An Introduction to Minimax Theorems and Their Applications to Differential Equations (Nonconvex Optimization and Its Applications, V. 52)


An Introduction to Minimax Theorems and Their Applications to Differential Equations (Nonconvex Optimization and Its Applications, V. 52)

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  • 製本 Hardcover:ハードカバー版/ページ数 269 p.
  • 言語 ENG
  • 商品コード 9780792368328
  • DDC分類 515.35

Full Description

This text is meant to be an introduction to critical point theory and its ap- plications to differential equations. It is designed for graduate and postgrad- uate students as well as for specialists in the fields of differential equations, variational methods and optimization. Although related material can be the treatment here has the following main purposes: found in other books, * To present a survey on existing minimax theorems, * To give applications to elliptic differential equations in bounded do- mains and periodic second-order ordinary differential equations, * To consider the dual variational method for problems with continuous and discontinuous nonlinearities, * To present some elements of critical point theory for locally Lipschitz functionals and to give applications to fourth-order differential equa- tions with discontinuous nonlinearities, * To study homo clinic solutions of differential equations via the varia- tional method. The Contents of the book consist of seven chapters, each one divided into several sections. A bibliography is attached to the end of each chapter.In Chapter I, we present minimization theorems and the mountain-pass theorem of Ambrosetti-Rabinowitz and some of its extensions. The con- cept of differentiability of mappings in Banach spaces, the Fnkhet's and Gateaux derivatives, second-order derivatives and general minimization the- orems, variational principles of Ekeland [EkI] and Borwein & Preiss [BP] are proved and relations to the minimization problem are given. Deformation lemmata, Palais-Smale conditions and mountain-pass theorems are consid- ered.

Table of Contents

Preface                                            ix
Minimization and Mountain-Pass Theorems 1 (50)
Differential Calculus for Mappings in 2 (9)
banach Spaces
Variational Principles and Minimization 11 (11)
Deformation Theorems and Palais-Smale 22 (13)
Mountain-Pass Theorems 35 (16)
Saddle-Point and Linking Theorems 51 (30)
Saddle-Point Theorems 52 (9)
Local Linking and Three Critical Points 61 (11)
Linking of Deformation Type and Generalized 72 (9)
Saddle-Point Theorems
Applications To Elliptic Problems In Bounded 81 (32)
Neumann Problem for Semilinear Second-order 83 (10)
Elliptic Equations
Existence Results for Hammerstein Integral 93 (8)
Equations with Positive Kernel
Nontrivial Solutions of Hammerstein 101(12)
Integral Equations with Indefinite Kernel
Periodic Solutions For Some Second-Order 113(26)
Differential Equations
The Quadratic Form I 114(12)
Periodic Solutions of Equation (E) 126(13)
Dual Variational Method and Applications 139(34)
Legendre--Fenchel Transform and Duality 141(8)
Applications to Problems for Semilinear 149(7)
Fourth-order Differential Equations with
Continuous Nonlinearity
Applications to Problems for Semilinear 156(17)
Fourth-order Differential Equations with
Discontinuous Nonlinearity
Minimax Theorems For Locally Lipschitz 173(34)
Functionals and Applications
Generalized Gradients 174(10)
Mountain-Pass Theorems for Locally 184(13)
Lipschitz Functionals
Applications to Differential Equations with 197(10)
Discontinuous Nonlinearities
Homoclinic Solutions of Differential Equations 207(58)
Preliminaries on Dynamical Systems 209(5)
Positive Homoclinic Solutions for a Class 214(13)
of Second-Order Differential Equations
Homoclinic Solutions for the Extended 227(6)
Fisher-Kolmogorov Equations
Nontrivial Solutions to the Semilinear 233(7)
Schrodinger Equation on Rn
Semilinear Schrodinger Equations in 240(6)
Strip-like Domains
Nontrivial Solutions to a Semilinear 246(19)
Equation relative to a Dirichlet Form
Notations 265(2)
Index 267