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Existence and multiplicity of periodic solutions to differential equations with attractive singularities

Part of: Qualitative theory Boundary value problems

Published online by Cambridge University Press:  12 April 2021

José Godoy ,
Robert Hakl Open the ORCID record for Robert Hakl [Opens in a new window]  and
Xingchen Yu
José Godoy
Affiliation:
Institute of Mathematics, Czech Academy of Sciences, Žižkova 22, 616 62 Brno, Czech Republic (godoy@math.cas.cz; hakl@ipm.cz)
Robert Hakl
Affiliation:
Institute of Mathematics, Czech Academy of Sciences, Žižkova 22, 616 62 Brno, Czech Republic (godoy@math.cas.cz; hakl@ipm.cz)
Xingchen Yu
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing210044, P.R. China (yuxingchen0@yeah.net)
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Abstract

The existence and multiplicity of T-periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree.

Keywords

periodic solutionsupper and lower functionsmultiplicity results

MSC classification

Primary: 34C25: Periodic solutions
Secondary: 34B18: Positive solutions of nonlinear boundary value problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 402 - 427
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Benson, J. D., Chicone, C. C. and Critser, J. K.. A general model for the dynamics of cell volume, global stability and optimal control. J. Math. Biol. 63 (2011), 339–359.CrossRefGoogle ScholarPubMed
Bereanu, C. and Mawhin, J.. Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian. J. Diff. Equ. 243 (2007), 536–557.CrossRefGoogle Scholar
Bevc, V., Palmer, J. L. and Süsskind, C.. On the design of the transition region of axisymmetric, magnetically focused beam valves. J. British Inst. Radio Eng. 18 (1958), 696–708.Google Scholar
Chu, J., Torres, P. J. and Wang, F.. Twist periodic solutions for differential equations with acombined attractive-repulsive singularity. J. Math. Anal. Appl. 437 (2016), 1070–1083.CrossRefGoogle Scholar
De Coster, C. and Habets, P.. Two-point boundary value problems: lower and upper solutions (Amsterdam: Elsevier, 2006).Google Scholar
Fonda, A. and Sfecci, A.. On a singular periodic Ambrosetti-Prodi problem. Nonlinear Anal. 149 (2017), 146–155.CrossRefGoogle Scholar
Godoy, J., Morales, N. and Zamora, M.. Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case.. Discrete Contin. Dyn. Syst. 39 (2019), 4137–4156.CrossRefGoogle Scholar
Greiner, W.. Classical mechanics: point particles and relativity (New York: Springer, 2004).Google Scholar
Habets, P. and Sanchez, L.. Periodic solutions of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990), 1135–1144.Google Scholar
Hakl, R. and Torres, P. J.. On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Diff. Equ. 248 (2010), 111–126.CrossRefGoogle Scholar
Hakl, R., Torres, P. J. and Zamora, M.. Periodic solutions of singular second order differential equations: upper and lower functions. Nonlinear Anal. 74 (2011), 7078–7093.CrossRefGoogle Scholar
Hakl, R. and Zamora, M.. On the open problems connected to the results of Lazer and Solimini. Proc. R. Soc. Edinburgh Sect. A Math. 144 (2014), 109–118.CrossRefGoogle Scholar
Hakl, R. and Zamora, M.. Existence and uniqueness of a perodic solution to an indefinite attractive singular equation. Ann. Mat. Pura Appl. 195 (2016), 995–1009.Google Scholar
Hakl, R. and Zamora, M.. Periodic solutions to the Liénard type equations with phase attractive singularities. Bound. Value Probl. 47 (2013), doi: 10.1186/1687-2770-2013-47, 1–20.Google Scholar
Hakl, R. and Zamora, M.. Existence and multiplicity of periodic solutions to indefinite singular equations having a non-monotone term with two singularities. Adv. Nonlinear Stud. 19 (2019), 317–332.CrossRefGoogle Scholar
Hernández, J. A.. A general model for the dynamics of the cell volume. Bull. Math. Biol. 69 (2007), 1631–1648.CrossRefGoogle ScholarPubMed
Lazer, A. C. and Solimini, S.. On periodic solutions of nonlinear differential equations with singularities. Proc. Amer. Math. Soc. 99 (1987), 109–114.CrossRefGoogle Scholar
Lomtatidze, A.. Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations. Mem. Diff. Equ. Math. Phys. 67 (2016), 1–129.Google Scholar
Lu, S. and Yu, X.. Periodic solutions for second order differential equations with indefinite singularities. Adv. Nonlinear Anal. 9 (2020), 994–1007.Google Scholar
Lu, S. and Yu, X.. Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type. Bound. Value Probl. 101 (2018). doi: 10.1186/s13661-018-1020-0.Google Scholar
Martínez-Amores, P. and Torres, P. J.. Dynamics of a periodic differential equation with a singular nonlinearity of attractive type. J. Math. Anal. Appl. 202 (1996), 1027–1039.CrossRefGoogle Scholar
Mawhin, J.. Topological degree and boundary value problems for nonlinear differential equations. In Topological methods for ordinary differential equations (Montecatini Terme, 1991). Lecture Notes in Mathematics (ed. Furi, M. and Zecca, P.), vol. 1537, pp. 74–142 (Berlin, Germany: Springer, 1993).CrossRefGoogle Scholar
Mawhin, J.. The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the ϕ-Laplacian. J. Eur. Math. Soc. 8 (2006), 375–388.CrossRefGoogle Scholar
Montesinos, G. D., Perez-García, V. M. and Torres, P. J.. Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers. Phys. D 191 (2004), 193–210.CrossRefGoogle Scholar
Rachůnková, I., Staněk, S. and Tvrdý, M.. Solvability of nonlinear singular problems for ordinary differential equations. Contemporary Mathematics and its Applications, vol. 5 (New York: Hindawi Publishing Corporation, 2008).Google Scholar
Rachůnková, I. and Tvrdý, M. Nonlinear systems of differential inequalities and solvability of certain boundary value problems. J. Inequal. Appl. 6 (2001), 199–226.Google Scholar
Rachůnková, I., Tvrdý, M. and Vrkoč, I.. Existence of nonnegative and nonpositive solutions for second–order periodic boundary–value problems. J. Diff. Equ. 176 (2001), 445–469.CrossRefGoogle Scholar
Torres, P. J.. Periodic oscillations of a model for membrane permeability with fluctuating environmental conditions. J. Math. Biol. 71 (2015), 57–68.Google Scholar
Torres, P. J.. Mathematical models with singularities — a zoo of singular creatures (Paris: Atlantis Press, 2015)Google Scholar
Yu, X. and Lu, S.. A multiplicity result for periodic solutions of Liénard equations with an attractive singularity. Appl. Math. Comput. 346 (2019), 183–192.Google Scholar