Introduction

Fractional differential systems have many applications in modeling of physical and chemical processes and in engineering [3, 14, 19], and have been of great interest recently. In its turn, mathematical aspects of studies on fractional differential systems were discussed by many authors, see the text books [5, 15] and papers [1, 6, 8, 13, 16, 17, 20, 21, 23–26]. A survey concerning the studies on solvability of two-point or four-point boundary value problems for fractional differential systems was given in [11].

In this paper, we discuss the existence of positive solutions of the following four-point integral type boundary value problem for the multi-term fractional differential system

$$\left\{ \begin{array}{l} D_{0^+}^{\alpha }u(t)+p(t)f(t,v(t),D_{0^+}^{n}v(t))=0,a.e., t\in (0,1), \\ D_{0^+}^{\beta }v(t)+q(t)g(t,u(t),D_{0^+}^{m}u(t))=0,a.e., t\in (0,1),\\ \lim \limits _{t\rightarrow 0}t^{2-\alpha }u(t)-au(\xi )=\int _0^1\phi _1(t,v(t),D_{0^+}^{n}v(t))\mathrm{{d}}t,\\ u(1)-bu(\eta )=\int _0^1\psi _1(t,v(t),D_{0^+}^{n}v(t))\mathrm{{d}}t,\\ \lim \limits _{t\rightarrow 0}t^{2-\beta }v(t)-cv(\xi )=\int _0^1\phi _2(t,u(t),D_{0^+}^{m}u(t))\mathrm{{d}}t,\\ v(1)-dv(\eta )=\int _0^1\psi _2(t,u(t),D_{0^+}^{m}u(t))\mathrm{{d}}t, \end{array} \right. $$
(1)

where

  1. (i)

    \(1< \alpha ,\beta \le 2\), \(\alpha -1<m<\alpha \) and \(\beta -1<n<\beta \), \(D_{0^+}^*\) is the standard Riemann–Liouville differential derivative of order \(*>0\) with the starting point 0,

  2. (ii)

    \(0<\xi \le \eta <1\) and \(a,b,c,d\ge 0\),

  3. (iii)

    \(p,q:(0,1)\rightarrow \mathrm{I}\!\mathrm{R}\), p satisfies that there exist numbers \(k_1,l_1\) such that \(k_1>-1\), \(\alpha -m+l_1>0\), \(2+k_1+l_1>0\) and \(|p(t)|<t^{k_1}(1-t)^{l_1}\) for \(t\in (0,1)\), q satisfies that there exist numbers \(k_2,l_2\) such that \(k_2>-1\), \(\beta -n+l_2>0\), \(2+k_2+l_2>0\) and \(|q(t)|<t^{k_2}(1-t)^{l_2}\) for \(t\in (0,1)\), with \(p(t)\not \equiv 0\) and \(q(t)\not \equiv 0\) on (0, 1),

  4. (iv)

    \(f,g,\phi _i,\psi _i:(0,1)\times [0,+\infty )\times \mathrm{I}\!\mathrm{R}\rightarrow [0,+\infty )\), f is a strong \((n,\beta )\)-Carathéory function and g is a strong \((m,\alpha )\)-Carathéory function with \(f(t,0,0)\not \equiv 0\) and \(g(t,0,0)\not \equiv 0\) on (0, 1), \(\phi _1,\psi _1\) are \((n,\beta )\)-Carathéory functions and \(\phi _2,\psi _2\) are \((m,\alpha )\)-Carathéory functions.

A pair of functions (x, y) is called a solution of BVP (1) if \(x,y\in C^0(0,1]\) and x, y satisfy all equations in (1). We obtain the results on solutions of BVP(1) using Schauder’s fixed point theorem in Banach spaces. The salient features of this study are as follows:

  1. (a)

    the fractional differential equations in (1) are multi-term ones and their nonlinearities depend on the lower order fractional derivatives with order greater than \(\alpha -1\) and \(\beta -1\);

  2. (b)

    instead of the condition \(u(0)=0,v(0)=0\) we consider integral boundary conditions which are more suitable as \(D_{0^+}^\alpha x(t)=0\) with \(\alpha \in (1,2)\) implies \(x(t)=ct^{\alpha -1}\) and obviously x is not continuous at \(t=0\) while \(\lim \nolimits _{t\rightarrow 0}t^{2-\alpha }x(t)\) exists;

  3. (c)

    BVP(1) is a generalized form of known ones in references [4, 7, 9, 10, 21], the positive solutions of BVP(1) obtained are unbounded (discontinuous at \(t=0\)) which are different from those ones (continuous on [0,1]) in [1, 8, 23, 24];

  4. (d)

    this paper is a complement of [11] in which the existence of positive solutions of BVP(1) was studied under the assumptions \(m\in (0,\alpha -1]\) and \(n\in (0,\beta -1]\) while \(m\in (\alpha -1,\alpha ),n\in (\beta -1,\beta )\) are supposed in this paper.

The remainder of this paper is arranged as follows: in Sect. 2, we present preliminary results; in Sect. 3, the main results are presented; and two examples are given in Sect. 4 to illustrate the main results.

Preliminary results

For the convenience of readers, we present here the necessary definitions from fixed point theory and fractional calculus theory.

Definition 2.1

[2] Let X be a Banach space. An operator \(T:X\rightarrow X\) is completely continuous if it is continuous and maps bounded sets into pre-compact sets (or relatively compact sets).

Definition 2.2

[15] The left Riemann–Liouville fractional integral (left forward) of order \(\alpha >0\) of a function \(f:(0,\infty )\rightarrow \mathrm{I}\!\mathrm{R}\) is given by

$$\begin{aligned} I_{0+}^{\alpha }f(t)=\frac{1}{\Gamma (\alpha )}\int _0^t(t-s)^{\alpha -1}f(s)\mathrm{{d}}s, \quad t>0 \end{aligned}$$

provided that the right-hand side exists.

Definition 2.3

[15] The left Riemann–Liouville fractional derivative (left farward) of order \(\alpha >0\) of a continuous function \(f:(0,\infty )\rightarrow \mathrm{I}\!\mathrm{R}\) is given by

$$\begin{aligned} D_{0^+}^{\alpha }f(t)=\frac{1}{\Gamma (n-\alpha )}\frac{\mathrm{{d}}^{n}}{\mathrm{{d}}t^{n}} \int _{0}^{t}\frac{f(s)}{(t-s)^{\alpha -n+1}}\mathrm{{d}}s,\quad t>0 \end{aligned}$$

where \(n-1<\alpha < n\), provided that the right-hand side exists.

Definition 2.4

\(h:(0,1)\times \mathrm{I}\!\mathrm{R}\times \mathrm{I}\!\mathrm{R}\rightarrow \mathrm{I}\!\mathrm{R}\) is called a \((m,\alpha )-\)Carathédory function if it satisfies

  1. (i)

    \(t\rightarrow h\left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \) is measurable on (0, 1) for all \((x,y)\in \mathrm{I}\!\mathrm{R}^2\),

  2. (ii)

    \((x,y)\rightarrow h\left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \) is continuous for a.e. \(t\in (0,1)\),

  3. (iii)

    for each \(r > 0\), there exists nonnegative number \(M_r\) such that \(|u|,|v|\le r\) imply

    $$\begin{aligned} \left| h \left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \right| \le M_r, \quad a.e. t \quad \in (0,1). \end{aligned}$$

Definition 2.5

\(h:(0,1)\times \mathrm{I}\!\mathrm{R}\times \mathrm{I}\!\mathrm{R}\rightarrow \mathrm{I}\!\mathrm{R}\) is called a \((m,\alpha )\)-Carathédory function if it satisfies

  1. (i)

    \(t\rightarrow h (t, t^{\alpha -2}x,t^{2+m-\alpha }y)\) is measurable on (0, 1) for all \((x,y)\in \mathrm{I}\!\mathrm{R}^2\),

  2. (ii)

    \((x,y)\rightarrow h (t, t^{\alpha -2}x,t^{2+m-\alpha }y)\) is continuous for a.e. \(t\in (0,1)\),

  3. (iii)

    for each \(r > 0\), there exists nonnegative function \(\phi _r\in L^1(0,1)\) such that \(|u|,|v|\le r\) imply

    $$\begin{aligned} \left| h \left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \right| \le \phi _r(t), \quad a.e. t\quad \in (0,1).\end{aligned}$$

Lemma 2.1

[15] Let \(n-1\le \alpha < n,\) \(u\in C^0(0,\infty )\bigcap L^1(0,\infty ).\) Then

$$\begin{aligned} I_{0+}^{\alpha }D_{0+}^{\alpha }u(t)=u(t)+C_{1}t^{\alpha -1}+C_{2}t^{\alpha -2}+\cdots +C_{n}t^{\alpha -n}, \end{aligned}$$

where \(C_{i}\in R\), \(i=1,2,\ldots n\)

Choose

$$X=\left\{ x:(0,1]\rightarrow \mathrm{I}\!\mathrm{R}\begin{array}{c} x,\;D_{0^+}^mx\in C^0(0,1]\; \mathrm{{ the \;following \;limits \;exist }} \\ \lim \limits _{t\rightarrow 0}t^{2-\alpha }x(t),\;\lim \limits _{t\rightarrow 0}t^{2+m-\alpha }D_{0^+}^mx(t)\end{array}\right\} $$

with the norm

$$\begin{aligned} ||x||=\Vert x\Vert _X=\max \left\{ \sup \limits _{t\in (0,1]}t^{2-\alpha }|x(t)|,\sup \limits _{t\in (0,1]}t^{2+m-\alpha }|D_{0^+}^mx(t)|\right\} \end{aligned}$$

for \(x\in X.\) It is easy to show that X is a real Banach space.

Choose

$$Y=\left\{ y:(0,1]\rightarrow \mathrm{I}\!\mathrm{R}\begin{array}{c} y,\;D_{0^+}^ny\in C^0(0,1]\; \mathrm{{ the \;following \;limits \;exist }} \\ \lim \limits _{t\rightarrow 0}t^{2-\beta }y(t),\;\lim \limits _{t\rightarrow 0}t^{2+n-\beta }D_{0^+}^ny(t)\end{array}\right\} $$

with the norm

$$\begin{aligned} ||y||=\Vert y\Vert _Y=\max \left\{ \sup \limits _{t\in (0,1]}t^{2-\beta }|y(t)|,\sup \limits _{t\in (0,1]}t^{2+n-\beta }|D_{0^+}^ny(t)|\right\} \end{aligned}$$

for \(y\in Y.\) It is easy to show that Y is a real Banach space.

Thus, \((X\times Y,||\cdot ||)\) is Banach space with the norm defined by

$$\begin{aligned} ||(x,y)||=\max \{||x||=||x||_X,\;||y||=||y||_Y\}\quad \mathrm{{ for }}\quad (x,y)\in X\times Y. \end{aligned}$$

For a function \(x:(0,1]\rightarrow \mathrm{I}\!\mathrm{R},\) a number m and a function \(F:(0,1)\times \mathrm{I}\!\mathrm{R}^2\rightarrow \mathrm{I}\!\mathrm{R},\) denote \(F_{m,x}(t)=F(t,x(t),D_{0^+}^m x(t)).\)

Denote

$$\begin{aligned} \mu _1&=a\xi ^{\alpha -1},\quad \upsilon _1=1-a\xi ^{\alpha -2},\quad \omega _1=1-b\eta ^{\alpha -1},\quad \lambda _1=1-b\eta ^{\alpha -2},\quad \Delta =\mu _1\lambda _1+\upsilon _1\omega _1,\\ \mu _2&=c\xi ^{\beta -1},\quad \upsilon _2=1-c\xi ^{\beta -2},\quad \omega _2=1-d\eta ^{\beta -1},\quad \lambda _2=1-d\eta ^{\beta -2},\quad \nabla =\mu _2\lambda _2+\upsilon _2\omega _2. \end{aligned}$$
(2)

Lemma 2.2

(Lemma 2.6 in [11]) Suppose that \(\Delta \not =0\) and

(B0) \(h\in C^0(0,1)\) and there exist \(k>-1\) and \(l\le 0\) such that \(2+l+k>0\) and \(|h(t)|\le t^k(1-t)^l\) for all \(t\in (0,1)\).

Then \(x\in X\) is a solution of problem

$$ \left\{ \begin{array}{ll} D^{\alpha }x(t)+h(t)=0,0<t<1, \\ \\ \lim \limits _{t\rightarrow 0}t^{2-\alpha }x(t)-ax(\xi )=M,\; x(1)-bx(\eta )=N \end{array}\right.$$
(3)

if and only if \(x\in X\) satisfies

$$\begin{aligned} x(t)&=\frac{\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2}}{\Delta }N+\frac{\omega _1t^{\alpha -2}-\lambda _1t^{\alpha -1} }{\Delta }M \nonumber \\&\quad -\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}h(s)\mathrm{{d}}s +\frac{\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2}}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}h(s)\mathrm{{d}}s \nonumber \\&\quad - \frac{b\upsilon _1t^{\alpha -1}+b\mu _1t^{\alpha -2}}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}h(s)\mathrm{{d}}s +\frac{a\lambda _1t^{\alpha -1}-a\omega _1t^{\alpha -2}}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}h(s)\mathrm{{d}}s. \end{aligned}$$
(4)

Lemma 2.3

(Lemma 2.7 in [11]) Suppose that \(\nabla \not =0\) and (B0) holds. Then \(y\in Y\) is a solution of problem

$$ \left\{ \begin{array}{ll} D^{\beta }y(t)+h(t)=0,0<t<1, \\ \\ \lim \limits _{t\rightarrow 0}t^{2-\beta }y(t)-cy(\xi )=M,\; y(1)-dy(\eta )=N \end{array}\right. $$
(5)

if and only if \(y\in Y\) satisfies

$$\begin{aligned} y(t)&=\frac{\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2}}{\nabla }N+\frac{\omega _2t^{\beta -2}-\lambda _2t^{\beta -1} }{\nabla }M \nonumber \\&\quad -\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}h(s)\mathrm{{d}}s +\frac{\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2}}{\nabla }\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )}h(s)\mathrm{{d}}s \nonumber \\&\quad - \frac{d\lambda _2t^{\beta -1}+d\mu _2t^{\beta -2}}{\nabla }\int _0^\eta \frac{(\eta -s)^{\beta -1}}{\Gamma (\beta )}h(s)\mathrm{{d}}s +\frac{c\lambda _2t^{\beta -1}-c\omega _2t^{\beta -2}}{\nabla }\int _0^\xi \frac{(\xi -s)^{\beta -1}}{\Gamma (\beta )}h(s)\mathrm{{d}}s. \end{aligned}$$
(6)

Define the operator T on \(X\times Y,\) for \((x,y)\in X\times Y,\) by \(T(x,y)(t)=((T_1y)(t),(T_2x)(t))\) with

$$\begin{aligned} (T_1y)(t)= & {} \frac{\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2}}{\Delta }\int _0^1{\psi _1}_{n,y}(s)\mathrm{{d}}s+\frac{\omega _1t^{\alpha -2}-\lambda _1t^{\alpha -1} }{\Delta }\int _0^1{\phi _1}_{n,y}(s)\mathrm{{d}}s\\&-\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s +\frac{\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2}}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\\&- \frac{b\upsilon _1t^{\alpha -1}+b\mu _1t^{\alpha -2}}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,x}(s)\mathrm{{d}}s +\frac{a\lambda _1t^{\alpha -1}-a\omega _1t^{\alpha -2}}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s \end{aligned}$$

and

$$\begin{aligned} (T_2x)(t)= & {} \frac{\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2}}{\nabla }\int _0^1{\psi _2}_{m,x}(s)\mathrm{{d}}s+\frac{\omega _2t^{\beta -2}-\lambda _2 t^{\beta -1} }{\nabla }\int _0^1{\phi _2}_{m,x}(s)\mathrm{{d}}s\\&-\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}q(s)g_{m,x}(s)\mathrm{{d}}s +\frac{\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2}}{\nabla }\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )}q(s)g_{m,x}(s)\mathrm{{d}}s\\&- \frac{d\upsilon _2t^{\beta -1}+d\mu _2t^{\beta -2}}{\nabla }\int _0^\eta \frac{(\eta -s)^{\beta -1}}{\Gamma (\beta )}q(s)g_{m,x}(s)\mathrm{{d}}s +\frac{c\lambda _2t^{\beta -1}-c\omega _2t^{\beta -2}}{\nabla }\int _0^\xi \frac{(\xi -s)^{\beta -1}}{\Gamma (\beta )}q(s)g_{m,x}(s)\mathrm{{d}}s. \end{aligned}$$

By Lemmas 2.2 and 2.3, we have that \((x,y)\in X\times Y\) is a solution of BVP(8) if and only if \((x,y)\in X\times Y\) is a fixed point of T.

Lemma 2.4

Suppose that (i)–(iv) defined in Sect. 1 hold, \(\Delta \not =0\) and \(\nabla \not =0.\) Then \(T:X\times Y\rightarrow X\times Y\) is completely continuous.

Proof

We will prove that both \(T_1\) and \(T_2\) are completely continuous. The proof of the completeness of \(T_1\) is divided into four steps and similarly we can prove that \(T_2\) is completely continuous.

Step 1 Suppose that \(\alpha -1<m<\alpha \). We prove that both \(T_1: Y\rightarrow X\) is well defined.

For \(y\in Y\), there exits \(r>0\) such that

$$\begin{aligned} \max \left\{ \sup \limits _{t\in (0,1]}t^{2-\beta }|y(t)|,\;\sup \limits _{t\in (0,1]}t^{2+n-\beta }|D_{0^+}^ny(t)||\right\} <r. \end{aligned}$$

Then (iii) and (iv) imply that there exists a number \(M_r>0\) and \(\phi _0,\psi _0\in L^1(0,1)\) such that

$$\begin{aligned} |f(t,y(t),D_{0^+}^ny(t)|&=\left| f\left( t,t^{\beta -2}t^{2-\beta }y(t),t^{\beta -n-2}t^{2+n-\beta }D_{0^+}^ny(t)\right) \right| \le M_r, \nonumber \\ |\phi _1(t,y(t),D_{0^+}^ny(t)|&\le \phi _0(t),\; |\psi _1(t,y(t),D_{0^+}^ny(t)|\le \psi _0(t) \end{aligned}$$
(7)

for all \(t\in (0,1)\). Then

$$\begin{aligned} \left| \int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\right|\le & {} \int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\\le & {} M_r\int _0^t\frac{(t-s)^{\alpha +l_1-1}}{\Gamma (\alpha )}s^{k_1}\mathrm{{d}}s\\= & {}\; M_rt^{\alpha +k_1+l_1}\int _0^1\frac{(1-w)^{\alpha +l_1-1}}{\Gamma (\alpha )}w^{k_1}\mathrm{{d}}w=\frac{\mathbf{B}(\alpha +l_1,k_1+1)M_r}{\Gamma (\alpha )}t^{\alpha +k_1+l_1},\\&\left| \int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}p(s)f_{n,y}(s)\mathrm{{d}}s\right| \le \int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\\le & {} M_r\int _0^t\frac{(t-s)^{\alpha -m+l_1-1}}{\Gamma (\alpha -m)}s^{k_1}\mathrm{{d}}s\\= & {}\; M_rt^{\alpha -m+k_1+l_1}\int _0^1\frac{(1-w)^{\alpha -m+l_1-1}}{\Gamma (\alpha )}w^{k_1}\mathrm{{d}}w=\frac{\mathbf{B}(\alpha -m+l_1,k_1+1)M_r}{\Gamma (\alpha -m)}t^{\alpha -m+k_1+l_1}. \end{aligned}$$

On the other hand, note \(D_{0^+}^mt^\mu =\frac{\Gamma (\mu +1)}{\Gamma (\mu +1-m)}t^{\mu -m}\), \(\Gamma (0)=\infty \) with \(\frac{1}{\Gamma (0)}=0\), we have

$$\begin{aligned} t^{2-\alpha }(T_1y)(t)= & {} \frac{\upsilon _1t+\mu _1}{\Delta }\int _0^1{\psi _1}_{n,y}(s)\mathrm{{d}}s+\frac{\omega _1-\lambda _1t}{\Delta }\int _0^1{\phi _1}_{n,y}(s)\mathrm{{d}}s\\&- t^{2-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s +\frac{\upsilon _1t+\mu _1}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\\&- \frac{b\upsilon _1t+b\mu _1}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,x}(s)\mathrm{{d}}s +\frac{a\lambda _1t-a\omega _1}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s,\\ t^{2+m-\alpha } D_{0^+}^m(T_1y)(t)= & {} \frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t+ \mu _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^1{\psi _1}_{n,y}(s)\mathrm{{d}}s\\&+\frac{\omega _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)} -\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t }{\Delta }\int _0^1{\phi _1}_{n,y}(s)\mathrm{{d}}s -t^{2+m-\alpha } \int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}p(s)f_{n,y}(s)\mathrm{{d}}s\\&+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t+\mu _1 \frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\\&- \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t+b\mu _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,x}(s)\mathrm{{d}}s\\&+\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t-a\omega \omega _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s. \end{aligned}$$

It is easy to show that \(T_1y\in X\). So \(T_1: Y\rightarrow X\) is well defined.

Step 2 Suppose that \(\alpha -1<m<\alpha \). Prove that \(T_1\) is continuous.

Let \(\{y_i\in Y\}\) be a sequence such that \(y_i\rightarrow y_0\) as \(i\rightarrow +\infty \) in Y. Then there exists \(r>0\) such that

$$\begin{aligned} \max \left\{ \sup \limits _{t\in (0,1]}t^{2-\beta }|y_i(t)|,\;\sup \limits _{t\in (0,1]}t^{2+n-\beta }|D_{0^+}^ny_i(t)|\right\} \le r \end{aligned}$$

holds for \(i=0,1,2,\ldots \).

Then (iii) and (iv) imply that there exists a number \(M_r>0\) and \(\phi _0,\psi _0\in L^1(0,1)\) such that

$$\begin{aligned} |f(t,y_i(t),D_{0^+}^ny_i(t)|&=\left| f\left( t,t^{\beta -2}t^{2-\beta }y_i(t),t^{\beta -n-2}t^{2+n-\beta }D_{0^+}^ny_i(t)\right) \right| \le M_r,\nonumber \\ |\phi _1(t,y_i(t),D_{0^+}^ny_i(t)|&\le \phi _0(t),\nonumber \\ |\psi _1(t,y_i(t),D_{0^+}^ny_i(t)|&\le \psi _0(t) \end{aligned}$$
(8)

for all \(t\in (0,1)\). By a direct computation, we get \((T_1y_i)(t)\) and \(D_{0^+}^m (T_1y_i)(t)\). One sees that

$$\begin{aligned} & t^{2-\alpha }\left| \int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y_i}(s)\mathrm{{d}}s-\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y_i}(s)\mathrm{{d}}s\right| \\ &\quad \le 2t^{2-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\ &\quad =\frac{2M_r\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}t^{2+k_1+l_1}\le \frac{2M_r\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )},\\ & t^{2+m-\alpha }\left| \int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}p(s)f_{n,y_i}(s)\mathrm{{d}}s-\int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m )}p(s)f_{n,y_i}(s)\mathrm{{d}}s\right| \\ &\quad \le 2M_rt^{2+m-\alpha }\int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s= 2M_rt^{2+m-\alpha }t^{\alpha -m+k_1+l_1}\int _0^1\frac{(1-w)^{\alpha -m+l_1-1}}{\Gamma (\alpha -m)}w^{k_1}\mathrm{{d}}s\\ &\quad \le \frac{2M_r\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}t^{2+k_1+l_1}\le \frac{2M_r\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}. \end{aligned}$$

We can show using the dominant convergence theorem that \(T_1y_i\rightarrow T_1y_0\) as \(i\rightarrow +\infty \). Then \(T_1\) is continuous.

Now we prove that \(T_1\) maps bounded sets in Y into relatively compact sets in X. Let \(\Omega \subset Y\) be a bounded subset. Then there exists \(r>0\) such that

$$\begin{aligned} \max \left\{ \sup \limits _{t\in (0,1]}t^{2-\beta }|y(t)|,\;\sup \limits _{t\in (0,1]}t^{2+n-\beta }|D_{0^+}^ny(t)|\right\} \le r \end{aligned}$$

holds for all \(y\in \Omega. \) Then (iii) and (iv) imply that there exists a number \(M_r>0\) and \(\phi _0,\psi _0\in L^1(0,1)\) such that

$$\begin{aligned} |f(t,y(t),D_{0^+}^ny(t)|&=\left| f\left( t,t^{\beta -2}t^{2-\beta }y(t),t^{\beta -n-2}t^{2+n-\beta }D_{0^+}^ny(t)\right) \right| \le M_r,\nonumber \\ |\phi _1(t,y(t),D_{0^+}^ny(t)|&\le \phi _0(t),\; \; |\psi _1(t,y(t),D_{0^+}^ny(t)|\le \psi _0(t) \end{aligned}$$
(9)

for all \(t\in (0,1)\).

Step 3 Suppose that \(\alpha -1<m<\alpha \). Prove that \(\{T_1y:y\in \Omega \}\) is a bounded set in X.

Similar to Step 1 and Step 2, we can show that

$$\begin{aligned} t^{2-\alpha }|(T_1y)(t)|& \le \frac{|1-a\xi ^{\alpha -2}|+|a|\xi ^{\alpha -1}}{|\Delta | }\int _0^1{\psi _0}(s)\mathrm{{d}}s+\frac{|1-b\eta ^{\alpha -1}|+|1-b\eta ^{\alpha -2}|}{|\Delta | }\int _0^1{\phi _0}(s)\mathrm{{d}}s\\&\quad +\frac{M_r\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )} +\frac{|1-a\xi ^{\alpha -2}|+|a|\xi ^{\alpha -1}}{|\Delta |}\frac{2M_r\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&\quad + \frac{|b(1-a\xi ^{\alpha -2})|+|ab|\xi ^{\alpha -1}}{|\Delta |}\eta ^{-2-k_1-l_1}\frac{M_r\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&\quad +\frac{|a(1-b\eta ^{\alpha -2})|+|a(1-b\eta ^{\alpha -1})|}{|\Delta |}\xi ^{-2-k_1-l_1}\frac{M_r\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )},\\ t^{2+m-\alpha } D_{0^+}^m(T_1y)(t)&\le \frac{|1-a\xi ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+ |a|\xi ^{\alpha -1}\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{|\Delta |}\int _0^1{\psi _0}(s)\mathrm{{d}}s\\&\quad +\frac{|1-b\eta ^{\alpha -1}|\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|} +|1-b\eta ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)} }{|\Delta | }\int _0^1{\phi _0}(s)\mathrm{{d}}s\\&\quad \times \frac{M_r\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}\\&\quad +\frac{|1-a\xi ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+|a|\xi ^{\alpha -1}\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{|\Delta | }\frac{M_r\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}\\&\quad + \frac{|b(1-a\xi ^{\alpha -2})|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+|ab|\xi ^{\alpha -1}\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{|\Delta | }\eta ^{m-2-k_1-l_1}\frac{M_r\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}\\& \quad +\frac{|a(1-b\eta ^{\alpha -2})|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+|a(1-b\eta ^{\alpha -1})|\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{|\Delta |}\xi ^{m-2-k_1-l_1}\frac{M_r\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}. \end{aligned}$$

So \(T_1\) maps bounded sets into bounded sets in X.

Step 4 Suppose that \(\alpha -1<m<\alpha \). Prove that \(\{T_1y:y\in \Omega \}\) is a relatively compact set in X.

We prove first that both \(\{t^{2-\alpha }(T_1y)(t):y\in \Omega \}\) and \(\{t^{2+m-\alpha }D_{0^+}^m(T_1y)(t):y\in \Omega \}\) are equi-continuous on (0, 1]. By the definition of \(T_1\), it suffices to show that both

$$\begin{array}{ll} &{}\left\{ t^{2-\alpha }\int _0^t(t-s)^{\alpha -1}p(s)f_{n,y}(s)\mathrm{{d}}s:y\in \Omega \right\} \hbox { and }\\ &{}\left\{ t^{2+m-\alpha }\int _0^t(t-s)^{\alpha -m-1}p(s)f_{n,y}(s)\mathrm{{d}}s:y\in \Omega \right\} \end{array}$$

are equi-continuous on (0, 1] (we can prove that the other parts of \(\{t^{2-\alpha }(T_1y)(t):y\in \Omega \}\) and \(\{t^{2+m-\alpha }D_{0^+}^m(T_1y)(t):y\in \Omega \}\) are equi-continuous on (0, 1] similar to [1]). Then, we prove that both \(\{t^{2-\alpha }(T_1y)(t):y\in \Omega \}\) and \(\{t^{2+m-\alpha }D_{0^+}^n(T_1y)(t):y\in \Omega \}\) are equi-convergent as \(t\rightarrow 0\). By the definition of \(T_1\), it suffices to show that both

$$\begin{array}{ll} &{}\left\{ t^{2-\alpha }\int _0^t(t-s)^{\alpha -1}p(s)f_{n,y}(s)\mathrm{{d}}s:y\in \Omega \right\} \hbox { and}\\ &{}\left\{ t^{2+m-\alpha }\int _0^t(t-s)^{\alpha -m-1}p(s)f_{n,y}(s)\mathrm{{d}}s:y\in \Omega \right\} \end{array}$$

are equi-convergent as \(t\rightarrow 0\).

First, let \(t_1,t_2\in [e,f]\subset (0,1]\) with \(t_1<t_2\), \(0<e<f\le 1\), and \(y\in \Omega \). Then we have

$$\begin{aligned}&\left| t_1^{2-\alpha }\int _0^{t_1}(t_1-s)^{\alpha -1}p(s)f_{n,y}(s) \mathrm{{d}}s-t_2^{2-\alpha }\int _0^{t_2}(t_2-s)^{\alpha -1}p(s)f_{n,y}(s)\mathrm{{d}}s\right| \\&\quad \le |t_1^{2-\alpha }-t_2^{2-\alpha }|\int _0^{t_2}(t_2-s)^{\alpha -1}|p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\qquad + t_1^{2-\alpha }\int _{t_1}^{t_2}(t_2-s)^{\alpha -1}|p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\qquad +t_1^{2-\alpha }\int _0^{t_1}|(t_1-s)^{\alpha -1}-(t_2-s)^{\alpha -1}||p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\quad \le M_r|t_1^{2-\alpha }-t_2^{2-\alpha }|\int _0^{t_2}(t_2-s)^{\alpha -1}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s + M_rt_1^{2-\alpha }\int _{t_1}^{t_2}(t_2-s)^{\alpha -1}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\\&\qquad +M_rt_1^{2-\alpha }\int _0^{t_1}|(t_1-s)^{\alpha -1}-(t_2-s)^{\alpha -1}|s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\\&\quad \le M_r|t_1^{2-\alpha }-t_2^{2-\alpha }|t_2^{\alpha +k_1+l_1}{} \mathbf{B}(\alpha +l_1,k_1+1) + M_rt_1^{2-\alpha }t_2^{\alpha +l_1+k_1}\int _{\frac{t_1}{t_2}}^{1}(1-w)^{\alpha +l_1-1}w^{k_1}\mathrm{{d}}w\\&\qquad +M_rt_2^{2-\alpha }\int _0^{1}[(t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}]s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\\&\quad \rightarrow 0\hbox { uniformly in }\Omega \hbox { as }t_1\rightarrow t_2\hbox { on }[e,f]. \end{aligned}$$

Second, let \(t_1,t_2\in [e,f]\subset (0,1]\) with \(0<e\le t_1<t_2\le f\le 1\) and \(y\in \Omega \). Then we have

$$\begin{aligned}&\left| t_1^{2+m-\alpha }\int _0^{t_1}(t_1-s)^{\alpha -m-1}p(s)f_{n,y}(s)\mathrm{{d}}s-t_2^{2+m-\alpha }\int _0^{t_2}(t_2-s)^{\alpha -m-1}p(s)f_{n,y}(s)\mathrm{{d}}s\right| \\&\quad \le |t_1^{2+m-\alpha }-t_2^{2+m-\alpha }|\int _0^{t_2}(t_2-s)^{\alpha -m-1}|p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\quad + t_1^{2+m-\alpha }\int _{t_1}^{t_2}(t_2-s)^{\alpha -m-1}|p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\qquad +t_1^{2+m-\alpha }\int _0^{t_1}|(t_1-s)^{\alpha -m-1}-(t_2-s)^{\alpha -m-1}||p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\quad \le M_r |t_1^{2+m-\alpha }-t_2^{2+m-\alpha }|\int _0^{t_2}(t_2-s)^{\alpha -m-1}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\\&\qquad + M_rt_1^{2+m-\alpha }\int _{t_1}^{t_2}(t_2-s)^{\alpha -m-1}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\\&\qquad +M_rt_1^{2+m-\alpha }\int _0^{t_1}[(t_1-s)^{\alpha -m-1}-(t_2-s)^{\alpha -m-1}]s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\\&\quad \le M_r |t_1^{2+m-\alpha }-t_2^{2+m-\alpha }|t_2^{\alpha -m+k_1+l_1}{} \mathbf{B}(\alpha -m+l_1,k_1+1)\\&\qquad + M_rt_1^{2+m-\alpha }t^{\alpha -m+k_1+l_1}_2\int _{\frac{t_1}{t_2}}^{1}(1-w)^{\alpha -m-1+l_1}w^{k_1}\mathrm{{d}}w\\&\qquad +M_rt_1^{2+m-\alpha }\int _0^{t_1}(t_1-s)^{\alpha -m-1}s^{k_1}(t_1-s)^{l_1}\mathrm{{d}}s\\&\qquad -M_rt_1^{2+m-\alpha }\int _0^{t_1}(t_2-s)^{\alpha -m-1}s^{k_1}(t_2-s)^{l_1}\mathrm{{d}}s\\&\quad \le M_r |t_1^{2+m-\alpha }-t_2^{2+m-\alpha }|t_2^{\alpha -m+k_1+l_1}{} \mathbf{B}(\alpha -m+l_1,k_1+1)\\&\qquad + M_r\max \{e^{\alpha -m+k_1+l_1},f^{\alpha -m+k_1+l_1}\}\int _{\frac{t_1}{t_2}}^{1}(1-w)^{\alpha -m-1+l_1}w^{k_1}\mathrm{{d}}w\\&\qquad +M_rt_1^{2+m-\alpha }t_1^{\alpha -m+k_1+l_1}\int _0^{1}(1-w)^{\alpha -m+l_1-1}w^{k_1}\mathrm{{d}}w\\&\qquad -M_rt_1^{2+m-\alpha }t_2^{\alpha -m+k_1+l_1}\int _0^{\frac{t_1}{t_2}}(1-w)^{\alpha -m+l_1-1}w^{k_1}\mathrm{{d}}w\\&\quad \le M_r |t_1^{2+m-\alpha }-t_2^{2+m-\alpha }|t_2^{\alpha -m+k_1+l_1}{} \mathbf{B}(\alpha -m+l_1,k_1+1)\\&\qquad + M_r\max \{e^{\alpha -m+k_1+l_1},f^{\alpha -m+k_1+l_1}\}\int _{\frac{t_1}{t_2}}^{1}(1-w)^{\alpha -m-1+l_1}w^{k_1}\mathrm{{d}}w\\&\qquad +M_rt_1^{2+m-\alpha }\left( t_1^{\alpha -m+k_1+l_1}-t_2^{\alpha -m+k_1+l_1}\right) \int _0^{1}(1-w)^{\alpha -m+l_1-1}w^{k_1}\mathrm{{d}}w\\&\qquad +M_rt_1^{2+m-\alpha }t_2^{\alpha -m+k_1+l_1}\int _{\frac{t_1}{t_2}}^1(1-w)^{\alpha -m+l_1-1}w^{k_1}\mathrm{{d}}w\\&\quad \rightarrow 0\hbox { uniformly in }\Omega \hbox { as }t_1\rightarrow t_2\hbox { on }[e,f]. \end{aligned}$$

Third, we have

$$\begin{aligned}&\left| t^{2-\alpha }(T_1y)(t)-\left( \frac{a\xi ^{\alpha -1}}{\Delta }\int _0^1{\psi _1}_{n,y}(s)\mathrm{{d}}s+ \frac{(1-b\eta ^{\alpha -1})}{\Delta }\int _0^1{\phi _1}_{n,y}(s)\mathrm{{d}}s\right. \right. \\&\quad +\frac{a\xi ^{\alpha -1}}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s - \frac{ab\xi ^{\alpha -1}}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,x}(s)\mathrm{{d}}s\\&\quad \left. \left. -\frac{a(1-b\eta ^{\alpha -1})}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\right) \right| \\&\quad \le \frac{|1-a\xi ^{\alpha -2}|t}{|\Delta | }\int _0^1{\psi _0}\mathrm{{d}}s+\frac{|1-b\eta ^{\alpha -2}|t}{|\Delta | }\int _0^1{\phi _0}(s)\mathrm{{d}}s\\&\qquad + t^{2-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}|p(s)f_{n,y}(s)|\mathrm{{d}}s +\frac{|1-a\xi ^{\alpha -2}|t}{|\Delta |}\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}|p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\qquad + \frac{|b(1-a\xi ^{\alpha -2})|t}{|\Delta | }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}|p(s)f_{n,x}(s)|\mathrm{{d}}s\\&\qquad +\frac{|a(1-b\eta ^{\alpha -2})|t}{|\Delta |}\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}|p(s)f_{n,y}(s)|\mathrm{{d}}s\\&\quad \le \frac{|1-a\xi ^{\alpha -2}|t}{|\Delta | }\int _0^1{\psi _0}\mathrm{{d}}s+\frac{|1-b\eta ^{\alpha -2}|t}{|\Delta | }\int _0^1{\phi _0}(s)\mathrm{{d}}s\\&\qquad + t^{2+k_1+l_1}\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r +\frac{|1-a\xi ^{\alpha -2}|t}{|\Delta |}\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r\\&\qquad + \frac{|b(1-a\xi ^{\alpha -2})|t}{|\Delta | }\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r +\frac{|a(1-b\eta ^{\alpha -2})|t}{|\Delta |}\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r\\&\quad \rightarrow 0\hbox { uniformly on }\Omega \hbox { as }t\rightarrow 0. \end{aligned}$$

Fourth, we have

$$\begin{aligned}&\left| t^{2+m-\alpha } D_{0^+}^m(T_1y)(t)-\left( \frac{ a\xi ^{\alpha -1}\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^1{\psi _1}_{n,y}(s)\mathrm{{d}}s \right. \right. \\&\quad +\frac{(1-b\eta ^{\alpha -1})\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)} }{\Delta }\int _0^1{\phi _1}_{n,y}(s)\mathrm{{d}}s +\frac{a\xi ^{\alpha -1}\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\\&\quad - \frac{ab\xi ^{\alpha -1}\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,x}(s)\mathrm{{d}}s\\&\quad -\left. \left. \frac{a(1-b\eta ^{\alpha -1})\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}p(s)f_{n,y}(s)\mathrm{{d}}s\right) \right| \\&\quad \le \frac{|1-a\xi ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta | }\int _0^1{\psi _0}(s)\mathrm{{d}}s +\frac{|1-b\eta ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t }{|\Delta | }\int _0^1{\phi _0}(s)\mathrm{{d}}s\\&\quad +t^{2+m-\alpha } \int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\&\quad +\frac{|1-a\xi ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta |}\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\&\quad + \frac{|b(1-a\xi ^{\alpha -2})|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta | }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\&\quad +\frac{|a(1-b\eta ^{\alpha -2})|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta | }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}M_r\mathrm{{d}}s\\&\quad \le \frac{|1-a\xi ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta | }\int _0^1{\psi _0}(s)\mathrm{{d}}s +\frac{|1-b\eta ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t }{|\Delta | }\int _0^1{\phi _0}(s)\mathrm{{d}}s\\&\quad +t^{2+k_1+l_1} \frac{\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}M_r +\frac{|1-a\xi ^{\alpha -2}|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta |}\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r\\&\quad + \frac{|b(1-a\xi ^{\alpha -2})|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta | }\eta ^{\alpha -2}\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r\\&\quad +\frac{|a(1-b\eta ^{\alpha -2})|\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}t}{|\Delta | }\xi ^{\alpha -2}\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}M_r\\&\quad \rightarrow 0\hbox { uniformly on }\Omega \hbox { as }t\rightarrow 0. \end{aligned}$$

Therefore, \(T_1\Omega \) is relatively compact.

From above discussion, \(T_1\) is completely continuous. The proof is completed. \(\square \)

Define

$$\begin{array}{l} G(t,s)=\frac{1}{\Gamma (\alpha )\Delta }\left\{ \begin{array}{l} \begin{array}{l}(\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2})(1-s)^{\alpha -1}\\ +(\lambda _1at^{\alpha -1}-\omega _1at^{\alpha -2})(\xi -s)^{\alpha -1}\\ -(\upsilon _1bt^{\alpha -1}+b\mu _1t^{\alpha -2})(\eta -s)^{\alpha -1}\\ -\left( \mu _1\lambda _1+\omega _1\upsilon _1\right) (t-s)^{\alpha -1},\end{array}0\le s\le \min \{t,\xi \},\\ \\ \begin{array}{l}(\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2})(1-s)^{\alpha -1}\\ -(\upsilon _1bt^{\alpha -1}+b\mu _1t^{\alpha -2})(\eta -s)^{\alpha -1} \\ -\left( \mu _1\lambda _1+\omega _1\upsilon _1\right) (t-s)^{\alpha -1},\end{array} \begin{array}{l}\xi<s\le \min \{t,\eta \},\end{array} \\ \\ \begin{array}{l}(\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2})(1-s)^{\alpha -1}\\ -(\upsilon _1bt^{\alpha -1}+b\mu _1t^{\alpha -2})(\eta -s)^{\alpha -1} ,\end{array} \begin{array}{l} \max \{t,\xi \}<s\le \eta ,\end{array}\\ \\ \begin{array}{l}(\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2})(1-s)^{\alpha -1}\\ +(\lambda _1at^{\alpha -1}-\omega _1at^{\alpha -2})(\xi -s)^{\alpha -1} \\ -(\upsilon _1bt^{\alpha -1}+b\mu _1t^{\alpha -2})(\eta -s)^{\alpha -1},\end{array} t<s\le \xi ,\\ \\ \begin{array}{l}(\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2})(1-s)^{\alpha -1}\\ -\left( \mu _1\lambda _1+\omega _1\upsilon _1\right) (t-s)^{\alpha -1},\end{array}\eta<s\le t,\\ \\ (\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2})(1-s)^{\alpha -1},\max \{\eta ,t\}<s\le 1, \end{array} \right. \end{array}$$

and

$$\begin{array}{l} H(t,s)=\frac{1}{\Gamma (\beta )\nabla }\left\{ \begin{array}{l} \begin{array}{l}(\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2})(1-s)^{\beta -1}\\ +(\lambda _2ct^{\beta -1}-\omega _2ct^{\beta -2})(\xi -s)^{\beta -1}\\ -\left( \mu _2\lambda _2+\omega _2\upsilon _2\right) (t-s)^{\beta -1},\end{array}0\le s\le \min \{t,\xi \},\\ \\ \begin{array}{l}(\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2})(1-s)^{\beta -1}\\ -(\upsilon _2\mathrm{{d}}t^{\beta -1}+d\mu _2t^{\beta -2})(\eta -s)^{\beta -1} \\ -\left( \mu _2\lambda _2+\omega _2\upsilon _2\right) (t-s)^{\beta -1},\end{array} \begin{array}{l}\xi<s\le \min \{t, \eta \},\end{array} \\ \\ \begin{array}{l}(\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2})(1-s)^{\beta -1}\\ -(\upsilon _2\mathrm{{d}}t^{\beta -1}+d\mu _2t^{\beta -2})(\eta -s)^{\beta -1} ,\end{array} \begin{array}{l} \max \{t,\xi \}<s\le \eta ,\end{array}\\ \\ \begin{array}{l} (\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2})(1-s)^{\beta -1}\\ +(\lambda _2ct^{\beta -1}-\omega _2ct^{\beta -2})(\xi -s)^{\beta -1}\\ -(\upsilon _2\mathrm{{d}}t^{\beta -1}+d\mu _2t^{\beta -2})(\eta -s)^{\beta -1},\end{array}t<s\le \xi ,\\ \\ \begin{array}{l}(\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2})(1-s)^{\beta -1}\\ -\left( \mu _2\lambda _2+\omega _2\upsilon _2\right) (t-s)^{\beta -1},\end{array}\eta<s\le t,\\ \\ (\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2})(1-s)^{\beta -1},\max \{\eta ,t\}<s\le 1. \end{array} \right. \end{array}$$

Now, we rewrite

$$\begin{aligned} (T(x,y))(t)&=((T_1y)(t),(T_2x)(t))\\ &=\left( \frac{\upsilon _1t^{\alpha -1}+\mu _1t^{\alpha -2}}{\Delta }\int _0^1{\psi _1}_{n,y}(s){\mathrm d}s+\frac{\omega _1t^{\alpha -2}-\lambda _1t^{\alpha -1} }{\Delta }\int _0^1{\phi _1}_{n,y}(s){\mathrm d}s+\int _0^1G(t,s)p(s)f_{n,y}(s){\mathrm d}s,\right. \\ &\left. \frac{\upsilon _2t^{\beta -1}+\mu _2t^{\beta -2}}{\nabla }\int _0^1{\psi _2}_{m,x}(s){\mathrm d}s+\frac{\omega _2t^{\beta -2}-\lambda _2 t^{\beta -1} }{\nabla }\int _0^1{\phi _2}_{m,x}(s){\mathrm d}s+\int _0^1H(t,s)g_{m,x}(s){\mathrm d}s\right) . \end{aligned}$$

Lemma 2.5

(Lemma 2.9 in [11]) Suppose that \(a,b,c,d\ge 0,\) and

$$\begin{array}{l} \Delta>0,\;\;0\le a<\frac{1}{\xi ^{\alpha -2}(1-\xi )},\;\;0\le b<\frac{1}{\eta ^{\alpha -1}},\\ \\ \nabla >0, \;\;0\le c<\frac{1}{\xi ^{\beta -2}(1-\xi )},\;\;0\le d<\frac{1}{\eta ^{\beta -1}}. \end{array}$$
(10)

Then

$$\begin{array}{l} G(t,s)\ge 0 \;\mathrm{for\; all }\;t,s\in (0,1),\;\;H(t,s)\ge\; 0 \; \mathrm{for \;all }\;t,s\in (0,1). \end{array} $$
(11)

Main results

In this section, we prove existence result on solutions of BVP(1). Let \(\mu _i,\upsilon _i,\omega _i,\lambda _i(i=1,2)\) and \(\Delta ,\nabla \) be defined by (10). For \(\Phi \in L^1(0,1)\), denote \(||\Phi ||_1=\int _0^1|\Phi (s)|\mathrm{{d}}s\). The following assumption will be used in the main theorem.

A function \(\Phi :[0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) is called a bi-increasing function if both \(u\rightarrow \Phi (u,v)\) and \(v\rightarrow \Phi (u,v)\) are increasing. We now list the following assumption:

(B1) there exist \(\overline{\phi }_i,\overline{\psi }_i\in L^1(0,1)(i=1,2)\) and bi-increasing functions \(\Phi ,\Psi ,\Phi _i,\Psi _i(i=1,2)\) such that

$$\begin{aligned} \left| f\left( t,\frac{u}{t^{2-\beta }},\frac{v}{t^{2+n-\beta }}\right) \right| &\le \Phi (|u|,|v|),t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| g\left( t,\frac{u}{t^{2-\alpha }},\frac{v}{t^{2+m-\alpha }}\right) \right| &\le \Psi (|u|,|v|),t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \phi _1\left( t,\frac{u}{t^{2-\beta }},\frac{v}{t^{2+n-\beta }}\right) \right| &\le \overline{\phi }_1(t)\Phi _1(|u|,|v|),t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \psi _1\left( t,\frac{u}{t^{2-\beta }},\frac{v}{t^{2+n-\beta }}\right) \right| &\le \overline{\psi }_1(t)\Psi _1(|u|,|v|),t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \phi _2\left( t,\frac{u}{t^{2-\alpha }},\frac{v}{t^{2+m-\alpha }}\right) \right| &\le \overline{\phi }_2(t)\Phi _2(|u|,|v|),t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \psi _2\left( t,\frac{u}{t^{2-\alpha }},\frac{v}{t^{2+m-\alpha }}\right) \right| &\le \overline{\psi }_2(t)\Psi _2(|u|,|v|),t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R}. \end{aligned}$$

For ease expression, denote

$$\begin{aligned} M_1= & {} \left[ \frac{\upsilon _1+\mu _1}{\Delta }||+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+ \mu _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }\right] ||\overline{\psi }_1||_1,\\ N_1= & {} \left[ \frac{\omega _1+\lambda _1}{\Delta }||+\frac{\omega _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)} +\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)} }{\Delta }\right] ||\overline{\phi }_1||_1,\\ Q_1= & {} \left[ 1+\frac{\upsilon _1+\mu _1}{\Delta }+ \frac{b\upsilon _1+b\mu _1}{\Delta }\eta ^{\alpha +k_1+l_1} +\frac{a\lambda _1+a\omega _1}{\Delta }\xi ^{\alpha +k_1+l_1}\right] \frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+\frac{\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+ \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+b\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\eta ^{\alpha +k_1+l_1}{} \mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+a\omega _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\xi ^{\alpha +k_1+l_1}\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )} , \end{aligned}$$

and

$$\begin{aligned} M_2= & {} \left[ \frac{\upsilon _2+\mu _2}{\nabla }||+\frac{\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+ \mu _2\frac{\Gamma (\beta -1)}{\Gamma (\beta -n-1)}}{\nabla }\right] ||\overline{\psi }_2||_1,\\ N_2= & {} \left[ \frac{\omega _2+\lambda _2}{\nabla }||+\frac{\omega _2\frac{\Gamma (\beta -1)}{\Gamma (\beta -n-1)} +\lambda _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)} }{\nabla }\right] ||\overline{\phi }_2||_1,\\ Q_2= & {} \left[ 1+\frac{\upsilon _2+\mu _2}{\nabla }+ \frac{c\upsilon _2+d\mu _2}{\nabla }\eta ^{\beta +k_2+l_2} +\frac{c\lambda _2+c\omega _2}{\nabla }\xi ^{\beta +k_2+l_2}\right] \frac{\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+\frac{\mathbf{B}(\beta -n+l_2,k_2+1)}{\Gamma (\beta -n)}+\frac{\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+\mu _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+ \frac{d\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+d\mu _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\eta ^{\beta +k_2+l_2}{} \mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+\frac{c\lambda _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+c\omega _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\xi ^{\beta +k_2+l_2}\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )} . \end{aligned}$$

Theorem 3.1

Suppose that (12) holds, (i)–(iv) defined in Sect. 1 and (B1) hold. Then BVP(1) has at least one positive solution if

$$\begin{array}{l} M_1\Psi _1(r_2,r_2)+N_1\Phi _1(r_2,r_2)+Q_1\Phi (r_2,r_2)\le r_1,\\ \\ M_2\Psi _2(r_1,r_1)+N_2\Phi _1(r_1,r_1)+Q_2\Psi (r_1,r_1)\le r_2 \end{array} $$
(12)

has a solution \((r_1,r_2)\) satisfying \(r_1>0,r_2>0\).

Proof

From Lemmas 2.2 and 2.3, we know that (x, y) is a solution of BVP(1) if and only if (x, y) is a fixed point of T. From Lemma 2.4, \(T:X\times Y\rightarrow X\times Y\) is completely continuous. By Lemma 2.5 and (i)–(iv), (x, y) is a positive solution if (x, y) is a solution of BVP(1).

To get a fixed point of T, we apply the Schauder’s fixed point theorem. We should define a closed convex bounded subset \(\Omega \) of E such that \(T(\Omega )\subseteq \Omega \). It is easy to see that \( \Omega =\{(x,y)\in E:||x||\le r_1,||y||\le r_2\}\) is a closed convex bounded subset \(\Omega \) of E.

For \((x,y)\in \Omega \), we get \( ||x||\le r_1,\;\;||y||\le r_2. \) Furthermore, we have

$$\begin{aligned} \left| f\left( t,y(t),D_{0^+}^ny(t)\right) \right| &\le \Phi (t^{2-\beta }|y(t)|,t^{2+n-\beta }|D_{0^+}^ny(t)|)\le \Phi (r_2,r_2),t\in (0,1),\\ \left| g\left( t,x(t),D_{0^+}^mx(t)\right) \right| &\le \Psi (t^{2-\alpha }|x(t)|,t^{2+m-\alpha }|D_{0^+}^mx(t)|)\le \Psi (r_1,r_1),t\in (0,1),\\ \left| \phi _1\left( t,y(t),D_{0^+}^ny(t)\right) \right| &\le \overline{\phi }_1(t)\Phi _1(r_2,r_2),t\in (0,1),\\ \left| \psi _1\left( t,y(t),D_{0^+}^ny(t)\right) \right| &\le \overline{\psi }_1(t)\Psi _1(r_2,r_2),t\in (0,1),\\ \left| \phi _2\left( t,x(t),D_{0^+}^mx(t)\right) \right| &\le \overline{\phi }_2(t)\Phi _2(r_1,r_1),t\in (0,1),\\ \left| \psi _2\left( t,x(t),D_{0^+}^mx(t)\right) \right| &\le \overline{\psi }_2(t)\Psi _2(r_1,r_1),t\in (0,1). \end{aligned}$$

By the definition of T, we have

$$\begin{aligned} &t^{2-\alpha }|(T_1y)(t)|\le \frac{\upsilon _1+\mu _1}{\Delta }||\overline{\psi }_1||_1\Psi _1(r_2,r_2) +\frac{\omega _1+\lambda _1}{\Delta }||\overline{\phi }_1||_1\Phi _1(r_2,r_2)\\ &\quad +t^{2-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2) +\frac{\upsilon _1+\mu _1}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2)\\ &\quad+ \frac{b\upsilon _1+b\mu _1}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2) +\frac{a\lambda _1+a\omega _1}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2) \\ &\qquad \le \frac{\upsilon _1+\mu _1}{\Delta }||\overline{\psi }_1||_1\Psi _1(r_2,r_2) +\frac{\omega _1+\lambda _1}{\Delta }||\overline{\phi }_1||_1\Phi _1(r_2,r_2)\\ &\quad +\left[ 1+\frac{\upsilon _1+\mu _1}{\Delta }+ \frac{b\upsilon _1+b\mu _1}{\Delta }\eta ^{\alpha +k_1+l_1}\right. \left. +\frac{a\lambda _1+a\omega _1}{\Delta }\xi ^{\alpha +k_1+l_1}\right] \frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\Phi (r_2,r_2) \end{aligned}$$

and similarly we get

$$\begin{aligned} t^{2+m-\alpha }|D_{0^+}^m(T_1y)(t)|\le & {} \frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+ \mu _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }||\overline{\psi }_1||_1\Psi _1(r_2,r_2)\\&+\frac{\omega _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)} +\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)} }{\Delta }||\overline{\phi }_1||_1\Phi _1(r_2,r_2)\\&+ t^{2+m-\alpha }\int _0^t\frac{(t-s)^{\alpha -m-1}}{\Gamma (\alpha -m)}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2)\\&+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2)\\&+ \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+b\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\int _0^\eta \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s)^{l_1}\mathrm{{d}}s\Phi (r_2,r_2)\\&+\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+a\omega _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\int _0^\xi \frac{(\xi -s)^{\alpha -1}}{\Gamma (\alpha )}s^{k_1}(1-s) ^{l_1}\mathrm{{d}}s\Phi (r_2,r_2)\\\le & {} \frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+ \mu _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }||\overline{\psi }_1||_1\Psi _1(r_2,r_2)\\&+\frac{\omega _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)} +\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)} }{\Delta }||\overline{\phi }_1||_1\Phi _1(r_2,r_2)\\&+ \left[ \frac{\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\right. \\&\left. + \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+b\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\eta ^{\alpha +k_1+l_1}{} \mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\right. \\&\left. +\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+a\omega _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\xi ^{\alpha +k_1+l_1}\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}{\Gamma (\alpha )}\right] \Phi (r_2,r_2). \end{aligned}$$

We get

$$\begin{aligned} ||T_1y||\le & {} \frac{\upsilon _1+\mu _1}{\Delta }||\overline{\psi }_1||_1\Psi _1(r_2,r_2) +\frac{\omega _1+\lambda _1}{\Delta }||\overline{\phi }_1||_1\Phi _1(r_2,r_2)\\&+\left[ 1+\frac{\upsilon _1+\mu _1}{\Delta }+ \frac{b\upsilon _1+b\mu _1}{\Delta }\eta ^{\alpha +k_1+l_1}\right. \left. +\frac{a\lambda _1+a\omega _1}{\Delta }\xi ^{\alpha +k_1+l_1}\right] \frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\Phi (r_2,r_2)\\&+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+ \mu _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)}}{\Delta }||\overline{\psi }_1||_1\Psi _1(r_2,r_2)\\&+\frac{\omega _1\frac{\Gamma (\alpha -1)}{\Gamma (\alpha -m-1)} +\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)} }{\Delta }||\overline{\phi }_1||_1\Phi _1(r_2,r_2)\\&+ \left[ \frac{\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\right. \\&\left. + \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+b\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\eta ^{\alpha +k_1+l_1}{} \mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\right. \\&\left. +\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+a\omega _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\xi ^{\alpha +k_1+l_1}\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}{\Gamma (\alpha )}\right] \Phi (r_2,r_2)\\= & {} M_1\Psi _1(r_2,r_2)+N_1\Phi _1(r_2,r_2)+Q_1\Phi (r_2,r_2). \end{aligned}$$

Similarly, we get

$$\begin{aligned} ||T_2x||\le M_2\Psi _2(r_1,r_1)+N_2\Phi _1(r_1,r_1)+Q_2\Psi (r_1,r_1). \end{aligned}$$

Since (13) has positive solution \(r_1>0,r_2>0\), we choose \(\Omega =\{(x,y)\in E:||x||\le r_1,||y||\le r_2\}\). Then we get \(T(\Omega )\subset \Omega \). Hence, the Schauder’s fixed point theorem implies that T has a fixed point \((x,y)\in \Omega \). So (x, y) is a positive solution of BVP(1).

The proof of Theorem 3.1 is completed. \(\square \)

Theorem 3.2

Suppose

(B2) there exists \(\overline{\phi }_i,\overline{\psi }_i\in L^1(0,1)(i=1,2)\) and nonnegative constants \(M_\Phi ,M_\Psi ,M_{\Phi _i},M_{\Psi _i}(i=1,2)\) such that

$$\begin{aligned} \left| f\left( t,\frac{u}{t^{2-\beta }},\frac{v}{t^{2+n-\beta }}\right) \right| &\le M_ \Phi ,t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| g\left( t,\frac{u}{t^{2-\alpha }},\frac{v}{t^{2+m-\alpha }}\right) \right| &\le M_\Psi ,t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \phi _1\left( t,\frac{u}{t^{2-\beta }},\frac{v}{t^{2+n-\beta }}\right) \right| &\le \overline{\phi }_1(t)M_{\Phi _1},t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \psi _1\left( t,\frac{u}{t^{2-\beta }},\frac{v}{t^{2+n-\beta }}\right) \right| &\le \overline{\psi }_1(t)M_{\Psi _1},t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \phi _2\left( t,\frac{u}{t^{2-\alpha }},\frac{v}{t^{2+m-\alpha }}\right) \right| &\le \overline{\phi }_2(t)M_{\Phi _2},t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \left| \psi _2\left( t,\frac{u}{t^{2-\alpha }},\frac{v}{t^{2+m-\alpha }}\right) \right| &\le \overline{\psi }_2(t)M_{\Psi _2},t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R}. \end{aligned}$$

Then BVP(1) has at least one positive solution.

Proof

Let \(M_i,N_i,Q_i(i=1,2)\) be defined in Theorem 3.1. Choose \(\Phi (u,v)=M_\Phi ,\) \(\Psi (u,v)=M_\Psi ,\) \(\Phi _i(u,v)=M_{\Phi _i}\) and \(\Psi _i(u,v)=M_{\Psi _i}(i=1,2)\). We see that (13) has positive solution

$$\begin{aligned} r_1=M_1M_{\Psi _1}+N_1M_{\Phi _1}+Q_1M_\Phi ,\;\;r_2= M_2M_{\Psi _2}+N_2M_{\Phi _1}+Q_2M_\Psi . \end{aligned}$$

The results follows from Theorem 3.1 directly. \(\square \)

Numerical examples

In this section, we present two examples for the illustration of our main result (Theorems 3.1 and 3.2).

Example 4.1

We consider the following boundary value problem

$$\left\{ \begin{array}{l} D_{0^+}^{\frac{19}{10}}u(t)+t^{-\frac{1}{10}}(1-t)^{-\frac{17}{20}}f(t,v(t),D_{0^+}^{\frac{39}{40}}v(t))=0,\quad t\in (0,1), \\ D_{0^+}^{\frac{39}{20}}v(t)+t^{-\frac{1}{10}}(1-t)^{-\frac{13}{20}}g(t,u(t),D_{0^+}^{\frac{19}{20}}u(t))=0,\quad t\in (0,1),\\ \lim \limits _{t\rightarrow 0}t^{\frac{1}{5}}u(t)-\frac{1}{2}u(1/2)=0,\\ u(1)-\frac{1}{2}u(3/4)=0,\\ \lim \limits _{t\rightarrow 0}t^{\frac{1}{9}}v(t)-\frac{1}{2}v(1/2)=0.\\ v(1)-\frac{1}{2}v(3/4)=0, \end{array} \right. $$
(13)

Then

  1. (i)

    BVP(13) has at least one positive solution if there exists a constant \(H>0\) such that

    $$\begin{array}{l} |f(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \\ |g(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R}. \end{array}$$
  2. (ii)

    BVP(13) has at least one positive solution if

    $$\begin{array}{l} |f(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)|\le c_1+b_1|u|^{\epsilon _1}+a_1|v|^{\delta _1},\;c_1,b_1,a_1\ge 0,\;\epsilon _1,\delta _1>0,\\ \\ |g(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)|\le c_2+b_2|u|^{\sigma _1}+a_2|v|^{\gamma _1},\;c_2,b_2,a_2\ge 0,\;\sigma _1,\gamma _1>0 \end{array} $$

and one of the followings holds:

  1. (a)

    \(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}<1\);

  2. (b)

    \(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}=1\) with \((38.1089b_1)^{1/\sigma _1}34.0678b_2<1\) or \(38.1089b_1(34.0678b_2)^{1/\tau _1}<1\)

  3. (c)

    \(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}>1\) for sufficiently small \(b_1,a_1,b_2,a_2\).

Proof

Corresponding to BVP(1), we have \(\alpha =\frac{19}{10},\beta =\frac{39}{20}\), \(m=\frac{19}{20}\) and \(n=\frac{39}{40}\), \(\xi =\frac{1}{2},\) \(\eta =\frac{3}{4}\), \(a=b=c=d=\frac{1}{2}\) and \(\phi _i(t,u,v)=\psi _i(t,u,v)\equiv 0(i=1,2)\) and \(p(t)=t^{-\frac{1}{10}}(1-t)^{-\frac{17}{20}},\) \(q(t)=t^{-\frac{1}{10}}(1-t)^{-\frac{13}{20}}.\)

It is easy to see that (i)–(iv) hold with \(k_1=-\frac{1}{10}=k_2\), and \(l_1=-\frac{17}{20}\), \(l_2=-\frac{13}{20}\). One sees that \(k_1>-1\), \(\alpha -m+l_1>0\), \(2+k_1+l_1>0\), \(k_2>-1\), \(\beta -n+l_2>0\), \(2+k_2+l_2>0\). Hence, (i)-(iv) defined in Sect. 1 hold.

By direct calculation using Matlab7, we find that

$$\begin{array}{l} \mu _1=\frac{1}{2}\left( \frac{1}{2}\right) ^{\frac{9}{10}},\quad \upsilon _1=1-\frac{1}{2}\left( \frac{1}{2}\right) ^{-\frac{1}{10}},\;\;\omega _1=1-\frac{1}{2}\left( \frac{3}{4}\right) ^{\frac{9}{10} }, \quad \lambda _1=1-\frac{1}{2}\left( \frac{3}{4} \right) ^{-\frac{1}{10}},\\ \mu _2=\frac{1}{2}\left( \frac{1}{2}\right) ^{\frac{19}{20}},\quad \upsilon _2=1-\frac{1}{2}\left( \frac{1}{2}\right) ^{-\frac{1}{20}},\;\;\omega _2=1-\frac{1}{2}\left( \frac{3}{4}\right) ^{\frac{19}{20} },\quad \lambda _2=1-\frac{1}{2}\left( \frac{3}{4}\right) ^{-\frac{1}{20}} \end{array} $$

and

$$\begin{array}{l} \Delta =\mu _1\lambda _1+\upsilon _1\omega _1>0,\;\;0\le a<\frac{1}{\xi ^{\alpha -2}(1-\xi )},\;\;0\le b<\frac{1}{\eta ^{\alpha -1}},\\ \\ \nabla =\mu _2\lambda _2+\upsilon _2\omega _2>0, \;\;0\le c<\frac{1}{\xi ^{\beta -2}(1-\xi )},\;\;0\le d<\frac{1}{\eta ^{\beta -1}}. \end{array} $$
  1. (i)

    By

    $$\begin{array}{ll} &{}|f(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \\ &{}|g(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \\ &{}\phi _1(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)=\psi _1(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)=\phi _2(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)\\ &{}=\psi _2(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)=0, \end{array} $$

    It follows from Theorem 3.2 that BVP(13) has at least one positive solution.

  2. (ii)

    One sees that (B1) holds with

    $$\begin{aligned}\Phi (u,v)&=c_1+b_1|u|^{\epsilon _1}+a_1v^{\delta _1},\\ \Psi (u,v)&=c_2+b_2u^{\sigma _1}+a_2v^{\gamma _1},\\ \Phi _i(u,v)&=\Psi _i(u,v)=0\; (i=1,2)\end{aligned}$$

    . Furthermore, we have by direct computation (use Mathlab7.0) that

    $$\begin{aligned} Q_1= & {} \left[ 1+\frac{\upsilon _1+\mu _1}{\Delta }+ \frac{b\upsilon _1+b\mu _1}{\Delta }\eta ^{\alpha +k_1+l_1} +\frac{a\lambda _1+a\omega _1}{\Delta }\xi ^{\alpha +k_1+l_1}\right] \frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+\frac{\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+ \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+b\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\eta ^{\alpha +k_1+l_1}{} \mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+a\omega _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\xi ^{\alpha +k_1+l_1}\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\simeq 67.8769\le 68 , \end{aligned}$$

    and

    $$\begin{aligned} Q_2= & {} \left[ 1+\frac{\upsilon _2+\mu _2}{\nabla }+ \frac{c\upsilon _2+d\mu _2}{\nabla }\eta ^{\beta +k_2+l_2} +\frac{c\lambda _2+c\omega _2}{\nabla }\xi ^{\beta +k_2+l_2}\right] \frac{\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+\frac{\mathbf{B}(\beta -n+l_2,k_2+1)}{\Gamma (\beta -n)}+\frac{\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+\mu _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+ \frac{d\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+d\mu _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\eta ^{\beta +k_2+l_2}{} \mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+\frac{c\lambda _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+c\omega _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\xi ^{\beta +k_2+l_2}\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\simeq 56.4653\le 57 . \end{aligned}$$

    One sees that inequality system (13) has positive solutions if

    $$\begin{array}{l} 68[c_1+b_1r_2^{\epsilon _1}+a_1r_2^{\delta _1}]\le r_1,\\ \\ 57[c_2+b_2r_1^{\sigma _1}+a_2r_1^{\gamma _1}]\le r_2 \end{array} $$
    (14)

    has positive solutions. One sees that if

    $$\begin{array}{l} 68[c_1+(b_1+a_1)r_2^{\max \{\epsilon _1,\delta _1\}}]\le r_1,\\ \\ 57[c_2+(b_2+a_2)r_1^{\max \{\sigma _1,\gamma _1\}}]\le r_2 \end{array} \qquad \qquad \qquad \qquad (14)' $$

    has positive solution \((r_1,r_2)\) with \(r_1>1,r_2>1\), then (14) has positive solution \((\max \{1,r_1\},\max \{1,r_2\}\).

  3. (ii)-(a)

    \(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}<1\). It is easy to see that (14) has positive a positive solution \((r_1,r_2)\) with \(r_1>0,r_2>0\). It follows from Theorem 3.1 that BVP(13) has at least one solution if one of the followings holds:

  4. (ii)-(b)

    \(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}=1\). One sees that (14)\('\) becomes

    $$\begin{array}{l} 68[c_1+(b_1+a_1)r_2]\le r_1,\;\; 57[c_2+(b_2+a_2)r_1]\le r_2. \end{array}$$

    It is easy to see that the latest inequality system holds for sufficiently large \(r_1',r_2'>0\) if \(68\times 57(a_1+b_1)(a_2+b_2)<1\). Hence (15) has positive solution \((\max \{1,r_1'\},\max \{1,r_2'\}\). Then BVP(1) has positive solution by Theorem 3.1.

  5. (ii)-(c)

    \(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}>1\). By

    $$\begin{aligned} \lim \limits _{(a_1,b_1,c_1)\rightarrow (0,0,0)}Q_1[c_1+b_1|u|^{\epsilon _1}+a_1v^{\delta _1}]=\lim \limits _{(a_2,b_2,c_2)\rightarrow (0,0,0)}Q_2[c_2+b_2u^{\sigma _1}+a_2v^{\gamma _1}]=0, \end{aligned}$$

    we know that (15) has positive solution \((r_1,r_2)\) with \(r_i>0\). Then Theorem 3.1 implies that BVP(1) has at least one positive solution if \(a_1,b_1,c_1,a_2,b_2,c_2\) are sufficiently small. The proof is completed.

\(\square \)

Example 4.2

We consider the following boundary value problem

$$\left\{ \begin{array}{l} D_{0^+}^{\frac{19}{10}}u(t)+t^{-\frac{1}{2}}(1-t)^{-\frac{1}{5}}f(t,v(t),D_{0^+}^{\frac{39}{40}}v(t))=0,\quad t\in (0,1), \\ D_{0^+}^{\frac{39}{20}}v(t)+t^{-\frac{1}{2}}(1-t)^{\frac{1}{10}}g(t,u(t),D_{0^+}^{\frac{19}{20}}u(t))=0,\quad t\in (0,1),\\ \lim \limits _{t\rightarrow 0}t^{\frac{1}{5}}u(t)-\frac{1}{2}u(1/2)=A,\;\; u(1)-\frac{1}{2}u(3/4)=B,\\ \lim \limits _{t\rightarrow 0}t^{\frac{1}{9}}v(t)-\frac{1}{2}v(1/2)=C,\;\; v(1)-\frac{1}{2}v(3/4)=D, \end{array} \right. $$
(15)

where

$$\begin{aligned} f(t,u,v) &= t^2+\frac{b_1t^{\frac{1}{20}}u^{\epsilon _1}+a_1t^{\frac{41}{40}}v^{\delta _1}}{\sqrt{2}\sqrt{b_1^2t^{\frac{1}{10}}u^{2\epsilon _1}+a_1^2t^{\frac{41}{20}}v^{2\delta _1}+1}},\;_1,b_1\ge 0,\;\epsilon _1,\delta _1>0,\\ g(t,u,v) &= 4t^5+\frac{b_2t^{\frac{1}{10}}u^{\sigma _1}+a_2t^{\frac{21}{20}}v^{\gamma _1}}{\sqrt{2}\sqrt{b_2^2t^{\frac{1}{5}}u^{2\sigma _1}+a_2^2t^{\frac{21}{10}}v^{2\gamma _1}+1}},\;a_2,b_2\ge 0,\;\sigma _1,\gamma _1>0. \end{aligned}$$

Then BVP(15) has at least one positive solution for sufficiently small \(a_i,b_i (i=1,2)\).

Proof

Corresponding to BVP(1), we have \(\alpha =\frac{19}{10},\beta =\frac{39}{20}\), \(m=\frac{19}{20}\) and \(n=\frac{39}{40}\), \(a=b=c=d=\frac{1}{2}\) and \(\phi _1(t,u,v)=A,\psi _1(t,u,v)=B,\phi _2(t,u,v)=C,\psi _2(t,u,v)=D\) and \(p(t)=t^{-\frac{1}{2}}(1-t)^{-\frac{1}{5}},\) \(q(t)=t^{-\frac{1}{2}}(1-t)^{\frac{1}{10}}.\)

It is easy to see that (i)–(iv) hold with \(k_1=-\frac{1}{10}=k_2\), and \(l_1=-\frac{1}{5}\), \(l_2=-\frac{1}{10}\). One sees that \(k_1>-1\), \(\alpha -m+l_1>0\), \(2+k_1+l_1>0\), \(k_2>-1\), \(\beta -n+l_2>0\), \(2+k_2+l_2>0\). One sees \(m>\alpha -1\), \(n>\beta -1\).

Then similar to Example 4.1, we know that BVP(15) has at least one positive solution by Theorem 3.2. \(\square \)

Conclusions

In this paper, we establish sufficient conditions for the existence of positive solutions of four-point integral type boundary value problems for singular fractional differential systems. We allow the nonlinearities p(t)f(t, x, y) and q(t)g(t, x, y) in fractional differential equations to be singular at \(t=0\). Both f and g may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. This paper contributes within the domain of fractional differential equations. The methods can be applied to solve other kinds of four-point integral type boundary value problems for singular fractional differential systems.

In [12, 22], authors studied the existence of positive solutions of two-point boundary value problems for fractional order elastic beam equations. One can discuss the following boundary value problem for nonlinear singular coupled fractional order elastic beam equations of the form

$$\left\{ \begin{array}{l} D_{0^+}^{\alpha }u(t)=f(t,v(t),v'(t),v''(t)),\; t\in (0,1), \\ D_{0^+}^\beta v(t)=g(t.u(t),u'(t),u''(t)),\;t\in (0,1),\\ \lim \limits _{t\rightarrow 0}t^{4-\alpha }u(t)=\lim \limits _{t\rightarrow 0}t^{4-\alpha }u'(t)=0,\\ u(1)=u'(1)=0,\\ \lim \limits _{t\rightarrow 0}t^{4-\beta }v(t)=\lim \limits _{t\rightarrow 0}t^{4-\beta }v'(t)=0,\\ v(1)=v'(1)=0, \end{array}\right. $$
(16)

where \(3<\alpha ,\beta \le 4\), \(D_{0^+}^{*}\) (\(D^*\) for short) is the Riemann–Liouville fractional derivative of order \(*\), and \(f,g:(0,1)\times [0,\infty )\times \mathrm{I}\!\mathrm{R}^2\rightarrow [0,\infty )\) is continuous. f, g depend on the lower order fractional derivatives \(u',v'\) and \(u'',v''\) and may be singular at \(t=0\) and \(t=1\), f, g are non-Carathéodory functions.