Discrete conjugate boundary value problems
Consider the discrete boundary value problem \[ \begin{aligned} & (-1)^{n-p}\Delta^n y(i-p)=f(i,y(i)),\quad i\in J_p,\\ & y(0)= \dots y(p-1)=0,\\ & y(T+p+1)=\dots=y(T+n)=0,\end{aligned} \] where \(T\in\{1,2,\dots\}\), \(J_p=\{p,p+1,\dots,T+p\}\), \(1\leq p\leq n-1\), \(n\geq 2\) and \(y:I_n=\{0,1,\dots,T+n\}\to\mathbb{R}\). If \(f:J_p\times [0,\infty)\to [0,\infty)\) is a continuous, nondecreasing function, if there exists a continuous, nondecreasing function \(\psi:[0,\infty)\to [0,\infty)\) such that \(\psi>0\) on \((0,\infty)\) and a function \(g:J_p\to[0,\infty)\) such that \[ f(k,u)\leq g(k)\psi(u) \text{ for all }u>0\text{ and }k\in J_p, \] and if \(\sup_{c\in (0,\infty)}(c/\psi(c))>Q\), where \(Q\) denotes certain constants, then the problem under consideration has a nonnegative solution. An analogous result is obtained for singular problems.
scientific article; zbMATH DE number 1428512