Indirect abelian theorems and a linear Volterra equation

where k and xo are real, c _ 0, and a(t) satisfies (H1) a(t) E C(O, oo) n L,(O, 1). a(t) is nonnegative and nonincreasing, limt, , a(t) =0, and O<a(O+)<oo; (H2) a(t) is convex downward; i.e., for 0<8< 1 and 0< tl < t3 <cc, ea(tl) + (1 -e)a(t3) a(t2), where t2 = etl + (1 -e)t3. By a familiar theorem on Volterra equations, (1.1) has a unique solution in C'[O, oo). We define u(t) as the solution of (1.1) with k=O, xo= 1, and we let w(t) = ft u(Q) dJ. It is easily checked that the solution of (1.1) is given by xou(t) + kw(t). Treating (1.1) as a special case of a nonlinear equation, Levin proved in [5] that if a(t) E C [0, oo), a(t) X a(O), and (-1)ka(k)(t) _ O for O < t < oo, k =O, 1, 2, 3, then