We study persistence of asymptotic properties of a semigroup T ( ·) on a Banach space X under small, time-varying, closed perturbations B(t) which are relatively bounded with respect to the generator of T ( ·). More precisely, it is assumed that the orbits T (·)x belong to a so-called homogeneous subspace E of BU C (R+, X ) (e.g., the space of almost periodic functions) and that a Miyadera-type estimate holds with a small constant. Then the mild solutions of the perturbed Cauchy problem u′ (t) = (A + B(t))u(t), u(0) = x, also belong to E. Further results of a somewhat different nature are proved in the case in which A(t) depends on t, too. The theorems are applied to delay equations.
``The asymptotic behaviour of perturbed evolution families'',
CASARINO, VALENTINA;
2002
Abstract
We study persistence of asymptotic properties of a semigroup T ( ·) on a Banach space X under small, time-varying, closed perturbations B(t) which are relatively bounded with respect to the generator of T ( ·). More precisely, it is assumed that the orbits T (·)x belong to a so-called homogeneous subspace E of BU C (R+, X ) (e.g., the space of almost periodic functions) and that a Miyadera-type estimate holds with a small constant. Then the mild solutions of the perturbed Cauchy problem u′ (t) = (A + B(t))u(t), u(0) = x, also belong to E. Further results of a somewhat different nature are proved in the case in which A(t) depends on t, too. The theorems are applied to delay equations.Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
