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Birkhäuser

Positive Solutions to Indefinite Problems

A Topological Approach

  • Book
  • © 2018

Overview

Authors:
  1. Guglielmo Feltrin

    1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Torino, Italy

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  • 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

Part of the book series: Frontiers in Mathematics (FM)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xxix
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  2. Superlinear Indefinite Problems

    1. Front Matter

      Pages 1-1
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    2. Dirichlet Boundary Conditions

      • Guglielmo Feltrin
      Pages 3-37

    3. More General Nonlinearities f (t, s)

      • Guglielmo Feltrin
      Pages 39-67

    4. Neumann and Periodic Boundary Conditions: Existence Results

      • Guglielmo Feltrin
      Pages 69-99

    5. Neumann and Periodic Boundary Conditions: Multiplicity Results

      • Guglielmo Feltrin
      Pages 101-130

    6. Subharmonic Solutions and Symbolic Dynamics

      • Guglielmo Feltrin
      Pages 131-168

  3. Super-sublinear Indefinite Problems

    1. Front Matter

      Pages 169-169
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    2. Existence Results

      • Guglielmo Feltrin
      Pages 171-194

    3. High Multiplicity Results

      • Guglielmo Feltrin
      Pages 195-237

    4. Subharmonic Solutions and Symbolic Dynamics

      • Guglielmo Feltrin
      Pages 239-254

  4. Future Perspectives

    1. Front Matter

      Pages 255-255
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    2. Further Results and Unexplored Problems

      • Guglielmo Feltrin
      Pages 257-268

  5. Appendices

    1. Front Matter

      Pages 269-269
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    2. Leray–Schauder Degree for Locally Compact Operators on Open Possibly Unbounded Sets

      • Guglielmo Feltrin
      Pages 271-275

    3. Mawhin’s Coincidence Degree

      • Guglielmo Feltrin
      Pages 277-282

    4. Maximum Principles and a Change of Variable

      • Guglielmo Feltrin
      Pages 283-289

  6. Back Matter

    Pages 291-304
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Keywords

About this book

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.

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)

Authors and Affiliations

  • Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Torino, Italy

    Guglielmo Feltrin

Bibliographic Information

  • Book Title: Positive Solutions to Indefinite Problems

  • Book Subtitle: A Topological Approach

  • Authors: Guglielmo Feltrin

  • Series Title: Frontiers in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-94238-4

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2018

  • Softcover ISBN: 978-3-319-94237-7Published: 05 December 2018

  • eBook ISBN: 978-3-319-94238-4Published: 23 November 2018

  • Series ISSN: 1660-8046

  • Series E-ISSN: 1660-8054

  • Edition Number: 1

  • Number of Pages: XXIX, 304

  • Topics: Ordinary Differential Equations, Operator Theory

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