A generalization of leighton's variational theorem

摘要

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)∈C2and 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 functionalimplies that all solutions of our equation will be bounded in absolute value by the number (m/K)1/(K-1), whereon some sub-interval of [α, β]. In the linear case, K= 1, we show that the negative sign of the above functional and the conditionimply the vanishing of all solutions on [α, β]