{2} \biggl(t,\frac{x_{1}}{(t-t_{i})^{2-\alpha}},\frac {x_{2}}{(t-t_{i})^{2-\alpha+\delta}},x_{3} \biggr)\biggr\vert \le M_{J2},\quad i\in \mathbb {N}_{1}^{m}. \end{aligned}$$
Then BVP (6) has at least one solution in \(X\times Y\).

Proof

In Theorem 3.1, choose \(\phi_{0}(t)=\psi_{0}(t)=0\), \(\sigma _{1}=\sigma_{2}=\sigma_{3}=\tau_{1}=\tau_{2}=\tau_{3}=0\), \(A_{1}=M_{f}\), \(B_{1}=M_{g}\), \(C_{1}=M_{I1}\), \(D_{1}=M_{J1}\), \(E_{1}=M_{I2}\), \(F_{1}=M_{J2}\), and \(A_{2}=A_{3}=B_{2}=B_{3}=C_{2}=C_{3}=D_{2}=D_{3}=E_{2}=E_{3}=F_{2}=F_{3}=0\). It is easy to see that (H1) and (H2) hold. We get Theorem 3.2 from Theorem 3.1. The proof is completed. □

4 An example

In this section, we present an example to illustrate main theorems.

Example 4.1

Consider the following boundary value problem for the impulsive multi-term fractional differential equation:
$$ \textstyle\begin{cases} D_{0^{+}}^{\frac{3}{2}}u(t)=t^{-\frac{1}{4}}(1-t)^{-\frac{1}{4}} (a_{0}+a_{1}(t-t_{i})^{\frac{\tau}{3}}[v(t)]^{\tau}+a_{2}(t-t_{i})^{\frac{2\tau }{3}}[D_{0^{+}}^{\frac{1}{3}} v(t)]^{\tau}\\ \hphantom{D_{0^{+}}^{\frac{3}{2}}u(t)=}{}+a_{3}[D_{0^{+}}^{\frac {2}{3}}v(t)]^{\tau}), \quad t\in(t_{i},t_{i+1}),i\in \mathbb {N}_{0}^{10}, \\ D_{0^{+}}^{\frac{5}{3}}v(t)=t^{-\frac{1}{6}}(1-t)^{-\frac{1}{6}} (b_{0}+b_{1}(t-t_{i})^{\frac{\sigma}{2}}[u(t)]^{\sigma}+b_{2}(t-t_{i})^{\frac {5\sigma }{8}}[D_{0^{+}}^{\frac{1}{8}} u(t)]^{\sigma}\\ \hphantom{D_{0^{+}}^{\frac{5}{3}}v(t)=}{}+b_{3}[D_{0^{+}}^{\frac {1}{2}}u(t)]^{\sigma}), \quad t\in(t_{i},t_{i+1}),i\in \mathbb {N}_{0}^{10}, \\ D_{0^{+}}^{\frac{1}{2}}u(0)=0, \quad\quad u(1)=0,\quad\quad D_{0^{+}}^{\frac {2}{3}}v(0)=0,\quad\quad v(1)=0, \\ \lim _{t\to t_{i}^{+}}(t-t_{i})^{\frac{1}{2}}u(t)=I_{1i},\quad\quad\lim _{t\to t_{i}^{+}}D_{0^{+}}^{\frac{1}{2}}u(t)=J_{1i},\quad i\in \mathbb {N}_{1}^{10}, \\ \lim _{t\to t_{i}^{+}}(t-t_{i})^{\frac{1}{2}}u(t)=I_{2i},\quad\quad\lim _{t\to t_{i}^{+}}D_{0^{+}}^{\frac{1}{2}}u(t)=J_{2i},\quad i\in \mathbb {N}_{1}^{10}, \end{cases} $$
(34)
where \(a_{i},b_{i}\in \mathbb {R}\) (\(i=0,1,2,3\)), \(I_{1i},I_{2i},J_{1i},J_{2i}\in \mathbb {R}\) (\(i\in \mathbb {N}_{1}^{m}\)) and \(\sigma\ge0\), \(\tau\ge0\), \(0=t_{0}< t_{1}=\frac {1}{11}<\cdots<t_{10}=\frac{1}{2}<t_{11}=1\), \(\mathbb {N}_{0}^{10}=\{0,1,2,\ldots ,10\}\), and \(\mathbb {N}_{1}^{10}=\{1,2,\ldots,10\}\).
Corresponding to BVP (6), we see that \(\alpha=\frac{3}{2}\), \(\beta=\frac {5}{3}\), \(\delta=\frac{1}{8}\), \(\theta=\frac{1}{3}\), \(p(t)=t^{-\frac {1}{4}}(1-t)^{-\frac{1}{4}}\), \(q(t)= t^{-\frac{1}{6}}(1-t)^{-\frac {1}{6}}\), with \(k_{1}=l_{1}=-\frac{1}{4}\), \(k_{2}=l_{2}=-\frac{1}{6}\). We find that \(2+k_{i}+l_{i}>0\) and
$$\begin{aligned}& f(t,y_{1},y_{2},y_{3})=a_{0}+a_{1}(t-t_{i})^{\frac{\tau}{3}}y_{1}^{\tau}+a_{2}(t-t_{i})^{\frac{2\tau }{3}}y_{2}^{\tau}+a_{3}y_{3}^{\tau},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{10}, \\& g(t,x_{1},x_{2},x_{3})=b_{0}+b_{1}(t-t_{i})^{\frac{\sigma}{2}}x_{1}^{\sigma}+b_{2}(t-t_{i})^{\frac{5\sigma }{8}}x_{2}^{\sigma}+b_{3}x_{3}^{\sigma},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{10}, \\& I_{1}(t_{i},y_{1},y_{2},y_{3})=I_{1},\quad\quad J_{1}(t_{i},y_{1},y_{2},y_{3})=J_{1},\quad i \in \mathbb {N}_{1}^{10}, \\& I_{2}(t_{i},x_{1},x_{2},x_{3})=I_{2},\quad\quad J_{2}(t_{i},x_{1},x_{2},x_{3})=J_{2},\quad i \in \mathbb {N}_{1}^{10}. \end{aligned}$$
One sees that (a)-(d), (H1), (H2) hold with
$$\begin{aligned}& \psi_{0}(t)=a_{0},\quad\quad\phi _{0}(t)=b_{0},\quad\quad I_{1i}=I_{1},\quad\quad J_{1i}=J_{1},\quad\quad I_{2i}=I_{2},\quad\quad J_{2i}=J_{2}, \\& A_{i}=|a_{i}|,\quad\quad B_{i}=|b_{i}|,\quad i=1,2,3, \\& C_{i}=D_{i}=E_{i}=F_{i}=0,\quad i=1,2,3, \\& \sigma_{1}=\sigma_{2}=\sigma_{3}=\sigma=\tau,\quad\quad \tau_{1}=\tau_{2}=\tau_{3}=\tau =\sigma. \end{aligned}$$
It is easy to get
$$\begin{aligned} \Phi(t) =&A_{0} \int_{0}^{t}\frac{(t-s)^{\frac{1}{2}}}{\Gamma(3/2)}s^{-\frac {1}{4}}(1-s)^{-\frac{1}{4}}\,ds \\ &{}- \Biggl[A_{0} \int_{0}^{1}\frac{(1-s)^{\frac{1}{2}}}{\Gamma(3/2)}s^{-\frac {1}{4}}(1-s)^{-\frac{1}{4}}\,ds+ \sum _{j=1}^{10}\frac{(1-t_{j})^{\frac{1}{2}}}{\Gamma (3/2)}J_{1}+ \sum _{j=1}^{10} (1-t_{j})^{-\frac{1}{2}}I_{1} \Biggr]t^{-\frac{1}{2}} \\ &{}+\sum _{j=1}^{i}(t-t_{j})^{-\frac{1}{2}}I_{1}+ \sum _{j=1}^{i}\frac{J_{1} }{\Gamma(3/2)}(t-t_{j})^{\frac{1}{2}},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{10}, \end{aligned}$$
and
$$\begin{aligned} \Psi(t) =&B_{0} \int_{0}^{t}\frac{(t-s)^{\frac{2}{3}}}{\Gamma(5/3)}s^{-\frac {1}{6}}(1-s)^{-\frac{1}{6}}\,ds \\ &{}- \Biggl[B_{0} \int_{0}^{1}\frac{(1-s)^{\frac{2}{3}}}{\Gamma(5/3)}s^{-\frac {1}{6}}(1-s)^{-\frac{1}{6}}\,ds+ \sum _{j=1}^{10}I_{2}(1-t_{j})^{-\frac{1}{3}} +\sum _{j=1}^{10}\frac{J_{2}}{\Gamma(5/3)}(1-t_{j})^{\frac {2}{3}} \Biggr]t^{-\frac{1}{3}} \\ &{}+\sum _{j=1}^{i}I_{2}(t-t_{j})^{-\frac{1}{3}}+ \sum _{j=1}^{i}\frac{J_{2} }{\Gamma(5/3)}(t-t_{j})^{\frac{2}{3}},\quad t \in(t_{i},t_{i+1}],i\in \mathbb {N}_{0}^{m}. \end{aligned}$$
By direct computation, using Mathlab 7.0, we have
$$\begin{aligned}& P_{1}=\frac{2\mathbf{B}(1/4,3/4)}{\Gamma(3/2)}+ \frac{\mathbf{B}(1/8,3/4)}{\Gamma(11/8)}+\frac{\Gamma(1/2)}{\Gamma (3/8)} \frac{\mathbf{B}(5/4,3/4)}{\Gamma(3/2)}+\mathbf{B}(3/4,3/4)< 22.2, \\& Q_{1}=\frac{2\mathbf{B}(3/2,5/6)}{\Gamma(5/3)}+ \frac{\mathbf{B}(7/6,5/6)}{\Gamma(4/3)}+\frac{\Gamma(2/3)}{\Gamma (1/3)} \frac{\mathbf{B}(3/2,5/6)}{\Gamma(5/3)}+\mathbf{B}(5/6,5/6)< 5.1, \\& \overline{P_{j}}=P_{1}A_{j}=P_{1}|a_{j}| \le22.2|a_{j}|, \\& \overline{Q_{j}}=Q_{1}B_{j}=Q_{1}|b_{j}| \le5.1|b_{j}|. \end{aligned}$$
One sees that
$$ 5.1\sum _{j=1}^{3}|b_{j}| \Biggl(22.2{ \sum _{j=1}^{3}|a_{j}|} \Biggr)^{1/\sigma}< 1\quad\hbox{or}\quad 22.2\sum _{j=1}^{3}|a_{j}| \Biggl(5.1{\sum _{j=1}^{3}|b_{j}|} \Biggr)^{1/\tau}< 1 $$
(35)
implies
$$ \sum _{j=1}^{3}\overline{Q_{j}} \Biggl({\sum _{j=1}^{3}\overline {P_{j}}} \Biggr)^{1/\sigma}< 1\quad\hbox{or}\quad\sum _{j=1}^{3} \overline{P_{j}} \Biggl({\sum _{j=1}^{3} \overline{Q_{j}}} \Biggr)^{1/\tau}< 1,\quad\hbox{respectively}. $$
(36)
Furthermore
$$ \begin{aligned} &5.1\frac{\sigma\tau-1}{\|\Psi\|}\sum _{j=1}^{3}|b_{j}| \Biggl[\|\Phi\| +22.2 \biggl(\frac{\sigma\tau\|\Psi\|}{\sigma\tau-1} \biggr)^{\sigma}\sum _{j=1}^{3}|a_{j}| \Biggr]^{\tau}\le1\quad\hbox{or } \\ &22.2\frac{\sigma\tau-1}{\|\Phi\|}\sum _{j=1}^{3}|a_{j}| \Biggl[\|\Psi\| +5.1 \biggl(\frac{\tau\sigma\|\Phi\|}{\sigma\tau-1 } \biggr)^{\sigma}\sum _{j=1}^{3}|b_{j}| \Biggr]^{\sigma}\le1 \end{aligned} $$
(37)
implies
$$ \begin{aligned} &\frac{\sigma\tau-1}{\|\Psi\|}\sum _{j=1}^{3} \overline{Q_{j}} \Biggl[\| \Phi\|+ \biggl(\frac{\sigma\tau\|\Psi\|}{\sigma\tau-1} \biggr)^{\sigma}\sum _{j=1}^{3} \overline{P_{j}} \Biggr]^{\tau}\le1\quad\hbox{or} \\ &\frac{\sigma\tau-1}{\|\Phi\|}\sum _{j=1}^{3} \overline{P_{j}} \Biggl[\| \Psi\|+ \biggl(\frac{\tau\sigma\|\Phi\|}{\sigma\tau-1 } \biggr)^{\sigma}\sum _{j=1}^{3} \overline{Q_{j}} \Biggr]^{\sigma}\le 1,\quad\hbox{respectively}. \end{aligned} $$
(38)
It follows from Theorem 3.1 that BVP (10) has at least one solution if
  1. (i)

    \(\sigma\tau\in[0,1)\) or

     
  2. (ii)

    \(\sigma\tau=1\) with (35) or

     
  3. (iii)

    \(\sigma\tau>1\) with (37).

     

Remark 4.1

It is easy to see that BVP (34) has at least one solution for any \(a_{i}\), \(b_{i}\) (\(i=0,1,2,3\)), \(I_{1i}\), \(J_{1i}\), \(I_{2i}\), \(J_{2i}\) (\(i=1,2,3\)) if \(\sigma\tau <1\). Since sufficiently small \(|a_{i}|\), \(|b_{i}|\) (\(i=1,2,3\)) and fixed \(a_{0}\), \(b_{0}\), \(I_{1i}\), \(J_{1i}\), \(I_{2i}\), \(J_{2i}\) (\(i=1,2,3\)) imply (35) holds, BVP (34) has at least one solution for any \(a_{0}\), \(b_{0}\), \(I_{1i}\), \(J_{1i}\), \(I_{2i}\), \(J_{2i}\) (\(i=1,2,3\)) and sufficiently small \(|a_{i}|\), \(|b_{i}|\) (\(i=1,2,3\)) if \(\sigma\tau=1\). Since sufficiently small \(|a_{i}|\), \(|b_{i}|\) (\(i=1,2,3\)) and fixed \(a_{0}\), \(b_{0}\), \(I_{1i}\), \(J_{1i}\), \(I_{2i}\), \(J_{2i}\) (\(i=1,2,3\)) imply (37) holds, BVP (34) has at least one solution for any \(a_{0}\), \(b_{0}\), \(I_{1i}\), \(J_{1i}\), \(I_{2i} \), \(J_{2i}\) (\(i=1,2,3\)) and sufficiently small \(|a_{i}|\), \(|b_{i}|\) (\(i=1,2,3\)) if \(\sigma\tau>1\).

Declarations

Acknowledgements

The authors would like to thank the referees and the editors for their careful reading and some useful comments on improving the presentation of this paper.

Supported by the Natural Science Foundation of Guangdong province (No. S2011010001900) and Natural science research project for colleges and universities of Guangdong Province (No: 2014KTSCX126).

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

Authors’ Affiliations

(1)
Department of Computer, Guangdong Police College
(2)
Department of Mathematics, Guangdong University of Finance and Economics

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