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

✍ Scribed by H.B. Thompson

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 185 KB
• Volume: 53
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(02)00297-3

✦ Synopsis

We consider discrete two-point boundary value problems of the form D 2 y k+1 =f(kh; y k ; Dy k ), for k = 1; : : : ; n -1; (0; 0) = G((y0; yn); (Dy1; Dyn)), where Dy k = (y k -y k-1 )=h and h = 1=n. This arises as a ÿnite di erence 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 = (g 0 ; g 1 ) 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.