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Math Help - Levitan's result?

  1. #1
    Senior Member bkarpuz's Avatar
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    Exclamation 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
    Quote 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
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  2. #2
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by bkarpuz View Post
    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)
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