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Nodal properties of radial solutions for a class of polyharmonic equations
Optimal constants for two point boundary value problems
1.  Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada 
2.  Department of Computation Science, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China 
$u''(t) + \mug(t)u(t)$ = 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.
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