Existence results for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions

Abstract

By means of coincidence degree theory, we present the existence of solutions of a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions. This paper enriches and extends some existing literature. Finally, an example is given to support our results.

1 Introduction

In this paper, we study the existence of solutions for higher-order nonlinear fractional differential equations with p-Laplacian operator:

$$ \textstyle\begin{cases} D_{0+}^{\beta_{1}} \phi_{p} ( D_{0+}^{\alpha_{1}} u(t) )=f (t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) ), \quad t\in(0,1),\\ D_{0+}^{\beta_{2}} \phi_{p} ( D_{0+}^{\alpha_{2}} v(t) )=g (t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) ), \quad t\in(0,1),\\ u'(0)= \cdots=u^{(n-1)}(0)=D_{0+}^{\alpha_{1}}u(0)=0, \quad \quad u(0)=\sum_{i=1}^{\infty}a_{i}u(\xi_{i}),\\ v'(0)= \cdots=v^{(n-1)}(0)=D_{0+}^{\alpha_{2}}v(0)=0,\quad \quad v(0)=\sum_{i=1}^{\infty}b_{i}v(\eta_{i}), \end{cases} $$

(1.1)

where the p-Laplacian operator is defined as $\phi_{p}(s)= \vert s \vert ^{p-2}s$, $p>1$, $\phi_{q}(s)=\phi _{p}^{-1}(s)$, $\frac{1}{p}+\frac{1}{q}=1$, $0< \beta_{1}, \beta_{2} <1$, $n-1<\alpha_{1},\alpha_{2}<n$, $0<\xi_{1}<\xi_{2}<\cdots<\xi_{i}<\cdots<1$, $0<\eta_{1}<\eta_{2}<\cdots<\eta_{i}<\cdots<1$, $\sum_{i=1}^{\infty}a_{i}= \sum_{i=1}^{\infty}b_{i}=1$, $\sum_{i=1}^{\infty} \vert a_{i} \vert <\infty$, $\sum_{i=1}^{\infty} \vert b_{i} \vert <\infty$, $D_{0+}^{\alpha_{1}}$, $D_{0+}^{\beta_{1}}$, $D_{0+}^{\alpha_{2}}$, $D_{0+}^{\beta_{2}}$ denote the Caputo fractional derivatives and $f,g:[0,1]\times\mathbb{R}^{n }\rightarrow \mathbb{R}$ are continuous.

The theory of fractional differential equations is a branch of differential equation theory, which occurs more frequently in different research areas and engineering, such as fluid mechanics, control system, viscoelasticity, chemistry, electromagnetic, etc. (see [1–5]). In the last few decades, many authors devoted their attention to the study of resonant boundary value problems for nonlinear fractional differential equations, see [6–19]. Meanwhile, some important results relative to the existence of solutions for a coupled system of fractional differential equations with p-Laplacian operator at resonance have been obtained, see [11–16].

In [15], Hu et al. considered the two-point boundary value problem for nonlinear fractional differential equations with p-Laplacian operator at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{\beta} \phi_{p}( D_{0^{+}}^{\alpha} u(t))=f(t,v(t),D_{0^{+}}^{\delta }u(t)), \quad t\in(0,1),\\ D_{0^{+}}^{\gamma} \phi_{p}( D_{0^{+}}^{\delta} v(t))=g(t,u(t),D_{0^{+}}^{\alpha}u(t)), \quad t\in(0,1),\\ D_{0^{+}}^{\alpha}u(0)=D_{0^{+}}^{\alpha}u(1)= D_{0^{+}}^{\delta}v(0)=D_{0^{+}}^{\delta}v(1)=0 , \end{cases} $$

where $\phi_{p}(s)= \vert s \vert ^{p-2}s$, $p>1$ is the p-Laplacian operator, $0<\alpha,\beta<1$, $1<\alpha+\beta<2$, $D_{0^{+}}^{\alpha}$, $D_{0^{+}}^{\beta}$ $D_{0^{+}}^{\gamma}$ $D_{0^{+}}^{\delta}$ denote the Caputo fractional derivatives and $f,g:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}$ are continuous.

In [16], Cheng et al. considered the two-point boundary value problem for nonlinear fractional p-Laplacian differential equations with $\operatorname{Ker} L=n\geq2$:

$$ \textstyle\begin{cases} D_{0^{+}}^{\gamma} \phi_{p}( D_{0^{+}}^{\alpha} u(t))=f(t,v(t) ), \quad t\in (0,1),\\ D_{0^{+}}^{\gamma} \phi_{p}( D_{0^{+}}^{\beta} v(t))=g(t,u(t) ), \quad t\in (0,1),\\ D_{0^{+}}^{\alpha}u(0)=D_{0^{+}}^{\alpha}u(1)= D_{0^{+}}^{\beta}v(0)=D_{0^{+}}^{\beta}v(1)=0 , \end{cases} $$

where $\phi_{p}(s)= \vert s \vert ^{p-2}s$, $p>1$ is the p-Laplacian operator, $0<\gamma<1$, $n-1<\alpha,\beta<n$, $D_{0^{+}}^{\alpha}$, $D_{0^{+}}^{\beta}$, $D_{0^{+}}^{\gamma}$ denote the Caputo fractional derivatives and $f,g:[0,1]\times\mathbb{R} \rightarrow \mathbb{R}$ are continuous.

In recent years, the subject of infinite-point boundary value problems of fractional differential equations which can extend many previous results have attracted more attention. Most of the results are mainly at nonresonance. For the resonance case, the existing results of fractional differential equations with infinite-point boundary value problems are few. We refer the reader to [20–23] and the references cited therein.

From the above work, we see that recent study on a coupled system of fractional p-Laplacian differential equations is mainly at two-point boundary value problem. The theory for fractional p-Laplacian differential equations with multi-point and even infinite-point at resonance has yet been sufficiently developed. To the best of our knowledge, this is the first paper to study higher order fractional differential equations with p-Laplacian and infinite-point boundary value conditions at resonance. Motivated by the works above, we consider the existence of solutions for BVP (1.1).

The rest of this paper is organized as follows. In Section 2, we give some necessary notations, definitions and lemmas. In Section 3, we study the existence of solutions of (1.1) by the coincidence degree theory due to Mawhin [24]. Finally, an example is given to illustrate our results in Section 4.

2 Preliminaries

We present the necessary definitions and lemmas from fractional calculus theory that will be used to prove our main theorems.

Definition 2.1

[1]

The Riemann-Liouville fractional integral of order $\alpha>0$ of a function $f:(0,\infty)\to\mathbb{R}$ is given by

$$I_{0+}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha -1}f(s)\,ds, $$

provided that the right-hand side is pointwise defined on $(0,\infty)$.

Definition 2.2

[1]

The Caputo fractional derivative of order $\alpha>0$ of a function $f\in \mathit{AC}^{n-1}[0,1]$ is given by

$$D_{0+}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \int_{0}^{t}\frac {f^{(n)}(s)}{(t-s)^{\alpha-n+1}}\,ds, $$

where $n-1<\alpha\leq n$, provided that the right-hand side is pointwise defined on $(0,\infty)$.

Lemma 2.1

[1]

Let $n-1<\alpha\leq n$, $u\in \mathit{AC}^{n-1}[0,1]$, then

$$I_{0+}^{\alpha}D_{0+}^{\alpha}u(t)=u(t)+c_{0}+ c_{1}t+\cdots+c_{n-1}t^{n-1}, $$

where $c_{i}\in\mathbb{R}$, $i=0,1,\dots, n-1$.

Lemma 2.2

[1]

If $\beta>0$, $\alpha+\beta>0$, then the equation

$$I_{0+}^{\alpha}I_{0+}^{\beta}f(x)=I_{0+}^{\alpha+\beta}f(x) $$

is satisfied for an integrable function f.

Lemma 2.3

[23]

For any $u,v \geq0$, then

$$\begin{aligned}& \phi_{p}(u+v) \leq\phi_{p}(u)+\phi_{p}(v) \quad \textit{if } p< 2; \\& \phi_{p}(u+v) \leq2^{p-1} \bigl(\phi_{p}(u)+ \phi_{p}(v) \bigr) \quad \textit{if } p\geq2. \end{aligned}$$

Firstly, we briefly recall some definitions on the coincidence degree theory. For more details, see [14].

Let Y, Z be real Banach spaces, $L:\operatorname{dom} L \subset Y \to Z$ be a Fredholm map of index zero and $P:Y\to Y$, $Q:Z\to Z$ be continuous projectors such that

$$\operatorname{Ker} L =\operatorname{Im}P, \quad\quad \operatorname{Im}L= \operatorname{Ker} Q, \quad\quad Y=\operatorname{Ker} L \oplus\operatorname{Ker} P, \quad\quad Z=\operatorname{Im}L \oplus\operatorname{Im}Q. $$

It follows that

$$L|_{\operatorname{dom}L \cap\operatorname{Ker} P}:\quad \operatorname{dom}L \cap\operatorname{Ker} P \to \operatorname{Im}L $$

is invertible. We denote the inverse of this map by $K_{P}$.

If Ω is an open bounded subset of Y, the map N will be called L-compact on Ω̅ if $QN(\overline{\Omega})$ is bounded and $K_{P,Q}N=K_{P}(I-Q)N:\overline{\Omega}\to Y$ is compact.

Theorem 2.1

Let L be a Fredholm operator of index zero and N be L-compact on Ω̅. Suppose that the following conditions are satisfied:

• (1) $Lx \neq\lambda Nx$ for each $(x,\lambda) \in [(\operatorname{dom}L\backslash\operatorname{Ker} L) \cap \partial\Omega ]\times (0,1)$;

• (2) $Nx \notin\operatorname{Im} L$ for each $x \in\operatorname{Ker} L \cap\partial\Omega$;

• (3) $\deg(JQN|_{\operatorname{Ker} L}, \Omega\cap\operatorname{Ker} L, 0)\neq0$, where $Q:Z\to Z$ is a continuous projection as above with $\operatorname{Im}L= \operatorname{Ker} Q$ and $J:\operatorname{Im}Q\to \operatorname{Ker} L$ is any isomorphism.

Then the equation $Lx = Nx$ has at least one solution in $\operatorname{dom}L\cap\overline{\Omega}$.

3 Main results

In this section, we begin to prove the existence of solutions to problem (1.1). Consider the functions $\phi_{1}(z) = \sum_{i=1}^{\infty} a_{i}\xi_{i}^{z}$, $\phi_{2}(z) = \sum_{i=1}^{\infty} b_{i}\eta _{i}^{z}$, $z\in[0,\infty)$. According to $\sum_{i=1}^{\infty} \vert a_{i} \vert <\infty$, $\sum_{i=1}^{\infty} \vert b_{i} \vert <\infty$, one has the series are (uniformly) convergent and thus $\phi _{1}$, $\phi_{2}$ are continuous on $[0,\infty)$.

The following assumption will be used in our main results:

$(\mathrm{H}_{0})$ :

There exist $z_{0}$, $\tilde{z}_{0}$ with $z_{0}\geq\alpha_{1}$, $\tilde{z}_{0}\geq \alpha_{2}$ such that $\phi_{1}(z_{0})\cdot\phi_{2}(\tilde{z}_{0})\neq0$.

The following lemma is fundamental in the proofs of our main results.

Lemma 3.1

Problem (1.1) is equivalent to the following equation:

$$ \textstyle\begin{cases} D_{0+}^{\alpha_{1}} u(t)=\phi_{q} [I_{0+}^{\beta_{1}}f (t,v(t),D_{0 +}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) ) ],\quad t\in(0,1), \\ D_{0+}^{\alpha_{2}} v(t)=\phi_{q} [I_{0+}^{\beta_{2}}g (t,u(t),D_{0 +}^{\alpha_{1}-1}u(t), \ldots,D_{0+}^{\alpha_{1}-(n-1)}u(t) ) ],\quad t\in(0,1), \\ u'(0)= \cdots=u^{(n-1)}(0)=0, \quad \quad u(0)=\sum_{i=1}^{\infty}a_{i}u(\xi _{i}),\\ v'(0)= \cdots=v^{(n-1)}(0)=0, \quad \quad v(0)=\sum_{i=1}^{\infty}b_{i}v(\eta_{i}). \end{cases} $$

(3.1)

Proof

By Lemma 2.1, $D_{0+}^{\beta_{1}} \phi_{p}( D_{0+}^{\alpha_{1}} u(t))=f (t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) )$ has the following solution:

$$\phi_{p} \bigl( D_{0+}^{\alpha_{1}}u(t) \bigr)=I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) \bigr)+c, \quad c\in\mathbb{R}. $$

Substituting $t=0$ into the above formula, by $D_{0^{+}}^{\alpha _{1}}u(0)=0$, we obtain $c=0$. Then we have

$$\begin{aligned} \phi_{p} \bigl( D_{0+}^{\alpha_{1}}u(t) \bigr)=I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) \bigr). \end{aligned}$$

(3.2)

Applying the operator $\phi_{q}$ to the both sides of (3.2) respectively, we have

$$D_{0+}^{\alpha_{1}} u(t)=\phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]. $$

By a similar argument, we have

$$D_{0+}^{\beta_{2}} \phi_{p} \bigl( D_{0+}^{\alpha_{2}} v(t) \bigr)=g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) $$

is equivalent to

$$D_{0+}^{\alpha_{2}} v(t)=\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0^{+}}^{\alpha_{1}-1}u(t), \ldots,D_{0+}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]. $$

Therefore, BVP (1.1) is rewritten by (3.1)

It is easy to verify that equation (1.1) has a solution $(u,v)$ if and only if $(u,v)$ solves equation (3.1). □

Let $E=C[0,1]$ with the norm $\Vert x \Vert _{\infty}=\max_{0 \le t \le1} \vert x(t) \vert$. Now, we set $X_{1} =\{u(t): u(t), D_{0+}^{\alpha_{1}-i}u(t)\in E, i=1,2,\ldots,n-1\}$ with the norm

$${{ \Vert u \Vert }_{X_{1}}}=\max \bigl\{ \Vert u \Vert _{\infty}, \bigl\Vert D_{0+}^{\alpha _{1}-1}u \bigr\Vert _{\infty},\ldots, \bigl\Vert D_{0+}^{\alpha_{1}-(n-1)}u \bigr\Vert _{\infty}\bigr\} $$

and $X_{2} =\{v(t): v(t), D_{0+}^{\alpha_{2}-i}v(t)\in E, i=1,2,\ldots,n-1\}$ with the norm

$${{ \Vert v \Vert }_{X_{2}}}=\max \bigl\{ \Vert v \Vert _{\infty}, \bigl\Vert D_{0+}^{\alpha_{2}-1}v \bigr\Vert _{\infty},\ldots, \bigl\Vert D_{0+}^{\alpha_{2}-(n-1)}v \bigr\Vert _{\infty}\bigr\} . $$

Let $Y=X_{1}\times X_{2}$ with the norm $\Vert (u,v) \Vert _{Y}=\max\{ \Vert u \Vert _{X_{1}}, \Vert v \Vert _{X_{2}} \}$ and $Z=E\times E$ with the norm $\Vert (x,y) \Vert _{Z}=\max \{ \Vert x \Vert _{\infty}, \Vert y \Vert _{\infty}\}$.

Clearly, X and Y are Banach spaces.

Define the linear operator $L_{1}:\operatorname{dom}L_{1}\rightarrow E$ by setting

$$\operatorname{dom}L_{1}= \Biggl\{ u\in X_{1}\bigg|u'(0)= \cdots=u^{(n-1)}(0)=0, u(0)=\sum_{i=1}^{\infty}a_{i}u(\xi_{i}) \Biggr\} $$

and

$$L_{1}u=D_{0 +}^{\alpha_{1}}u, \quad u\in\operatorname{dom}L_{1}. $$

Define the linear operator $L_{2}$ from $\operatorname{dom}L_{2} \rightarrow E$ by setting

$$\operatorname{dom}L_{2}= \Biggl\{ v\in X_{2}\bigg|v'(0)= \cdots=v^{(n-1)}(0))=0, v(0)=\sum_{i=1}^{\infty}b_{i}v(\eta_{i}) \Biggr\} $$

and

$$L_{2}v=D_{0 +}^{\alpha_{2}}v, \quad v\in\operatorname{dom}L_{2}. $$

Define the operator $L: \operatorname{dom}L \rightarrow Z$ with

$$\operatorname{dom}L= \bigl\{ (u,v)\in Y|u\in\operatorname{dom}L_{1}, v \in\operatorname{dom}L_{2} \bigr\} $$

and

$$L(u,v)=(L_{1}u,L_{2}v). $$

Let $N:Y\to Z$ be the Nemytskii operator

$$N(u,v)=(N_{1}v,N_{2}u), $$

where $N_{1}:X\to E$ is defined by

$$N_{1}v(t)=\phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr] $$

and $N_{2}:X\to E$ is defined by

$$N_{2}u(t)=\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0 +}^{\alpha_{1}-1}u(t), \ldots,D_{0+}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]. $$

Then BVP (3.1) can be written as $L(u,v)=N(u,v)$.

Lemma 3.2

L is defined as above, then

$$\begin{aligned} &\operatorname{Ker}L= \bigl\{ (u,v)\in X: (u,v)=(c_{0},d_{0}), c_{0},d_{0} \in \mathbb{R} \bigr\} , \end{aligned}$$

(3.3)

$$\begin{aligned} &\operatorname{Im} L = \Biggl\{ (x,y)\in Z: \sum_{i=1}^{\infty}a_{i}I_{0+}^{\alpha _{1}}x(\xi_{i})=0; \sum _{i=1}^{\infty}b_{i}I_{0+}^{\alpha_{2}}y( \eta_{i})=0 \Biggr\} . \end{aligned}$$

(3.4)

Proof

For $(u,v)\in\operatorname{Ker}L$, then $L_{1}u=L_{2}v=0$. By Lemma 2.1, the equation $D_{0 +}^{\alpha_{1}}u(t)=0$ has solution

$$u(t)=c_{0}+ c_{1}t+\cdots+c_{n-1}t^{n-1} . $$

In view of $u^{(i)}(0)=0$, $i=1,2,\ldots,n-1$, we get $c_{i}=0$, $i=1,2,\ldots,n-1$. Then $u(t)=c_{0}$. Similarly, for $v\in\operatorname{Ker}L_{2}$, we have $v(t)=d_{0}\in\mathbb{R}$. Thus, we obtain (3.3).

Next we prove that (3.4) holds. Let $(x,y)\in\operatorname{Im}L$, so there exists $(u,v)\in\operatorname{dom}L$ such that $x(t)=D_{0 +}^{\alpha _{1}}u(t)$, $y(t)=D_{0 +}^{\alpha_{2}}v(t)$. By Lemma 2.1, we have

$$u(t) =I_{0 +}^{\alpha_{1} }x(t)+\sum_{i=0}^{n-1}{{c}_{i}}t^{i}, \quad\quad v(t) =I_{0 +}^{\alpha_{2}}y(t)+\sum _{i=0}^{n-1}{{d}_{i}}t^{i},\quad c_{i},d_{i} \in \mathbb{R}. $$

In view of $u^{(i)}(0)=v^{(i)}(0)=0$, $i=1,2,\ldots,n-1$, we get $c_{i}=d_{i}=0$, $i=1,2,\ldots, n-1$. Hence,

$$u(t)=I_{0 +}^{\alpha_{1}}x(t)+c_{0},\quad\quad v(t)=I_{0 +}^{\alpha _{2}}y(t)+d_{0}. $$

According to $u(0)=\sum_{i=1}^{\infty}a_{i}u(\xi_{i})$ and $v(0)=\sum_{i=1}^{\infty}b_{i}v(\eta_{i})$, we have

$$ \begin{gathered} u(0)= I_{0 +}^{\alpha_{1}}x(0)+c_{0}= \sum_{i=1}^{\infty}a_{i}u( \xi_{i})=\sum_{i=1}^{\infty}a_{i} \bigl(I_{0 +}^{\alpha_{1} }x(\xi_{i})+{{c}_{0}} \bigr)=\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x( \xi_{i})+{c}_{0}, \\ v(0)=I_{0 +}^{\alpha_{2} }y(0)+d_{0}=\sum _{i=1}^{\infty}b_{i}v(\xi_{i})=\sum _{i=1}^{\infty}b_{i} \bigl(I_{0 +}^{\alpha_{2}}y(\eta_{i})+{{c}_{0}} \bigr)=\sum_{i=1}^{\infty}b_{i}I_{0 +}^{\alpha_{2}}y( \eta_{i})+{d}_{0}, \end{gathered} $$

that is,

$$\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x( \xi_{i})=0,\qquad\sum_{i=1}^{\infty}b_{i}I_{0 +}^{\alpha_{2}}y(\eta_{i})=0. $$

On the other hand, suppose that $(x,y)$ satisfies the above equations. Let $u(t)=I_{0 +}^{\alpha_{1}}x(t)$ and $v(t)=I_{0 +}^{\alpha_{2}}y(t)$, we can prove $(u ,v )\in\operatorname{dom} L$ and $L (u ,v )=(x,y)$. Then (3.4) holds. □

Lemma 3.3

The mapping $L:\operatorname{dom} L \subset Y\rightarrow Z$ is a Fredholm operator of index zero.

Proof

The linear continuous projector operator $P(u,v)=(P_{1}u,P_{2}v)$ can be defined as

$${{P}_{1}}u=u(0), \qquad {{P}_{2}}v=v(0). $$

Obviously, $P_{1}^{2}=P_{1}$ and $P_{2}^{2}=P_{2}$.

It is clear that

$$\operatorname{Ker}P= \bigl\{ (u,v): u(0)=0, v(0)=0 \bigr\} . $$

It follows from $(u,v)=(u,v)-P(u,v)+P(u,v)$ that $Y=\operatorname{Ker}P+ \operatorname{Ker}L$. For $(u,u)\in\operatorname{Ker} L\cap\operatorname{Ker}P$, then $u=c_{0}$, $v=d_{0}$, $c_{0},d_{0}\in\mathbb{R}$. Furthermore, by the definition of KerP, we have $c_{0}= d_{0}=0$. Thus, we get

$$Y=\operatorname{Ker}L\oplus\operatorname{Ker}P. $$

By $(\mathrm{H}_{0})$, the linear operator $Q(x,y)=(Q_{1}x,Q_{2}y)$ can be defined as

$$\begin{aligned}& Q_{1}x(t) =t^{\theta_{1}}\cdot\frac{\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi_{i})}{\sum_{i=1}^{\infty}a_{i}(I_{0 +}^{\alpha_{1}}t^{\theta _{1}})(\xi_{i})}=t^{\theta_{1}} \cdot\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi _{1}(z_{0})\Gamma(1+\theta_{1} )}\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi _{i}), \\& Q_{2}y(t) =t^{\theta_{2}}\cdot\frac{\sum_{i=1}^{\infty}b_{i}I_{0 +}^{\alpha _{2}}y(\eta_{i})}{\sum_{i=1}^{\infty}b_{i}(I_{0 +}^{\alpha_{2}}t^{\theta_{2}})(\eta _{i})}=t^{\theta_{2}} \cdot\frac{\Gamma(1+\alpha_{2}+\theta_{2} )}{\phi_{2}(\tilde {z}_{0})\Gamma(1+\theta_{2} )}\sum_{i=1}^{\infty}b_{i}I_{0 +}^{\alpha_{2}}y(\eta_{i}), \end{aligned}$$

where $\theta_{1}=z_{0}-\alpha_{1}$, $\theta_{2}=\tilde{z}_{0}-\alpha_{2}$.

Obviously, $Q(x,y)= (Q_{1}x(t),Q_{2}y(t) )\cong\mathbb{R}^{2}$.

For $x(t)\in E$, we have

$$\begin{aligned} {{Q}_{1}} \bigl({{Q}_{1}}x(t) \bigr) =&\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )} \sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha _{1}}x( \xi_{i})\cdot Q_{1} \bigl(t^{\theta_{1}} \bigr) \\ =&\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi_{i})\cdot t^{\theta_{1}}\cdot \frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )} \cdot\sum_{i=1}^{\infty}a_{i} \bigl(I_{0 +}^{\alpha_{1}}t^{\theta_{1}} \bigr) ( \xi_{i}) \\ =&\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi_{i})\cdot t^{\theta_{1}} \\ &{}\cdot \frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}\cdot\sum_{i=1}^{\infty}a_{i}\frac{\Gamma(1+\theta_{1} ){\xi_{i}}^{\alpha _{1}+\theta_{1}}}{{\Gamma(1+\alpha_{1}+\theta_{1} )}} \\ =&\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi_{i})\cdot t^{\theta_{1}}\cdot \frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}\cdot\sum_{i=1}^{\infty}a_{i}\frac{\Gamma(1+\theta_{1} ){\xi _{i}}^{z_{0}}}{{\Gamma(1+\alpha_{1}+\theta_{1} )}} \\ =&\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi_{i})\cdot t^{\theta_{1}}\cdot \frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )} \cdot\frac{\phi_{1}(z_{0})\Gamma(1+\theta_{1} )}{\Gamma(1+\alpha_{1}+\theta _{1} )} \\ =&t^{\theta_{1}}\cdot\frac{\Gamma(1+\alpha_{1}+\theta_{1} )}{\phi _{1}(z_{0})\Gamma(1+\theta_{1} )}\sum_{i=1}^{\infty}a_{i}I_{0 +}^{\alpha_{1}}x(\xi _{i}) \\ =&{{Q}_{1}}x(t). \end{aligned}$$

Similarly, $Q_{2}^{2}=Q_{2}$, that is to say, the operator Q is idempotent. It follows from $(x,y)=(x,y)-Q(x,y)+Q(x,y)$ that $Z=\operatorname{Im}L+ \operatorname{Im}Q$. Moreover, by $\operatorname{Ker}Q=\operatorname{Im}L$ and $Q_{2}^{2}=Q_{2}$, we get $\operatorname{Im}L\cap\operatorname{Im}Q=\{(0,0)\}$. Hence,

$$Z=\operatorname{Im}L\oplus\operatorname{Im}Q. $$

Now, $\operatorname{Ind}L = \operatorname{\operatorname{dim} \operatorname{Ker}} L - \operatorname{\operatorname{codim} \operatorname{Im}}L = 0$, so L is a Fredholm mapping of index zero. □

For every $(u,v)\in Y$,

$$\begin{aligned} \bigl\Vert P(u,v) \bigr\Vert _{Y} = \max \bigl\{ \Vert P_{1}u \Vert _{X_{1}}; \Vert P_{2}v \Vert _{X_{2}} \bigr\} =\max \bigl\{ \bigl\vert {u} (0) \bigr\vert ; \bigl\vert {v} (0) \bigr\vert \bigr\} . \end{aligned}$$

(3.5)

Furthermore, the operator $K_{P}:\operatorname{Im}L\to \operatorname {dom}L \cap\operatorname{Ker} P$ can be defined

$$K_{P}(x,y)= \bigl(I_{0 +}^{\alpha_{1}} x,I_{0 +}^{\alpha_{2}}y \bigr). $$

For $(x,y)\in\operatorname{Im} L$, we have

$$\begin{aligned} LK_{P}(x,y)=L \bigl(I_{0 +}^{\alpha_{1}}x,I_{0 +}^{\alpha_{2}}y \bigr)= \bigl(D_{0 +}^{\alpha_{1}}I_{0 +}^{\alpha_{1}}x, D_{0 +}^{\alpha_{2}}I_{0 +}^{\alpha_{2}}y \bigr) =(x,y). \end{aligned}$$

(3.6)

On the other hand, for $(u,v)\in\operatorname{dom} L \cap\operatorname{Ker} P$, according to Lemma 2.1, we have

$$\begin{aligned}& I_{0 +}^{\alpha_{1}}L_{1}u(t)=I_{0 +}^{\alpha_{1}} D_{0 +}^{\alpha_{1}}u(t)=u(t)+{{c}_{0}}+{{c}_{1}}t+ \cdots +{{c}_{n-1}} {{t}^{n-1}}, \\& I_{0 +}^{\alpha_{2}}L_{2}v(t)=I_{0 +}^{\alpha_{2}}D_{0 +}^{\alpha_{2}} v(t)=v(t)+{{d}_{0}}+{{d}_{1}}t+\cdots+{{d}_{n-1}} {{t}^{n-1}}. \end{aligned}$$

By the definitions of domL and KerP, one has $u^{(i)}(0)=v^{(i)}(0)$, $i=0,1,\ldots,n-1$, which implies that $c_{i}=d_{i}$, $i=0,1,\ldots,n-1$. Thus, we obtain

$$\begin{aligned} K_{p}L(x,y)= \bigl(I_{0 +}^{\alpha_{1}}D_{0 +}^{\alpha_{1}} x,I_{0 +}^{\alpha_{2}}D_{0 +}^{\alpha_{2}}y \bigr)=(x,y). \end{aligned}$$

(3.7)

Combining (3.6) and (3.7), we get $K_{P}=(L_{\operatorname{dom} L \cap\operatorname{Ker} P })^{-1}$.

For $(x,y)\in\operatorname{Im} L$, we have

$$\begin{aligned} \bigl\Vert K_{P}(x,y) \bigr\Vert _{Y} &= \bigl\Vert \bigl(I_{0 +}^{\alpha_{1}}x,I_{0 +}^{\alpha_{2}}y \bigr) \bigr\Vert _{Y} =\max \bigl\{ \bigl\Vert I_{0 +}^{\alpha_{1}} x \bigr\Vert _{X_{1}}; \bigl\Vert I_{0 +}^{\alpha_{2}} y \bigr\Vert _{X_{2}} \bigr\} \\ &\leq\max \bigl\{ \max \bigl\{ \bigl\Vert I_{0 +}^{\alpha_{1}}x \bigr\Vert _{\infty}, \bigl\Vert D_{0+}^{\alpha_{1}-1}I_{0 +}^{\alpha_{1}}x \bigr\Vert _{\infty},\ldots, \bigl\Vert D_{0+}^{\alpha_{1}-(n-1)}I_{0^{+}}^{\alpha _{1}}x \bigr\Vert _{\infty}\bigr\} ; \\ &\quad{} \max \bigl\{ \bigl\Vert I_{0 +}^{\alpha_{2}} y \bigr\Vert _{\infty}, \bigl\Vert D_{0+}^{\alpha_{2}-1}I_{0 +}^{\alpha_{2}} y \bigr\Vert _{\infty},\ldots, \bigl\Vert D_{0+}^{\alpha_{2}-(n-1)}I_{0 +}^{\alpha_{2}} y \bigr\Vert _{\infty}\bigr\} \bigr\} \\ &=\max \bigl\{ \Vert x \Vert _{\infty}; \Vert y \Vert _{\infty}\bigr\} . \end{aligned}$$

(3.8)

Again, for $(u,v)\in\Omega_{1}$, $(u,v)\in\operatorname{dom}(L)\setminus \operatorname{Ker}(L)$, then $(I-P)(u,v)\in\operatorname{dom}L\cap\operatorname{Ker}P$ and $LP(u,v)=(0,0)$, thus from (3.8) we have

$$\begin{aligned} \bigl\Vert (I-P) (u,v) \bigr\Vert _{Y} &= \bigl\Vert K_{P}L(I-P) (u,v) \bigr\Vert _{Y}= \bigl\Vert K_{P}(L_{1}u,L_{2}v) \bigr\Vert _{Y} \\ &\leq\max \bigl\{ \Vert N_{1}v \Vert _{\infty}; \Vert N_{2}u \Vert _{\infty}\bigr\} . \end{aligned}$$

(3.9)

By similar arguments as in [11] or [12], we have the following lemma. We omit the proof of it.

Lemma 3.4

$K_{P}(I-Q)N:Y\rightarrow Y$ is completely continuous.

For simplicity of notation, we set

$$a=\frac{1}{\Gamma(\alpha_{1}+1)}; \quad\quad b= \biggl[\frac{1}{\Gamma(\beta _{1}+1)} \biggr]^{q-1};\quad\quad \tilde{a}=\frac{1}{\Gamma(\alpha_{2}+1)}; \quad\quad \tilde{b}= \biggl[\frac{1}{\Gamma(\beta_{2}+1)} \biggr]^{q-1}. $$

Theorem 3.1

Assume that $(\mathrm{H}_{0})$ and the following conditions hold.

• (H1) There exist nonnegative functions $\psi(t),\tilde{\psi}(t),\varphi _{i}(t),\tilde{\varphi}_{i}(t) \in E$, $i=1,2,\ldots,n-1$, such that for $t\in[0,1]$, $(u_{1},u_{2},\ldots,u_{n}),(v_{1},v_{2},\ldots,v_{n}) \in\mathbb {R}^{n}$, one has

  $$\begin{aligned}& \bigl\vert f(t,u_{1},u_{2},\ldots,u_{n}) \bigr\vert \leq \psi(t)+ \varphi_{1}(t) \vert u_{1} \vert ^{p-1} +\cdots+\varphi _{n-1}(t) \vert u_{n} \vert ^{p-1}, \\& \bigl\vert g(t,v_{1},v_{2},\ldots,v_{n}) \bigr\vert \leq\tilde{\psi }(t)+ \tilde{\varphi}_{1}(t) \vert v_{1} \vert ^{p-1} +\cdots +\tilde{\varphi}_{n-1}(t) \vert v_{n} \vert ^{p-1}. \end{aligned}$$

• (H2) There exists $A>0$ such that if $\vert u \vert >A$ or $\vert v \vert >A$, $\forall t\in[0,1]$, one has

  $$\begin{aligned} &u\cdot \Biggl[ \sum_{i=1}^{\infty}a_{i}\phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]\big| _{t=\xi_{i}} \Biggr]>0, \\ &v\cdot \Biggl[ \sum_{i=1}^{\infty}b_{i}\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]\big|_{t=\eta_{i}} \Biggr]>0, \end{aligned}$$

  or

  $$\begin{aligned} &u\cdot \Biggl[ \sum_{i=1}^{\infty}a_{i}\phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]\big|_{t=\xi_{i}} \Biggr]< 0, \\ &v\cdot \Biggl[ \sum_{i=1}^{\infty}b_{i}\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]\big|_{t=\eta_{i}} \Biggr]< 0. \end{aligned}$$

Then BVP (3.1) has at least a solution in X provided that

$$\begin{aligned}& \max \bigl\{ 2^{q-1} \tilde{a}\tilde{b}\tilde{c}+2^{q-1}bc , 2^{q-1}abc +2^{q-1} \tilde{b}\tilde{c}, \\& \quad 2^{q-1}abc +2^{q-1}bc, 2^{q-1} \tilde{a}\tilde{b}\tilde{c}+2^{q-1} \tilde{b}\tilde{c} \bigr\} < 1 \quad \textit{for } p< 2, \end{aligned}$$

(3.10)

$$\begin{aligned}& \max \{ \tilde{a}\tilde{b}\tilde{c}+bc , abc+\tilde{b}\tilde{c}, abc+bc, \tilde{a}\tilde{b}\tilde{c}+\tilde{b}\tilde{c} \} < 1 \quad \textit{for }p\geq2, \end{aligned}$$

(3.11)

where $c= (\sum_{i=1}^{n-1} \Vert \varphi_{i}(t) \Vert _{\infty})^{q-1}$ and $\tilde{c}= (\sum_{i=1}^{n-1} \Vert \tilde{\varphi}_{i}(t) \Vert _{\infty})^{q-1}$.

Proof

According to the definitions of $N_{1}$ and $N_{2}$, we have the following inequalities.

For $1< p\leq2$, one has

$$\begin{aligned} \Vert N_{1}v \Vert _{\infty}&= \bigl\Vert \phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr] \bigr\Vert _{\infty} \\ &=\max \bigl\vert I_{0+}^{\beta_{1}}f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0 +}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr\vert ^{q-1} \\ &\leq \Biggl\vert \frac{1}{\Gamma(\beta_{1}+1)} \Biggl[ \Vert \psi \Vert \infty+ \Vert v \Vert ^{p-1}_{X_{2}}\cdot\sum _{i=1}^{n-1} \bigl\Vert \varphi_{i}(t) \bigr\Vert _{\infty}\Biggr] \Biggr\vert ^{q-1} \\ & \leq2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ 2^{q-1}b \Biggl(\sum_{i=1}^{n-1} \bigl\Vert \varphi_{i}(t) \bigr\Vert _{\infty}\Biggr)^{q-1}\cdot \Vert v \Vert _{X_{2}} \\ &=2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ 2^{q-1}bc\cdot \Vert v \Vert _{X_{2}} \end{aligned}$$

(3.12)

and

$$\begin{aligned} \Vert N_{2}u \Vert _{\infty}&= \bigl\Vert \phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0+}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr] \bigr\Vert _{\infty} \\ &=\max \bigl\vert I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0+}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr\vert ^{q-1} \\ &\leq \Biggl\vert \frac{1}{\Gamma(\beta_{2}+1)} \Biggl[ \Vert \tilde {\psi} \Vert _{\infty}+ \Vert u \Vert ^{p-1}_{X_{1}}\cdot\sum _{i=1}^{n-1} \bigl\Vert \varphi_{i}(t) \bigr\Vert _{\infty}\Biggr] \Biggr\vert ^{q-1} \\ & \leq2^{q-1}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{b} \Biggl(\sum _{i=1}^{n-1} \bigl\Vert \tilde{\varphi}_{i}(t) \bigr\Vert _{\infty}\Biggr)^{q-1}\cdot \Vert u \Vert _{X_{1}} \\ &= 2^{q-1}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{b}\tilde{c}\cdot \Vert u \Vert _{X_{1}}. \end{aligned}$$

(3.13)

By the similar proof of (3.12) and (3.13), one has

$$\begin{aligned}& \Vert N_{1}v \Vert _{\infty}\leq b \Vert \psi \Vert ^{q-1} _{\infty}+ bc\cdot \Vert v \Vert _{X_{2}} \quad \text{for }p \geq2 , \end{aligned}$$

(3.14)

$$\begin{aligned}& \Vert N_{2}u \Vert _{\infty}\leq \tilde{b} \Vert \tilde {\psi} \Vert ^{q-1} _{\infty}+ \tilde{b}\tilde{c}\cdot \Vert u \Vert _{X_{1}} \quad \text{for }p\geq 2. \end{aligned}$$

(3.15)

Let

$$\Omega_{1}= \bigl\{ (u,v)\in\operatorname{dom}L\setminus \operatorname{Ker}L: L(u,v)= \lambda N(u,v),\lambda\in(0,1) \bigr\} . $$

First, we give a proof that for $1< p\leq2$, $\Omega_{1}$ is bounded.

Let $L(u,v)=\lambda N(u,v)\in\operatorname{Im}L=\operatorname{Ker}Q$, that is, $L_{1}u=\lambda N_{1}v\in\operatorname{Ker}Q_{1}$ and $L_{2}v=\lambda N_{2}u\in \operatorname{Ker}Q_{2}$. By the definition of $\operatorname{Ker}Q_{1}$ and $\operatorname{Ker}Q_{2}$, we have

$$\begin{aligned} &\sum_{i=1}^{\infty}a_{i}\cdot \lambda \phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]_{t=\xi_{i}} =0, \\ & \sum_{i=1}^{\infty}b_{i}\cdot \lambda\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]_{t=\eta_{i}} =0. \end{aligned}$$

According to (H2), there exist $t_{0},t_{1}\in(0,1)$ such that $\vert u(t_{0}) \vert \leq A$ and $\vert v(t_{1}) \vert \leq A$. Again, $L_{1}u=\lambda N_{1}v$, $u\in\operatorname{dom}L_{1}\setminus\operatorname{Ker}L_{1}$, that is, $D_{0^{+}}^{\alpha_{1}}u=\lambda N_{1}v$, we have

$$u(t)=\frac{\lambda}{\Gamma ( \alpha_{1} )} \int_{0}^{t}{{{ ( t-s )}^{\alpha_{1} -1}} \phi_{q} \bigl[I_{0+}^{\beta_{1}} f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr]}\,ds+c_{0}. $$

Substituting $t=t_{0}$ into the above equation, we get

$$u(t_{0})=\frac{\lambda}{\Gamma ( \alpha_{1} )} \int _{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha_{1} -1}}\phi_{q} \bigl[I_{0+}^{\beta_{1}} f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr]}\,ds+c_{0}. $$

So, we obtain

$$\begin{aligned} u(t)-u(t_{0})&=\frac{\lambda}{\Gamma ( \alpha_{1} )} \int _{0}^{t}{{{ ( t-s )}^{\alpha_{1} -1}} \phi_{q} \bigl[I_{0+}^{\beta_{1}} f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr]}\,ds \\ &\quad{} -\frac{\lambda}{\Gamma ( \alpha_{1} )} \int_{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha_{1} -1}}\phi_{q} \bigl[I_{0+}^{\beta_{1}} f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr]}\,ds . \end{aligned}$$

Together with $\vert u (t_{0}) \vert \leq A$ and (3.12), we have

$$\begin{aligned} \bigl\vert {{u} }(0) \bigr\vert &\le \bigl\vert u(t_{0}) \bigr\vert + \biggl\vert \frac{\lambda}{\Gamma ( \alpha_{1} )} \int_{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha_{1} -1}} \phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr] }\,ds \biggr\vert \\ &\le A+\frac{1}{\Gamma ( \alpha_{1} )} \int _{0}^{{{t}_{0}}}{{ ( {{t}_{0}}-s )}^{\alpha_{1} -1}} \bigl\vert \phi_{q} \bigl[I_{0+}^{\beta _{1}}f \bigl(s,v(s),D_{0+}^{\alpha_{2}-1}v(s), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(s) \bigr) \bigr] \bigr\vert \,ds \\ &= A+\frac{1}{\Gamma ( \alpha_{1} )}\cdot \bigl(2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ 2^{q-1}bc\cdot \Vert v \Vert _{X_{2}} \bigr) \cdot \int_{0}^{{{t}_{0}}} (t_{0}-s )^{\alpha_{1}-1} \,ds \\ &\leq A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}abc\cdot \Vert v \Vert _{X_{2}}. \end{aligned}$$

(3.16)

Similarly, by (3.13), we obtain

$$\begin{aligned} \bigl\vert {{v} }(0) \bigr\vert \le& \bigl\vert v(t_{0}) \bigr\vert \\ &{}+ \biggl\vert \frac{\lambda}{\Gamma ( \alpha_{2} )} \int_{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha_{2} -1}} \phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(s,u(s),D_{0+}^{\alpha_{1}-1}u(s), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(s) \bigr) \bigr] }\,ds \biggr\vert \\ \le &A+\frac{1}{\Gamma ( \alpha_{2} )} \int _{0}^{{{t}_{0}}}{{ ( {{t}_{0}}-s )}^{\alpha_{2} -1}} \bigl\vert \phi_{q} \bigl[I_{0+}^{\beta _{2}}g \bigl(s,u(s),D_{0+}^{\alpha_{1}-1}u(s), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(s) \bigr) \bigr] \bigr\vert \,ds \\ =& A+\frac{1}{\Gamma ( \alpha_{2} )}\cdot \bigl( 2^{q-1}\tilde{b} \Vert \tilde{ \psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{b}\tilde{c} \cdot \Vert u \Vert _{X_{1}} \bigr) \cdot \int_{0}^{{{t}_{0}}} (t_{0}-s )^{\alpha_{2}-1} \,ds \\ \leq &A+2^{q-1}\tilde{a}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{a}\tilde{b}\tilde{c} \cdot \Vert u \Vert _{X_{1}}. \end{aligned}$$

(3.17)

For $(u,v)\in\Omega_{1}$, by (3.5) and (3.9), we have

$$\begin{aligned} \bigl\Vert (u,v) \bigr\Vert _{Y} &= \bigl\Vert P(u,v)+(I-P) (u,v) \bigr\Vert _{Y} \leq \bigl\Vert P(u,v) \bigr\Vert _{Y}+ \bigl\Vert (I-P) (u,v) \bigr\Vert _{Y} \\ &\leq\max \bigl\{ \bigl\vert {{u}}(0) \bigr\vert + \Vert N_{1}v \Vert _{\infty}; \bigl\vert {{u}}(0) \bigr\vert + \Vert N_{2}u \Vert _{\infty}; \\ & \quad{} \bigl\vert {{v}}(0) \bigr\vert + \Vert N_{1}v \Vert _{\infty}; \bigl\vert {{v}}(0) \bigr\vert + \Vert N_{2}u \Vert _{\infty}\bigr\} . \end{aligned}$$

The following proof is divided into four cases.

Case 1. $\Vert (u,v) \Vert _{Y} \leq \vert {{u} }(0) \vert + \Vert N_{1}v \Vert _{\infty}$.

By (3.12) and (3.16), we have

$$\begin{aligned} \Vert v \Vert _{X_{2}}&\leq \bigl\Vert (u,v) \bigr\Vert _{Y} \leq \bigl\vert {{u} }(0) \bigr\vert + \Vert N_{1}v \Vert _{\infty} \\ & \leq A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}abc\cdot \Vert v \Vert _{X_{2}} +2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ 2^{q-1}bc\cdot \Vert v \Vert _{X_{2}} \\ & = A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ \bigl(2^{q-1}abc + 2^{q-1}bc \bigr)\cdot \Vert v \Vert _{X_{2}}. \end{aligned}$$

According to (3.10), we can derive

$$\Vert v \Vert _{X_{2}} \leq\frac{A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}}{1-(2^{q-1}abc + 2^{q-1}bc)}:=M_{1}. $$

Thus, $\Omega_{1}$ is bounded.

Case 2. $\Vert (u,v) \Vert _{Y} \leq \vert {{u} }(0) \vert + \Vert N_{2}u \Vert _{\infty}$.

By (3.13) and (3.16), we have

$$\begin{aligned} \bigl\Vert (u,v) \bigr\Vert _{Y} &\leq \bigl\vert {{u} }(0) \bigr\vert + \Vert N_{2}u \Vert _{\infty} \\ &\leq A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}abc\cdot \Vert v \Vert _{X_{2}} +2^{q-1} \tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{b}\tilde{c}\cdot \Vert u \Vert _{X_{1}} \\ &= A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+2^{q-1}abc\cdot \Vert v \Vert _{X_{2}} + 2^{q-1} \tilde{b}\tilde{c}\cdot \Vert u \Vert _{X_{1}} \\ &\leq A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ \bigl(2^{q-1}abc + 2^{q-1} \tilde{b}\tilde{c} \bigr)\cdot \bigl\Vert (u,v) \bigr\Vert _{Y}. \end{aligned}$$

By (3.10), we can derive

$$\bigl\Vert (u,v) \bigr\Vert _{Y}\leq\frac{A+ 2^{q-1}ab \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}}{1-2^{q-1}abc- 2^{q-1} \tilde{b}\tilde{c} }:=M_{2}. $$

Then $\Omega_{1}$ is bounded.

Case 3. $\Vert (u,v) \Vert _{Y} \leq \vert {{v}}(0) \vert + \Vert N_{1}v \Vert _{\infty}$.

According to (3.12) and (3.17), we have

$$\begin{aligned} \bigl\Vert (u,v) \bigr\Vert _{Y} &\leq \bigl\vert {{v}}(0) \bigr\vert + \Vert N_{1}v \Vert _{\infty} \\ &\leq A+2^{q-1}\tilde{a}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{a}\tilde{b}\tilde{c} \cdot \Vert u \Vert _{X_{1}} +2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ 2^{q-1}bc\cdot \Vert v \Vert _{X_{2}} \\ &= A+2^{q-1}\tilde{a}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+2^{q-1} \tilde {a}\tilde{b}\tilde{c} \cdot \Vert u \Vert _{X_{1}} + 2^{q-1}bc\cdot \Vert v \Vert _{X_{2}} \\ &\leq A+2^{q-1}\tilde{a}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}+ \bigl( 2^{q-1} \tilde{a}\tilde{b} \tilde{c} + 2^{q-1}bc \bigr) \cdot \bigl\Vert (u,v) \bigr\Vert _{Y}. \end{aligned}$$

By (3.10), we have

$$\bigl\Vert (u,v) \bigr\Vert _{Y} \leq \frac{A+2^{q-1}\tilde{a}\tilde {b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1}b \Vert \psi \Vert ^{q-1} _{\infty}}{1- ( 2^{q-1} \tilde{a}\tilde{b}\tilde{c} + 2^{q-1}bc ) }:=M_{3}. $$

Then $\Omega_{1}$ is bounded.

Case 4. $\Vert (u,v) \Vert _{Y} \leq \vert {{v}}(0) \vert + \Vert N_{2}u \Vert _{\infty}$.

According to (3.13) and (3.17), we have

$$\begin{aligned} \Vert u \Vert _{X_{1}} &\leq \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \bigl\vert {{v}}(0) \bigr\vert + \Vert N_{2}u \Vert _{\infty}\\ &\leq A+2^{q-1}\tilde{a}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{a}\tilde{b}\tilde{c} \cdot \Vert u \Vert _{X_{1}} +2^{q-1}\tilde{b} \Vert \tilde{ \psi} \Vert ^{q-1} _{\infty}+ 2^{q-1} \tilde{b} \tilde{c} \cdot \Vert u \Vert _{X_{1}} \\ &= A+2^{q-1}\tilde{a}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+2^{q-1}\tilde{b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+ \bigl(2^{q-1} \tilde{a} \tilde{b}\tilde{c} + 2^{q-1} \tilde{b}\tilde{c} \bigr)\cdot \Vert u \Vert _{X_{1}}. \end{aligned}$$

By (3.10), we get

$$\Vert u \Vert _{X_{1}} \leq\frac{A+2^{q-1}\tilde{a}\tilde {b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}+2^{q-1}\tilde {b} \Vert \tilde{\psi} \Vert ^{q-1} _{\infty}}{ 1- (2^{q-1} \tilde{a}\tilde{b}\tilde{c} + 2^{q-1} \tilde{b}\tilde{c} )}:=M_{4}. $$

Then $\Omega_{1}$ is bounded.

Therefore, we have proved that $\Omega_{1}$ is bounded for $1< p\leq2$. By similar arguments as the above proof, according to (3.11), (3.14) and (3.15), we can prove that $\Omega_{1}$ is also bounded for $p>2$. We omit the proof of it.

Let

$$\Omega_{2}= \bigl\{ (u,v)\in\operatorname{Ker}L:N(u,v)\in \operatorname{Im}L \bigr\} . $$

Let $(u,v)\in\operatorname{Ker}L$, so we have $u=c_{0}$, $v=d_{0}$. In view of $N(u,v)=(N_{1}v,N_{2}u)\in\operatorname{Im}L=\operatorname{Ker}Q$, we have $Q_{1}(N_{1}v)=0$, $Q_{2}(N_{2}u)=0$, that is,

$$\begin{aligned} & \sum_{i=1}^{\infty}a_{i} \phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]\big|_{t=\xi_{i}} =0, \\ & \sum_{i=1}^{\infty}b_{i} \phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]\big|_{t=\eta_{i}} =0. \end{aligned}$$

By (H2), there exist constants $t_{0},t_{1}\in[0,1]$ such that

$$\bigl\vert u(t_{0}) \bigr\vert = \vert c_{0} \vert \leq A,\qquad \bigl\vert v( t_{1}) \bigr\vert = \vert d_{0} \vert \leq A. $$

Therefore, $\Omega_{2}$ is bounded.

Let

$$\Omega_{3}= \bigl\{ (u,v)\in\operatorname{Ker}L:\lambda(u,v)+(1- \lambda) QN(u,v)=(0,0),\lambda\in[0,1] \bigr\} . $$

For $(u,v)\in\operatorname{Ker}L$, so we have $u=c_{0}$ and $v=d_{0}$. By the definition of the set $\Omega_{3}$, we have

$$\begin{aligned} \lambda c_{0} +(1-\lambda)Q_{1}N_{1}(d_{0})=0, \qquad \lambda d_{0} +(1-\lambda)Q_{2}N_{2}(c_{0})=0. \end{aligned}$$

(3.18)

If $\lambda=0$, similar to the proof of the boundedness of $\Omega_{2}$, we have $\vert c_{0} \vert \leq A$ and $\vert d_{0} \vert \leq A$. If $\lambda=1$, we have $c_{0}=d_{0}=0$. If $\lambda\in(0,1)$, we also have $\vert c_{0} \vert \leq A$ and $\vert d_{0} \vert \leq A$. Otherwise, if $\vert c_{0} \vert >A$ or $\vert d_{0} \vert >A$, in view of the first part of (H2), we obtain

$$\begin{aligned} \lambda c^{2}_{0} +(1-\lambda)c _{0}\cdot Q_{1}N_{1}(d_{0})>0,\quad \quad \lambda d^{2}_{0} +(1-\lambda)d_{0}\cdot Q_{2}N_{2}(c_{0})>0, \end{aligned}$$

which contradict (3.18). Thus, $\Omega_{3}$ is bounded.

If the second part of (H2) holds, then we can prove that the set

$$\Omega_{3}'= \bigl\{ (u,v)\in\operatorname{Ker}L:- \lambda(u,v)+(1-\lambda) QN(u,v)=(0,0),\lambda\in[0,1] \bigr\} $$

is bounded.

Finally, let Ω to be a bounded open set of Y such that $\bigcup_{i=1}^{3}{\overline{\Omega}_{i}}\subset\Omega$. By Lemma 3.4, N is L-compact on Ω. Then, by the above arguments, we get

• (1) $L(u,v)\neq\lambda N(u,v)$, for every $(u,v)\in[(\operatorname{dom}L\setminus{Ker}L)\cap\partial\Omega]\times(0,1)$;

• (2) $N(u,v)\notin\operatorname{Im}L$ for every $(u,v)\in\operatorname{Ker}L\cap \partial\Omega$;

• (3) Let $H((u,v),\lambda)=\pm\lambda I(u,v)+(1-\lambda)JQN(u,v)$, where I is the identical operator. Via the homotopy property of degree, we obtain that

  $$\begin{aligned} \deg (JQ N|_{\operatorname{Ker} L},\Omega\cap\operatorname{Ker} L,0 ) &= \deg \bigl(H( \cdot,0), \Omega\cap\operatorname{Ker} L,0 \bigr) \\ &= \deg \bigl(H(\cdot,1),\Omega\cap\operatorname{Ker} L,0 \bigr) \\ &= \deg (I,\Omega\cap\operatorname{Ker} L,0 ) \\ &=1\neq0. \end{aligned}$$

Applying Theorem 2.1, we conclude that $Lu=Nu$ has at least one solution in $\operatorname{dom}L\cap\overline{\Omega}$. □

4 Example

Let us consider the following fractional differential equations with p-Laplacian operator at resonance:

$$ \textstyle\begin{cases} D_{0+}^{0.6} \phi_{3} ( D_{0+}^{2.6} u(t) )=f (t,v(t),D_{0+}^{1.8}v(t),D_{0^{+}}^{0.8}v(t) ), \quad t\in(0,1),\\ D_{0+}^{0.7} \phi_{3} ( D_{0+}^{2.8} u(t) )=f (t,u(t),D_{0+}^{1.6}u(t),D_{0^{+}}^{0.6}u(t) ), \quad t\in(0,1),\\ u'(0)=u''(0)=D_{0+}^{2.6}u(0)=0, \qquad u(0)=\sum_{i=1}^{\infty}\frac {1}{2^{i}}u(\frac{1}{2i}),\\ v'(0)=v''(0)=D_{0+}^{2.8}v(0)=0, \qquad v(0)=\sum_{i=1}^{\infty}\frac {2}{3^{i}}u(\frac{1}{3i}), \end{cases} $$

(4.1)

where

$$\begin{aligned}& f(t,x_{1},x_{2},x_{3} ) =\frac{t}{10} + \frac{1}{10}x_{1}^{2}+\frac { \vert \sin x_{2} \vert }{20}+ \frac{ \vert \arctan x_{3} \vert }{10\pi}, \\& g(t,y_{1},y_{2},y_{3} ) =\frac{t^{2}}{20} + \frac{1}{20}y_{1}^{2}+\frac{\cos^{2} y _{2} }{20}+ \frac{e^{- \vert y_{3} \vert }}{40}. \end{aligned}$$

Corresponding to BVP (1.1), we have that $\alpha_{1}=2.6$, $\beta _{1}=0.6$, $\alpha_{2}=2.8$, $\beta_{2}=0.7$, $n=3$, $p=3$, $q=1.5$, $a= ({\Gamma(\alpha_{1}+1)})^{-1}= ({\Gamma(3.6)})^{-1}\approx0.269$, $b= ({\Gamma(\beta_{1}+1)})^{1-q}=({\Gamma(1.6)})^{-0.5}\approx1.058$, $\tilde{a}= ({\Gamma(\alpha_{2}+1)})^{-1}= ({\Gamma(3.8)})^{-1}\approx 0.213$, $\tilde{b}= ({\Gamma(\beta_{2}+1)})^{1-q}=({\Gamma (1.7)})^{-0.5}\approx1.049$, $a_{i}=\frac{1}{2^{i}}$, $\xi_{i}=\frac{1}{2i}$, $b_{i}=\frac{2}{3^{i}}$, $\eta _{i}=\frac{1}{3i}$, $i=1,2,\ldots$ . Then we have $\sum_{i=1}^{\infty}a_{i}=\sum_{i=1}^{\infty} \vert a_{i} \vert =\sum_{i=1}^{\infty}b_{i}=\sum_{i=1}^{\infty} \vert b_{i} \vert =1$. Taking $z_{0}=\tilde{z}_{0}=3$, we have

$$\phi_{1}(z_{0})\phi_{2}(\tilde{z}_{0}) = \sum_{i=1}^{\infty} a_{i} \xi_{i}^{z_{0}} \cdot\sum_{i=1}^{\infty} b_{i}\eta_{i}^{\tilde{z}_{0}}= \sum _{i=1}^{\infty}\frac{1}{2^{i}} \biggl( \frac{1}{2i} \biggr)^{3} \cdot\sum_{i=1}^{\infty} \frac{2}{3^{i}} \biggl(\frac{1}{3i} \biggr)^{3}\neq0 , $$

which implies that $(\mathrm{H}_{0})$ holds.

By a simple proof, we have

$$\begin{aligned} & \bigl\vert f(t,x_{1},x_{2},x_{3} ) \bigr\vert = \biggl\vert \frac {t}{10} +\frac{1}{10}x_{1}^{2}+ \frac{ \vert \sin x_{2} \vert }{20}+\frac{ \vert \arctan x_{3} \vert }{10\pi} \biggr\vert \leq\frac{1}{5 } + \frac{1}{10}x_{1}^{2}, \\ & \bigl\vert g(t,y_{1},y_{2},y_{3} ) \bigr\vert = \biggl\vert \frac{t^{2}}{20} +\frac{1}{20}y_{1}^{2}+ \frac{\cos^{2} y _{2} }{20}+\frac{e^{- \vert y_{3} \vert }}{40} \biggr\vert \leq\frac{1}{8} + \frac{1}{20}x_{1}^{2}. \end{aligned}$$

Choose $\psi(t)=\frac{1}{5}$, $\varphi_{1}(t)= \frac{1}{10}$, $\varphi _{2}=\varphi_{3}=0$, $\tilde{\psi}(t)=\frac{1}{8}$, $\tilde{\varphi}_{1}(t)= \frac{1}{20}$, $\tilde{\varphi}_{2}=\tilde{\varphi}_{3}=0$, then we have (H1) of Theorem 3.1 is satisfied.

By a simple computation, we have $c= (\sum_{i=1}^{n-1} \Vert \varphi_{i}(t) \Vert _{\infty})^{q-1}=(\varphi_{1})^{q-1}=\sqrt{0.1}\approx0.316$, $\tilde {c}= (\sum_{i=1}^{n-1} \Vert \tilde{\varphi}_{i}(t) \Vert _{\infty})^{q-1}= (\tilde{\varphi}_{1})^{q-1}=\sqrt{0.05}\approx0.224$, $\tilde{a}\tilde{b}\tilde{c}+bc\approx0.287$, $abc+\tilde{b}\tilde {c}\approx0.298$, $abc+bc\approx0.301$, $\tilde{a}\tilde{b}\tilde {c}+\tilde{b}\tilde{c}\approx0.240$. So, (3.11) holds.

In addition, by choosing $A=1$, we have if $u> 1$, or $v> 1$, then f, g are positive functions. So, the first inequality of (H2) is satisfied.

Thus, all the conditions of Theorem 3.1 are satisfied; consequently, its conclusion implies that problem (4.1) has a solution on $[0, 1]$.

5 Conclusion

In this paper, we have obtained the existence of solutions for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions at resonance. We base our analysis on the known coincidence degree theory. The issue on the existence of solutions of infinite-point boundary value problems is interesting. As applications, an example is presented to illustrate the main results. In the future, we will consider the positive solutions for the fractional infinite-point boundary value problems at resonance.