一类离散三点边值共振问题解的存在性

运用紧向量场方程的解集连通理论为三点边值共振问题 Δ²u(t-1)=f(t, u(t)),t∈T, u(0)=εΔu(0), u(T+1)=αu(η) 发展上下解方法, 其中f: T×R→R 连续,T为固定的正整数, T:=｛1, 2,…,T｝, ε∈［0,∞), α∈(0,∞), η∈T 均为固定的常数, 且满足 α(η+ε)=T+1+ε。

Existence of solutions for a discrete three-point boundary value problems at resonance

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.