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Stepanov-like C(n)-pseudo almost automorphy and applications to some nonautonomous higher-order differential equations

Diagana T., Nelson V., N'Guerekata G. M.

Opuscula Mathematica

2012

Vol. 32, no. 3

455-471

In this paper we introduce and study a new concept called Stepanov-like C(n)-pseudo almost automorphy, which generalizes in a natural fashion both the notions of C(n)-pseudo almost periodicity and that of C(n)-pseudo almost automorphy recently introduced in the literature by the authors. Basic properties of these new functions are investigated. Furthermore, we study and obtain the existence of C(N+m)-pseudo almost automorphic solutions to some nonautonomous higher-order systems of differential equations with Stepanov-like C(m)-pseudo almost automorphic coefficients.

On some perturbations of some abstract differential equations

Diagana T., Nguerekata G. M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2003

[Z] 43(2)

201-206

We are concerned with the existence of almost automorphic mild solutions to the perturbed abstract differential equations of the form x(t) = (A + B)x(t) (*), where A is a unbounded linear operator generator of a Co-group of bounded operators and B is a bounded linear operator acting in a Banach space X. Under suitable conditions we show that every bounded mild solution to (*) is almost automorphic. Meanwhile under a relatively compactness hypothesis of the range of a solution, we show that such a solution is also almost periodic.