Existence of subharmonic solutions for non-quadratic second-order Hamiltonian systems

Abstract

In this paper, some existence theorems are obtained for subharmonic solutions of second-order Hamiltonian systems with linear part under non-quadratic conditions. The approach is the minimax principle. We consider some new cases and obtain some new existence results.

MSC:34C25, 58E50, 70H05.

1 Introduction and main results

Consider the second-order Hamiltonian system

$$\ddot{u}(t)+\mathit{Au}(t)+\mathrm{\nabla }F(t,u(t))=0\text{a.e. }t\in \mathbb{R},$$

(1.1)

where A is an $N\times N$ symmetric matrix and $F:\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is T-periodic in t and satisfies the following assumption:

Assumption (A)′ $F(t,x)$ is measurable in t for every $x\in {\mathbb{R}}^N$ and continuously differentiable in x for a.e. $t\in [0,T]$, and there exist $a\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and $b:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ which is T-periodic and $b\in L^p(0,T;{\mathbb{R}}^{+})$ with $p>1$ such that

$$|F(t,x)|\le a(|x|)b(t),|\mathrm{\nabla }F(t,x)|\le a(|x|)b(t)$$

for all $x\in {\mathbb{R}}^N$ and a.e. $t\in [0,T]$.

When $A=0$, system (1.1) reduces to the second-order Hamiltonian system

$$\ddot{u}(t)+\mathrm{\nabla }F(t,u(t))=0\text{a.e. }t\in \mathbb{R}.$$

(1.2)

There have been many existence results for system (1.2) (for example, see [1–7] and references therein). In 1978, Rabinowitz [6] obtained the nonconstant periodic solutions for system (1.2) under the following AR-condition: there exist $\mu >2$ and $L>0$ such that

$$0<\mu F(t,x)\le (\mathrm{\nabla }F(t,x),x),\mathrm{\forall }|x|\ge L,t\in [0,T].$$

From then on, the condition has been used extensively in the literature; see [8–12] and the references therein. In [13], Fei also obtained the existence of nonconstant solutions for system (1.2) under a kind of new superquadratic condition. Subsequently, Tao and Tang [14] gave the following more general one than Fei’s: there exist $\theta >2$ and $\mu >\theta -2$ such that

$$\underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{F(t,x)}{{|x|}^{\theta }}<\mathrm{\infty }\text{uniformly for a.e. }t\in [0,T],$$

(1.3)

$$\underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{(\mathrm{\nabla }F(t,x),x)-2F(t,x)}{{|x|}^{\mu }}>0\text{uniformly for a.e. }t\in [0,T].$$

(1.4)

They also considered the existence of subharmonic solutions and obtained the following result.

Theorem A (See [14], Theorem 2)

Suppose that F satisfies

• (A) $F(t,x)$ is measurable in t for every $x\in {\mathbb{R}}^N$ and continuously differentiable in x for a.e. $t\in [0,T]$, and there exist $a\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and $b\in L^1(0,T;{\mathbb{R}}^{+})$ such that

  $$|F(t,x)|\le a(|x|)b(t),|\mathrm{\nabla }F(t,x)|\le a(|x|)b(t)$$

for all $x\in {\mathbb{R}}^N$ and a.e. $t\in [0,T]$. Assume that (1.3), (1.4) and the following conditions hold:

$$F(t,x)\ge 0,\mathrm{\forall }(t,x)\in [0,T]\times {\mathbb{R}}^N,$$

(1.5)

$$\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^2}=0\mathit{\text{uniformly for a.e.}}t\in [0,T],$$

(1.6)

$$\underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^2}>\frac{2{\pi }^2}{T^2}\mathit{\text{uniformly for a.e.}}t\in [0,T].$$

(1.7)

Then system (1.2) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$.

Recently, Ma and Zhang [15] considered the following p-Laplacian system:

$${({|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t))}^{\prime }+\mathrm{\nabla }F(t,u(t))=0\text{a.e. }t\in [0,T],$$

(1.8)

where $p>1$. By using some techniques, they obtained the following more general result than Theorem A.

Theorem B (See [15], Theorem 1)

Suppose that F satisfies (A), (1.3) and (1.4) with 2 replaced by p, (1.5) and the following condition:

$$\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^p}=0<\underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^p}\mathit{\text{uniformly for a.e.}}t\in [0,T].$$

(1.9)

Then system (1.8) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$.

When $A=m^2{\omega }^2{I}_N$, where $\omega =2\pi /T$ and $I_N$ is the unit matrix of order N. Ye and Tang [16] obtained the following result.

Theorem C (See [16], Theorem 2)

Suppose that $A=m^2{\omega }^2{I}_N$, F satisfies (A), (1.3), (1.4), (1.5), (1.6) and the following conditions:

$$\underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^2}>\frac{1+2m}{2}{\omega }^2\mathit{\text{uniformly for a.e.}}t\in [0,T].$$

Then system (1.1) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$.

Recently, in [17], we considered a more general case than that in [16]. We considered the case that A only has 0 or $l_i^2{\omega }^2$ as its eigenvalues, where $\omega =2\pi /T$, $l_i\in \mathbb{N}$, $i=1,\dots ,r$ and $0\le r\le N$. In [17], we used the following condition which presents some advantages over (1.3) and (1.4):

• (H) there exist positive constants m, ζ, η and $\nu \in [0,2)$ such that

  $$(2+\frac{1}{\zeta +\eta {|x|}^{\nu }})F(t,x)\le (\mathrm{\nabla }F(t,x),x),x\in {\mathbb{R}}^N,|x|>m\text{ a.e. }t\in [0,T].$$

In this paper, we consider some new cases which can be seen as a continuance of our work in [17].

Next, we state our main results. Assume that $r\in \mathbb{N}\cup \{0\}$ and $r\le N$. Let ${\lambda }_i>0$ ($i\in \{1,\dots ,r\}$) and $-{\lambda }_i<0$ ($i\in \{r+s+1,\dots ,N\}$) be the positive and negative eigenvalues of A, respectively, where r and s denote the number of positive eigenvalues and zero eigenvalues of A (counted by multiplicity), respectively. Moreover, we denote by q the number of negative eigenvalues of A (counted by multiplicity). We make the following assumption:

Assumption (A0) A has at least one nonzero eigenvalue and all positive eigenvalues are not equal to $l^2{\omega }^2$ for all $l\in \mathbb{N}$, where $\omega =2\pi /T$, that is, ${\lambda }_i\ne l^2{\omega }^2$ ($i=1,\dots ,r$) for all $l\in \mathbb{N}$.

The Assumption (A0) implies that one can find $l_i\in {\mathbb{Z}}^{+}:=\{0,1,2,\dots \}$ such that

$$l_i^2{\omega }^2<{\lambda }_i<{(l_i+1)}^2{\omega }^2,i=1,\dots ,r.$$

(1.10)

For the sake of convenience, we set

$$\begin{array}{r}{\lambda }_{i^{+}}=max\{{\lambda }_i|i=1,\dots ,r\},{\lambda }_{i^{-}}=min\{{\lambda }_i|i=1,\dots ,r\},\\ {\lambda }_{i_{+}}=max\{{\lambda }_i|i=r+s+1,\dots ,N\},{\lambda }_{i_{-}}=min\{{\lambda }_i|i=r+s+1,\dots ,N\}.\end{array}$$

Then

$$i^{+},i^{-}\in \{1,\dots ,r\},i_{+},i_{-}\in \{r+s+1,\dots ,N\}.$$

Corresponding to (1.10), we know that there exist $l_{i^{+}},l_{i^{-}}\in {\mathbb{Z}}^{+}$ such that

$$l_{i^{+}}^2{\omega }^2<{\lambda }_{i^{+}}<{(l_{i^{+}}+1)}^2{\omega }^2,l_{i^{-}}^2{\omega }^2<{\lambda }_{i^{-}}<{(l_{i^{-}}+1)}^2{\omega }^2.$$

Moreover, set

$$h_i={(l_i+1)}^2{\omega }^2-{\lambda }_i,i=1,\dots ,r,$$

and let $h_{i_0}={min}_{i\in \{1,\dots ,r\}}\{h_i\}$. Then $i_0\in \{1,\dots ,r\}$. Corresponding to (1.10), there exists $l_{i_0}\in {\mathbb{Z}}^{+}$ such that

$$l_{i_0}^2{\omega }^2<{\lambda }_{i_0}<{(l_{i_0}+1)}^2{\omega }^2.$$

(1.11)

Theorem 1.1 Assume that (A0) holds and F satisfies (A)′, (1.5) and the following conditions.

(H1) For some $k\in \mathbb{N}$, assume that k satisfies

$${(l_i+1-\frac{1}{k})}^2{\omega }^2\le {\lambda }_i<{(l_i+1)}^2{\omega }^2\mathit{\text{for all}}i\in \{1,\dots ,r\}.$$

(1.12)

(H2) There exist positive constants m, ζ, η and $\nu \in [0,2)$ such that

$$(2+\frac{1}{\zeta +\eta {|x|}^{\nu }})F(t,x)\le (\mathrm{\nabla }F(t,x),x),x\in {\mathbb{R}}^N,|x|>m,\mathit{\text{a.e.}}t\in [0,T].$$

(H3) Assume that one of the following cases holds:

• (1) when $r>0$, $s>0$ and $r+s=N$, there exist $L_k>0$ and ${\beta }_k>min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2k^2}\}$ such that

  $$F(t,x)\ge {\beta }_k{|x|}^2,\mathrm{\forall }x\in {\mathbb{R}}^N,|x|>L_k,\mathit{\text{a.e.}}t\in [0,T],$$

  (1.13)

where $l_{i_0}$ and ${\lambda }_{i_0}$ are defined by (1.11);

• (2) when $r>0$, $s>0$ and $r+s<N$, there exist $L_k>0$ and ${\beta }_k>min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2k^2},\frac{{\lambda }_{i_{-}}}{2}\}$ such that (1.13) holds;

• (3) when $r>0$, $s=0$ and $r+s<N$, there exist $L_k>0$ and ${\beta }_k>min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\lambda }_{i_{-}}}{2}\}$ such that (1.13) holds;

• (4) when $r>0$, $s=0$ and $r=N$, there exist $L_k>0$ and ${\beta }_k>\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2}$ such that (1.13) holds;

• (5) when $r=0$, $s>0$ and $s<N$, there exist $L_k>0$ and ${\beta }_k>min\{\frac{{\omega }^2}{2k^2},\frac{{\lambda }_{i_{-}}}{2}\}$ such that (1.13) holds;

• (6) when $r=0$, $s=0$ and $q=N$, there exist $L_k>0$ and ${\beta }_k>\frac{{\lambda }_{i_{-}}}{2}$ such that (1.13) holds;

(H4) there exist $l_k>0$ and ${\alpha }_k<\frac{{\sigma }_k}{2}$ such that

$$F(t,x)\le {\alpha }_k{|x|}^2\mathit{\text{for all}}|x|\le l_k\mathit{\text{and a.e.}}t\in [0,T],$$

where

$$\begin{array}{c}{\sigma }_k=min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\omega }^2}{{\omega }^2+k^2}\}\mathit{\text{if}}(\mathrm{H}3)\text{ (1) }\mathit{\text{holds}};\hfill \\ {\sigma }_k=min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\omega }^2}{{\omega }^2+k^2},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\}\mathit{\text{if}}(\mathrm{H}3)\text{ (2) }\mathit{\text{holds}};\hfill \\ {\sigma }_k\equiv \sigma =min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\}\mathit{\text{if}}(\mathrm{H}3)\text{ (3) }\mathit{\text{holds}};\hfill \\ {\sigma }_k\equiv \sigma =\underset{i\in \{1,\dots ,N\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\}\mathit{\text{if}}(\mathrm{H}3)\text{ (4) }\mathit{\text{holds}};\hfill \\ {\sigma }_k=min\{\frac{{\omega }^2}{{\omega }^2+k^2},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\}\mathit{\text{if}}(\mathrm{H}3)\text{ (5) }\mathit{\text{holds}};\hfill \\ {\sigma }_k\equiv \sigma =\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\mathit{\text{if}}(\mathrm{H}3)\text{ (6) }\mathit{\text{holds}},\hfill \end{array}$$

where σ implies that ${\sigma }_k$ is independent of k. Then system (1.1) has a nonzero kT-periodic solution. Especially, for cases (H3)(1) and (H3)(4), system (1.1) has a nonconstant kT-periodic solution.

Remark 1.1 For cases (H3)(1)-(H3)(4), from (1.10) and (1.12), it is easy to see that the number of $k\in \mathbb{N}$ satisfying (1.12) is finite. Let $m\in K$ be the maximum integer satisfying (1.12), where

$$K=\{k\in \mathbb{N}\mid k\text{ satisfies }(\text{1.12})\}.$$

Then $K=\{1,2,\dots ,m\}$. Hence, Theorem 1.1 implies that system (1.1) has nonzero kT-periodic solutions ($k=1,2,\dots ,m$). For cases (H3)(5) and (H3)(6), since $r=0$, (1.12) holds for every $k\in \mathbb{N}$. Hence, Theorem 1.1 implies that system (1.11) has nonzero kT-periodic solutions for every $k\in \mathbb{N}$.

Remark 1.2 In [18], Costa and Magalhães studied the first-order Hamiltonian system

$$-J\dot{u}(t)+\mathit{Au}+\mathrm{\nabla }H(t,u)=0\text{a.e. }t\in [0,T].$$

(1.14)

They obtained that system (1.14) has a $T=2\pi$ periodic solution under the following non-quadraticity conditions:

$$\underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{(x,\mathrm{\nabla }H(t,x))-2H(t,x)}{{|x|}^{\mu }}\ge a>0\text{uniformly for a.e. }t\in [0,2\pi ],$$

(1.15)

and the so-called asymptotic noncrossing conditions

$${\lambda }_{k-1}<\underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{2H(t,x)}{{|x|}^2}\le \underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{2H(t,x)}{{|x|}^2}\le {\lambda }_k\text{uniformly for a.e. }t\in [0,2\pi ],$$

where ${\lambda }_{k-1}<{\lambda }_k$ are consecutive eigenvalues of the operator $L=-Jd/dt-A$. Moreover, they also obtained system (1.14) has a nonzero $T=2\pi$ periodic solution under (1.15) and the called crossing conditions

$$\begin{array}{r}H(t,u)\ge \frac{1}{2}{\lambda }_{k-1}{|x|}^2\text{for all }(t,u)\in [0,2\pi ]\times {\mathbb{R}}^{2N},\\ \underset{|x|\to 0}{lim\hspace{0.17em}sup}\frac{2H(t,x)}{{|x|}^2}\le \alpha <{\lambda }_k<\beta \le \underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{2H(t,x)}{{|x|}^2}\text{uniformly for }t\in [0,2\pi ].\end{array}$$

One can also establish the similar results for the second-order Hamiltonian system (1.1). Some related contents can be seen in [19]. It is worth noting that in [18] and [19], ${\lambda }_{k-1}<{\lambda }_k$ are consecutive eigenvalues of the operator $L=-Jd/dt-A$ or $-d^2/d{t}^2+A$. In our Theorem 1.1 and Theorem 1.2, we study the existence of subharmonic solutions for system (1.1) from a different perspective. ${\lambda }_i$ ($i\in \{1,\dots ,r\}$) in our theorems are the eigenvalues of the matrix A. Obviously, it is much easier to seek the eigenvalue of a matrix. In Section 4, we present an interesting example satisfying our Theorem 1.1 but not satisfying the theorem in [19].

Theorem 1.2 Suppose that (A0) holds and F satisfies (A)′, (1.5), (H2) and the following conditions:

(H3)′ when $r=0$, $s>0$ and $s<N$, there exist $L>0$ and $\beta >\frac{{\omega }^2}{2}$ such that

$$F(t,x)\ge \beta {|x|}^2,\mathrm{\forall }x\in {\mathbb{R}}^N,|x|>L,\mathit{\text{a.e.}}t\in [0,T];$$

(1.16)

(H4)′

$$\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^2}=0\mathit{\text{uniformly for a.e.}}t\in [0,T].$$

Then system (1.1) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$.

In the final theorem, we present a result about the existence of subharmonic solutions for system (1.8). Using a condition like (H2) and similar to the argument of Remark 1.1 in [17], we can improve Theorem B.

Theorem 1.3 Suppose that F satisfies (A), (1.5) and the following conditions:

(H5) there exist positive constants m, ζ, η and $\nu \in [0,p)$ such that

$$(p+\frac{1}{\zeta +\eta {|x|}^{\nu }})F(t,x)\le (\mathrm{\nabla }F(t,x),x),x\in {\mathbb{R}}^N,|x|>m\mathit{\text{a.e.}}t\in [0,T];$$

(H6)

$$\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^p}=0<\underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^p}\mathit{\text{uniformly for a.e.}}t\in [0,T].$$

Then system (1.8) has a sequence of distinct nonconstant periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$.

2 Some preliminaries

Let

$$H_{kT}^1=\{u:\mathbb{R}\to {\mathbb{R}}^N|u\text{ be absolutely continuous},u(t)=u(t+kT)\text{ and }\dot{u}\in L^2([0,kT])\}.$$

Then $H_{kT}^1$ is a Hilbert space with the inner product and the norm defined by

$$〈u,v〉={\int }_0^{kT}(u(t),v(t))dt+{\int }_0^{kT}(\dot{u}(t),\dot{v}(t))dt$$

and

$$\parallel u\parallel ={[{\int }_0^{kT}{|u(t)|}^{2}dt+{\int }_0^{kT}{|\dot{u}(t)|}^{2}dt]}^{1/2}$$

for each $u,v\in H_{kT}^1$. Let

$$\overline{u}=\frac{1}{kT}{\int }_0^{kT}u(t)dt\text{and}\tilde{u}(t)=u(t)-\overline{u}.$$

Then one has

$$\begin{array}{r}{\parallel \tilde{u}\parallel }_{\mathrm{\infty }}^2\le \frac{kT}{12}{\int }_0^{kT}{|\dot{u}(t)|}^{2}dt\text{(Sobolev’s inequality)},\\ {\parallel \tilde{u}\parallel }_{L^2}^2\le \frac{k^2{T}^2}{4{\pi }^2}{\int }_0^{kT}{|\dot{u}(t)|}^{2}dt\text{(Wirtinger’s inequality)}\end{array}$$

(see Proposition 1.3 in [1]).

Lemma 2.1 If $u\in H_{kT}^1$, then

$${\parallel u\parallel }_{\mathrm{\infty }}\le \sqrt{\frac{12+k^2{T}^2}{12kT}}\parallel u\parallel ,$$

where ${\parallel u\parallel }_{\mathrm{\infty }}={max}_{t\in [0,kT]}|u(t)|$.

Proof Fix $t\in [0,kT]$. For every $\tau \in [0,kT]$, we have

$$u(t)=u(\tau )+{\int }_{\tau }^t\dot{u}(s)ds.$$

(2.1)

Set

$$\varphi (s)=\{\begin{array}{cc}s-t+\frac{kT}{2},\hfill & t-kT/2\le s\le t,\hfill \\ t+\frac{kT}{2}-s,\hfill & t\le s\le t+kT/2.\hfill \end{array}$$

Integrating (2.1) over $[t-kT/2,t+kT/2]$ and using the Hölder inequality, we obtain

$$\begin{array}{rcl}kT|u(t)|& =& |{\int }_{t-kT/2}^{t+kT/2}u(\tau )d\tau +{\int }_{t-kT/2}^{t+kT/2}{\int }_{\tau }^t\dot{u}(s)dsd\tau |\\ \le & {\int }_{t-kT/2}^{t+kT/2}|u(\tau )|d\tau +{\int }_{t-kT/2}^t{\int }_{\tau }^t|\dot{u}(s)|dsd\tau +{\int }_t^{t+kT/2}{\int }_t^{\tau }|\dot{u}(s)|dsd\tau \\ =& {\int }_{t-kT/2}^{t+kT/2}|u(\tau )|d\tau +{\int }_{t-kT/2}^t(s-t+\frac{kT}{2})|\dot{u}(s)|ds\\ +{\int }_t^{t+kT/2}(t+\frac{kT}{2}-s)|\dot{u}(s)|ds\\ =& {\int }_{t-kT/2}^{t+kT/2}|u(\tau )|d\tau +{\int }_{t-kT/2}^{t+kT/2}\varphi (s)|\dot{u}(s)|ds\\ \le & {(kT)}^{1/2}{\left({\int }_{t-kT/2}^{t+kT/2}{|u(\tau )|}^{2}d\tau \right)}^{1/2}\\ +{\left({\int }_{t-kT/2}^{t+kT/2}{[\varphi (s)]}^{2}ds\right)}^{1/2}{\left({\int }_{t-kT/2}^{t+kT/2}{|\dot{u}(s)|}^{2}ds\right)}^{1/2}\\ =& {(kT)}^{1/2}{\left({\int }_{t-kT/2}^{t+kT/2}{|u(\tau )|}^{2}d\tau \right)}^{1/2}+\frac{{(kT)}^{3/2}}{2\sqrt{3}}{\left({\int }_{t-kT/2}^{t+kT/2}{|\dot{u}(s)|}^{2}ds\right)}^{1/2}\\ \le & {(kT+\frac{{(kT)}^3}{12})}^{1/2}{({\int }_{t-kT/2}^{t+kT/2}{|u(\tau )|}^{2}d\tau +{\int }_{t-kT/2}^{t+kT/2}{|\dot{u}(s)|}^{2}ds)}^{1/2}\\ =& {(kT+\frac{{(kT)}^3}{12})}^{1/2}{({\int }_0^{kT}{|u(\tau )|}^{2}d\tau +{\int }_0^{kT}{|\dot{u}(s)|}^{2}ds)}^{1/2}.\end{array}$$

Hence, we have

$${\parallel u\parallel }_{\mathrm{\infty }}\le {(\frac{1}{kT}+\frac{kT}{12})}^{1/2}{({\int }_0^{kT}{|u(s)|}^{2}ds+{\int }_0^{kT}{|\dot{u}(s)|}^{2}ds)}^{1/2}.$$

The proof is complete. □

Lemma 2.2 (see [[17], Lemma 2.2])

Assume that $F=F(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is T-periodic in t, $F(t,x)$ is measurable in t for every $x\in {\mathbb{R}}^N$ and continuously differentiable in x for a.e. $t\in [0,T]$. If there exist $a\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and $b\in L^p([0,T],{\mathbb{R}}^{+})$ ($p>1$) such that

$$|\mathrm{\nabla }F(t,x)|\le a(|x|)b(t),\mathrm{\forall }x\in {\mathbb{R}}^N,\mathit{\text{a.e.}}t\in [0,T],$$

(2.2)

then

$$c(u)={\int }_0^{kT}F(t,u(t))dt$$

is weakly continuous and uniformly differentiable on bounded subsets of $H_{kT}^1$.

Remark 2.1 In [[17], Lemma 2.2], $F\in C^1(\mathbb{R},{\mathbb{R}}^N)$. In fact, in its proof, it is not essential that F is continuously differentiable in t.

We use Lemma 2.3 below due to Benci and Rabinowitz [20] to prove our results.

Lemma 2.3 (see [20] or [[5], Theorem 5.29])

Let E be a real Hilbert space with $E=E_1\oplus E_2$ and $E_2=E_1^{\perp }$. Suppose that $\phi \in C^1(E,\mathbb{R})$ satisfies (PS)-condition, and

(I₁) $\phi (u)=1/2(\mathrm{\Phi }u,u)+b(u)$, where $\mathrm{\Phi }u={\mathrm{\Phi }}_1{P}_{1}u+{\mathrm{\Phi }}_2{P}_{2}u$ and ${\mathrm{\Phi }}_i:E_i\to E_i$ bounded and self-adjoint, $i=1,2$;

(I₂) $b^{\prime }$ is compact, and

(I₃) there exists a subspace $\tilde{E}\subset E$ and sets $S\subset E$, $Q\subset \tilde{E}$ and constants $\alpha >\beta$ such that

• (i) $S\subset E_1$ and $\phi {|}_S\ge \alpha$,

• (ii) Q is bounded and $\phi {|}_{\partial Q}\le \beta$,

• (iii) S and ∂Q link.

Then φ possesses a critical value $c\ge \alpha$ which can be characterized as

$$c=\underset{h\in \mathrm{\Gamma }}{inf}\underset{u\in Q}{sup}\phi (h(1,u)),$$

where

$$\mathrm{\Gamma }\equiv \{h\in C([0,1]\times E,E)|h\mathit{\text{satisfies the following}}({\mathrm{\Gamma }}_1)\text{-}({\mathrm{\Gamma }}_3)\},$$

$({\mathrm{\Gamma }}_1)$ $h(0,u)=u$,

$({\mathrm{\Gamma }}_2)$ $h(t,u)=u$ for $u\in \partial Q$, and

$({\mathrm{\Gamma }}_3)$ $h(t,u)=e^{\theta (t,u)\mathrm{\Phi }}u+K(t,u)$, where $\theta \in C([0,1]\times E,\mathbb{R})$ and K is compact.

Remark 2.2 As shown in [21], a deformation lemma can be proved with replacing the usual (PS)-condition with condition (C), and it turns out that Lemma 2.3 holds true under condition (C). We say φ satisfies condition (C), i.e., for every sequence $\{u_n\}\subset H_T^1$, $\{u_n\}$ has a convergent subsequence if $\phi (u_n)$ is bounded and $(1+\parallel u_n\parallel )\parallel {\phi }^{\prime }(u_n)\parallel \to 0$ as $n\to \mathrm{\infty }$.

3 Proofs of theorems

Proof of Theorem 1.1 It follows from Assumption (A)′ that the functional ${\phi }_k$ on $H_{kT}^1$ given by

$${\phi }_k(u)=\frac{1}{2}{\int }_0^{kT}{|\dot{u}(t)|}^{2}dt-\frac{1}{2}{\int }_0^{kT}(\mathit{Au}(t),u(t))dt-{\int }_0^{kT}F(t,u(t))dt$$

is continuously differentiable. Moreover, one has

$$〈{\phi }_k^{\prime }(u),v〉={\int }_0^{kT}[(\dot{u}(t),\dot{v}(t))-(\mathit{Au}(t),v(t))-(\mathrm{\nabla }F(t,u(t)),v(t))]dt$$

for $u,v\in H_{kT}^1$ and the solutions of system (1.1) correspond to the critical points of ${\phi }_k$ (see [1]).

Obviously, there exists an orthogonal matrix Q such that

$$Q^{\tau }AQ=B=\left(\begin{array}{c}{\lambda }_1\\ \ddots \\ {\lambda }_r\hfill \\ 0\hfill \\ \ddots \hfill \\ 0\hfill \\ -{\lambda }_{r+s+1}\hfill \\ \ddots \hfill \\ -{\lambda }_N\hfill \end{array}\right)$$

(3.1)

Let $u=Qw$. Then by (1.1),

$$Q\ddot{w}(t)+AQw(t)+\mathrm{\nabla }F(t,Qw(t))=0\text{a.e. }t\in \mathbb{R}.$$

Furthermore

$$\ddot{w}(t)+Q^{-1}AQw(t)+Q^{-1}\mathrm{\nabla }F(t,Qw(t))=0\text{a.e. }t\in \mathbb{R},$$

that is,

$$\ddot{w}(t)+Bw(t)+Q^{-1}\mathrm{\nabla }F(t,Qw(t))=0\text{a.e. }t\in \mathbb{R}.$$

(3.2)

Let $G(t,w)=F(t,Qw)$ and then $\mathrm{\nabla }G(t,w)=Q^{-1}\mathrm{\nabla }F(t,Qw(t))$. Let

$${\psi }_k(w)=\frac{1}{2}{\int }_0^{kT}{|\dot{w}(t)|}^{2}dt-\frac{1}{2}{\int }_0^{kT}(Bw(t),w(t))dt-{\int }_0^{kT}G(t,w(t))dt.$$

Then the critical points of ${\psi }_k$ correspond to solutions of system (3.2). It is easy to verify that ${\phi }_k(u)={\psi }_k(w)$ and G satisfies all the conditions of Theorem 1.1 and Theorem 1.2 if F satisfies them. Hence, w is the critical point of ${\psi }_k$ if and only if $u=Qw$ is the critical point of ${\phi }_k$. Therefore, we only need to consider the special case that $A=B$ is the diagonal matrix defined by (3.1). We divide the proof into six steps.

Step 1: Decompose the space $H_{kT}^1$. Let

$$I_N=\left(\begin{array}{c}1\\ 1\\ \ddots \hfill \\ 1\hfill \end{array}\right)=(e_1,e_2,\dots ,e_N).$$

Note that

$$H_{kT}^1\subset \{\sum _{i=0}^{\mathrm{\infty }}(c_{i}cosi{k}^{-1}\omega t+d_{i}sini{k}^{-1}\omega t)|c_i,d_i\in {\mathbb{R}}^N,i=0,1,2\cdots \}.$$

Define

$$\begin{array}{c}{H}_{kT}^{-}=\{u\in H_{kT}^1|u=u(t)=\sum _{i=1}^r{e}_i\sum _{j=0}^{k{l}_i}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),c_{ij},d_{ij}\in \mathbb{R}\},\hfill \\ H_{kT}^0=\{u\in H_{kT}^1|u=u(t)=\sum _{i=r+1}^{r+s}{e}_i\sum _{j=0}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),c_{ij},d_{ij}\in \mathbb{R}\},\hfill \\ H_{kT}^{+}=\{u\in H_{kT}^1|u=u(t)=\sum _{i=1}^r{e}_i\sum _{j=k{l}_i+1}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t)\hfill \\ \phantom{H_{kT}^{+}=}+\sum _{i=r+s+1}^N{e}_i\sum _{j=0}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),c_{ij},d_{ij}\in \mathbb{R}\}.\hfill \end{array}$$

Then $H_{kT}^{-}$, $H_{kT}^0$ and $H_{kT}^{+}$ are closed subsets of $H_{kT}^1$ and

• (1)
  $$H_{kT}^1=H_{kT}^{-}\oplus H_{kT}^0\oplus H_{kT}^{+};$$
• (2)
  $$\begin{array}{c}{P}_k(u,v)=0,\mathrm{\forall }u\in H_{kT}^{-},v\in H_{kT}^0\oplus H_{kT}^{+},\text{ or}\hfill \\ P_k(u,v)=0,\mathrm{\forall }u\in H_{kT}^0,v\in H_{kT}^{-}\oplus H_{kT}^{+},\text{ or}\hfill \\ P_k(u,v)=0,\mathrm{\forall }u\in H_{kT}^{+},v\in H_{kT}^{-}\oplus H_{kT}^0,\hfill \end{array}$$

where

$$P_k(u,v)={\int }_0^{kT}[(\dot{u}(t),\dot{v}(t))-(\mathit{Au}(t),v(t))]dt,\mathrm{\forall }u,v\in H_{kT}^1.$$

Let

$$\begin{array}{c}{H}_{kT}^{01}=\{u\in H_{kT}^0|u=\sum _{i=r+1}^{r+s}{c}_{i0}{e}_i,c_{i0}\in \mathbb{R}\},\hfill \\ H_{kT}^{02}=\{u\in H_{kT}^0|u=u(t)=\sum _{i=r+1}^{r+s}{e}_i\sum _{j=1}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),c_{ij},d_{ij}\in \mathbb{R}\},\hfill \\ H_{kT}^{+1}=\{u\in H_{kT}^{+}|u=u(t)=\sum _{i=1}^r{e}_i\sum _{j=k{l}_i+1}^{k{l}_i+k-1}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),c_{ij},d_{ij}\in \mathbb{R}\},\hfill \\ H_{kT}^{+2}=\{u\in H_{kT}^{+}|u=u(t)=\sum _{i=1}^r{e}_i\sum _{j=k{l}_i+k}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t)\hfill \\ \phantom{H_{kT}^{+2}=}+\sum _{i=r+s+1}^N{e}_i\sum _{j=0}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),c_{ij},d_{ij}\in \mathbb{R}\}.\hfill \end{array}$$

Then

$$H_{kT}^0=H_{kT}^{01}\oplus H_{kT}^{02},H_{kT}^{+}=H_{kT}^{+1}\oplus H_{kT}^{+2},H_{kT}^1=H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{02}\oplus H_{kT}^{+1}\oplus H_{kT}^{+2}$$

and

$$P_k(u,v)=0,\mathrm{\forall }u\in H_{kT}^{+1},\mathrm{\forall }v\in H_{kT}^{+2}.$$

Remark 3.1 When $k=1$, it is easy to see $H_T^{+1}=\{0\}$.

Step 2: Let

$$q_k(u)=\frac{1}{2}{\int }_0^{kT}[{|\dot{u}(t)|}^2-(\mathit{Au}(t),u(t))]dt.$$

Next we consider the relationship between $q_k(u)$ and $\parallel u\parallel$ on those subspaces defined above. We only consider the case that (H3)(2) holds. For others, the conclusions are easy to be seen from the argument of this case.

• (a) For $\mathrm{\forall }u\in H_{kT}^{-}$, since

  $$u=u(t)=\sum _{i=1}^r{e}_i\sum _{j=0}^{k{l}_i}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),$$

then

$$\begin{array}{rcl}{q}_k(u)& =& \frac{1}{2}{\int }_0^{kT}[{|\dot{u}(t)|}^2-(\mathit{Au}(t),u(t))]dt\\ =& \frac{1}{2}{\int }_0^{kT}\left[\right(\sum _{i=1}^r{e}_i\sum _{j=0}^{k{l}_i}j{k}^{-1}\omega (d_{ij}cosj{k}^{-1}\omega t-c_{ij}sinj{k}^{-1}\omega t),\\ \sum _{i=1}^r{e}_i\sum _{j=0}^{k{l}_i}j{k}^{-1}\omega (d_{ij}cosj{k}^{-1}\omega t-c_{ij}sinj{k}^{-1}\omega t))\\ -(\sum _{i=1}^{r}A{e}_i\sum _{j=0}^{k{l}_i}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),\\ \sum _{i=1}^r{e}_i\sum _{j=0}^{k{l}_i}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t)\left)\right]dt\\ =& \frac{1}{2}\sum _{i=1}^r{\int }_0^{kT}\{{\left[\sum _{j=0}^{k{l}_i}j{k}^{-1}\omega (d_{ij}cosj{k}^{-1}\omega t-c_{ij}sinj{k}^{-1}\omega t)\right]}^2\\ -{\lambda }_i{\left[\sum _{j=0}^{k{l}_i}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t)\right]}^2\}dt\\ =& \frac{kT}{4}\sum _{i=1}^r\sum _{j=0}^{k{l}_i}[{\left(j{k}^{-1}\omega \right)}^2-{\lambda }_i](c_{ij}^2+d_{ij}^2)\end{array}$$

and

$${\parallel u\parallel }^2={\int }_0^{kT}({|\dot{u}(t)|}^2+{|u(t)|}^2)dt=\frac{kT}{2}\sum _{i=1}^r\sum _{j=0}^{k{l}_i}[{\left(j{k}^{-1}\omega \right)}^2+1](c_{ij}^2+d_{ij}^2).$$

Let

$$\delta =\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{\lambda }_i-{(l_i\omega )}^2}{{(l_i\omega )}^2+1}\right\}>0.$$

Then

$$q_k(u)\le -\frac{\delta }{2}{\parallel u\parallel }^2,\mathrm{\forall }u\in H_{kT}^{-}.$$

(3.3)

Remark 3.2 Obviously, if one of (H3)(5) and (H3)(6) holds, then $H_{kT}^{-}=\{0\}$. Hence,

$$q_k(u)=0,\mathrm{\forall }u\in H_{kT}^{-}.$$

• (b) For $\mathrm{\forall }u\in H_{kT}^{+2}\oplus H_{kT}^{02}$, let

  $$u=u(t)=u_1(t)+u_2(t)+u_3(t),$$

where

$$\begin{array}{c}{u}_1(t)=\sum _{i=1}^r{e}_i\sum _{j=k{l}_i+k}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),\hfill \\ u_2(t)=\sum _{i=r+s+1}^N{e}_i\sum _{j=0}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),\hfill \\ u_3(t)=\sum _{i=r+1}^{r+s}{e}_i\sum _{j=1}^{\mathrm{\infty }}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t).\hfill \end{array}$$

Then

$$\begin{array}{rcl}{q}_k(u)& =& \frac{1}{2}{\int }_0^{kT}[{|\dot{u}(t)|}^2-(\mathit{Au}(t),u(t))]dt\\ =& \frac{1}{2}{\int }_0^{kT}[({\dot{u}}_1(t)+{\dot{u}}_2(t)+{\dot{u}}_3(t),{\dot{u}}_1(t)+{\dot{u}}_2(t)+{\dot{u}}_3(t))\\ -(A{u}_1(t)+A{u}_2(t)+A{u}_3(t),u_1(t)+u_2(t)+u_3(t))]dt\\ =& \frac{1}{2}{\int }_0^{kT}[({\dot{u}}_1(t),{\dot{u}}_1(t))+({\dot{u}}_2(t),{\dot{u}}_2(t))+({\dot{u}}_3(t),{\dot{u}}_3(t))\\ -(A{u}_1(t),u_1(t))-(A{u}_2(t),u_2(t))-(A{u}_3(t),u_3(t))]\\ =& \frac{kT}{4}[\sum _{i=1}^r\sum _{j=k{l}_i+k}^{\mathrm{\infty }}{\left(j{k}^{-1}\omega \right)}^2(c_{ij}^2+d_{ij}^2)+\sum _{i=r+s+1}^N\sum _{j=0}^{\mathrm{\infty }}{\left(j{k}^{-1}\omega \right)}^2(c_{ij}^2+d_{ij}^2)\\ +\sum _{i=r+1}^{r+s}\sum _{j=1}^{\mathrm{\infty }}{\left(j{k}^{-1}\omega \right)}^2(c_{ij}^2+d_{ij}^2)\\ -\sum _{i=1}^r{\lambda }_i\sum _{j=k{l}_i+k}^{\mathrm{\infty }}(c_{ij}^2+d_{ij}^2)+\sum _{i=r+s+1}^N{\lambda }_i\sum _{j=0}^{\mathrm{\infty }}(c_{ij}^2+d_{ij}^2)]\\ =& \frac{kT}{4}\{\sum _{i=1}^r\sum _{j=k{l}_i+k}^{\mathrm{\infty }}[{\left(j{k}^{-1}\omega \right)}^2-{\lambda }_i](c_{ij}^2+d_{ij}^2)+\sum _{i=r+s+1}^N\sum _{j=0}^{\mathrm{\infty }}[{\left(j{k}^{-1}\omega \right)}^2+{\lambda }_i](c_{ij}^2+d_{ij}^2)\\ +\sum _{i=r+1}^{r+s}\sum _{j=1}^{\mathrm{\infty }}{\left(j{k}^{-1}\omega \right)}^2(c_{ij}^2+d_{ij}^2)\}\end{array}$$

and

$$\begin{array}{rcl}{\parallel u\parallel }^2& =& {\int }_0^{kT}({|\dot{u}(t)|}^2+{|u(t)|}^2)dt\\ =& \frac{kT}{2}\{\sum _{i=1}^r\sum _{j=k{l}_i+k}^{\mathrm{\infty }}[{\left(j{k}^{-1}\omega \right)}^2+1](c_{ij}^2+d_{ij}^2)\\ +\sum _{i=r+s+1}^N\sum _{j=0}^{\mathrm{\infty }}[{\left(j{k}^{-1}\omega \right)}^2+1](c_{ij}^2+d_{ij}^2)\\ +\sum _{i=r+1}^{r+s}\sum _{j=1}^{\mathrm{\infty }}[{\left(j{k}^{-1}\omega \right)}^2+1](c_{ij}^2+d_{ij}^2)\}.\end{array}$$

Since for fixed $i\in \{1,\dots ,r\}$,

$$f(j)=\frac{{(j{k}^{-1}\omega )}^2-{\lambda }_i}{{(j{k}^{-1}\omega )}^2+1}\text{and}g(j)=\frac{{(j{k}^{-1}\omega )}^2}{{(j{k}^{-1}\omega )}^2+1}$$

are strictly increasing on $j\in \mathbb{N}$,

$$f(j)\ge f(k{l}_i+k)=\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}>0,\mathrm{\forall }j\ge k{l}_i+k$$

and

$$g(j)\ge g(1)=\frac{{(k^{-1}\omega )}^2}{{(k^{-1}\omega )}^2+1}=\frac{{\omega }^2}{{\omega }^2+k^2}>0.$$

Moreover, it is easy to verify that

$$\frac{{(j{k}^{-1}\omega )}^2+{\lambda }_i}{{(j{k}^{-1}\omega )}^2+1}\ge \frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}},\mathrm{\forall }j\in \mathbb{N}\cup \{0\},i=r+s+1,\dots ,N.$$

Let

$${\sigma }_k=min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\omega }^2}{{\omega }^2+k^2},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\}.$$

Then

$$q_k(u)\ge \frac{{\sigma }_k}{2}{\parallel u\parallel }^2,\mathrm{\forall }u\in H_{kT}^{+2}\oplus H_{kT}^{02}.$$

(3.4)

Remark 3.3 From the above discussion, it is easy to see the following conclusions:

• (i) if (H3)(1) holds, then (3.4) holds with

  $${\sigma }_k=min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\omega }^2}{{\omega }^2+k^2}\};$$

• (ii) if (H3)(2) holds, then (3.4) holds with

  $${\sigma }_k=min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\omega }^2}{{\omega }^2+k^2},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\};$$

• (iii) if (H3)(3) holds, then (3.4) holds with

  $${\sigma }_k\equiv \sigma =min\{\underset{i\in \{1,\dots ,r\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\};$$

• (iv) if (H3)(4) holds, then (3.4) holds with

  $${\sigma }_k\equiv \sigma =\underset{i\in \{1,\dots ,N\}}{min}\left\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\right\};$$

• (v) if (H3)(5) holds, then (3.4) holds with

  $${\sigma }_k=min\{\frac{{\omega }^2}{{\omega }^2+k^2},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\};$$

• (vi) if (H3)(6) holds, then (3.4) holds with

  $${\sigma }_k\equiv \sigma =\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}.$$

• (c) For $\mathrm{\forall }u\in H_{kT}^{+1}$, since

  $$\begin{array}{c}u=\sum _{i=1}^r{e}_i\sum _{j=k{l}_i+1}^{k{l}_i+k-1}(c_{ij}cosj{k}^{-1}\omega t+d_{ij}sinj{k}^{-1}\omega t),\hfill \\ q_k(u)=\frac{kT}{4}\sum _{i=1}^r\sum _{j=k{l}_i+1}^{k{l}_i+k-1}[{\left(j{k}^{-1}\omega \right)}^2-{\lambda }_i](c_{ij}^2+d_{ij}^2)\hfill \end{array}$$

and

$${\parallel u\parallel }^2=\frac{kT}{2}\sum _{i=1}^r\sum _{j=k{l}_i+1}^{k{l}_i+k-1}[{\left(j{k}^{-1}\omega \right)}^2+1](c_{ij}^2+d_{ij}^2).$$

Obviously, when $k=1$, $u=0$. So $q_1(u)=0$. When $k>1$, it follows from

$${(l_i+1-\frac{1}{k})}^2{\omega }^2\le {\lambda }_i<{(l_i+1)}^2{\omega }^2,\mathrm{\forall }i\in \{1,\dots ,r\}$$

that

$$q_k(u)\le 0,\mathrm{\forall }u\in H_{kT}^{+1}.$$

(3.5)

• (d) Obviously, for $\mathrm{\forall }u\in H_{kT}^{01}$, we have

  $$q_k(u)=0,\mathrm{\forall }u\in H_{kT}^{01}.$$

  (3.6)

Step 3: Assume that (H3)(2) holds. We prove that there exist ${\rho }_k>0$ and $b_k>0$ such that

$${\phi }_k(u)\ge b_k>0,\mathrm{\forall }u\in (H_{kT}^{+2}\oplus H_{kT}^{02})\cap \partial B_{{\rho }_k}.$$

Let

$$C_k=\sqrt{\frac{12+k^2{T}^2}{12kT}}.$$

Choosing ${\rho }_k=min\{1,l_k/C_k\}>0$ and $b_k=(\frac{{\sigma }_k}{2}-{\alpha }_k){\rho }_k^2>0$, by Lemma 2.1, (H4) and (3.4), we have, for all $u\in (H_{kT}^{+2}\oplus H_{kT}^{02})\cap \partial B_{{\rho }_k}$,

$$\begin{array}{rcl}{\phi }_k(u)& \ge & \frac{1}{2}{\int }_0^{kT}{|\dot{u}(t)|}^{2}dt-\frac{1}{2}{\int }_0^{kT}(\mathit{Au}(t),u(t))dt-{\int }_0^{kT}F(t,u(t))dt\\ \ge & \frac{{\sigma }_k}{2}{\parallel u\parallel }^2-{\alpha }_k{\int }_0^{kT}{|u(t)|}^{2}dt\\ \ge & (\frac{{\sigma }_k}{2}-{\alpha }_k){\parallel u\parallel }^2\\ =& (\frac{min\{{min}_{i\in \{1,\dots ,r\}}\{\frac{{(l_i+1)}^2{\omega }^2-{\lambda }_i}{{(l_i+1)}^2{\omega }^2+1}\},\frac{{\omega }^2}{{\omega }^2+k^2},\frac{{\lambda }_{i_{-}}}{1+{\lambda }_{i_{+}}}\}}{2}-{\alpha }_k){\rho }_k^2.\end{array}$$

For cases (H3)(1) and (H3)(3)-(H3)(6), correspondingly, by (H4) and Remark 3.3, similar to the above argument, we can also obtain that

$${\phi }_k(u)\ge (\frac{{\sigma }_k}{2}-{\alpha }_k){\rho }_k^2>0,\mathrm{\forall }u\in (H_{kT}^{+2}\oplus H_{kT}^{02})\cap \partial B_{{\rho }_k}.$$

Step 4: Let

$$Q_k=\{s{h}_k|s\in [0,s_1]\}\oplus (B_{s_2}\cap (H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1})),$$

where $h_k\in H_{kT}^{+2}\oplus H_{kT}^{02}$, $s_1$ and $s_2$ will be determined later. In this step, we prove ${\phi }_k{|}_{\partial Q_k}\le 0$. We only consider the case that F satisfies (H3)(2). For other cases, the results can be seen easily from the argument of case (H3)(2).

Assume that F satisfies (H3)(2). Let

$$d_k=min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2k^2},\frac{{\lambda }_{i_{-}}}{2}\}.$$

Case (i): if

$$d_k:=d=\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},$$

then we choose

$$h_k(t)=sin(l_{i_0}+1)\omega t\cdot e_{i_0},\mathrm{\forall }t\in \mathbb{R}.$$

Obviously, $h_k\in H_{kT}^{+2}$ and ${\dot{h}}_k(t)=(l_{i_0}+1)\omega cos(l_{i_0}+1)\omega t\cdot e_{i_0}$, $\mathrm{\forall }t\in \mathbb{R}$. Then

$${\parallel h_k\parallel }_{L^2}^2=\frac{kT}{2},{\parallel {\dot{h}}_k\parallel }_{L^2}^2=\frac{kT{(l_{i_0}+1)}^2{\omega }^2}{2}.$$

By (H3)(2), (1.5) and the periodicity of F, we have

$$F(t,x)\ge {\beta }_k{|x|}^2-{\beta }_k{\hat{L}}_k^2=(d+{\epsilon }_{0k}){|x|}^2-{\beta }_k{\hat{L}}_k^2,\mathrm{\forall }x\in {\mathbb{R}}^N,\text{ a.e. }t\in [0,kT],$$

(3.7)

where ${\epsilon }_{0k}={\beta }_k-d>0$ and ${\hat{L}}_k>max\{1,L_k\}$. Since $H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}$ is the finite dimensional space, there exists a constant $K_{1k}>0$ such that

$$K_{1k}{\parallel u\parallel }^2\le {\parallel u\parallel }_{L^2}^2\le {\parallel u\parallel }^2,\mathrm{\forall }u\in H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}.$$

(3.8)

By (3.3), (3.5), (3.6), (3.7) and (3.8), we know that for all $s>0$ and $u=u^{-}+u^{01}+u^{+1}\in H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}$,

$$\begin{array}{rcl}{\phi }_k(s{h}_k+u)& \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}{\int }_0^{kT}{|{\dot{h}}_k(t)|}^{2}dt-\frac{{\lambda }_{i_0}{s}^2}{2}{\int }_0^{kT}{|h_k(t)|}^{2}dt\\ -{\int }_0^{kT}F(t,s{h}_k(t)+u(t))dt\\ \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}\cdot \frac{kT{(l_{i_0}+1)}^2{\omega }^2}{2}-\frac{{\lambda }_{i_0}{s}^2}{2}\cdot \frac{kT}{2}\\ -(d+{\epsilon }_{0k}){\int }_0^{kT}{|s{h}_k(t)+u(t)|}^{2}dt+{\beta }_k{\hat{L}}_k^{2}kT\\ =& -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}\cdot \frac{kT{(l_{i_0}+1)}^2{\omega }^2}{2}-\frac{{\lambda }_{i_0}{s}^2}{2}\cdot \frac{kT}{2}\\ -(d+{\epsilon }_{0k})(s^2{\parallel h_k\parallel }_{L^2}^2+{\parallel u\parallel }_{L^2}^2)+{\beta }_k{\hat{L}}_k^{2}kT\\ \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+(\frac{kT{(l_{i_0}+1)}^2{\omega }^2}{4}-\frac{{\lambda }_{i_0}kT}{4}-\frac{dkT}{2}-\frac{kT{\epsilon }_{0k}}{2})s^2\\ -(d+{\epsilon }_{0k}){\parallel u\parallel }_{L^2}^2+{\beta }_k{\hat{L}}_k^{2}kT\\ \le & -\frac{kT{\epsilon }_{0k}}{2}{s}^2-{\epsilon }_{0k}{\parallel u\parallel }_{L^2}^2+{\beta }_k{\hat{L}}_k^{2}kT\\ \le & -\frac{kT{\epsilon }_{0k}}{2}{s}^2-{\epsilon }_{0k}{K}_{1k}{\parallel u\parallel }^2+{\beta }_k{\hat{L}}_k^{2}kT.\end{array}$$

(3.9)

Hence,

$${\phi }_k(s{h}_k+u)\le 0,\text{either }s\ge s_1\text{ or }\parallel u\parallel \ge s_2,$$

where

$$s_1=\sqrt{\frac{2{\beta }_k{\hat{L}}_k^2}{{\epsilon }_{0k}}},s_2=\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}{K}_{1k}}}.$$

Case (ii): if $d_k={\omega }^2/(2k^2)$, then we choose

$$h_k(t)=sin{k}^{-1}\omega t\cdot e_{r+1}\in H_{kT}^{02},\mathrm{\forall }t\in \mathbb{R}.$$

Then

$${\dot{h}}_k(t)=\frac{\omega }{k}cos{k}^{-1}\omega t\cdot e_{r+1},\mathrm{\forall }t\in \mathbb{R},$$

and

$$(A{h}_k,h_k)=0,{\parallel h_k\parallel }_{L^2}^2=\frac{kT}{2},{\parallel {\dot{h}}_k\parallel }_{L^2}^2=\frac{T{\omega }^2}{2k}.$$

(3.10)

By (H3)(2), (1.5) and the periodicity of F, we have

$$F(t,x)\ge {\beta }_k{|x|}^2-{\beta }_k{\hat{L}}_k^2=(\frac{{\omega }^2}{2k^2}+{\epsilon }_{0k}^{\prime }){|x|}^2-{\beta }_k{\hat{L}}_k^2,\mathrm{\forall }x\in {\mathbb{R}}^N,\text{ a.e. }t\in [0,T],$$

(3.11)

where ${\hat{L}}_k>max\{1,L_k\}$ and ${\epsilon }_{0k}^{\prime }={\beta }_k-\frac{{\omega }^2}{2k^2}$. By (3.3), (3.5), (3.6), (3.8) and (3.11), we know that for all $s>0$ and $u=u^{-}+u^{01}+u^{+1}\in H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}$,

$$\begin{array}{rcl}{\phi }_k(s{h}_k+u)& \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}{\int }_0^{kT}{|{\dot{h}}_k(t)|}^{2}dt-{\int }_0^{kT}F(t,s{h}_k(t)+u)dt\\ \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}\cdot \frac{T{\omega }^2}{2k}-(\frac{{\omega }^2}{2k^2}+{\epsilon }_{0k}^{\prime }){\int }_0^{kT}{|s{h}_k(t)+u(t)|}^{2}dt\\ +{\beta }_k{\hat{L}}_k^{2}kT\\ =& -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}\cdot \frac{T{\omega }^2}{2k}-(\frac{{\omega }^2}{2k^2}+{\epsilon }_{0k}^{\prime })(s^2{\parallel h_k\parallel }_{L^2}^2+{\parallel u\parallel }_{L^2}^2)\\ +{\beta }_k{\hat{L}}_k^{2}kT\\ =& -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+(\frac{T{\omega }^2}{4k}-\frac{T{\omega }^2}{4k}-\frac{kT{\epsilon }_{0k}^{\prime }}{2})s^2-(\frac{{\omega }^2}{2k^2}+{\epsilon }_{0k}^{\prime }){\parallel u\parallel }_{L^2}^2\\ +{\beta }_k{\hat{L}}_k^{2}kT\\ \le & -\frac{kT{\epsilon }_{0k}^{\prime }}{2}{s}^2-{\epsilon }_{0k}^{\prime }{\parallel u\parallel }_{L^2}^2+{\beta }_k{\hat{L}}_k^{2}kT\\ \le & -\frac{kT{\epsilon }_{0k}^{\prime }}{2}{s}^2-{\epsilon }_{0k}^{\prime }{K}_{1k}{\parallel u\parallel }^2+{\beta }_k{\hat{L}}_k^{2}kT.\end{array}$$

Hence,

$${\phi }_k(s{h}_k+u)\le 0,\text{either }s\ge s_1\text{ or }\parallel u\parallel \ge s_2,$$

where

$$s_1=\sqrt{\frac{2{\beta }_k{\hat{L}}_k^2}{{\epsilon }_{0k}^{\prime }}},s_2=\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}^{\prime }{K}_{1k}}}.$$

Case (iii): if $d_k={\lambda }_{i_{-}}/2$, then we choose

$$h_k=\frac{1}{\sqrt{kT}}\cdot e_{i_{-}}\in H_{kT}^{+2}.$$

Then

$${\dot{h}}_k=0,(A{h}_k,h_k)=-{\lambda }_{i_{-}}(h_k,h_k),{\parallel h_k\parallel }_{L^2}^2=1.$$

By (H3)(2), (1.5) and the periodicity of F, we have

$$F(t,x)\ge {\beta }_k{|x|}^2-{\beta }_k{\hat{L}}_k^2=(\frac{{\lambda }_{i_{-}}}{2}+{\epsilon }_{0k}^{″}){|x|}^2-{\beta }_k{\hat{L}}_k^2,\mathrm{\forall }x\in {\mathbb{R}}^N,\text{ a.e. }t\in [0,kT],$$

(3.12)

where ${\hat{L}}_k>max\{\sqrt{1+\frac{1}{T}},L_k\}$ and ${\epsilon }_{0k}^{″}={\beta }_k-{\lambda }_{i_{-}}/2$. By (3.3), (3.5), (3.6), (3.8) and (3.12), for all $s>0$ and $u=u^{-}+u^{01}+u^{+1}\in H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}$, we have

$$\begin{array}{rcl}{\phi }_k(s{h}_k+u)& \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{s^2}{2}{\int }_0^{kT}{|{\dot{h}}_k(t)|}^{2}dt+\frac{{\lambda }_{i_{-}}{s}^2}{2}{\int }_0^{kT}{|h_k(t)|}^{2}dt\\ -{\int }_0^{kT}F(t,s{h}_k(t)+u(t))dt\\ \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{{\lambda }_{i_{-}}{s}^2}{2}-(\frac{{\lambda }_{i_{-}}}{2}+{\epsilon }_{0k}^{″}){\int }_0^{kT}{|s{h}_k(t)+u(t)|}^{2}dt+{\beta }_k{\hat{L}}_k^{2}kT\\ =& -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+\frac{{\lambda }_{i_{-}}{s}^2}{2}-(\frac{{\lambda }_{i_{-}}}{2}+{\epsilon }_{0k}^{″})(s^2{\parallel h_k\parallel }_{L^2}^2+{\parallel u\parallel }_{L^2}^2)+{\beta }_k{\hat{L}}_k^{2}kT\\ =& -\frac{\delta }{2}{\parallel u^{-}\parallel }^2+(\frac{{\lambda }_{i_{-}}}{2}-\frac{{\lambda }_{i_{-}}}{2}-{\epsilon }_{0k}^{″})s^2-(\frac{{\lambda }_{i_{-}}}{2}+{\epsilon }_{0k}^{″}){\parallel u\parallel }_{L^2}^2+{\beta }_k{\hat{L}}_k^{2}kT\\ \le & -{\epsilon }_{0k}^{″}{s}^2-{\epsilon }_{0k}^{″}{K}_{1k}{\parallel u\parallel }^2+{\beta }_k{\hat{L}}_k^{2}kT.\end{array}$$

Hence,

$${\phi }_k(s{e}_k+u)\le 0,\text{either }s\ge s_1\text{ or }\parallel u\parallel \ge s_2,$$

where

$$s_1=\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}^{″}}},s_2=\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}^{″}{K}_{1k}}}.$$

Combining cases (i), (ii) and (iii), if we let

$$\begin{array}{c}{s}_1=max\{\sqrt{\frac{2{\beta }_k{\hat{L}}_k^2}{{\epsilon }_{0k}}},\sqrt{\frac{2{\beta }_k{\hat{L}}_k^2}{{\epsilon }_{0k}^{\prime }}},\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}^{″}}}\},\hfill \\ s_2=max\{\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}{K}_{1k}}},\sqrt{\frac{{\beta }_k{\hat{L}}_k^{2}kT}{{\epsilon }_{0k}^{\prime }{K}_{1k}}},\sqrt{\frac{{\beta }_k{\hat{L}}_k^2}{{\epsilon }_{0k}^{″}{K}_{1k}}}\},\hfill \end{array}$$

then

$${\phi }_k(s{h}_k+u)\le 0,\text{either }s\ge s_1\text{ or }\parallel u\parallel \ge s_2.$$

(3.13)

By (1.5), (3.3), (3.5) and (3.6), for all $u\in H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}$, we have

$$\begin{array}{rcl}{\phi }_k(u)& =& \frac{1}{2}{\int }_0^{kT}{|\dot{u}(t)|}^{2}dt-\frac{1}{2}{\int }_0^{kT}(\mathit{Au}(t),u(t))dt-{\int }_0^{kT}F(t,u(t))dt\\ \le & -\frac{\delta }{2}{\parallel u^{-}\parallel }^2\\ \le & 0.\end{array}$$

(3.14)

Thus, it follows from (3.13) and (3.14) that $\phi {|}_{\partial Q_k}\le 0<b_k$.

Step 5: We prove that ${\phi }_k$ satisfies (C)-condition in $H_{kT}^1$. The proof is similar to that in Theorem 1.1 in [17]. We omit it.

Step 6: We claim that ${\phi }_k$ has a nontrivial critical point $u_k\in H_{kT}^1$ such that ${\phi }_k(u_k)\ge b_k>0$. Especially, we claim that, for cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, (1.5) implies that $u_k$ is nonconstant.

In fact, it is easy to see that

$$\begin{array}{rcl}{q}_k(u)& =& \frac{1}{2}{\parallel u\parallel }^2-\frac{1}{2}{\int }_0^{kT}((A+I)u(t),u(t))dt\\ =& \frac{1}{2}〈(I-K)u,u)〉,\end{array}$$

where $K:H_{kT}^1\to H_{kT}^1$ is the linear self-adjoint operator defined, using the Riesz representation theorem, by

$${\int }_0^{kT}((A+I)u(t),v(t))dt=〈(Ku,v)〉,\mathrm{\forall }u,v\in H_T^1.$$

The compact imbedding of $H_{kT}^1$ into $C([0,kT];{\mathbb{R}}^N)$ implies that K is compact. In order to use Lemma 2.3, we let $\mathrm{\Phi }=I-K$ and define ${\mathrm{\Phi }}_i:E_i\to E_i$, $i=1,2$ by

$$〈{\mathrm{\Phi }}_{i}u,v〉=〈(I-K)u,v〉,u,v\in E_i,$$

where $E_1=H_{kT}^{+2}\oplus H_{kT}^{02}$ and $E_2=H_{kT}^{-}\oplus H_{kT}^{01}\oplus H_{kT}^{+1}$. Since K is a self-adjoint compact operator, it is easy to see that ${\mathrm{\Phi }}_i$ ($i=1,2$) are bounded and self-adjoint. Let

$$b(u)=-{\int }_0^{kT}F(t,u(t))dt.$$

Assumption (A)′ and Lemma 2.2 imply that b is weakly continuous and is uniformly differentiable on bounded subsets of $E=H_{kT}^1$. Furthermore, by standard theorems in [22], we conclude that $b^{\prime }$ is compact. Let $S_k=(H_{kT}^{+2}\oplus H_{kT}^{02})\cap \partial B_{{\rho }_k}$. Then $S_k$ and $\partial Q_k$ link. Hence, by Step 1-Step 5, Lemma 2.3 and Remark 2.2, there exists a critical point $u_k\in H_{kT}^1$ such that ${\phi }_k(u_k)\ge b_k>0$, which implies that $u_k$ is nonzero. For cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, it follows from (1.5) that $u_k$ is nonconstant. The proof is complete.  □

Proof of Theorem 1.2 Obviously, when $r=0$, $s>0$ and $s<N$, (H1) holds for any $k\in \mathbb{N}$. Moreover, since (H3)′ implies that (H3)(5) and (H4)′ implies that (H4), system (1.1) has kT-periodic solution for every $k\in \mathbb{N}$.

Let $d=\frac{{\omega }^2}{2}$. Like the argument of case (ii) in the proof of Theorem 1.1, choose

$$e_k(t)=sin{k}^{-1}\omega t{e}_{r+1}\in H_{kT}^{02},\mathrm{\forall }t\in \mathbb{R}.$$

By (H3)′, (1.5) and the T-periodicity of F, we have

$$F(t,x)\ge \beta {|x|}^2-\beta L^2=(\frac{{\omega }^2}{2}+{\epsilon }_1){|x|}^2-\beta L^2,\mathrm{\forall }x\in {\mathbb{R}}^N,\text{ a.e. }t\in [0,kT],$$

(3.15)

where ${\epsilon }_1=\beta -\frac{{\omega }^2}{2}$. In the proof of Theorem 1.1, if we replace (3.15) with (3.11), then we obtain

$${\phi }_k(s{e}_k+u)\le 0,\text{either }s\ge s_1\text{ or }\parallel u\parallel \ge s_2,$$

where

$$s_1=\sqrt{\frac{2\beta L^2}{{\epsilon }_1}}=\sqrt{\frac{2\beta L^2}{\beta -\frac{{\omega }^2}{2}}},s_2=\sqrt{\frac{\beta L^{2}kT}{{\epsilon }_1{K}_{1k}}}.$$

Note that $s_1$ is independent of k. Hence, if $u_k$ is the critical point of ${\phi }_k$, then it follows from (3.3), (3.5), (3.6), the definitions of critical value c in Lemma 2.3 and $Q_k$ that

$$\begin{array}{rcl}{\phi }_k(u_k)& \le & \underset{u\in Q_k}{sup}{\phi }_k(u)\\ \le & \underset{s\in [0,s_1]}{sup}\{\frac{s^2}{2}{\int }_0^{kT}{|{\dot{e}}_k(t)|}^{2}dt-\frac{s^2}{2}{\int }_0^{kT}(A{e}_k(t),e_k(t))dt\}\\ \le & \frac{s_1^2}{2}{\int }_0^{kT}{|{\dot{e}}_k(t)|}^{2}dt\\ =& \frac{\beta L^{2}T{\omega }^2}{2k(\beta -\frac{{\omega }^2}{2})}\\ \le & \frac{\beta L^{2}T{\omega }^2}{2(\beta -\frac{{\omega }^2}{2})}:=M.\end{array}$$

(3.16)

Hence, ${\phi }_k(u_k)$ is bounded for any $k\in \mathbb{N}$.

Obviously, we can find $k_1\in \mathbb{N}/\{1\}$ such that $k_1>\frac{M}{b_1}$, then we claim that $u_k$ is distinct from $u_1$ for all $k\ge k_1$. In fact, if $u_k=u_1$ for some $k\ge k_1$, it is easy to check that

$${\phi }_k(u_k)=k{\phi }_1(u_1)\ge k{b}_1.$$

Then by (3.16), we have $k_1\le k\le \frac{M}{b_1}$, a contradiction. We also can find $k_2>max\{k_1,\frac{k_{1}M}{b_{k_1}}\}$ such that $u_{k_{1}k}\ne u_{k_1}$ for all $k\ge \frac{k_2}{k_1}$. Otherwise, if $u_{k_{1}k}=u_{k_1}$ for some $k\ge k_1$, we have ${\phi }_{k_{1}k}(u_{k_{1}k})=k{\phi }_{k_1}(u_{k_1})\ge k{b}_{k_1}$. Then by (3.16), we have $\frac{k_2}{k_1}\le k\le \frac{M}{b_{k_1}}$, a contradiction. In the same way, we can obtain that system (1.1) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$. The proof is complete. □

Proof of Theorem 1.3 Except for verifying (C) condition, the proof is the same as in Theorem B (that is Theorem 1 in [15]). To verify (C) condition, we only need to prove the sequence $\{u_n\}$ is bounded if $\phi (u_n)$ is bounded and $\parallel {\phi }^{\prime }(u_n)\parallel (\parallel 1+\parallel u_n\parallel )\to 0$ as $n\to +\mathrm{\infty }$. Other proofs are the same as in [15]. The proof of boundedness of $\{u_n\}$ is essentially the same as in Theorem 1.1 in [17] except that 2 is replaced by p, $H_{kT}^1$ by

$$W_{kT}^{1,p}=\{u:\mathbb{R}\to {\mathbb{R}}^N|u\text{ is absolutely continuous},u(t)=u(t+T)\text{ and }\dot{u}\in L^p([0,T])\}$$

equipped with the norm

$$\parallel u\parallel ={({\int }_0^{kT}{|u(t)|}^{p}dt+{\int }_0^{kT}{|\dot{u}(t)|}^{p}dt)}^{1/p},$$

and

$$F(t,x)\ge {\beta }_k{|x|}^2,\mathrm{\forall }x\in {\mathbb{R}}^N,|x|>L$$

by

$$F(t,x)\ge \epsilon {|x|}^p,\mathrm{\forall }x\in {\mathbb{R}}^N,|x|>L$$

for some $\epsilon >0$. So, we omit the details. □

4 Examples

Example 4.1 Let $T=2\pi$ and

$$A=\left(\begin{array}{ccccc}7.5& 0& 0& 0& 0\\ 0& 7.4& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& -3& 0\\ 0& 0& 0& 0& -4\end{array}\right).$$

Then $\omega =1$, $r=2$, ${\lambda }_1=7.5$, ${\lambda }_2=7.4$, ${\lambda }_3=0$, ${\lambda }_4=-3$, ${\lambda }_5=-4$, ${\lambda }_{i_{+}}=4$ and ${\lambda }_{i_{-}}=3$. Obviously, the matrix A satisfies Assumption (A0) and $l_1=l_2=2$ such that

$$l_i^2{\omega }^2<{\lambda }_i<{(l_i+1)}^2{\omega }^2,i=1,2.$$

It is easy to verify that (H1) holds with $k=1,2,3$. Let

$$F(t,x)\equiv \frac{4}{63k^2}{|x|}^2({11}^{\frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}}-\frac{1}{2})\text{a.e. }t\in [0,T].$$

Then $F(t,x)\ge 0$ for all $x\in {\mathbb{R}}^N$ and a.e. $t\in [0,T]$ and

$$\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^2}=\frac{2}{63k^2}\text{uniformly for a.e. }t\in [0,T],$$

(4.1)

$$\underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^2}=\frac{2}{3k^2}\text{uniformly for a.e. }t\in [0,T].$$

(4.2)

It is easy to verify that

$$(\mathrm{\nabla }F(t,x),x)-2F(t,x)=\frac{6ln11}{63k^2}{|x|}^2\cdot {11}^{\frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}}\cdot \frac{{|x|}^{3/2}}{{(1+{|x|}^{3/2})}^2}.$$

Choose $\xi =1$, $\eta =1$ and $\nu =3/2$. Moreover, obviously, there exists $m>0$ such that $\frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}>\frac{2}{3}$. Then

$$\frac{(\mathrm{\nabla }F(t,x),x)-2F(t,x)}{\frac{F(t,x)}{\xi +\eta {|x|}^{\nu }}}=\frac{\frac{3}{2}ln11\cdot {11}^{\frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}}\cdot \frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}}{{11}^{\frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}}-\frac{1}{2}}>ln11>1.$$

Hence, (H2) holds.

When $k=1$,

$$min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2k^2},\frac{{\lambda }_{i_{-}}}{2}\}=\frac{1}{2}\text{and}{\sigma }_1=0.15.$$

By (4.2), we can find $L_1>0$ such that

$$F(t,x)\ge (\frac{2}{3}-\frac{1}{10}){|x|}^2=\frac{17}{30}{|x|}^2,\mathrm{\forall }|x|>L_1,\text{ and a.e. }t\in [0,T].$$

Let ${\beta }_1=\frac{17}{30}$. Then (H3)(2) holds with $k=1$. Moreover, by (4.1), we can find $l_1>0$ such that

$$F(t,x)\le (\frac{2}{63}+\frac{23}{2520}){|x|}^2\approx 0.0409{|x|}^2,\mathrm{\forall }|x|\le l_1\text{ and a.e. }t\in [0,T].$$

Let ${\alpha }_1=0.0409$. Then (H4) holds. By Theorem 1.1, we obtain that system (1.1) has a T-periodic solution.

When $k=2$,

$$min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2k^2},\frac{{\lambda }_{i_{-}}}{2}\}=\frac{1}{8}\text{and}{\sigma }_2=0.15.$$

By (4.2), we can find $L_2>0$ such that

$$F(t,x)\ge (\frac{1}{6}-\frac{1}{100}){|x|}^2\approx 0.1567{|x|}^2,\mathrm{\forall }|x|>L_2\text{ and a.e. }t\in [0,T].$$

Let ${\beta }_2=0.1567$. Then (H3)(2) holds with $k=2$. Moreover, by (4.1), we can find $l_2>0$ such that

$$F(t,x)\le (\frac{1}{126}+\frac{1}{1000}){|x|}^2\approx 0.00894{|x|}^2,\mathrm{\forall }|x|\le l_2\text{ and a.e. }t\in [0,T].$$

Let ${\alpha }_2=0.00894$. Then (H4) holds. Note that $\frac{1}{6}<\frac{1}{2}=min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2},\frac{{\lambda }_{i_{-}}}{2}\}$. So, when $k=2$, by Theorem 1.1, we cannot judge that system (1.1) has a T-periodic solution. However, we can obtain that system (1.1) has a 2T-periodic solution.

When $k=3$,

$$min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2k^2},\frac{{\lambda }_{i_{-}}}{2}\}=\frac{1}{18}\text{and}{\sigma }_3=0.1.$$

By (4.2), we can find $L_3>0$ such that

$$F(t,x)\ge (\frac{2}{27}-\frac{1}{100}){|x|}^2\approx 0.0641{|x|}^2,\mathrm{\forall }|x|>L_3\text{ and a.e. }t\in [0,T].$$

Let ${\beta }_3=0.0641$. Then (H3)(2) holds with $k=3$. Moreover, by (4.1), we can find $l_3>0$ such that

$$F(t,x)\le (\frac{2}{567}+\frac{1}{1000}){|x|}^2\approx 0.00453{|x|}^2,\mathrm{\forall }|x|\le l_3\text{ and a.e. }t\in [0,T].$$

Let ${\alpha }_3=0.00453$. Then (H4) holds. Note that $\frac{2}{27}<\frac{1}{8}=min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2\times 2^2},\frac{{\lambda }_{i_{-}}}{2}\}<\frac{1}{2}=min\{\frac{{(l_{i_0}+1)}^2{\omega }^2-{\lambda }_{i_0}}{2},\frac{{\omega }^2}{2},\frac{{\lambda }_{i_{-}}}{2}\}$. So, when $k=3$, by Theorem 1.1, we cannot judge that system (1.1) has T-periodic solution and 2T-periodic solution. However, we can obtain that system (1.1) has a 3T-periodic solution. It is easy to verify that Example 4.1 does not satisfy the theorem in [19] even if $k=1$.

Example 4.2 Let

$$A=\left(\begin{array}{ccc}0& 0& 0\\ 0& -1& 0\\ 0& 0& -2\end{array}\right)$$

and

$$F(t,x)\equiv \frac{2{\pi }^2}{T^2}{|x|}^2(e^{\frac{{|x|}^{3/2}}{1+{|x|}^{3/2}}}-1)\text{a.e. }t\in [0,T].$$

Then

$$\begin{array}{c}\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^2}=0\text{uniformly for a.e. }t\in [0,T],\hfill \\ \underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^2}=\frac{2{\pi }^2}{T^2}(e-1)\text{uniformly for a.e. }t\in [0,T].\hfill \end{array}$$

Obviously, (A0), (A)′, (1.5), (H3)′ and (H4)′ hold. Let $\xi =1$, $\eta =1$ and $\nu =\frac{3}{2}$. Similar to the argument in Example 4.1, we obtain (H2) also holds. Then by Theorem 1.2, system (1.1) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$.

Example 4.3 Let $p=4$ and

$$F(t,x)\equiv {|x|}^p(e^{{|x|}^p}-1)={|x|}^4(e^{{|x|}^4}-1)\text{a.e. }t\in [0,T].$$

Then (1.5) holds and

$$\underset{|x|\to 0}{lim}\frac{F(t,x)}{{|x|}^4}=0,\underset{|x|\to \mathrm{\infty }}{lim}\frac{F(t,x)}{{|x|}^4}=+\mathrm{\infty }\text{uniformly for a.e. }t\in [0,T].$$

Let $\xi =1$, $\eta =1$ and $\nu =1/2$. Then it is easy to obtain that there exists $m>1$ such that (H5) holds. By Theorem 1.3, system (1.8) has a sequence of distinct periodic solutions with period $k_{j}T$ satisfying $k_j\in \mathbb{N}$ and $k_j\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$. It is easy to see that Example 4.3 does not satisfy (1.3). Hence, Theorem 1.3 improved Theorem B.