Abstract
We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.
Similar content being viewed by others
References
Arnaudies, J. M. and Fraysse, H.: Cours de mathématiques: Analyse, tome 2, Dunod, Paris, 1996.
Avantaggiati, A., Bruno, G. and Iannacci, R.: A functional approach to B q-a.p. spaces and L ∞ Fourier expansions, Rend. Math. Appl. (7) 13 (1993), 199-228.
Avantaggiati, A., Bruno, G. and Iannacci, R.: The Hausdorff-Young theorem for almost periodic functions and some applications, Nonlinear Anal. 25(1) (1995), 61-87.
Belley, J.-M., Fournier, G. and Saadi Drissi, K.: Solutions presque-périodiques des équations différentielles du type pendule forcé, Acad. Roy. Belg. Bull. Cl. Sci. (6) 3(10-11) (1992), 173-186.
Berger, M. and Gostiaux, B.: Géometrie différentielle, Armand Colin, Paris, 1972.
Berger, M. S. and Zhang, L.: A new method for large quasiperiodic nonlinear oscillations with fixed frequencies for the nondissipative conservative systems, (I), Comm. Appl. Nonlinear Anal. 2(2) (1995), 79-106.
Berger, M. S. and Zhang, L.: A new method for large quasiperiodic nonlinear oscillations with fixed frequencies for the nondissipative second order conservative systems of second type, Comm. Appl. Nonlinear Anal. 3(1) (1996), 25-49.
Besicovitch, A. S.: Almost Periodic Functions, Cambridge Univ. Press, Cambridge, 1932.
Blot, J.: Une approche variationnelle des orbites quasi-périodiques des systèmes hamiltoniens, Ann. Sci. Math. Québec 13(2) (1989), 7-32.
Blot, J.: Une méthode hilbertienne pour les trajectoires presque-périodiques, Notes C.R. Acad. Sci. Paris Série I 313 (1991), 487-490.
Blot, J.: Almost periodic oscillations of forced second order hamiltonian systems, Ann. Fac. Sci. Toulouse 13(3) (1991), 351-363.
Blot, J.: Almost periodically forced pendulum, Funkc. Ekv. 36(2) (1993), 235-250.
Blot, J.: Oscillations presque-périodiques forcées d'équations d'Euler-Lagrange, Bull. Soc. Math. France 122 (1994), 285-304.
Bourbaki, N.: Topologie générale, Chapitres 5 à 10, Hermann, Paris, 1974.
Broer, A. W., Huitema, G. B. and Sevryuk, M. B.: Quasi-Periodic Motions in Families of Dynamical Systems, Lecture Notes in Math. 1645, Springer-Verlag, Berlin, 1996.
Chou, C. C.: Séries de Fourier et théorie des distributions, French edn, Éditions Scientifiques, Beijing, 1983.
Cieutat, P.: Private communication, 1996.
Corduneanu, C.: Almost Periodic Functions, 2nd English edn, Chelsea, New York, 1989.
Dhombres, J.: Moyennes, In: J.-P. Bertrandias et al. (eds), Espaces de Marcinkiewicz, corrélations, mesures, systèmes dynamiques, Masson, Paris, 1987.
Dieudonné, J.: Éléments d'analyse, tome 2, 2nd rev. French edn, Gauthier-Villars, Paris, 1974.
Dunford, N. and Schwartz, J. T.: Linear Operators, Part I: General Theory, Interscience, New York, 1958.
Halmos, P. R.: Measure Theory, Springer-Verlag, New York, 1974.
Hardy, G. H., Littlewood, J. E. and Polya, G.: Inequalities, Cambridge Univ. Press, Cambridge, 1952.
Hörmander, L. V.: The Analysis of Linear Partial Differential Operators, I, 2nd edn, Springer-Verlag, Berlin, 1990.
Iannacci, R., Bersani, A. M., Dell'acqua, G. and Santucci, P.: Embedding theorems for Sobolev-Besicovitch spaces of almost periodic functions, Z. Anal. Anwendungen 17(2) (1998), 443-457.
Fink, A. M.: Almost Periodic Differential Equations, Lecture Notes in Math. 377, Springer-Verlag, Berlin, 1974.
Krasnoselskii, M. A.: Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, 1964.
Mawhin, J.: Remarques sur les solutions bornées ou presque-périodiques de l'équation du pendule forcé, Proc. Conf. G. Fournier, July 1997.
Mawhin, J.: Bounded and almost periodic solutions of nonlinear differential equations: Variational vs nonvariational approach, Preprint, 1998.
Mawhin, J. and Willem, M.: Critical Points Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.
Nečas, J.: Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967.
Pankov, A. A.: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, English edn, Kluwer Acad. Publ., Dordrecht, 1990.
Percival, I. C.: Variational principles for invariant tori and cantori, Amer. Inst. Phys. Conf. Proc. 57 (1979), 302-310.
Rohlin, V. A. and Fuchs, D. B.: Premier cours de topologie: chapitres géométriques, French edn, Mir, Moscow, 1981.
Samoilenko, A. M.: Elements of the Theory of Multi-Frequency Oscillations, English edn, Kluwer Acad. Publ., Dordrecht, 1991.
Schwartz, L.: Théorie des distributions, Hermann, Paris, 1966.
Stein, E. M. and Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, NJ, 1971.
Vo Khac, K.: Distributions, analyse de Fourier, opérateurs aux dérivées partielles, tome 2, Vuibert, Paris, 1966.
Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-Verlag, New York, 1975.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Blot, J., Pennequin, D. Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations. Acta Applicandae Mathematicae 65, 83–113 (2001). https://doi.org/10.1023/A:1010631520978
Issue Date:
DOI: https://doi.org/10.1023/A:1010631520978