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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.
List of contents
Introduction.- Part I - Superlinear indefinite problems.- Dirichlet boundary conditions.- More general nonlinearities f(t; s).- Neumann and periodic conditions: existence results.- Neumann and periodic conditions: multiplicity results.- Subharmonic solutions and symbolic dynamics.- Part II - Super-sublinear indefinite problems.- Existence results.- High multiplicity results.- Subharmonic solutions and symbolic dynamics.- Part III - Appendices.- Leray-Schauder degree for locally compact operators.- Mawhin's coincidence degree.- Maximum principles and a change of variable.- Bibliography.
Summary
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.
Additional text
“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)
Report
"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)
