232 Homogeneous Difference Schen1es- Stability of the first kind operator equations with respect to coefficients.
Here we give the general formulation of the concept of stability with respect
to coefficients for the difference scheme in Section 1 by having recourse to
an operator equation of the first kind
( 11) Au = f, f EH,
where A is a linear operator acting from Hilbert space H into H, A: H ---+
H, f E H is a given vector, 1l E H is the unknown vector.
Problem ( 11) is said to be well-posed if there exists a unique solution
of equation (11) for any f EH and this solution continuously depends on
the right-hand side f, so that(12) II u - u 11(1) < !vlo II j - f 11(2)'
where u is a solution of equation (11) with perturbed right-hand side f
( 13) Au= f,
here II · ll(l) and 11 · 11( 2 ) are some suitable norms on the set H.
A case in point that in the statement of problem (11) we must specify
not only the right-hand side, but also the operator A. If, for instance, A is
a differential or difference operator, the coefficients of the equation should
be known in advance.
It is natural to require a solution of problem (11) to depend con-
tinuously not only on perturbations of the right-hand side, but also on
perturbations of the operator A (for instance, on the coefficients of a dif-
ference operator). As in the case of difference sche1nes arising in Section 1,
this property of operator equations is to be understood as stability with
respect to coefficients or co-stability of an operator equation.
The stability of a solution to equation (11) with respect to perturba-
tions of the right-hand side f and perturbations of the operator A is called
strong stability. "The proble1n statement here is as follows: with regard
to the equations
- (14) Au= f, Au= f,
where A and A are linear operators, whose domains coincide with the entire
space H, f and J are arbitrary vectors of the space H, it is required to
estimate the perturbations of a solution
(15) z=u-u