Bifurcation of positive solutions of a nonlinear discrete fourth-order boundary value problem

Abstract

Let ${T \in \mathbb{N}}$ be an integer with T ≥ 4. We give a global description of the branches of positive solutions of the nonlinear boundary value problem of fourth-order difference equation of the form

$$\begin{array}{lll}\Delta^4 u(t-2)&=&f(t,u(t),\Delta^2u(t-1)),\quad t\in \{2,\ldots, T\},\\u(0)=&u(T+2)=\Delta^2u(0)=\Delta^2u(T)=0,\end{array}$$

that is not necessarily linearizable. Our approach is based on Krein–Rutman theorem, topological degree theory, and global bifurcation techniques.