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Full Description
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.
Contents
0 Overview.- 1 Compactness in Metric Spaces.- 2 Completely Continuous Operators on Banach Spaces.- 3 Continuous Solutions of Integral Equations via Schauder's Theorem.- 4 The Leray-Schauder Principle and Applications.- 5 Existence Theory in LP Spaces.- 6 Positive Self-Adjoint Operators in Hilbert Spaces.- 7 The Fréchet Derivative and Critical Points of Extremum.- 8 The Mountain Pass Theorem and Critical Points of Saddle Type.- 9 Nontrivial Solutions of Abstract Hammerstein Equations.- 10 The Discrete Continuation Principle.- 11 Monotone Iterative Methods.- 12 Quadratically Convergent Methods.
