Eigenvalues of a third order BVP subject to functional BCs
DOI:
https://doi.org/10.12775/TMNA.2025.041Keywords
Eigenvalue, eigenfunction, functional boundary condition, Birkhoff-Kellogg theoremAbstract
We discuss the existence of eigenvalues for a third order boundary value problem subject to functional boundary conditions and higher order derivative dependence in the nonlinearities. We prove the existence of positive and negative eigenvalues and provide a localization of the corresponding eigenfunctions in terms of their norm. The methodology involves a version of the classical Birkhoff-Kellogg theorem. We illustrate the applicability of the theoretical results in an example.References
D.R. Anderson and J.M. Davis, Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. Math. Anal. Appl. 267 (2002), 135–157.
J. Appell, E. De Pascale and A. Vignoli, Nonlinear Spectral and Theory, Walter de Gruyter & Co., Berlin, 2004.
A. Cabada, L. López-Somoza and F. Minhós, Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem, J. Nonlinear Sci. Appl. 10 (2017), 5445–5463.
A. Calamai and G. Infante, An affine Birkhoff–Kellogg type result in cones with applications to functional differential equations, Math. Meth. Appl. Sci. 46 (2023), 11897–11905.
J.R. Graef and J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. 71 (2009), 1542–1551.
J. Graef and B. Yang, Multiple positive solutions to a three point third order boundary value problem, Discrete Contin. Dyn. Syst. vol. suppl. (2005), 337–344.
J. Graef and B. Yang, Positive solutions of a third order nonlocal boundary value problem, Discrete Contin. Dyn. Syst. Ser. S. 1 (2008), 89–97.
G. Infante, On the solvability of a parameter-dependent cantilever-type BVP, Appl. Math. Lett. 132 (2022), 108090.
G. Infante and P. Pietramala, A third order boundary value problem subject to nonlinear boundary conditions, Math. Bohem. 135 (2010), 113–121.
S. Smirnov, Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type, Nonlinear Anal. Model. Control 26 (2021), 914–927.
S. Smirnov, Existence of sign-changing solutions for a third–order boundary value problem with nonlocal conditions of integral type, Topol. Methods Nonlinear Anal. 62 (2023), 377–384.
S. Smirnov, Multiplicity of positive solutions for a third-order boundary value problem with nonlocal conditions of integral type, Miskolc Math. Notes 25 (2024), 967–975.
G. Szajnowska and M. Zima, Positive solutions to a third order nonlocal boundary value problem with a parameter, Opuscula Math. 44 (2024), 267–283.
G. Szajnowska and M. Zima, A fixed point index approach to a third order nonlocal boundary value problem, Topol. Methods Nonlinear Anal. 66 (2025), no. 2, 673–686.
J.R.L. Webb, Higher order non-local (n − 1, 1) conjugate type boundary value problems, AIP Conf. Proc. 1124 (2009), 332–341.
J.R.L. Webb, Compactness of nonlinear integral operators with discontinuous and with singular kernels, J. Math. Anal. Appl. 509 (2022), paper no. 126000, 17 pp.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
