438 Stability Theory of Differe11ce Sche1nes
Theore1n 5 Let A = A* > 0, R = R* > 0 and B = B* > 0, A and R be
constant operators and R > ~A. Then for schen1e (1) the estimate holds:
(46) II Yn+l 112 < II Y111^2 + ~ t 7 /l<tJkll~-1 ·
k=l
The fact that the operator B is self-adjoint is kept in mind in the
estimation of the expression
( 4 7)
which is involved in identity (31). Putting E = 2 and substituting this
estimate into (31), we obtain the inequality
( 48)
implying the desired estimate. Note that the operator B inay depend on t:
B = B(t) In that case we must write ll<tJkll~-i, where B-;;^1 = B-^1 (tk)·
k
- Schemes with variable operators. If operators A and R depend on the
variable t, the extra property of the Lipschitz continuity of A and R with
respect to the variable t is needed in this connection:
(49) I ((A(t) - A(t - 7)) x, x) I < 7 c 3 (A(t - 7) x, x)
for all x E Handt = 27, ... , ( n 0 - l )7, where c 3 = const > 0 is independent
of h and 7; the condition imposed on the operator R is analogous. In this
case the compound norm II Y(t+7) II= llY(t+7)11(i) depends on the variable
t:
(50)
l
llY(t + 7)11~i) = 4 (A(t) (y(t + 7) + y(t)), y(t + 7) + y(t))
- 72 ( ( R( t) - % A( t)) Yt ( t), Yt ( t)) ,
( 51)
1
llY(t)ll~t-r) = 4 (A(t - 7) (y(t) + y(t - 7)), y(t) + y(t - 7))
+ 72 ( (R(t - 7) - % A(t - 7)) Yr(t), Yr(t)).