Existence of multiple solutions for finite difference approximations to second-order boundary value problems

• Authors: H. B. Thompson
• Venue: Nonlinear Analysis: Theory, Methods & Applications
• Year: 2003

We consider discrete two-point boundary value problems of the form D2yk+1=f(kh, yk, Dyk), for k = 1,...,n - 1, (0,0) = G((y0, yn);(Dy1,Dyn)), where Dyk = (yk - yk-1)/h and h = 1/n. This arises as a finite difference approximation to y'' = f(x, y, y'), x ∈[0,1], (0,0)= G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g0,g1) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0.