Difference schemes as operator equations. General formulations 129Remark 1 Convergence conditions (27) imply that(29) lim II Yh 11( 1 ) =II u ll(l) ·
lhl--+0 hIndeed, with the aid of the triangle inequality and the relation uh = P~^1 )u
one can write downII Yh ll(h) = II ( Yh - uh)+ tth ll(h)
< II Yh - uh ll(h) +II uh ll(h)
<II Yh - uh ll(h) +II P~l) u ll(h) ·
Passing to the limit as I h I --+ 0 and taking into account (27) and (26), we
obtainlhl--+0 lim II Yh 11(^1 h ) <II u 11(^1 ) ·
A similar reasoning yieldsII u ll(l) < lhl--+0 lim II Yh 11( (^1) h ) '
giving (29).
Remark 2 The sequence { Yh } can converge only to a single element
u E 5(l). On the contrary, let there exist two limit elements ii, u E 5(l),
ii f- u ( { Yh} converges to each of them), that is,
vVe are going to show that u f- 1i by considering the difference
and appealing to the triangle inequality
Passing to the limit as I h I --+ 0 and taking into account the convergence of
Yh to both ii and u and the norm concordance condition (26), we deduce
that 11 ii - u 11(1) = 0, implying that ii= u.