Comptes Rendus
Ordinary differential equations/Partial differential equations
Almost automorphic evolution equations with compact almost automorphic solutions
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1071-1077.

We prove that some almost automorphic evolution equations carry compact almost automorphic solutions. Moreover, we show that the almost automorphy of the coefficients is not necessary to obtain almost automorphic solutions. This improves the assumptions and the conclusion of a result of M. Zaki (Ann. Mat. Pura Appl. (4) 101 (1) (1974) 91–114), which gives the nature of solutions with relatively compact range for some almost automorphic evolution equations in Banach spaces. We note that many results in the literature can be improved in this direction.

Nous montrons que certaines équations d'évolution presque automorphes possèdent des solutions compactes presque automorphes. De plus, nous montrons que la presque automorphie des coefficients n'est pas nécessaire pour obtenir des solutions presque automorphes. Cela améliore les hypothèses et la conclusion d'un résultat de M. Zaki (Ann. Mat. Pura Appl. (4) 101 (1) (1974) 91–114), qui donne la nature des solutions avec image relativement compacte pour certaines équations d'évolution presque automorphes dans les espaces de Banach. Nous notons que de nombreux résultats dans la littérature peuvent être améliorés dans cette direction.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.10.001

Brahim Es-sebbar 1

1 Université Cadi-Ayyad, Faculté des sciences Semlalia, Département de mathématiques, BP 2390, Marrakesh, Morocco
@article{CRMATH_2016__354_11_1071_0,
     author = {Brahim Es-sebbar},
     title = {Almost automorphic evolution equations with compact almost automorphic solutions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1071--1077},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.10.001},
     language = {en},
}
TY  - JOUR
AU  - Brahim Es-sebbar
TI  - Almost automorphic evolution equations with compact almost automorphic solutions
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1071
EP  - 1077
VL  - 354
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crma.2016.10.001
LA  - en
ID  - CRMATH_2016__354_11_1071_0
ER  - 
%0 Journal Article
%A Brahim Es-sebbar
%T Almost automorphic evolution equations with compact almost automorphic solutions
%J Comptes Rendus. Mathématique
%D 2016
%P 1071-1077
%V 354
%N 11
%I Elsevier
%R 10.1016/j.crma.2016.10.001
%G en
%F CRMATH_2016__354_11_1071_0
Brahim Es-sebbar. Almost automorphic evolution equations with compact almost automorphic solutions. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1071-1077. doi : 10.1016/j.crma.2016.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.001/

[1] L. Amerio; G. Prouse Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, 1971

[2] J.P. Aubin Applied Functional Analysis, John Wiley & Sons, 2011

[3] S. Bochner Abstrakte fastperiodische Funktionen, Acta Math., Volume 61 (1933) no. 1, pp. 149-184

[4] S. Bochner Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA, Volume 52 (1964) no. 4, pp. 907-910

[5] R. Cooke Almost periodicity of bounded and compact solutions of differential equations, Duke Math. J., Volume 36 (1969), pp. 273-276

[6] T. Diagana; G.M. N'Guérékata Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., Volume 86 (2007) no. 6, pp. 723-733

[7] B. Es-sebbar; K. Ezzinbi Almost periodicity and almost automorphy for some evolution equations using Favard's theory in uniformly convex Banach spaces, Semigroup Forum (2016) (in press) | DOI

[8] A. Fink Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer-Verlag, Berlin, New York, 1974

[9] J.A. Goldstein Convexity, boundedness, and almost periodicity for differential equations in Hillbert space, Int. J. Math. Math. Sci., Volume 2 (1979) no. 1, pp. 1-13

[10] A. Haraux Asymptotic behavior for two-dimensional, quasi-autonomous, almost-periodic evolution equations, J. Differ. Equ., Volume 66 (1987) no. 1, pp. 62-70

[11] J.L. Massera The existence of periodic solutions of systems of differential equations, Duke Math. J., Volume 17 (1950) no. 4, pp. 457-475

[12] A.A. Pankov Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Springer, 1990

[13] S. Zaidman Quasi-periodicità per una equazione operazionale del primo ordine, Rend. Accad. Naz. Lincei, Volume 35 (1963), pp. 152-157

[14] S. Zaidman On some almost-periodic functions, Ann. Univ. Ferrara, Volume 14 (1969) no. 1, pp. 29-34

[15] S. Zaidman Remarks on differential equations with Bohr–Neugebauer property, J. Math. Anal. Appl., Volume 38 (1972) no. 1, pp. 167-173

[16] S. Zaidman Bohr–Neugebauer theorem for operators of finite rank in Hilbert spaces, Not. Am. Math. Soc., Volume 21 (1974) no. 7

[17] M. Zaki Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl. (4), Volume 101 (1974) no. 1, pp. 91-114

Cited by Sources:

Comments - Policy