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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