The Largest Class of Hereditary Systems Defining a <i>C</i>₀ 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 C 0 on the product space M p = R n × L p (-h, 0), 1 ≦ p &lt; ∞ , 0 &lt; h ≦ + ∞ ( 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] ⌒ R → R n 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 &lt; + ∞ ) 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) ∊ R n , x:[–h, 0] ⌒ R → R n is defined as x t (θ) = x(t + θ) , C(–h, 0) is the space of bounded continuous functions [– h , 0] ⌒ R → R n and L:C (– h , 0) → R n is a continuous linear map.