一阶离散周期问题的多解性

Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.

Multiple Solutions for Discrete First-Order Periodic Problems

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.