Ambrosetti-Prodi type results of the nonlinear first-order periodic problem
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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}t0; f, g\in C({\mathbb{R}},[0,\infty)). 当\,u0\,时, f(u)0, 当\,u\geqslant0\,时, 0l\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\, ss_{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; ss_{1}\,时, 该问题至少有两个周期解.
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关键词:
- Ambrosetti-Prodi问题 /
- 上下解方法 /
- 拓扑度
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}t0; f, g\in C({\mathbb{R}},[0,\infty)), and f(u)0 foru0, 0l\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 ss_{1}, s=s_{1}, ss_{1}, respectively. -
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