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Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems. (English) Zbl 1164.34341
Ukr. Mat. Zh. 60, No. 2, 240-259 (2008) and Ukr. Math. J. 60, No. 2, 277-298 (2008).
Summary: We present existence principles for the nonlocal boundary-value problem \[ (\phi(u^{(p-1)}))'=g(t,u,\dots,u^{(p-1)}), \alpha_k(u)=0, 1\leq k\leq p-1, \] where \(p\geq 2\), \(\phi:\mathbb R\to\mathbb R\) is an increasing and odd homeomorphism, \(g\) is a Carathéodory function that is either regular or has singularities in its space variables, and \(\alpha_k: C^{p-1}[0,T]\to\mathbb R\) is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems \[ \begin{gathered} (-1)^n(\phi(u^{(2n-1)}))'=f(t,u,\dots,u^{(2n-1)}), \\ u^{(2k)}(0)=0, a_ku^{(2k)}(T)+ b_ku^{(2k+1)}(T)=0, 1\leq k\leq n-1, \end{gathered} \] is given.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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