Ebook: Methods in Nonlinear Integral Equations

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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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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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.
Content:
Front Matter....Pages i-xiv
Overview....Pages 1-9
Front Matter....Pages 11-11
Compactness in Metric Spaces....Pages 13-23
Completely Continuous Operators on Banach Spaces....Pages 25-34
Continuous Solutions of Integral Equations via Schauder’s Theorem....Pages 35-41
The Leray-Schauder Principle and Applications....Pages 43-60
Existence Theory in L p Spaces....Pages 61-76
Front Matter....Pages 83-83
Positive Self-Adjoint Operators in Hilbert Spaces....Pages 85-96
The Fr?chet Derivative and Critical Points of Extremum....Pages 97-110
The Mountain Pass Theorem and Critical Points of Saddle Type....Pages 111-127
Nontrivial Solutions of Abstract Hammerstein Equations....Pages 129-144
Front Matter....Pages 149-149
The Discrete Continuation Principle....Pages 151-162
Monotone Iterative Methods....Pages 163-194
Quadratically Convergent Methods....Pages 195-210
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.
Content:
Front Matter....Pages i-xiv
Overview....Pages 1-9
Front Matter....Pages 11-11
Compactness in Metric Spaces....Pages 13-23
Completely Continuous Operators on Banach Spaces....Pages 25-34
Continuous Solutions of Integral Equations via Schauder’s Theorem....Pages 35-41
The Leray-Schauder Principle and Applications....Pages 43-60
Existence Theory in L p Spaces....Pages 61-76
Front Matter....Pages 83-83
Positive Self-Adjoint Operators in Hilbert Spaces....Pages 85-96
The Fr?chet Derivative and Critical Points of Extremum....Pages 97-110
The Mountain Pass Theorem and Critical Points of Saddle Type....Pages 111-127
Nontrivial Solutions of Abstract Hammerstein Equations....Pages 129-144
Front Matter....Pages 149-149
The Discrete Continuation Principle....Pages 151-162
Monotone Iterative Methods....Pages 163-194
Quadratically Convergent Methods....Pages 195-210
Back Matter....Pages 217-218
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