Quasi-periodicity of bounded solutions to some periodic evolution equations

Alain HARAUX; Mitsuharu ÔTANI

doi:10.2969/jmsj/04220277

Alain HARAUX, Mitsuharu ÔTANI

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JOURNAL FREE ACCESS

1990 Volume 42 Issue 2 Pages 277-294

DOI https://doi.org/10.2969/jmsj/04220277

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Published: 1990 Received: March 27, 1989 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Hiroshi Fujita on the occasion of his sixtieth birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Analyse Numérique, T. 55-65, 5^ème étage Université Pierre et Marie Curie
2) Department of Mathematics Faculty of Science Tokai University

Right :
1) Analyse Numérique, 5^ème étage Université Pierre et Marie Curie
2) Department of Mathematics Faculty of Science Tokai University

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, J. Nonlinear Analysis, 3 (1978), 329-353.
2) V. Barbu, Weak solutions for nonlinear functional equations in Banach spaces, Ann. Mat. Pura. Appl., 87-110 (1970).
3) P.M. Batchelder, An Introduction to Linear Difference Equations, Dover, New-York, 1967.
4) P. Benilan and H. Brezis, Solutions faibles d'équations d'évolution dans les espaces de Hilbert, Ann. Inst. Fourier, 22 (1972), 311-329.
5) H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Lecture Notes, 5, North-Holland, Amsterdam, London, 1972.
6) M.G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94.
7) C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Funct. Anal., 12 (1973), 97-103.
8) A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer, 1974.
9) A. Haraux, Equations d'évolution non linéaires: solutions bornées et périodiques, Ann. Inst. Fourier, 28 (1978), 202-220.
10) A. Haraux, Behavior at infinity for dissipative systems with forcing term in Hilbert space, in “Trends in applications of pure Mathematics to Mechanics”. 3 (ed. R.J. Knops), Pitman Pub. Ltd., 1980, pp. 120-125.
11) A. Haraux, Asymptotic behavior of trajectories for some non autonomous, almost periodic processes, J. Diff. Eq., 49 (1983), 473-483.
12) A. Haraux, A simple almost periodicity criterion and applications, J. Diff. Eq., 66 (1987), 51-61.
13) A. Haraux, Asymptotic behavior for two-dimensional, quasi-autonmous, almost periodic evolution equations, J. Diff. Eq., 66 (1987), 62-70.
14) N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Ed. Chiba Univ., 30 (1981), 1-87.
15) N. Kenmochi and M. Otani, Instability of periodic solutions of some evolution equations governed by time-dependent subdifferential operators, Proc. Japan Acad., 61 (1985), 4-7.
16) N. Kenmochi and M. Ôtani, Asymptotic behavior of periodic systems generated by time-dependent subdifferential operators, Funkcial. Ekvac., 29 (1986), 219-236.
17) R. Martin, Generating an evolution system in a class of uniformly convex Banach spaces, J. Funct. Anal., 11 (1972), 62-76.
18) K.S. Miller, Linear Difference Equations, Mathematics monographs series, Benjamin, New-York, Amsterdam, 1968.
19) N.E. Nördlund, Leçons sur les Équations Linéaires aux Différences Finies, Gauthier-Villars, Paris, 1929.
20) Y. Yamada, On nonlinear evolution equations generated by the subdifferentials, J. Fac. Sci. Univ. Tokyo, 23(1976), 491-515.

Right : [1] H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, J. Nonlinear Analysis, 3 (1978), 329-353.
[2] V. Barbu, Weak solutions for nonlinear functional equations in Banach spaces, Ann. Mat. Pura. Appl., 87-110 (1970).
[3] P. M. Batchelder, An Introduction to Linear Difference Equations, Dover, New-York, 1967.
[4] P. Benilan and H. Brezis, Solutions faibles d'équations d'évolution dans les espaces de Hilbert, Ann. Inst. Fourier, 22 (1972), 311-329.
[5] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Lecture Notes, 5, North-Holland, Amsterdam, London, 1972.
[6] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94.
[7] C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Funct. Anal., 12 (1973), 97-103.
[8] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer, 1974.
[9] A. Haraux, Equations d'évolution non linéaires: solutions bornées et périodiques, Ann. Inst. Fourier, 28 (1978), 202-220.
[10] A. Haraux, Behavior at infinity for dissipative systems with forcing term in Hilbert space, in “Trends in applications of pure Mathematics to Mechanics”. 3 (ed. R.J. Knops), Pitman Pub. Ltd., 1980, pp. 120-125.
[11] A. Haraux, Asymptotic behavior of trajectories for some non autonomous, almost periodic processes, J. Diff. Eq., 49 (1983), 473-483.
[12] A. Haraux, A simple almost periodicity criterion and applications, J. Diff. Eq., 66 (1987), 51-61.
[13] A. Haraux, Asymptotic behavior for two-dimensional, quasi-autonmous, almost periodic evolution equations, J. Diff. Eq., 66 (1987), 62-70.
[14] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Ed. Chiba Univ., 30 (1981), 1-87.
[15] N. Kenmochi and M. Otani, Instability of periodic solutions of some evolution equations governed by time-dependent subdifferential operators, Proc. Japan Acad., 61 (1985), 4-7.
[16] N. Kenmochi and M. Ôtani, Asymptotic behavior of periodic systems generated by time-dependent subdifferential operators, Funkcial. Ekvac., 29 (1986), 219-236.
[17] R. Martin, Generating an evolution system in a class of uniformly convex Banach spaces, J. Funct. Anal., 11 (1972), 62-76.
[18] K. S. Miller, Linear Difference Equations, Mathematics monographs series, Benjamin, New-York, Amsterdam, 1968.
[19] N. E. Nördlund, Leçons sur les Équations Linéaires aux Différences Finies, Gauthier-Villars, Paris, 1929.
[20] Y. Yamada, On nonlinear evolution equations generated by the subdifferentials, J. Fac. Sci. Univ. Tokyo, 23(1976), 491-515.

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