where A : D(A) [subset or equal to] X [right arrow] X is the infinitesimal generator of

strongly continuous semigroup of bounded linear operators T(t), t > 0.

[{[B.sub.t]}.sub.t[greater than or equal to]0] is a

strongly continuous semigroup. Proof.

One can verify that the family [P.sup.w] = [([P.sup.w.sub.t]).sub.t[greater than or equal to] 0] forms a

strongly continuous semigroup of contraction operators on [L.sup.2]([OMEGA]).

Denote by [DELTA] = [[summation].sup.m.sub.j=1]([[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.j]) the Laplace operator, with domain D([DELTA]) = [W.sup.1,2.sub.0](Y) [intersection] [W.sup.2,2.sub.0](Y), which generates a

strongly continuous semigroup [mathematical expression not reproducible], where [W.sup.1,2.sub.0](Y) and [W.sup.2,2.sub.0](Y) are the Sobolev spaces with compactly supported sets.

where x(*) the state of the system (1.1) takes values in a Banach space X (state space), the unbounded operator (A,D(A)) generates a [C.sub.0]-semigroup (

strongly continuous semigroup) [(T(t)).sub.t[greater than or equal to]0] on X, B is unbounded linear control operators in the sense that it is a bounded linear operator from X to a larger Banach space V [contains] X, i.e.

Motivated by the works of Mustari and Taylor [15,17], we have recently begun to study the mean ergodic theorem under the framework of RN modules [18] to obtain the mean ergodic theorem in the sense of convergence in probability, where we proved the mean ergodic theorem for a

strongly continuous semigroup of random unitary operators defined on complete random inner product modules (briefly, complete RIP modules).

Then A generates a

strongly continuous semigroup that is analytic, and resolvent operator R(t) can be extracted from this analytic semigroup(see [17]).

where A:D(A) [subset] X [right arrow] X is the infinitesimal generator of a

strongly continuous semigroup [{T(t)}.sub.t[greater than or equal to]0], and B: X [right arrow] [??](H, X) is a bounded linear operator.

i) The two-parameter (nonlinear) semiflow [sigma] is continuous if and only if it induces a

strongly continuous semigroup [{T(t)}.sub.t[greater than or equal to]0] on X by the formula:

where A generates a

strongly continuous semigroup ([e.sup.tA])t [greater than or equal to] 0 on an infinite dimensional Banach space Y (state space) whose norm will be denoted by [parallel].[parallel], B : D(B)([subset] Y) [arrow right] Y is a possibly unbounded linear operator and the control function u(*) denotes the scalar control.

After a short introduction to the theory of

strongly continuous semigroups on linear operators on Banach spaces, the authors introduce abstract delay equations on a Banach space X and discuss examples.

van Casteren, Generators of

Strongly Continuous Semigroups, Research Notes in Math., Pitman Publishing Limited, London, 1985.