between such functions as u and ü (the representative function agreeing with m on p¡ and approaching u in the norm) the proof runs as follows: From (23), there is, for any e>0, a 50 such that, for §<S0, \\w— 5ä_1M|| <£ where we write Au = w, Sw = u. On multiplying w' = (w — SPu) by Ss one has, from (24) ||Säw —m|| =||Sjh/|| 5SM'||w'|| < M'e which gives (25) and the scheme is convergent. As in P...

Uniform boundedness, ultimate boundedness, uniform stability, and asymptotic stability are studied by analyzing a Liapunov function satisfying v′(t) ≤ −αv(t) + √ v(t) ∫ t # ω(t, s) √ v(s)ds, t ≥ t0 ≥ 0, (# = 0 or −∞). The results are then applied to integrodifferential equations x′(t) = A(t) [ x(t) + ∫ t # F (t, s)x(s)ds ] , t ≥ t0 ≥ 0, (# = 0 or −∞), in real Hilbert space with unbounded linear...