ABSTRACT
Since C(D) c Loo(D), another criterion for the strong continuity of the operator function (13.34) in C(D) follows from Theorems 12.9 and 13.7:
Theorem 13.8. Suppose that the operator function (13.34) takes its values in £p(C). Then it is strongly continuous if and only if the junction (13.39) is bounded on J x D for the each bounded interval J C lR, the function 'P ..-. c( 'P, " .) is continuous as a function with values in C(D), and the functions
'P ..-.lu 1('P,',',T)dT, 'P ..-.1'1 m('P,.,.,u)du, 'P ..-.lu LV n('P,"" T, u) dudT
are continuous for each U E [a, b), 11 E [c, d), and (u, v) ED, respectively, as functions with values in C(D).
