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October  2018, 38(10): 4819-4835. doi: 10.3934/dcds.2018211

Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

Pablo Amster 1, and  Manuel Zamora 2,,

1. 

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, 1428, Buenos Aires, Argentina
2. 

Departamento de Matemática, Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad de Oviedo, C/ Federico García Lorca, n18, Oviedo, Spain

* Corresponding author: Manuel Zamora

Received  February 2017 Revised  December 2017 Published  July 2018

Fund Project: The first author is supported by projects UBACyT 20020120100029BA and CONICET PIP11220130100006CO, and the second author was supported by FONDECYT, project no. 11140203.

TWe prove the existence of
$T$
-periodic solutions for the second order non-linear equation
$ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = h(t)g(u), $
where the non-linear term
$g$
 has two singularities and the weight function
$h$
 changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.
Keywords: Singular differential equations, indefinite singularity, periodic solutions, degree theory, Leray-Schauder continuation theorem.
Mathematics Subject Classification: Primary: 34C25, 34B18; Secondary: 34B30.
Citation: Pablo Amster, Manuel Zamora. Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4819-4835. doi: 10.3934/dcds.2018211
References:
[1]

J. C. Alexander, A primer on connectivity. In proceeding of the conference in fixed point theory, Fadell, E.- Fournier, G. Editors, Springer-Verlag Lecture Notes in Mathematics, 886 (1981), 455-488. Google Scholar

[2]

C. Bereanu, D. Gheorghe and M. Zamora, Periodic solutions for singular perturbations of the singular $\phi-$Laplacian operator, Commun. Contemp. Math., 15(2013), 1250063 (22 pages). doi: 10.1142/S0219199712500630.  Google Scholar

[3]

C. Bereanu, D. Gheorghe and M. Zamora, Non-resonant boundary value problems with singular $\phi$-Laplacian operators, Nonlinear Differential Equations and Applications, 20 (2013), 1365-1377.  doi: 10.1007/s00030-012-0212-z.  Google Scholar

[4]

C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbations of singular $\phi$-Laplacians, Rend. Lincei Mat. Appl., 22 (2011), 89-111.  doi: 10.4171/RLM/589.  Google Scholar

[5]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differ. Equ., 243 (2007), 536-557.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[6]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differ. Integral Equ., 23 (2010), 801-810.   Google Scholar

[7]

A. V. Borisov and I. S. Mamaev, The restricted two body problem in constant curvature spaces, Celestial Mech. Dynam. Astronom, 96 (2006), 1-17.  doi: 10.1007/s10569-006-9012-2.  Google Scholar

[8]

A. V. Borisov, I. S. Mamaev and A. A. Kilin, Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn., 9 (2004), 265-279.  doi: 10.1070/RD2004v009n03ABEH000280.  Google Scholar

[9]

A. Boscaggin and F. Zanolin, Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl., 194 (2015), 451-478.  doi: 10.1007/s10231-013-0384-0.  Google Scholar

[10]

J. L. Bravo and P. J. Torres, Periodic solutions of a singular equation with indefinite weight, Adv. Nonlinear Stud., 10 (2010), 927-938.  doi: 10.1515/ans-2010-0410.  Google Scholar

[11]

A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar

[12]

P. Fitzpatrick, M. Martelli, J. Mawhin and R. Nussbaum, Topological methods for ordinary differential equations, Lecture Notes in Mathematics, 1537 (1991), 1-209, Springer-Verlag, ISBN 3-540-56461-6.  doi: 10.1007/BFb0085073.  Google Scholar

[13]

A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311.  doi: 10.1137/0524074.  Google Scholar

[14]

R. Hakl and M. Zamora, On the open problems connected to the results of Lazer and Solimini, Proc. Roy. Soc. Edinb., Sect. A. Math., 144 (2014), 109-118.  doi: 10.1017/S0308210512001862.  Google Scholar

[15]

R. Hakl and M. Zamora, Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere, Canadian J. Math., 70 (2018), 173-190.  doi: 10.4153/CJM-2016-050-1.  Google Scholar

[16]

E. H. Hutten, Relativistic (non-linear) oscillator, Nature, 203 (1965), 892. doi: 10.1038/205892a0.  Google Scholar

[17]

A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 99 (1987), 109-114.  doi: 10.1090/S0002-9939-1987-0866438-7.  Google Scholar

[18]

L. A. Mac-Coll, Theory of the relativistic oscillator, Am. J. Phys., 25 (1957), 535-538.   Google Scholar

[19]

P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equations with $\phi$-Laplacian, Commun. Contemp. Math., 13 (2011), 283-293.  doi: 10.1142/S0219199711004208.  Google Scholar

[20]

A. J. Ureña, Periodic solutions of singular equations, Topological Methods in Nonlinear Analysis, 47 (2016), 55-72.   Google Scholar

[21]

G. T. Whyburn, Topological Analysis, Princeton Univ. Press, 1958.  Google Scholar

[22]

M. Zamora, New periodic and quasi-periodic motions of a relativistic particle under a planar central force field with applications to scalar boundary periodic problems, J. Qualitative Theory of Differential Equations, 31 (2013), 1-16.   Google Scholar

show all references

References:
[1]

J. C. Alexander, A primer on connectivity. In proceeding of the conference in fixed point theory, Fadell, E.- Fournier, G. Editors, Springer-Verlag Lecture Notes in Mathematics, 886 (1981), 455-488. Google Scholar

[2]

C. Bereanu, D. Gheorghe and M. Zamora, Periodic solutions for singular perturbations of the singular $\phi-$Laplacian operator, Commun. Contemp. Math., 15(2013), 1250063 (22 pages). doi: 10.1142/S0219199712500630.  Google Scholar

[3]

C. Bereanu, D. Gheorghe and M. Zamora, Non-resonant boundary value problems with singular $\phi$-Laplacian operators, Nonlinear Differential Equations and Applications, 20 (2013), 1365-1377.  doi: 10.1007/s00030-012-0212-z.  Google Scholar

[4]

C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbations of singular $\phi$-Laplacians, Rend. Lincei Mat. Appl., 22 (2011), 89-111.  doi: 10.4171/RLM/589.  Google Scholar

[5]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differ. Equ., 243 (2007), 536-557.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[6]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differ. Integral Equ., 23 (2010), 801-810.   Google Scholar

[7]

A. V. Borisov and I. S. Mamaev, The restricted two body problem in constant curvature spaces, Celestial Mech. Dynam. Astronom, 96 (2006), 1-17.  doi: 10.1007/s10569-006-9012-2.  Google Scholar

[8]

A. V. Borisov, I. S. Mamaev and A. A. Kilin, Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn., 9 (2004), 265-279.  doi: 10.1070/RD2004v009n03ABEH000280.  Google Scholar

[9]

A. Boscaggin and F. Zanolin, Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl., 194 (2015), 451-478.  doi: 10.1007/s10231-013-0384-0.  Google Scholar

[10]

J. L. Bravo and P. J. Torres, Periodic solutions of a singular equation with indefinite weight, Adv. Nonlinear Stud., 10 (2010), 927-938.  doi: 10.1515/ans-2010-0410.  Google Scholar

[11]

A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar

[12]

P. Fitzpatrick, M. Martelli, J. Mawhin and R. Nussbaum, Topological methods for ordinary differential equations, Lecture Notes in Mathematics, 1537 (1991), 1-209, Springer-Verlag, ISBN 3-540-56461-6.  doi: 10.1007/BFb0085073.  Google Scholar

[13]

A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311.  doi: 10.1137/0524074.  Google Scholar

[14]

R. Hakl and M. Zamora, On the open problems connected to the results of Lazer and Solimini, Proc. Roy. Soc. Edinb., Sect. A. Math., 144 (2014), 109-118.  doi: 10.1017/S0308210512001862.  Google Scholar

[15]

R. Hakl and M. Zamora, Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere, Canadian J. Math., 70 (2018), 173-190.  doi: 10.4153/CJM-2016-050-1.  Google Scholar

[16]

E. H. Hutten, Relativistic (non-linear) oscillator, Nature, 203 (1965), 892. doi: 10.1038/205892a0.  Google Scholar

[17]

A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 99 (1987), 109-114.  doi: 10.1090/S0002-9939-1987-0866438-7.  Google Scholar

[18]

L. A. Mac-Coll, Theory of the relativistic oscillator, Am. J. Phys., 25 (1957), 535-538.   Google Scholar

[19]

P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equations with $\phi$-Laplacian, Commun. Contemp. Math., 13 (2011), 283-293.  doi: 10.1142/S0219199711004208.  Google Scholar

[20]

A. J. Ureña, Periodic solutions of singular equations, Topological Methods in Nonlinear Analysis, 47 (2016), 55-72.   Google Scholar

[21]

G. T. Whyburn, Topological Analysis, Princeton Univ. Press, 1958.  Google Scholar

[22]

M. Zamora, New periodic and quasi-periodic motions of a relativistic particle under a planar central force field with applications to scalar boundary periodic problems, J. Qualitative Theory of Differential Equations, 31 (2013), 1-16.   Google Scholar

Figure 1.  The figure illustrates a possible behaviour of any $T-$periodic solution $u$ of (20) when $t_*$ is included on $[\bar{\ell}_i, \bar{\ell}_i')$ (Case Ⅰ.) or if $t_*\in (\bar{\ell_i}, \bar{\ell}_i']$ (Case Ⅱ.).
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Figure 2.  The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) when $t_*$ is included on $[a_i, a_i']$, distinguishing if $t_*\in [a_i, \bar{\ell}_i)$ (Case Ⅰ. a)) or $t_*\in (\bar{\ell}_i', a_i']$ (Case Ⅰ. b)).
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Figure 3.  The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) on the interval $[\bar{\ell}_i', \ell_i']$, assuming that $u(a_i')<\alpha+2\delta$.
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Figure 4.  The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) on the interval $[\bar{\ell}_i', \ell_i']$, assuming that $u(a_i')\geq \alpha+2\delta$.
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