关键词: 共振问题; 上下解; 解集连通理论; 存在性

Abstract:

The existence of solutions are discussed for the secondorder difference equation threepoint boundary value problems
Δ²u(t-1)=f(t, u(t)),t∈T,
u(0)=εΔu(0), u(T+1)=αu(η),
where f: T×R→R is continuous, T is a fixed positive integer, T:=｛1, 2,…,T｝. And ε∈［0,∞), α∈(0,∞), η∈T  are given constants such that α(η+ε)=T+1+ε. The proof of our main results is based on the method of lower and upper solutions using the connectivity properties of the solution sets of parameterized families of compact vector fields.

Key words: resonance problem; lower and upper solutions; connectivity of the solution sets; existence