Nonlinear Differential Equations

Nonlinear Differential Equations

1st Edition - January 1, 1980

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  • Authors: Svatopluk Fucik, Alois Kufner
  • eBook ISBN: 9781483278377

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Studies in Applied Mathematics, 2: Nonlinear Differential Equations focuses on modern methods of solutions to boundary value problems in linear partial differential equations. The book first tackles linear and nonlinear equations, free boundary problem, second order equations, higher order equations, boundary conditions, and spaces of continuous functions. The text then examines the weak solution of a boundary value problem and variational and topological methods. Discussions focus on general boundary conditions for second order ordinary differential equations, minimal surfaces, existence theorems, potentials of boundary value problems, second derivative of a functional, convex functionals, Lagrange conditions, differential operators, Sobolev spaces, and boundary value problems. The manuscript examines noncoercive problems and vibrational inequalities. Topics include existence theorems, formulation of the problem, vanishing nonlinearities, jumping nonlinearities with finite jumps, rapid nonlinearities, and periodic problems. The text is highly recommended for mathematicians and engineers interested in nonlinear differential equations.

Table of Contents

  • Preface

    List of Symbols

    Chapter I. Some Examples to Begin with

    Section 1. Various Notations. Linear Equations

    Section 2. Nonlinear Equations

    Section 3. Nonlinear Systems

    Section 4. Further Nonlinear Problems

    Section 5. A Free Boundary Problem. The Plate Equation

    Chapter II. Introduction

    Section 6. Second Order Equations

    Section 7. Higher Order Equations

    Section 8. Spaces of Continuous Functions. Solution of a Differential Equation

    Section 9. Boundary Conditions

    Section 10. Solution of a Boundary Value Problem

    Section 11. On an Integral Identity

    Chapter III. The Weak Solution of a Boundary Value Problem

    Section 12. The Carathéodory Property and the Němyckiĭ Operators

    Section 13. Sobolev Spaces

    Section 14. Differential Operators

    Section 15. Boundary Value Problems

    Section 16. Various Generalizations

    Section 17. Regularity of the Weak Solution

    Chapter IV. The Variational Method

    Section 18. First Derivative of a Functional

    Section 19. Potentials of Boundary Value Problems

    Section 20. The Euler Necessary Condition

    Section 21. Second Derivative of a Functional

    Section 22. Lagrange Conditions

    Section 23. Convex Functionals

    Section 24. Weak Convergence and Weak Compactness

    Section 25. Reflexive Spaces

    Section 26. Existence Theorems

    Section 27. Minimal Surfaces

    Section 28. Excursion on Numerical Methods

    Chapter V. The Topological Method

    Section 29. Existence Theorems

    Section 30. The Brouwer and the Leray-Schauder Degree of a Mapping

    Section 31. General Boundary Conditions for Second Order Ordinary Differential Equations

    Section 32. Summary of Chapters IV and V. Some Additional Remarks

    Chapter VI. Noncoercive Problems

    Section 33. Vanishing Nonlinearities. Regular Case

    Section 34. Vanishing Nonlinearities. Singular Case

    Section 35. Jumping Nonlinearities with Finite Jumps

    Section 36. Jumping Nonlinearities with Infinite Jumps

    Section 37. Rapid Nonlinearities

    Section 38. Periodic Problems

    Chapter VII. Variational Inequalities

    Section 39. Formulation of the Problem

    Section 40. More on the Definition of the Solution of a Variational Inequality

    Section 41. Examples

    Section 42. Some Special Results

    Section 43. Existence Theorems



Product details

  • No. of pages: 360
  • Language: English
  • Copyright: © Elsevier 1980
  • Published: January 1, 1980
  • Imprint: Elsevier
  • eBook ISBN: 9781483278377

About the Authors

Svatopluk Fucik

Alois Kufner

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