Multiple periodic solutions of differential delay systems with $2k$ lags

1.   Introduction

Delay differential equations naturally appear in many fields of science and engineering. Such equations have been proposed as models for a variety of physiological processes and conditions including production of blood cells, respiration and cardiac arrhythemias. In recent years, the existence of positive periodic solutions for delay differential equations has been received considerable attention. J. Kaplan and J. Yorke ^[10] gave conditions for the existence of $4-$ or $6-$ periodic solutions of equation

while $n = 1$ and $n = 2$, respectively. After then, lots of results are achieved for this equation in general cases ^{[1,2,3,4,5,6,7,8,9,10,11,12]}. W. Ge ^[4,5,6] proved several existence theorems of periodic solutions for (1.1) in the general case by use of the fixed-point theorem in cone. After transforming (1.1) into a Hamiltonian system and applying a theorem given by Mawhin and Willem ^[13], J.Li and X. He ^[11,12] proved a theorem for the existence of several periodic solutions under the condition $xf(x) > 0 \; (xf(x) < 0)$ for $x\neq 0$. G. Fei ^[1,2] studied the multiplicity of periodic solutions of (1.1) and proved theorems for the cases $n = 2k-1$ and $n = 2k$ by using the index theory. In his work, the way to construct the functional for the second case is quite different from that for the first case.

Applying $S^{1}$ index theory Z. Guo and J. Yu ^[8] gave a theorem for the multiple periodic solutions of the delay-differential system with one lag in the form

And by applying the same index theory, in ^[9], they obtained results on the multiplicity of $2(n+1)$-periodic solutions to the delay differential system

when $n = 2k-1$. After then B. Zheng and Z. Guo studied (1.3) while $n = 2k$ and gave two criteria for the multiplicity of $2(2k+1)$-periodic solutions. The theorems given in both^[9] and ^[14] contain a condition to calculate the following function

for symmetric matrices A and B, respectively. This is a hard task to be fulfilled since it is not definite for the problem how large the j must be to ensure $\Psi_{j}(A, B) = 0$.

In ^[7], we researched the same problem for system (1.3) when $n = 2k-1$ and by constructing a new functional we gave criteria for the multiplicity of periodic solutions only based on the eigenvalues of symmetric matrices A and B.

In this paper we are to study the multiplicity of $2(2k+1)$-periodic solutions for system (1.3) when $n = 2k$, i.e.,

with the conditions $F\in C^{1}(\mathbb{R}^{N}, \mathbb{R}), \nabla F(-x) = -F(x), F(0) = 0$,

At the same time, assume there are real symmetric matrices $A_{\infty}$ and $A_{0}$ such that

In section 2, some notations and variation structure for system (1.4) are introduced. Meanwhile, a lemma about the relation between the periodic solutions of system(1.4) and the critical points of the functional $\Phi$ are also proved in section 2. Section 3 presents and proves the main results of this paper. Before the proof of the main results some lemmas about the calculation of the differential of functional $\Phi$ are proven. Two examples are given in section 4 to illustrate the applications of the main results.

2.   $2(2k+1)$-periodic solution of (1.4) and variation structure

Firstly, we consider a linear space which consists of all the $2(2k+1)$-periodic functions satisfying (1.5) in $\mathbb{R}^{N}$, i.e.

$\begin{array}{ll} H^{\frac{1}{2}} = \{ & x(t) = \sum\limits_{i = 1}^{\infty}(a_i \cos \frac{(2i-1)\pi}{2k+1}t + b_i \sin \frac{(2i-1)\pi}{2k+1}t):a_i, b_i \in \mathbb{R}^N, \\ & \sum\limits_{i = 1}^{\infty}(2i-1)(|a_i|^2+|b_i|^2) < \infty\} \end{array}$

Since our goal is to find the $2(2k+1)$-periodic solutions for system (1.4) with condition (1.5), we discuss now the special requirement for a solution of system (1.4) in $H^{\frac{1}{2}}$. After then, we are able to choose a suitable Hilbert space for our problem. Assume $\sum\limits_{i = m}^{n}a_i = 0$ if $n < m.$

If $x(t)$ is a $2(2k+1)$-periodic solution of (1.4) satisfying (1.5), then, $x'(t-l) = \sum\limits^{l-1}_{i = 0}\nabla F(x(t-i)) - \sum\limits^{2k}_{i = l+1}\nabla F(x(t-i)), \; l = 0, 1, \cdots, 2k.$ Hence,

From the equation above, one has $\sum\limits^{2k}_{l = 0}(-1)^l x(t-l) = c.$ However, $c = \sum\limits^{2k}_{l = 0}(-1)^l x(t-l-2k -1) = -\sum\limits^{2k}_{l = 0}(-1)^l x(t-l) = -c$, which means $c = 0$. Then,

By Fourier's expansion theory and (1.5), it holds that

Substituting the above expression into (2.2) we have

Then suppose

For $x\in X$, define $P:X \rightarrow X$:

and denote

Define

It is obvious that $(X, \|\cdot\|_X)$ is a Hilbert space and $P: X \rightarrow X$ is an invertible and self-adjoint operator. Define

Therefore $z_1(t) = x(t) \in X$. Furthermore, let

Then,

Denote $H(z) = \sum\limits_{i = 0}^{2k-1}F(x(t-i))+ F(\sum\limits_{i = 0}^{2k-1}(-1)^{i+1}x(t-i))$, then,

where $x(t-2k) = \sum\limits_{i = 0}^{2k-1}(-1)^{i+1}x(t-i)$, $z_l = x(t - l + 1), l = 1, 2, \cdots, 2k$, and from (2.2),

Define $\Phi : Z \rightarrow \mathbb{R}$ as a functional of $z$:

where

$\hat{P} = (P, P, \cdots, P): Z \rightarrow Z$. Then

By Mawhin-Willem's theorem [13, Theorem 1.4], $\Phi$ is continuously differentiable and

where $z(t) = (x(t), x(t-1), \cdots, x(t-2k+1)), v(t) = (y(t), y(t-1), \cdots, y(t-2k+1))$.

Denote $\Phi'(z) = (\Phi'_1(z), \Phi'_2(z), \cdots, \Phi'_{2k}(z))$, where

If a point $z\in Z$ satisfies $\Phi'(z) = 0$, then $z$ is called a critical point of $\Phi$. The following lemma reveals the relation between the $2(2k+1)$-periodic solutions of system (1.4) and the critical points of $\Phi$. Based on this lemma we can discuss the multiplicity of critical points of $\Phi$ instead of discussing that of $2(2k+1)$-periodic solutions of system (1.4). In the following, denote $x(t)$ as the first N components of $z(t)$ in Z and set $\sum\limits_{i = 0}^{-1} x'(t-i) = 0$.

Lemma 2.1 If $x \in X$, $z = (x(t), x(t-1), \cdots, x(t-2k+1)) \in Z$, then the following propositions are equivalent to each other.

1) $x(t)$ is one of the solutions of system (1.4),

2) $z(t)$ satisfies $\Phi_1'(z) = 0$, i.e.

3) $z(t)$ is one of the critical points of $\Phi$ on Z, i.e., $z(t)$ satisfies

$\begin{array}{r} \sum\limits_{i = 0}^{l-1}(-1)^{i+l} x'(t-i)+\sum\limits_{i = l+1}^{2k-1}(-1)^{i+l+1} x'(t-i)-\nabla F(x(t-l))+(-1)^{l}\nabla F(x(t-2k))\\ = 0, \quad \quad l = 0, 1, \cdots, 2k-1 \quad \quad(2.8 - l) \end{array}$

Proof. 1)$\Rightarrow$ 2).

Suppose $x(t)$ is a solution of system (1.4).

By the periodicity of $x(t)$ and (1.5), one has

$l = 1, 2, \cdots, 2k-1$, $x(t-2k) = \sum\limits_{i = 0}^{2k-1}(-1)^{i+1}x(t-i)$. Multiplying $(-1)^{l+1}$ to the equalities $(2.9-l)$ and summing them up, we have

which means (2.7) holds.

2)$\Rightarrow$ 3).

System (2.7) is the same as $(2.8 - 0)$.

From (2.7) one has

i.e.,

Then (2.7) and (2.10) yield

which means $(2.8 - 1)$.

Suppose $(2.8 - l)$ holds. Then,

Multiplying (2.7)with $(-1)^{l+1}$ and add it to (2.11), one can obtain $(2.8 - (l+1))$.

3) $\Rightarrow$ 1).

Summing the equalities from $(2.8 -1)$ to $(2.8 - (2k-1))$, one has

which implies $x(t)$ is a solution of system (1.4).

Denote $\hat{P_i}:Z \rightarrow Z(i)$, $Z(i) = \{(x(t), x(t-1), \cdots, x(t-2k +1)):x(t) = a_i \cos \frac{(2i-1)\pi}{2k+1}t + b_i \sin \frac{(2i-1)\pi}{2k+1}t, a_i, b_i \in \mathbb{R}^N\}, \; i \neq (2l+1)(2k+1)$; $Z((2l+1)(2k+1)) = 0, \quad l\in \mathbb{N}^+ \cup \{0\}.$ Furthermore, let $P_i = \sum\limits_{j = 1}^{i}\hat{P}_j:Z \rightarrow \sum\limits_{j = 1}^{i}Z(i), \quad i = 1, 2, \cdots.$ Then, one has $LP_i = P_iL.$

Let $-G'(z) = (P^{-1}(\nabla F(x(t)) - \nabla F(x(t-2k))), P^{-1}(\nabla F(x(t-1)) + \nabla F(x(t-2k))), \cdots, P^{-1}(\nabla F(x(t-l)) + (-1)^{l+1}\nabla F(x(t-2k))), \cdots, P^{-1}(\nabla F(x(t-2k+1)) - \nabla F(x(t-2k)))).$ Then $\Phi'(z) = Lz + G'(z).$ L is a bounded self-adjoint operator and $G'(z): Z \rightarrow X^{2k}$ is compact since $(G'z)(t)$ is differentiable with respect to t.

Denote $B_\infty, B_0 \in \mathbb{R}^{2kN \times 2kN}$ as

and for $z(t) = (x(t), x(t-1), \cdots, x(t-2k +1)),$

Define $\delta z(t) = T_{2kN} z(t): Z \rightarrow Z$, where

is a $2kN \times 2kN$ matrix, I is a $N \times N$ unit matrix and $O$ is a $N \times N$ zero matrix.

From (1.5) and (2.2), one has $\delta z(t) = z(t+1) \; {\rm{and}}\; \delta^{2k+1}z(t) = -z(t).$ Then, $\{1, \delta, \delta^2, \cdots, \delta^{4k+1}\}$ is a Lie group with $\delta^{4k+2} = 1,$ where 1 stands for the identity transform in Z.

Now we have

i.e., with respect to the Lie group $\{1, \delta, \delta^2, \cdots, \delta^{4k+1}\}$, $\Phi$ is invariant and $\Phi'$ is $\delta-$equivariant.

In order to prove our results we shall apply the following lemma.

Lemma 2.2 [2, Lemma 2.4]

Assume there are two $S^1$-invariant linear subspaces, $Z^+$ and $Z^- \subset Z$, and $r > 0$ such that

(a) $Z^+ \cup Z^-$ is a closed and finite codimension in Z,

(b) $\hat{L}(Z^-) \subset Z^-$ with $\hat{L} = L - P^{-1}B_\infty$ or $\hat{L} = L - P^{-1}B_0,$

(c) there exists $c_\infty \in \mathbb{R}$ such that

(d) there exists $c_0 \in \mathbb{R}$ such that

(e) $\Phi$ satisfies (P.S)$_c$-condition for $c_0 \leq c \leq c_\infty.$

Then $\Phi$ possesses at least $m = \frac{1}{2}[\dim(Z^- \cap Z^+) - {\rm{codim}}_Z(Z^- \cup Z^+)]$ different critical orbits in $\Phi^{-1}([c_0, c_\infty])$ provided $m > 0$.

Remark 2.1 (P.S)$_c$-condition in (e) can be replaced by (P.S)-condition since (P.S)-condition implies (P.S)$_c$-condition for each $c \in \mathbb{R}$.

Remark 2.2 If (P.S)$_c$-condition in (e) is replaced by (P.S)-condition, (c) and (d) can be changed into

($c'$) there exists $c_\infty \in \mathbb{R}$ such that

($d'$) there exists $c_0 \in \mathbb{R}$ such that

if $\Phi(z)$ is replaced by $-\Phi(z)$.

Remark 2.3 When $Z = \bigoplus\limits_{i = 1}^{\infty}Z(i)$, $Z^+(i) = Z^+ \cap Z(i)$, $Z^-(i) = Z^- \cap Z(i)$, we have $\dim Z(i) = 2N$ and

3.   Main results

Suppose $\{\alpha_j: j = 1, 2, \cdots, N\}$ are the eigenvalues of $A_\infty$ and $\{\beta_j: j = 1, 2, \cdots, N\}$ are the eigenvalues of $A_0$, $d_j, e_j \in \mathbb{R}^N$ are the eigenvectors of $A_\infty$ and $A_0$ with respect to $\alpha_j$ and $\beta_j$, respectively. Assume $\{d_1, d_2, \cdots, d_N\}$ and $\{e_1, e_2, \cdots, e_N\}$ are two orthogonal bases of $\mathbb{R}^N$.

Lemma 3.1 If $z \in Z(i), \; i\neq l(k+1)$, then there is $\gamma = \frac{\pi}{2k+1}$ such that

Proof. Denote $|\cdot|$ as the norm in $\mathbb{R}^N$, then

It follows that

$\begin{array}{l} \langle Lz, z \rangle\\ = \sum\limits_{0 \leq m < l \leq 2k-1}(-1)^{m+l+1}\int_{0}^{2(2k+1)}[(x(t-m), x'(t-l))-(x'(t-m), x(t-l))]dt \\ = 2\sum\limits_{0 \leq m < l \leq 2k-1}(-1)^{m+l+1}\int_{0}^{2(2k+1)}(x(t-m), x'(t-l))dt\\ {\stackrel{j = l-m}{ = }} 2\sum\limits_{j = 1}^{2k-1}\sum\limits_{l = j}^{2k-1}(-1)^{2l-j+1}\int_{0}^{2(2k+1)}(x(t-l+j), x'(t-l))dt\\ = 2\sum\limits_{j = 1}^{2k-1}(-1)^{j+1}(2k-j)\int_{0}^{2(2k+1)}(x(t+j), x'(t))dt\\ = 2\sum\limits_{j = 1}^{2k-1}(2k-j)(-1)^{j+1}(2i-1)\gamma(2k+1)(|a|^2+|b|^2)\sin j(2i-1)\gamma\\ = 2(2i-1)\gamma(2k+1)\sum\limits_{j = 1}^{2k-1}(2k-j)(-1)^{j+1}\sin j(2i-1)\gamma(|a|^2+|b|^2). \end{array}$

Since

one has

Lemma 3.1 is proved.

Denote $\mathbb{E}_j = \{\lambda e_j : \lambda \in \mathbb{R}\}$, $\mathbb{D}_j = \{\lambda d_j : \lambda \in \mathbb{R}\}$, $j = 1, 2, \cdots, N$, $l \geq 0$ and

then $\mathbb{R}^N = \sum\limits_{j = 1}^{N} \mathbb{D}_j = \sum\limits_{j = 1}^{N} \mathbb{E}_j$. Suppose

and

therefore,

Denote

It follows from (3.1) and (3.2) that for $z_1(t) = x(t),$ one has

and

and

Denote

Then

From (3.1) and Lemma 3.1 ^[7], we have the following lemmas.

Lemma 3.2 $\hat{L}(Z_{\infty}^{-}) \subset Z_{\infty}^{-}$, $\hat{L}(Z_{\infty}^{0}) \subset Z_{\infty}^{0}$, $\hat{L}(Z_{\infty}^{+}) \subset Z_{\infty}^{+}$, $\hat{L}(Z_{0}^{-}) \subset Z_{0}^{-}$, $\hat{L}(Z_{0}^{0}) \subset Z_{0}^{0}$, $\hat{L}(Z_{0}^{+}) \subset Z_{0}^{+}$, ($\hat{L} = L - \hat{P}^{-1} B_{\infty}$ or $\hat{L} = L - \hat{P}^{-1} B_0$).

Lemma 3.3 All the subspaces of Z,

are of finite codimensions in Z.

Denote $\Gamma_\infty^+ = \{\alpha_j > 0: \; {\text{there are}} \; l\geq 0, \; i\in \{1, 2, \cdots, k\} \; {\text{such that}} \; \alpha_j = \gamma (2l(2k+1)+2i-1)\tan\frac{(2i-1)\gamma}{2}\}, \; \Gamma_\infty^- = \{\alpha_j < 0: \; {\text{there are}} \; l\geq 0, \; i\in \{1, 2, \cdots, k\} \; {\text{such that}} \; \alpha_j = -\gamma ((2l+1)(2k+1)+2(k+1-i))\tan\frac{(2i-1)\gamma}{2}\}, \; \Gamma_0^+ = \{\beta_j > 0: \; {\text{there are}} \; l\geq 0, \; i\in \{1, 2, \cdots, k\} \; {\text{such that}} \; \beta_j = \gamma (2l(2k+1)+2i-1)\tan\frac{(2i-1)\gamma}{2}\}, \; \Gamma_0^- = \{\beta_j < 0: \; {\text{there are}} \; l\geq 0, \; i\in \{1, 2, \cdots, k\} \; {\text{such that}} \; \beta_j = -\gamma ((2l+1)(2k+1)+2(k+1-i))\tan\frac{(2i-1)\gamma}{2}\}, \; \Gamma_\infty = \Gamma_\infty^+ \cup \Gamma_\infty^-, \; \Gamma_0 = \Gamma_0^+ \cup \Gamma_0^-$.

For $i\in \{1, 2, \cdots, k\}$ and $l \geq 0$, denote

$\eta_\infty^+ = \sharp\{(l, i): \; {\text{there is}} \; \alpha_j \in \Gamma_\infty^+ \; {\text{such that}} \; \gamma (2l(2k+1)+2i-1)\tan\frac{(2i-1)\gamma}{2} = \alpha_j\},$

$\eta_\infty^- = \sharp\{(l, i): \; {\text{there is}} \; \alpha_j \in \Gamma_\infty^- \; {\text{such that}} \; -\gamma ((2l+1)(2k+1)+2(k+1-i))\tan\frac{(2i-1)\gamma}{2} = \alpha_j\},$

$\eta_0^+ = \sharp\{(l, i): \; {\text{there is}} \; \beta_j \in \Gamma_0^+ \; {\text{such that}} \; \gamma (2l(2k+1)+2i-1)\tan\frac{(2i-1)\gamma}{2} = \beta_j\},$

$\eta_0^- = \sharp\{(l, i): \; {\text{there is}} \; \beta_j \in \Gamma_0^- \; {\text{such that}} \; -\gamma ((2l+1)(2k+1)+2(k+1-i))\tan\frac{(2i-1)\gamma}{2} = \beta_j\}.$

Let $\mathbb{D} = \bigoplus\{\mathbb{D}_j:\alpha_j\in \Gamma_{\infty}\}$ and $\Pi:\mathbb{R}^N\rightarrow \mathbb{D}$ be an orthogonal projection with $\Pi\mathbb{R}^N = \mathbb{D}$. Assume

($A_1$) $F \in C^1(\mathbb{R}^N, \mathbb{R})$ satisfies (1.4) and $F(-x) = F(x), \; F(0) = 0,$

($A_{2}$) there exist $M$ and $r\in C^0(\mathbb{R}^{+}, \mathbb{R}^{+})$ with $r(\infty) = +\infty$, $\frac{r(s)}{s}\rightarrow 0$ as $s\rightarrow \infty$, such that $|F(x)-\frac{1}{2}(A_\infty x, x)| \geq -M+r(\mid \Pi x \mid)$ whenever $x\in \cup \{\mathbb{D}_j: \alpha_j\in \Gamma_\infty\},$

($A_{3}^{\pm}$) $\pm[F(x)-\frac{1}{2}(A_\infty x, x)] > 0, \; {\rm{as}} \; |x|\rightarrow \infty,$

($A_{4}^{\pm}$) $\pm[F(x)-\frac{1}{2}(A_0 x, x)] > 0, 0 < |x| \ll 1.$

By a standard argument as the proof of Lemma 2.1 ^[13] and Lemma 3.3 ^[8], we have

Lemma 3.4 Assume ($A_{1}$) and ($A_{2}$) hold. Then, $\Phi(x)$ defined by (2.4) satisfies (P.S)-condition.

Lemma 3.5 Suppose ($A_1$) and ($A_2$) hold. Then there is $I > 0$ such that

if

Now we give the main results of this paper.

Theorem 3.1 Assume ($A_{1}$) and ($A_2$) hold. Then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-(2k+1)) = -x(t)$ provided $m > 0$.

Corollary 3.1 Suppose ($A_{1}$) and ($A_2$) hold.

ⅰ) If $\Gamma_\infty^+ \cup \Gamma_\infty^- = \emptyset$, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅱ) If $\Gamma_0^+ \cup \Gamma_0^- = \emptyset$, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅲ) If $\Gamma_0^+ \cup \Gamma_0^- = \Gamma_\infty^+ \cup \Gamma_\infty^- = \Gamma_0^+ \cup \Gamma_0^- = \emptyset$, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

Corollary 3.1 can be directly obtained from Theorem 3.1.

Theorem 3.2 Suppose ($A_{1}$) and ($A_2$) hold.

ⅰ) If ($A_{3}^{+}$) holds, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅱ) If ($A_{3}^{-}$) holds, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅲ) If ($A_{4}^{+}$) holds, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅳ) If ($A_{4}^{-}$) holds, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅴ) If ($A_{3}^{+}$), ($A_{4}^{-}$) hold, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅵ) If ($A_{3}^{-}$), ($A_{4}^{+}$) hold, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅶ) If ($A_{3}^{+}$), ($A_{4}^{+}$) hold, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

ⅷ) If ($A_{3}^{-}$), ($A_{4}^{-}$) hold, then system (1.4) possesses at least

different $2(2k+1)$-periodic orbits satisfying $x(t-2k-1) = -x(t)$ provided that $m > 0$.

Proof of Theorem 3.1.

Suppose $m = {\rm{ind}}(A_{0}) - {\rm{ind}}(A_{\infty})-\eta_{0}^{-}-\eta_{\infty}^{+} > 0$.

Let $Z^{+} = Z_{\infty}^{-}$ and $Z^{-} = Z_{0}^{+}, \forall z \in Z^+$. There is $\sigma > 0$ such that

Then,

Therefore,

which implies that there is $c_0 \in \mathbb{R}$ such that

At the same time, there are $c_\infty \in \mathbb{R}^-$ and $\gamma > 0$ such that

Suppose $c_0 < c_\infty$. Then conditions (c) and (d) in Lemma 2.2 are satisfied.

Since Lemmas 2.1 and 3.2–3.4 imply (P.S)-condition as well as conditions (a) and (b) in Lemma 2.2 hold under the requirement of Theorem 3.1, we only need to compute $m$.

From Lemma 3.5 we have

Then,

Denote $I_1 = \sharp \{l(2k+1)+i \leq I: l \geq 0, i\in \{1, 2, \cdots, k\}\}, \; I_2 = \sharp \{l(2k+1)+2(k+1)-i \leq I: l \geq 0, i\in \{1, 2, \cdots, k\}\},$ and ${\rm{ind}}A_\infty^+ = \sum\limits_{\alpha_j > 0}{\rm{ind}}\alpha_j, \; {\rm{ind}}A_\infty^- = \sum\limits_{\alpha_j < 0}{\rm{ind}}\alpha_j, \; {\rm{ind}}A_0^+ = \sum\limits_{\beta_j > 0}{\rm{ind}}\beta_j, \; {\rm{ind}}A_0^- = \sum\limits_{\beta_j < 0}{\rm{ind}}\beta_j$, then,

Obviously,

$\sum\limits_{\stackrel{l(2k+1)+i \leq I}{i\in \{1, 2, \cdots, k\}, l\geq 0}} \dim Z_0^-(l(2k+1)+i) = 2 {\rm{ind}} A_0^+,$

$\begin{array}{l} \sum\limits_{\stackrel{l(2k+1)+i \leq I}{i\in \{1, 2, \cdots, k\}, l\geq 0}} \dim Z_\infty^+(l(2k+1)+i) \\ = \sum\limits_{\stackrel{l(2k+1)+i \leq I}{i\in \{1, 2, \cdots, k\}, l\geq 0}}[2N-\dim Z_\infty^-(l(2k+1)+i)- \dim Z_\infty^0(l(2k+1)+i)]\\ = 2NI_1- 2{\rm{ind}}A_\infty^+-2\eta_\infty^+, \end{array}$

$\begin{array}{l} \sum\limits_{\stackrel{l(2k+1)+2(k+1)-i \leq I}{i\in \{1, 2, \cdots, k\}, l\geq 0}} \dim Z_0^-(l(2k+1)+2(k+1)-i) \\ = \sum\limits_{\stackrel{l(2k+1)+2(k+1)-i \leq I}{i\in \{1, 2, \cdots, k\}, l\geq 0}}[2N-\dim Z_0^0- \dim Z_0^+]\\ = 2NI_2+ 2{\rm{ind}}A_0^–2\eta_0^-. \end{array}$

Therefore,

Theorem 3.1 is proved.

Proof of Theorem 3.2.

Since the proof of Theorem 3.2 for each case is similar, we prove the theorem only for case (ⅱ).

Without loss of generality, suppose

and denote $Z^+ = Z_\infty^+ + Z_\infty^0, \; Z^- = Z_0^-.$

From $(A_3^-)$, there is $M > 0$ such that

Then, for $z \in Z_\infty^+ + Z_\infty^0$,

which implies that condition (d) in Lemma 2.2 holds.

From (3.8), one has

$\begin{array}{rl} m = & \frac{1}{2}\sum\limits_{\stackrel{1 \leq i \leq I}{i \neq l+k+1}} [\dim Z^+(i) + \dim Z^-(i) -2N]\\ = & \frac{1}{2}\sum\limits_{\stackrel{1 \leq i \leq I}{i \neq l(2k+1)+k+1}} [\dim Z_\infty^+(i) + \dim Z_\infty^0(i) + \dim Z_0^-(i) -2N]\\ = & \frac{1}{2}\sum\limits_{\stackrel{1 \leq i \leq I}{i \neq l(2k+1)+k+1}} [\dim Z_\infty^+(i) + \dim Z_\infty^0(i) -2N]+ \frac{1}{2}\sum\limits_{\stackrel{1 \leq i \leq I}{i \neq l(2k+1)+k+1}}\dim Z_\infty^0(i) \\ = & {\rm{ind}} A_0 -{\rm{ind}} A_\infty -\eta_\infty^+ -\eta_0^- +\frac{1}{2}\dim Z_\infty^0\\ = & {\rm{ind}} A_0 -{\rm{ind}} A_\infty -\eta_\infty^+ -\eta_0^- +\eta_\infty^++\eta_\infty^-\\ = & {\rm{ind}} A_0 -{\rm{ind}} A_\infty -\eta_0^- +\eta_\infty^-. \end{array}$

Theorem 3.2 is proved.

4.   Examples

Example 4.1 Consider the number of 6-periodic solutions of the following system

where $x = (x_1, x_2)\in \mathbb{R}^2$,

Obviously, $F \in C^1(\mathbb{R}^2, \mathbb{R}), \; F(0) = 0, \; F(-x) = F(x)$,

and

The eigenvalues of $A_\infty$ and $A_0$ are

respectively. Since $k = 1$, one has

Therefore,

At the same time,

and all the conditions of Theorem 3.2 (vi) are satisfied. Then system (4.1) has at least

different 6-periodic orbits satisfying $x(t-3) = -x(t)$.

Example 4.2 Let $N = 2$. We discuss the multiplicity of 6-periodic solutions of system (4.1), where $x = (x_1, x_2)\in \mathbb{R}^2$,

Then

where

$A_\infty$ and $A_0$ have their eigenvalues $\alpha_1 = -\frac{10\pi}{3\sqrt{3}}, \; \alpha_2 = \frac{20\pi}{3\sqrt{3}},$ and $\beta_1 = -\frac{10\pi}{\sqrt{3}}, \; \beta_2 = -\frac{10\pi}{3\sqrt{3}},$ respectively.

So

and then

On the other hand, we have

Therefore,

and system (4.1) has at least 9 different 6-periodic orbits satisfying $x(t-3) = -x(t)$ by Theorem 3.1.

Acknowledgments

Supported by R & D Program of Beijing Municipal Education Commission (No. KM202011417010), Academic Research Projects of Beijing Union University (No. ZB10202004), Scientific Research Projects of Beijing Union University (No. WZ10201902).

Conflict of interest

The authors declare that there is no conflicts of interest regarding the publication of this paper.