July  2019, 39(7): 4137-4156. doi: 10.3934/dcds.2019167

Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case

1. 

Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y aplicaciones (GISDA), Universidad del Bo-Bo, Casilla 5-C, Concepción, Chile

2. 

Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y aplicaciones (GISDA), C/Federico Garca Lorca, n°18, Oviedo, Spain

Received  October 2018 Published  April 2019

Fund Project: This research was partially supported by FONDECYT

We analyze the existence of T−periodic solutions to the second-order indefinite singular equation
$ u'' = \beta \frac{{h\left( t \right)}}{{{{\cos }^2}u}} $
which depends on a positive parameter β > 0. Here, h is a sign-changing function with h = 0 and where the nonlinear term of the equation has two singularities. For the first time, the degenerate case is studied, displaying an unexpected feature which contrasts with the results known in the literature for indefinite singular equations.
Citation: José Godoy, Nolbert Morales, Manuel Zamora. Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4137-4156. doi: 10.3934/dcds.2019167
References:
[1]

P. Amster and M. Zamora, Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity, Discrete and Continuous Dynamical Systems, 38 (2018), 4819-4835. doi: 10.3934/dcds.2018211. Google Scholar

[2]

J. AndradeN. DávilaE. Pérez-Chavela and C. Vidal, Dynamics and regularization of the Kepler problem on surfaces of constant curvature, Canadian J. Math., 69 (2017), 961-991. doi: 10.4153/CJM-2016-014-5. Google Scholar

[3]

C. Bereanu and M. Zamora, Periodic solutions for indefinite singular perturbations of the relativistic acceleration, Proc. Roy. Soc. Edinb., Sect. A. Math., 148 (2018), 703-712. doi: 10.1017/S0308210518000239. Google Scholar

[4]

A. BoscagginW. Dambrosio and D. Papini, Multiple positive solutions to elliptic boundary blow-up problems, J. Differential Equations, 262 (2017), 5990-6017. doi: 10.1016/j.jde.2017.02.025. Google Scholar

[5]

A. Boscaggin and M. Garrione, Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities, J. Dyn. Diff. Equat., 28 (2016), 167-187. doi: 10.1007/s10884-015-9430-5. Google Scholar

[6]

A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differential Equations, 252 (2012), 2900-2921. doi: 10.1016/j.jde.2011.09.011. Google Scholar

[7]

A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results nad complex dynamics, J. Differential Equations, 252 (2012), 2922-2950. doi: 10.1016/j.jde.2011.09.010. Google Scholar

[8]

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

[9]

A. BoscagginG. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: A topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinb., Sect. A. Math., 146 (2015), 449-474. doi: 10.1017/S0308210515000621. Google Scholar

[10]

A. BoscagginG. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree, Transaction of American Mathematical Society, 370 (2018), 791-845. doi: 10.1090/tran/6992. Google Scholar

[11]

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

[12]

G. J. Butler, Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations, 22 (1976), 467-477. doi: 10.1016/0022-0396(76)90041-3. Google Scholar

[13]

C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper solutions, Elsevier, Amsterdam, The Netherlands, 2006. Google Scholar

[14]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7. Google Scholar

[15]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963. doi: 10.1016/j.jde.2015.02.032. Google Scholar

[16]

G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear inde- finite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291. doi: 10.1016/j.jde.2017.01.009. Google Scholar

[17]

J. Godoy and M. Zamora, Periodic solutions for indefinite singular equations with applications to the weak case, to appear in Proc. Roy. Soc. Edinb., Sect. A. Math..Google Scholar

[18]

J. Godoy and M. Zamora, A general result to the existence of a periodic solution to an indefinite equation with a weak singularity, J. Dyn. Diff. Equat., (2018), 1–18. doi: 10.1007/s10884-018-9704-9. Google Scholar

[19]

R. Hakl and P. J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 248 (2010), 111-126. doi: 10.1016/j.jde.2009.07.008. Google Scholar

[20]

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

[21]

R. Hakl and M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differential Equations, 263 (2017), 451-469. doi: 10.1016/j.jde.2017.02.044. Google Scholar

[22]

R. Hakl and M. Zamora, Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-Monotone Term with two Singularities, Advanced Nonlinear Studies, 2018. doi: 10.1515/ans-2018-2018. Google Scholar

[23]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. Google Scholar

[24]

I. Kiguradze and A. Lomtatidze, Periodic solutions of nonautonomous ordinary differential equations, Monatsh. Math., 159 (2010), 235-252. doi: 10.1007/s00605-009-0138-7. Google Scholar

[25]

J. Leray and J. Schauder, Periodic solutions of nonautonomous ordinary differential equations, Topologie et équations Fonctionnelles (French), Ann. Sci. École Norm. Sup., 51 (1934), 45-78. doi: 10.24033/asens.836. Google Scholar

[26]

J. Mawhin, Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1997), 179-200. doi: 10.12775/TMNA.1997.008. Google Scholar

[27]

J. MawhinC. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic potentials, Commun. Contemp. Math., 2 (2000), 87-126. doi: 10.1142/S0219199700000074. Google Scholar

[28]

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91. doi: 10.1515/ans-2004-0105. Google Scholar

[29]

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

[30]

A. J. Ureña, A counterexample for singular equations with indefinite weight, Adv. Nonlinear Stud., 17 (2017), 497-516. doi: 10.1515/ans-2016-6017. Google Scholar

[31]

P. Waltman, A counterexample for singular equations with indefinite weight, J. Math. Anal. Appl., 10 (1965), 439-441. doi: 10.1016/0022-247X(65)90138-1. Google Scholar

show all references

References:
[1]

P. Amster and M. Zamora, Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity, Discrete and Continuous Dynamical Systems, 38 (2018), 4819-4835. doi: 10.3934/dcds.2018211. Google Scholar

[2]

J. AndradeN. DávilaE. Pérez-Chavela and C. Vidal, Dynamics and regularization of the Kepler problem on surfaces of constant curvature, Canadian J. Math., 69 (2017), 961-991. doi: 10.4153/CJM-2016-014-5. Google Scholar

[3]

C. Bereanu and M. Zamora, Periodic solutions for indefinite singular perturbations of the relativistic acceleration, Proc. Roy. Soc. Edinb., Sect. A. Math., 148 (2018), 703-712. doi: 10.1017/S0308210518000239. Google Scholar

[4]

A. BoscagginW. Dambrosio and D. Papini, Multiple positive solutions to elliptic boundary blow-up problems, J. Differential Equations, 262 (2017), 5990-6017. doi: 10.1016/j.jde.2017.02.025. Google Scholar

[5]

A. Boscaggin and M. Garrione, Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities, J. Dyn. Diff. Equat., 28 (2016), 167-187. doi: 10.1007/s10884-015-9430-5. Google Scholar

[6]

A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differential Equations, 252 (2012), 2900-2921. doi: 10.1016/j.jde.2011.09.011. Google Scholar

[7]

A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results nad complex dynamics, J. Differential Equations, 252 (2012), 2922-2950. doi: 10.1016/j.jde.2011.09.010. Google Scholar

[8]

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

[9]

A. BoscagginG. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: A topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinb., Sect. A. Math., 146 (2015), 449-474. doi: 10.1017/S0308210515000621. Google Scholar

[10]

A. BoscagginG. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree, Transaction of American Mathematical Society, 370 (2018), 791-845. doi: 10.1090/tran/6992. Google Scholar

[11]

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

[12]

G. J. Butler, Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations, 22 (1976), 467-477. doi: 10.1016/0022-0396(76)90041-3. Google Scholar

[13]

C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper solutions, Elsevier, Amsterdam, The Netherlands, 2006. Google Scholar

[14]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7. Google Scholar

[15]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963. doi: 10.1016/j.jde.2015.02.032. Google Scholar

[16]

G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear inde- finite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291. doi: 10.1016/j.jde.2017.01.009. Google Scholar

[17]

J. Godoy and M. Zamora, Periodic solutions for indefinite singular equations with applications to the weak case, to appear in Proc. Roy. Soc. Edinb., Sect. A. Math..Google Scholar

[18]

J. Godoy and M. Zamora, A general result to the existence of a periodic solution to an indefinite equation with a weak singularity, J. Dyn. Diff. Equat., (2018), 1–18. doi: 10.1007/s10884-018-9704-9. Google Scholar

[19]

R. Hakl and P. J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 248 (2010), 111-126. doi: 10.1016/j.jde.2009.07.008. Google Scholar

[20]

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

[21]

R. Hakl and M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differential Equations, 263 (2017), 451-469. doi: 10.1016/j.jde.2017.02.044. Google Scholar

[22]

R. Hakl and M. Zamora, Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-Monotone Term with two Singularities, Advanced Nonlinear Studies, 2018. doi: 10.1515/ans-2018-2018. Google Scholar

[23]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. Google Scholar

[24]

I. Kiguradze and A. Lomtatidze, Periodic solutions of nonautonomous ordinary differential equations, Monatsh. Math., 159 (2010), 235-252. doi: 10.1007/s00605-009-0138-7. Google Scholar

[25]

J. Leray and J. Schauder, Periodic solutions of nonautonomous ordinary differential equations, Topologie et équations Fonctionnelles (French), Ann. Sci. École Norm. Sup., 51 (1934), 45-78. doi: 10.24033/asens.836. Google Scholar

[26]

J. Mawhin, Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1997), 179-200. doi: 10.12775/TMNA.1997.008. Google Scholar

[27]

J. MawhinC. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic potentials, Commun. Contemp. Math., 2 (2000), 87-126. doi: 10.1142/S0219199700000074. Google Scholar

[28]

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91. doi: 10.1515/ans-2004-0105. Google Scholar

[29]

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

[30]

A. J. Ureña, A counterexample for singular equations with indefinite weight, Adv. Nonlinear Stud., 17 (2017), 497-516. doi: 10.1515/ans-2016-6017. Google Scholar

[31]

P. Waltman, A counterexample for singular equations with indefinite weight, J. Math. Anal. Appl., 10 (1965), 439-441. doi: 10.1016/0022-247X(65)90138-1. Google Scholar

Figure 1.  Open balls ${\frak{B}}_i$ and connected sets ${\frak{C}}_i$ for i = 1, 2
Figure 2.  A numerical approximation to a closed orbit of the equation (3) for β = 0.01, with h(t) = sin99 t
Figure 3.  A numerical approximation to a closed orbit of the equation (3) for β = 0.01, with h(t) = sin99 t
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