具有脉冲的非线性耦合积分–微分系统的周期性

Periodic and Asymptotically Periodic Solutions on Nonlinear Coupled Integro-Differential Systems with Impulses

摘要: 该文研究了具有脉冲的非线性耦合积分—微分系统的周期性。利用Schauder不动点定理，证明了具有脉冲的非线性耦合积分—微分系统至少存在一个周期解和一个渐近周期解，我们的结果推广和改进了相关文献的结果。

Abstract: In this paper, we study the existence of periodic and asymptotically periodic solutions for a coupled nonlinear Volterra integro-differential equation with impulses. By using Schauder’s fixed point theorem, we obtain that the system has at least one periodic solution and an asymptotically periodic solution.

1. 引言

脉冲效应指在一定时间内状态突然发生改变，脉冲微分方程广泛应用于种群生物学，医学，工程，化学等领域 [1] [2] [3] [4] [5] 。耦合积分-微分方程也广泛应用于生物和环境科学的许多领域。该文中，我们研究一类具有脉冲的非线性耦合积分-微分方程周期解和渐近周期解的存在性。

在文献 [6] [7] [8] [9] 中，作者研究了Volterra积分–微分方程线性系统渐近周期解的存在性。在文献 [10] 中，作者考虑了无脉冲时周期解和渐近周期解的存在性，该文改进了文献 [10] 中的一些条件，进而得到周期解和渐近周期解的存在性结果。

该文考虑下面的耦合Volterra脉冲微分方程

$\{\begin{array}{l}{x}^{\prime }\left(t\right)=h_1\left(t\right)x\left(t\right)+h_2\left(t\right)y\left(t\right)+{\displaystyle \underset{-\infty }{\overset{t}{\int }}a\left(t,s\right)f\left(x\left(s\right),y\left(s\right)\right)},t\ne t_k\\ y^{\prime }\left(t\right)=p_1\left(t\right)y\left(t\right)+p_2\left(t\right)x\left(t\right)+{\displaystyle \underset{-\infty }{\overset{t}{\int }}b\left(t,s\right)g\left(x\left(s\right),y\left(s\right)\right)},t\ne t_k\\ x\left(t_k^{+}\right)=x\left(t_k\right)+I_k\left(x\left(t_k\right)\right),t=t_k\\ y\left(t_k^{+}\right)=y\left(t_k\right)+J_k\left(y\left(t_k\right)\right),t=t_k\end{array}$ (1.1)

其中a，b，f，g， $h_i,p_i,i=1,2$ ， $I_k,J_k,k\in N^{+}$ 是连续函数。

假设存在一个正常数 $T>0$ ，使得对 $\forall t,s\in R$ 有

$\begin{array}{l}a\left(t+T,s+T\right)=a\left(t,s\right),b\left(t+T,s+T\right)=b\left(t,s\right),\\ p_i\left(t+T\right)=p_i\left(t\right),h_i\left(t+T\right)=h_i\left(t\right),i=1,2\end{array}$ (1.2)

且存在常数 $q>0$ ，使得

$t_{k+q}=t_k+T,I_{k+q}(\cdot )=I_k(\cdot ),J_{k+q}(\cdot )=J_k(\cdot ),k\in N^{+}$ (1.3)

此外，假设

${\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}\ne 0,{\displaystyle \underset{0}{\overset{T}{\int }}{p}_1\left(s\right)\text{d}s}\ne 0$ (1.4)

定义 $P_T=\left\{\left(\phi ,\psi \right)\left(t+T\right)=\left(\phi ,\psi \right)\left(t\right)\right\}$ ，其中 $\phi ,\psi$ 是实值连续函数。定义范数

$\Vert \left(x,y\right)\Vert =\max \left\{\underset{t\in \left[0,T\right]}{\max }\left|x\left(t\right)\right|,\underset{t\in \left[0,T\right]}{\max }\left|y\left(t\right)\right|\right\}$

那么， $P_T$ 是Banach空间。

引理1.1 假设(1.2)~(1.4)成立，如果 $\left(x,y\right)\in P_T$ ，那么 $x\left(t\right)$ 和 $y\left(t\right)$ 是(1.1)的一个周期解当且仅当

$x\left(t\right)=\{\begin{array}{l}{x}^{\ast }\left(t\right),t\in \left[0,T\right]\\ x^{\ast }\left(t-nT\right),t\in \left[nT,\left(n+1\right)T\right]\end{array}$

$y\left(t\right)=\{\begin{array}{l}{y}^{\ast }\left(t\right),t\in \left[0,T\right]\\ y^{\ast }\left(t-nT\right),t\in \left[nT,\left(n+1\right)T\right]\end{array}$

其中

$x^{*}\left(t\right)={\displaystyle \underset{0}{\overset{T}{\int }}{G}_1\left(t,s\right)\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }{G}_1\left(t,t_k\right)I_k\left(x\left(t_k\right)\right)}$

$G_1\left(t,s\right)=\frac{1}{1-{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}\{\begin{array}{l}{\text{e}}^{{\displaystyle \underset{s}{\overset{t}{\int }}{h}_1\left(u\right)\text{d}u}},0\le s<t\le T\\ {\text{e}}^{{\displaystyle \underset{s}{\overset{t+T}{\int }}{h}_1\left(u\right)\text{d}u}},0\le t\le s\le T\end{array}$

$y^{*}\left(t\right)={\displaystyle \underset{0}{\overset{T}{\int }}{G}_2\left(t,s\right)\left(p_2\left(s\right)x\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }{G}_2\left(t,t_k\right)J_k\left(y\left(t_k\right)\right)}$

$G_2\left(t,s\right)=\frac{1}{1-{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{p}_1\left(s\right)\text{d}s}}}\{\begin{array}{l}{\text{e}}^{{\displaystyle \underset{s}{\overset{t}{\int }}{p}_1\left(u\right)\text{d}u}},0\le s<t\le T\\ {\text{e}}^{{\displaystyle \underset{s}{\overset{t+T}{\int }}{p}_1\left(u\right)\text{d}u}},0\le t\le s\le T\end{array}$

证明：设 $\left(x,y\right)\in P_T$ 是(1.1)的解，对(1.1)的第一个方程从 $\left[0,t\right]$ 积分得

$\begin{array}{c}x\left(t\right){\text{e}}^{-{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}-x\left(0\right)={\displaystyle \underset{0}{\overset{t}{\int }}{\text{e}}^{-{\displaystyle \underset{0}{\overset{s}{\int }}{h}_1(u)du}}}\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s\\ +{\displaystyle \underset{0<t_k<t}{\sum }{\text{e}}^{-{\displaystyle \underset{0}{\overset{t_k}{\int }}{h}_1\left(s\right)\text{d}s}}{I}_k\left(x\left(t_k\right)\right)}\end{array}$

由 $x\left(0\right)=x\left(T\right)$ 得

$\begin{array}{c}{x}^{*}\left(t\right)=x\left(0\right){\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(u\right)\text{d}u}}+{\displaystyle \underset{0}{\overset{t}{\int }}{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(u\right)\text{d}u}}}\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s+{\displaystyle \underset{0<t_k<T}{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}{I}_k\left(x\left(t_k\right)\right)}\\ =\frac{1}{1-{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}{\displaystyle \underset{0}{\overset{t}{\int }}{\text{e}}^{{\displaystyle \underset{s}{\overset{t}{\int }}{h}_1\left(u\right)\text{d}u}}}\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s\\ +\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}{1-{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}{\displaystyle \underset{t}{\overset{T}{\int }}{\text{e}}^{{\displaystyle \underset{s}{\overset{t}{\int }}{h}_1\left(u\right)\text{d}u}}}\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s\\ +\frac{1}{1-{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}{\displaystyle \underset{0<t_k<t}{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}{I}_k\left(x\left(t_k\right)\right)}+\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}{1-{\text{e}}^{{\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}}}{\displaystyle \underset{t<t_k<T}{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}{I}_k\left(x\left(t_k\right)\right)}\\ ={\displaystyle \underset{0}{\overset{T}{\int }}{G}_1\left(t,s\right)\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }{G}_1\left(t,t_k\right)I_k\left(x\left(t_k\right)\right)}\end{array}$

同理可证，

$y^{*}\left(t\right)={\displaystyle \underset{0}{\overset{T}{\int }}{G}_2\left(t,s\right)\left(p_2\left(s\right)x\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}b(s,u)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }{G}_2\left(t,t_k\right)J_k\left(y\left(t_k\right)\right)}$

反之亦然，因此，引理1.1得证。

2. 周期解

定理2.1 (Schuder不动点定理)设X是Banach空间，K是X中有界闭凸子集。如果 $T:K\to K$ 是全连续的，则T在K中有一个不动点。

本文给出以下假设。

假设存在正常数 $L_1,L_2,L_1^{\ast },L_2^{\ast },M_1,M_2,N_1,N_2,b_k,c_k,\left(k\in N^{+}\right)$ ，使得

$\left|f\left(x,y\right)\right|\le M_1,\left|g\left(x,y\right)\right|\le M_2$ (2.1)

${\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)h_2\left(s\right)\right|\text{d}s}\le L_1,{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)p_2\left(s\right)\right|\text{d}s}\le L_2$ (2.2)

${\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\text{d}u\text{d}s}\le N_1,{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|b\left(s,u\right)\right|\text{d}u\text{d}s}\le N_2$ (2.3)

$\{\begin{array}{l}\left|I_k\left(x\right)\right|\le b_k\left|x\right|,\left|{\displaystyle \underset{k=1}{\overset{p}{\sum }}{G}_1\left(t,t_k\right)b_k}\right|\le L_1^{\ast },t\in \left[0,T\right]\\ \left|J_k\left(x\right)\right|\le c_k\left|y\right|,\left|{\displaystyle \underset{k=1}{\overset{p}{\sum }}{G}_2\left(t,t_k\right)c_k}\right|\le L_2^{\ast },t\in \left[0,T\right]\end{array}$ (2.4)

设 ${\Omega }_M=\left\{\left(x,y\right):\left(x,y\right)\in P_T,\Vert \left(x,y\right)\Vert \le M\right\}$ 是 $P_T$ 的子集，则 ${\Omega }_M$ 是 $P_T$ 中的有界闭子集。

定义算子 $F:{\Omega }_M\to P_T$

$F\left(x,y\right)=(F_1\left(x,y\right)\left(t\right),F_2\left(x,y\right)(t))$

其中

$F_1\left(x,y\right)\left(t\right)={\displaystyle \underset{0}{\overset{T}{\int }}{G}_1\left(t,s\right)\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }{G}_1\left(t,t_k\right)I_k\left(x\left(t_k\right)\right)}$

$F_2\left(x,y\right)\left(t\right)={\displaystyle \underset{0}{\overset{T}{\int }}{G}_2\left(t,s\right)\left(p_2\left(s\right)x\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }{G}_2\left(t,t_k\right)J_k\left(y\left(t_k\right)\right)}$

定理2.2 假设(1.2)~(1.4)，(2.1)~(2.4)成立，且

$L_1+L_1^{\ast }<1,L_2+L_2^{\ast }<1$ (2.5)

则(1.1)至少有一个T-周期解。

证明：若 $x\left(t\right),y\left(t\right)$ 是(1.1)的周期解，由引理1.1可知， $F\left(x,y\right)\left(t+T\right)=F\left(x,y\right)\left(t\right)$ 。设

$M=\max \{\frac{N_1{M}_1}{1-L_1-L_1^{\ast }},\frac{N_2{M}_2}{1-L_2-L_2^{\ast }}\}$

对于 $\left(x,y\right)\in {\Omega }_M,t\in \left[0,T\right]$ 有

$\begin{array}{c}\left|F_1\left(x,y\right)\left(t\right)\right|\le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)h_2\left(s\right)\right|\left|y\left(s\right)\right|\text{d}s+}{\displaystyle \underset{0<t_k<T}{\sum }\left|G_1\left(t,t_k\right)\right|\left|I_k\left(x\left(t_k\right)\right)\right|}\\ +{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\left|f\left(x\left(u\right),y\left(u\right)\right)\right|\text{d}u\text{d}s}\\ \le M{L}_1+N_1{M}_1+M{L}_1^{\ast }\le M\end{array}$

$\begin{array}{c}\left|F_2\left(x,y\right)\left(t\right)\right|\le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)p_2\left(s\right)\right|\left|x\left(s\right)\right|\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }\left|G_2\left(t,t_k\right)\right|\left|J_k\left(y\left(t_k\right)\right)\right|}\\ +{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|b\left(s,u\right)\right|\left|g\left(x\left(u\right),y\left(u\right)\right)\right|\text{d}u\text{d}s}\\ \le M{L}_2+N_2{M}_2+M{L}_2^{\ast }\le M\end{array}$

所以， $F\left({\Omega }_M\right)\subseteq {\Omega }_M$ 。

下证F连续。取 ${\Omega }_M$ 中的一个序列 $\left\{\left(x^n,y^n\right)\right\}$ ，且

$\underset{n\to \infty }{\lim }\Vert \left(x^n,y^n\right)-\left(x,y\right)\Vert =0$

因为 ${\Omega }_M$ 是闭的，我们有 $\left(x,y\right)\in {\Omega }_M$ 。由F的定义，有

$\begin{array}{l}\Vert F\left(x^n,y^n\right)-F\left(x,y\right)\Vert \\ =\max \left\{\underset{t\in \left[0,T\right]}{\max }\left|F_1\left(x^n,y^n\right)\left(t\right)-F_1\left(x,y\right)\left(t\right)\right|,\underset{t\in \left[0,T\right]}{\max }\left|F_2\left(x^n,y^n\right)\left(t\right)-F_2\left(x,y\right)\left(t\right)\right|\right\}\end{array}$

$\begin{array}{l}\left|F_1\left(x^n,y^n\right)\left(t\right)-F_1\left(x,y\right)\left(t\right)\right|\\ =|{\displaystyle \underset{0}{\overset{T}{\int }}{G}_1\left(t,s\right)\left(h_2\left(s\right)y^n\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x^n\left(u\right),y^n\left(u\right)\right)\text{d}u}\right)\text{d}s}\\ -{\displaystyle \underset{0}{\overset{T}{\int }}{G}_1\left(t,s\right)\left(h_2\left(s\right)y\left(s\right)+{\displaystyle \underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u}\right)\text{d}s}\\ +{\displaystyle \underset{0<t_k<T}{\sum }{G}_1\left(t,t_k\right)\left(I_k\left(x^n\left(t_k\right)\right)-I_k\left(x\left(t_k\right)\right)\right)}|\\ \le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)\right|\left({\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\left|f\left(x^n\left(u\right),y^n\left(u\right)\right)-f\left(x\left(u\right),y\left(u\right)\right)\right|\text{d}u}\right)\text{d}s}\\ +{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)h_2\left(s\right)\right|\left|y^n\left(s\right)-y\left(s\right)\right|\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }\left|G_1\left(t,t_k\right)\right|\left|I_k\left(x^n\left(t_k\right)\right)-I_k\left(x\left(t_k\right)\right)\right|}\end{array}$

由 $f,I_k$ 的连续性和Lebesgue控制收敛定理，有

$\underset{n\to \infty }{\lim }\underset{t\in \left[0,T\right]}{\max }\left|F_1\left(x^n,y^n\right)\left(t\right)-F_1\left(x,y\right)\left(t\right)\right|=0$

同理可证，

$\underset{n\to \infty }{\lim }\underset{t\in \left[0,T\right]}{\max }\left|F_2\left(x^n,y^n\right)\left(t\right)-F_2\left(x,y\right)\left(t\right)\right|=0$

所以，F是连续映射。因为 $F\left({\Omega }_M\right)\subseteq {\Omega }_M$ ，所以 $F\left({\Omega }_M\right)$ 是一致有界的。易证对任意的 $\left(x,y\right)\in {\Omega }_M,t\ne t_k$ 存在一个常数 $L>0$ ，使得 $\left|\frac{\text{d}}{\text{d}t}{F}_1\left(x,y\right)\left(t\right)\right|\le L,\left|\frac{\text{d}}{\text{d}t}{F}_2\left(x,y\right)\left(t\right)\right|\le L$ ，即 $\left|\frac{\text{d}}{\text{d}t}F\left(x,y\right)\left(t\right)\right|\le L$ 。所以， $F\left({\Omega }_M\right)\subseteq {\Omega }_M$ 是拟等度连续的。由Arzela-Ascoli’s定理， $F\left({\Omega }_M\right)$ 是列紧的。综上，F是全连续的。

由定理2.1可知，F在 ${\Omega }_M$ 中有一个不动点，即存在 $\left(x,y\right)\in {\Omega }_M$ ，使得 $\left(x,y\right)=F\left(x,y\right)$ 。由引理1.1可知，(1.1)有一个T-周期解。

定理2.3 假设(1.2)~(1.4)，(2.3)~(2.5)成立，且存在连续函数 $G,W$ ，及正常数 $Q_1,Q_2$ 使得

$\left|f\left(x,y\right)\right|\le Q_{1}W\left(\left|y\right|\right),\frac{W\left(u\right)}{u}\le \frac{1-L_1-L_1^{\ast }}{N_1{Q}_1},u>0$

$\left|g\left(x,y\right)\right|\le Q_{2}G\left(\left|x\right|\right),\frac{G\left(u\right)}{u}\le \frac{1-L_2-L_2^{\ast }}{N_2{Q}_2},u>0$

那么(1.1)至少有一个T-周期解。

证明：令

$M=\max \left\{\frac{W\left(M\right)N_1{Q}_1}{1-L_1-L_1^{\ast }},\frac{G\left(M\right)N_2{Q}_2}{1-L_2-L_2^{\ast }}\right\}$

对 $\left(x,y\right)\in {\Omega }_M,t\in \left[0,T\right]$ 有

$\begin{array}{c}\left|F_1\left(x,y\right)\left(t\right)\right|\le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)h_2\left(s\right)\right|\left|y\left(s\right)\right|\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }\left|G_1\left(t,t_k\right)\right|\left|I_k\left(x\left(t_k\right)\right)\right|}\\ +{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\left|f\left(x\left(u\right),y\left(u\right)\right)\right|\text{d}u\text{d}s}\\ \le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)h_2\left(s\right)\right|\left|y\left(s\right)\right|\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }\left|G_1\left(t,t_k\right)\right|\left|I_k\left(x\left(t_k\right)\right)\right|}\\ +Q_1{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_1\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|W\left(\left|y\left(u\right)\right|\right)\text{d}u\text{d}s}\\ \le M{L}_1+M{L}_1^{\ast }+\frac{1-L_1-L_1^{\ast }}{N_1{Q}_1}{N}_1{Q}_1\left|y\left(u\right)\right|\\ \le M{L}_1+M{L}_1^{\ast }+\left(1-L_1-L_1^{\ast }\right)M=M\end{array}$

类似地，

$\begin{array}{c}\left|F_2\left(x,y\right)\left(t\right)\right|\le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)p_2\left(s\right)\right|\left|x\left(s\right)\right|\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }\left|G_2\left(t,t_k\right)\right|\left|J_k\left(y\left(t_k\right)\right)\right|}\\ +{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|b\left(s,u\right)\right|\left|g\left(x\left(u\right),y\left(u\right)\right)\right|\text{d}u\text{d}s}\\ \le {\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)p_2\left(s\right)\right|\left|x\left(s\right)\right|\text{d}s}+{\displaystyle \underset{0<t_k<T}{\sum }\left|G_2\left(t,t_k\right)\right|\left|J_k\left(y\left(t_k\right)\right)\right|}\\ +Q_2{\displaystyle \underset{0}{\overset{T}{\int }}\left|G_2\left(t,s\right)\right|}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|b\left(s,u\right)\right|G\left(\left|x\left(u\right)\right|\right)\text{d}u\text{d}s}\\ \le M{L}_2+M{L}_2^{\ast }+\frac{1-L_2-L_2^{\ast }}{N_2{Q}_2}{N}_2{Q}_2\left|x\left(u\right)\right|\\ \le M{L}_2+M{L}_2^{\ast }+\left(1-L_2-L_2^{\ast }\right)M=M\end{array}$

其余的证明类似定理2.2的证明。

3. 渐近周期解

定义3.1 一个函数 $x\left(t\right)$ 称为渐近T-周期的，如果存在两个函数 $x_1\left(t\right)$ 和 $x_2\left(t\right)$ ，使得 $x\left(t\right)=x_1\left(t\right)+x_2\left(t\right)$ ，其中 $x_1\left(t\right)$ 是T-周期的， $\underset{t\to \infty }{\lim }{x}_2\left(t\right)=0$ 。

假设 $h_1\left(t\right),p_1\left(t\right)$ 是T-周期的，且

${\displaystyle \underset{0}{\overset{T}{\int }}{h}_1\left(s\right)\text{d}s}=0,{\displaystyle \underset{0}{\overset{T}{\int }}{p}_1\left(s\right)\text{d}s}=0$ (3.1)

从而存在常数 $m_k,M_k,k=1,2$ ，使得

$m_1\le {\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}\le M_1^{\ast },m_2\le {\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}\le M_2^{\ast }$ (3.2)

假设存在正数A，B使得

${\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\text{d}u\text{d}s}}\le A,{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|b\left(s,u\right)\right|\text{d}u\text{d}s}}\le B$ (3.3)

另外，假设

${\displaystyle \underset{t}{\overset{\infty }{\int }}\left|h_2\left(s\right)\right|\text{d}s}\le M_3^{\ast },{\displaystyle \underset{t}{\overset{\infty }{\int }}\left|p_2\left(s\right)\right|\text{d}s}\le M_4^{\ast }$ (3.4)

$b_k\ge 0,c_k\ge 0,{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{b}_k<\infty },{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{c}_k<\infty }$ (3.5)

定理3.1 假设(2.4)，(3.1)~(3.5)成立，存在正常数 ${\bar{Q}}_1,{\bar{Q}}_2$ ，使得

$\left|f\left(x,y\right)\right|\le {\bar{Q}}_1\left|y\right|,\left|g\left(x,y\right)\right|\le {\bar{Q}}_2\left|x\right|$

且

$\begin{array}{l}1-M_3^{\ast }{M}_1^{\ast }{m}_1^{-1}-M_1^{\ast }{m}_1^{-1}A{\bar{Q}}_1-M_1^{\ast }{M}_5^{\ast }>0,\\ 1-M_4^{\ast }{M}_2^{\ast }{m}_2^{-1}-M_2^{\ast }{m}_2^{-1}B{\bar{Q}}_2-M_2^{\ast }{M}_6^{\ast }>0\end{array}$

则系统(1.1)有渐近T-周期解 $\left(x,y\right)$ ，满足

$x\left(t\right)=x_1\left(t\right)+x_2\left(t\right),y\left(t\right)=y_1\left(t\right)+y_2(t)$

其中，

$x_1\left(t\right)=c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}},y_1\left(t\right)=c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}},t\in R$

$c_1,c_2$ 为固定的非零常数，

$\underset{t\to \infty }{\lim }{x}_2\left(t\right)=\underset{t\to \infty }{\lim }{y}_2\left(t\right)=0$ .

证明：定义 $\begin{array}{l}{H}_T=\{\left(\phi ,\psi \right):\phi ={\phi }_1+{\phi }_2,\psi ={\psi }_1+{\psi }_2,\left({\phi }_1,{\psi }_1\right)\left(t+T\right)=\left({\phi }_1,{\psi }_1\right)\left(t\right),\\ \left({\phi }_2,{\psi }_2\right)\left(t\right)\to \left(0,0\right),t\to \infty \}\end{array}$ 。定义范数

$\Vert \left(x,y\right)\Vert =\max \left\{\underset{t\in \left[0,T\right]}{\max }\left|x\left(t\right)\right|,\underset{t\in \left[0,T\right]}{\max }\left|y\left(t\right)\right|\right\}$

则 $H_T$ 是一个Banach空间。记 ${\Omega }_{M^{*}}=\left\{\left(x,y\right):\left(x,y\right)\in H_T,\Vert \left(x,y\right)\Vert \le M^{*}\right\}$ ，则 ${\Omega }_{M^{*}}$ 是 $H_T$ 中的有界闭凸子集。

对 $\left(x,y\right)\in {\Omega }_{M^{*}}$ ，定义算子 $E:{\Omega }_{M^{*}}\to H_T$

$E\left(x,y\right)\left(t\right)=(E_1\left(y\right)\left(t\right),E_2\left(x\right)(t))$

其中

$\begin{array}{l}{E}_1\left(y\right)\left(t\right)\\ =c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}-{\displaystyle \underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{h}_1\left(l\right)\text{d}l}}}{h}_2\left(s\right)y\left(s\right)\text{d}s}-{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{h}_1\left(l\right)\text{d}l}}}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s}}\\ -{\displaystyle \underset{t<t_k<\infty }{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}{I}_k\left(x\left(t_k\right)\right)}\end{array}$

$\begin{array}{l}{E}_2\left(x\right)\left(t\right)\\ =c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}-{\displaystyle \underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{p}_1\left(l\right)\text{d}l}}}{p}_2\left(s\right)x\left(s\right)\text{d}s}-{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{p}_1\left(l\right)\text{d}l}}}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s}}\\ -{\displaystyle \underset{t<t_k<\infty }{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}{J}_k\left(y\left(t_k\right)\right)}\end{array}$

下证映射E在 ${\Omega }_{M^{*}}$ 中有一个不动点。由(3.5)可知，存在正常数 $M_5^{\ast },M_6^{\ast }$ 使得

$\left|{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{b}_k}\right|\le M_5^{\ast },\left|{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{c}_k}\right|\le M_6^{\ast }$

设

$M^{*}=\left\{\frac{c_1{M}_1^{\ast }}{1-M_3^{\ast }{M}_1^{\ast }{m}_1^{-1}-M_1^{\ast }{m}_1^{-1}A{\bar{Q}}_1-M_1^{\ast }{M}_5^{\ast }},\frac{c_2{M}_2^{\ast }}{1-M_4^{\ast }{M}_2^{\ast }{m}_2^{-1}-M_2^{\ast }{m}_2^{-1}B{\bar{Q}}_2-M_2^{\ast }{M}_6^{\ast }}\right\}$

$\begin{array}{l}\left|E_1\left(y\right)\left(t\right)-c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)ds}}\right|\\ \le M^{*}{M}_3^{\ast }{M}_1^{\ast }{m}_1^{-1}+M_1^{\ast }{m}_1^{-1}{\bar{Q}}_1{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\left|x\right|\text{d}u\text{d}s}}+{\displaystyle \underset{t<t_k<\infty }{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}\left|I_k\left(x\left(t_k\right)\right)\right|}\\ \le M^{*}{M}_3^{\ast }{M}_1^{\ast }{m}_1^{-1}+M_1^{\ast }{m}_1^{-1}{\bar{Q}}_1\Vert x\Vert {\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\text{d}u\text{d}s}}+\left|{\displaystyle \underset{t<t_k<\infty }{\sum }{b}_k}\right|M_1^{\ast }{M}^{*}\\ \le M^{*}{M}_3^{\ast }{M}_1^{\ast }{m}_1^{-1}+M_1^{\ast }{m}_1^{-1}A{\bar{Q}}_1{M}^{*}+M_1^{\ast }{M}^{*}{M}_5^{\ast }\end{array}$

同理，

$\left|E_2\left(x\right)\left(t\right)-c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}\right|\le M^{*}{M}_4^{\ast }{M}_2^{\ast }{m}_2^{-1}+M_2^{\ast }{m}_2^{-1}B{\bar{Q}}_2{M}^{*}+M_2^{\ast }{M}^{*}{M}_6^{\ast }$

因此，

$\left|E_1\left(y\right)\left(t\right)\right|\le M^{*}{M}_3^{\ast }{M}_1^{\ast }{m}_1^{-1}+M_1^{\ast }{m}_1^{-1}A{\bar{Q}}_1{M}^{*}+M_1^{\ast }{M}^{*}{M}_5^{\ast }+c_1{M}_1^{\ast }\le M^{*}$

$\left|E_2\left(x\right)\left(t\right)\right|\le M^{*}{M}_4^{\ast }{M}_2^{\ast }{m}_2^{-1}+M_2^{\ast }{m}_2^{-1}B{\bar{Q}}_2{M}^{*}+M_2^{\ast }{M}^{*}{M}_6^{\ast }+c_2{M}_2^{\ast }\le M^{*}$

设

$x_1\left(t\right)=c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}},y_1\left(t\right)=c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}},t\in R$

$\begin{array}{c}{x}_2\left(t\right)=-{\displaystyle \underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{h}_1\left(l\right)\text{d}l}}}{h}_2\left(s\right)y\left(s\right)\text{d}s}-{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{h}_1\left(l\right)\text{d}l}}}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s}}\\ -{\displaystyle \underset{t<t_k<\infty }{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}{I}_k\left(x\left(t_k\right)\right)}\end{array}$

$\begin{array}{c}{y}_2\left(t\right)=-{\displaystyle \underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{p}_1\left(l\right)\text{d}l}}}{p}_2\left(s\right)x\left(s\right)\text{d}s}-{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(l\right)\text{d}l}}}{{\text{e}}^{{\displaystyle \underset{0}{\overset{s}{\int }}{p}_1\left(l\right)\text{d}l}}}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s}}\\ -{\displaystyle \underset{t<t_k<\infty }{\sum }{\text{e}}^{{\displaystyle \underset{t_k}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}{J}_k\left(y\left(t_k\right)\right)}\end{array}$

计算可知，

$x_1\left(t+T\right)=c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t+T}{\int }}{h}_1\left(s\right)\text{d}s}}=c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}{\text{e}}^{{\displaystyle \underset{t}{\overset{t+T}{\int }}{h}_1\left(s\right)\text{d}s}}=c_1{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{h}_1\left(s\right)\text{d}s}}$

$y_1\left(t+T\right)=c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t+T}{\int }}{p}_1\left(s\right)\text{d}s}}=c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}{\text{e}}^{{\displaystyle \underset{t}{\overset{t+T}{\int }}{p}_1\left(s\right)\text{d}s}}=c_2{\text{e}}^{{\displaystyle \underset{0}{\overset{t}{\int }}{p}_1\left(s\right)\text{d}s}}$

由(3.4)~(3.5)可知，

$\begin{array}{l}\underset{t\to \infty }{\lim }\left|x_2\left(t\right)\right|\le M^{*}{M}_1^{\ast }{m}_1^{-1}\underset{t\to \infty }{\lim }{\displaystyle \underset{t}{\overset{\infty }{\int }}\left|h_2\left(s\right)\right|\text{d}s}+M_1^{\ast }{m}_1^{-1}{\bar{Q}}_1{M}^{*}\underset{t\to \infty }{\lim }{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|a\left(s,u\right)\right|\text{d}u\text{d}s}}\\ +M^{*}{M}_1^{\ast }\underset{t\to \infty }{\lim }\left|{\displaystyle \underset{t<t_k<\infty }{\sum }{b}_k}\right|=0\end{array}$

$\begin{array}{l}\underset{t\to \infty }{\lim }\left|y_2\left(t\right)\right|\le M^{*}{M}_2^{\ast }{m}_2^{-1}\underset{t\to \infty }{\lim }{\displaystyle \underset{t}{\overset{\infty }{\int }}\left|p_2\left(s\right)\right|\text{d}s}+M_2^{\ast }{m}_2^{-1}{\bar{Q}}_2{M}^{*}\underset{t\to \infty }{\lim }{\displaystyle \underset{t}{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{s}{\int }}\left|b\left(s,u\right)\right|\text{d}u\text{d}s}}\\ +M^{*}{M}_2^{\ast }\underset{t\to \infty }{\lim }\left|{\displaystyle \underset{t<t_k<\infty }{\sum }{c}_k}\right|=0\end{array}$

因此， $\underset{t\to \infty }{\lim }{x}_2\left(t\right)=0,\underset{t\to \infty }{\lim }{y}_2\left(t\right)=0$ 。故 $E\left({\Omega }_{M^{*}}\right)\subseteq {\Omega }_{M^{*}}$ 。

类似定理2.2可证E是全连续的。因此，由Schauder不动点定理，存在一个不动点 $\left(x,y\right)\in {\Omega }_{M^{*}}$ ，使得

$E\left(x,y\right)\left(t\right)=\left(E_1\left(y\right)\left(t\right),E_2\left(x\right)\left(t\right)\right)=(x\left(t\right),y(t))$

易知该不动点是(1.1)的解。所以 $\left(x,y\right)$ 是(1.1)的解，定理3.2得证。

例3.1 考虑下面的系统

$\{\begin{array}{l}{x}^{\prime }\left(t\right)=\cos \left(t\right)x\left(t\right)+\frac{2t}{{\left(t^2+10\right)}^2}y\left(t\right)+{\displaystyle \underset{-\infty }{\overset{t}{\int }}{\text{e}}^{-6t+4s}\sin \left(x\left(t\right)+y\left(t\right)\right)\text{d}s},t\ne t_k\\ y^{\prime }\left(t\right)=\cos \left(t\right)y\left(t\right)+\frac{2t}{{\left(t^2+10\right)}^2}x\left(t\right)+{\displaystyle \underset{-\infty }{\overset{t}{\int }}{\text{e}}^{-6t+4s}\cos \left(x\left(t\right)+y\left(t\right)\right)\text{d}s},t\ne t_k\\ x\left(t_k^{+}\right)=x\left(t_k\right)+{\left(\frac{1}{12}\right)}^k\left(x\left(t_k\right)\right),t=t_k\\ y\left(t_k^{+}\right)=y\left(t_k\right)+{\left(\frac{1}{13}\right)}^k\left(y\left(t_k\right)\right),t=t_k\end{array}$

经过简单的计算知，定理3.1中的条件均满足，因此，该系统有一个渐近2π-周期解。

基金项目

国家自然科学基金(No. 11571088, No. 11471109)；

浙江省自然科学项目(No. LY14A010024)；

湖南省教育厅项目(No. 14A098)。