# Thread: Levitan's result?

1. ## Levitan's result?

Dear Friends,

I need help with a result due to Levitan.
Please see the last paragraph of page 4 of the following paper: http://www.emis.de/journals/EJDE/2007/163/rath.pdf
Originally Posted by Author
Next, we show that $BS:=\{Bf:\ f\in S\}$ is relatively compact.
It suffices to show that the family of functions $BS$ is uniformly bounded and equicontinuous on $[T_{0},\infty)$.
The uniform boundedness is obvious.
For the equicontinuity, according to Levitan’s result we only need to show that, for any given $\varepsilon>0$, $[T_{0},\infty)$ can be decomposed into finite subintervals in such a way that on each subinterval all functions of the
family have change of amplitude less than $\varepsilon$.
Above, $S$ is the set of continuous functions bounded below and above by $3(1-p)/5$ (here $p$ is a positive constant less than $1$) and $1$, respectively, and that $B:S\to C([T_{0},\infty),\mathbb{R})$.

I really need help about finding a book reference for Levitan's result.

Many thanks!

bkarpuz

2. Originally Posted by bkarpuz
Dear Friends,

I need help with a result due to Levitan.
Please see the last paragraph of page 4 of the following paper: http://www.emis.de/journals/EJDE/2007/163/rath.pdf

Above, $S$ is the set of continuous functions bounded below and above by $3(1-p)/5$ (here $p$ is a positive constant less than $1$) and $1$, respectively, and that $B:S\to C([T_{0},\infty),\mathbb{R})$.

I really need help about finding a book reference for Levitan's result.

Many thanks!

bkarpuz
Book reference.
Agarwal, Ravi P.; Bohner, Martin; Li, Wan-Tong,
Nonoscillation and Oscillation: Theory for Functional Differential Equations,
Monographs and Textbooks in Pure and Applied Mathematics, 267. Marcel Dekker, Inc., New York, 2004. (see Remark 1.4.20 on pp. 7)