TITLE: Asymptotic Fixed Points, Part II
ABSTRACT: Continuing the earlier seminar, I will give nonstandard proofs for
one or more (depending on time) results like the
one below (which consolidates and generalizes a number of recent results in
the area).
Suppose
$(X,d)$ is a complete metric space,
$T:Xto X$ is continuous,
$phi, phi_n:[0,infty)to[0,infty)$, and
$phi_n$ converges to $phi$ uniformly on the range of $d$,
$phi$ is semicontinuous and satisfies $phi(s)0$,
$d(T^nx,T^ny)lephi_n(d(x,y))$ for all $x,yin X$ and $ninmathbb N$.
Moreover, suppose that for any $x,y$, $phi(t)lesssim t$ on infinite
elements of
$${^*d(T^nx,T^my) : m, n text{ hyperintegers}}.$$
Then $T$ has a unique fixed point $x_infty$, and for every $xin X, limlimits_{ntoinfty}T^n(x)=x_infty$. Moreover, this convergence is
uniform on bounded subsets of $X$.