Spectral Characterization of the Constant Sign Derivatives of Green’s Function Related to Two Point Boundary Value Conditions
- Alberto Cabada 1
- Lucía López-Somoza 1
- Mouhcine Yousfi 1
-
1
Universidade de Santiago de Compostela
info
ISSN: 1575-5460
Ano de publicación: 2025
Volume: 24
Número: 1
Tipo: Artigo
Outras publicacións en: Qualitative theory of dynamical systems
Resumo
In this paper we will consider a n-th order linear operator Tn[M], depending on a real parameter M, coupled to different two-point boundary conditions, and we will study the set of parameters for which certain partial derivatives of the related Green’s function are of constant sign. We will do it without using the explicit expression of the Green’s function. In particular, the set of parameters for which the derivatives of the Green’s function have constant sign will be an interval whose extremes are characterized as the first eigenvalues of the studied operator related to suitable boundary conditions. As a consequence of the main result, we will be able to give sufficient conditions to ensure that the derivatives of the Green’s function cannot be nonpositive (nonnegative). These characterizations and the obtained results can be used to deduce the existence of solutions of nonlinear problems under additional conditions on the nonlinear part. In order to illustrate the obtained results, some examples are given.
Referencias bibliográficas
- 1. Almenar, P., Jódar, L.: Solvability of a class of N-th order linear focal problems. Math. Model. Anal. 22(n4), 528–547 (2017)
- 2. Almenar, P., Jódar, L.: The sign of the green function of an n-th order linear boundary value problem. Mathematics 8(673), 22 (2020)
- 3. Almenar, P., Jódar, L.: Estimation of the smallest eigenvalue of an nth-order linear boundary value problem. Math. Methods Appl. Sci. 44(6), 4491–4514 (2021)
- 4. Almenar, P., Jódar, L.: The principal eigenvalue of some nth order linear boundary value problems. Bound. Value Probl. 84, 16 (2021)
- 5. Almenar, P., Jódar, L.: New results on the sign of the Green function of a two-point n-th order linear boundary value problem. Bound. Value Probl. 50, 22 (2022)
- 6. Butler, G., Erbe, L.: Integral comparison theorems and extremal points for linear differential equations. J. Differ. Equ. 47, 214–226 (1983)
- 7. Cabada, A.: Green’s Functions in the Theory of Ordinary Differential Equations. Springer Briefs in Mathematics. Springer, New York (2014)
- 8. Cabada, A., López-Somoza, L.: An explicit formula of the parameter dependence of the partial derivatives of the Green’s functions related to arbitrary two-point boundary conditions. arXiv preprint arXiv:2405.17320
- 9. Cabada, A., López-Somoza, L., Minhós, F.: Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem. J. Nonlinear Sci. Appl. 10, 5445–5463 (2017)
- 10. Cabada, A., Saavedra, L.: The eigenvalue characterization for the constant sign Green’s functions of (k, n − k) problems. Bound. Value Probl. 44, 35 (2016)
- 11. Cabada, A., Saavedra, L.: Disconjugacy characterization by means of spectral (k; n − k) problems. Appl. Math. Lett. 52, 21–29 (2016)
- 12. Cabada, A., Saavedra, L.: Characterization of constant sign Green’s function for a two-point boundaryvalue problem by means of spectral theory. Electron. J. Differ. Equ. 2017(146), 1–95 (2017)
- 13. Cabada, A., Saavedra, L.: Existence of solutions for nth-order nonlinear differential boundary value problems by means of fixed point theorems. Nonlinear Anal. Real World Appl. 42, 180–206 (2018)
- 14. Coppel, W.A.: Disconjugacy. In: Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)
- 15. De Coster, C., Habets, P.: Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering. Elsevier, Amsterdam (2006)
- 16. Elias, U.: Eventual disconjugacy of y(n) + μ p(x) y = 0 for every μ. Arch. Math. (Brno) 40(2), 193–200 (2004)
- 17. Elias, U.: Green’s functions for a nondisconjugate differential operator. J. Differ. Equ. 37, 319–350 (1980)
- 18. Eloe, P.W., Henderson, J.: Focal point characterizations and comparisons for right focal differential operators. J. Math. Anal. Appl. 181, 22–34 (1994)
- 19. Eloe, P.W., Ridenhour, J.: Sign properties of Green’s functions for a family of two-point boundary value problems. Proc. Am. Math. Soc. 120(2), 443–452 (1994)
- 20. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
- 21. Guo, D., Lakshmikantham, L.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)
- 22. Krasnosel’ski˘ı, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)
- 23. Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985)
- 24. Nehari, Z.: Disconjugate linear differential operators. Trans. Am. Math. Soc. 129, 500–516 (1967)
- 25. Peterson, A.: Green’s functions for focal type boundary value problems. Rocky Mt. J. Math. 9, 721–732 (1979)
- 26. Peterson, A.: Focal Green’s functions for fourth-order differential equations. J. Math. Anal. Appl. 75, 602–610 (1980)
- 27. Simons, W.: Some disconjugacy criteria for selfadjoint linear differential equations. J. Math. Anal. Appl. 34, 445–463 (2017)
- 28. Torres, P.J.: Existence of one-signed periodic solutions of some second-order differential equations via Krasnoselskii fixed point theorem. J. Differ. Equ. 190(2), 643–662 (2003)
- 29. Zettl, A.: A constructive characterization of disconjugacy. Bull. Am. Math. Soc. 81(1), 145–147 (1975)