INTEGRO-DIFFERENTIAL EQUATIONS 101 2* Existence and uniqueness of a strong solution of the homogeneous problem (I)* Let A be a closed linear operator on a Banach space X to itself with domain &(A) dense in 36 and let @(3£) be the Banach algebra of all bounded linear transformations on X to itself. We choose A such that the resolvent R(X, A) for n = 1, 2, and some real numbers M > 0 and /3 Ξ> 0 satisfies ( 5 ) || i2(λ, A)" || ^ Λf(λ £ ) — for λ > β . By the Hille-Yosida-Phillips theorem [4, Theorem 2.1] this implies that A generates a semi-group of class (Co) of linear operators on the semi-module [0, oo) to ( i ) T(t, + ί8) T(^) Γ(ί2), ίx, t2 e [0, oo), ( 6 ) 0 T(t) and A commute on &r(A) [1, Theorem 10.3.3] and for x e &(A) T(i)x is strongly continuously differentiable in [0, oo) and is the unique solution [1, Corollary to Theorem 23.8.1] of the differential equation dT(t)x/dt = AT(t)x with initial condition T(0)x — x. Instead of this we first investigate the homogeneous integro-differential equation ( 7 ) dU(t)x/dt = AU(t)x + [B(t s)U(s)xds Jo for U(t)x e &(A), t ^ 0 where the initial condition is U(0)x = x. We take B(t) as a strongly continuous family of operators on [0, oo) to We have now the following theorem: THEOREM 1. Let B(t) be a strongly continuous function of [0, oo) to G?(£) with Mt = M \ || J5(s) || ds. Then it exists a unique Jo one-parameter family of bounded linear operators U(t) on [0, oo) to ©(X) satisfying ( i) U(t) is strongly continuous on [0, oo). (ii) For xe£^(A) U(t)x is strongly continuously differentiable in [0, oo) and (iii) is the unique solution of the integro-diff erential equation (7) with (iv) U(0) = I. (v) U(t) has the representation (8) ±