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Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model
作者姓名:Xian-yi Li  De-ming Zhu
作者单位:Xian-yi Li,De-ming Zhu Department of Mathematics,East China Normal University,Shanghai 200062,China Department of Mathematics and Physics,Nanhua University,Hengyang 421001,China
基金项目:National Natural Science Foundation of China (Grant No.10071022),Mathematical Tianyuan Foudation of China (Grant No.TY10026002-01-05-03) & Shanghai Priority Academic Research.
摘    要:By using the continuation theorem of Mawhin's coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition modelwhere ri and r2 are continuous w-periodic functions in R+=0,∞) with ,ai,ci(i =1,2) are positive continuous w-periodic functions in R+=0,∞),bi (i = 1,2) is nonnegative continuous w-periodic function in R+=0,∞), w and T are positive constants. Ki,Lt ∈ C(-T,0], (01 88)) and Ki(s)ds = 1,ds - 1. i = 1,2. Some known results are improved and extended.

关 键 词:全局存在性  周期解  分布时滞  竞争模型  叠和度  延拓定理

Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model
Xian-yi Li,De-ming Zhu.Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model[J].Acta Mathematicae Applicatae Sinica,2003,19(3):491-498.
Authors:Xian-yi?Li  Email author" target="_blank">De-ming?ZhuEmail author
Institution:(1) Department of Mathematics, East China Normal University, Shanghai, 200062, China;(2) Department of Mathematics and Physics, Nanhua University, Hengyang, 421001, China
Abstract:By using the continuation theorem of Mawhin’s coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition model
$$
\left\{ \begin{aligned}
  & {u}'{\left( t \right)}{\kern 1pt}  = {\kern 1pt} u{\left( t \right)}{\left {r_{1} {\left( t \right)} - a_{1} {\left( t \right)}u{\left( t \right)} - b_{1} {\left( t \right)}{\int_{ - T}^0 {L_{1} {\left( s \right)}u{\left( {t + s} \right)}ds - c_{1} {\left( t \right)}{\int_{ - T}^0 {K_{1} {\left( s \right)}v{\left( {t + s} \right)}ds} }} }} \right]}, \\
  & {v}'{\left( t \right)} = v{\left( t \right)}{\left {r_{2} {\left( t \right)} - a_{2} {\left( t \right)}v{\left( t \right)} - b_{2} {\left( t \right)}{\int_{ - T}^0 {L_{2} {\left( s \right)}v{\left( {t + s} \right)}ds - c_{2} {\left( t \right)}{\int_{ - T}^0 {K_{2} {\left( s \right)}u{\left( {t + s} \right)}ds} }} }} \right]} \\ 
  \end{aligned}  \right.
$$
, where r 1 and r 2 are continuous ω-periodic functions in R + = 0,∞) with $$
{\int_0^\omega  {r_{i} {\left( t \right)}dt > 0,a_{i} ,c_{i} {\left( {i = 1,2} \right)}} }
$$ are positive continuous ω-periodic functions in R + = 0,∞), b i (i = 1, 2) is nonnegative continuous ω-periodic function in R + = 0,∞), ω and T are positive constants, $$
K_{i} ,L_{i}  \in C{\left( {{\left { - T,0} \right]},(0,\infty )} \right)}
$$ and $$
{\int_{ - T}^0 {K_{i} {\left( s \right)}ds = 1,{\kern 1pt} {\int_{ - T}^0 {L_{i} {\left( s \right)}ds = 1,{\kern 1pt} i = 1,2} }} }
$$ . Some known results are improved and extended. Supported by National Natural Science Foundation of China (Grant No. 10071022), Mathematical Tianyuan Foundation of China (Grant No. TY10026002-01-05-03) & Shanghai Priority Academic Research.
Keywords:Global existence  positive periodic solution  coincidence degree  distributed delay model
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