Functional Differential Equations

1. The Eigenvalues of a Linear Autonomous Equation. Decomposition of C.

   • Jack K. Hale

   Pages 98-103

2. Decomposing C with the Adjoint Equation

   • Jack K. Hale

   Pages 104-111

3. Estimates on the Complementary Subspace

   • Jack K. Hale

   Pages 112-115

4. An Example

   • Jack K. Hale

   Pages 116-119

5. The Decomposition in the Variation of Constants Formula

   • Jack K. Hale

   Pages 120-124

6. Forced Linear Systems

   • Jack K. Hale

   Pages 125-130

7. The Saddle Point Property

   • Jack K. Hale

   Pages 131-141

8. A Fixed Point Theorem for Cones

   • Jack K. Hale

   Pages 142-151

9. A Periodicity Theorem for Functional Equations

   • Jack K. Hale

   Pages 152-155

10. The Equation \({\rm{\dot x}}\left( {\rm{t}} \right) = - \alpha {\rm{x}}\left( {{\rm{t}} - 1} \right)\left[ {{\rm{1}} + {\rm{x}}\left( {\rm{t}} \right)} \right]\)

    • Jack K. Hale

    Pages 156-161

11. The Equation \({\rm{\dot x}}\left( {\rm{t}} \right) = - \alpha {\rm{x}}\left( {{\rm{t}} - 1} \right)\left[ {{\rm{l}} - {\rm{x}}^2 \left( {\rm{t}} \right)} \right]\)

    • Jack K. Hale

    Pages 162-163

12. The Equation \({\rm{\ddot x}}\left( {\rm{t}} \right) + {\rm{f}}\left( {{\rm{x}}\left( {\rm{t}} \right){\rm{\dot x}}\left( {\rm{t}} \right)} \right) + {\rm{g}}\left( {{\rm{x}}\left( {{\rm{t}} - {\rm{r}}} \right)} \right) = 0\)

    • Jack K. Hale

    Pages 164-176

13. The “Adjoint” Equation for General Linear Systems

    • Jack K. Hale

    Pages 177-181

14. The True Adjoint of a Linear System

    • Jack K. Hale

    Pages 182-186

15. Boundary Value Problems

    • Jack K. Hale

    Pages 187-195

16. Linear Periodic Systems. General Theory

    • Jack K. Hale

    Pages 196-202

17. Decomposition of Linear Periodic Systems

    • Jack K. Hale

    Pages 203-212

18. Nondegenerate Periodic Orbits

    • Jack K. Hale

    Pages 213-220

19. Notes and Remarks

    • Jack K. Hale

    Pages 221-226

20. Back Matter

    Pages 227-239

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It is hoped that these notes will serve as an introduction to the subject of functional differential equations. The topics are very selective and represent only one particular viewpoint. Complementary material dealing with extensions of closely related topics are given in the notes at the end. A short bibliography is appended as source material for further study. The author is very grateful to the Mathematics Department at UCLA for having extended the invitation to give a series of lectures on functional differ­ ential equations during the Applied Mathematics Year, 1968-1969. The extreme interest and sincere criticism of the members of the audience were a constant source of inspiration in the preparation of the lectures as well as the notes. Except for Sections 6, 32, 33, 34 and some other minor modifications, the notes represent the material covered in two quarters at UCLA. The author wishes to thank Katherine McDougall and Sandra Spinacci for their excellent preparation of the text. The author is also indebted to Eleanor Addison for her work on the drawings and to Dr. H. T. Banks for his careful proofreading of this material. Jack K. Hale Providence March 4, 1971 v TABLE OF CONTENTS 1. INTRODUCTION •••••.•..••.•••••••••.•••..•.••••••.••••••.••.••.•••.••• 1 2 • A GENERAL INITIAL VALUE PROBLEM 11 3 • EXISTENCE 13 4. CONTINUATION OF SOLUTIONS 16 CONTINUOUS DEPENDENCE AND UNIQUENESS 21 5.

• Boundary value problem
• Eigenvalue
• differential equation
• functional equation
• stability

• Division of Applied Mathematics, Brown University, Providence, USA

  Jack K. Hale

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