高承华.一阶离散周期问题的多解性[J].数学研究及应用,2014,34(3):323~331
一阶离散周期问题的多解性
Multiple Solutions for Discrete First-Order Periodic Problems
投稿时间:2012-10-25  最后修改时间:2014-01-13
DOI:10.3770/j.issn:2095-2651.2014.03.009
中文关键词:  一阶离散周期问题  Landesman-Lazer 条件  Leray-Schauder 度  分歧  存在性.hauder 度  分歧  存在性.
英文关键词:discrete first-order periodic problems  Landesman-Lazer type condition  Leray-Schauder degree  bifurcation  existence.
基金项目:国家自然科学基金(Grant Nos.11326127; 11101335), 甘肃省高等学校科研项目(Grant No.2013A-001),西北师范大学青年教师科研提升计划(Grant No.NWNU-LKQN-11-23).
作者单位
高承华 西北师范大学大学数学系, 甘肃 兰州 730070 
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中文摘要:
      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.
英文摘要:
      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.
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