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Discrete Nonselfadjoint Second-Order Two-Point Problems at Resonance

  • Ruyun MaEmail author
  • Man Xu
  • Dongliang Yan
Article
  • 19 Downloads

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 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouPeople’s Republic of China

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