Sestelo R. Degree Theory for Discontinuous Operators. Applications...2021

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Sestelo R. Degree Theory for Discontinuous Operators. Applications...2021

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Textbook in PDF format This unique book contains a generalization of the Leray-Schauder degree theory which applies for wide and meaningful types of discontinuous operators. The discontinuous degree theory introduced in the first section is subsequently used to prove new, applicable, discontinuous versions of many classical fixed-point theorems such as Schauder’s. Finally, readers will find in this book several applications of those discontinuous fixed-point theorems in the proofs of new existence results for discontinuous differential problems. Written in a clear, expository style, with the inclusion of many examples in each chapter, this book aims to be useful not only as a self-contained reference for mature researchers in nonlinear analysis but also for graduate students looking for a quick accessible introduction to degree theory techniques for discontinuous differential equations. Preface Introduction Degree Theory for a Class of Discontinuous Operators A Topological Degree for Discontinuous Operators Basic Properties of the Degree Fixed Point Index for Discontinuous Operators Fixed Point Theorems for Some Discontinuous Operators Schauder Type Fixed Point Theorems Krasnosel'skiĭ's Compression–Expansion Type Fixed Point Theorems in Cones Krasnosel'skiĭ Type Fixed Point Theorems in Cones for Monotone Operators A Generalization of Leggett–Williams' Three-Solutions Theorem A Vectorial Version of Krasnosel'skiĭ's Fixed Point Theorem First Order Problems Existence Result for First Order Scalar Problems with Functional Initial Conditions Existence Results for Non-autonomous Systems Discontinuous First-Order Functional Boundary Value Problems An Application to Second-Order Problems with Functional Boundary Conditions Second Order Problems and Lower and Upper Solutions Existence Results on Bounded Domains Existence Results via Well-Ordered Lower and Upper Solutions Existence of Extremal Solutions Between the Lower and Upper Solutions Existence Results via Non-ordered Lower and Upper Solutions Multiplicity Results Existence Results on Unbounded Domains Existence Results on the Half Line Extremal Solutions Between the Lower and Upper Solutions Positive Solutions for Second and Higher Order Problems Second Order Problems with Sturm–Liouville Boundary Conditions Second Order Systems Multiplicity Result to a Three-Point Problem Positive Solutions to a One Dimensional Beam Equation Existence Results A Multiplicity Result A Degree Theory for Multivalued Operators References