Subexponential solutions of scalar linear integro-differential equations with delay

This paper considers the asymptotic behaviour of solutions of the scalar linear convolution integro-differential equation with delay x0(t) = − n Xi=1 aix(t − i) + Z t 0 k(t − s)x(s) ds, t > 0, x(t) = (t), − t 0, where = max1in i. In this problem, k is a non-negative function in L1(0,1)\C[0,1), i 0, ai > 0 and is a continuous function on [−, 0]. The kernel k is subexponential in the sense that limt!1 k(t)(t)−1 > 0 where is a positive subexponential function. A consequence of this is that k(t)et ! 1 as t ! 1 for every > 0