Let D be a nonempty subset of a real Banach space X. A
sequence (Tn)n ≥ 0 of self maps of D is called almost
asymptotically nonexpansive if there exist sequences {kn} and
{εn} of positive numbers with limn→∞ kn=1 and limn→ ∞ εn=0
such that
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|| Ti+lx−Tj+ly||2 ≤ kl2||Tix−Tjy||2+ εl2 for all i,j,l ≥ 0 |
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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 {kn}
of positive numbers with limn→ ∞ kn=1 such
that ||Tn x−Tn y|| ≤ kn|| 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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