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: |
[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.
![]() ![]() |