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Optimal constants for two point boundary value problems

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  • 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.

    Mathematics Subject Classification: Primary: 34B18; Secondary: 34B15, 34B16, 47H10, 47H30.

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