Multiple Solutions for Discrete First-Order Periodic Problems

DOI：10.3770/j.issn:2095-2651.2014.03.009

 作者 单位 高承华 西北师范大学大学数学系, 甘肃 兰州 730070

设$T>1$是一个整数, $\mathbb{T}=\{0,1,2,\ldots,T-1\}$. 考虑一阶离散周期边值问题$$\Delta u(t)-a(t)u(t)=\lambda u(t)+f(u(t-\tau(t)))-h(t),~~t\in\mathbb{T},$$$$u(0)=u(T)$$ 周期解的存在性, 其中, $\Delta u(t)=u(t+1)-u(t)$,\ $a:\mathbb{T}\to\mathbb{R}$ 并且满足\ $\prod^{T-1}_{t=0}(1+a(t))=1$, $\tau:\mathbb{T}\to\mathbb{Z}$, $t-\tau(t)\in\mathbb{T}$,$t\in\mathbb{T}$, $f:\mathbb{R}\to\mathbb{R}$ 连续并且满足Landesman-Lazer条件,$h:\mathbb{T}\to\mathbb{R}$.本文所用的主要工具是Rabinowitz全局分歧定理和Leray-Schauder\ 度理论.

Let $T>1$ be an integer, $\mathbb{T}=\{0,1,2,\ldots,T-1\}$. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems $$\Delta u(t)-a(t)u(t)=\lambda u(t)+f(u(t-\tau(t)))-h(t),~~t\in\mathbb{T},$$ $$u(0)=u(T),$$ where $\Delta u(t)=u(t+1)-u(t)$, $a:\mathbb{T}\to\mathbb{R}$ and satisfies $\prod^{T-1}_{t=0}(1+a(t))=1$, $\tau:\mathbb{T}\to\mathbb{Z}$ $t-\tau(t)\in\mathbb{T}$ for $t\in\mathbb{T}$, $f:\mathbb{R}\to\mathbb{R}$ is continuous and satisfies Landesman-Lazer type condition and $h:\mathbb{T}\to\mathbb{R}$. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.