Methods in Nonlinear Integral Equations

By: R Precup

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Hardcover | 31 August 2002

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236 Pages

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Methods in Nonlinear Integral Equations presents several extremely fruitful methods for the analysis of systems and nonlinear integral equations. They include: fixed point methods (the Schauder and Leray-Schauder principles), variational methods (direct variational methods and mountain pass theorems), and iterative methods (the discrete continuation principle, upper and lower solutions techniques, Newton's method and the generalized quasilinearization method). Many important applications for several classes of integral equations and, in particular, for initial and boundary value problems, are presented to complement the theory. Special attention is paid to the existence and localization of solutions in bounded domains such as balls and order intervals. The presentation is essentially self-contained and leads the reader from classical concepts to current ideas and methods of nonlinear analysis.

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"This book deals with several methods of nonlinear analysis for the investigation of nonlinear integral equations ... . Necessary abstract results of nonlinear analysis ... are provided. ... new points of view, extensions and applications are presented. ... The presentation is self-contained and therefore should be useful to find the current ideas and methods." (V. Lakshmikantham, Zentralblatt MATH, Vol. 1060 (11), 2005)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Differential Calculus & Equations Integral Calculus & Equations Optimisation Linear Programming

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Published: 29th September 2011

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Preface	p. xi
Notation	p. xiii
Overview	p. 1
Fixed Point Methods	p. 11
Compactness in Metric Spaces	p. 13
Hausdorff's Theorem	p. 13
The Ascoli-Arzela Theorem	p. 15
The Frechet-Kolmogorov Theorem	p. 18
Completely Continuous Operators on Banach Spaces	p. 25
Completely Continuous Operators	p. 25
Brouwer's Fixed Point Theorem	p. 28
Schauder's Fixed Point Theorem	p. 32
Continuous Solutions of Integral Equations via Schauder's Theorem	p. 35
The Fredholm Integral Operator	p. 35
The Volterra Integral Operator	p. 37
An Integral Operator with Delay	p. 39
The Leray-Schauder Principle and Applications	p. 43
The Leray-Schauder Principle	p. 43
Existence Results for Fredholm Integral Equations	p. 45
Existence Results for Volterra Integral Equations	p. 50
The Cauchy Problem for an Integral Equation with Delay	p. 53
Periodic Solutions of an Integral Equation with Delay	p. 55
Existence Theory in L[superscript p] Spaces	p. 61
The Nemytskii Operator	p. 61
The Fredholm Linear Integral Operator	p. 64
The Hammerstein Integral Operator	p. 68
Hammerstein Integral Equations	p. 69
Volterra-Hammerstein Integral Equations	p. 72
References: Part I	p. 77
Variational Methods	p. 83
Positive Self-Adjoint Operators in Hilbert Spaces	p. 85
Adjoint Operators	p. 85
The Square Root of a Positive Self-Adjoint Operator	p. 88
Splitting of Linear Operators in L[superscript p] Spaces	p. 90
The Frechet Derivative and Critical Points of Extremum	p. 97
The Frechet Derivative. Examples	p. 97
Minima of Lower Semicontinuous Functionals	p. 103
Application to Hammerstein Integral Equations	p. 107
The Mountain Pass Theorem and Critical Points of Saddle Type	p. 111
The Ambrosetti-Rabinowitz Theorem	p. 111
Flows and Generalized Pseudo-Gradients	p. 117
Schechter's Bounded Mountain Pass Theorem	p. 121
Nontrivial Solutions of Abstract Hammerstein Equations	p. 129
Nontrivial Solvability of Abstract Hammerstein Equations	p. 129
Nontrivial Solutions of Hammerstein Integral Equations	p. 132
A Localization Result for Nontrivial Solutions	p. 137
References: Part II	p. 145
Iterative Methods	p. 149
The Discrete Continuation Principle	p. 151
Perov's Theorem	p. 151
The Continuation Principle for Contractive Maps on Generalized Metric Spaces	p. 154
Hammerstein Integral Equations with Matrix Kernels	p. 159
Monotone Iterative Methods	p. 163
Ordered Banach Spaces	p. 163
Fixed Point Theorems for Monotone Operators	p. 167
Monotone Iterative Technique for Fredholm Integral Equations	p. 172
Minimal and Maximal Solutions of a Delay Integral Equation	p. 177
Methods of Upper and Lower Solutions for Equations of Hammerstein Type	p. 184
Quadratically Convergent Methods	p. 195
Newton's Method	p. 196
Generalized Quasilinearization for an Integral Equation with Delay	p. 201
References: Part III	p. 211
Index	p. 217
Table of Contents provided by Ingram. All Rights Reserved.

Methods in Nonlinear Integral Equations

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