Positivity 12 (2008), 167–183
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010167-17, published online October 29, 2007
Modulus Semigroups and Perturbation Classes
for Linear Delay Equations in L
Hendrik Vogt and J¨urgen Voigt
Dedicated to the memory of H. H. Schaefer
Abstract. In this paper we study C
-semigroups on X × L
(−h, 0; X)
associated with linear differential equations with delay, where X is a Banach
space. In the case that X is a Banach lattice with order continuous norm, we
describe the associated modulus semigroup, under minimal assumptions on
the delay operator. Moreover, we present a new class of delay operators for
which the delay equation is well-posed for p in a subinterval of [1, ∞).
Mathematical Subject Classiﬁcation (2000). 47D06; 34K06; 47B60.
Keywords. Functional differential equation, delay equation, modulus semigroup,
perturbation theory, domination, Banach lattice.
We treat two topics arising in connection with the Cauchy problem for the linear
u(0) = x, u
with initial values x ∈ X, f ∈ L
(−h, 0; X), where X is a Banach space, 1 p<∞,
and 0 <h ∞. (For a function u :(−h, ∞) → X, we recall the notation
(θ):=u(t + θ)(−h<θ<0),
for t 0.) The foundations for treating this problem in the context of C
semigroups on X × L
(−h, 0; X) have been presented in ; we also refer to .
One of the topics concerns the question of the kind of operators L that are
allowed in (DE). In the previous papers it was assumed that L is associated with
a function η :[−h, 0] →L(X) of bounded variation (cf. Example 5.1). Then the
problem (DE) could be treated in X ×L
(−h, 0; X) for any p ∈ [1, ∞). We present