Existence of solutions to a coupled system of fractional differential equations with infinite-point boundary value conditions at resonance

Abstract

By means of coincidence degree theory, we present an existence result for the solution of a higher-order coupled system of nonlinear fractional differential equations with infinite-point boundary conditions at resonance.

1 Introduction

In this paper, we investigate the existence of solutions for the following higher-order coupled fractional differential equation with infinite-point boundary value conditions:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=f (t,v(t),v'(t), \ldots,v^{(n-1)}(t) ),\quad 0< t< 1, \\ D_{0^{+}}^{\beta}v(t)=g (t,u(t),u'(t) ,\ldots,u^{(n-1)}(t) ),\quad 0< t< 1, \\ u'(0)= \cdots=u^{(n-1)}(0)=0, \qquad u(1)=\sum_{i=1}^{\infty}a_{i}u(\xi _{i}), \\ v'(0)= \cdots=v^{(n-1)}(0)=0, \qquad v(1)=\sum_{i=1}^{\infty}b_{i}v(\eta_{i}), \end{cases} $$

(1.1)

where $n-1<\alpha,\beta<n$, $n\geq2$, $0<\xi_{1}<\xi_{2}<\cdots<\xi_{i}<\xi _{i+1}<\cdots<1$, $0<\eta_{1}<\eta_{2}<\cdots<\eta_{i}<\eta_{i+1}<\cdots<1$, $0< a_{i},b_{i}<1$; $D_{0^{+}}^{\alpha}$ and $D_{0^{+}}^{\beta}$ denote the Caputo fractional derivatives, f, g are given continuous functions, and

$$\sum_{i=1}^{\infty}a_{i}=1,\qquad \sum_{i=1}^{\infty}b_{i}=1, $$

which implies that BVP (1.1) is at resonance.

During the past decades, fractional differential equations have attracted considerable interest because of their wide applications in various sciences, such as physics, mechanics, chemistry, engineering, electromagnetic, control, etc. (see [1–4]). In recent years, boundary value problems of fractional differential equations or systems of fractional differential equations at resonance have been discussed in some papers, such as [5–10]. Most of the results on the existence of solutions for fractional boundary value problems at resonance are concerned with finite points. For example, Wang et al. [5] discussed the following coupled system of fractional 2m-point boundary value problem at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=f(t,v(t),D_{0^{+}}^{\beta-1}v(t), D_{0^{+}}^{\beta-2}v(t)), \quad 0< t< 1, \\ D_{0^{+}}^{\beta}v(t)=g(t,u(t),D_{0^{+}}^{\alpha-1}u(t), D_{0^{+}}^{\alpha-2}u(t)),\quad 0< t< 1, \\ I_{0^{+}}^{3-\alpha}u(t)|_{t=0}=0, \qquad D_{0^{+}}^{\alpha-2}u(1)= {\sum_{i=1}^{m}{{{a }_{i}}D_{0^{+}}^{\alpha-2}u(\xi_{i})}}, \\ I_{0^{+}}^{3-\beta}v(t)|_{t=0}=0,\qquad D_{0^{+}}^{\beta-2}v(1)= {\sum_{i=1}^{m}{{{c }_{i}}D_{0^{+}}^{\beta-2}v(\gamma_{i})}}, \\ u(1)=\sum_{i=1}^{m} b_{i}u(\eta_{i}), \qquad v(1)=\sum_{i=1}^{m} d_{i}v(\delta_{i}), \end{cases} $$

where $2<\alpha,\beta\leq3$, $0<\xi_{1}<\cdots<\xi_{m}<1$, $0<\eta_{1}<\cdots<\eta_{m}<1$, $0<\gamma_{1}<\cdots<\gamma_{m}<1$, $0<\delta_{1}<\cdots<\delta_{m}<1$, $a_{i},b_{i},c_{i},d_{i}\in\mathbb{R}$, $f,g:[0,1]\times\mathbb{R}^{3}\rightarrow\mathbb{R}$, f, g satisfy the Carathéodory conditions, $D_{0^{+}}^{\alpha}$, $I_{0^{+}}^{\alpha}$ are standard Riemann-Liouville fractional operators.

In [6], Liu et al. discussed the following boundary value problem for a coupled system of fractional differential equations at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=f(t,v(t),D_{0^{+}}^{p}v(t)),\quad 0< t< 1, \\ D_{0^{+}}^{\beta}v(t)=g(t,u(t),D_{0^{+}}^{q}u(t)),\quad 0< t< 1, \\ u(0)=0,\qquad D_{0^{+}}^{p}u(1)={\sum_{i=1}^{m-2}{{{a }_{i}}D_{0^{+}}^{p}u(\xi_{i})}}, \\ v(0)=0, \qquad D_{0^{+}}^{q}v(1)={\sum_{i=1}^{m-2}{{{b }_{i}}D_{0^{+}}^{q}v(\eta_{i})}}, \end{cases} $$

where $1<\alpha,\beta\leq2$, $0< p,q<1$, $\alpha-p-1,\beta-q-1\geq0$, $a_{i},b_{i}\geq0$, $0<\xi_{i},\eta_{i}<1$ ($i=1,2,\ldots,m-2$), ${\sum_{i=1}^{m-2 }{{{a }_{i}}\xi_{i}^{\alpha-p-1 }}}={\sum_{i=1}^{m-2}{{{b }_{i}}\eta_{i}^{\beta-q-1 }}}=1$; $D_{0^{+}}^{\alpha }$, $D_{0^{+}}^{\beta}$ are standard Riemann-Liouville fractional derivatives.

Very recently, the infinite-point boundary value problems of fractional differential equations have been discussed by researchers, whose excellent results extend many previous results; see [11–14].

In 2015, Zhang [11] considered the existence of positive solutions of the following nonlinear fractional differential equation with infinite-point boundary value conditions:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)+q(t)f(t,u(t))=0,\quad 0< t< 1, \\ u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0, \\ v^{(i)}(1)={\sum_{j=1}^{\infty}{{{\alpha}_{j}}u(\xi_{j})}}, \end{cases} $$

where $\alpha>2$, $n-1<\alpha\leq n$, $a_{j}\geq0$, $0<\xi_{1}<\xi_{2}<\cdots <\xi_{j}<\cdots<1$ ($j=1,2,\ldots$), $\Delta-{\sum_{j=1}^{\infty }{{{\alpha}_{j}}\xi_{j}^{\alpha-1 }}}>0$, $\Delta=(\alpha-1)(\alpha -2)\cdots(\alpha-i)$, $i\in[1,n-2]$ is a fixed integer, $D_{0^{+}}^{\alpha }$ is the standard Riemann-Liouville fractional derivative.

In [14], Ge et al. considered the existence of solutions of the following nonlinear fractional differential equation with infinitely many points boundary value problems at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}x_{1}(t)=f_{1} (t,x_{1}(t), D_{0^{+}}^{\beta-1}x_{2}(t) ), \\ D_{0^{+}}^{\beta}x_{2}(t)=f_{2} (t,x_{2}(t), D_{0^{+}}^{\alpha-1}x_{1}(t) ), \\ x_{1}(0)=0,\qquad \lim_{t\rightarrow\infty}D_{0^{+}}^{\alpha-1}x_{1}(t)= \sum_{i=1}^{+\infty}\gamma_{i}x_{1}(\eta_{i}), \\ x_{2}(0)=0,\qquad \lim_{t\rightarrow\infty}D_{0^{+}}^{\beta-1}x_{2}(t)= \sum_{i=1}^{+\infty}\sigma_{i}x_{2}(\xi_{i}), \end{cases} $$

where $1<\alpha,\beta\leq2$, $0<\eta_{1}<\eta_{2}<\cdots<\eta_{i}<\cdots$, $0<\xi_{1}<\xi_{2}<\cdots<\xi_{i}<\cdots$, $\lim_{i\rightarrow\infty}\eta_{i}=\infty$, $\lim_{i\rightarrow\infty}\xi_{i}=\infty$, and $\sum_{i=1}^{\infty}|\gamma_{i}|\eta_{i}^{\alpha}<\infty$, $\sum_{i=1}^{\infty}|\sigma_{i}|\xi_{i}^{\beta}<\infty$. Here, $f_{1},f_{2}:[0,+\infty)\times\mathbb{R} \times\mathbb{R} \rightarrow\mathbb{R}$ satisfy the Carathéodory conditions, $D_{0^{+}}^{\alpha}$, $D_{0^{+}}^{\beta}$ are the standard Riemann-Liouville fractional derivatives.

From the above work, we note that it is meaningful and interesting to study the existence of solutions for fractional boundary value problems with infinite-point boundary conditions. Although fractional boundary value problems at resonance have been studied by some authors, to the best of our knowledge, fractional differential equations subject to infinite points at resonance have not been studied till now. Motivated by the work above, we considered 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 [14]. 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 continuous function $f:(0,\infty)\to\mathbb{R}$ 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 $n-1<\alpha\leq n$, $u\in C(0,1)\cap L^{1}(0,1)$, then

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

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

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 a continuous function f.

First of all, 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

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

It follows that

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

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 $(x,\lambda) \in [(\operatorname{dom}L\setminus\operatorname{Ker} L) \cap \partial\Omega ]\times (0,1)$;

• (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 paper, we will always suppose the following condition holds:

• (H1) ${\sum_{i=1}^{\infty}{{{a }_{i}}\xi_{i}^{\alpha }}}\neq1$, ${\sum_{i=1}^{\infty}{{{b }_{i}}\eta_{i}^{\beta}}}\neq1$.

Denote by E the Banach space $E=C[0,1]$ with the norm $\|u\|_{\infty}=\max_{0 \le t \le1}|u(t)|$. We denote a Banach space $X =\{u(t)|u^{(i)}(t)\in E, i=1,2,\ldots,n-1\}$ with the norm ${{\Vert u \Vert }_{X}}=\max \{{{\Vert u \Vert }_{\infty}},{{ \Vert {{u}'} \Vert }_{\infty}},\ldots,{{ \Vert {{u}^{(n-1)}} \Vert }_{\infty}}\}$. Let $Y=X\times X$ be endowed with the norm $\|(u,v)\|_{Y}=\max\{\|u\|_{X},\|v\|_{X} \}$, and $Z=E\times E$ is a Banach space with the norm defined by $\|(x,y)\|_{Z}=\max \{\|x\|_{\infty},\|y\|_{\infty}\}$.

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

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

and

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

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

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

and

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

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 Nemytski operator

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

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

$$N_{1}v(t)=f \bigl(t,v(t),v'(t),v''(t), \ldots,v^{(n-1)}(t) \bigr), $$

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

$$N_{2}u(t)=g \bigl(t,u(t),u'(t),u''(t), \ldots,u^{(n-1)}(t) \bigr). $$

Then BVP (1.1) can be written as $L(u,v)=N(u,v)$.

Lemma 3.1

L is defined as above, then

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

(3.1)

$$\begin{aligned}& \operatorname{Im} L = \Biggl\{ (x,y)\in Y \Big|I_{0^{+}}^{\alpha }x(1)- \sum_{i=1}^{\infty}a_{i}I_{0^{+}}^{\alpha}x( \xi_{i})=0, I_{0^{+}}^{\beta }y(1)-\sum _{i=1}^{\infty}b_{i}I_{0^{+}}^{\beta}y( \eta_{i})=0 \Biggr\} . \end{aligned}$$

(3.2)

Proof

By Lemma 2.1, the equation $D_{0^{+}}^{\alpha}u(t)=0$ has the solution

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

Combining with $u^{(i)}(0)=0$, $i=1,2,\ldots,n-1$, one has $c_{i}=0$, $i=1,2,\ldots,n-1$. Then $u(t)=c_{0}$. Similarly, for $v\in\operatorname{Ker}L_{2}$, we have $v(t)=d_{0}$. Thus, we obtain (3.1).

Next we prove (3.2) holds. Let $(x,y)\in\operatorname{Im}L$, so there exists $(u,v)\in\operatorname{dom}L$ such that $x(t)=D_{0^{+}}^{\alpha }u(t)$, $y(t)=D_{0^{+}}^{\beta}v(t)$. By Lemma 2.1, we have

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

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, we have

$$u(t)=I_{0^{+}}^{\alpha}x(t)+c_{0}, \qquad v(t)=I_{0^{+}}^{\beta}y(t)+d_{0}. $$

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

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

that is,

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

On the other hand, suppose $(x,y)$ satisfies the above equations. Let $u(t)=I_{0^{+}}^{\alpha}x(t)$ and $v(t)=I_{0^{+}}^{\beta}y(t)$, we can prove $(u(t),v(t) )\in\operatorname{dom} L$ and $L (u(t),v(t) )=(x,y)$. Then (3.2) holds. □

Lemma 3.2

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

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

The linear operator $Q(x,y)=(Q_{1}x,Q_{2}y)$ can be defined as

$$\begin{aligned} Q_{1}x(t) =\frac{\Gamma(1+\alpha)}{1-\sum_{i=1}^{\infty}{{{a }_{i}}\xi_{i}^{\alpha}}} \Biggl[I_{0^{+}}^{\alpha }x(1)- \sum_{i=1}^{\infty}a_{i}I_{0^{+}}^{\alpha}x( \xi_{i}) \Biggr], \\ Q_{2}y(t) =\frac{\Gamma(1+\beta)}{1-\sum_{i=1}^{\infty }{{{b}_{i}}\eta_{i}^{\beta}}} \Biggl[I_{0^{+}}^{\beta}y(1)- \sum_{i=1}^{\infty}b_{i}I_{0^{+}}^{\beta}y( \eta_{i}) \Biggr]. \end{aligned}$$

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

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

$${{Q}_{1}} \bigl({{Q}_{1}}x(t) \bigr)={{Q}_{1}}x(t) \cdot\frac{\Gamma (1+\alpha)}{1-\sum_{i=1}^{\infty}{{{a }_{i}}\xi_{i}^{\alpha }}} \Biggl[ \bigl(I_{0^{+}}^{\alpha}1 \bigr)|_{t=1}-\sum_{i=1}^{\infty}a_{i} \bigl(I_{0^{+}}^{\alpha}1 \bigr)|_{t=\xi_{i}} \Biggr]={{Q}_{1}}x(t). $$

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

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

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

(3.3)

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

$$ K_{P}(x,y)= \bigl(I_{0^{+}}^{\alpha}x,I_{0^{+}}^{\beta}y \bigr). $$

(3.4)

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

$$LK_{P}(x,y)=L\bigl(I_{0^{+}}^{\alpha}x,I_{0^{+}}^{\beta}y \bigr)= \bigl(D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}x,D_{0^{+}}^{\beta}I_{0^{+}}^{\beta}y \bigr) =(x,y). $$

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

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

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

$$ K_{p}L(x,y)= \bigl(I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha} x,I_{0^{+}}^{\beta}D_{0^{+}}^{\beta}y \bigr)=(x,y). $$

(3.5)

Combining (3.4) and (3.5), we get $K_{P}=(L_{\operatorname{dom} L \cap\operatorname{Ker} P })^{-1}$.

For simplicity of notation, we set $a=\frac{1}{\Gamma(\alpha-n+2)}$, $b=\frac{1}{\Gamma(\beta-n+2)}$.

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

$$\begin{aligned} \bigl\Vert K_{P}(x,y)\bigr\Vert _{Y} &= \bigl\Vert \bigl(I_{0^{+}}^{\alpha}x,I_{0^{+}}^{\beta}y \bigr)\bigr\Vert _{Y} =\max \bigl\{ \bigl\Vert I_{0^{+}}^{\alpha}x\bigr\Vert _{X}; \bigl\Vert I_{0^{+}}^{\beta}y \bigr\Vert _{X} \bigr\} \\ &\leq\max \biggl\{ \frac{1}{\Gamma(\alpha-n+2)}\Vert x\Vert _{\infty}; \frac {1}{\Gamma(\beta-n+2)}\Vert y\Vert _{\infty}\biggr\} \\ &=\max\bigl\{ a\Vert x\Vert _{\infty}; b\Vert y\Vert _{\infty}\bigr\} . \end{aligned}$$

(3.6)

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 (3.6), we have

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

(3.7)

With a similar proof to [15], we have the following lemma.

Lemma 3.3

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

Theorem 3.1

Assume (H1) and the following conditions hold:

• (H2) There exist nonnegative functions $\varphi_{i}(t),\psi_{i}(t)\in E$, $i=0,1,\ldots,n$, such that, for all $t\in[0,1]$, $(u_{1},u_{2},\ldots,u_{n}),(v_{1},v_{2},\ldots,v_{n})\in\mathbb {R}^{n}$, one has

  $$\begin{aligned}& \bigl\vert f(t,u_{1},u_{2},\ldots,u_{n})\bigr\vert \leq\varphi_{0}(t)+ \varphi _{1}(t)\vert u_{1}\vert +\varphi_{2}(t)\vert u_{2}\vert + \cdots+\varphi_{n}(t)\vert u_{n}\vert , \\& \bigl\vert g(t,v_{1},v_{2},\ldots,v_{n})\bigr\vert \leq\psi_{0}(t)+ \psi_{1}(t)\vert v_{1} \vert +\psi _{2}(t)\vert v_{2}\vert +\cdots+ \psi_{n}(t)\vert v_{n}\vert . \end{aligned}$$

• (H3) There exists $A>0$ such that, for $(u,u',\ldots,u^{(n-1)})$, $(v,v',\ldots,v^{(n-1)})$, if $|u |>A$ or $|v |>A$, $\forall t\in[0,1]$, one has

  $$\begin{aligned}& u\cdot \Biggl[I_{0^{+}}^{\alpha}f\bigl(t,v,v',\ldots ,v^{(n-1)}\bigr)|_{t=1}-\sum_{i=1}^{\infty}a_{i}I_{0^{+}}^{\alpha}f\bigl(t,v,v',\ldots ,v^{(n-1)}\bigr)|_{t=\xi_{i}} \Biggr]>0, \\& v\cdot \Biggl[I_{0^{+}}^{\beta}g\bigl(t,u,u', \ldots,u^{(n-1)}\bigr)|_{t=1} -\sum_{i=1}^{\infty}b_{i}I_{0^{+}}^{\beta}g\bigl(t,u,u', \ldots,u^{(n-1)}\bigr)|_{t=\eta _{i}} \Biggr]>0, \end{aligned}$$

  or

  $$\begin{aligned}& u\cdot \Biggl[I_{0^{+}}^{\alpha}f\bigl(t,v,v', \ldots,v^{(n-1)}\bigr)|_{t=1} -\sum_{i=1}^{\infty}a_{i}I_{0^{+}}^{\alpha}f\bigl(t,v,v',\ldots ,v^{(n-1)}\bigr)|_{t=\xi_{i}} \Biggr]< 0, \\& v\cdot \Biggl[I_{0^{+}}^{\beta}g\bigl(t,u,u', \ldots,u^{(n-1)}\bigr)|_{t=1} -\sum_{i=1}^{\infty}b_{i}I_{0^{+}}^{\beta}g\bigl(t,u,u',\ldots ,u^{(n-1)}\bigr)|_{t=\eta_{i}} \Biggr]< 0. \end{aligned}$$

Then BVP (1.1) has at least a solution in Y provided that

$$ \max \Biggl\{ 2a{\sum_{i=1}^{n} \|\varphi_{i}\|_{\infty}}, a\sum_{i=1}^{n}{{ \|\varphi_{i}\|_{\infty}}}+ b\sum_{i=1}^{n}{{ \|\psi_{i}\|_{\infty}}}, 2b\sum_{i=1}^{n}{{ \|\psi_{i} \|_{\infty}}} \Biggr\} < 1. $$

(3.8)

Proof

Let

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

For $L(u,v)=\lambda N(u,v)\in\operatorname{Im}L=\operatorname{Ker}Q$, by the definition of KerQ, hence

$$\begin{aligned}& I_{0^{+}}^{\alpha}f\bigl(t,v,v',\ldots,v^{(n-1)} \bigr)|_{t=1} -\sum_{i=1}^{\infty}a_{i}I_{0^{+}}^{\alpha}f\bigl(t,v,v', \ldots,v^{(n-1)}\bigr)|_{t=\xi _{i}}=0, \\& I_{0^{+}}^{\beta}g\bigl(t,u,u',\ldots,u^{(n-1)} \bigr)|_{t=1} -\sum_{i=1}^{\infty}b_{i}I_{0^{+}}^{\beta}g\bigl(t,u,u',\ldots ,u^{(n-1)}\bigr)|_{t=\eta_{i}}=0. \end{aligned}$$

From (H3), there exist $t_{0},t_{1}\in(0,1)$ such that $|u(t_{0})|\leq A$ and $| v(t_{1})| \leq A$. According to $L_{1}u=\lambda N_{1}v$, $u\in\operatorname{dom}L_{1}\setminus\operatorname{Ker}L_{1}$, that is, $D_{0^{+}}^{\alpha}u=\lambda N_{1}v$, we have

$$u(t)=\frac{\lambda}{\Gamma ( \alpha )} \int_{0}^{t}{{{ ( t-s )}^{\alpha-1}} f \bigl( s,v(s),v'(s),\ldots,{{v}^{(n-1)}}(s) \bigr)} \,ds+c_{0}. $$

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

$$u(t_{0})=\frac{\lambda}{\Gamma ( \alpha )} \int _{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha-1}} f \bigl( s,v(s),v'(s),\ldots,{{v}^{(n-1)}}(s) \bigr)}\,ds+c_{0}. $$

Furthermore, we get

$$\begin{aligned} u(t)-u(t_{0}) =&\frac{\lambda}{\Gamma ( \alpha )} \int _{0}^{t}{{{ ( t-s )}^{\alpha-1}} f \bigl( s,v(s),v'(s),\ldots,{{v}^{(n-1)}}(s) \bigr)}\,ds \\ &{} -\frac{\lambda}{\Gamma ( \alpha )} \int _{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha-1}} f \bigl( s,v(s),v'(s),\ldots,{{v}^{(n-1)}}(s) \bigr)}\,ds . \end{aligned}$$

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

$$\begin{aligned} \bigl\vert {{u} }(0) \bigr\vert &\le\bigl\vert u(t_{0})\bigr\vert +\biggl\vert \frac{\lambda}{\Gamma ( \alpha )} \int_{0}^{t_{0}}{{{ ( t_{0}-s )}^{\alpha-1}} f \bigl( s,v(s),v'(s),\ldots,{{v}^{(n-1)}}(s) \bigr)}\,ds \biggr\vert \\ &\le A+\frac{1}{\Gamma ( \alpha )} \int _{0}^{{{t}_{0}}}{{ ( {{t}_{0}}-s )}^{\alpha-1}}\bigl\vert f \bigl( s,v(s),v'(s), \ldots,{{v}^{(n-1)}}(s) \bigr)\bigr\vert \,ds \\ &\leq A+\frac{1}{\Gamma ( \alpha )} \int_{0}^{{{t}_{0}}}{{{ ( {{t}_{0}}-s )}^{\alpha -1}}} \Biggl(\varphi_{0}(t)+{\sum _{i=1}^{n}\varphi_{i}(t){\bigl\vert v^{(i-1)} \bigr\vert }} \Biggr)\,ds \\ &\leq A+\frac{1}{\Gamma ( \alpha )} \Biggl(\bigl\Vert \varphi_{0}(t)\bigr\Vert _{\infty}+{\sum_{i=1}^{n}\bigl\Vert \varphi_{i}(t)\bigr\Vert _{\infty}{\bigl\Vert v^{(i-1)} \bigr\Vert _{\infty}}} \Biggr)\cdot \int_{0}^{{{t}_{0}}} (t_{0}-s )^{\alpha-1} \,ds \\ &\leq A+\frac{1}{\Gamma(\alpha+1 )}\bigl\Vert \varphi_{0}(t)\bigr\Vert _{\infty}+\frac {1}{\Gamma(\alpha+1 )}{\sum_{i=1}^{n} \bigl\Vert \varphi_{i}(t)\bigr\Vert _{\infty}{\bigl\Vert v^{(i-1)} \bigr\Vert _{\infty}}} \\ &\leq A+\frac{1}{\Gamma(\alpha+1 )}\bigl\Vert \varphi_{0}(t)\bigr\Vert _{\infty}+\Vert v\Vert _{X}\cdot \frac{1}{\Gamma(\alpha+1 )}{\sum _{i=1}^{n}\bigl\Vert \varphi_{i}(t) \bigr\Vert _{\infty}} \\ &\leq A+a\bigl\Vert \varphi_{0}(t)\bigr\Vert _{\infty}+ \Vert v\Vert _{X}\cdot a{\sum_{i=1}^{n} \bigl\Vert \varphi_{i}(t)\bigr\Vert _{\infty}}. \end{aligned}$$

(3.9)

By similar arguments, we obtain

$$ \bigl\vert {{v} }(0) \bigr\vert \le A+b\bigl\Vert \psi_{0}(t)\bigr\Vert _{\infty}+\|u\|_{X}\cdot b{ \sum_{i=1}^{n}\bigl\Vert \psi_{i}(t)\bigr\Vert _{\infty}}. $$

(3.10)

For $(u,v)\in\Omega_{1}$, by (3.7), we have

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

The following proof is divided into four cases.

Case 1. $\|(u,v)\|_{Y} \leq \vert {{u} }(0) \vert +a\|N_{1}v\|_{\infty}$. By (3.9) and (H₁), we have

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y}&\leq \bigl\vert {{u} }(0) \bigr\vert +a\Vert N_{1}v\Vert _{\infty} \\ &\leq A+a\Vert \varphi_{0} \Vert _{\infty}+\Vert v\Vert _{X}\cdot a{\sum_{i=1}^{n} \Vert \varphi_{i} \Vert _{\infty}} +a \bigl\Vert f \bigl(t,v , \ldots,v^{(N-1)} \bigr)\bigr\Vert _{\infty} \\ &\leq A+ a\Vert \varphi_{0} \Vert _{\infty}+\Vert v \Vert _{X}\cdot a{\sum_{i=1}^{n} \Vert \varphi_{i} \Vert _{\infty}} +a \Biggl(\Vert \varphi_{0} \Vert _{\infty}+\Vert v\Vert _{X} \cdot {\sum_{i=1}^{n}\Vert \varphi_{i} \Vert _{\infty}} \Biggr) \\ &= A+2a \Biggl(\Vert \varphi_{0} \Vert _{\infty}+\Vert v \Vert _{X}\cdot {\sum_{i=1}^{n} \Vert \varphi_{i} \Vert _{\infty}} \Biggr). \end{aligned}$$

(3.11)

According to (3.11) and the definition of $\|(u,v)\|_{Y}$, we can derive

$$\Vert v\Vert _{X}\leq\bigl\Vert (u,v)\bigr\Vert _{Y}\leq A+2a \Biggl(\Vert \varphi_{0} \Vert _{\infty}+\Vert v\Vert _{X}\cdot {\sum _{i=1}^{n}\Vert \varphi_{i} \Vert _{\infty}} \Biggr). $$

By (3.8), we have

$$\|v\|_{X}\leq\frac{A+2a \|\varphi_{0} \|_{\infty}}{1-2a\sum_{i=1}^{n}\| \varphi_{i} \|_{\infty}}:=M. $$

From (3.11), we see that $\Omega_{1}$ is bounded.

Case 2. $\|(u,v)\|_{Y} \leq | {{v} }(0) |+b\|N_{2}u\|_{\infty}$. Similar to the above argument, we can also prove that $\Omega_{1}$ is bounded. Here, we omit it.

Case 3. $\|(u,v)\|_{Y} \leq | {{u} }(0) |+b\|N_{2}u\|_{\infty}$. From (3.10) and (H2), we obtain

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y}&\leq \bigl\vert {{u} }(0) \bigr\vert +b\Vert N_{2}u\Vert _{\infty}\\ &\leq A+a\Vert \varphi_{0} \Vert _{\infty}+\Vert v\Vert _{X}\cdot a{\sum_{i=1}^{n} \Vert \varphi_{i} \Vert _{\infty}} +b\bigl\Vert g \bigl(t,u,u',\ldots,u^{(n-1)}\bigr)\bigr\Vert _{\infty}\\ &\leq A+a\Vert \varphi_{0} \Vert _{\infty}+\Vert v\Vert _{X}\cdot a{\sum_{i=1}^{n} \Vert \varphi_{i} \Vert _{\infty}} +b \Biggl(\Vert \psi_{0} \Vert _{\infty}+\Vert u\Vert _{X}\cdot {\sum_{i=1}^{n}\Vert \psi_{i} \Vert _{\infty}} \Biggr) \\ &\leq A+a\Vert \varphi_{0} \Vert _{\infty}+b\Vert \psi_{0} \Vert _{\infty}+ \Biggl[a{\sum _{i=1}^{n}\Vert \varphi_{i} \Vert _{\infty}} + b{\sum_{i=1}^{n}\Vert \psi_{i} \Vert _{\infty}} \Biggr]\cdot\bigl\Vert (u,v)\bigr\Vert _{Y}. \end{aligned}$$

By (3.8), we get

$$\bigl\| (u,v)\bigr\| _{Y}\leq \Biggl[1-a{\sum_{i=1}^{n} \|\varphi_{i} \|_{\infty}}-b{\sum_{i=1}^{n} \|\psi_{i} \|_{\infty}} \Biggr]^{-1} \bigl(A+a\| \varphi_{0} \|_{\infty}+b\|\psi_{0} \|_{\infty}\bigr):=M, $$

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

Case 4. $\|(u,v)\|_{\infty}\leq | {{v} }(0) |+a\|N_{1}v\|_{\infty}$. We can prove that $\Omega_{1}$ is bounded too. The proof is similar to the case 2. Here, we omit it.

According the above arguments, we have proved that $\Omega_{1}$ is bounded.

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,

$$\begin{aligned}& I_{0^{+}}^{\alpha}f(t,d_{0} ,0,\ldots,0)|_{t=1} -\sum_{i=1}^{\infty}a_{i}I_{0^{+}}^{\alpha}f(t,d_{0} ,0,\ldots,0)|_{t=\xi_{i}}=0, \\& I_{0^{+}}^{\beta}g(t,c_{0},0,\ldots,0)|_{t=1} -\sum_{i=1}^{\infty}b_{i}I_{0^{+}}^{\beta}g(t,c_{0},0, \ldots,0)|_{t=\eta_{i}}=0. \end{aligned}$$

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

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

(3.12)

If $\lambda=0$, similar to the proof of the boundedness of $\Omega_{2}$, we have $|c_{0}|\leq A$ and $|d_{0}|\leq A$. If $\lambda=1$, we have $c_{0}=d_{0}=0$. If $\lambda\in(0,1)$, we also have $|c_{0}|\leq A$ and $|d_{0}|\leq A$. Otherwise, if $|c_{0}|>A$ or $|d_{0}|>A$, in view of the first part of (H3), we obtain

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

which contradict (3.12). Thus, $\Omega_{3}$ is bounded.

If the second part of (H₃) holds, then we can prove 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 be a bounded open set of Y, such that $\bigcup_{i=1}^{3}{\overline{\Omega}_{i}}\subset\Omega$. By Lemma 3.3, N is L-compact on Ω. Then by the above arguments, we get:

• (1) $Lu\neq\lambda Nu$, 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

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

Applying Theorem 2.1, we conclude that $L(u,v)=N(u,v)$ has at least one solution in $\operatorname{dom}L\cap\overline{\Omega}$. □

4 Example

Let us consider the following coupled system of fractional differential equations at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{2.4}u(t)=f(t,v,v',v''),\quad 0< t< 1, \\ D_{0^{+}}^{2.7}v(t)=g(t,u,u',u''),\quad 0< t< 1, \\ u'(0)= u''(0)=0=v'(0)=v''(1), \\ u(1)=\sum_{i=1}^{\infty}\frac{1}{2^{i}}u(\frac{1}{2i}),\qquad v(1)=\sum_{i=1}^{\infty}\frac{2}{3^{i}}v(\frac{1}{2i+1}), \end{cases} $$

(4.1)

where

$$\begin{aligned}& f(t,x_{1},x_{2},x_{3} )=\frac{t}{2} + \frac{1}{15}e^{-|x_{1}|}+\sin^{2} x_{2} + \cos x_{3}+1, \\& g(t,y_{1},y_{2},y_{3} )=t +\frac{\arctan y_{1}}{25}+ \frac{\sin(y_{2}+y_{3})}{30}+\frac{1}{2}. \end{aligned}$$

Corresponding to BVP (1.1), we have $\alpha=2.4$, $\beta=2.7$, $n=3$, $a= ({\Gamma(\alpha-n+2)})^{-1}= ({\Gamma(1.4)})^{-1}\approx1.13$, $b= ({\Gamma(\beta-n+2)})^{-1}=({\Gamma(1.7)})^{-1}\approx1.10$, $a_{i}=\frac{1}{2^{i}}$, $b_{i}=\frac{2}{3^{i}}$, $\xi_{i}=\frac{1}{2i}$, $\eta _{i}=\frac{1}{2i+1}$, $i=1,2,\ldots$ . We can get

$${\sum_{i=1}^{\infty}{{{a }_{i}} \xi_{i}^{\alpha}}}={\sum_{i=1}^{\infty }{{ \frac{1}{2^{i}}} {\frac{1}{(2i)^{2.4 }}}}}\approx0.106\neq1,\qquad {\sum _{i=1}^{\infty}{{{b }_{i}}\eta_{i}^{\beta}}}= {\sum_{i=1}^{\infty}{{\frac{2}{3^{i}}} \frac{1}{(2i+1)^{2.7 }}}} \approx0.037\neq1, $$

which implies (H1) holds. We choose $\varphi_{0}(t)=\frac{t}{2}+4$, $\psi _{0}(t)=t+2$, $\varphi_{i}=\psi_{i}=0$, $i=1,2,3$. Then we can verify (H2) and (3.8) hold. Take $A=12$, then the condition (H3) holds. Hence, from Theorem 3.1, BVP (4.1) has at least one solution.