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\title{Invertibility of Functional Operators
 with\\ Slowly Oscillating non-Carleman Shifts} 
\author{A. Karlovich} 
\date{}
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\maketitle 
We prove criteria for the invertibility in Lebesgue spaces $L^p, 1<p<\infty$,
of a binomial functional operator of the form
\[
A=aI-bW_\alpha
\]
where $a$ and $b$ are continuous functions on $(0,1)$, $I$ is the
identity operator, $W_\alpha$ is the shift operator, $W_\alpha
f=f\circ\alpha$, generated by a non-Carleman shift
$\alpha:[0,1]\to[0,1]$, which has only two fixed points $0,1$. We
suppose that the coefficients $a,b$ slowly oscillate at $0$ and $1$,
and $\alpha$ is a slowly oscillating shift, that is, $\log\alpha'$
is bounded and continuous on $(0,1)$ and
\[
\lim_{t\to j}(t-j)\frac{d}{dt}\left(\frac{\alpha(t)-\alpha(j)}{t-j}\right)=0,
\quad
j\in\{0,1\}.
\]
The main difficulty connected with slow oscillation is overcome using 
the method of limit operators.

These results are obtained in collaboration with 
Yuri Karlovich and Amarino Lebre.
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