Optimal constants for two point boundary value problems

Abstract

The upper and lower bounds of the smallest positive characteristic value $\mu_1$ of a linear differential equation of the form $u''(t) + \mug(t)u(t)$ = 0 a.e. on [0, 1], subject to the general separated boundary conditions (BCs) are estimated. It is shown that $m$ < $\mu_1$ < $M(a, b)$, where $m$ and $M(a, b)$ are computable definite integrals related to the kernels arising from the above boundary value problems. The mimimum values for $M(a, b)$ are discussed when $g \stackrel{-}{=}$ 1 and $g(s) = 1/s^\alpha (\alpha > 0)$ for some of these BCs. All of these values obtained here are useful in studying the existence of nonzero positive solutions for the nonlinear differential equations of the form $u''(t) + g(t)f(t, u(t)) = 0$ a.e. on [0, 1], subject to the above BCs.