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Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian

Abstract

This paper deals with the existence of infinitely many solutions for a class of impulsive fractional boundary value problems with p-Laplacian. Based on a variant fountain theorem, the existence of infinitely many nontrivial high or small energy solutions is obtained. In addition, two examples are worked out to illustrate the effectiveness of the main results.

1 Introduction

Consider the following nonlinear impulsive fractional boundary value problem (BVP, for short):

$$ \left \{ \textstyle\begin{array}{l} D_{T^{-}}^{\alpha}\Phi_{p}({}^{c}D_{0^{+}}^{\alpha }u(t))+ \vert u(t) \vert ^{p-2}u(t)=f(t,u(t)),\quad t\in[0,T],t\neq t_{j},\\ \Delta(D_{T^{-}}^{\alpha-1}\Phi_{p}({}^{c}D_{0^{+}}^{\alpha }u))(t_{j})=I_{j}(u(t_{j})),\quad j=1,2,\dots,m,\\ u(0)=u(T)=0, \end{array}\displaystyle \right . $$
(1.1)

where \(\alpha\in(\frac{1}{p}, 1]\), \(p>1\), \(\Phi_{p}(s)= |s|^{p-2}s\), \(D_{T^{-}}^{\alpha}\) represents the right Riemann–Liouville fractional derivative of order α and \({}^{c}D_{0^{+}}^{\alpha}\) represents the left Caputo fractional derivative of order α, \(0=t_{0}< t_{1}<\cdots<t_{m+1}=T\) and

$$\begin{gathered} \Delta\bigl(D_{T^{-}}^{\alpha-1}\Phi_{p} \bigl({}^{c}D_{0^{+}}^{\alpha }u\bigr)\bigr) (t_{j})=D_{T^{-}}^{\alpha-1}\Phi_{p} \bigl({}^{c}D_{0^{+}}^{\alpha }u\bigr) \bigl(t_{j}^{+} \bigr)-D_{T^{-}}^{\alpha-1}\Phi_{p}\bigl({}^{c}D_{0^{+}}^{\alpha }u \bigr) \bigl(t_{j}^{-}\bigr), \\ D_{T^{-}}^{\alpha-1}\Phi_{p}\bigl({}^{c}D_{0^{+}}^{\alpha }u \bigr) \bigl(t_{j}^{+}\bigr)=\lim_{t\rightarrow t_{j}^{+}}D_{T^{-}}^{\alpha-1} \Phi _{p}\bigl({}^{c}D_{0^{+}}^{\alpha}u\bigr) (t), \\ D_{T^{-}}^{\alpha-1}\Phi_{p}\bigl({}^{c}D_{0^{+}}^{\alpha }u \bigr) \bigl(t_{j}^{-}\bigr)=\lim_{t\rightarrow t_{j}^{-}}D_{T^{-}}^{\alpha-1} \Phi _{p}\bigl({}^{c}D_{0^{+}}^{\alpha}u\bigr) (t).\end{gathered} $$

\(f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}\) and \(I_{j}:\mathbb {R}\rightarrow\mathbb{R}\) are continuous.

Fractional differential equations have gained importance because of their numerous applications in various fields such as chemical physics, neural network model, signal processing and control, mechanics and engineering, fractal theory, and so on. For details, see [15] and the references therein. Recently, the existence and multiplicity of solutions for nonlinear fractional differential equations have been studied extensively by using the theory of coincidence degree, some fixed point theorems, upper-lower solution method, monotone iterative method, etc. [69]. It should be noted that the critical point theory and variational methods have proved to be a very effective approach in dealing with the existence and multiple solutions for fractional boundary value problems, see [1017].

On the other hand, the impulsive differential equation is used to describe the dynamics of processes in which sudden, discontinuous jumps occur. It has numerous applications in many fields such as population dynamics, ecology, optimal control, economics, and so on. For details, see [1821] and the references therein. Recently, many authors have studied the existence of solutions for impulsive fractional boundary value problem by using variational methods and critical point theory, see [2230].

For example, Heidarkhani et al. [26] and [28] studied the following impulsive nonlinear fractional boundary value problem:

$$ \left \{ \textstyle\begin{array}{l} D_{T^{-}}^{\alpha} ({}^{c}D_{0^{+}}^{\alpha}u(t))+a(t)u(t)=\lambda f(t,u(t))+h(u(t)),\quad t\in[0,T],t\neq t_{j},\\ \Delta(D_{T^{-}}^{\alpha-1}({}^{c}D_{0^{+}}^{\alpha}u))(t_{j})=\mu I_{j}(u(t_{j})),\quad j=1,2,\dots,m,\\ u(0)=u(T)=0. \end{array}\displaystyle \right . $$
(1.2)

Based on variational methods and critical point theory, they obtained the existence results of infinitely many classical solutions and three solutions for problem (1.2). In particular, Rodríguez-López and Tersian [22] established one and three solutions for problem (1.2) when \(h(u(t))\equiv0\). In [25], Heidarkhani and Salari obtained the existence of two and three weak solutions for a class of nonlinear impulsive fractional systems by applying variational methods.

Furthermore, the p-Laplacian often occurs in non-Newtonian fluid theory, nonlinear elastic mechanics, and so on. So, the impulsive fractional boundary value problem with p-Laplacian is worth considering. For instance, in [31], basing on the mountain pass theorem and minimax methods, the existence of multiple solutions for BVP(1.1) is obtained.

To the best of our knowledge, there are fewer results on the existence and multiplicity of solutions for impulsive fractional boundary value problem with p-Laplacian. Inspired by the above references, we apply variant fountain theorems to study the existence of infinitely many small or high energy solutions for BVP (1.1). The main new features presented in this paper are as follows. Firstly, the main results of this paper are different from those in the aforementioned references, and extend the results obtained in [31]. Secondly, the main tool of this paper is variant fountain theorems, which is different from the aforementioned papers. Thirdly, the assumed conditions in this paper are easier to verify than those in [31]. Finally, two examples are worked out to demonstrate the effectiveness of our results.

For convenience, we list the following assumptions.

\((H_{1})\) :

\(I_{j}(u)\) (\(j=1,2,\dots,m\)) are odd about u and satisfy \(\int^{u}_{0}I_{j}(s)\,ds\geq0 \) for all \(u\in\mathbb{R}\).

\((H_{2})\) :

There exist \(b_{j}>0\) and \(\gamma_{j}\in (p-1,+\infty)\) such that

$$\bigl\vert I_{j}(u) \bigr\vert \leq b_{j} \vert u \vert ^{\gamma_{j}}. $$
\((H_{3})\) :

There exist \(b_{j}>0\), \(\mu> p\), and \(\gamma_{j}\in (p-1,\mu-1)\) such that

$$\bigl\vert I_{j}(u) \bigr\vert \leq b_{j} \vert u \vert ^{\gamma_{j}},\qquad I_{j}(u)u\leq\mu \int ^{u}_{0}I_{j}(s)\,ds. $$
\((F_{1})\) :

There exist \(\eta\in(p-1,p)\) and \(b(t)\in L^{\frac {p}{p-\eta}}[0,T]\) with \(b(t)\geq0\) such that

$$\bigl\vert f(t,u) \bigr\vert \leq b(t) \bigl(1+ \vert u \vert ^{\eta-1}\bigr), \quad\forall(t,u) \in[0,T]\times \mathbb{R}. $$
\((F_{2})\) :

There exist \(\sigma\in(p-1,\eta)\) and \(d>0\) such that

$$\lim_{ \vert u \vert \rightarrow\infty}\frac{F(t,u)}{ \vert u \vert ^{\sigma}}>d, \quad\text{uniformly for } t\in[0,T], \text{ where } F(t,u)= \int_{0}^{u}f(t,s)\,ds. $$
\((F_{3})\) :

\(\lim_{|u|\rightarrow0}\frac{f(t,u)}{|u|^{p-1}}=0\), uniformly for \(t\in[0,T]\).

\((F_{4})\) :

\(F(t,u)\geq0\), \(\forall(t,u) \in[0,T]\times\mathbb{R}\).

\((F_{5})\) :

\(F(t,-u)= F(t,u)\), \(\forall(t,u) \in[0,T]\times \mathbb{R}\).

\((F_{6})\) :

There exist constants \(\theta_{1}>0\), \(\theta _{2}>0\), and \(q>p\) such that

$$\bigl\vert f(t,u) \bigr\vert \leq\theta_{1} \vert u \vert ^{p-1}+\theta_{2} \vert u \vert ^{q-1}\quad \text{for all }t \in[0,T], u\in\mathbb{R}. $$
\((F_{7})\) :

There exists \(\mu>p\) such that

$$-\mu F(t,u)+uf(t,u)\geq0 \quad\text{for all }t\in[0,T], u\in\mathbb{R}. $$

Here are our main results.

Theorem 1.1

Assume that \((H_{1})\)\((H_{2})\) and \((F_{1})\)\((F_{5})\) hold. Then BVP (1.1) possesses infinitely many small energy solutions \(u_{k}\in E\setminus\{0\}\) satisfying

$$\begin{gathered} \frac{1}{p} \int_{0}^{T}\bigl( \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u_{k}(t) \bigr\vert ^{p}+ \bigl\vert u_{k}(t) \bigr\vert ^{p} \bigr)\,dt\\\quad{}+\sum^{m}_{j=1} \int ^{u_{k}(t_{j})}_{0}I_{j}(s)\,ds- \int_{0}^{T}F\bigl(t,u_{k}(t)\bigr)\, dt \rightarrow0^{-} \quad\textit{as } k\rightarrow\infty.\end{gathered} $$

Theorem 1.2

Assume that \((H_{1})\), \((H_{3})\) and \((F_{4})\)\((F_{7})\) hold. Then BVP (1.1) possesses infinitely many high energy solutions \(u^{k}\in E\setminus\{0\}\) satisfying

$$\begin{gathered} \frac{1}{p} \int_{0}^{T}\bigl( \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u^{k}(t) \bigr\vert ^{p}+ \bigl\vert u^{k}(t) \bigr\vert ^{p} \bigr)\,dt\\\quad{}+\sum^{m}_{j=1} \int ^{u^{k}(t_{j})}_{0}I_{j}(s)\,ds- \int_{0}^{T}F\bigl(t,u^{k}(t)\bigr)\, dt \rightarrow\infty \quad\textit{as } k\rightarrow\infty.\end{gathered} $$

The rest of this paper is organized as follows. Section 2 contains some preliminary results. In Sect. 3, we apply variant fountain theorems to prove the existence of infinitely many small or high energy solutions for BVP (1.1). In Sect. 4, two examples are presented to illustrate the main results.

2 Preliminaries

To obtain multiple solutions for BVP (1.1), it is necessary to introduce several definitions and preliminary lemmas which are used further in this paper.

Let \(\mathrm{AC}[a,b]\) be the space of absolutely continuous functions on \([a,b]\).

Definition 2.1

([10])

Let f be a function defined on \([a, b]\) and \(0<\alpha\leq1\). The left and right Riemann–Liouville fractional integrals of order α for the function f are defined by

$$\begin{gathered} D_{a^{+}}^{-\alpha}f(t)= \frac{1}{\Gamma(\alpha)} \int _{a}^{t}(t-s)^{\alpha-1}f(s)\,ds,\quad t \in[a,b], \\ D_{b^{-}}^{-\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int _{t}^{b}(s-t)^{\alpha-1}f(s)\,ds,\quad t\in[a,b],\end{gathered} $$

while the right-hand side is pointwise defined on \([a,b]\).

Definition 2.2

([10])

Let \(f\in \mathrm{AC}[a,b]\) and \(0<\alpha\leq1\). The left and right Riemann–Liouville fractional derivatives of order α for the function f are defined by

$$\begin{gathered} D_{a^{+}}^{\alpha}f(t)= \frac{d}{\,dt}D_{a^{+}}^{\alpha-1}f(t)= \frac {1}{\Gamma(1-\alpha)} \frac{d}{\,dt} \int_{a}^{t}(t-s)^{-\alpha }f(s)\,ds,\quad t\in[a,b], \\ D_{b^{-}}^{\alpha}f(t)=-\frac{d}{\,dt}D_{b^{-}}^{\alpha -1}f(t)=- \frac{1}{\Gamma(1-\alpha)}\frac{d}{\,dt} \int _{t}^{b}(s-t)^{-\alpha}f(s)\,ds, \quad t \in[a,b].\end{gathered} $$

Definition 2.3

([10])

Let \(f\in \mathrm{AC}[a,b]\) and \(0<\alpha\leq1\). The left and right Caputo fractional derivatives of order α for the function f are defined by

$$\begin{gathered} {}^{c}D_{a^{+}}^{\alpha}f(t)= D_{a^{+}}^{\alpha-1}f'(t)= \frac {1}{\Gamma(1-\alpha)} \int_{a}^{t}(t-s)^{-\alpha}f'(s) \,ds,\quad t\in[a,b], \\ {}^{c}D_{b^{-}}^{\alpha}f(t)=-D_{b^{-}}^{\alpha-1}f'(t)=- \frac {1}{\Gamma(1-\alpha)} \int_{t}^{b}(s-t)^{-\alpha}f'(s) \,ds,\quad t\in[a,b].\end{gathered} $$

In particular, when \(\alpha=1\), we have \({}^{c}D_{a^{+}}^{1}f(t)=f'(t)\) and \({}^{c}D_{b^{-}}^{1}f(t)=-f'(t)\).

Lemma 2.4

([32])

  1. (1)

    If \(u \in L^{p}[a,b]\), \(v \in L^{p}[a,b]\), and \(p\geq1\), \(q\geq1\), \(\frac{1}{p}+\frac{1}{q}\leq1+\alpha\) or \(p\neq1\), \(q\neq 1\), \(\frac{1}{p}+\frac{1}{q}= 1+\alpha\), then

    $$\int^{b}_{a}\bigl(D_{a^{+}}^{-\alpha}u(t) \bigr)v(t)\,dt= \int ^{b}_{a}u(t) \bigl(D_{b^{-}}^{-\alpha}v(t) \bigr)\,dt. $$
  2. (2)

    If \(0<\alpha\leq1\), \(u\in \mathrm{AC}[a,b]\), and \(v \in L^{p}[a,b]\) (\(1\leq p<\infty\)), then

    $$\int^{b}_{a}u(t) \bigl({}^{c}D_{a^{+}}^{\alpha}v(t) \bigr)\,dt=D_{b^{-}}^{\alpha -1}u(t)v(t)\big|^{t=b}_{t=a}+ \int^{b}_{a}D_{b^{-}}^{\alpha}u(t)v(t) \,dt. $$

Denote

$$\Vert u \Vert _{L^{p}}=\biggl( \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\,dt\biggr)^{\frac{1}{p}}, \qquad \Vert u \Vert _{\infty}=\max _{t\in[0,T]} \bigl\vert u(t) \bigr\vert . $$

Definition 2.5

Let \(0<\alpha\leq1\), \(1< p<\infty\). The fractional derivative space \(E_{0} ^{\alpha,p}\) is defined by the closure of \(C_{0}^{\infty}([0,T],\mathbb{R})\) with respect to the norm

$$ \Vert u \Vert _{E_{0} ^{\alpha,p}}=\biggl( \int_{0}^{T}\biggl( \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u(t) \bigr\vert ^{p} \,dt+ \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\biggr)\,dt\biggr)^{\frac{1}{p}},\quad \forall u\in E_{0} ^{\alpha,p}. $$
(2.1)

Remark 2.6

  1. (1)

    \(E_{0} ^{\alpha,p}\) is a reflexive and separable Banach space.

  2. (2)

    For any \(u\in E_{0} ^{\alpha,p}\), we have \(u\in L^{p}([0,T],\mathbb{R})\), \({}^{c}D_{a^{+}}^{\alpha}u\in L^{p}([0,T],\mathbb{R})\), and \(u(0)=u(T)=0\).

Lemma 2.7

([32])

Let \(0<\alpha\leq1\) and \(1< p<\infty\). For any \(u\in E_{0} ^{\alpha,p}\), we have

$$ \Vert u \Vert _{L^{p}}\leq\frac{T^{\alpha}}{\Gamma(\alpha+1)} \bigl\Vert {}^{c}D_{0^{+}}^{\alpha}u \bigr\Vert _{L^{p}}. $$
(2.2)

In addition, for \(\frac{1}{p}<\alpha\leq1\) and \(\frac{1}{p}+\frac {1}{q}= 1\), we have

$$ \Vert u \Vert _{\infty}\leq\frac{T^{\alpha-\frac{1}{p}}}{\Gamma(\alpha )(\alpha q-q +1)^{\frac{1}{q}}} \bigl\Vert {}^{c}D_{0^{+}}^{\alpha}u \bigr\Vert _{L^{p}}. $$
(2.3)

Remark 2.8

According to Lemma 2.7, it is easy to see that the norm of \(E_{0} ^{\alpha,p}\) defined in (2.1) is equivalent to the following norm:

$$ \Vert u \Vert _{\alpha,p}=\biggl( \int_{0}^{T} \bigl\vert {}^{c}D_{0^{+}}^{\alpha}u(t) \bigr\vert ^{p}\, dt\biggr)^{\frac{1}{p}},\quad \forall u\in E_{0} ^{\alpha,p}. $$
(2.4)

Lemma 2.9

([32])

Let \(\frac{1}{p}<\alpha\leq 1\). If the sequence \(\{u_{k}\}\) converges weakly to u in \(E_{0} ^{\alpha,p}\), i.e., \(u_{k}\rightharpoonup u\), then \({u_{k}}\rightarrow u\) in \(C[0,T]\), i.e., \(\|u-u_{k}\|_{\infty}\rightarrow0\) as \(k\rightarrow\infty\).

In the following, we denote \(E=E_{0} ^{\alpha,p}\), \(\|u\|=\|u\|_{E_{0} ^{\alpha,p}}\), \(\|u\|_{p}=\|u\|_{L^{p}}\) for convenience.

Definition 2.10

A function

$$u\in\biggl\{ u\in \mathrm{AC}[0,T]: \int_{t_{j}}^{t_{j+1}}\bigl( \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u(t) \bigr\vert ^{p}+ \bigl\vert u(t) \bigr\vert ^{p}\bigr)\,dt< \infty,j=0,1,\ldots,m \biggr\} $$

is called a classical solution of BVP (1.1) if

  1. (1)

    u satisfies (1.1).

  2. (2)

    The limits \(D_{T^{-}}^{\alpha-1}\Phi _{p}({}^{c}D_{0^{+}}^{\alpha}u)(t_{j}^{+})\), \(D_{T^{-}}^{\alpha-1}\Phi _{p}({}^{c}D_{0^{+}}^{\alpha}u)(t_{j}^{-})\) exist.

Definition 2.11

A function \(u\in E\) is a weak solution of BVP (1.1) if

$$ \begin{aligned}[b] & \int_{0}^{T} \bigl\vert {}^{c}D_{0^{+}}^{\alpha}u(t) \bigr\vert ^{p-2} \bigl({}^{c}D_{0^{+}}^{\alpha}u(t) \bigr) \bigl({}^{c}D_{0^{+}}^{\alpha}v(t)\bigr)\,dt+ \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p-2}u(t)v(t)\,dt \\ &\quad+\sum^{m}_{j=1}I_{j}\bigl(u(t_{j})\bigr)v(t_{j})- \int_{0}^{T}f\bigl(t,u(t)\bigr)v(t)\, dt=0,\quad \forall u\in E.\end{aligned} $$
(2.5)

The energy functional \(J: E\rightarrow\mathbb{R}\) associated with BVP (1.1) is defined by

$$\begin{aligned}[b] J(u)={}&\frac{1}{p} \int_{0}^{T}\bigl( \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u(t) \bigr\vert ^{p}+ \bigl\vert u(t) \bigr\vert ^{p}\bigr)\,dt\\&+\sum ^{m}_{j=1} \int^{u(t_{j})}_{0}I_{j}(s)\, ds- \int_{0}^{T}F\bigl(t,u(t)\bigr)\,dt , \quad\forall u\in E.\end{aligned} $$
(2.6)

It is easy to see that \(J\in C^{1}(E,\mathbb{R})\), and

$$ \begin{aligned}[b] \bigl(J'(u), v\bigr)={}& \int_{0}^{T} \bigl\vert {}^{c}D_{0^{+}}^{\alpha}u(t) \bigr\vert ^{p-2} \bigl({}^{c}D_{0^{+}}^{\alpha}u(t) \bigr) \bigl({}^{c}D_{0^{+}}^{\alpha}v(t)\bigr)\,dt+ \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p-2}u(t)v(t)\,dt \\ &+\sum^{m}_{j=1}I_{j}\bigl(u(t_{j})\bigr)v(t_{j})- \int_{0}^{T}f\bigl(t,u(t)\bigr)v(t)\, dt \\ ={}&0,\quad \forall u\in E.\end{aligned} $$
(2.7)

Moreover, the critical points of J correspond to the weak solutions of BVP (1.1).

Lemma 2.12

([31])

If \(u\in E \) is a weak solution of BVP (1.1), then u is a classical solution of BVP (1.1).

To prove our main results, we need the following two variant fountain theorems in [33].

Let X be a Banach space with the norm \(\|\cdot\|\) and \(X=\overline {\bigoplus_{j\in\mathbb{N}}X_{j}}\) with \(\operatorname{dim} X_{j}<\infty\) for each \(j\in\mathbb{N}\). Set \(W_{k}=\bigoplus_{j=1}^{k}X_{j}\), \(Z_{k}=\overline{\bigoplus_{j=k}^{\infty}X_{j}}\), \(B_{k}=\{u\in W_{k}: \|u\|\leq\rho_{k}\}\), \(S_{k}=\{u\in Z_{k}: \|u\|=r_{k}\}\), where \(\rho_{k}>r_{k}>0\).

Consider a family of \(C^{1}\) functionals \(J_{\lambda}: X\rightarrow \mathbb{R}\) defined by

$$J_{\lambda}(u)=A(u)-\lambda B(u), \quad\lambda\in[1,2], $$

where \(A, B: X\rightarrow\mathbb{R}\) are two functions.

Lemma 2.13

([33])

Assume that the functional \(J_{\lambda}\) defined above satisfies:

\((B_{1})\) :

\(J_{\lambda}\) maps bounded sets into bounded sets uniformly for \(\lambda\in[1,2]\), and \(J_{\lambda}(-u)=J_{\lambda }(u)\) for all \((\lambda,u)\in[1,2]\times X\);

\((B_{2})\) :

\(B(u)\geq0\) for all \(u\in X\), \(B(u)\rightarrow\infty \) as \(\|u\| \rightarrow\infty\) on any finite dimensional subspace of X;

\((B_{3})\) :

there exist \(\rho_{k}>r_{k}>0\) such that

$$a_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|=\rho_{k}}J_{\lambda}(u)\geq 0,\qquad b_{k}(\lambda)=\max_{u\in W_{k},\|u\|=r_{k}}J_{\lambda}(u)< 0,\quad \forall\lambda\in[1,2] $$

and

$$d_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|\leq\rho_{k}}J_{\lambda }(u) \rightarrow0 \quad \textit{as } k \rightarrow\infty\textit{ uniformly for }\lambda\in[1,2]. $$

Then there exist \(\lambda_{n}\rightarrow1\), \(u_{n}(\lambda_{n})\in W_{n}\) such that

$$J'_{\lambda_{n}}|_{W_{n}}\bigl(u(\lambda_{n}) \bigr)=0, \qquad J_{\lambda _{n}}\bigl(u(\lambda_{n})\bigr)\rightarrow c_{k} \quad \textit{as } n\rightarrow\infty, $$

where \(c_{k}\in[d_{k}(2), b_{k}(1)]\). In particular, if \(\{u(\lambda _{n})\}\) has a convergent subsequence for every k, then \(J_{1}\) has infinitely many nontrivial critical points \(\{u_{k}\}\in X\setminus\{ 0\}\) satisfying \(J_{1}(u_{k})\rightarrow0^{-}\) as \(k\rightarrow \infty\).

Lemma 2.14

([33])

Assume that the functional \(J_{\lambda}\) defined above satisfies

\((A_{1})\) :

\(J_{\lambda}\) maps bounded sets into bounded sets uniformly for \(\lambda\in[1,2]\), and \(J_{\lambda}(-u)=J_{\lambda }(u)\) for all \((\lambda,u)\in[1,2]\times X\);

\((A_{2})\) :

\(B(u)\geq0\) for all \(u\in X\), \(A(u)\rightarrow\infty \) or \(B(u)\rightarrow\infty\) as \(\|u\| \rightarrow\infty\); or

\((A_{3})\) :

\(B(u)\leq0\) for all \(u\in X\), \(B(u)\rightarrow-\infty \) as \(\|u\| \rightarrow\infty\);

\((A_{4})\) :

there exist \(\rho_{k}>r_{k}>0\) such that

$$b_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|=r_{k}}J_{\lambda }(u)>a_{k}( \lambda)=\max_{u\in W_{k},\|u\|=\rho_{k}}J_{\lambda}(u),\quad \forall\lambda\in[1,2]. $$

Then

$$b_{k}(\lambda)\leq c_{k}(\lambda)=\inf _{\gamma\in\Gamma_{k}}\max_{u\in B_{k}}J_{\lambda}\bigl( \gamma(u)\bigr),\quad \forall\lambda\in[1,2], $$

where \(\Gamma_{k}= \{ \gamma\in C( B_{k}, X ): \gamma\textit{ is odd}, \gamma|_{\partial B_{k}}=\mathrm{id}\}\). Moreover, for almost every \(\lambda \in[1,2]\), there exists a sequence \(\{u_{n}^{k}(\lambda)\}\) such that

$$\sup_{n} \bigl\Vert u_{n}^{k}( \lambda) \bigr\Vert < \infty,\qquad J'_{\lambda }\bigl(u_{n}^{k}( \lambda)\bigr)\rightarrow0\quad \textit{and}\quad J_{\lambda}\bigl(u_{n}^{k}( \lambda )\bigr)\rightarrow c_{k}(\lambda) \quad \textit{as } n\rightarrow\infty. $$

As E is a separable and reflexive Banach space, then there exist \(\{ e_{j}\}_{j=1}^{\infty}\subset E\) and \(\{e_{j}^{\ast}\}_{j=1}^{\infty }\subset E^{\ast}\) such that

$$\begin{gathered} E=\overline{\operatorname{span}\{e_{j}\}} ,\qquad E^{\ast}=\overline{\operatorname{span}\bigl\{ e_{j}^{\ast}\bigr\} }, \\ \bigl(e_{i}^{\ast},e_{i}\bigr)=1,\qquad \bigl(e_{j}^{\ast},e_{i}\bigr)=0\quad (i\neq j).\end{gathered} $$

Define \(X_{j}= \operatorname{span}\{e_{j}\}\), \(W_{k}=\bigoplus_{j=1}^{k}X_{j}\), \(Z_{k}=\overline{\bigoplus_{j=k}^{\infty}X_{j}}\). In order to apply Lemma 2.13 and Lemma 2.14 to prove the existence of infinitely many solutions of BVP (1.1), we define A, B, and \(J_{\lambda}\) on a fractional derivative space E by

$$A(u)=\frac{1}{p} \Vert u \Vert ^{p}+\sum ^{m}_{j=1} \int^{u(t_{j})}_{0}I_{j}(s)\, ds,\qquad B(u)= \int_{0}^{T}F\bigl(t,u(t)\bigr)\,dt, $$

and

$$\begin{aligned} J_{\lambda}(u)&=A(u)-\lambda B(u)\\&=\frac{1}{p} \Vert u \Vert ^{p}+\sum^{m}_{j=1} \int^{u(t_{j})}_{0}I_{j}(s)\,ds-\lambda \int _{0}^{T}F\bigl(t,u(t)\bigr)\,dt,\quad\forall u\in E, \lambda\in[1,2].\end{aligned} $$

3 Proof of the main results

In order to complete the proof of our main results, it is necessary to give the following two lemmas. Because of using similar arguments to the proofs of Lemma 3.2 and Lemma 3.5 in [15], we omit the proving processes for convenience.

Lemma 3.1

Let H be any finite dimensional subspace of E. Then there exists a constant \(\varepsilon_{0}>0\) such that

$$ \operatorname{meas} \bigl\{ t\in[0,T]: \bigl\vert u(t) \bigr\vert \geq \varepsilon_{0} \Vert u \Vert \bigr\} \geq\varepsilon _{0},\quad \forall u\in H\setminus\{0\}. $$
(3.1)

Lemma 3.2

Let \(\alpha_{r}(k)=\sup_{u\in Z_{k},\|u\|=1}\|u\|_{r}\) with \(r\geq p\). Then \(\alpha_{r}(k)\rightarrow0\) as \(k\rightarrow\infty\).

Now we are ready to prove Theorem 1.1 and Theorem 1.2.

Proof of Theorem 1.1

By \((F_{1})\) and \((F_{3})\), for \(\forall\varepsilon>0\), there exists \(\delta_{\varepsilon} >0\) such that

$$ \bigl\vert F(t,u) \bigr\vert \leq\varepsilon \vert u \vert ^{p}+\delta_{\varepsilon}b(t) \vert u \vert ^{\eta }. $$
(3.2)

Combining (3.2), \((H_{2})\), and Lemma 2.7, it is easily seen that \(J_{\lambda}\) maps bounded sets into bounded sets uniformly for \(\lambda\in[1,2]\). It follows from \((H_{1})\) and \((F_{5})\) that \(J_{\lambda }(-u)=J_{\lambda}(u)\) for all \((\lambda,u)\in[1,2]\times E\). Thus, condition \((B_{1})\) holds.

Next, we verify condition \((B_{2})\).

According to \((F_{4})\), \(B(u)\geq0\) is obvious. By \((F_{2})\), there exists \(M>0\) such that

$$ F(t,u)\geq d \vert u \vert ^{\sigma} \quad\text{for all }\vert u \vert >M. $$
(3.3)

Assume that H is a finite dimensional subspace of E. According to Lemma 3.1, there exists \(\varepsilon_{0}>0\) such that (3.1) holds. Then

$$\operatorname{meas}({D}_{u})\geq\varepsilon_{0},\quad \forall u\in H\setminus\{0\}, $$

where \({D}_{u} = \{t\in[0,T]:|u(t)|\geq\varepsilon_{0}\|u\|\}\). Hence, for any \(u\in H\) with \(\|u\|\geq\frac{M}{\varepsilon_{0}}\), by (3.3), we get

$$\begin{aligned} B(u)&= \int_{0}^{T}F\bigl(t,u(t)\bigr)\,dt \\ &\geq \int_{{D}_{u}}d \bigl\vert u(t) \bigr\vert ^{\sigma}\,dt \\ &\geq d\varepsilon_{0}^{\sigma} \Vert u \Vert ^{\sigma}\operatorname{meas}( {D}_{u}) \\ &\geq d\varepsilon_{0}^{1+\sigma} \Vert u \Vert ^{\sigma}.\end{aligned} $$

This means that \(B(u)\rightarrow\infty\) as \(\|u\|\rightarrow\infty \) on any finite dimensional subspace. Hence, condition \((B_{2})\) holds.

In the end, we claim that condition \((B_{3})\) holds.

For \(u\in Z_{k}\), by (3.2), Hölder’s inequality, and \((H_{1})\), we have

$$\begin{aligned} J_{\lambda}(u)&\geq\frac{1}{p} \Vert u \Vert ^{p}- \int_{0}^{T}\bigl(\varepsilon \vert u \vert ^{p}+\delta_{\varepsilon}b(t) \vert u \vert ^{\eta} \bigr)\,dt \\ &\geq\frac{1}{2p} \Vert u \Vert ^{p}-\delta \bigl\Vert b(t) \bigr\Vert _{\frac{p}{p-\eta}} \Vert u \Vert ^{\eta}_{p}, \end{aligned} $$

where we choose \(\varepsilon= \frac{1}{2p}\). According to the definition of \(\alpha_{r}(k)\) in Lemma 3.2, we have

$$\Vert u \Vert ^{\eta}_{p}\leq\alpha_{p}^{\eta}(k) \Vert u \Vert ^{\eta}, \quad\forall u\in Z_{k}. $$

Hence,

$$J_{\lambda}(u)\geq\frac{1}{2p} \Vert u \Vert ^{p}- \delta \bigl\Vert b(t) \bigr\Vert _{\frac {p}{p-\eta}}\alpha_{p}^{\eta}(k) \Vert u \Vert ^{\eta}. $$

Choose \(\rho_{k}=(4p\delta\|b(t)\|_{\frac{p}{p-\eta}}\alpha _{p}^{\eta}(k))^{\frac{1}{p-\eta}}\). Then \(\rho_{k}\rightarrow 0^{+}\) as \(k\rightarrow\infty\). Therefore,

$$a_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|=\rho_{k}}J_{\lambda}(u)\geq \frac{1}{4p}\rho_{k}^{p}>0. $$

In addition, for \(\lambda\in[1,2]\) and \(u\in Z_{k}\) with \(\|u\|\leq \rho_{k}\), we have

$$J_{\lambda}(u)\geq-\delta \bigl\Vert b(t) \bigr\Vert _{\frac{p}{p-\eta}} \alpha _{p}^{\eta}(k)\rho_{k}^{\eta} \rightarrow0^{+},\quad k\rightarrow\infty. $$

So,

$$d_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|\leq\rho_{k}}J_{\lambda }(u) \rightarrow0\quad\text{as }k \rightarrow\infty. $$

By \((F_{1})\)\((F_{3})\), we have

$$ F(t,u)\geq d \vert u \vert ^{\sigma}-\varepsilon \vert u \vert ^{p}-\delta_{\varepsilon }b(t) \vert u \vert ^{\eta}. $$
(3.4)

If \(u\in W_{k}\), by the equivalence of any norm in a finite dimensional space, (3.4), \((H_{2})\), Lemma 2.7, and Hölder’s inequality, we get

$$\begin{aligned} J_{\lambda}(u)&=\frac{1}{p} \Vert u \Vert ^{p}+\sum ^{m}_{j=1} \int ^{u(t_{j})}_{0}I_{j}(s)\,ds-\lambda \int_{0}^{T}F\bigl(t,u(t)\bigr)\,dt \\ &\leq\frac{1}{p} \Vert u \Vert ^{p}+\sum ^{m}_{j=1}\frac{b_{j}}{\gamma _{j}+1} \bigl\vert u(t_{j}) \bigr\vert ^{\gamma_{j}+1}-d \int_{0}^{T} \vert u \vert ^{\sigma}\, dt+\varepsilon \int_{0}^{T} \vert u \vert ^{p}\,dt+ \delta_{\varepsilon} \int _{0}^{T}b(t) \vert u \vert ^{\eta} \,dt \\ &\leq \Vert u \Vert ^{p}+ M \Vert u \Vert ^{\gamma_{j}+1}- \delta_{1} \Vert u \Vert ^{\sigma }+\delta \bigl\Vert b(t) \bigr\Vert _{\frac{p}{p-\eta}} \Vert u \Vert ^{\eta}.\end{aligned} $$

Choose \(r_{k}>0\) small enough and \(r_{k}<\rho_{k}\) such that

$$b_{k}(\lambda)=\max_{u\in W_{k},\|u\|=r_{k}}J_{\lambda}(u)< 0. $$

This guarantees that condition \((B_{3})\) holds.

Consequently, by Lemma 2.13, for every \(k\in\mathbb{N}\), there exist \(\lambda_{n}\rightarrow1\), \(u_{n}(\lambda_{n})\in W_{n}\) such that

$$J'_{\lambda_{n}}|_{W_{n}}\bigl(u(\lambda_{n}) \bigr)=0,\qquad J_{\lambda _{n}}\bigl(u(\lambda_{n})\bigr)\rightarrow c_{k}\in\bigl[d_{k}(2), b_{k}(1)\bigr] \quad\text{as }n \rightarrow\infty. $$

For simplicity, denote \(u(\lambda_{n})\) by \(u_{n}\). Now we show that \(\{u_{n}\}\) has a strong convergent subsequence for every \(k\in\mathbb{N}\). In fact, by (3.2), \((H_{1})\), Hölder’s inequality, and Lemma 2.7, we get

$$\begin{aligned} \Vert u_{n} \Vert ^{p}&= pJ_{\lambda_{n}}(u_{n})-p \sum^{m}_{j=1} \int ^{u_{n}(t_{j})}_{0}I_{j}(s)\,ds+p \lambda_{n} \int_{0}^{T}F(t,u_{n})\, dt \\ &\leq pc_{k}+p\lambda_{n}\delta_{2} \Vert u_{n} \Vert ^{p}+p\lambda _{n}\delta \bigl\Vert b(t) \bigr\Vert _{\frac{p}{p-\eta}} \Vert u_{n} \Vert ^{\eta}.\end{aligned} $$

This means that \(\{u_{n}\}\) is bounded in E. Without loss of generality, we may assume \({ u_{n}\rightharpoonup u}\) in E. Since \(\{e_{j}\}\) is a completely orthonormal basis of E, \(W_{n}= L(e_{1}, e_{2},\ldots, e_{n})\), \(u=\sum_{j=1}^{\infty}(e_{j},u)e_{j}\). Let \(P_{n}: E\mapsto W_{n}\) be the orthogonal projection operator. We know that \(P_{n}u= \sum_{j=1}^{n}(e_{j},u)e_{j}\) and \(P_{n}u\rightarrow u \) in E as \(n\rightarrow\infty\). Therefore \(u_{n}-P_{n}u \rightharpoonup0 \) in E as \(n\rightarrow \infty\). Moreover, it follows from \(J'_{1}(u)\in E^{\ast}\) that

$$ \bigl( J'_{1}(u), u_{n}-P_{n}u \bigr)\rightarrow0,\quad n\rightarrow\infty. $$
(3.5)

Also, since \(J'_{1}\in C(E\rightarrow E^{\ast})\) and \(P_{n}u\rightarrow u \) in E, we have

$$ \bigl(J'_{1}(P_{n}u)-J'_{1}(u), u_{n}-P_{n}u\bigr)\rightarrow0,\quad n\rightarrow \infty. $$
(3.6)

Therefore, by (3.5) and (3.6), we get

$$ \bigl(J'_{1}(P_{n}u), u_{n}-P_{n}u \bigr)\rightarrow0,\quad n\rightarrow\infty. $$
(3.7)

Note that \(P_{n}u_{n}= u_{n}\) and \((J'_{\lambda _{n}}(P_{n}u_{n}),P_{n}(u_{n}-u))=0\) since \(u_{n}\in W_{n}\) and \(J'_{\lambda_{n}}|_{W_{n}}(u_{n})=0\). By the continuity of f, \(I_{j}\), and Lemma 2.9, it is easily seen that

$$\begin{gathered} \lambda_{n} \int_{0}^{T}f(t,u_{n}) (u_{n}-P_{n}u)\,dt\rightarrow0, \quad n\rightarrow\infty. \\ \int_{0}^{T}f(t,P_{n}u) (u_{n}-P_{n}u)\,dt\rightarrow0, \quad n\rightarrow \infty. \\ \sum^{m}_{j=1}\bigl[I_{j} \bigl(u_{n}(t_{j})\bigr)-I_{j} \bigl(P_{n}u(t_{j})\bigr)\bigr]\bigl(u_{n}(t_{j})- P_{n}u(t_{j})\bigr)\rightarrow0, \quad n\rightarrow\infty.\end{gathered} $$

Set

$$\begin{gathered} \begin{aligned}\psi_{1}={}& \int_{0}^{T}\bigl[ \bigl\vert {}^{c}D_{0^{+}}^{\alpha}u_{n}(t) \bigr\vert ^{p-2} \bigl({}^{c}D_{0^{+}}^{\alpha}u_{n}(t) \bigr)- \bigl\vert {}^{c}D_{0^{+}}^{\alpha }P_{n}u(t) \bigr\vert ^{p-2}\bigl({}^{c}D_{0^{+}}^{\alpha}P_{n}u(t) \bigr)\bigr]\\&\times \bigl[{}^{c}D_{0^{+}}^{\alpha}u_{n}(t)-{}^{c}D_{0^{+}}^{\alpha}P_{n}u(t) \bigr]\,dt,\end{aligned} \\ \psi_{2}= \int _{0}^{T}\bigl( \bigl\vert u_{n}(t) \bigr\vert ^{p-2}u_{n}- \bigl\vert P_{n}u(t) \bigr\vert ^{p-2}P_{n}u(t)\bigr) \bigl(u_{n}(t)-P_{n}u(t)\bigr)\, dt.\end{gathered} $$

Therefore,

$$\begin{aligned} \psi_{1}+\psi_{2}={}&\bigl(J'_{\lambda _{n}}(u_{n}),u_{n}-P_{n}u \bigr)-\bigl(J_{1}'(P_{n}u),u_{n}-P_{n}u \bigr) \\&-\sum^{m}_{j=1}\bigl[I_{j} \bigl(u_{n}(t_{j})\bigr)-I_{j} \bigl(P_{n}u(t_{j})\bigr)\bigr]\bigl(u_{n}(t_{j})- P_{n}u(t_{j})\bigr) \\ &+\lambda_{n} \int_{0}^{T}f(t,u_{n}) (u_{n}-P_{n}u)\,dt- \int _{0}^{T}f(t,P_{n}u) (u_{n}-P_{n}u)\,dt\\\rightarrow{}&0.\end{aligned} $$

In what follows, we prove \(\|u_{n}- P_{n}u\|\rightarrow0 \) in two cases.

Case 1: \(p\geq2\).

According to the following inequality (see [34], Lemma 4.2)

$$\bigl( \vert x \vert ^{p-2}x- \vert y \vert ^{p-2}y \bigr) (x-y)\geq\omega \vert x-y \vert ^{p}, $$

there exist \(\omega_{1}>0\), \(\omega_{2}>0\) such that

$$\begin{aligned}& \psi_{1}\geq\omega_{1} \int_{0}^{T} \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u_{n}(t)-{}^{c}D_{0^{+}}^{\alpha}P_{n}u(t) \bigr\vert ^{p}\,dt, \end{aligned}$$
(3.8)
$$\begin{aligned}& \psi_{2}\geq\omega_{2} \int_{0}^{T} \bigl\vert u_{n}(t)-P_{n}u(t) \bigr\vert ^{p}\, dt. \end{aligned}$$
(3.9)

Combining (3.8) and (3.9), we get

$$\psi_{1}+\psi_{2}\geq M_{1} \Vert u_{n}-P_{n}u \Vert ^{p}, $$

where \(M_{1}=\min\{\omega_{1},\omega_{2}\}\). Thus, \(\|u_{n}- P_{n}u\|\rightarrow0\).

Case 2: \(1< p<2\).

According to the following inequality (see [34], Lemma 4.2)

$$\bigl(\bigl( \vert x \vert ^{p-2}x- \vert y \vert ^{p-2}y\bigr) (x-y)\bigr)^{\frac{p}{2}}\bigl( \vert x \vert ^{p}+ \vert y \vert ^{p}\bigr)^{\frac {2-p}{2}}\geq\omega \vert x-y \vert ^{p}, $$

there exist positive numbers \(\omega_{3}\) and \(\omega_{4}\) such that

$$\begin{aligned}& \psi_{1}\geq\omega_{3} \int_{0}^{T}\frac{ \vert {}^{c}D_{0^{+}}^{\alpha }u_{n}(t)-{}^{c}D_{0^{+}}^{\alpha }P_{n}u(t) \vert ^{2}}{( \vert {}^{c}D_{0^{+}}^{\alpha }u_{n}(t) \vert + \vert {}^{c}D_{0^{+}}^{\alpha}P_{n}u(t) \vert )^{2-p}}\,dt, \end{aligned}$$
(3.10)
$$\begin{aligned}& \psi_{2}\geq\omega_{4} \int_{0}^{T}\frac { \vert u_{n}(t)-P_{n}u(t) \vert ^{2}}{( \vert u_{n}(t) \vert + \vert P_{n}u(t) \vert )^{2-p}}\,dt. \end{aligned}$$
(3.11)

By Hölder’s inequality, we get

$$\begin{gathered} \int_{0}^{T} \bigl\vert u_{n}(t)-P_{n}u(t) \bigr\vert ^{p}\,dt\\\quad \leq\biggl( \int_{0}^{T}\frac { \vert u_{n}(t)-P_{n}u(t) \vert ^{2}}{( \vert u_{n}(t) \vert + \vert P_{n}u(t) \vert )^{2-p}}\,dt \biggr)^{\frac {p}{2}}\biggl( \int_{0}^{T}\bigl( \bigl\vert u_{n}(t) \bigr\vert + \bigl\vert P_{n}u(t) \bigr\vert \bigr)^{p}\,dt\biggr)^{\frac {2-p}{2}} \\ \quad\leq M_{2}\bigl( \Vert u_{n} \Vert ^{p}_{p}+ \Vert P_{n}u \Vert ^{p}_{p}\bigr)^{\frac {2-p}{2}}\biggl( \int_{0}^{T}\frac { \vert u_{n}(t)-P_{n}u(t) \vert ^{2}}{( \vert u_{n}(t) \vert + \vert P_{n}u(t) \vert )^{2-p}}\,dt \biggr)^{\frac {p}{2}},\end{gathered} $$

where \(M_{2}=2^{\frac{(p-1)(2-p)}{2}}\). Therefore,

$$\begin{aligned}[b] &\int_{0}^{T}\frac { \vert u_{n}(t)-P_{n}u(t) \vert ^{2}}{( \vert u_{n}(t) \vert + \vert P_{n}u(t) \vert )^{2-p}}\,dt\\&\quad\geq M_{2}^{-\frac{2}{p}}\biggl( \int_{0}^{T} \bigl\vert u_{n}(t)-P_{n}u(t) \bigr\vert ^{p}\,dt\biggr)^{\frac {2}{p}}\bigl( \Vert u_{n} \Vert ^{p}_{p}+ \Vert P_{n}u \Vert ^{p}_{p}\bigr)^{\frac{p-2}{p}}.\end{aligned} $$
(3.12)

Similarly, we have

$$ \begin{aligned}[b] &\int_{0}^{T}\frac{ \vert {}^{c}D_{0^{+}}^{\alpha }u_{n}(t)-{}^{c}D_{0^{+}}^{\alpha }P_{n}u(t) \vert ^{2}}{( \vert {}^{c}D_{0^{+}}^{\alpha }u_{n}(t) \vert + \vert {}^{c}D_{0^{+}}^{\alpha}P_{n}u(t) \vert )^{2-p}}\,dt \\&\quad\geq M_{2}^{-\frac{2}{p}}\biggl( \int_{0}^{T} \bigl\vert {}^{c}D_{0^{+}}^{\alpha }u_{n}(t)-{}^{c}D_{0^{+}}^{\alpha}P_{n}u(t) \bigr\vert ^{p}\,dt\biggr)^{\frac {2}{p}} \\ &\qquad{}\cdot\bigl( \bigl\Vert {}^{c}D_{0^{+}}^{\alpha}u_{n} \bigr\Vert ^{p}_{p}+ \bigl\Vert {}^{c}D_{0^{+}}^{\alpha}P_{n}u \bigr\Vert ^{p}_{p}\bigr)^{\frac{p-2}{p}}. \end{aligned} $$
(3.13)

Combining (3.10)–(3.13), we get

$$\psi_{1}+\psi_{2}\geq M_{3n} \Vert u_{n}-P_{n}u \Vert ^{2}, $$

where \(M_{3n}= M_{2}^{-\frac{2}{p-1}}\min\{\omega_{3}(\| {}^{c}D_{0^{+}}^{\alpha}u_{n}\|^{p}_{p}+\|{}^{c}D_{0^{+}}^{\alpha }P_{n}u\|^{p}_{p})^{\frac{p-2}{p}}, \omega_{4}(\|u_{n}\|^{p}_{p}+\|P_{n}u\|^{p}_{p})^{\frac{p-2}{p}}\}\).

If \(\{u_{n}\}\) has a subsequence (still relabeled \(\{u_{n}\}\) for convenience) such that \(\|u_{n}\|\rightarrow0\) as \(n\rightarrow \infty\), it is easy to see that \(\|u_{n}- P_{n}u\|\rightarrow0\). On the other hand, if \(\inf_{n\geq1}\|u_{n}\| >0\), by the boundedness of \(\{u_{n}\}\) in E, there exists \(M_{3}>0\) such that \(M_{3n}\geq M_{3}>0\). Then \({\|u_{n}- P_{n}u\|\rightarrow0}\).

So, \(u_{n}- P_{n}u\rightarrow0 \) in E as \(n\rightarrow\infty\), which means that \(u_{n}\rightarrow u \) in E as \(n\rightarrow\infty \). By Lemma 2.13, we know that \(J=J_{1}\) has infinitely many nontrivial critical points \(u_{k}\). Consequently, BVP (1.1) has infinitely many small energy solutions. □

Proof of Theorem 1.2

For any \(\varepsilon>0\), it follows from \((F_{6})\) that there exist positive numbers \(\theta_{3}\) and \(\theta_{4}\) such that

$$ \bigl\vert F(t,u) \bigr\vert \leq\theta_{3} \vert u \vert ^{p}+\theta_{4} \vert u \vert ^{q}. $$
(3.14)

Combining (3.14), \((H_{3})\), and Lemma 2.7, it is easily seen that \(J_{\lambda}\) maps bounded sets into bounded sets uniformly for \(\lambda\in[1,2]\). By \((H_{1})\) and \((F_{5})\), \(J_{\lambda }(-u)=J_{\lambda}(u)\) for all \((\lambda,u)\in [1,2]\times E\). Thus, condition \((A_{1})\) holds. Assumption \((F_{4})\) means that \(B(u)\geq0\). Condition \((A_{2})\) holds for the fact that \(A(u)\geq\frac{1}{p}\|u\|^{p}\rightarrow\infty\) as \(n\rightarrow\infty\) and \(B(u)\geq0\).

In what follows, we verify condition \((A_{4})\). For this sake, we need to prove that there exist two sequences \(\rho_{k}>r_{k}>0\) such that

$$\begin{aligned}& b_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|=r_{k}}J_{\lambda}(u)>0,\quad \forall\lambda\in[1,2], \end{aligned}$$
(3.15)
$$\begin{aligned}& a_{k}(\lambda)=\max_{u\in W_{k},\|u\|=\rho_{k}}J_{\lambda}(u)< 0,\quad \forall\lambda\in[1,2]. \end{aligned}$$
(3.16)

First, we prove that (3.15) is true.

For \(u \in Z_{k}\), by (3.14), \((H_{1})\), and the definition of \(\alpha _{r}(k)\) in Lemma 3.2, we have

$$\begin{aligned} J_{\lambda}(u)&\geq \frac{1}{p} \Vert u \Vert ^{p}-2 \theta_{3} \Vert u \Vert ^{p}_{p}-2 \theta_{4} \Vert u \Vert ^{q}_{q} \\ &\geq\frac{1}{p} \Vert u \Vert ^{p}-2\theta_{3} \alpha_{p}^{p}(k) \Vert u \Vert ^{p}-2 \theta_{4}\alpha_{q}^{q}(k) \Vert u \Vert ^{q}.\end{aligned} $$

Choose \(r_{k}=\frac{1}{\alpha_{p}(k)+\alpha_{q}(k)}\). Then \(r_{k}\rightarrow\infty\) as \(k\rightarrow\infty\). For any \(u\in Z_{k}\) with \(\|u\|=r_{k}\), we know

$$\begin{aligned} J_{\lambda}(u)&\geq \frac{1}{p} \Vert u \Vert ^{p}-2 \theta_{3}\frac{\alpha _{p}^{p}(k)}{ \vert \alpha_{p}(k)+\alpha_{q}(k) \vert ^{p}}-2\theta_{4}\frac {\alpha_{q}^{q}(k)}{ \vert \alpha_{p}(k)+\alpha_{q}(k) \vert ^{q}} \\ &\geq\frac{1}{p}r_{k}^{p}-2\theta_{3}-2 \theta_{4} \\ &> 0.\end{aligned} $$

Therefore,

$$b_{k}(\lambda)=\inf_{u\in Z_{k},\|u\|=r_{k}}J_{\lambda}(u)>0,\quad \forall\lambda\in[1,2]. $$

Next, we prove that (3.16) is true.

By \((F_{7})\), there exists \(\delta_{3}>0\) such that

$$ F(t,u)\geq\delta_{3} \vert u \vert ^{\mu}, \quad\text{for all }t \in[0,T], u\in\mathbb {R}. $$
(3.17)

According to (3.17), \((H_{3})\), and Lemma 2.7, we have

$$\begin{aligned} J_{\lambda}(u)&= \frac{1}{p} \Vert u \Vert ^{p}+\sum ^{m}_{j=1} \int ^{u(t_{j})}_{0}I_{j}(s)\,ds-\lambda \int_{0}^{T}F\bigl(t,u(t)\bigr)\,dt \\ &\leq \frac{1}{p} \Vert u \Vert ^{p}+\sum ^{m}_{j=1}\frac{b_{j}}{\gamma _{j}+1} \bigl\vert u(t_{j}) \bigr\vert ^{\gamma_{j}+1}-\delta_{3} \int_{0}^{T} \vert u \vert ^{\mu}\, dt \\ &\leq \frac{1}{p} \Vert u \Vert ^{p}+\delta_{4} \Vert u \Vert ^{\gamma_{j}+1}-\delta _{5} \Vert u \Vert ^{\mu}.\end{aligned} $$

Hence, one can take \(\rho_{k}>\gamma_{k}\) large enough such that

$$a_{k}(\lambda)=\max_{u\in W_{k},\|u\|=\rho_{k}}J_{\lambda}(u)< 0. $$

Until now, all the conditions of Lemma 2.14 hold. Hence, for \(\lambda \in[1,2]\), there exists a sequence \(\{u_{n}^{k}(\lambda)\} _{n=1}^{\infty}\) such that

$$\begin{gathered} \sup_{n} \bigl\Vert u_{n}^{k}( \lambda) \bigr\Vert < \infty,\\ J'_{\lambda }\bigl(u_{n}^{k}( \lambda)\bigr)\rightarrow0,\qquad J_{\lambda}\bigl(u_{n}^{k}( \lambda )\bigr)\rightarrow c_{k}(\lambda)=\inf_{\gamma\in\Gamma_{k}} \max_{u\in B_{k}}J_{\lambda}\bigl(\gamma(u)\bigr),\quad n\rightarrow \infty.\end{gathered} $$
(3.18)

Furthermore,

$$\begin{gathered} c_{k}(\lambda)\geq b_{k}(\lambda)\geq\frac{1}{p}r_{k}^{p}-2 \theta _{3}-2\theta_{4}:=\overline{b_{k}} \rightarrow\infty,\quad k\rightarrow \infty, \\ c_{k}(\lambda)\leq\max_{u\in B_{k}}J_{1}(u):= \overline{c_{k}}.\end{gathered} $$

Thus,

$$ \overline{b_{k}}\leq c_{k}(\lambda)\leq \overline{c_{k}},\quad \lambda \in[1,2]. $$
(3.19)

Choose a sequence \(\lambda_{m}\rightarrow1\) such that (3.18) holds. Using similar arguments of the proof of Theorem 1.1, we can show that \(\{u_{n}^{k}(\lambda_{m})\}_{n=1}^{\infty}\) possesses a strong convergent subsequence. Thus, we suppose that \(u_{n}^{k}(\lambda_{m})\rightarrow u^{k}(\lambda _{m})\) in E as \(n\rightarrow\infty\). By (3.18) and (3.19), we can get

$$J'_{\lambda_{m}}\bigl(u^{k}(\lambda_{m}) \bigr)=0,\qquad J_{\lambda _{m}}\bigl(u^{k}(\lambda_{m})\bigr) \in[\overline{b_{k}}, \overline{c_{k}}] \quad\text{for }k\geq k_{1}. $$

In the following we prove that \(\{u^{k}(\lambda_{m})\}_{m=1}^{\infty }\) is bounded.

From \((H_{3})\) and \((F_{6})\), we have

$$\begin{gathered} \mu J_{\lambda_{m}}\bigl(u^{k}(\lambda_{m})\bigr)- \bigl(J'_{\lambda _{m}}\bigl(u^{k}(\lambda_{m}) \bigr),u^{k}(\lambda_{m})\bigr) \\ \quad=\biggl(\frac{\mu}{p}-1\biggr) \bigl\Vert u^{k}( \lambda_{m}) \bigr\Vert ^{p}+\sum ^{m}_{j=1}\biggl(\mu \int^{u^{k}(\lambda_{m})(t_{j})}_{0}I_{j}(s) \,ds-I_{j}\bigl(u^{k}(\lambda _{m}) (t_{j})\bigr)u^{k}(\lambda_{m}) (t_{j})\biggr) \\ \qquad{} +\lambda_{m} \int_{0}^{T}\bigl[f\bigl(t,u^{k}( \lambda_{m})\bigr)u^{k}(\lambda _{m})-\mu F \bigl(t,u^{k}(\lambda_{m})\bigr)\bigr]\,dt \\ \quad\geq\biggl(\frac{\mu}{p}-1\biggr) \bigl\Vert u^{k}( \lambda_{m}) \bigr\Vert ^{p}. \end{gathered} $$

Therefore, \(\{u^{k}(\lambda_{m})\}_{m=1}^{\infty}\) is bounded in E. Similar arguments of the proof of Theorem 1.1 show that \(u^{k}(\lambda _{m})\rightarrow u^{k}\) in E as \(m\rightarrow\infty\) (\(k\geq k_{1}\)). Then \(u^{k}\) is a critical point of \(J=J_{1}\) with \(I(u^{k})\in[\overline {b_{k}},\overline{c_{k}}]\). According to \(\overline{b_{k}}\rightarrow\infty\) as \(k\rightarrow \infty\), we know that BVP (1.1) has infinitely many nontrivial high energy solutions. □

4 Examples

In this section, two examples are given to illustrate our results.

Example 4.1

Consider the following nonlinear impulsive fractional boundary value problem:

$$ \left \{ \textstyle\begin{array}{l} D_{T^{-}}^{0.8}\Phi _{4}({}^{c}D_{0^{+}}^{0.8}u(t))+ \vert u(t) \vert u(t)=f(t,u),\quad t\in[0,T],t\neq t_{1},\\ \Delta(D_{T^{-}}^{-0.2}\Phi _{4}({}^{c}D_{0^{+}}^{0.8}u))(t_{1})=u^{5}(t_{1}),\\ u(0)=u(T)=0, \end{array}\displaystyle \right . $$
(4.1)

where

$$f(t,u)= \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{7}{2}e^{t} \vert u \vert ^{4}, & \vert u \vert \leq1, \\ \frac{7}{2}e^{t} \vert u \vert ^{\frac{5}{2}}, & \vert u \vert >1. \end{array}\displaystyle \right . $$

Choose \(p=4\), \(\alpha=0.8 \in(\frac{1}{4},1]\), and \(I_{1}(u)=u^{5}(t_{1})\). It is easy to show that assumption \((H_{1})\) holds. Take \(b_{1}=2\), \(\gamma_{1}=5\in(3,+\infty)\). From this we can see that assumption \((H_{2})\) holds.

Moreover,

$$\bigl\vert f(t,u) \bigr\vert \leq4 e^{t}\bigl(1+ \vert u \vert ^{\frac{5}{2}}\bigr). $$

Choose \(\eta=\frac{7}{2}\in(3,4) \) and \(b(t)= 4e^{t}\). This means that assumption \((F_{1})\) is satisfied.

Take \(\sigma=\frac{13}{4}\in(3,\frac{7}{2})\). By a simple calculation, one has \(F(t,u)=e^{t}|u|^{\frac{7}{2}}\) and \(\lim_{|u|\rightarrow\infty}\frac{e^{t}|u|^{\frac{7}{2}}}{ |u|^{\frac{13}{4}}}\rightarrow\infty\). Therefore, assumption \((F_{2})\) holds.

In addition, \(\lim_{|u|\rightarrow0}\frac{\frac {7}{2}e^{t}|u|^{4}}{|u|^{3}}=0 \) implies that assumption \((F_{3})\) holds.

Finally, it is easy to see that \((F_{4})\) and \((F_{5})\) hold. Consequently, BVP (4.1) has infinitely many small energy solutions by Theorem 1.1.

Example 4.2

Consider the following nonlinear fractional impulsive boundary value problem:

$$ \left \{ \textstyle\begin{array}{l} D_{T^{-}}^{0.8}\Phi_{\frac {5}{2}}({}^{c}D_{0^{+}}^{0.8}u(t))+ \vert u(t) \vert ^{\frac {1}{2}}u(t)=5 \vert u \vert ^{4}\ln( \vert u \vert +1)+\frac{ \vert u \vert ^{5}}{ \vert u \vert +1},\quad t\in[0,T],t\neq t_{1},\\ \Delta(D_{T^{-}}^{-0.2}\Phi_{\frac {5}{2}}({}^{c}D_{0^{+}}^{0.8}u))(t_{1})=u^{3}(t_{1}),\\ u(0)=u(T)=0. \end{array}\displaystyle \right . $$
(4.2)

First, choose \(p=\frac{5}{2}\), \(\alpha=0.8\in(\frac{2}{5},1]\), and \(I_{1}(u)=u^{3}(t_{1})\). From this one can see that assumption \((H_{1})\) holds. Taking \(b_{1}=2\), \(\mu=5>\frac{5}{2}\), and \(\gamma_{1}=3\in(\frac{3}{2},4)\) means that assumption \((H_{3})\) holds.

Next, a simple calculation shows that

$$F(t,u)= \vert u \vert ^{5}\ln\bigl( \vert u \vert +1\bigr),\qquad {-}5F(t,u)+ uf(t,u)=\frac{ \vert u \vert ^{5}}{ \vert u \vert +1}\geq0. $$

Hence, assumption \((F_{7})\) holds.

Finally, it is easy to show that \((F_{4})\)\((F_{6})\) hold. Consequently, BVP (4.2) has infinitely many high energy solutions by Theorem 1.2.

Abbreviations

BVP:

boundary value problem

\(\mathrm{AC}{[a,b]}\) :

the space of absolutely continuous functions on \([a,b]\)

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Acknowledgements

The authors wish to thank anonymous referees for their valuable suggestions.

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This work was supported by the National Natural Science Foundation of China (No. 11671237), and by Shandong Provincial Natural Science Foundation of China (No. ZR2013AM005).

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Wang, Y., Liu, Y. & Cui, Y. Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian. Bound Value Probl 2018, 94 (2018). https://doi.org/10.1186/s13661-018-1012-0

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