Abstract
We provide sufficient conditions for the existence of one positive solution for a fourth--order beam equation with a discontinuous nonlinear term. Also a multiplicity result is established. They are based on a recent generalization of the Krasnosel’skiĭ fixed point theorem in cones.
Similar content being viewed by others
References
Bonanno, G., Bisci, G.M.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, 670675 (2009)
Bonanno, G., Buccellato, S.M.: Two point boundary value problems for the Sturm-Liouville equation with highly discontinuous nonlinearities. Taiwan. J. Math. 14(5), 2059–2072 (2010)
Cabada, A., Cid, J.A., Infante, G.: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 2013, 125 (2013)
Cid, J.A., Franco, D., Minhós, F.: Positive fixed points and fourth-order equations. Bull. Lond. Math. Soc. 41, 72–78 (2009)
Drábek, P., Holubová, G., Matas, A., Nećesal, P.: Nonlinear models of suspension bridges: discussion of the results. Appl. Math. 48, 497–514 (2003)
Drábek, P., Holubová, G.: Positive and negative solutions of one-dimensional beam equation. Appl. Math. Lett. 51, 1–7 (2016)
Figueroa, R., Infante, G.: A Schauder-type theorem for discontinuous operators with applications to second-order BVPs. Fixed Point Theory Appl. 2016, 53 (2016)
Figueroa, R., López Pouso, R., Rodríguez-López, J.: A version of Krasnosel’skiĭ’s compression-expansion fixed point theorem in cones for discontinuous operators with applications. Topol. Methods Nonlinear Anal. 51, 493–510 (2018)
Filippov, A.F.: Differential equations with discontinuous righthand sides. Kluwer Academic, Dordrecht (1988)
Fitzpatrick, P.M., Petryshyn, W.V.: Fixed point theorems and the fixed point index for multivalued mappings in cones. J. London Math. Soc. 2(11), 75–85 (1975)
Hu, S.: Differential equations with discontinuous right-hand sides. J. Math. Anal. Appl. 154, 377–390 (1991)
López Pouso, R.: Schauder’s fixed-point theorem: new applications and a new version for discontinuous operators. Bound. Value Probl. 2012, 92 (2012)
Webb, J.R.L., Infante, G., Franco, D.: Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions. Proc. R. Soc. Edinb. A 148, 427–446 (2008)
Acknowledgements
Partially supported by Xunta de Galicia under grants ED481A-2017/178 and ED431C-2019/2, Spain.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Syakila Ahmad.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rodríguez-López, J. Positive Solutions of a Discontinuous One-Dimensional Beam Equation. Bull. Malays. Math. Sci. Soc. 44, 2357–2370 (2021). https://doi.org/10.1007/s40840-020-01072-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-01072-w
Keywords
- Fourth-order problem
- Positive solution
- Krasnosel’skiĭ theorem
- Discontinuous differential equations
- Multiplicity result