\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Manuel Zamora

    * Corresponding author: Manuel Zamora
The first author is supported by projects UBACyT 20020120100029BA and CONICET PIP11220130100006CO, and the second author was supported by FONDECYT, project no. 11140203.
Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 34C25, 34B18; Secondary: 34B30.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • 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] 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. 
  • 加载中

Figures(4)

SHARE

Article Metrics

HTML views(914) PDF downloads(293) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return