A priori bounds and upper and lower solutions for nonlinear second-order boundary-value problems

Abstract

y” = f(h y, r’), (1.1) r(a) kL(Y’(a)) = 0, (1.2) Y(b) i92(3”(b)) = 0, (1.3) where we assume throughout that f(t, y, y’) is continuous on [a, b] x Rg, and g,( y’) and g,( y’) are continuous and nondecreasing in (--CO, co). Our method originates from the observation that if y(t) is a solution to (l.l)--(1.3) with ~rlax[,,,~y(t) = y(t,) M > 0 and a --I#( y, y’) on an appropriate set and (A) The function $((x, x’) is positive, continuous, and satisfies a locai f,ipschitz condition on H”. 291 Copyright