带参数的一阶周期边值问题正解的存在性及多解性
Existence and multiplicity of positive solutions of first order periodic boundary value problems with parameter

摘要： 研究了一阶周期边值问题{u'(t)+a(t)u(t)=λf(t,u(t)), t∈［0,T］,u(0)=u(T)正解的个数与参数λ的关系, 其中λ>0, a∈C(R, ［0,+∞))且∫T0a(θ)dθ>0, f∈C(［0,T］×［0,+∞),(0,+∞))以及f∞=limu→∞ inf(f(t,u))/u=∞对任意的t∈［0,T］一致成立。 运用上下解方法及拓扑度理论, 获得存在λ*>0, 当λ>λ*时, 该问题不存在正解, λ=λ* 时, 该问题恰有一个正解; 0 * 时, 该问题至少存在两个正解。
Abstract: We study the relationship λ between and the number of positive solutions of first order periodic boundary value problems{u'(t)+a(t)u(t)=λf(t,u(t)), t∈［0,T］,u(0)=u(T),where λ is a positive parameter, a∈C(R, ［0,+∞))且∫T0a(θ)dθ>0, f∈C(［0,T］×［0,+∞),(0,+∞))and f∞=limu→∞ inf(f(t,u))/u=∞ uniformly for t∈［0,T］. By using the method of the upper and lower solutions and topological degree techniques, we obtain that the problem has no positive solution, exactly one positive solution and at least two positive solutions, when λ>λ*, λ=λ*, 0 *, respectively