Stability with respect to coefficients 235Theorem Let u be a so~1tion to equation (11) and u be a solution to
equation (14), where A, A and Ao are self-adjoint positive operators for
~hich the inverse operators exist. If condition (18) and the inequality
A > c 1 A 0 , c 1 > 0 hold, then the estimates are valid:(21) llu-ullA < llf-Jll,4- (^1) +cxll!ll,4- (^1) ,
(22)
The first summand on the right-hand side of (21) is the value of pertur-
bations of the right-hand side f and the second one involves the coefficient
ex, which is the value of a relative perturbation of the operator.
Example Let H be the set of all grid functions defined on w h = {xi =
ih, 0 < i < N} and vanishing for i = 0 and i = N. We refer to the
difference operators
Ay = -(ay 5 ;):r: + dy,
Ay = -(ayx):r: + cZy,
Aoy = -Yxx ·
a > c 1 > 0, d > 0,
a > cl > 0, d > 0'
By introducing the inner product_in the usual way and applying Green's
formulae we get the inequalities A ·> c 1 Ao and A > c 1 Ao.
with Chapter 2, Section 5,
(
N-1 (N-1 )2)1/2
ll!llA;;-' < II f 11(-1) = i~l h t:i hf1.: '
With these relations established, estimate (22) reduces to
1 - Cl'
II Z,i; II< - II f - f 11(-1) + - II f 11(-1)
c1 c1
or, on account of the inequality II z lie<~ II zx II,
In conformity
f EH.
. 1 - Cl'
llzllc = ll:Y-Yllc <- (^2) c1 llf-!11(-1)+-llfll(-l)' C1
In concluding this chapter we clarify the meaning of condition (18) by
observing that
(1-ex) ((a,y;J + (d,y^2 )) < (a,y;] + (d,y^2 ) < (1 +ex) ((a,y;] + (d,y^2 )).
From such reasoning it seems clear that the fulfilment of (18) is ensured by
the inequalities
I a - a I< cxa,