1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schemes 443

Substituting (75) into the right-hand side of the inequality

n

II Yn+l II < L T II Gn+l,s II

s=l

we obtain for a solution of proble1n ( 63) with y(O) = y( T) = 0 the estimate

t

(76) II Y(t + r) II< J2 (cr 1 + cr2) ~ T II <p(t') II,

t'=r

which in combination with (71) implies (69).

Theorem 8 If A(t) = A*(t) > 0 is a positive operator and conditions (68)

are satisfied, then for a solution of problem (63) the inequality

1 [ t ] 1/2

(77) II Y(t + r) II< II Y(r) II+ y!2 t~ T ll<p(t')llA-'(t')

holds, where II Y(t + r) II is given by relation (70).

To prove this assertion the estimates

are incorporatred in identity (18) for scheme (65).
Applying Thebrem 3 to scheme (65) with a constant positive operator
A, it is plain to derive under conditions (68) the estimate

(78) II Y(t + r) II< II Y(r) II+ +M2 r<t':St max (II A-^1 <p(t') II+ llA-^1 <pr(t')ll).

Note that estimate (60) holds true for scheme (63) if A(t) = A*(t) > 0,
A(t) being Lipschitz continuous int, and