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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.
List of contents
1. Introduction.- 2. A General Initial Value Problem.- 3. Existence.- 4. Continuation of Solutions.- 5. Continuous Dependence and Uniqueness.- 6. Backward Continuation.- 7. Caratheodory Conditions.- 8. Remarks on the Map Defined by Solutions.- 9. Autonomous Systems.- 10. Definitions of Stability.- 11. Sufficient Conditions for Stability of General Systems.- 12. Sufficient Conditions for Instability.- 13. Stability in Autonomous Systems.- 14. An Example of Levin and Nohel.- 15. An Equation of Volterra.- 16. Nonhomogeneous Linear Systems.- 17. The "Adjoint" Equation and Representation of Solutions.- 18. Stability of Perturbed Systems.- 19. Linear Autonomous Equations. The Semigroup and Infinitesimal Generator.- 20. The Eigenvalues of a Linear Autonomous Equation. Decomposition of C.- 21. Decomposing C with the Adjoint Equation.- 22. Estimates on the Complementary Subspace.- 23. An Example.- 24. The Decomposition in the Variation of Constants Formula.- 25. Forced Linear Systems.- 26. The Saddle Point Property.- 27. A Fixed Point Theorem for Cones.- 28. A Periodicity Theorem for Functional Equations.- 29. The Equation $${rm{dot x}}left( {rm{t}} right) = - alpha {rm{x}}left( {{rm{t}} - 1} right)left[ {{rm{l}} + {rm{x}}left( {rm{t}} right)} right]$$.- 30. The Equation $${rm{dot x}}left( {rm{t}} right) = - alpha {rm{x}}left( {{rm{t}} - 1} right)left[ {{rm{l}} + {rm{x}}^2 left( {rm{t}} right)} right]$$.- 31. The Equation $${rm{ddot x}}left( {rm{t}} right) + {rm{f}}left( {{rm{x}}left( {rm{t}} right){rm{dot x}}left( {rm{t}} right)} right) + {rm{g}}left( {{rm{x}}left( {{rm{t}} - {rm{r}}} right)} right) = 0$$.- 32. The "Adjoint" Equation for General Linear Systems.- 33. The True Adjoint of a Linear System.- 34. Boundary Value Problems.- 35. Linear Periodic Systems. General Theory.- 36. Decomposition of Linear Periodic Systems.- 37. Nondegenerate Periodic Orbits.- 38. Notes and Remarks.
