Theory and Modern Applications

# A resonant boundary value problem for the fractional p-Laplacian equation

## Abstract

The purpose of this paper is to study the solvability of a resonant boundary value problem for the fractional p-Laplacian equation. By using the continuation theorem of coincidence degree theory, we obtain a new result on the existence of solutions for the considered problem.

## 1 Introduction

In this paper, we establish an existence theorem of solutions for the following resonant boundary value problem with p-Laplacian operator:

\begin{aligned} \left \{ \textstyle\begin{array}{l} {}_{0}^{c}D_{t}^{\beta}\phi_{p}({}_{0}^{c}D_{t}^{\alpha}x)=f(t,x,{}_{0}^{c}D_{t}^{\alpha}x),\quad t\in[0,1],\\ x(0)=0, \qquad {}_{0}^{c}D_{t}^{\alpha}x(0)={}_{0}^{c}D_{t}^{\alpha}x(1), \end{array}\displaystyle \right . \end{aligned}
(1.1)

where $$0<\alpha,\beta\leq1$$ are constants, $${}_{0}^{c}D_{t}^{\alpha}$$ is a Caputo fractional derivative, $$f:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}$$ is a continuous function, $$\phi_{p}:\mathbb{R}\rightarrow \mathbb{R}$$ is a p-Laplacian operator defined by

$$\phi_{p}(s)=|s|^{p-2}s\quad (s\neq0), \qquad\phi_{p}(0)=0,\quad p>1.$$

Obviously, $$\phi_{p}$$ is invertible and its inverse operator is $$\phi_{q}$$, where $$q>1$$ is a constant such that $$1/p+1/q=1$$.

Fractional calculus is a generalization of ordinary differentiation and integration, and fractional differential equations appear in various fields (see ). Recently, because of the intensive development of fractional calculus theory and its applications, the initial and boundary value problems (BVPs for short) of fractional differential equations have gained popularity (see  and the references therein).

In , by using the coincidence degree theory for Fredholm operators, the authors considered the existence of solutions for BVP (1.1). Notice that $${}_{0}^{c}D_{t}^{\beta}\phi_{p}({}_{0}^{c}D_{t}^{\alpha})$$ is nonlinear, and so it is not a Fredholm operator. Thus there is a gap in the proof of the main result, and we fix this gap in the present paper.

## 2 Preliminaries

For convenience of the reader, we will introduce some necessary basic knowledge about fractional calculus theory (see [2, 4]).

### Definition 2.1

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

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

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

### Definition 2.2

The Caputo fractional derivative of order $$\alpha>0$$ of a continuous function $$u:(0,+\infty)\rightarrow \mathbb{R}$$ is given by

\begin{aligned} {}_{0}^{c}D_{t}^{\alpha}u &={}_{0}I_{t}^{n-\alpha}\frac{\mbox{d}^{n}u}{\mbox{d}t^{n}} \\ &=\frac{1}{\Gamma(n-\alpha)} \int_{0}^{t}(t-s)^{n-\alpha-1}u^{(n)}(s)\,ds, \end{aligned}

where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined in $$(0,+\infty)$$.

### Lemma 2.1

See 

Let $$\alpha>0$$. Assume that $$u,{}_{0}^{c}D_{t}^{\alpha}u\in L([0,T],\mathbb{R})$$. Then the following equality holds:

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

where $$c_{i}\in{\mathbb{R}}$$, $$i=0,1,\ldots,n-1$$, here n is the smallest integer greater than or equal to α.

Next we present some notations and an abstract existence result (see ).

Let X, Y be real Banach spaces, $$L: \operatorname{dom}L\subset X\rightarrow Y$$ be a Fredholm operator with index zero, and $$P: X\rightarrow X$$, $$Q:Y\rightarrow Y$$ be projectors such that

\begin{aligned}& \operatorname{Im}P=\operatorname{Ker}L,\qquad \operatorname{Ker}Q= \operatorname{Im}L, \\& X=\operatorname{Ker}L\oplus\operatorname{Ker}P,\qquad Y=\operatorname{Im}L\oplus \operatorname{Im}Q. \end{aligned}

It follows that

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

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

If Ω is an open bounded subset of X such that $$\operatorname{dom}L\cap \overline{\Omega}\neq\varnothing$$, then the map $$N:X\rightarrow Y$$ will be called L-compact on Ω̅ if $$QN(\overline{\Omega})$$ is bounded and $$K_{P}(I-Q)N:\overline{\Omega}\rightarrow X$$ is compact.

### Lemma 2.2

See 

Let $$L:\operatorname{dom}L\subset X\rightarrow Y$$ be a Fredholm operator of index zero and $$N:X\rightarrow Y$$ be L-compact on Ω̅. Assume that the following conditions are satisfied:

1. (1)

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

2. (2)

$$Nx\notin\operatorname{Im}L$$ for every $$x\in\operatorname{Ker}L\cap\partial\Omega$$,

3. (3)

$$\operatorname{deg}(QN|_{\operatorname{Ker}L},\Omega\cap\operatorname{Ker}L,0)\neq0$$, where $$Q:Y\rightarrow Y$$ is a projection such that $$\operatorname{Im}L=\operatorname{Ker}Q$$.

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

In this paper, we let $$Z=C([0,1],\mathbb{R})$$ with the norm $$\|z\| _{\infty}=\max_{t\in[0,1]}|z(t)|$$ and take

$$X= \bigl\{ x=(x_{1},x_{2})^{\top}|x_{1},x_{2} \in Z \bigr\}$$

with the norm

$$\|x\|_{X}=\max\bigl\{ \|x_{1}\|_{\infty}, \|x_{2} \|_{\infty}\bigr\} .$$

By means of the linear functional analysis theory, we can prove that X is a Banach space.

## 3 Main result

We will establish the existence theorem of solutions for BVP (1.1).

### Theorem 3.1

Let $$f:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}$$ be continuous. Assume that

$$(H_{1})$$ :

there exist nonnegative functions $$a,b,c\in Z$$ such that

$$\big|f(t,u,v)\big|\leq a(t)+b(t)|u|^{p-1}+c(t)|v|^{p-1},\quad \forall(t,u,v)\in [0,1]\times\mathbb{R}^{2},$$
$$(H_{2})$$ :

there exists a constant $$B>0$$ such that

$$vf(t,u,v)>0\ (\textit{or } < 0), \quad\forall t\in[0,1],u\in\mathbb{R},|v|>B.$$

Then BVP (1.1) has at least one solution provided that

$$\gamma:=\frac{2}{\Gamma(\beta+1)} \biggl(\frac{\|b\|_{\infty}}{(\Gamma(\alpha+1))^{p-1}}+\|c\|_{\infty}\biggr)< 1.$$

Consider BVP of the linear differential system as follows:

\begin{aligned} \left \{ \textstyle\begin{array}{l} {}_{0}^{c}D_{t}^{\alpha}x_{1}=\phi_{q} (x_{2}), \quad t\in[0,1],\\ {}_{0}^{c}D_{t}^{\beta}x_{2}=f(t,x_{1},\phi_{q} (x_{2})),\quad t\in[0,1],\\ x_{1}(0)=0, \qquad x_{2}(0)=x_{2}(1). \end{array}\displaystyle \right . \end{aligned}
(3.1)

Obviously, if $$x=(x_{1},x_{2})^{\top}$$ is a solution of BVP (3.1), then $$x_{1}$$ must be a solution of BVP (1.1). Therefore, to prove BVP (1.1) has solutions, it suffices to show that BVP (3.1) has solutions.

Define the operator $$L:\operatorname{dom}L\subset X\rightarrow X$$ by

$$Lx=\binom{{}_{0}^{c}D_{t}^{\alpha}x_{1}}{{}_{0}^{c}D_{t}^{\beta}x_{2}},$$
(3.2)

where

$$\operatorname{dom}L= \bigl\{ x\in X|{}_{0}^{c}D_{t}^{\alpha}x_{1},{}_{0}^{c}D_{t}^{\beta}x_{2}\in Z, x_{1}(0)=0, x_{2}(0)=x_{2}(1) \bigr\} .$$

Let $$N:X\rightarrow X$$ be the Nemytskii operator defined by

$$Nx(t)=\binom{\phi_{q}(x_{2}(t))}{f(t,x_{1}(t),\phi_{q}(x_{2}(t)))}, \quad\forall t\in[0,1].$$
(3.3)

Then BVP (3.1) is equivalent to the following operator equation:

$$Lx=Nx,\quad x\in\operatorname{dom}L.$$

Now, in order to prove Theorem 3.1, we give some lemmas.

### Lemma 3.1

Let L be defined by (3.2), then

\begin{aligned}& \operatorname{Ker}L=\bigl\{ x\in X|x_{1}(t)=0, x_{2}(t)=c, \forall t\in[0,1],c\in \mathbb{R}\bigr\} , \end{aligned}
(3.4)
\begin{aligned}& \operatorname{Im}L= \bigl\{ y\in X|{_{0}I_{t}^{\beta}}y_{2}(1)=0 \bigr\} . \end{aligned}
(3.5)

### Proof

By Lemma 2.1, the equation $$Lx=0$$ has solutions

$$x_{1}(t)=c_{1},\quad\quad x_{2}(t)=c_{2},\quad c_{1},c_{2}\in\mathbb{R}.$$

Thus, from the boundary value condition $$x_{1}(0)=0$$, one has that (3.4) holds.

Let $$y\in\operatorname{Im}L$$, then there exists a function $$x\in\operatorname{dom}L$$ such that $$y_{2}={}_{0}^{c}D_{t}^{\beta}x_{2}$$. So, by Lemma 2.1, we have

$$x_{2}(t)=c+{}_{0}I_{t}^{\beta}y_{2}(t),\quad c\in\mathbb{R}.$$

Hence, from the boundary value condition $$x_{2}(0)=x_{2}(1)$$, we get (3.5).

On the other hand, suppose that $$y\in X$$ satisfies $${}_{0}I_{t}^{\beta}y_{2}(1)=0$$. Let $$x_{1}={}_{0}I_{t}^{\alpha}y_{1}$$, $$x_{2}={}_{0}I_{t}^{\beta}y_{2}(t)$$, then $$x=(x_{1},x_{2})^{\top}\in\operatorname{dom}L$$ and $$Lx=y$$. That is, $$y\in\operatorname{Im}L$$. The proof is complete. □

### Lemma 3.2

Let L be defined by (3.2), then L is a Fredholm operator of index zero. And the projectors $$P:X\rightarrow X$$, $$Q:X\rightarrow X$$ can be defined as

\begin{aligned}& Px(t)=\binom{0}{x_{2}(0)}, \quad\forall t\in[0,1], \\& Qy(t)=\binom{0}{\Gamma(\beta+1){}_{0}I_{t}^{\beta}y_{2}(1)}, \quad\forall t\in[0,1]. \end{aligned}

Furthermore, the operator $$K_{P}:\operatorname{Im}L\rightarrow\operatorname{dom}L\cap\operatorname{Ker}P$$ can be written as

$$K_{P}y=\binom{{}_{0}I_{t}^{\alpha}y_{1}}{{}_{0}I_{t}^{\beta}y_{2}}.$$

### Proof

For any $$y \in X$$, one has

\begin{aligned} Q^{2}y&=Q\binom{0}{\Gamma(\beta+1){}_{0}I_{t}^{\beta}y_{2}(1)} \\ &=\binom{0}{\Gamma(\beta+1){}_{0}I_{t}^{\beta}y_{2}(1)\cdot\Gamma(\beta +1){}_{0}I_{t}^{\beta}1(1)} \\ &=Qy. \end{aligned}
(3.6)

Let $$y^{*}=y-Qy$$, then we get from (3.6) that

\begin{aligned} {}_{0}I_{t}^{\beta}y^{*}_{2}(1)&={}_{0}I_{t}^{\beta}y_{2}(1)-{}_{0}I_{t}^{\beta}(Qy_{2}) (1) \\ &=\frac{1}{\Gamma(\beta+1)}\bigl((Qy_{2}) (t)-\bigl(Q^{2}y_{2} \bigr) (t)\bigr) \\ &=0, \end{aligned}

which yields $$y^{*}\in\operatorname{Im}L$$. So $$X=\operatorname{Im}L+\operatorname{Im}Q$$. Since $$\operatorname{Im}L\cap\operatorname{Im}Q=\{(0,0)^{\top}\}$$, we have $$X=\operatorname{Im}L\oplus \operatorname{Im}Q$$. Hence

$$\operatorname{dim}\operatorname{Ker}L=\operatorname{dim}\operatorname{Im}Q= \operatorname{codim}\operatorname{Im}L=1.$$

Thus L is a Fredholm operator of index zero.

For $$y\in\operatorname{Im}L$$, by the definition of operator $$K_{P}$$, we have

\begin{aligned} LK_{P}y&=\binom{{}_{0}^{c}D_{t}^{\alpha}{}_{0}I_{t}^{\alpha}y_{1}}{{}_{0}^{c}D_{t}^{\beta}{}_{0}I_{t}^{\beta}y_{2}} \\ &=y. \end{aligned}
(3.7)

On the other hand, for $$x\in\operatorname{dom}L\cap\operatorname{Ker}P$$, one has

$$x_{1}(0)=x_{2}(0)=x_{2}(1)=0.$$

Thus, from Lemma 2.1, we get

\begin{aligned} K_{P}Lx(t)&=\binom{{}_{0}I_{t}^{\alpha}{}_{0}^{c}D_{t}^{\alpha}x_{1}(t)}{{}_{0}I_{t}^{\beta}{}_{0}^{c}D_{t}^{\beta}x_{2}(t)} \\ &=\binom{x_{1}(t)-x_{1}(0)}{x_{2}(t)-x_{2}(0)} \\ &=x(t). \end{aligned}
(3.8)

Hence, combining (3.7) with (3.8), we know $$K_{P}= (L|_{\operatorname{dom}L\cap\operatorname{Ker}P} )^{-1}$$. The proof is complete. □

### Lemma 3.3

Let N be defined by (3.3). Assume $$\Omega\subset X$$ is an open bounded subset such that $$\operatorname{dom}L\cap \overline{\Omega}\neq\varnothing$$, then N is L-compact on Ω̅.

### Proof

From the continuity of $$\phi_{q}$$ and f, we obtain $$K_{P}(I-Q)N$$ is continuous in X and $$QN(\overline{\Omega})$$, $$K_{P}(I-Q)N(\overline{\Omega})$$ are bounded. Moreover, there exists a constant $$T>0$$ such that

$$\big\| (I-Q)Nx\big\| _{X}\leq T, \quad\forall x\in\overline{\Omega}.$$
(3.9)

Thus, in view of the Arzelà-Ascoli theorem, we need only to prove $$K_{P}(I-Q)N(\overline{\Omega})\subset X$$ is equicontinuous.

For $$0\leq t_{1}< t_{2}\leq1$$, $$x\in\overline{\Omega}$$, one has

\begin{aligned} &\big|K_{P}(I-Q)Nx(t_{2})-K_{P}(I-Q)Nx(t_{1})\big| \\ &\quad=\binom{{}_{0}I_{t}^{\alpha}((I-Q)Nx)_{1}(t_{2})-{}_{0}I_{t}^{\alpha}((I-Q)Nx)_{1}(t_{1})}{ {}_{0}I_{t}^{\beta}((I-Q)Nx)_{2}(t_{2})-{}_{0}I_{t}^{\beta}((I-Q)Nx)_{2}(t_{1})}. \end{aligned}

From (3.9), we have

\begin{aligned} &\big|{}_{0}I_{t}^{\alpha}\bigl((I-Q)Nx \bigr)_{1}(t_{2})-{}_{0}I_{t}^{\alpha}\bigl((I-Q)Nx\bigr)_{1}(t_{1})\big| \\ &\quad=\frac{1}{\Gamma(\alpha)} \biggl\vert \int_{0}^{t_{2}}(t_{2}-s)^{\alpha-1} \bigl((I-Q)Nx\bigr)_{1}(s)\,ds \\ &\qquad - \int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1} \bigl((I-Q)Nx\bigr)_{1}(s)\,ds \biggr\vert \\ &\quad\leq\frac{T}{\Gamma(\alpha)} \biggl\{ \int_{0}^{t_{1}}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1} \bigr]\,ds + \int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}\,ds \biggr\} \\ &\quad=\frac{T}{\Gamma(\alpha+1)}\bigl[t_{1}^{\alpha}-t_{2}^{\alpha} +2(t_{2}-t_{1})^{\alpha}\bigr]. \end{aligned}

Since $$t^{\alpha}$$ is uniformly continuous on $$[0,1]$$, we get $$(K_{P}(I-Q)N(\overline{\Omega}))_{1}\subset Z$$ is equicontinuous. A similar proof can show that $$(K_{P}(I-Q)N(\overline{\Omega}))_{2}\subset Z$$ is also equicontinuous. Hence, we obtain $$K_{P}(I-Q)N:\overline{\Omega }\rightarrow X$$ is compact. The proof is complete. □

Finally, we give the proof of Theorem 3.1.

### Proof of Theorem 3.1

Let

$$\Omega_{1}=\bigl\{ x\in\operatorname{dom}L\backslash \operatorname{Ker}L|Lx=\lambda Nx, \lambda \in(0,1)\bigr\} .$$

For $$x\in\Omega_{1}$$, we have $$x_{1}(0)=0$$ and $$Nx\in\operatorname{Im}L$$. So, by Lemma 2.1, we get

$$x_{1}={}_{0}I_{t}^{\alpha}{}_{0}^{c}D_{t}^{\alpha}x_{1}.$$

Thus one has

$$\big|x_{1}(t)\big|\leq\frac{1}{\Gamma(\alpha+1)}\big\| {}_{0}^{c}D_{t}^{\alpha}x_{1} \big\| _{\infty}, \quad\forall t\in[0,1].$$

That is,

\begin{aligned} \|x_{1}\|_{\infty}\leq\frac{1}{\Gamma(\alpha+1)} \big\| {}_{0}^{c}D_{t}^{\alpha}x_{1}\big\| _{\infty}. \end{aligned}
(3.10)

From $$Nx\in\operatorname{Im}L$$ and (3.5), we obtain

\begin{aligned} 0&={}_{0}I_{t}^{\beta}(Nx)_{2}(1) \\ &=\frac{1}{\Gamma(\beta)} \int_{0}^{1}(1-s)^{\beta-1}f \bigl(s,x_{1}(s),\phi_{q}\bigl(x_{2}(s)\bigr) \bigr)\,ds. \end{aligned}

Then, by the integral mean value theorem, there exists a constant $$\xi \in(0,1)$$ such that

$$f\bigl(\xi,x_{1}(\xi),\phi_{q}\bigl(x_{2}(\xi) \bigr)\bigr)=0.$$

So, by $$(H_{2})$$, we have $$|x_{2}(\xi)|\leq B^{p-1}$$. From Lemma 2.1, we get

$$x_{2}(t)=x_{2}(\xi)-{}_{0}I_{t}^{\beta}{}_{0}^{c}D_{t}^{\beta}x_{2}( \xi)+{}_{0}I_{t}^{\beta}{}_{0}^{c}D_{t}^{\beta}x_{2}(t),$$

which together with

$$\bigl\vert {}_{0}I_{t}^{\beta}{}_{0}^{c}D_{t}^{\beta}x_{2}(t) \bigr\vert \leq\frac{1}{\Gamma(\beta +1)} \bigl\Vert {}_{0}^{c}D_{t}^{\beta}x_{2} \bigr\Vert _{\infty}, \quad\forall t\in[0,1]$$

yields

$$\|x_{2}\|_{\infty}\leq B^{p-1}+ \frac{2}{\Gamma(\beta+1)} \bigl\Vert {}_{0}^{c}D_{t}^{\beta}x_{2} \bigr\Vert _{\infty}.$$
(3.11)

From $$Lx=\lambda Nx$$, one has

\begin{aligned}& {}_{0}^{c}D_{t}^{\alpha}x_{1}= \lambda\phi_{q}(x_{2}), \end{aligned}
(3.12)
\begin{aligned}& {}_{0}^{c}D_{t}^{\beta}x_{2}= \lambda f\bigl(t,x_{1},\phi_{q}(x_{2})\bigr). \end{aligned}
(3.13)

By (3.12), we have

$$\big\| {}_{0}^{c}D_{t}^{\alpha}x_{1} \big\| _{\infty}\leq\|x_{2}\|_{\infty}^{q-1},$$

which together with (3.10) yields

$$\|x_{1}\|_{\infty}\leq\frac{1}{\Gamma(\alpha+1)} \|x_{2}\|_{\infty}^{q-1}.$$
(3.14)

By (3.13) and $$(H_{1})$$, we obtain

$$\bigl\Vert {}_{0}^{c}D_{t}^{\beta}x_{2} \bigr\Vert _{\infty}\leq \|a\|_{\infty}+\|b\|_{\infty}\|x_{1}\|_{\infty}^{p-1}+\|c\|_{\infty}\|x_{2}\|_{\infty},$$

which together with (3.11) and (3.14) yields

\begin{aligned} \bigl\Vert {}_{0}^{c}D_{t}^{\beta}x_{2} \bigr\Vert _{\infty}&\leq\|a\|_{\infty}+\frac{\Gamma(\beta+1)\gamma}{2} \|x_{2}\|_{\infty} \\ &\leq\|a\|_{\infty}+\frac{\Gamma(\beta+1)\gamma B^{p-1}}{2}+\gamma \bigl\Vert {}_{0}^{c}D_{t}^{\beta}x_{2} \bigr\Vert _{\infty}. \end{aligned}
(3.15)

Since $$\gamma<1$$, we get from (3.15) that there exists a constant $$M_{0}>0$$ such that

$$\bigl\Vert {}_{0}^{c}D_{t}^{\beta}x_{2} \bigr\Vert _{\infty}\leq M_{0}.$$

Thus, combining (3.11) with (3.14), we have

\begin{aligned}& \|x_{2}\|_{\infty}\leq B^{p-1}+\frac{2M_{0}}{\Gamma(\beta+1)}:=M_{1}, \\& \|x_{1}\|_{\infty}\leq\frac{M_{1}^{q-1}}{\Gamma(\alpha+1)}:=M_{2}. \end{aligned}

Hence

$$\|x\|_{X}\leq\max\{M_{1}, M_{2}\}:=M,$$

which means $$\Omega_{1}$$ is bounded.

Let

$$\Omega_{2}=\{x\in\operatorname{Ker}L|Nx\in\operatorname{Im}L\}.$$

For $$x\in\Omega_{2}$$, we have $${}_{0}I_{t}^{\beta}(Nx)_{2}(1)=0$$ and $$x_{1}(t)=0$$, $$x_{2}(t)=c$$, $$c\in\mathbb{R}$$. Thus one has

$$\int_{0}^{1}(1-s)^{\beta-1}f\bigl(s,0, \phi_{q}(c)\bigr)\,ds=0,$$

which together with $$(H_{2})$$ yields $$|c|\leq B^{p-1}$$. Hence

$$\|x\|_{X}\leq\max\bigl\{ 0, B^{p-1}\bigr\} =B^{p-1},$$

which means $$\Omega_{2}$$ is bounded.

By $$(H_{2})$$, one has

$$\phi_{p}(v)f(t,u,v)>0,\quad \forall t\in[0,1], u\in \mathbb{R}, |v|>B$$
(3.16)

or

$$\phi_{p}(v)f(t,u,v)< 0, \quad\forall t\in[0,1], u\in \mathbb{R}, |v|>B.$$
(3.17)

When (3.16) is true, let

$$\Omega_{3}=\bigl\{ x\in\operatorname{Ker}L|\lambda x+(1-\lambda)QNx=0, \lambda\in [0,1]\bigr\} .$$

For $$x\in\Omega_{3}$$, we have $$x_{1}(t)=0$$, $$x_{2}(t)=c$$, $$c\in\mathbb{R}$$ and

$$\lambda c +(1-\lambda)\beta \int_{0}^{1}(1-s)^{\beta-1}f\bigl(s,0, \phi_{q}(c)\bigr)\,ds=0.$$
(3.18)

If $$\lambda=0$$, we get from (3.16) that $$|c|\leq B^{p-1}$$. If $$\lambda\in(0,1]$$, we assume $$|c|>B^{p-1}$$. Thus, by (3.16), we obtain

$$\lambda c^{2} +(1-\lambda)\beta \int_{0}^{1}(1-s)^{\beta-1}\phi_{p} \bigl(\phi_{q}(c)\bigr)f\bigl(s,0,\phi_{q}(c)\bigr)\,ds>0,$$

which contradicts (3.18). Hence, $$\Omega_{3}$$ is bounded.

When (3.17) is true, let

$$\Omega'_{3}=\bigl\{ x\in\operatorname{Ker}L|{-}\lambda x+(1-\lambda)QNx=0, \lambda\in [0,1]\bigr\} .$$

A similar proof can show $$\Omega'_{3}$$ is also bounded.

Set

$$\Omega=\bigl\{ x\in X|\|x\|_{X}< \max\bigl\{ M,B^{p-1}\bigr\} +1 \bigr\} .$$

Clearly, $$\Omega_{1}\cup\Omega_{2}\cup\Omega_{3}\subset\Omega$$ (or $$\Omega _{1}\cup\Omega_{2}\cup\Omega'_{3}\subset\Omega$$). It follows from Lemma 3.2 and 3.3 that L (defined by (3.2)) is a Fredholm operator of index zero and N (defined by (3.3)) is L-compact on Ω̅. Moreover, based on the above proof, the conditions (1) and (2) of Lemma 2.2 are satisfied. Define the operator $$H:\overline{\Omega}\times[0,1]\rightarrow X$$ by

$$H(x,\lambda)=\pm\lambda x+(1-\lambda)QNx.$$

Then, from the above proof, we have

$$H(x,\lambda)\neq0,\quad \forall x\in\partial\Omega\cap\operatorname{Ker}L.$$

Thus, by the homotopy property of degree, we get

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

Hence, condition (3) of Lemma 2.2 is also satisfied.

Therefore, by using Lemma 2.2, the operator equation $$Lx=Nx$$ has at least one solution in $$\operatorname{dom}L\cap\overline{\Omega}$$. Namely, BVP (1.1) has at least one solution in X. The proof is complete. □

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## Acknowledgements

This work was supported by the Natural Science Research Foundation of Colleges and Universities in Anhui Province (KJ2016A648).

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Correspondence to Bo Zhang.

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