Since the left hand side is the member of a

bounded sequence, while the right hand side approaches +[infinity], we have a contradiction.

(ii) In addition, if {[x.sub.n]} c E is a

bounded sequence such that [lim.sub.n[right arrow][infinity]] dist([x.sub.n], T([x.sub.n])) = 0, where dist(a, B) is the distance from a point a e X to the set B e C(X), then for all Banach limits [mu],

Let ([x.sub.n]) [subset] X be a

bounded sequence. It is well known and easy to see that ([x.sub.n]) defines an operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] through the equality

where [f.sub.n] (t,x, [lambda]) = f (*, *,[lambda]) * [n.sup.2][omega](nt, nx) is a regularization of the flux f via the standard non-negative mollifier [omega] [member of] [C.sup.[infinity].sub.c][((-1,1).sup.2]) with total mass one, and ([u.sup.0.sub.n]) is a

bounded sequence of functions converging strongly in [L.sup.1.sub.loc] (R) toward [u.sub.0] By multiplying (1.4) by sgn([u.sub.n]t,x) - [lambda]) we get after standard manipulations (see also formula (2.8) of the next section):

Assume here and after that x = ([x.sub.k]) be a sequence such that [x.sub.k] [not equal to] 0 for all k [member of] N and ([p.sub.k]) be the

bounded sequence of strictly positive real numbers with [supp.sub.k] = H and M = max{1, H}.

For any

bounded sequence g(n) defined on Z, define [g.sup.u] = [sup.sub.n[member of]Z] g(n), [g.sup.l] = [inf.sub.n[member of]Z] g(n).

In other words [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a

bounded sequence.

where {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} are sequences in [0,1] such that [a.sub.n] + [b.sub.n] + [c.sub.n] = 1 and {[u.sub.n]} is a

bounded sequence in D(T).

Let [lambda] [member of] [OMEGA] and let x = ([x.sub.k]) be a

bounded sequence in X.

In other words (M([[|[x.sub.k]|.sup.1/k]/[rho]])) is a

bounded sequences. [[conjunction].sub.M] is called the Orlicz space of

bounded sequence.

Then for every

bounded sequence {[{[[omega].sub.j]}.sub.j[member of]J], for all [lambda] [member of] [0,1] and all f [member of] H, we have

Let [bar.x] = [([x.sub.n]).sub.n] [subset or equal to] X be an f--statistically

bounded sequence, then the set [[GAMMA].sup.f.sub.[bar.x]] is bounded.