A

strongly continuous vector function y : I [right arrow] X is called a weak solution of the evolution equation

(ii) Let [([R.sub.1](t)).sub.t[member of][0,[tau])] [subset or equal to] L(Y,X) be

strongly continuous. Then it is said that A is a subgenerator of a (local, if [tau] < [infinity]) mild (a, k)-regularized [C.sub.1] -existence family [([R.sub.1](t)).sub.t[member of][0,[tau])] iff (25) holds.

Lately, there has been an increasing interest in the study of the exponential dichotomy of the exponentially bounded,

strongly continuous cocycles over continuous flows on a locally compact metric space [THETA] and acting on a Banach space X (see for instance [7]).

A is the infinitesimal generator of a

strongly continuous cosine family of bounded linear operators [(C(t)).sub.t[member of]R] on X.

where A generates a

strongly continuous, not necessarily compact, semigroup [(T(t)).sub.t[greater than or equal to]0] in the Banach space x.

The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a

strongly continuous unitary flow.

[3] if A is a closed linear operator with domain D(A) defined on a Banach space E and [alpha] > 0, the we say that A is the generator of an [alpha]-resolvent family if there exists [omega] [greater than or equal to] 0 and a

strongly continuous function [S.sub.[alpha]] : [R.sub.+] [right arrow]L(E) such that {[[lambda].sub.[alpha]] : Re([lambda]) > [omega]} [subset] [rho](A)) ([rho](A) being the resolvent set of A) and

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).

The derivative [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and it is

strongly continuous on 0 [less than or equal to] [theta] [less than or equal to] t [less than or equal to] T.

Among his topics are

strongly continuous semigroups, compatible and regular systems, stabilization and detection, and time discrete systems.

So, X is weakly compact and, at the same time, [PHI] is weakly continuous, being affine and

strongly continuous.

Levine (1960) introduced

strongly continuous functions in the widely read Classroom Notes section of the American Mathematical Monthly.