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

Abstract

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 C0 on the product space Mp = Rn × Lp(-h, 0), 1 ≦ p < ∞, 0 < h ≦ + ∞ (R is the field of real numbers and Lp(– h, 0) is the space of equivalence classes of Lebesgue measurable maps x:[ – h, 0] ⌒ R → Rn which are p-integrable in [ –h, 0] ⌒R.) Our results extend and complete those of [4] and [15], [16] for linear hereditary differential equations possessing “finite memory” (h < + ∞ ) and those of [14], [5] and [6] in the “infinite memory case (h = + ∞ )”.Consider the autonomous linear hereditary differential equation(1.1)where x(t) ∊ Rn, x:[–h, 0] ⌒ R → Rn is defined as xt(θ) = x(t + θ), C(–h, 0) is the space of bounded continuous functions [–h, 0] ⌒ R → Rn and L:C(–h, 0) → Rn is a continuous linear map.