# Discrete Nonselfadjoint Second-Order Two-Point Problems at Resonance

• Ruyun Ma
• Man Xu
• Dongliang Yan
Article

## Abstract

Let $$T > 2$$ be an integer, $$\mathbb {T}=\{1, 2,\ldots ,T\}$$. We are considered with the discrete nonlinear two-point boundary value problem at resonance:
\begin{aligned} \begin{aligned}&\mathcal {L} u(j)=\nu _1 u(j)+g(u(j))-e(j),\ \ j\in \mathbb {T}, \\&u(0)=u(T+1)=0,&\quad \quad (P)\\ \end{aligned} \end{aligned}
where
\begin{aligned} \mathcal {L}u(j)=\left\{ \begin{array}{ll} -\triangle ^2 u(j-1)+b(j)\Delta u(j)+a_0(j) u(j), &{}\quad j\in \mathbb {T},\\ \quad 0,&{}\quad j\in \{0, T+1\},\\ \end{array} \right. \end{aligned}
$$b, e: \mathbb {T}\rightarrow \mathbb {R}$$, $$a_0:\mathbb {T}\rightarrow [0, \infty )$$, $$\nu _1$$ is the principal eigenvalue of $$\mathcal {L}$$. We show that there exists a constant $$d_0 > \nu _1$$, depending only on b and $$a_0$$, such that if
\begin{aligned} \underset{|\xi |\rightarrow \infty }{\lim \sup } \; \frac{g(\xi )}{\xi }<d_0-\nu _1, \end{aligned}
and
\begin{aligned} {\overline{g}}(-\infty )\sum ^T_{j=1} \Theta ^*(j)<\sum ^T_{j=1} \Theta ^*(j)h(j)<{\underline{g}}(\infty )\sum ^T_{j=1} \Theta ^*(j), \end{aligned}
then (P) has at least one solution. Here, $$\underline{g}(\infty )=\liminf \nolimits _{\xi \rightarrow \infty } g(\xi )$$, $$\overline{g}(-\infty )=\limsup \nolimits _{\xi \rightarrow -\infty } g(\xi )$$, and $$\Theta ^*$$ is an eigenfunction of $$\mathcal {L}^*$$ corresponding to the principle eigenvalue $$\nu _1$$.

## Keywords

Nonselfadjoint linear operator discrete problems at resonance Landesman–Lazer condition principal eigenvalue Brouwer degree

## Mathematics Subject Classification

39A10 34B27 34B10 34B05 34A40

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