Abstract
In this paper, using variational methods and two critical point theorems, we prove the existence of two intervals for a parameter for which a nonlinear equation of sixth-order admits three weak solutions.
Similar content being viewed by others
References
Atay, M.T., Kartal, S., Hadjian, A.: Computation of eigenvalues of Sturm–Liouville Problems using Homotopy perturbation method. Int. J. Nonlinear Sci. Numer. Simul. 11, 105–112 (2010)
Averna, D., Giovannelli, N., Tornatore, E.: Existence of three solutions for a mixed boundary value problem with the Sturm–Liouville equation. Bull. Korean Math. Soc. 49, 1213–1222 (2012)
Bonanno, G.: A critical points theorem and nonlinear differential problems. J. Global Optim. 28, 249–258 (2004)
Bonanno, G., Bella, B.D.: A fourth-order boundary value problem for a Sturm–Liouville type equation. Appl. Math. Comput. 217, 3635–3640 (2010)
Bonanno, G., Candito, P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031–3059 (2008)
Bonanno, G., Candito, P., ÓRegan, D.: Existence of nontrivial solutions for sixth-order differential equations. Mathematics 9, 9 (2021). https://doi.org/10.3390/math9161852
Bonanno, G., Livrea, R.: A sequence of positive solutions for sixth-order ordinary nonlinear differential problems. Electron. J. Qual. Theory Differ. Equ. 20, 1–17 (2021)
El-Gamel, M., Sameeh, M.: An efficient technique for finding the eigenvalues of fourth-order Sturm–Liouville problems. Appl. Math. 3, 920–925 (2012)
Heidarkhani, S.: Existence of solutions for a two-point boundary-value problem of a fourth-order Sturm–Liouville type. Electron. J. Differ. Equ. 84, 1–15 (2012)
Heidarkhani, S.: Non-trivial solutions for two-point boundary-value problems of fourth-order Sturm–Liouville type equations. Electron. J. Differ. Equ. 27, 1–9 (2012)
Ge, W., Ren, J.: New existence theorems of positive solutions for Sturm–Liouville boundary value problems. Appl. Math. Comput. 148, 631–644 (2004)
Greenberg, L., Marletta, M.: Oscillation theory and numerical solution of sixth order Sturm–Liouville problems. SIAM J. Numer. Anal. 35, 2070–2098 (1998)
Gutierrez, R.H., Laura, P.A.: Vibrations of non-uniform rings studied by means of the differential quadrature method. J. Sound Vib. 185, 507–513 (1995)
Lesnic, D., Attili, B.S.: An efficient method for sixth-order Sturm–Liouville problems. Int. J. Sci. Technol. 2, 109–114 (2007)
Li, Y.: On the existence and nonexistence of positive solutions for nonlinear Sturm–Liouville boundary value problems. J. Math. Anal. Appl. 304, 74–86 (2005)
Perera, U., Böckmann, C.: Solutions of direct and inverse even-order Sturm–Liouville problems using magnus expansion. Mathematics 7, 24 (2019). https://doi.org/10.3390/math7060544
Rattana, A., Böckmann, C.: Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four. J. Comput. Appl. Math. 249, 144–156 (2013)
Shokooh, S.: Variational techniques for a system of Sturm–Liouville equations. J. Elliptic Parabol. Equ. (2023). https://doi.org/10.1007/s41808-023-00217-9
Taher, A.S., Malek, A.: An efficient algorithm for solving high order Sturm–Liouville problems using variational iteration method. Fixed Point Theory 14, 193–210 (2013)
Taher, A.S., Malek, A., Momeni-Masuleh, S.: Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems. Appl. Math. Model. 37, 4634–4642 (2013)
Tian, Y., Ge, W.G.: Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 72, 277–287 (2009)
Tian, Y., Liu, X.: Applications of variational methods to Sturm–Liouville boundary value problem for fourth-order impulsive differential equations. Math. Methods Appl. Sci. 37, 95–105 (2014)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. II. Berlin, Germany (1990)
Zhao, Y., Chen, H.: Multiplicity of solutions to two-point boundary value problems for a second-order impulsive differential equation. Appl. Math. Comput. 206, 925–931 (2008)
Zhao, Y., Li, H., Zhang, Q.: Existence results for an impulsive Sturm–Liouville boundary value problems with mixed double parameters. Bound. Value Probl. 150, 1–17 (2015)
Acknowledgements
The author would like to expresses his sincere gratitude to the referee for reading this paper carefully and specially for valuable comments concerning improvement of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shokooh, S. Existence of solutions for a sixth-order nonlinear equation. Rend. Circ. Mat. Palermo, II. Ser 72, 4251–4271 (2023). https://doi.org/10.1007/s12215-023-00901-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-023-00901-8