Global behavior and nontrivial solutions for discrete Sturm–Liouville problems with eigenparameter-dependent boundary condition

Abstract

In this article, we consider a Dancer-type unilateral global bifurcation for the discrete Sturm-Liouville problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\nabla (p(t)\Delta u(t))+q(t)u(t)=\lambda h(t)u(t)+g(t,u(t),\lambda ),\ \ \ \ t\in [1,T]_Z,\\ \quad b_1u(0)=d_1\Delta u(0),\\ \quad (c\lambda +b_2)u(T+1)=(c\lambda +d_2)\nabla u(T+1), \end{array}\right. \end{aligned}$$

where $\lambda \in {\mathbb {R}}$ is a spectral parameter, $T>1$ is an integer, Z is the set of integers, for $a, b\in Z$ with $a<b$, $[a, b]_{Z}:=\{a, a+1,\ldots , b\}$. Under natural hypotheses on g, we prove that $(\lambda _{k,h},0)$ is a bifurcation point of the above problem. As applications, we determine the interval of $\lambda$, in which there exist nontrivial solutions for the discrete Sturm-Liouville problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\nabla (p(t)\Delta u(t))+q(t)u(t)=\lambda h(t)u(t)+\lambda f(t,u(t)),\ \ \ \ t\in [1,T]_Z,\\ \quad b_1u(0)=d_1\Delta u(0),\\ \quad (c\lambda +b_2)u(T+1)=(c\lambda +d_2)\nabla u(T+1), \end{array}\right. \end{aligned}$$

where $p:[0,T]_Z\rightarrow (0,+\infty )$, $q:[1,T]_Z\rightarrow {\mathbb {R}}$, $h:[1,T]_Z\rightarrow (0,+\infty )$, $f\in C([1,T]_Z\times {\mathbb {R}}, {\mathbb {R}})$. We prove that, under some suitable assumptions on nonlinear terms, there exist two unbounded continua of nontrivial solutions of the above problem which bifurcate from the line of trivial solutions or from infinity, respectively.