ABSTRACT
Observing that in case k = 0 the initial value problem for (8.1) has the explicit solution ~(t, s) = e(t, r, s )Ip(s), we may apply the variationsof-constants formula and prove the following
Theorem 8.1. Let Ip EX, c and k be measurable, and the corresponding operators C and K be locally integrable and strongly continuous in £(X). If f E Ct(X), each solution of the i;'itial value problem for (8.1) is also a solution of the fized point equation (8.2) ~(t,s) = L~(t,s)+get,s). Conversely, every solution ~ E Ct(X) of (8.2) actually belongs to Cf(X) and solves the initial value problem for (8.1). To solve the fixed point equation (8.2), we assume that c and k are degenerated in that sense that L degenerates to
L~(t,s)=! 16 exp (1t cU,s)~) k(T,,,,u)~(T,u)do-dT n 1'16=E Jt. CJ;(t, T,u)b;(t, S)~(T, u) do-dT.
