GLOBAL STRUCTURE OF POSITIVE SOLUTIONS FOR SECOND ORDER DISCRETE PERIODIC BOUNDARY VALUE PROBLEM WITH INDEFINITE WEIGHT

Abstract

We show the global structure of positive solutions for second order periodic boundary value problem

$$\{\begin{array}{l}-{\mathrm{\Delta }}^{2}u(t-1)=\lambda a(t)g(u(t)),t\in {\mathbb{N}}_1^T,\hfill \\ u(0)=u(T),u(1)=u(T+1),\hfill \end{array}$$

where ${\mathbb{N}}_1^T=\{1,2,\dots ,T\},T\ge 3$ is an integer, is a parameter, $g:[0,\infty )\to [0,\infty )$ is a continuous function with $g(0)=0$ and $a:{\mathbb{N}}_1^T\to ℝ$ is sign-changing. Depending on the behavior of $g$ near $0$ and $\infty$, we obtain that there exist such that above problem has at least two positive solutions for and no solution for $\lambda \in (0,{\lambda }_0)$. The proof of our main results is based upon bifurcation technique.