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A unified approach to nonlocal boundary value problems. (English) Zbl 1203.34033
Ladde, G. S. (ed.) et al., Dynamic systems and applications. Vol. 5. Proceedings of the 5th international conference, Morehouse College, Atlanta, GA, USA, May 30–June 2, 2007. Atlanta, GA: Dynamic Publishers (ISBN 1-890888-01-6). 510-515 (2008).
Summary: We show how the existence of multiple positive solutions of nonlinear second-order differential equations of the general form \[ u''(t)+p(t)u'(t)+q(t)u(t)+g(t)f(t,u(t))=0,\quad t\in(0,1), \] subject to various nonlocal boundary conditions, can be established with a unified approach. The nonlocal boundary conditions are of the general form \[ au(0)-bu'(0)=\alpha[u],\quad cu(1)+du'(1)=\beta[u], \] where \(\alpha[u],\beta[u]\) are linear functionals given by Stieltjes integrals but they are not assumed to be positive for all \(u\geq 0\). The well-known multi-point BVPs are special cases, and we can allow some coefficients to have opposite signs.
For the entire collection see [Zbl 1203.35008].

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
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