Classes of stable three-layer schernes^449assuming that(85) A(t)=A(t)>O, D(t)=D(t)>O, B(t)>O,
(86) A(t) and D(t) are Lipschitz continuous int with constant c 3.
Theorem 6 implies that scheme (84) under conditions (85)-(86) and the
extra restriction(87)
1 + E 2
D > 4 TAwhere a number c: > 0 is independent of T and h, is stable with respect to
the initial data and a solution of problen1 (84a) adn1its the estimate(88)where lvf 1 = M 1 (c 3 ,c:,t 0 ) > 0 is independent ofr, h, n and
(89) llYn+ill(n)^2 = 4 1 llYn + Yn+1llA(t^2
71 )- ((n(tn)- :
2
A(t,J) Yt,n,Yt,n),(90)The constant M 1 equals 1 if the operators A and D are independent oft.
In order to turn from (88) to the estimates in the spaces HA and Hn,
we shall need yet some bilateral estimates of the functional II Yn+I 11·
Lemma 3 Under conditions (85) and (87) one has(91) llYn+1 /l(n) < llYnllA(tn) + l/Yi(t,JllD(tn)'
(92) llYn+1 ll(n) > ~ /IYn+1 llA(t 71 )'
(93) llYn+1 ll(n) > ~ ~ (llYn+1llA(tn) + llYt(tn)//n(t,.)) ·