As shown in Example 23, there is an infinite dimension Banach space E and weakly
absolutely continuous function y : I [right arrow] E which is nowhere weakly differentiable (hence the CFWD of y does not exist).
From the fact that X is compact, we obtain that f is
absolutely continuous function on X; thus, given [member of] > 0, there exists [delta] > 0 such that
an
absolutely continuous function which satisfies (5.1) almost everywhere.
The following identity for an
absolutely continuous function f: [a, b] [right arrow] R holds (see [14]):
It can be seen w'(t) is
absolutely continuous function in [0,1].
Then there exists an
absolutely continuous function p and [mu] [member of] [C.sup.*] (0,1) such that
It is easy to see that for any locally
absolutely continuous function f: [a, b] [right arrow] R, we have the identity
A solution of problem (1.1), (1.2) is an
absolutely continuous function, which satisfies that equation on (0, [infinity]) almost everywhere and condition (1.2).
Lemma 2.2 Let p [member of] [1, [infinity]) and let g : R [right arrow] C be a locally
absolutely continuous function with g and g' belonging to [L.sup.p](R).
Then for every ([t.sub.0]; [y.sub.0]) [member of] GrK the multivalued Cauchy problem (2.1) has a viable solution y : [[t.sub.0]; a] [right arrow] [R.sup.n] which is an
absolutely continuous function.
for x [member of] [a, b] where f : [a, b] [right arrow] R is an
absolutely continuous function on [a, b].
We will denote by [H.sub.w] the set of all
absolutely continuous functions f defined on [0, [infinity]) satisfying f (0) = 0 and