Double solutions of boundary value problems for 2\(m^{\text{th}}\)-order differential equations and difference equations
The authors prove the existence of two positive solutions for the two-point boundary value problem \[ (-1)^m\,y^{(2m)}=f(y),\qquad 0\leq t\leq 1,\quad y^{(2i)}(0)=y^{(2i+1)},\qquad i=0,1,\dots,m-1, \] where \(f:\mathbb{R}\to [0,\,\infty)\) is continuous. The arguments are based on a double fixed-point theorem due to \textit{R. I. Avery} and \textit{J. Henderson} in [Commun. Appl. Nonlinear Anal. 8, 27--36 (2001; Zbl 1014.47025)]. They also apply the double fixed-point theorem to the boundary value problem for the finite difference equation \[ \begin{aligned}&(-1)^m\,\Delta^{2m}u(k)=g(u(k)),\qquad k=0,1,\dots,N,\\ &\Delta^{2i}u(0)=\Delta^{2i+1}u(N+1),\qquad i=0,1,\dots,m-1, \end{aligned} \] where \(g:\mathbb{R}\to [0,\,\infty)\) is continuous, and obtain an analogous existence result of two positive solutions.
scientific article; zbMATH DE number 2082201