92 Basic Concepts of the Theory of Difference Sche111eswhich implies that(8)From such 1·easoning it seems clear that the solution of the Cauchy problern
decreases along with increasing t:(9) II u(t) II< II Uo II for t > 0.
Of special interest is Euler's scheme for the Cauchy problem(10) Yj+1 T - Yj + A Yj = O ' j = 0, 1, ... ' Yo = Uo,
where Yj = y(tj) and tj = jr. We seek its solution as a sum
n.
( 11) Yj = L ak qi ek ,
k=lso that Yo = Uo = L~= 1 ak ek and ak = ( Uo , ek). We note in passing that
expression (11) satisfies equation (10) only ifor, what amounts to the same,It follows from the foregoing that q k = 1 - T >. k.
By rearranging the norm asit is easy to derive the chain of the relations
. n.
II Yj 112 <max I Yi 12 La~= max I qi 12 II Yo 112 <II Yo 112,
k k=l k
which are valid under the conditions maxk I qk I < 1 and 11 Yj 112 > 11Yo11^2 if
mink I qk I > 1. Being an alternative of (9), the inequality
(12) II Yj II < II Yo II