A generalization of leighton's variational theorem

Abstract

This paper offers the following generalization of the well known variational theorem of Leighton. Considering the non-linear equation , we show that following an arbitrary choice of function u(x)∊C² and of G(u(x)), G(u(α)) = G(u(β)) = 0; G(u(x)) > 0 for all x in [α, β] the negative sign of the functional implies that all solutions of our equation will be bounded in absolute value by the number (m/K)^1/(K-1), where

on some sub-interval of [α, β]. In the linear case, K= 1, we show that the negative sign of the above functional and the condition imply the vanishing of all solutions on [α, β]