Nonlinear Differential Equations in Ordered Spaces

Nonlinear Differential Equations in Ordered Spaces

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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 more

Product details

  • Hardback | 336 pages
  • 160 x 237.5 x 19.3mm | 726.57g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 2003.
  • 1584880686
  • 9781584880684

Review quote

"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 Reviewsshow more

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 Analysisshow more