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Periodic solutions for a second-order differential equation with indefinite weak singularity

Part of: Boundary value problems Qualitative theory

Published online by Cambridge University Press:  15 January 2019

José Godoy  and
Manuel Zamora
José Godoy
Affiliation:
Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile ([email protected]; [email protected])
Manuel Zamora
Affiliation:
Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile ([email protected]; [email protected])
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Abstract

As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation

$${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$
are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

Keywords

Singular differential equationsweak-indefinite singularityperiodic solutionsdegree theoryLeray-Schauder continuation theorem

MSC classification

Primary: 34C25: Periodic solutions
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 34B30: Special equations (Mathieu, Hill, Bessel, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1135 - 1152
Copyright
Copyright © Royal Society of Edinburgh 2019 

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References

1 Berestycki, H., Capuzzo-Dolcetta, I. and Nirenberg, L.. Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2 (1995), 553–572.Google Scholar
2Boscaggin, A. and Garrione, M.. Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities. J. Dyn. Diff. Equ. 28 (2016), 167–187.Google Scholar
3Boscaggin, A. and Zanolin, F.. Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight. J. Differ. Equ. 252 (2012), 2900–2921.Google Scholar
4Boscaggin, A. and Zanolin, F.. Second-order ordinary differential equations with indefinite weight the Neumann boundary value problem. Ann. Mat. Pura Appl. (4) 194 (2015), 451–478.Google Scholar
5Boscaggin, A., Feltrin, G. and Zanolin, F.. Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, preprint.Google Scholar
6Bravo, J. L. and Torres, P. J.. Periodic solutions of a singular equation with indefinite weight. Adv. Nonlinear Stud. 10 (2010), 927–938.Google Scholar
7De Figueiredo, D. G., Gossez, J.-P. and Ubilla, P.. Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199 (2003), 452–467.Google Scholar
8Deimling, K.. Nonlinear functional analysis (Berlin Heidelberg: Springer-Verlag, 1985).Google Scholar
9Hakl, R. and Torres, P. J.. On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ. 248 (2010), 111–126.Google Scholar
10Hakl, R. and Zamora, M.. Periodic solutions of an indefinite singular equation arising from the Kepler's problem on the sphere. Canadian J. Math., http://dx.doi.org/10.4153/CJM-2016-050-1.Google Scholar
11Hakl, R. and Zamora, M.. Periodic solutions to second-order indefinite singular equations. J. Differ. Equ. 263 (2017), 451–469.Google Scholar
12Leray, J. and Schauder, J.. Topologie et équations fonctionnelles (French), Ann. Sci. École Norm. Sup. 51 (1934), 45–78.Google Scholar
13Mawhin, J.. Leray-Schauder continuation theorems in the absence of a priori bounds. Topol. Meth. Nonlinear Anal. 9 (1997), 179–200.Google Scholar
14Mawhin, J., Rebelo, C. and Zanolin, F.. Continuation theorems for Ambrosetti-Prodi type periodic potentials. Commun. Contemp. Math. 2 (2000), 87–126.Google Scholar
15Ureña, A. J.. Periodic solutions of singular equations. Topol. Meth. Nonlinear Anal. 47 (2016), 55–72.Google Scholar
16Ureña, A. J.. A counterexample for singular equations with indefinite weight. Adv. Nonlinear Stud. 17 (2017), 497–516.Google Scholar