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© 2018

Positive Solutions to Indefinite Problems

A Topological Approach

Benefits

  • Deals with new, challenging problems in nonlinear analysis and solves several open problems and questions

  • Gives a good overview of existing methods and presents new ideas and results as well

  • Proposes open problems, research ideas and suggestions

Book

Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
    Pages i-xxix
  2. Superlinear Indefinite Problems

    1. Front Matter
      Pages 1-1
    2. Guglielmo Feltrin
      Pages 3-37
    3. Guglielmo Feltrin
      Pages 39-67
    4. Guglielmo Feltrin
      Pages 131-168
  3. Super-sublinear Indefinite Problems

    1. Front Matter
      Pages 169-169
    2. Guglielmo Feltrin
      Pages 171-194
    3. Guglielmo Feltrin
      Pages 195-237
    4. Guglielmo Feltrin
      Pages 239-254
  4. Future Perspectives

    1. Front Matter
      Pages 255-255
    2. Guglielmo Feltrin
      Pages 257-268
  5. Appendices

    1. Front Matter
      Pages 269-269
    2. Guglielmo Feltrin
      Pages 277-282
    3. Guglielmo Feltrin
      Pages 283-289
  6. Back Matter
    Pages 291-304

About this book

Introduction

This book is devoted to the study of positive solutions to indefinite problems. The monograph intelligibly provides an extensive overview of topological methods and introduces new ideas and results. Sticking to the one-dimensional setting, the author shows that compelling and substantial research can be obtained and presented in a penetrable way.

In particular, the book focuses on second order nonlinear differential equations. It analyzes the Dirichlet, Neumann and periodic boundary value problems associated with the equation and provides existence, nonexistence and multiplicity results for positive solutions. The author proposes a new approach based on topological degree theory that allows him to answer some open questions and solve a conjecture about the dependence of the number of positive solutions on the nodal behaviour of the nonlinear term of the equation. The new technique developed in the book gives, as a byproduct, infinitely many subharmonic solutions and globally defined positive solutions with chaotic behaviour. Furthermore, some future directions for research, open questions and interesting, unexplored topics of investigation are proposed.

Keywords

indefinite equations superlinear problems super-sublinear problems existence results multiplicity results subharmonic solutions degree theory

Authors and affiliations

  1. 1.Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”Politecnico di TorinoTorinoItaly

Bibliographic information

  • Book Title Positive Solutions to Indefinite Problems
  • Book Subtitle A Topological Approach
  • Authors Guglielmo Feltrin
  • Series Title Frontiers in Mathematics
  • Series Abbreviated Title Frontiers in Mathematics
  • DOI https://doi.org/10.1007/978-3-319-94238-4
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-94237-7
  • eBook ISBN 978-3-319-94238-4
  • Series ISSN 1660-8046
  • Series E-ISSN 1660-8054
  • Edition Number 1
  • Number of Pages XXIX, 304
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Ordinary Differential Equations
    Operator Theory
  • Buy this book on publisher's site

Reviews

“The book gives a complete overview of indefinite problems, starting from the more classical results in the literature up to the very recent and novel ones. It has the advantage of being self-contained, with the prerequisites recalled in the appendices and the proofs throughout the book are provided in full detail.” (Andrea Tellini, Mathematical Reviews, November, 2019)

“This book would be suitable for graduate students and young researchers willing to learn more on this fashionable topic.” (Gennaro Infante, zbMATH 1426.34002, 2020)