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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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