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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1071-1077 (novembre 2016)
Doi : 10.1016/j.crma.2016.10.001
Received : 11 August 2016 ;  accepted : 3 October 2016
Almost automorphic evolution equations with compact almost automorphic solutions
Sur une classe d'équations d'évolution presque automorphes possédant des solutions compactes presque automorphes
 

Brahim Es-sebbar
 Université Cadi-Ayyad, Faculté des sciences Semlalia, Département de mathématiques, BP 2390, Marrakesh, Morocco 

Abstract

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.

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Résumé

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.

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