Research ArticleOpen AccessGoogle Scholar indexed
Exponential Dichotomy and Eberlein-Weak Almost Periodic Solutions
Départment de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakesh, Morocco
Laboratoire L. M. C.; Départment de Mathématiques et Informatique, Faculté Polydisciplinaire-Safi (FPS)Université Cadi Ayyad, Safi, Morocco
Départment de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakesh, Morocco
- 1 Départment de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakesh, Morocco
- 2 Laboratoire L. M. C.; Départment de Mathématiques et Informatique, Faculté Polydisciplinaire-Safi (FPS)Université Cadi Ayyad, Safi, Morocco
- 3 Départment de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakesh, Morocco
Applied Mathematics·Volume 03 (2012)·Pages 969–975·Published 27 September 2012·DOI10.4236/am.2012.39144
Copy link · social · email
Abstract
We give sufficient conditions ensuring the existence and uniqueness of an Eberlein-weakly almost periodic solution to the following linear equation dx/dt(t) = A(t)x(t) + f(t) in a Banach space X, where (A(t)) t ∈□ is a family of infinitesimal generators such that for all t ∈□, A(t + T) = A(t) for some T > 0, for which the homogeneuous linear equation dx/dt(t) = A(t)x(t) is well posed, stable and has an exponential dichotomy, and f:□ →X is Eberlein-weakly amost periodic.
KeywordsBounded SolutionsAlmost Periodic and Eberlein Weak Almost Periodic FunctionsExponential DichotomyLinear Differential Equations
- C. Corduneanu, “Almost Periodic Functions,” Wiley, New York, 1968.
- A. M. Fink, “Almost Periodic Differential Equations,” Springer-Verlag, New York, 1974.
- C. Zhang, “Almost Periodic Type Functions and Ergodicity,” Science Press, Beijing; Kluwer Academic Publishers, Dordrecht, 2003.
- E. Ait Dads, “Contribution à l’existence de Solutions Presque Périodiques d'une équation Fonctionnelle non Linéaire,” Thèse d’Etat, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakech, 1994.
- E. Ait Dads and K. Ezzinbi, “Existence of Positive Pseudo-Almost-Periodic Solutions for Some Nonlinear Infinite Delay Integral Equations Arising in Epedimic Problems,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 41, No. 1-2, 2002, pp. 1-13.
- E. Ait Dads and K. Ezzinbi, “Pseudo-Almost-Periodic Solutions for Some Delay Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 201, No. 3, 1996, pp. 840-850. doi:10.1006/jmaa.1996.0287
- W. F. Eberlein, “Eberlein Weak Almost Periodicity and Differential Equations in Banach Spaes,” Ph.D. Thesis, Universitat Essen, Germany, 1992.
- W. M. Ruess and W. H. Summers, “Weak Almost Periodicity and the Strongly Ergodic Limit Theorem for Conraction Semigroups,” Israel Journal of Mathematics, Vol. 64, 1988, pp. 139-157.
- W. M. Ruess and W. H. Summers, “Weak Almost Periodicity and the Strongly Ergodic Limit Theorem for Periodic Evolution Systems,” Journal of Functional Analysis, Vol. 94, No. 1, 1990, pp. 177-195. doi:10.1016/0022-1236(90)90033-H
- W. M. Ruess and W. H. Summers, “Weak Almost Periodic Semigroups of Operators,” Pacific Journal of Mathematics, Vol. 143, 1990, pp. 175-193.
- W. M. Ruess; W. H. Summers, “Integration of Asymptotically Almost Periodic Functions and Weak Asymptotic Almost Periodicity,” Dissertationes Mathematicae, Vol. 279, 1989.
- J. Kreulich, “Weakly Almost-Periodic Solutions of Evolution Equations in Banach Spaces,” Differential Integral Equations, Vol. 5, No. 9, 2011, pp. 1005-1027.
- J. Kreulich, “Eberlein-Weakly Almost-Periodicity Sand Differential Equations in Banach Spaces,” Ph.D. Thesis, University Essen, Germany, 1992.
- E. Ait Dads; K. Ezzinbi and S. Fatajou, “Weakly Almost Periodic Solutions for Some Differential Equations in a Banach Space,” Nonlinear Studies, Vol. 4, No. 2, 1997, pp. 157-170.