A class of second-order nonlinear difference equations. I: Extremum principles and approximation of solutions (Q1067117)

The following difference equation arising from numerical methods for solving nonlinear boundary value problems in differential equations is considered: \(-\phi_{n+1}+2\phi_ n-\phi_{n-1}=f(x_ n,\phi_ n)\), \(n=1,2,...,N\) with \(\phi_ 0=\phi_{N+1}=0\). \(\phi_ n=\phi (x_ n)\), where \(0=x_ 0<x_ 1<...<x_ N<x_{N+1}=1\) is a partition of [0,1] by equidistant points. Further it is assumed that \(f(x_ n,\phi_ n)<0\) for \(\phi_ n>0\) and f is a differentiable function of \(\phi_ n\) such that \((\partial f/\partial \phi_ n)(x_ n,\phi_ n)\leq 0\) for all \(x_ n\), \(\phi_ n.\) First it is shown that if there is a solution \(\{\phi_ n\}\) it is unique and \(\phi_ n<0\) for \(n=1,2,...,N\). Then the problem of finding a solution is formulated as a variational problem with associated maximum and minimum principles. The extremum principles provide a basis for approximating the solution. (For part II, cf. the following review).