IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4613-8112-9_4.html
   My bibliography  Save this book chapter

Investigation of a Class of Nonlinear Integral Equations and Applications to Nonlinear Network Theory

In: Theory and Applications of Some New Classes of Integral Equations

Author

Listed:
  • A. G. Ramm

    (University of Michigan, Department of Mathematics)

Abstract

Nonlinear oscillations in some networks can be described by the operator equation Au + Fu = J, where A is an unbounded linear operator on a Hilbert or Banach space, F is a nonlinear operator, and B = A + F is monotone. This is true for the general passive one-loop network, consisting of an e.m.f. E(t), and arbitrary linear passive stable one-port L, and a nonlinear one-port N with a monotone voltage-current characteristic i = Fu, where i is the current through N and u is the voltage on N. The theory presented below makes it possible to study nonlinear oscillations qualitatively, including questions of existence, uniqueness, stability in the large, convergence, and calculation of stationary regimes by means of an iterative process. Our assumptions concerning the network are more general than those usually adopted in the literature. No assumptions concerning the “smallness” of the nonlinearity or filter property of the linear one-port are made. Our results for nonlinear networks of the class defined above are final in the sense that if we omit the assumption concerning passivity of the network the results will not hold.

Suggested Citation

  • A. G. Ramm, 1980. "Investigation of a Class of Nonlinear Integral Equations and Applications to Nonlinear Network Theory," Springer Books, in: Theory and Applications of Some New Classes of Integral Equations, chapter 0, pages 156-189, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8112-9_4
    DOI: 10.1007/978-1-4613-8112-9_4
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-1-4613-8112-9_4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.