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

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.
Citation: Pablo Amster, Manuel Zamora. Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 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.

[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.

[3]

C. BereanuD. 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.

[4]

C. BereanuP. 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.

[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.

[6]

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

[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.

[8]

A. V. BorisovI. 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.

[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.

[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.

[11]

A. CapiettoJ. 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.

[12]

P. FitzpatrickM. MartelliJ. 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.

[13]

A. FondaR. 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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[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. 

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.

[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.

[3]

C. BereanuD. 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.

[4]

C. BereanuP. 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.

[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.

[6]

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

[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.

[8]

A. V. BorisovI. 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.

[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.

[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.

[11]

A. CapiettoJ. 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.

[12]

P. FitzpatrickM. MartelliJ. 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.

[13]

A. FondaR. 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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[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. 

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.).
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)).
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 $u(a_i')<\alpha+2\delta$.
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 $u(a_i')\geq \alpha+2\delta$.
[1]

Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015

[2]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[3]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248

[4]

Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010

[5]

José Godoy, Nolbert Morales, Manuel Zamora. Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4137-4156. doi: 10.3934/dcds.2019167

[6]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252

[7]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure and Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029

[8]

Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557

[9]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[10]

Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1383-1395. doi: 10.3934/dcdsb.2019232

[11]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[12]

Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4587-4612. doi: 10.3934/dcdsb.2020304

[13]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301

[14]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[15]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[16]

Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453

[17]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113

[18]

M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure and Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411

[19]

Zuzana Došlá, Mauro Marini, Serena Matucci. Global Kneser solutions to nonlinear equations with indefinite weight. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3297-3308. doi: 10.3934/dcdsb.2018252

[20]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (275)
  • HTML views (130)
  • Cited by (0)

Other articles
by authors

[Back to Top]