Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. P. Agarwal, D. O’Regan, and S. Staněk, “General existence principles for nonlocal boundary-value problems with π-Laplacian and their applications,” Abstr. Anal. Appl., 30 (2006).
I. Rachůnková, S. Staněk, and M. Tvrdý, “Singularities and Laplacians in boundary-value problems for nonlinear ordinary differential equations,” in: A. Caňada, P. Drábek, and A. Fonda (editors), Handbook of Differential Equations. Ordinary Differential Equations, Vol. 3, Elsevier (2006), pp. 607–723.
R. P. Agarwal and P. J. Y. Wong, “Existence of solutions for singular boundary-value problems for higher-order differential equations,” Rend. Semin. Mat. Fis. Milano, 65, 249–264 (1995).
I. Rachůnková and S. Staněk, “Sturm-Liouville and focal higher-order BVPs with singularities in phase variables,” Georg. Math. J., 10, 165–191 (2003).
C. Zhao, Y. Yuan, and Y. Liu, “A necessary and sufficient condition for the existence of positive solutions to singular boundary-value problems for higher-order differential equations,” Electron. J. Different. Equat., No. 8, 1–19 (2006).
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin (1985).
R. G. Bartle, A Modern Theory of Integration, AMS, Providence, RI (2001).
S. Lang, Real and Functional Analysis, Springer, New York (1993).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-008-0058-z