Almost-periodic solutions in various metrics of higher-order differential equations with a nonlinear restoring term

Andres, Jan   ; Bersani, Alberto Maria   ; Radová, Lenka

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 45 (2006) no. 1, pp. 7-29

Abstract (VO)

Almost-periodic solutions in various metrics (Stepanov, Weyl, Besicovitch) of higher-order differential equations with a nonlinear Lipschitz-continuous restoring term are investigated. The main emphasis is focused on a Lipschitz constant which is the same as for uniformly almost-periodic solutions treated in [A1] and much better than those from our investigations for differential systems in [A2], [A3], [AB], [ABL], [AK]. The upper estimates of $\varepsilon $ for $\varepsilon $-almost-periods of solutions and their derivatives are also deduced under various restrictions imposed on the constant coefficients of the linear differential operator on the left-hand side of the given equation. Besides the existence, uniqueness and localization of almost-periodic solutions and their derivatives are established.

Détail

Classification : 34C15, 34C27, 42A75
Keywords: Almost-periodic solutions; various metrics; higher-order differential equation; nonlinear restoring term; existence and uniqueness criteria

Bibliographie

[A1] Andres J.: Existence of two almost periodic solutions of pendulum-type equations. Nonlin. Anal. 37 (1999), 797–804. | MR | Zbl

[A2] Andres J.: Almost-periodic and bounded solutions of Carathéodory differential inclusions. Differential Integral Eqns 12, (1999), 887–912. | MR | Zbl

[A3] Andres J.: Bounded, almost-periodic and periodic solutions of quasi-linear differential inclusions. Lecture Notes in Nonlinear Anal. 2, (J. Andres, L. Górniewicz and P. Nistri, eds.), N. Copernicus Univ., Toruń, 1998, 35–50. | Zbl

[AB] Andres J., Bersani A. M.: Almost-periodicity problem as a fixed-point problem for evolution inclusions. Topol. Meth. Nonlin. Anal. 18 (2001), 337–350. | MR | Zbl

[ABG] Andres J., Bersani A. M., Grande R. F.: Hierarchy of almost-periodic function spaces. Rendiconti Mat. Appl. Ser. VII, 26, 2 (2006), 121–188. | MR | Zbl

[ABL] Andres J., Bersani A. M., Leśniak K.: On some almost-periodicity problems in various metrics. Acta Appl. Math. 65, 1-3 (2001), 35–57. | MR | Zbl

[AG] Andres J., Górniewicz L.: Topological Fixed Point Principles for Boundary Value Problems. : Kluwer, Dordrecht. 2003. | MR

[AK] Andres J., Krajc B.: Unified approach to bounded, periodic and almost periodic solutions of differential systems. Ann. Math. Sil. 11 (1997), 39–53. | MR | Zbl

[BFSD1] Belley J. M., Fournier G., Saadi Drissi K.: Almost periodic weak solutions to forced pendulum type equations without friction. Aequationes Math. 44 (1992), 100–108. | MR | Zbl

[BFSD2] Belley J. M., Fournier G., Saadi Drissi K.: Solutions faibles presque périodiques d’équation différentialle du type du pendule forcé. Acad. Roy. Belg. Bull. Cl. Sci. 6, 3 (1992), 173–186. | MR

[BFSD3] Belley J. M., Fournier G., Saadi Drissi K.: Solutions presque périodiques du systéme différential du type du pendule forcé. Acad. Roy. Belg. Bull. Cl. Sci. 6, 3 (1992), 265–278.

[BFH] Belley J. M., Fournier G., Hayes J.: Existence of almost periodic weak type solutions for the conservative forced perdulum equation. J. Diff. Eqns 124, (1996), 205–224. | MR

[D1] Danilov L. I.: Almost periodic solutions of multivalued maps. Izv. Otdela Mat. Inform. Udmurtsk. Gos. Univ. 1 (1993), Izhevsk, 16–78 (in Russian).

[D2] Danilov L. I.: Measure-valued almost periodic functions and almost periodic selections of multivalued maps. Mat. Sb. 188 (1997), 3–24 (in Russian); Sbornik: Mathematics 188 (1997), 1417–1438. | MR | Zbl

[D3] Danilov L. I.: On Weyl almost periodic solutions of multivalued maps. J. Math. Anal. Appl. 316, 1 (2006), 110–127. | MR

[DHS] Deimling K., Hetzer G., Wenxian Shen: Almost periodicity enforced by Coulomb friction. Advances Diff. Eqns 1, 2 (1996), 265–281. | MR

[DM] Dzurnak A., Mingarelli A. B.: Sturm-Liouville equations with Besicovitch almost periodicity. Proceed. Amer. Math. Soc. 106, 3 (1989), 647–653. | MR

[DS] Dolbilov A. M., Shneiberg I. Ya.: Almost periodic multifunctions and their selections. Sibirsk. Mat. Zh. 32 (1991), 172–175 (in Russian). | MR

[H] Haraux A.: Asymptotic behavior for two-dimensional, quasi-autonomous, almost-periodic evolution equations. J. Diff. Eqns 66 (1987), 62–70. | MR | Zbl

[HP] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis, Volume I: Theory. : Kluwer, Dordrecht. 1997. | MR

[Kh] Kharasakhal V. Kh.: Almost-Periodic Solutions of Ordinary Differential Equations. : Nauka, Alma-Ata. 1970 (in Russian). | MR

[KBK] Krasnosel’skii M. A., Burd V. Sh., Kolesov, Yu. S.: Nonlinear Almost Periodic Oscillations. : Nauka, Moscow. 1970 (in Russian); English translation: J. Wiley, New York, 1971. | MR

[Ku] Kunze M.: Non-Smooth Dynamical Systems. : Lect. Notes Math., Vol. 1744, Springer, Berlin. 2000. | MR

[L] Levitan B. M.: Almost Periodic Functions. : GITTL, Moscow. 1953 (in Russian). | MR

[LZ] Levitan B. M., Zhikov V. V.: Almost Periodic Functions, Differential Equations. : Cambridge Univ. Press, Cambridge. 1982. | MR

[P] Pankov A. A.: Bounded, Almost Periodic Solutions of Nonlinear Operator Differential Equations. : Kluwer, Dordrecht. 1990. | MR

[R] Radová L.: Theorems of Bohr–Neugebauer-type for almost-periodic differential equations. Math. Slovaca 54 (2004), 191–207. | MR | Zbl

[ZL] Zhikov V. V., Levitan B. M.: The Favard theory. Uspekhi Matem. Nauk. 32 (1977), 123–171 (in Russian); Russian Math. Surv. 32 (1977), 129–180. | MR