Adjoint semigroup theory for a Volterra integrodifferential system

Abstract

Communicated by Richard Miller, June 23, 1975 I. Introduction. In this note we announce some recent results concerning the semigroup theory for a class of linear Volterra integrodifferential systems. The system under consideration has previously been studied by Barbu and Grossman [2] and Miller [6], via semigroup methods. Although semigroup theory is employed in both of the above mentioned articles, it is important to note that the semigroup constructed in [6] differs greatly from the semigroup constructed in [2]. In particular, Miller is able to obtain certain stability results that do not hold for the semigroup constructed in [2]. However, we show that by an appropriate choice of the state space, Miller's semigroup may be considered as the "adjoint" semigroup (in the sense of Hille and Phillips [5]) to the semigroup constructed by Barbu and Grossman. We shall state the results without proofs. Proofs of the theorems will appear elsewhere (see [3]).

The Largest Class of Hereditary Systems Defining a C0 Semigroup on the Product Space

“…The object of this paper is to characterize the largest class of autonomous linear hereditary differential systems which generates a strongly continuous semigroup of class C 0 on the product space M v = R n X L p (-h, 0), 1 ^ p < oo, 0 < h ^ + oo (R is the field of real numbers and L p ( -h, 0) is the space of equivalence classes of Lebesgue measurable maps x: [ -h, 0] H R -• > R w which are ^-integrable in [ -h, 0] C\ R.) Our results extend and complete those of [4] and [15], [16] for linear hereditary differential equations possessing "finite memory" (h < + oo ) and those of [14], [5] and [6] in the "infinite memory case (h = + oo )".…”

“…where A T ^ 0 is an integer, a > 0 is a finite real, for an n X n matrix X" of functions in L l (-oo, 0). Properties of the adjoint semigroup and its relation to the semigroup of Barbu and Grossman [2] were announced in 1975 in [5]. Detailed proofs were pro-vided in 1976 in [6].…”