Opuscula Math. 32, no. 3 (2012), 455-471
http://dx.doi.org/10.7494/OpMath.2012.32.3.455

Opuscula Mathematica

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

Toka Diagana
Valerie Nelson
Gaston M. N'Guérékata

Abstract. 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.

Keywords: pseudo almost automorphic $$C^{(n)}$$-pseudo almost automorphy, Stepanov-like $$C^{(n)}$$-pseudo almost automorphy, exponential dichotomy.

Mathematics Subject Classification: 35B15, 34D09, 58D25, 42A75, 37L05.

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• Toka Diagana
• Howard University ,Department of Mathematics, 2441 6th Street, N.W. Washington, D.C. 20059, USA
• Valerie Nelson
• Howard University ,Department of Mathematics, 2441 6th Street, N.W. Washington, D.C. 20059, USA
• Gaston M. N'Guérékata
• Morgan State University, Department of Mathematics, Baltimore, MD 21251, USA
• Revised: 2011-11-28.
• Accepted: 2011-11-28.