Nonlinear Problems in Abstract Cones

Nonlinear Problems in Abstract Cones

1st Edition - January 28, 1988

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  • Authors: Dajun Guo, V. Lakshmikantham
  • eBook ISBN: 9781483261904

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Notes and Reports in Mathematics in Science and Engineering, Volume 5: Nonlinear Problems in Abstract Cones presents the investigation of nonlinear problems in abstract cones. This book uses the theory of cones coupled with the fixed point index to investigate positive fixed points of various classes of nonlinear operators. Organized into four chapters, this volume begins with an overview of the fundamental properties of cones coupled with the fixed point index. This text then employs the fixed point theory developed to discuss positive solutions of nonlinear integral equations. Other chapters consider several examples from integral and differential equations to illustrate the abstract results. This book discusses as well the fixed points of increasing and decreasing operators. The final chapter deals with the development of the theory of nonlinear differential equations in cones. This book is a valuable resource for graduate students in mathematics. Mathematicians and researchers will also find this book useful.

Table of Contents

  • Preface

    Chapter 1. Basic Properties of Cones

    1.0 Introduction

    1.1 Notarial Cones

    1.2 Regular and Fully Regular Cones

    1.3 Minlhedral and Strongly Minlhedral Cones

    1.4 Positive Linear Functional

    1.5 The e-Norm and Hilbert's Protective Metric

    1.6 Notes and Comments

    Chapter 2. Positive Fixed Point Theory

    2.0 Introduction

    2.1 Fixed Points of Monotone Operators

    2.2 Fixed Points of Concave and Convex Operators

    2.3 Fixed Points ο£ Cone Expansion and Compression

    2.4 Multiple Fixed Point Theorems

    2.5 Fixed Points of Domain Expansion and Compression

    2.6 Notes and Comments

    Chapter 3. Applications to Nonlinear Integal Equations

    3.0 Introduction

    3.1 integral Equations of Polynomial Type

    3.2 Eigenvalues and Elgenfunctions

    3.3 Some Nonlinear Integral Equations Arising in Science

    3.4 Infinitely Many Solutions Obtained by Variational Methods

    3.5 Notes and Comments

    Chapter 4. Applications to Nonlinear Differential Equations

    4.0 Introduction

    4.1 Differential Inequalities

    4.2 Flow-Invariant Sets

    4.3 Method of Upper and Lower Solutions

    4.4 Monotone Iterative Technique

    4.5 Method of Upper and Lower Quasi-Solutions

    4.6 Cone-Valued Lyapunov Functions and Stabltity Theory

    4.7 Notes and Comments


    A.1 Separation Theorems of Convex Sets

    A.2 Zorn's Lemma

    A.3 Leray-Schauder Degree

    A.4 Properties of Nemitskii Operators

    A.5 Extension Theorems


Product details

  • No. of pages: 286
  • Language: English
  • Copyright: © Academic Press 1988
  • Published: January 28, 1988
  • Imprint: Academic Press
  • eBook ISBN: 9781483261904

About the Authors

Dajun Guo

V. Lakshmikantham

Affiliations and Expertise

University of Texas at Arlington, USA

About the Editor

William F. Ames

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