On Nonautonomous Functional Differential Equations

Abstract

In this paper, we prove a theorem on boundary perturbation of nonautonomous Cauchy problerris arid then apply this result to show the existence and uniqueness of classical solutions of the nonautonomous, Banach space valued functional differential equation0 1999 Academic Press where each A(t) is a linear operator on X . Supposing dense domains D ( A ( t ) ) , one can characterize well-posedness of ( wlCP) by the following definition (see 191). DEFINITION 1.1. The problem ( d C P ) is called wellposed if there exists a unique, strongly continuous evolutionfamily {U(t, s)}, ~ ,~ ~ ~ of bounded, NONAUTONOMOUS FDE'S 159 linear operators on X satisfying 1. U(t, t ) = Id for all 0 I t I T , 2. U(t, s)U(s, r ) = U(t, r ) for all 0 I r I s I t I T , 3. ( t , s) -U(t, s) is strongly continuous, and for every s E [0, TI there exists a dense subspace Y, L X satisfying U(t, 3)x L D ( A ( t ) ) for every 0 I s I t I T such that d -U ( t , S ) X = A ( t ) U ( T , S ) X d t for all x E Y,.

Local manifolds for non-autonomous boundary Cauchy problems: existence and attractivity

“…It was shown in [6,Theorem 2.3] that under the assumptions (H2)-(H5) there exists an evolution family (U (t, s)) t s∈J generated by (A(t)) t∈J satisfying…”

“…The well-posedness of the linear boundary Cauchy problem (2) has been studied in [5] and [6]. In these works, the authors have shown the existence of an evolution family solution to this problem.…”

Non - Autonomous Inhomogeneous Boundary Cauchy Problems and Retarded Equations

“…Note that under the above hypotheses, the Cauchy problem associated to the family A 0 (•) is well-posed and its solutions are given by an evolution family (U (t, s)) 0≤s≤t≤T , see [7].…”

“…He has also showed the existence of classical solutions of (IBCP ) via a variation of constants formula. In the non-autonomous case, Kellermann [6] and Nguyen Lan [7] have showed the existence of an evolution family (U (t, s)) t≥s≥0 which provides the classical solution of the homogeneous boundary Cauchy problem. The aim of this paper is to show the well-posedness of the inhomogeneous problem (IBCP).…”