where k(t, s) and w0(t) are known non-negative functions and u(t) is an unknown non-negative function. For examples, one can refer to Bellman [l, pp. 35 ff.], Coddington and Levinson [2, pp. 37 ff.], Willett [3], and others. In order to obtain from (1.1) a genuine upper bound for u(t), i.e., an upper bound independent of u, it seems necessary to separate the variable t in k(t, s) from the integrand involving u(s). This can be done by assuming that k(t, s) is directly separable, i.e.,thatthereexistmeasurablefunctionsi)?(/)andw,(i) (? = 1,2, • • ■ ,n) such that