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

doi:10.12677/AAM.2019.81015

Advances in Applied Mathematics
Vol. 08 No. 01 ( 2019 ), Article ID: 28619 , 10 pages
10.12677/AAM.2019.81015

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

Qiufeng Chen, Jianli Li

●How to Cite this Article

School of Mathmatics and Statistics, Hunan Normal University, Changsha Hunan

Received: Jan. 2^nd, 2019; accepted: Jan. 17^th, 2019; published: Jan. 24^th, 2019

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.

Keywords:Impulsive Differential Equation, Schauder’s Fixed Point Theorem, Periodic Solutions, Asymptotic Periodic Solutions

具有脉冲的非线性耦合积分–微分系统的周期性

陈秋凤，李建利

湖南师范大学，数学与统计学院，湖南长沙

收稿日期：2019年1月2日；录用日期：2019年1月17日；发布日期：2019年1月24日

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。

关键词 :脉冲微分方程，Schauder不动点定理，周期解，渐近周期解

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

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

${\begin{cases} x^{'} (t) = h_{1} (t) x (t) + h_{2} (t) y (t) + \int_{- \infty}^{t} a (t, s) f (x (s), y (s)), t \neq t_{k} \\ y^{'} (t) = p_{1} (t) y (t) + p_{2} (t) x (t) + \int_{- \infty}^{t} b (t, s) g (x (s), y (s)), t \neq t_{k} \\ x (t_{k}^{+}) = x (t_{k}) + I_{k} (x (t_{k})), t = t_{k} \\ y (t_{k}^{+}) = y (t_{k}) + J_{k} (y (t_{k})), t = t_{k} \end{cases}$ (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

$\begin{array}{l} a (t + T, s + T) = a (t, s), b (t + T, s + T) = b (t, s), \\ p_{i} (t + T) = p_{i} (t), h_{i} (t + T) = h_{i} (t), i = 1, 2 \end{array}$ (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

$\int_{0}^{T} h_{1} (s) d s \neq 0, \int_{0}^{T} p_{1} (s) d s \neq 0$ (1.4)

定义 $P_{T} = {(φ, ψ) (t + T) = (φ, ψ) (t)}$ ，其中 $φ, ψ$ 是实值连续函数。定义范数

$‖ (x, y) ‖ = \max {\max_{t \in [0, T]} | x (t) |, \max_{t \in [0, T]} | y (t) |}$

那么， $P_{T}$ 是Banach空间。

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

$x (t) = {\begin{cases} x^{*} (t), t \in [0, T] \\ x^{*} (t - n T), t \in [n T, (n + 1) T] \end{cases}$

$y (t) = {\begin{cases} y^{*} (t), t \in [0, T] \\ y^{*} (t - n T), t \in [n T, (n + 1) T] \end{cases}$

其中

$x^{*} (t) = \int_{0}^{T} G_{1} (t, s) (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} G_{1} (t, t_{k}) I_{k} (x (t_{k}))$

$G_{1} (t, s) = \frac{1}{1 - e^{\int_{0}^{T} h_{1} (s) d s}} {\begin{cases} e^{\int_{s}^{t} h_{1} (u) d u}, 0 \leq s < t \leq T \\ e^{\int_{s}^{t + T} h_{1} (u) d u}, 0 \leq t \leq s \leq T \end{cases}$

$y^{*} (t) = \int_{0}^{T} G_{2} (t, s) (p_{2} (s) x (s) + \int_{- \infty}^{s} b (s, u) g (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} G_{2} (t, t_{k}) J_{k} (y (t_{k}))$

$G_{2} (t, s) = \frac{1}{1 - e^{\int_{0}^{T} p_{1} (s) d s}} {\begin{cases} e^{\int_{s}^{t} p_{1} (u) d u}, 0 \leq s < t \leq T \\ e^{\int_{s}^{t + T} p_{1} (u) d u}, 0 \leq t \leq s \leq T \end{cases}$

证明：设 $(x, y) \in P_{T}$ 是(1.1)的解，对(1.1)的第一个方程从 $[0, t]$ 积分得

$\begin{matrix} x (t) e^{- \int_{0}^{t} h_{1} (s) d s} - x (0) = \int_{0}^{t} e^{- \int_{0}^{s} h_{1} (u) d u} (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s \\ + \sum_{0 < t_{k} < t} e^{- \int_{0}^{t_{k}} h_{1} (s) d s} I_{k} (x (t_{k})) \end{matrix}$

由 $x (0) = x (T)$ 得

$\begin{matrix} x^{*} (t) = x (0) e^{\int_{0}^{t} h_{1} (u) d u} + \int_{0}^{t} e^{\int_{0}^{t} h_{1} (u) d u} (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} e^{\int_{t_{k}}^{t} h_{1} (s) d s} I_{k} (x (t_{k})) \\ = \frac{1}{1 - e^{\int_{0}^{T} h_{1} (s) d s}} \int_{0}^{t} e^{\int_{s}^{t} h_{1} (u) d u} (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s \\ + \frac{e^{\int_{0}^{T} h_{1} (s) d s}}{1 - e^{\int_{0}^{T} h_{1} (s) d s}} \int_{t}^{T} e^{\int_{s}^{t} h_{1} (u) d u} (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s \\ + \frac{1}{1 - e^{\int_{0}^{T} h_{1} (s) d s}} \sum_{0 < t_{k} < t} e^{\int_{t_{k}}^{t} h_{1} (s) d s} I_{k} (x (t_{k})) + \frac{e^{\int_{0}^{T} h_{1} (s) d s}}{1 - e^{\int_{0}^{T} h_{1} (s) d s}} \sum_{t < t_{k} < T} e^{\int_{t_{k}}^{t} h_{1} (s) d s} I_{k} (x (t_{k})) \\ = \int_{0}^{T} G_{1} (t, s) (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} G_{1} (t, t_{k}) I_{k} (x (t_{k})) \end{matrix}$

同理可证，

$y^{*} (t) = \int_{0}^{T} G_{2} (t, s) (p_{2} (s) x (s) + \int_{- \infty}^{s} b (s, u) g (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} G_{2} (t, t_{k}) J_{k} (y (t_{k}))$

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 $L_{1}, L_{2}, L_{1}^{*}, L_{2}^{*}, M_{1}, M_{2}, N_{1}, N_{2}, b_{k}, c_{k}, (k \in N^{+})$ ，使得

$| f (x, y) | \leq M_{1}, | g (x, y) | \leq M_{2}$ (2.1)

$\int_{0}^{T} | G_{1} (t, s) h_{2} (s) | d s \leq L_{1}, \int_{0}^{T} | G_{2} (t, s) p_{2} (s) | d s \leq L_{2}$ (2.2)

$\int_{0}^{T} | G_{1} (t, s) | \int_{- \infty}^{s} | a (s, u) | d u d s \leq N_{1}, \int_{0}^{T} | G_{2} (t, s) | \int_{- \infty}^{s} | b (s, u) | d u d s \leq N_{2}$ (2.3)

${\begin{cases} | I_{k} (x) | \leq b_{k} | x |, | \sum_{k = 1}^{p} G_{1} (t, t_{k}) b_{k} | \leq L_{1}^{*}, t \in [0, T] \\ | J_{k} (x) | \leq c_{k} | y |, | \sum_{k = 1}^{p} G_{2} (t, t_{k}) c_{k} | \leq L_{2}^{*}, t \in [0, T] \end{cases}$ (2.4)

设 $Ω_{M} = {(x, y) : (x, y) \in P_{T}, ‖ (x, y) ‖ \leq M}$ 是 $P_{T}$ 的子集，则 $Ω_{M}$ 是 $P_{T}$ 中的有界闭子集。

定义算子 $F : Ω_{M} \to P_{T}$

$F (x, y) = (F_{1} (x, y) (t), F_{2} (x, y) (t))$

其中

$F_{1} (x, y) (t) = \int_{0}^{T} G_{1} (t, s) (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} G_{1} (t, t_{k}) I_{k} (x (t_{k}))$

$F_{2} (x, y) (t) = \int_{0}^{T} G_{2} (t, s) (p_{2} (s) x (s) + \int_{- \infty}^{s} b (s, u) g (x (u), y (u)) d u) d s + \sum_{0 < t_{k} < T} G_{2} (t, t_{k}) J_{k} (y (t_{k}))$

定理2.2 假设(1.2)~(1.4)，(2.1)~(2.4)成立，且

$L_{1} + L_{1}^{*} < 1, L_{2} + L_{2}^{*} < 1$ (2.5)

则(1.1)至少有一个T-周期解。

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

$M = \max {\frac{N_{1} M_{1}}{1 - L_{1} - L_{1}^{*}}, \frac{N_{2} M_{2}}{1 - L_{2} - L_{2}^{*}}}$

对于 $(x, y) \in Ω_{M}, t \in [0, T]$ 有

所以， $F (Ω_{M}) \subseteq Ω_{M}$ 。

下证F连续。取 $Ω_{M}$ 中的一个序列 ${(x^{n}, y^{n})}$ ，且

$\lim_{n \to \infty} ‖ (x^{n}, y^{n}) - (x, y) ‖ = 0$

因为 $Ω_{M}$ 是闭的，我们有 $(x, y) \in Ω_{M}$ 。由F的定义，有

$\begin{array}{l} ‖ F (x^{n}, y^{n}) - F (x, y) ‖ \\ = \max {\max_{t \in [0, T]} | F_{1} (x^{n}, y^{n}) (t) - F_{1} (x, y) (t) |, \max_{t \in [0, T]} | F_{2} (x^{n}, y^{n}) (t) - F_{2} (x, y) (t) |} \end{array}$

$\begin{array}{l} | F_{1} (x^{n}, y^{n}) (t) - F_{1} (x, y) (t) | \\ = | \int_{0}^{T} G_{1} (t, s) (h_{2} (s) y^{n} (s) + \int_{- \infty}^{s} a (s, u) f (x^{n} (u), y^{n} (u)) d u) d s \\ - \int_{0}^{T} G_{1} (t, s) (h_{2} (s) y (s) + \int_{- \infty}^{s} a (s, u) f (x (u), y (u)) d u) d s \\ + \sum_{0 < t_{k} < T} G_{1} (t, t_{k}) (I_{k} (x^{n} (t_{k})) - I_{k} (x (t_{k}))) | \\ \leq \int_{0}^{T} | G_{1} (t, s) | (\int_{- \infty}^{s} | a (s, u) | | f (x^{n} (u), y^{n} (u)) - f (x (u), y (u)) | d u) d s \\ + \int_{0}^{T} | G_{1} (t, s) h_{2} (s) | | y^{n} (s) - y (s) | d s + \sum_{0 < t_{k} < T} | G_{1} (t, t_{k}) | | I_{k} (x^{n} (t_{k})) - I_{k} (x (t_{k})) | \end{array}$

由 $f, I_{k}$ 的连续性和Lebesgue控制收敛定理，有

$\lim_{n \to \infty} \max_{t \in [0, T]} | F_{1} (x^{n}, y^{n}) (t) - F_{1} (x, y) (t) | = 0$

同理可证，

$\lim_{n \to \infty} \max_{t \in [0, T]} | F_{2} (x^{n}, y^{n}) (t) - F_{2} (x, y) (t) | = 0$

所以，F是连续映射。因为 $F (Ω_{M}) \subseteq Ω_{M}$ ，所以 $F (Ω_{M})$ 是一致有界的。易证对任意的 $(x, y) \in Ω_{M}, t \neq t_{k}$ 存在一个常数 $L > 0$ ，使得 $| \frac{d}{d t} F_{1} (x, y) (t) | \leq L, | \frac{d}{d t} F_{2} (x, y) (t) | \leq L$ ，即 $| \frac{d}{d t} F (x, y) (t) | \leq L$ 。所以， $F (Ω_{M}) \subseteq Ω_{M}$ 是拟等度连续的。由Arzela-Ascoli’s定理， $F (Ω_{M})$ 是列紧的。综上，F是全连续的。

由定理2.1可知，F在 $Ω_{M}$ 中有一个不动点，即存在 $(x, y) \in Ω_{M}$ ，使得 $(x, y) = F (x, y)$ 。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

$| f (x, y) | \leq Q_{1} W (| y |), \frac{W (u)}{u} \leq \frac{1 - L_{1} - L_{1}^{*}}{N_{1} Q_{1}}, u > 0$

$| g (x, y) | \leq Q_{2} G (| x |), \frac{G (u)}{u} \leq \frac{1 - L_{2} - L_{2}^{*}}{N_{2} Q_{2}}, u > 0$

那么(1.1)至少有一个T-周期解。

证明：令

$M = \max {\frac{W (M) N_{1} Q_{1}}{1 - L_{1} - L_{1}^{*}}, \frac{G (M) N_{2} Q_{2}}{1 - L_{2} - L_{2}^{*}}}$

对 $(x, y) \in Ω_{M}, t \in [0, T]$ 有

$\begin{matrix} | F_{1} (x, y) (t) | \leq \int_{0}^{T} | G_{1} (t, s) h_{2} (s) | | y (s) | d s + \sum_{0 < t_{k} < T} | G_{1} (t, t_{k}) | | I_{k} (x (t_{k})) | \\ + \int_{0}^{T} | G_{1} (t, s) | \int_{- \infty}^{s} | a (s, u) | | f (x (u), y (u)) | d u d s \\ \leq \int_{0}^{T} | G_{1} (t, s) h_{2} (s) | | y (s) | d s + \sum_{0 < t_{k} < T} | G_{1} (t, t_{k}) | | I_{k} (x (t_{k})) | \\ + Q_{1} \int_{0}^{T} | G_{1} (t, s) | \int_{- \infty}^{s} | a (s, u) | W (| y (u) |) d u d s \\ \leq M L_{1} + M L_{1}^{*} + \frac{1 - L_{1} - L_{1}^{*}}{N_{1} Q_{1}} N_{1} Q_{1} | y (u) | \\ \leq M L_{1} + M L_{1}^{*} + (1 - L_{1} - L_{1}^{*}) M = M \end{matrix}$

类似地，

$\begin{matrix} | F_{2} (x, y) (t) | \leq \int_{0}^{T} | G_{2} (t, s) p_{2} (s) | | x (s) | d s + \sum_{0 < t_{k} < T} | G_{2} (t, t_{k}) | | J_{k} (y (t_{k})) | \\ + \int_{0}^{T} | G_{2} (t, s) | \int_{- \infty}^{s} | b (s, u) | | g (x (u), y (u)) | d u d s \\ \leq \int_{0}^{T} | G_{2} (t, s) p_{2} (s) | | x (s) | d s + \sum_{0 < t_{k} < T} | G_{2} (t, t_{k}) | | J_{k} (y (t_{k})) | \\ + Q_{2} \int_{0}^{T} | G_{2} (t, s) | \int_{- \infty}^{s} | b (s, u) | G (| x (u) |) d u d s \\ \leq M L_{2} + M L_{2}^{*} + \frac{1 - L_{2} - L_{2}^{*}}{N_{2} Q_{2}} N_{2} Q_{2} | x (u) | \\ \leq M L_{2} + M L_{2}^{*} + (1 - L_{2} - L_{2}^{*}) M = M \end{matrix}$

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 $x (t)$ 称为渐近T-周期的，如果存在两个函数 $x_{1} (t)$ 和 $x_{2} (t)$ ，使得 $x (t) = x_{1} (t) + x_{2} (t)$ ，其中 $x_{1} (t)$ 是T-周期的， $\lim_{t \to \infty} x_{2} (t) = 0$ 。

假设 $h_{1} (t), p_{1} (t)$ 是T-周期的，且

$\int_{0}^{T} h_{1} (s) d s = 0, \int_{0}^{T} p_{1} (s) d s = 0$ (3.1)

从而存在常数 $m_{k}, M_{k}, k = 1, 2$ ，使得

$m_{1} \leq e^{\int_{0}^{t} h_{1} (s) d s} \leq M_{1}^{*}, m_{2} \leq e^{\int_{0}^{t} p_{1} (s) d s} \leq M_{2}^{*}$ (3.2)

假设存在正数A，B使得

$\int_{0}^{\infty} \int_{- \infty}^{s} | a (s, u) | d u d s \leq A, \int_{0}^{\infty} \int_{- \infty}^{s} | b (s, u) | d u d s \leq B$ (3.3)

另外，假设

$\int_{t}^{\infty} | h_{2} (s) | d s \leq M_{3}^{*}, \int_{t}^{\infty} | p_{2} (s) | d s \leq M_{4}^{*}$ (3.4)

$b_{k} \geq 0, c_{k} \geq 0, \sum_{k = 1}^{\infty} b_{k} < \infty, \sum_{k = 1}^{\infty} c_{k} < \infty$ (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

$| f (x, y) | \leq {\bar{Q}}_{1} | y |, | g (x, y) | \leq {\bar{Q}}_{2} | x |$

且

$\begin{array}{l} 1 - M_{3}^{*} M_{1}^{*} m_{1}^{- 1} - M_{1}^{*} m_{1}^{- 1} A {\bar{Q}}_{1} - M_{1}^{*} M_{5}^{*} > 0, \\ 1 - M_{4}^{*} M_{2}^{*} m_{2}^{- 1} - M_{2}^{*} m_{2}^{- 1} B {\bar{Q}}_{2} - M_{2}^{*} M_{6}^{*} > 0 \end{array}$

则系统(1.1)有渐近T-周期解 $(x, y)$ ，满足

$x (t) = x_{1} (t) + x_{2} (t), y (t) = y_{1} (t) + y_{2} (t)$

其中，

$x_{1} (t) = c_{1} e^{\int_{0}^{t} h_{1} (s) d s}, y_{1} (t) = c_{2} e^{\int_{0}^{t} p_{1} (s) d s}, t \in R$

$c_{1}, c_{2}$ 为固定的非零常数，

$\lim_{t \to \infty} x_{2} (t) = \lim_{t \to \infty} y_{2} (t) = 0$ .

证明：定义 $\begin{array}{l} H_{T} = {(φ, ψ) : φ = φ_{1} + φ_{2}, ψ = ψ_{1} + ψ_{2}, (φ_{1}, ψ_{1}) (t + T) = (φ_{1}, ψ_{1}) (t), \\ (φ_{2}, ψ_{2}) (t) \to (0, 0), t \to \infty} \end{array}$ 。定义范数

$‖ (x, y) ‖ = \max {\max_{t \in [0, T]} | x (t) |, \max_{t \in [0, T]} | y (t) |}$

则 $H_{T}$ 是一个Banach空间。记 $Ω_{M^{*}} = {(x, y) : (x, y) \in H_{T}, ‖ (x, y) ‖ \leq M^{*}}$ ，则 $Ω_{M^{*}}$ 是 $H_{T}$ 中的有界闭凸子集。

对 $(x, y) \in Ω_{M^{*}}$ ，定义算子 $E : Ω_{M^{*}} \to H_{T}$

$E (x, y) (t) = (E_{1} (y) (t), E_{2} (x) (t))$

其中

$\begin{array}{l} E_{1} (y) (t) \\ = c_{1} e^{\int_{0}^{t} h_{1} (s) d s} - \int_{t}^{\infty} \frac{e^{\int_{0}^{t} h_{1} (l) d l}}{e^{\int_{0}^{s} h_{1} (l) d l}} h_{2} (s) y (s) d s - \int_{t}^{\infty} \int_{- \infty}^{s} \frac{e^{\int_{0}^{t} h_{1} (l) d l}}{e^{\int_{0}^{s} h_{1} (l) d l}} a (s, u) f (x (u), y (u)) d u d s \\ - \sum_{t < t_{k} < \infty} e^{\int_{t_{k}}^{t} h_{1} (s) d s} I_{k} (x (t_{k})) \end{array}$

$\begin{array}{l} E_{2} (x) (t) \\ = c_{2} e^{\int_{0}^{t} p_{1} (s) d s} - \int_{t}^{\infty} \frac{e^{\int_{0}^{t} p_{1} (l) d l}}{e^{\int_{0}^{s} p_{1} (l) d l}} p_{2} (s) x (s) d s - \int_{t}^{\infty} \int_{- \infty}^{s} \frac{e^{\int_{0}^{t} p_{1} (l) d l}}{e^{\int_{0}^{s} p_{1} (l) d l}} b (s, u) g (x (u), y (u)) d u d s \\ - \sum_{t < t_{k} < \infty} e^{\int_{t_{k}}^{t} p_{1} (s) d s} J_{k} (y (t_{k})) \end{array}$

下证映射E在 $Ω_{M^{*}}$ 中有一个不动点。由(3.5)可知，存在正常数 $M_{5}^{*}, M_{6}^{*}$ 使得

$| \sum_{k = 1}^{\infty} b_{k} | \leq M_{5}^{*}, | \sum_{k = 1}^{\infty} c_{k} | \leq M_{6}^{*}$

设

$M^{*} = {\frac{c_{1} M_{1}^{*}}{1 - M_{3}^{*} M_{1}^{*} m_{1}^{- 1} - M_{1}^{*} m_{1}^{- 1} A {\bar{Q}}_{1} - M_{1}^{*} M_{5}^{*}}, \frac{c_{2} M_{2}^{*}}{1 - M_{4}^{*} M_{2}^{*} m_{2}^{- 1} - M_{2}^{*} m_{2}^{- 1} B {\bar{Q}}_{2} - M_{2}^{*} M_{6}^{*}}}$

$\begin{array}{l} | E_{1} (y) (t) - c_{1} e^{\int_{0}^{t} h_{1} (s) d s} | \\ \leq M^{*} M_{3}^{*} M_{1}^{*} m_{1}^{- 1} + M_{1}^{*} m_{1}^{- 1} {\bar{Q}}_{1} \int_{t}^{\infty} \int_{- \infty}^{s} | a (s, u) | | x | d u d s + \sum_{t < t_{k} < \infty} e^{\int_{t_{k}}^{t} h_{1} (s) d s} | I_{k} (x (t_{k})) | \\ \leq M^{*} M_{3}^{*} M_{1}^{*} m_{1}^{- 1} + M_{1}^{*} m_{1}^{- 1} {\bar{Q}}_{1} ‖ x ‖ \int_{0}^{\infty} \int_{- \infty}^{s} | a (s, u) | d u d s + | \sum_{t < t_{k} < \infty} b_{k} | M_{1}^{*} M^{*} \\ \leq M^{*} M_{3}^{*} M_{1}^{*} m_{1}^{- 1} + M_{1}^{*} m_{1}^{- 1} A {\bar{Q}}_{1} M^{*} + M_{1}^{*} M^{*} M_{5}^{*} \end{array}$

同理，

$| E_{2} (x) (t) - c_{2} e^{\int_{0}^{t} p_{1} (s) d s} | \leq M^{*} M_{4}^{*} M_{2}^{*} m_{2}^{- 1} + M_{2}^{*} m_{2}^{- 1} B {\bar{Q}}_{2} M^{*} + M_{2}^{*} M^{*} M_{6}^{*}$

因此，

$| E_{1} (y) (t) | \leq M^{*} M_{3}^{*} M_{1}^{*} m_{1}^{- 1} + M_{1}^{*} m_{1}^{- 1} A {\bar{Q}}_{1} M^{*} + M_{1}^{*} M^{*} M_{5}^{*} + c_{1} M_{1}^{*} \leq M^{*}$

$| E_{2} (x) (t) | \leq M^{*} M_{4}^{*} M_{2}^{*} m_{2}^{- 1} + M_{2}^{*} m_{2}^{- 1} B {\bar{Q}}_{2} M^{*} + M_{2}^{*} M^{*} M_{6}^{*} + c_{2} M_{2}^{*} \leq M^{*}$

设

$x_{1} (t) = c_{1} e^{\int_{0}^{t} h_{1} (s) d s}, y_{1} (t) = c_{2} e^{\int_{0}^{t} p_{1} (s) d s}, t \in R$

$\begin{matrix} x_{2} (t) = - \int_{t}^{\infty} \frac{e^{\int_{0}^{t} h_{1} (l) d l}}{e^{\int_{0}^{s} h_{1} (l) d l}} h_{2} (s) y (s) d s - \int_{t}^{\infty} \int_{- \infty}^{s} \frac{e^{\int_{0}^{t} h_{1} (l) d l}}{e^{\int_{0}^{s} h_{1} (l) d l}} a (s, u) f (x (u), y (u)) d u d s \\ - \sum_{t < t_{k} < \infty} e^{\int_{t_{k}}^{t} h_{1} (s) d s} I_{k} (x (t_{k})) \end{matrix}$

$\begin{matrix} y_{2} (t) = - \int_{t}^{\infty} \frac{e^{\int_{0}^{t} p_{1} (l) d l}}{e^{\int_{0}^{s} p_{1} (l) d l}} p_{2} (s) x (s) d s - \int_{t}^{\infty} \int_{- \infty}^{s} \frac{e^{\int_{0}^{t} p_{1} (l) d l}}{e^{\int_{0}^{s} p_{1} (l) d l}} b (s, u) g (x (u), y (u)) d u d s \\ - \sum_{t < t_{k} < \infty} e^{\int_{t_{k}}^{t} p_{1} (s) d s} J_{k} (y (t_{k})) \end{matrix}$

计算可知，

$x_{1} (t + T) = c_{1} e^{\int_{0}^{t + T} h_{1} (s) d s} = c_{1} e^{\int_{0}^{t} h_{1} (s) d s} e^{\int_{t}^{t + T} h_{1} (s) d s} = c_{1} e^{\int_{0}^{t} h_{1} (s) d s}$

$y_{1} (t + T) = c_{2} e^{\int_{0}^{t + T} p_{1} (s) d s} = c_{2} e^{\int_{0}^{t} p_{1} (s) d s} e^{\int_{t}^{t + T} p_{1} (s) d s} = c_{2} e^{\int_{0}^{t} p_{1} (s) d s}$

由(3.4)~(3.5)可知，

$\begin{array}{l} \lim_{t \to \infty} | x_{2} (t) | \leq M^{*} M_{1}^{*} m_{1}^{- 1} \lim_{t \to \infty} \int_{t}^{\infty} | h_{2} (s) | d s + M_{1}^{*} m_{1}^{- 1} {\bar{Q}}_{1} M^{*} \lim_{t \to \infty} \int_{t}^{\infty} \int_{- \infty}^{s} | a (s, u) | d u d s \\ + M^{*} M_{1}^{*} \lim_{t \to \infty} | \sum_{t < t_{k} < \infty} b_{k} | = 0 \end{array}$

$\begin{array}{l} \lim_{t \to \infty} | y_{2} (t) | \leq M^{*} M_{2}^{*} m_{2}^{- 1} \lim_{t \to \infty} \int_{t}^{\infty} | p_{2} (s) | d s + M_{2}^{*} m_{2}^{- 1} {\bar{Q}}_{2} M^{*} \lim_{t \to \infty} \int_{t}^{\infty} \int_{- \infty}^{s} | b (s, u) | d u d s \\ + M^{*} M_{2}^{*} \lim_{t \to \infty} | \sum_{t < t_{k} < \infty} c_{k} | = 0 \end{array}$

因此， $\lim_{t \to \infty} x_{2} (t) = 0, \lim_{t \to \infty} y_{2} (t) = 0$ 。故 $E (Ω_{M^{*}}) \subseteq Ω_{M^{*}}$ 。

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 $(x, y) \in Ω_{M^{*}}$ ，使得

$E (x, y) (t) = (E_{1} (y) (t), E_{2} (x) (t)) = (x (t), y (t))$

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

例3.1 考虑下面的系统

${\begin{cases} x^{'} (t) = \cos (t) x (t) + \frac{2 t}{{(t^{2} + 10)}^{2}} y (t) + \int_{- \infty}^{t} e^{- 6 t + 4 s} \sin (x (t) + y (t)) d s, t \neq t_{k} \\ y^{'} (t) = \cos (t) y (t) + \frac{2 t}{{(t^{2} + 10)}^{2}} x (t) + \int_{- \infty}^{t} e^{- 6 t + 4 s} \cos (x (t) + y (t)) d s, t \neq t_{k} \\ x (t_{k}^{+}) = x (t_{k}) + {(\frac{1}{12})}^{k} (x (t_{k})), t = t_{k} \\ y (t_{k}^{+}) = y (t_{k}) + {(\frac{1}{13})}^{k} (y (t_{k})), t = t_{k} \end{cases}$

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)。

文章引用

陈秋凤,李建利. 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[J]. 应用数学进展, 2019, 08(01): 135-144. https://doi.org/10.12677/AAM.2019.81015

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