Discrete Lyapunov inequality conditions for partial difference equations
The authors obtain some necessary conditions for the existence of a nontrivial solution to some boundary value problems of second-order partial difference equations by means of a discrete maximum principle. The main result of the paper is embodied in the following Theorem 3.2. Given the boundary value problem \[ (1)\quad \Delta^ 2_ 1u(i-1,j)+\Delta^ 2_ 2u(i,j-1)+p(i,j)u(i,j)=0,\quad (i,j)\in S;\quad (2)\quad u(i,j)=0,\quad (i,j)\in \partial S, \] where S is a net, i.e., a connected and finite nonempty set of lattice points, \(\partial S\) is the exterior boundary of S, p(i,j)\(\geq 0\) for (i,j)\(\in S\), \(\Delta_ 1h(i,j)=h(i+1,j)-h(i,j)\), \(\Delta_ 2h(i,j)=h(i,j+1)-h(i,j)\). If (1)-(2) has a nontrivial solution then \[ \sum_{z\in S}p(z)\geq \mu (S)=\{\max_{z\in S}G(z| z)\}^{-1}, \] where G(z\(| w)\) with \(w=z\) being the Green function associated with (1)-(2), and the inequality is sharp. The above result is similar to a classical inequality proved by \textit{A. M. Lyapunov} [Problème général de a stabilité du mouvement (1947; M.R. 9, 34)] for boundary value problems of second-order ordinary differential equations. The maxima of the Green's functions are then calculated for nets of three different kinds. (Note that, Theorem 5.1 on page 235 should be Theorem 3.1, and in its statement the word ``circular'' should be replaced by ``straight''.) An open problem is suggested in the final section.
scientific article; zbMATH DE number 4165553