Electron. J. Diff. Eqns., Vol. 2003(2003), No. 62, pp. 1-15.

Non-autonomous retarded differential equations: the variation of constants formulas and the asymptotic behaviour

Said Boulite, Lahcen Maniar, & Mohammed Moussi

Abstract:
This paper is devoted to show a variation of constants formula for the operator solution to the non-autonomous retarded differential equation
$$ x'(t)=A(t)x(t)+L(t)x_t+f(t),\quad x_s=\varphi, \quad t\geq s\geq 0, $$
in terms of the inhomogeneous term $f$, which will allow us to study the asymptotic behaviour of this solution. We treat also the existence of fundamental solutions and the stability of semi-linear retarded differential equations.

Submitted November 4, 2002. Published May 29, 2003.
Math Subject Classifications: 47D06, 47A55, 34D05, 34G10.
Key Words: Non-autonomous retarded differential equations, variation of constants formula, Dyson-Phillips series, asymptotic almost periodicity.

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Said Boulite
Department of Mathematics, University Cadi Ayyad
Faculty of Sciences Semlalia
B.P. 2390, 40000 Marrakesh, Morocco
email: sboulite@ucam.ac.ma
Lahcen Maniar
Department of Mathematics, University Cadi Ayyad
Faculty of Sciences Semlalia
B.P. 2390, 40000 Marrakesh, Morocco
email: maniar@ucam.ac.ma
Mohammed Moussi
Department of Mathematics, Faculty of Sciences
University Mohammed I, 60000 Oujda, Morocco
email: moussi@sciences.univ-oujda.ac.ma

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