Existence of a positive solution for an $n$th order boundary value problem for nonlinear difference equations

Abstract

The $n$th order eigenvalue problem: $$\begin{array}{l}{\Delta }^{n}x\left(t\right)={\left(-1\right)}^{n-k}\lambda f\left(t,x\left(t\right)\right),t\in \left[0,T\right],\\ x\left(0\right)=x\left(1\right)=\cdots =x\left(k-1\right)=x\left(T+k+1\right)=\cdots =x\left(T+n\right)=0,\end{array}$$ is considered, where $n\ge 2$ and $k\in \left\{1,2,\dots ,n-1\right\}$ are given. Eigenvalues $\lambda$ are determined for $f$ continuous and the case where the limits $f_0\left(t\right)=\underset{n\to 0^{+}}{lim}\frac{f\left(t,u\right)}{u}$ and $f_{\infty }\left(t\right)=\underset{n\to \infty }{lim}\frac{f\left(t,u\right)}{u}$ exist for all $t\in \left[0,T\right]$. Guo′s fixed point theorem is applied to operators defined on annular regions in a cone.