带Φ-Laplacian 算子的差分方程周期边值问题正解集的全局结构

Global structure of positive solutions of periodic boundary value problem of difference equation involving Φ -Laplacian operator

本文运用区间分歧理论研究一类带Φ-Laplacian算子的差分方程周期边值问题-▽[?(Δu t)]+q tu t=λf(t,u t),t∈T,u 0=u T,u 1=u T+1正解集的全局结构,其中T={1,2,…,T},={0,1,…,T+1},T∈N且T>1,Δu t=u t+1-u t,▽u t=u t-u t-1,λ∈[0,∞)为一个参数,q t∈C(,(0,∞))且对于任意的t 0∈,q(t 0)>0,f∈C(×[0,∞),[0,∞)),f(t,0)=0,对于任意的s>0有f(t,s)>0且f(t,s)在s=0处不能线性化,?:(-1,1)→R,?(y)=y 1-y 2为一个递增的同胚映射,且?(0)=0.本文的主要结果推广和改进了Bereanu和Mawhin的工作.

In this paper,we study the global structure of the set of positive solutions of periodic boundary value problem of a class of difference equations involvingΦ-Laplacian operator-▽[?(Δu t)]+q tu t=λf(t,u t),t∈T,u 0=u T,u 1=u T+1 by using the method of bifurcation from an interval,where T={1,2,…,T},={0,1,…,T+1},T∈N and T>1,Δu t=u t+1-u t,▽u t=u t-u t-1,λ∈[0,∞)is a parameter;q t∈C(,(0,∞))q(t 0)>0 for t 0∈,f∈C(×[0,∞),[0,∞)),where f∈C(×[0,∞),[0,∞)),f(t,0)=0,f(t,s)>0 for s>0,and f(t,s)are not necessarily linearizable near 0?:(-1,1)→R,?(y)=y 1-y 2 is an increasing homeomorphism,?(0)=0.The main results extend and improve the corresponding ones of Bereanu and Mawhin.