Multiple periodic orbits of high-dimensional differential delay systems

Abstract

In this paper, we consider differential delay systems of the form

$$x'(t)=-\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr), $$

in which the coefficients of the nonlinear terms with different hysteresis have different signs. Such systems have not been studied before. The multiplicity of the periodic orbits is related to the eigenvalues of the limit matrix. The results provide a theoretical basis for the study of differential delay systems.

1 Introduction

We consider the asymptotically linear differential delay system

$$ x'(t)=-\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr), $$

(1.1)

where

$$ x\in R^{N},\qquad F\in C^{1} \bigl(R^{N},R \bigr), \qquad \nabla F(-x)=-\nabla F(x), $$

(1.2)

and there are real symmetric matrices $A_{0},A_{\infty}\in R^{N\times N}$ such that

$$ \nabla F(x)=A_{0}x+\circ \bigl( \vert x \vert \bigr),\quad \vert x \vert \rightarrow0, \qquad\nabla F(x)=A_{\infty}x+\circ \bigl( \vert x \vert \bigr), \quad \vert x \vert \rightarrow\infty. $$

(1.3)

In the past several decades, many papers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] have studied the existence of periodic solutions of delay differential equations. In 1974, Kaplan and Yorke [15] studied the multiple periodic solutions of the equations

$$ x'(t)=-f \bigl(x(t-1) \bigr) $$

(1.4)

and

$$ x'(t)=-f \bigl(x(t-1) \bigr)-f \bigl(x(t-2) \bigr) $$

(1.5)

by transforming them respectively into associated systems of ordinary differential equations and then making analysis by qualitative approaches. Meanwhile, they guessed that there should exist $2(n+1)$-periodic solutions to the equation

$$ x'(t)=-\sum_{i=1}^{n}f \bigl(x(t-i) \bigr), $$

(1.6)

where $f\in C^{0}(R,R)$ with $f(-x)=-f(x)$, $xf(x)>0$, $x\neq0$. This was proved in [17]. On the basis of this work, Fei [3, 4] studied the multiple periodic solutions of differential delay equations via Hamiltonian systems. Li and He [10,11,12] studied the multiple solutions by an asymptotically linear Hamiltonian system. Guo and Yu [13, 14] gave some multiple results for periodic solutions via critical point theory.

In this paper, our main purpose is to study system (1.1), in which the coefficients of nonlinear terms corresponding to different hysteresis have different signs, which is an extension of [3]. To construct the even functional, the variation structure here is much simpler since we do not transform system (1.1) into a $2kN$-dimensional system. At the same time, according to the variational method and the method of Kaplan–Yorke coupling system, we get an exact counting method of the number of 4k-periodic orbits. Moreover, our results are easier to examine by introducing the eigenvalues and eigenvectors of the matrices $A_{\infty}$ and $A_{0}$.

Let

$$\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{N} \quad\mbox{and}\quad \beta_{1}\leq\beta_{2}\leq\cdots\leq \beta_{N} $$

be the eigenvalues of $A_{0}$ and $A_{\infty}$, respectively, and let $u_{1},u_{2},\ldots,u_{N}$ and $v_{1},v_{2},\ldots,v_{N}$ be the corresponding unit eigenvectors in space. For convenience, we make the following assumptions:

$(f_{1})$:

F satisfies (1.2) and (1.3),

$(f_{2})$:

there are $M>0$ and a function $r\in C^{0}(R^{+},R^{+})$ satisfying $r(s)\rightarrow\infty$ and $r(s)/s\rightarrow0$ as $s\rightarrow\infty$ such that

$$\biggl\vert F(x)-\frac{1}{2}(A_{\infty} x,x) \biggr\vert >r \bigl( \vert x \vert \bigr)-M, $$

$(f_{3}^{\pm})$:

$\pm[F(x)-\frac{1}{2}(A_{\infty} x,x)]>0$, $|x|\rightarrow \infty$,

$(f_{4}^{\pm})$:

$\pm[F(x)-\frac{1}{2}(A_{0} x,x)]>0$, $0<|x|\ll1$.

2 Variational structure

Let

$$\begin{aligned}& \widehat{X}= \bigl\{ x\in L^{2}:x(t-2k)=-x(t) \bigr\} \\& \phantom{\widehat{X}}= \Biggl\{ \sum_{i=0}^{\infty}\biggl(a_{i}\cos \frac{(2i+1)\pi t}{2k}+b_{i}\sin \frac{(2i+1)\pi t}{2k} \biggr):a_{i},b_{i} \in R^{N} \Biggr\} , \\& X=\operatorname{cl} \Biggl\{ \sum_{i=0}^{\infty}\biggl(a_{i}\cos\frac{(2i+1)\pi t}{2k}+b_{i}\sin \frac{(2i+1)\pi t}{2k} \biggr): \\ & \phantom{X=}a_{i},b_{i}\in R^{N}, \sum _{i=0}^{\infty}(2i+1) \bigl(a_{i}^{2}+b_{i}^{2} \bigr)< \infty \Biggr\} \subset\widehat{X}, \end{aligned}$$

and define $P:X\rightarrow L^{2}$ by

$$\begin{aligned} Px(t) =&P \Biggl(\sum_{i=0}^{\infty}\biggl(a_{i}\cos\frac{(2i+1)\pi t}{2k}+b_{i}\sin \frac{(2i+1)\pi t}{2k} \biggr) \Biggr) \\ =&\sum_{i=0}^{\infty}(2i+1) \biggl(a_{i}\cos\frac{(2i+1)\pi t}{2k}+b_{i}\sin \frac{(2i+1)\pi t}{2k} \biggr) \end{aligned}$$

and the inverse of P as

$$P^{-1} x(t)=\sum_{i=0}^{\infty}\frac{1}{2i+1} \biggl(a_{i}\cos\frac{(2i+1)\pi t}{2k}+b_{i} \sin \frac{(2i+1)\pi t}{2k} \biggr). $$

Define

$$\begin{gathered} \langle x,y\rangle= \int_{0}^{4k} \bigl(Px(t),y(t) \bigr)\,dt, \quad \Vert x \Vert =\sqrt{\langle x,x\rangle} , \\ \langle x,y\rangle_{2}= \int_{0}^{4k} \bigl(x(t),y(t) \bigr)\,dt, \quad \Vert x \Vert _{2}=\sqrt {\langle x,x\rangle_{2}} . \end{gathered} $$

Then $(X,\|\cdot\|)$ is an $H^{\frac{1}{2}}_{4k}([0,4k],R^{N})$ space.

For system (1.1), define the following functional $\varPhi :X\rightarrow R$:

$$ \varPhi(x)=\frac{1}{2}\langle Lx,x\rangle+ \int _{0}^{4k}F \bigl(x(t) \bigr)\,dt, $$

(2.1)

where

$$ Lx=-P^{-1}\sum_{s=1}^{2k-1}x'(t-s). $$

(2.2)

Let

$$X(i)= \biggl\{ x(t)=a_{i}\cos\frac{(2i+1)\pi t}{2k}+b_{i}\sin \frac{(2i+1)\pi t}{2k}:a_{i},b_{i}\in R^{N} \biggr\} . $$

Then we have

$$ X=\sum_{h=0}^{\infty} \Biggl[\sum _{i=0}^{k-1} \bigl(X(2hk+i)+X(2hk+2k-i-1) \bigr) \Biggr]. $$

(2.3)

On the basis of Theorem 1.4 in [18], we can get that the differential of Φ satisfies

$$ \varPhi'(x)=Lx+K(x), $$

(2.4)

where $K(x)=P^{-1}\nabla F(x)$.

For convenience of further calculations, we can also make a more detailed division of space X by introducing the eigenvalues and eigenvectors mentioned before.

Suppose that

$$\begin{gathered} X_{0j}(i)= \biggl\{ x(t)= \biggl(a_{j}\cos \frac{(2i+1)\pi t}{2k}+b_{j}\sin \frac{(2i+1)\pi t}{2k} \biggr)u_{j}:a_{j},b_{j} \in R, j=1,2,\ldots,N \biggr\} , \\ X_{\infty j}(i)= \biggl\{ x(t)= \biggl(a_{j}\cos \frac{(2i+1)\pi t}{2k}+b_{j}\sin\frac{(2i+1)\pi t}{2k} \biggr)v_{j}:a_{j},b_{j} \in R, j=1,2,\ldots,N \biggr\} .\end{gathered} $$

Then we have

$$X(i)=\sum_{j=1}^{N}X_{\infty j}(i)= \sum_{j=1}^{N}X_{0j}(i) $$

and

$$X=\sum_{i=0}^{\infty}X(i)=\sum _{i=0}^{\infty}\sum_{j=1}^{N}X_{\infty j}(i)= \sum_{i=0}^{\infty}\sum _{j=1}^{N}X_{0j}(i). $$

Therefore from (2.2) we find that if

$$x(t)=\sum_{i=0}^{\infty} \biggl(a_{i} \cos\frac{(2i+1)\pi t}{2k}+b_{i}\sin\frac {(2i+1)\pi t}{2k} \biggr), $$

then

$$\begin{aligned} \langle Lx,x\rangle =&-\sum_{i=0}^{\infty}(2i+1) \pi \bigl(a_{i}^{2}+b_{i}^{2} \bigr) \cot\frac{(2i+1)\pi}{4k} \\ =&\sum_{h=0}^{\infty} \Biggl[-\sum _{i=0}^{k-1}(4hk+2i+1)\pi \bigl(a_{2hk+i}^{2} +b_{2hk+i}^{2} \bigr)\cot\frac{(2i+1)\pi}{4k} \\ &{}+\sum_{i=0}^{k-1}(4hk+4k-2i-1)\pi \bigl(a_{2hk+2k-i-1}^{2} +b_{2hk+2k-i-1}^{2} \bigr)\cot \frac{(2i+1)\pi}{4k} \Biggr]. \end{aligned}$$

On the other hand, when

$$\begin{aligned} x \in& X_{\infty j}=\sum_{i=0}^{\infty}X_{\infty j}(i) \\ =&\sum_{i=0}^{\infty} \biggl\{ x(t)= \biggl(a_{j}\cos\frac{(2i+1)\pi t}{2k}+b_{j}\sin\frac{(2i+1)\pi t}{2k} \biggr)v_{j}:a_{j},b_{j}\in R, j=1,2,\ldots,N \biggr\} , \end{aligned}$$

we have

$$ \begin{aligned} \bigl\langle P^{-1}A_{\infty}x,x \bigr\rangle &= \sum_{i=0}^{\infty}2k\beta_{j} \bigl(a_{i}^{2}+b_{i}^{2} \bigr) \\ &=\sum_{h=0}^{\infty} \Biggl[\sum _{i=0}^{k-1}2k\beta_{j} \bigl(a_{2hk+i}^{2}+b_{2hk+i}^{2} \bigr)+\sum _{i=0}^{k-1}2k\beta_{j} \bigl(a_{2hk+2k-i-1}^{2}+b_{2hk+2k-i-1}^{2} \bigr) \Biggr].\end{aligned} $$

Then we can get

$$\begin{aligned} & \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle \\ &\quad=2k\sum_{j=1}^{N}\sum _{h=0}^{\infty} \Biggl[\sum_{i=0}^{k-1} \biggl(-\frac{(4hk+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi}{4k} +\beta_{j} \biggr) \bigl(a_{2lk+i}^{2} +b_{2hk+i}^{2} \bigr) \\ &\qquad{}+\sum_{i=0}^{k-1} \biggl( \frac{(4hk+4k-2i-1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k} +\beta_{j} \biggr) \bigl(a_{2hk+2k-i-1}^{2} +b_{2hk+2k-i-1}^{2} \bigr) \Biggr]. \end{aligned}$$

Similarly,

$$\begin{aligned} & \bigl\langle \bigl(L+P^{-1}A_{0} \bigr)x,x \bigr\rangle \\ &\quad=2k\sum_{j=1}^{N}\sum _{h=0}^{\infty} \Biggl[\sum_{i=0}^{k-1} \biggl(-\frac{(4hk+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi}{4k} +\alpha_{j} \biggr) \bigl(a_{2hk+i}^{2} +b_{2hk+i}^{2} \bigr) \\ &\qquad{}+\sum_{i=0}^{k-1} \biggl( \frac{(4hk+4k-2i-1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k} +\alpha_{j} \biggr) \bigl(a_{2hk+2k-i-1}^{2} +b_{2hk+2k-i-1}^{2} \bigr) \Biggr]. \end{aligned}$$

3 Division of space X and lemmas

Let

$$\begin{aligned}& \begin{aligned} X_{\infty}^{+} ={}&\sum _{j=1}^{N} \biggl\{ X_{\infty j}(2hk+i):h\geq0,0 \leq i\leq k-1, -\frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+\beta_{j}>0 \biggr\} \\ &\cup\sum_{j=1}^{N} \biggl\{ X_{\infty j}(2hk+2k-i-1): \\ &h\geq0,0\leq i\leq k-1, \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+ \beta_{j}>0 \biggr\} ,\end{aligned} \\& \begin{aligned} X_{\infty}^{-} ={}&\sum _{j=1}^{N} \biggl\{ X_{\infty j}(2hk+i):h\geq0,0 \leq i\leq k-1, -\frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+\beta_{j}< 0 \biggr\} \\ &\cup\sum_{j=1}^{N} \biggl\{ X_{\infty j}(2hk+2k-i-1): \\ &h\geq0,0\leq i\leq k-1, \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+ \beta_{j}< 0 \biggr\} ,\end{aligned} \\& \begin{aligned} X_{0}^{+} ={}&\sum _{j=1}^{N} \biggl\{ X_{0j}(2hk+i):h\geq0,0 \leq i\leq k-1, -\frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+\alpha_{j}>0 \biggr\} \\ &\cup\sum_{j=1}^{N} \biggl\{ X_{0j}(2hk+2k-i-1): \\ &h\geq0,0\leq i\leq k-1, \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+ \alpha_{j}>0 \biggr\} ,\end{aligned} \\& \begin{aligned} X_{0}^{-} ={}&\sum _{j=1}^{N} \biggl\{ X_{0j}(2hk+i):h\geq0,0 \leq i\leq k-1, -\frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+\alpha_{j}< 0 \biggr\} \\ &\cup\sum_{j=1}^{N} \biggl\{ X_{0j}(2hk+2k-i-1): \\ &h\geq0,0\leq i\leq k-1, \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+ \alpha_{j}< 0 \biggr\} ,\end{aligned} \\& \begin{aligned} X_{\infty}^{0} ={}&\sum _{j=1}^{N} \biggl\{ X_{\infty j}(2hk+i):h\geq0,0 \leq i\leq k-1, -\frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+\beta_{j}=0 \biggr\} \\ &\cup\sum_{j=1}^{N} \biggl\{ X_{\infty j}(2hk+2k-i-1): \\ &h\geq0,0\leq i\leq k-1, \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+ \beta_{j}=0 \biggr\} ,\end{aligned} \\& \begin{aligned} X_{0}^{0} ={}&\sum _{j=1}^{N} \biggl\{ X_{0j}(2hk+i):h\geq0,0 \leq i\leq k-1, -\frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+\alpha_{j}=0 \biggr\} \\ &\cup\sum_{j=1}^{N} \biggl\{ X_{0j}(2hk+2k-i-1): \\ &h\geq0,0\leq i\leq k-1, \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k}+ \alpha_{j}=0 \biggr\} .\end{aligned} \end{aligned}$$

It is easy to see that $\dim X_{\infty}^{0}<\infty$ and $\dim X_{0}^{0}<\infty$.

Lemma 3.1

([3], Lemma 2.4)

SupposeXis a Hilbert space, $\varPhi:X\rightarrow R$is a differentiable functional, and $L:X\rightarrow X$is a linear operator. Then there are two closed $S^{1}$-invariant linear subspaces $X^{+}$and $X^{-}$such that

• (a) $X^{+}\cup X^{-}$is closed and of finite codimension inX,

• (b) $\widehat{L}(X^{-})\subset X^{-}$, $\widehat{L}=L+P^{-1}A_{0}$or $\widehat{L}=L+P^{-1}A_{\infty}$,

• (c) there exists $c_{0}\in R$such that

  $$\inf_{x\in X^{+}}\varPhi(x)\geq c_{0}, $$

• (d) there is $c_{\infty}\in R$such that

  $$\varPhi(x)\leq c_{\infty}< \varPhi(0)=0,\quad \forall x\in X^{-} \cap S_{r}=\bigl\{ x\in X^{-}:\|x\|=r\bigr\} , $$

• (e) Φsatisfies the $(P.S)_{c}$-condition for $c_{0}< c< c_{\infty }$, that is, every $\{x_{n}\}\subseteq X$satisfying $\varPhi (x_{n})\rightarrow c$and $\varPhi'(x_{n})\rightarrow0$has a convergent subsequence. ThenΦhas at least $\frac{1}{2}[\dim(X^{+}\cap X^{-})-\operatorname{codim}_{X}(X^{+}\cup X^{-})]$generally different critical orbits in $\varPhi^{-1}([c_{0},c_{\infty}])$if

  $$\bigl[\dim \bigl(X^{+}\cap X^{-} \bigr)- \operatorname{codim}_{X} \bigl(X^{+}\cup X^{-} \bigr) \bigr]>0. $$

Lemma 3.2

There exists $\sigma>0$ such that

$$ \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle > \sigma \Vert x \Vert ^{2},\quad x\in X_{\infty}^{+}, $$

(3.1)

and

$$ \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle < - \sigma \Vert x \Vert ^{2},\quad x\in X_{\infty}^{-} . $$

(3.2)

Proof

Let

$$\begin{aligned} X_{\infty j} =&\sum_{i=0}^{\infty}X_{\infty j}(i) \\ =& \Biggl\{ x(t)=\sum_{i=0}^{\infty} \biggl(a_{j}\cos\frac{(2i+1)\pi t}{2k}+b_{j}\sin\frac{(2i+1)\pi t}{2k} \biggr)v_{j}:a_{j},b_{j}\in R, j=1,2,\ldots,N \Biggr\} . \end{aligned}$$

Then $X=\sum_{j=1}^{N}X_{\infty j}$. We need to consider two cases, $\beta_{j}\geq0$ and $\beta_{j}< 0$. In the following part, we just give the proof for $\beta_{j}\geq0$, as the other case is similar.

For $\beta_{j}\geq0$, $i\in\{0,1,\ldots,k-1\}$, and $x\in X_{\infty j}$,

$$-\frac{(4hk+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi}{4k}+\beta_{j}>-\frac {(4h^{+}(i)k+2i+1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k} +\beta_{j}>0, $$

where $h^{+}(i)=\max \{h\in N:-\frac{(4hk+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi}{4k} +\beta_{j}>0 \}$, and

$$-\frac{(4hk+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi}{4k} +\beta_{j}< - \frac{(4h^{-}(i)k+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi }{4k}+\beta_{j}< 0, $$

where $h^{-}(i)=\min \{h\in N:-\frac{(4hk+2i+1)\pi}{2k}\cot\frac{(2i+1)\pi }{4k}+\beta_{j}<0 \}$.

Then we can choose

$$\begin{aligned} \sigma_{i} =&\min \biggl\{ -\frac{\pi}{2k}\cot\frac{(2i+1)\pi}{4k} + \frac{\beta_{j}}{4h^{+}(i)k+2i+1}, \\ & \frac{\pi}{2k}\cot\frac{(2i+1)\pi}{4k} -\frac{\beta_{j}}{4h^{-}(i)k+2i+1} \biggr\} \\ >&0, \end{aligned}$$

and let $\sigma_{j}=\min\{\sigma_{0},\sigma_{1},\ldots,\sigma_{k-1}\} >0$, and then let $\sigma=\min\{\sigma_{j}:j=1,2,\ldots,N\}$. The proof is over. □

Lemma 3.3

If $(f_{1})$and $(f_{2})$hold, then the functionalΦgiven by (2.1) satisfies the $(P.S)$-condition.

Proof

Let Π, Λ, and Γ be the orthogonal mappings from X to $X_{\infty}^{+}$, $X_{\infty}^{-}$, and $X_{\infty}^{0}$, respectively. From (1.3) we get

$$\bigl\vert \bigl\langle P^{-1} \bigl(\nabla F(x)-A_{\infty}x \bigr),x \bigr\rangle \bigr\vert < \frac {\sigma}{2} \Vert x \Vert ^{2}+\widetilde{M},\quad x\in X, $$

for some $\widetilde{M}>0$.

Suppose that $\{x_{n}\}\subset X$ is a subsequence such that $\varPhi '(x_{n})\rightarrow0$ and $\varPhi(x_{n})$ is bounded. Let $w_{n}=\varPi x_{n}$, $y_{n}=\varLambda x_{n}$, and $z_{n}=\varGamma x_{n}$. Then

$$ \varPi \bigl(L+P^{-1}A_{\infty} \bigr)= \bigl(L+P^{-1}A_{\infty} \bigr)\varPi, \varLambda \bigl(L+P^{-1}A_{\infty} \bigr)= \bigl(L+P^{-1}A_{\infty} \bigr)\varLambda. $$

(3.3)

From

$$\bigl\langle \varPhi'(x_{n}),x_{n} \bigr\rangle = \bigl\langle Lx_{n}+P^{-1}\nabla F(x_{n}),x_{n} \bigr\rangle = \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x_{n},x_{n} \bigr\rangle + \bigl\langle P^{-1} \bigl(\nabla F(x_{n})-A_{\infty} x_{n} \bigr),x_{n} \bigr\rangle $$

and (3.1) we have

$$\begin{aligned} \bigl\langle \varPi\varPhi'(x_{n}),x_{n} \bigr\rangle =& \bigl\langle \varPi \bigl(L+P^{-1}A_{\infty} \bigr)x_{n},x_{n} \bigr\rangle + \bigl\langle \varPi P^{-1} \bigl(\nabla F(x_{n})-A_{\infty} x_{n} \bigr),x_{n} \bigr\rangle \\ =& \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)w_{n},w_{n} \bigr\rangle + \bigl\langle \varPi P^{-1} \bigl(\nabla F(x_{n})-A_{\infty} x_{n} \bigr),w_{n} \bigr\rangle \\ >&\frac{\sigma}{2} \Vert w_{n} \Vert ^{2}- \widetilde{M}. \end{aligned}$$

Then we get that $w_{n}$ is bounded. Similarly, $y_{n}$ is bounded. Meanwhile, from $(f_{2})$ we get

$$\begin{aligned} \varPhi(x_{n}) =&\frac{1}{2} \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x_{n},x_{n} \bigr\rangle + \int_{0}^{4k} \biggl(F(x_{n})- \frac{1}{2}(A_{\infty}x_{n},x_{n}) \biggr)\,dt \\ \geq&\frac{1}{2} \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)w_{n},w_{n} \bigr\rangle + \frac{1}{2} \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)y_{n},y_{n} \bigr\rangle \\ &{} + \int_{0}^{4k}r \bigl( \bigl\vert x_{n}(t) \bigr\vert \bigr)\,dt-4kM. \end{aligned}$$

Then $\|z_{n}\|$ is bounded since $\varPhi(x_{n})$ is bounded. So, $\|x_{n}\|$ is bounded.

Furthermore, from (2.4) we have

$$\begin{aligned} (\varPi+N)\varPhi'(x_{n}) =&(\varPi+ \varLambda)Lx_{n}+( \varPi+N)K(x_{n}) \\ =&L(w_{n}+y_{n})+(\varPi+N)K(x_{n}). \end{aligned}$$

Then we can suppose without loss of generality that $K (x_{n})\rightarrow\eta$ because K is compact and $x_{n}$ is bounded. Then

$$ L|_{x_{\infty}^{+}+x_{\infty }^{-}}(w_{n}+y_{n})\rightarrow-(\varPi+N) \eta. $$

(3.4)

Meanwhile, we can easily see that the dimension of $X_{\infty}^{0}$ is finite, so we can suppose that $z_{n}\rightarrow\varphi$ as $z_{n}$ is bounded. Hence

$$x_{n}=z_{n}+w_{n}+y_{n}\rightarrow \varphi-(L|_{x_{\infty}^{+}+x_{\infty }^{-}})^{-1} (\varPi+\varLambda)\eta, $$

and the $(P.S)$-condition is proved. □

Lemma 3.4

Ifxis a critical point ofΦ, then it is a solution to system (1.4).

Proof

Suppose x is a critical point of Φ given by (2.1). Then $x(t)$ satisfies

$$ -\sum_{s=1}^{2k-1}x'(t-s)+\nabla F \bigl(x(t) \bigr)=0,\quad \mbox{a.e. } t\in[0,4k]. $$

(3.5)

Consequently,

Calculating (3.5_1) − (3.5_2) + (3.5_3) −⋯+ (3.5_$(2k-1)$), we have

$$x'(t)+\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr)=0, \quad\mbox{a.e. } t\in[0,4k], $$

that is,

$$x'(t)=-\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr),\quad \mbox{a.e. } t\in[0,4k], $$

and hence x is a solution of (1.1). □

4 Main results

Denote

$$\begin{aligned}& N(\alpha_{j})=\textstyle\begin{cases} -\sum_{i=0}^{k-1}\sharp \{h\geq0:0< \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k} < -\alpha_{j} \}, &\alpha_{j}< 0,\\ \sum_{i=0}^{k-1}\sharp \{h\geq0:0< \frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k} < \alpha_{j} \}, &\alpha_{j}\geq0, \end{cases}\displaystyle \\& N(\beta_{j})= \textstyle\begin{cases} -\sum_{i=0}^{k-1}\sharp \{h\geq0:0< \frac{(4hk+4k-2i-1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k} < -\beta_{j} \},& \beta_{j}< 0,\\ \sum_{i=0}^{k-1}\sharp \{h\geq0:0< \frac{(4hk+2i+1)\pi}{2k}\cot \frac{(2i+1)\pi}{4k} < \beta_{j} \},& \beta_{j}\geq0, \end{cases}\displaystyle \\& N^{0}(\alpha_{j-})=\sum_{i=0}^{k-1} \sharp \biggl\{ h\geq0:0< \frac {(4hk+4k-2i-1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k}=- \alpha_{j} \biggr\} ,\quad \alpha_{j}< 0, \\& N^{0}(\alpha_{j+})=\sum_{i=0}^{k-1} \sharp \biggl\{ h\geq0:0< \frac {(4hk+2i+1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k}= \alpha_{j} \biggr\} ,\quad \alpha_{j}\geq0, \\& N^{0}(\beta_{j-})=\sum_{i=0}^{k-1} \sharp \biggl\{ h\geq0:0< \frac {(4hk+4k-2i-1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k}=- \beta_{j} \biggr\} ,\quad \beta_{j}< 0, \\& N^{0}(\beta_{j+})=\sum_{i=0}^{k-1} \sharp \biggl\{ h\geq0:0< \frac {(4hk+2i+1)\pi}{2k} \cot\frac{(2i+1)\pi}{4k}= \beta_{j} \biggr\} ,\quad \beta_{j}\geq0, \end{aligned}$$

and

$$\begin{gathered} N(A_{\infty})=\sum_{j=1}^{N}N( \beta_{j}),\qquad N(A_{0})=\sum_{j=1}^{N}N( \alpha_{j}), \\ N^{0}(A_{\infty-})=\sum_{j=1}^{N}N^{0}( \beta_{j-}),\qquad N^{0}(A_{\infty +})=\sum _{j=1}^{N}N^{0}(\beta_{j+}), \\ N^{0}(A_{0-})=\sum_{j=1}^{N}N^{0}( \alpha_{j-}),\qquad N^{0}(A_{0+})=\sum _{j=1}^{N}N^{0}(\alpha_{j+}).\end{gathered} $$

Theorem 4.1

System (1.1) has at least

$$\begin{aligned} n={}&\max \bigl\{ N(A_{\infty})-N(A_{0})-N^{0}(A_{\infty-})-N^{0}(A_{0+}),\\& N(A_{0})-N(A_{\infty})-N^{0}(A_{0-})-N^{0}(A_{\infty+}) \bigr\} \\>{}&0\end{aligned} $$

4k-periodic orbits when $(f_{1})$and $(f_{2})$hold.

Proof

Without loss of generality, we suppose

$$n=N(A_{\infty})-N(A_{0})-N^{0}(A_{\infty-})-N^{0}(A_{0+}). $$

Then letting $X^{+}=X_{\infty}^{+}$ and $X^{-}=X_{0}^{-}$, we get

$$X\setminus \bigl(X^{+}\cup X^{-} \bigr)=X\setminus \bigl(X_{\infty}^{+}\cup X_{0}^{-} \bigr) \subseteq X_{\infty}^{0}\cup X_{0}^{0}\cup \bigl(X_{\infty }^{+}\cap X_{0}^{-} \bigr). $$

Obviously,

$$\operatorname{codim}_{X} \bigl(X^{+}+X^{-} \bigr) \leq\dim X_{\infty}^{0}+\dim X_{0}^{0}+ \dim \bigl(X_{\infty}^{+}\cap X_{0}^{-} \bigr)< \infty, $$

which means that the codimension of $(X^{+}\cup X^{-})$ is finite. For each $x\in X(i)$, we have $(L+P^{-1}A_{\infty})x\in X(i)$. The $(PS)$-condition is satisfied by Lemma 3.3. Moreover, from (1.3) we get $|F(x)-\frac{1}{2}(A_{\infty }x,x)|<\frac{1}{4}\sigma\|x\|^{2}+M_{1}$, $x\in R^{N}$, for some $M_{1}>0$, and from Lemma 3.2 we know that there exists $\sigma>0$ such that $\langle(L+P^{-1}A_{\infty})x,x \rangle>\sigma\|x\| ^{2}$, $x\in X_{\infty}^{+}$. Then

$$\begin{aligned} \varPhi(x) =&\frac{1}{2} \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle + \int _{0}^{4k} \biggl[F \bigl(x(t) \bigr)- \frac{1}{2}(A_{\infty}x,x) \biggr]\,dt \\ \geq&\frac{1}{2}\sigma \Vert x \Vert ^{2}- \frac{1}{4}\sigma \Vert x \Vert ^{2}-4kM_{1} \\ \geq&\frac{1}{4}\sigma \Vert x \Vert ^{2}-4kM_{1} \end{aligned}$$

for $x\in X^{+}$. Therefore there exists $c_{0}\in R$ such that

$$\inf_{x\in X^{+}}\varPhi(x)\geq c_{0}. $$

Similarly, we get that there exist $r,\sigma>0$ such that $|F(x)-\frac{1}{2}(A_{0}x,x)|<\frac{1}{4}\sigma\|x\|^{2}$, $\|x\|=r$. Then

$$\begin{aligned} \varPhi(x) =&\frac{1}{2} \bigl\langle \bigl(L+P^{-1}A_{0} \bigr)x,x \bigr\rangle + \int _{0}^{4k} \biggl[F \bigl(x(t) \bigr)- \frac{1}{2}(A_{0}x,x) \biggr]\,dt \\ \leq&-\frac{1}{2}\sigma \Vert x \Vert ^{2}+ \frac{1}{4}\sigma \Vert x \Vert ^{2} \\ \leq&-\frac{1}{4}\sigma \Vert x \Vert ^{2} \end{aligned}$$

for $x\in X^{-}$. This means that there exist $r>0$ and $c_{\infty}<0$, such that

$$\varPhi(x)\leq c_{\infty}< 0=\varPhi(0),\quad\forall x\in X^{-} \cap S_{r}= \bigl\{ x\in X: \Vert x \Vert =r \bigr\} . $$

On the other hand, for $i\in\{0,1,\ldots,k-1\}$,

$$\begin{gathered} \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle = \biggl(-\frac{\pi}{2k}\cot \frac{(2i+1)\pi}{4k} +\frac{\beta_{j}}{4hk+2i+1} \biggr) \Vert x \Vert ^{2},\quad x\in X(2hk+i), \\ \bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle = \biggl(\frac{\pi}{2k}\cot \frac{(2i+1)\pi}{4k} +\frac{\beta_{j}}{4hk+4k-2i-1} \biggr) \Vert x \Vert ^{2},\\\phantom{\bigl\langle \bigl(L+P^{-1}A_{\infty} \bigr)x,x \bigr\rangle =} x\in X(2hk+2k-i-1),\end{gathered} $$

and

$$\begin{gathered} \bigl\langle \bigl(L+P^{-1}A_{0} \bigr)x,x \bigr\rangle = \biggl(-\frac{\pi}{2k}\cot\frac {(2i+1)\pi}{4k} +\frac{\alpha_{j}}{4hk+2i+1} \biggr) \Vert x \Vert ^{2},\quad x\in X(2hk+i), \\ \bigl\langle \bigl(L+P^{-1}A_{0} \bigr)x,x \bigr\rangle = \biggl(\frac{\pi}{2k}\cot\frac {(2i+1)\pi}{4k} +\frac{\alpha_{j}}{4hk+4k-2i-1} \biggr) \Vert x \Vert ^{2},\\ \phantom{\bigl\langle \bigl(L+P^{-1}A_{0} \bigr)x,x \bigr\rangle =} x\in X(2hk+2k-i-1).\end{gathered} $$

Hence we have that

$$\begin{gathered}X_{\infty}^{+}(2hk+i)=X_{\infty}^{+} \cap X(2hk+i)=\emptyset, \\ X_{0}^{-}(2hk+2k-i-1)=X_{0}^{-}\cap X(2hk+2k-i-1)=\emptyset, \\ X_{0}^{-}(2hk+i)=X_{0}^{-}\cap X(2hk+i)=X(2hk+i), \\ X_{\infty}^{+}(2hk+2k-i-1)=X_{\infty}^{+}\cap X(2hk+2k-i-1)=X(2hk+2k-i-1),\end{gathered} $$

when $h\geq0$ is large enough. So there is $M>0$ such that

$$\dim \bigl(X_{\infty}^{+}(s)\cap X_{0}^{-}(r) \bigr)-\operatorname {codim}_{X} \bigl(X_{\infty}^{+}(s)+X_{0}^{-}(s) \bigr)=0,\quad s>M. $$

Then

$$\begin{aligned} n =&\frac{1}{2} \bigl[\dim \bigl(X^{+}\cap X^{-} \bigr)-\operatorname {codim}_{X} \bigl(X^{+}+X^{-} \bigr) \bigr] \\ =&\frac{1}{2} \bigl[\dim \bigl(X_{\infty}^{+}\cap X_{0}^{-} \bigr)-\operatorname {codim}_{X} \bigl(X_{\infty}^{+}+X_{0}^{-} \bigr) \bigr] \\ =&\frac{1}{2}\sum_{s=0}^{M} \bigl[ \dim \bigl(X_{\infty}^{+}(s)\cap X_{0}^{-}(s) \bigr)-\operatorname{codim}_{X(s)} \bigl(X_{\infty }^{+}(s)+X_{0}^{-}(s) \bigr) \bigr] \\ =&\frac{1}{2}\sum_{s=0}^{M} \bigl[ \dim X_{\infty}^{+}(s)+\dim X_{0}^{-}(s)-2N \bigr] \\ =&\frac{1}{2}\sum_{s=0}^{M} \bigl[ \dim X_{\infty}^{+}(s)+\dim X_{0}^{-}(s) \bigr]-N(M+1). \end{aligned}$$

Then we have

$$\begin{gathered} \sum_{s=0}^{M}\dim(X_{\infty}^{+}(s))=2 \textstyle\begin{cases} N(A_{\infty})\\\hspace{8pt}{}+N\times\sharp\{2hk+2k-i-1:0\leq2hk+2k-i-1\leq M\},& \beta _{j}\geq0, \\ N(A_{\infty})-N^{0}(A_{\infty-})\\\hspace{8pt}{}+N\times\sharp\{2hk+2k-i-1:0\leq 2hk+2k-i-1\leq M\},& \beta_{j}< 0, \end{cases}\displaystyle \\ \sum_{s=0}^{M}\dim(X_{0}^{-}(s))=2 \textstyle\begin{cases} -N(A_{0})-N^{0}(A_{0+})+N\times\sharp\{2hk+i:0\leq2hk+i\leq M\}, &\alpha _{j}\geq0, \\ -N(A_{0})+N\times\sharp\{2hk+i:0\leq2hk+i\leq M\}, &\alpha_{j}< 0, \end{cases}\displaystyle \end{gathered} $$

and

$$\begin{aligned}[b] \sum_{s=0}^{M} \bigl[\dim X_{\infty}^{+}(s)+\dim X_{0}^{-}(s) \bigr]={}& 2 \bigl[N(A_{\infty})-N(A_{0})-N^{0}(A_{\infty-})-N^{0}(A_{0+}) \bigr]\\&+2N(M+1).\end{aligned} $$

(4.1)

Therefore

$$n=N(A_{\infty})-N(A_{0})-N^{0}(A_{\infty-})-N^{0}(A_{0+}) . $$

 □

Theorem 4.2

System (1.1) possesses at least

$$n=N(A_{\infty})-N(A_{0})+N^{0}(A_{\infty+})+N^{0}(A_{0-})>0 $$

4k-periodic orbits when $(f_{1})$, $(f_{2})$, $(f_{3}^{+})$, and $(f_{4}^{-})$hold.

Proof

Let $X^{+}=X_{\infty}^{+}+X_{\infty}^{0}$ and $X^{-}=X_{-}^{0}+X_{0}^{0}$. The verification of conditions (a), (b), (c), (d), and (e) is similar to Theorem 4.1, so we can assume that (4.1) still holds. Let $X_{\infty}^{0}(i)=X_{\infty }^{0}\cap X(i)$ and $X_{0}^{0}(i)=X_{0}^{0}\cap X(i)$. Then

$$\begin{aligned} n&=\frac{1}{2}\sum_{i=0}^{M} \bigl[ \dim \bigl(X_{\infty}^{+}(i)\cap X_{0}^{-}(i) \bigr)-\operatorname{codim}_{X(i)} \bigl(X_{\infty }^{+}(i)+X_{0}^{-}(i) \bigr) \bigr]+ \bigl(\dim X_{\infty}^{0}+\dim X_{0}^{0} \bigr) \\ &=\frac{1}{2}\sum_{i=0}^{M} \bigl[ \dim X_{\infty}^{+}(i)+\dim X_{0}^{-}(i)-2N \bigr]+ \bigl(\dim X_{\infty}^{0}+\dim X_{0}^{0} \bigr) \\ &=\frac{1}{2}\sum_{i=0}^{M} \bigl[ \dim X_{\infty}^{+}(i)+\dim X_{0}^{-}(i) \bigr]-N(M+1)+ \bigl(\dim X_{\infty}^{0}+\dim X_{0}^{0} \bigr) \\ &=N(A_{\infty})-N(A_{0})-N^{0}(A_{\infty-})-N^{0}(A_{0+})\\ &\quad+ \bigl(N^{0}(A_{\infty+})+N^{0}(A_{0-})+N^{0}(A_{0+})+N^{0}(A_{0-}) \bigr) \\ &=N(A_{\infty})-N(A_{0})+N^{0}(A_{\infty+})+N^{0}(A_{0-}).\end{aligned} $$

 □

Theorem 4.3

System (1.1) possesses at least

$$n=N(A_{0})-N(A_{\infty})+N^{0}(A_{0+})+N^{0}(A_{\infty-})>0 $$

4k-periodic orbits when $(f_{1})$, $(f_{2})$, $(f_{3}^{-})$, and $(f_{4}^{+})$hold.

The proof is almost the same as that of Theorem 4.2, and we omit it.

5 Example

Assume that $F\in C^{1}(R^{2},R)$ satisfies

$$F(x)= \textstyle\begin{cases} \frac{3\pi}{2} x_{1}^{2}+\frac{\pi}{2} x_{2}^{2}+(2x_{1}^{2}+x_{2}^{2})^{\frac{2}{3}}, &\vert x \vert \gg1,\\ \frac{\pi}{2} x_{1}^{2}-\frac{3\pi}{2} x_{2}^{2}-3x_{1}^{\frac {12}{5}}-x_{2}^{\frac{8}{3}}, &\vert x \vert \ll1. \end{cases} $$

We are to discuss the multiplicity of 12-periodic solutions of the equation

$$ x'(t)=-\sum_{s=1}^{5}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr). $$

(5.1)

In this case, $k=3$, $\alpha_{1}=\pi$, $\alpha_{2}=-3\pi$, $\beta_{1}=3\pi$, $\beta_{2}=\pi$. Then

$$\begin{aligned}& \begin{aligned} N(\alpha_{1})={}&\sharp \biggl\{ h\geq0:0< \frac{(12h+1)\pi}{6}\cot \frac{\pi }{12}< \pi \biggr\} \\ &+\sharp \biggl\{ h\geq0:0< \frac{(12h+3)\pi}{6}\cot\frac{3\pi}{12}< \pi \biggr\} \\ &+\sharp \biggl\{ h\geq0:0< \frac{(12h+5)\pi}{6}\cot\frac{5\pi}{12}< \pi \biggr\} \\ ={}&4, \end{aligned} \\& \begin{aligned} N(\alpha_{2})={}&{-}\sharp \biggl\{ h\geq0:0< \frac{(12h+11)\pi}{6}\cot \frac{\pi }{12}< 3\pi \biggr\} \\ &-\sharp \biggl\{ h\geq0:0< \frac{(12h+9)\pi}{6}\cot\frac{3\pi}{12}< 3\pi \biggr\} \\ &-\sharp \biggl\{ h\geq0:0< \frac{(12h+7)\pi}{6}\cot\frac{5\pi}{12}< 3\pi \biggr\} \\ ={}&{-}7, \end{aligned} \\& \begin{aligned} N(\beta_{1})={}&\sharp \biggl\{ h\geq0:0< \frac{(12h+1)\pi}{6}\cot \frac{\pi }{12}< 3\pi \biggr\} \\ &+\sharp \biggl\{ h\geq0:0< \frac{(12h+3)\pi}{6}\cot\frac{3\pi}{12}< 3\pi \biggr\} \\ &+\sharp \biggl\{ h\geq0:0< \frac{(12h+5)\pi}{6}\cot\frac{5\pi}{12}< 3\pi \biggr\} \\={}&9, \end{aligned} \\& \begin{aligned} N(\beta_{2})={}&\sharp \biggl\{ h\geq0:0< \frac{(12h+1)\pi}{6}\cot \frac{\pi }{12}< \pi \biggr\} \\ &+\sharp \biggl\{ h\geq0:0< \frac{(12h+3)\pi}{6}\cot\frac{3\pi}{12}< \pi \biggr\} \\ &+\sharp \biggl\{ h\geq0:0< \frac{(12h+5)\pi}{6}\cot\frac{5\pi}{12}< \pi \biggr\} \\={}&4, \end{aligned} \\& N(A_{0})=N(\alpha_{1})+N(\alpha_{2})=-3,\quad \quad N(A_{\infty})=N( \beta _{1})+N(\beta_{2})=13, \\& N^{0}(\alpha_{+})=N^{0}(\beta_{-})=N^{0}( \alpha_{-})=N^{0}(\beta_{+})=0. \end{aligned}$$

According to Theorem 4.2, we get that Eq. (5.1) has at least 16 different 12-periodic orbits satisfying $x(t-6)=-x(t)$.