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Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

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We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α k: C p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.

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Staněk, S. Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems. Ukr Math J 60, 277–298 (2008). https://doi.org/10.1007/s11253-008-0058-z

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  • DOI: https://doi.org/10.1007/s11253-008-0058-z

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