非线性—阶周期问题的Ambrosetti-Prodi型结果

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

研究了一阶周期问题u'(t)=a(t)g(u(t)u(t)-b(t)f(u(t))+s,t∈R,u(t)=u(t+T)解的个数与参数s(s∈R)的关系,其中a∈C(R,[0,∞)),b∈C(R,(0,∞))均为T周期函数.∫0Ta(t)dt>0;_f,g∈C(R,[0,∞)).当u>0时,f(u)>0,当u≥0时,0<l≤g(u)<L<∞.运用上下解方法及拓扑度理论,获得结论:存在常数s_1∈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.

This paper shows the relationship between the parameter s and the number of solutions of the first-order periodic problem u＇（t）=a（t）g（u（t））u（t）-b（t）f（u（t））＋s,t∈R u（t）=u（t＋T）,where a ∈ C（R,[0,∞））,b ∈ C（R,（0,∞）） are T-periodic,∫0T a（t）dt 0;f,g ∈ C（R,[0,∞））,and f（u） 0 for u 0,0 l≤ g（u） L ∞ for u≥0.By using the method of upper and lower solutions and topological degree techniques,we prove that there exists s1 ∈R,such that the problem has zero,at least one or at least two periodic solutions when s s1,s=s1,s s1,respectively.