[Sur une classe d'équations d'évolution presque automorphes possédant des solutions compactes presque automorphes]
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.
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.
Accepté le :
Publié le :
@article{CRMATH_2016__354_11_1071_0,
author = {Es-sebbar, Brahim},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.10.001/}
}
TY - JOUR AU - Es-sebbar, Brahim 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2016.10.001/ DO - 10.1016/j.crma.2016.10.001 LA - en ID - CRMATH_2016__354_11_1071_0 ER -
%0 Journal Article %A Es-sebbar, Brahim %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 %U http://www.numdam.org/articles/10.1016/j.crma.2016.10.001/ %R 10.1016/j.crma.2016.10.001 %G en %F CRMATH_2016__354_11_1071_0
Es-sebbar, Brahim. Almost automorphic evolution equations with compact almost automorphic solutions. Comptes Rendus. Mathématique, Tome 354 (2016) no. 11, pp. 1071-1077. doi : 10.1016/j.crma.2016.10.001. http://www.numdam.org/articles/10.1016/j.crma.2016.10.001/
[1] Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, 1971
[2] Applied Functional Analysis, John Wiley & Sons, 2011
[3] Abstrakte fastperiodische Funktionen, Acta Math., Volume 61 (1933) no. 1, pp. 149-184
[4] Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA, Volume 52 (1964) no. 4, pp. 907-910
[5] Almost periodicity of bounded and compact solutions of differential equations, Duke Math. J., Volume 36 (1969), pp. 273-276
[6] Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., Volume 86 (2007) no. 6, pp. 723-733
[7] 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] Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer-Verlag, Berlin, New York, 1974
[9] 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] Asymptotic behavior for two-dimensional, quasi-autonomous, almost-periodic evolution equations, J. Differ. Equ., Volume 66 (1987) no. 1, pp. 62-70
[11] The existence of periodic solutions of systems of differential equations, Duke Math. J., Volume 17 (1950) no. 4, pp. 457-475
[12] Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Springer, 1990
[13] Quasi-periodicità per una equazione operazionale del primo ordine, Rend. Accad. Naz. Lincei, Volume 35 (1963), pp. 152-157
[14] On some almost-periodic functions, Ann. Univ. Ferrara, Volume 14 (1969) no. 1, pp. 29-34
[15] Remarks on differential equations with Bohr–Neugebauer property, J. Math. Anal. Appl., Volume 38 (1972) no. 1, pp. 167-173
[16] Bohr–Neugebauer theorem for operators of finite rank in Hilbert spaces, Not. Am. Math. Soc., Volume 21 (1974) no. 7
[17] Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl. (4), Volume 101 (1974) no. 1, pp. 91-114
Cité par Sources :
