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

DOI: 10.1186/s13662-017-1161-y

Accepted: 23 March 2017

Published: 4 April 2017

## 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.

### Keywords

resonant boundary value problem fractional p-Laplacian equation continuation theorem

34A08 34B15

## 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 [14]). 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 [515] and the references therein).

In [11], 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 [1]

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 [16]).

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 [16]

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. □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
School of Mathematical Sciences, Huaibei Normal University
(2)
Information College, Huaibei Normal University

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