The continuum branch of positive solutions for discrete simply supported beam equation with local linear growth condition

Abstract

Abstract In this paper, we obtain the global structure of positive solutions for nonlinear discrete simply supported beam equation Δ4u(t−2)=λf(t,u(t)),t∈T,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0, $$\begin{aligned}& \Delta ^{4}u(t-2)= \lambda f\bigl(t,u(t)\bigr),\quad t\in \mathbb{T}, \\& u(1)=u(T+1)=\Delta ^{2}u(0)=\Delta ^{2}u(T)=0, \end{aligned}$$ with f∈C(T×[0,∞),[0,∞)) $f\in C(\mathbb{T}\times [0,\infty ),[0,\infty ))$ satisfying local linear growth condition and f(t,0)=0 $f(t,0)=0$ uniformly for t∈T $t\in \mathbb{T}$, where T={2,…,T} $\mathbb{T}=\{2,\ldots,T\}$, λ>0 $\lambda >0$ is a parameter. The main results are based on the global bifurcation theorem.

Keywords

• Positive solutions
• Global structure
• Discrete simply supported beam
• Bifurcation