华东师范大学学报（自然科学版） >

2015 , Vol. 2015 >Issue 6: 53 - 58

DOI: https://doi.org/10.3969/j.issn.1000-5641.2015.06.008

应用数学与基础数学

非线性一阶周期问题的Ambrosetti-Prodi型结果

  • 马陆一
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  • 西北师范大学~~数学与统计学院, 兰州 730070)
马陆一, 男, 硕士研究生,研究方向为常微分方程边值问题. E-mail: maly0318@126.com.

收稿日期: 2014-11-03

  网络出版日期: 2015-12-23

基金资助

国家自然科学基金(11361054); 甘肃省自然科学基金(1208RJZA258)

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Ambrosetti-Prodi type results of the nonlinear first-order periodic problem

  • MA Lu-Yi
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Received date: 2014-11-03

  Online published: 2015-12-23

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摘要

研究了一阶周期问题\left\{\!\!\!\begin{array}{ll} u'(t)=a(t)g(u(t))u(t)-b(t)f(u(t))+s, t\in {\mathbb{R}},\\[2ex]
 u(t)=u(t+T)\end{array}\right.\eqno 解的个数与参数\,s\,(s\in{\mathbb{R}})\,的关系,其中\,a\in C({\mathbb{R}},[0,\infty)), b\in
C({\mathbb{R}},(0,\infty))\,均为\,T\,周期函数, \int_0^T a(t){\rmd}t>0; f, g\in C({\mathbb{R}},[0,\infty)). 当\,u>0\,时,
f(u)>0, 当\,u\geqslant0\,时, 0<l\leqslant g(u)<L<\infty.运用上下解方法及拓扑度理论, 获得结论:存在常数\,s_{1}\in{\mathbb{R}}, 当\, s<s_{1}\,时,
该问题没有周期解;  s=s_{1}\,时, 该问题至少有一个周期解; s>s_{1}\,时, 该问题至少有两个周期解.

关键词： Ambrosetti-Prodi问题;上下解方法; 拓扑度

本文引用格式

马陆一 . 非线性一阶周期问题的Ambrosetti-Prodi型结果[J]. 华东师范大学学报（自然科学版）, 2015 , 2015(6) : 53 -58 . DOI: 10.3969/j.issn.1000-5641.2015.06.008

Abstract

This paper shows the relationship between the parameter~s~and the number of solutions of the first-order
periodic problem \left\{\!\!\!\begin{array}{ll}  u'(t)=a(t)g(u(t))u(t)-b(t)f(u(t))+s,~~\ \ \ t\in {\mathbb{R}},\\[2ex]
 u(t)=u(t+T)\end{array}\right.\eqno  where a\in C({\mathbb{R}},[0,\infty)),~b\inC({\mathbb{R}},(0,\infty)) are T-periodic, \int_0^T a(t){\rm
d}t>0; f, g\in C({\mathbb{R}},[0,\infty)), and f(u)>0 foru>0, 0<l\leqslant g(u)<L<\infty for u\geqslant0. By using the
method of upper and lower solutions and topological degree techniques, we prove that there exists s_{1}\in{\mathbb{R}}, such
that the problem has zero, at least one or at least two periodicsolutions when  s<s_{1}, s=s_{1}, s>s_{1}, respectively.

Key words： Ambrosetti-Prodi problem; upper and lower solutions; topological degree

参考文献



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