Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDEs (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDEs covers all the basics of the subject of fixed-point theory and its applications with a strong focus on examples, proofs and practical problems, thus making it ideal as course material but also as a reference for self-study.Many problems in science lead to nonlinear equations T x + F x = x posed in some closed convex subset of a Banach space. In particular, ordinary, fractional, partial differential equations and integral equations can be formulated like these abstract equations. It is desirable to develop fixed-point theorems for such equations. In this book, the authors investigate the existence of multiple fixed points for some operators that are of the form T + F, where T is an expansive operator and F is a k-set contraction. This book offers the reader an overview of recent developments of multiple fixed-point theorems and their applications.About the AuthorsSvetlin G. Georgiev is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations and dynamic calculus on time scales.Khaled Zennir is assistant professor at Qassim University, KSA. He received his PhD in mathematics in 2013 from Sidi Bel Abbès University, Algeria. He obtained his Habilitation in mathematics from Constantine University, Algeria in 2015. His research interests lie in nonlinear hyperbolic partial differential equations: global existence, blow up and long-time behavior. Cover......Page 1 Half Title......Page 2 Series Page......Page 3 Title Page......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 8 1.1 Measures of Noncompactness......Page 10 1.2 The Brouwer Fixed-Point Theorem. The Schauder Fixed-Point Theorem......Page 20 1.3 Fixed Points of Strict Set Contractions......Page 29 1.4 The Kronecker Index......Page 34 1.5.1 Smooth Mappings......Page 43 1.5.2 Homotopy Invariance......Page 48 1.5.3 Continuous Mappings. Basic Properties......Page 51 1.5.4 The Case f:∂D⊂nR →Sn − 1.......Page 58 1.6 The Leray-Schauder Degree......Page 61 1.7 The Fixed-Point Index for Completely Continuous Mappings......Page 65 1.8 The Fixed-Point Index for Strict Set Contractions......Page 66 1.9 Multiple Fixed-Point Theorems......Page 73 2.1 Periodic Solutions for First Order ODEs......Page 92 2.2 BVPs for First Order ODEs......Page 111 2.3 BVPs for Second Order ODEs......Page 128 2.4 BVPs with Impulses......Page 132 3.1 Global Existence for a Class of Fractional-Differential Equations......Page 146 3.2 Multiple Solutions for a BVP of Nonlinear Riemann-Liouville Fractional Differential Equations......Page 162 3.3 Multiple Solutions for a BVP of Nonlinear Caputo Fractional Differential Equations......Page 175 4.1 Differentiability of Classical Solutions with Respect to the Initial Conditions of an IVP......Page 178 4.2 Local Existence of Classical Solutions for an IBVP......Page 191 4.3 Periodic Solutions......Page 209 4.4 Multiple Solutions for an IBVP with Robin Boundary Conditions......Page 222 5.1 Differentiability of Classical Solutions with Respect to the Initial Conditions for an IVP for a Class of Hyperbolic Equations......Page 230 5.2 Multiple Solutions for an IBVP with Robin Boundary Conditions......Page 247 5.3 Periodic Solutions......Page 254 6.1 Multiple Solutions for a BVP with Robin Boundary Conditions......Page 266 6.2 Existence and Smoothness of Navier-Stokes Equations......Page 273 Bibliography......Page 302 Index......Page 304 Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDEs covers all the basics of the subject of fixed-point theory and its applications with a strong focus on examples, proofs and practical problems, thus making it ideal as course material but also as a reference for self-study. Many problems in science lead to nonlinear equations T x + F x = x posed in some closed convex subset of a Banach space. In particular, ordinary, fractional, partial differential equations and integral equations can be formulated like these abstract equations. It is desirable to develop fixed-point theorems for such equations. In this book, the authors investigate the existence of multiple fixed points for some operators that are of the form T + F, where T is an expansive operator and F is a k-set contraction. This book offers the reader an overview of recent developments of multiple fixed-point theorems and their applications. About the Authors Svetlin G. Georgiev is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations and dynamic calculus on time scales. Khaled Zennir is assistant professor at Qassim University, KSA. He received his PhD in mathematics in 2013 from Sidi Bel Abbs University, Algeria. He obtained hisHabilitation in mathematics from Constantine University, Algeria in 2015. His research interests lie in nonlinear hyperbolic partial differential equations: global existence, blow up and long-time behavior The aim of this book is to present a clear and well-organized treatment of the concept behind the development of mathematics and solution techniques.