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Positive characteristic values and optimal constants for three-point boundary value problems. (English) Zbl 1129.34008

Agarwal, Ravi P. (ed.) et al., Proceedings of the international conference on differential and difference equations and their applications, Melbourne, FL, USA, August 1–5, 2005. New York: Hindawi Publishing Corporation (ISBN 977-5945-38-0/hbk). 623-633 (2006).
Summary: The smallest characteristic value \(\mu_1\) of the linear second-order differential equation of the form
\[ u''(t) +\mu_1 g(t)u(t) = 0,\text{ a.e. on }[0,1] \]
subject to the three-point boundary condition
\[ z(0) = 0,\quad \alpha z(\eta) = z(1),\quad 0 < \eta < 1,\text{ and }0 < \alpha < 1/\eta \] is investigated. The upper and lower bounds for \(\mu_1\) are provided, namely, \(m <\mu_1 < M(a,b)\), where \(m\) and \(M(a,b)\) are computable definite integrals related to the kernels arising from the above boundary value problem. When \(g\equiv 1\), the minimum values for \(M(a, b)\) for some \(a, b\in (0, 1]\) with \(a < b\) are discussed. All of these values obtained here are useful in studying the existence of nonzero positive solutions for some nonlinear three-point boundary value problems.
For the entire collection see [Zbl 1118.34001].

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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