Nonlinear Differential Equations in Ordered Spaces: 1st Edition (Hardback) book cover

Nonlinear Differential Equations in Ordered Spaces

1st Edition

By S. Carl, Seppo Heikkila

Chapman and Hall/CRC

336 pages

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pub: 2000-06-14
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Description

Extremality results proved in this Monograph for an abstract operator equation provide the theoretical framework for developing new methods that allow the treatment of a variety of discontinuous initial and boundary value problems for both ordinary and partial differential equations, in explicit and implicit forms. By means of these extremality results, the authors prove the existence of extremal solutions between appropriate upper and lower solutions of first and second order discontinuous implicit and explicit ordinary and functional differential equations. They then study the dependence of these extremal solutions on the data.

The authors begin by developing an existence theory for an abstract operator equation in ordered spaces and offer new tools for dealing with different kinds of discontinuous implicit and explicit differential equation problems. They present a unified approach to the existence of extremal solutions of quasilinear elliptic and parabolic problems and extend the upper and lower solution method to elliptic and parabolic inclusion of hemivariation type using variational and nonvariational methods.

Nonlinear Differential Equations in Ordered Spaces includes research that appears for the first time in book form and is designed as a source book for pure and applied mathematicians. Its self-contained presentation along with numerous worked examples and complete, detailed proofs also make it accessible to researchers in engineering as well as advanced students in these fields.

Reviews

"This is a very interesting monograph written by well-known experts on iterative techniques for differential and functional equations. It mainly contains results obtained by the authors during the last few years."

--Salvatore A. Marano in Mathematical Reviews

Table of Contents

OPERATOR EQUATIONS IN ORDERED SPACES AND FIRST APPLICATIONS

Operator Equations in Ordered Spaces

Applications to Differential Equations

EXTREMALITY RESULTS FOR FIRST ORDER DIFFERENTIAL EQUATIONS

Explicit Initial Value Problems

Explicit Boundary Value Problems

Explicit Functional Problems

Implicit Functional Problems

UNIQUENESS, COMPARISON AND WELL-POSEDNESS RESULTS FOR QUASILINEAR DIFFERENTIAL EQUATIONS

First Order Boundary Value Problems

First Order Initial Value Problems

Well-Posedness Results

Second Order Problems

SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

Explicit Sturm-Liouville Boundary Value Problems

Implicit Sturm-Liouville Boundary Value Problems

Convergence of Successive Approximations

Explicit Phi-Laplacian Problems

Implicit Phi-Laplacian Problems

EXTREMALITY RESULTS FOR QUASILINEAR PDE

Quasilinear Elliptic Boundary Value Problems

Quasilinear Parabolic Problems

Discontinuous Quasilinear Problems

DIFFERENTIAL INCLUSIONS OF HEMIVARIATIONAL INEQUALITY TYPE

Quasilinear Elliptic Inclusions with State-Dependent Subdifferentials

State-Dependent Subdifferentials Perturbed by Discontinuous Nonlinearities

Elliptic Inclusions with Generalized Gradients

Quasilinear Parabolic Inclusions with State-Dependent Subdifferentials

DISCONTINUOUS IMPLICIT ELLIPTIC AND PARABOLIC PROBLEMS

Statement of the Problem and Notations

Preliminaries

Main Results

Implicit Elliptic Problems

APPENDIX

Analysis in Ordered Spaces

Inequalities

Sobolev Spaces

Pseudomonotone and Quasilinear Elliptic Operators

Nonlinear First Order Evolution Equations

Nonsmooth Analysis

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics