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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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 |
TWe prove the existence of
| $T$ |
| $ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = h(t)g(u), $ |
where the non-linear term
| $g$ |
| $h$ |
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 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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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 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 $u(a_i')\geq \alpha+2\delta$ .
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