Let
C be a nonempty closed convex subset of a uniformly convex Banach space
E,
G a right reversible semitopological semigroup and
S={
S(
t) :
t∈
G} a continuous representation of
G as Lipschitzain self-mappings on
C. We consider the asymptoic behavior of an almost-orbit {
u(
t) :
t∈
G} of
S={
S(
t) : (
t)∈
G}. We show that if
E has a Fréchet differentiable norm and if lim
t sup
kt{≤}1, then the closed convex set
\underset{
s∈
G}∩\overline{
co}{
u(
t) :
t{≥}
s}∩
F(
S)
consists of at most one point, where
kt is the Lipschitzian constant of
S(
t). This result is applied to study the problem of weak convergence of the net {
u(
t) :
t∈
G}.
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