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2007, 2007(Special): 624-633. doi: 10.3934/proc.2007.2007.624

Optimal constants for two point boundary value problems


Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada


Department of Computation Science, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China

Received  September 2006 Revised  June 2007 Published  September 2007

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.

Citation: K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

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