Methods in Nonlinear Integral Equations

1. Front Matter

   Pages i-xiv

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2. Overview

   1. Overview

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      Pages 1-9

3. Fixed Point Methods

   1. Front Matter

      Pages 11-11

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   2. Compactness in Metric Spaces

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      Pages 13-23

   3. Completely Continuous Operators on Banach Spaces

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      Pages 25-34

   4. Continuous Solutions of Integral Equations via Schauder’s Theorem

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      Pages 35-41

   5. The Leray-Schauder Principle and Applications

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      Pages 43-60

   6. Existence Theory in L ^p Spaces

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      Pages 61-76

4. Variational Methods

   1. Front Matter

      Pages 83-83

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   2. Positive Self-Adjoint Operators in Hilbert Spaces

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      Pages 85-96

   3. The Fréchet Derivative and Critical Points of Extremum

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      Pages 97-110

   4. The Mountain Pass Theorem and Critical Points of Saddle Type

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      Pages 111-127

   5. Nontrivial Solutions of Abstract Hammerstein Equations

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      Pages 129-144

5. Iterative Methods

   1. Front Matter

      Pages 149-149

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   2. The Discrete Continuation Principle

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      Pages 151-162

   3. Monotone Iterative Methods

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      Pages 163-194

   4. Quadratically Convergent Methods

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      Pages 195-210

6. Back Matter

   Pages 217-218

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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.

• Boundary value problem
• Derivative
• Hilbert space
• Integral equation
• Newton's method
• compactness
• differential equation
• functional analysis
• linear optimization
• partial differential equation
• ordinary differential equations

From the reviews:

"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)

• Department of Applied Mathematics, Babeş-Bolyai University, Cluj, Romania

  Radu Precup

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