Let D be a nonempty subset of a real Banach space X. A
sequence (T_{n})_{n ≥ 0} of self maps of D is called almost
asymptotically nonexpansive if there exist sequences {k_{n}} and
{ε_{n}} of positive numbers with lim_{n→∞} k_{n}=1 and lim_{n→ ∞} ε_{n}=0
such that
 T_{i+lx}−T_{j+ly}^{2} ≤ k_{l}^{2}T_{ix}−T_{jy}^{2}+ ε_{l}^{2} for all i,j,l ≥ 0 

and all x,y in D.
First, in a Hilbert space, we show the existence of an extension
to such a sequence of self maps of D with a common fixed point.
A self map T of D is called symptotically nonexpansive if
there exists a sequence {k_{n}}
of positive numbers with lim_{n→ ∞} k_{n}=1 such
that T^{n} x−T^{n} y ≤ k_{n} x−y for all n ≥ 0 and
x,y in D. by introducing the notions of absolute and almost absolute
fixed points for T, we investigate the existence of such points
for such mappings in a Hilbert space.
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