Classes of stable three-layer schemes 439We modify the expression in the brackets on the right-hand side of
identity (18) by a simple observation that(A(y + y), y + :iJ) = (A(y + :iJ), y + :iJ) + r (Ar(y + :iJ), y + :iJ) ,
( ( R - i A) Yf, Yr) = ( ( R - i A) Yr, Yt) + r ( (R - t A) f Yr, Yr)
A=A(t-r), Ar = (A - A)/ r.
Also, it will be convenient to introduce more compact notations(52) J = J(t + r) + llY(t + r)ll~t), J = J(t) = llY(t)ll~t-r) ·
With these, identity ( 18) is recast as(53) 2r (Byo, t yo)+ t J = J + 2r (<tJ, yo)+ t T F,
(54)showing the new members to be sensible ones. If the operators R - A/4
and A satisfy condition ( 49), thenllFll<4 C3 ( A(y+:iJ),y+:iJ - ) +r^2 c 3 ( ( R.-4Afyf,Yf -^1 -) ) =c 31
and from identity (53) for R. > A/4 it follows that
(55)In the general case when each of the operators R.(t) and A(t) satisfies
condition ( 49), we obtainunder the constraint R. >^1 !' A, where E = const > 0 is independent of h
and r, since
for R. >^1 + 4 E A.