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Topological Methods in Nonlinear Analysis

Existence, localization and stability of limit-periodic solutions to differential equations involving cubic nonlinearities
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Existence, localization and stability of limit-periodic solutions to differential equations involving cubic nonlinearities

Authors

  • Jan Andres
  • Denis Pennequin https://orcid.org/0000-0002-5557-0006

Keywords

Limit-periodic solutions, differential equations, cubic nonlinearity, existence of solutions, localization, (in)stability, essentiality

Abstract

We will prove, besides other things like localization and (in)stability, that the differential equations $x'+x^3-\lambda x=\varepsilon r(t)$, $\lambda> 0$, and $x''+x^3-x=\varepsilon r(t)$, where $r\colon\mathbb{R}\to\mathbb{R}$ are uniformly limit-periodic functions, possess for sufficiently small values of $\varepsilon > 0$ uniformly limit-periodic solutions, provided $r$ in the first-order equation is strictly positive. As far as we know, these are the first nontrivial effective criteria, obtained for limit-periodic solutions of nonlinear differential equations, in the lack of global lipschitzianity restrictions. A simple illustrative example is also indicated for difference equations.

References

S. Ahmad and A. Tineo, Almost periodic solutions of second order systems, Appl. Anal. 63 (1996), no. 3–4 389–395.

A.I. Alonso, R. Obaya and R. Ortega, Differential equations with limit-periodic forcings, Proc. Amer. Math. Soc. 131 (2003), no. 3, 851–857.

J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003.

J. Andres and L. Górniewicz, On essential fixed points of compact mappings on arbitrary absolute neighborhood retracts and their application to multivalued fractals, Int. J. Bifurc. Chaos 26 (2016), no. 3, 1–11.

J. Andre, and D. Pennequin, Semi-periodic solutions of difference and differential equations, Bound. Value Probl. 2012 (2012), no. 141, 1–16.

J. Andres and D. Pennequin, Limit-periodic solutions of difference and differential systems without global lipschitzianity restrictions, J. Difference Equ. Appl. 2018 (2018), 1–21.

M.S. Berger and Y.Y. Chen, Forced quasiperiodic and almost periodic oscillations of nonlinear duffing equations, Nonlinear Anal. 19 (1992), no. 3, 249–257.

H. Bohr, Zur Theorie der fastperiodischen Funktionen, II Teil: Zusammenhang der fastperiodischen funktionen mit Funktionen von unendlichvielen Variablen; gleichmässige Approximation durch trigonometrische Summen, Acta Math. 46 (1925), no. 1–2, 101–214.

C.C. Conley and R.K. Miller, Asymptotic stability without uniform stability: almost periodic coefficients, J. Differential Equatio ns 1(1965), no. 3, 333–336.

T. Ding, Qualitative Theory of Ordinary Differential Equations. Dynamical Systems and Nonlinear Oscillations, Peking Univ. Ser. Math., vol. 3, World Scientific, Singapore, 2007.

G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung, Sammlung Vieweg, Braunschweig, 1918.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., vol. 42, Springer, Berlin, 1990.

J.K. Hale and H. Koçak, Dynamics and Bifurcations, Springer, Berlin, 1991.

A. Haraux, Stability and multiplicity of periodic or almost periodic solutions to scalar first order ODE, Anal. Appl. 4(2006), no. 3, 237–246.

J. Hong and C. Núñez, The almost periodic type difference equations, Math. Comput. Modelling 28 (1998), no. 12, 21–31.

R.A. Johnson, A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc. 87 (1981), no. 2, 199–205.

B.M. Levitan, Almost-Periodic Functions, G.I.T – T.L., Moscow, 1959. (Russian)

K.J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Report, vol. 1, 1991, pp. 265–306.

G. Seifert, Almost periodic solutions for limit periodic systems, SIAM J. Appl. Math. 22 (1972), no. 1, 38–44.

A. Tineo, First-order ordinary differential equations with several bounded separate solutions, J. Math. Anal. Appl. 225 (1998), no. 2, 359–372.

A. Tineo, A result of Ambrosetti–Prodi type for first-order ODEs with cubic nonlinearities, Part II , Ann. Mat. 182 (2003), no. 2, 129–141.

W. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica (N.S.) 13 (1997), no. 3, 373–380.

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Published

2019-07-13

How to Cite

1.
ANDRES, Jan and PENNEQUIN, Denis. Existence, localization and stability of limit-periodic solutions to differential equations involving cubic nonlinearities. Topological Methods in Nonlinear Analysis. Online. 13 July 2019. Vol. 54, no. 2B, pp. 887 - 906. [Accessed 7 July 2025].
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