Dichotomy and almost automorphic solution of difference system

• Main Authors: Samuel Castillo, Manuel Pinto
• Format: Article
• Language: English
• Published: University of Szeged 2013-06-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: almost automorphic sequences; banach space; massera type theorems
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2211
• ISSN: 1417-3875; 1417-3875

We study almost automorphic solutions of recurrence relations with values in a Banach space $V$ for quasilinear almost automorphic difference systems. Its linear part is a constant bounded linear operator $\varLambda$ defined on $V$ satisfying an exponential dichotomy. We study the existence of almost automorphic solutions of the non-homogeneous linear difference equation and to quasilinear difference equation. Assuming global Lipschitz type conditions, we obtain Massera type results for these abstract systems. The case where the eigenvalues $\lambda$ verify $\left|\lambda\right|=1$ is also treated. An application to differential equations with piecewise constant argument is given.