Functional Differential Equations (Applied Mathematical Sciences)

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. Front Matter....Pages i-ix Introduction....Pages 1-10 A General Initial Value Problem....Pages 11-12 Existence....Pages 13-15 Continuation of Solutions....Pages 16-20 Continuous Dependence and Uniqueness....Pages 21-23 Backward Continuation....Pages 24-29 Caratheodory Conditions....Pages 30-31 Remarks on the Map Defined by Solutions....Pages 32-42 Autonomous Systems....Pages 43-46 Definitions of Stability....Pages 47-50 Sufficient Conditions for Stability of General Systems....Pages 51-64 Sufficient Conditions for Instability....Pages 65-68 Stability in Autonomous Systems....Pages 69-71 An Example of Levin and Nohel....Pages 72-77 An Equation of Volterra....Pages 78-79 Nonhomogeneous Linear Systems....Pages 80-87 The “Adjoint” Equation and Representation of Solutions....Pages 88-90 Stability of Perturbed Systems....Pages 91-93 Linear Autonomous Equations. The Semigroup and Infinitesimal Generator....Pages 94-97 The Eigenvalues of a Linear Autonomous Equation. Decomposition of C.....Pages 98-103 Decomposing C with the Adjoint Equation....Pages 104-111 Estimates on the Complementary Subspace....Pages 112-115 An Example....Pages 116-119 The Decomposition in the Variation of Constants Formula....Pages 120-124 Forced Linear Systems....Pages 125-130 The Saddle Point Property....Pages 131-141 A Fixed Point Theorem for Cones....Pages 142-151 A Periodicity Theorem for Functional Equations....Pages 152-155 The Equation $${\rm{\dot x}}\left( {\rm{t}} \right) = - \alpha {\rm{x}}\left( {{\rm{t}} - 1} \right)\left[ {{\rm{1}} + {\rm{x}}\left( {\rm{t}} \right)} \right]$$ ....Pages 156-161 The Equation $${\rm{\dot x}}\left( {\rm{t}} \right) = - \alpha {\rm{x}}\left( {{\rm{t}} - 1} \right)\left[ {{\rm{l}} - {\rm{x}}^2 \left( {\rm{t}} \right)} \right]$$ ....Pages 162-163 The Equation $${\rm{\ddot x}}\left( {\rm{t}} \right) + {\rm{f}}\left( {{\rm{x}}\left( {\rm{t}} \right){\rm{\dot x}}\left( {\rm{t}} \right)} \right) + {\rm{g}}\left( {{\rm{x}}\left( {{\rm{t}} - {\rm{r}}} \right)} \right) = 0$$ ....Pages 164-176 The “Adjoint” Equation for General Linear Systems....Pages 177-181 The True Adjoint of a Linear System....Pages 182-186 Boundary Value Problems....Pages 187-195 Linear Periodic Systems. General Theory....Pages 196-202 Decomposition of Linear Periodic Systems....Pages 203-212 Nondegenerate Periodic Orbits....Pages 213-220 Notes and Remarks....Pages 221-226 Back Matter....Pages 227-239 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 differƯ 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 [by] J. Hale. Second Ed. Published In 1977 Under Title: Theory Of Functional Differential Equations. Bibliography: P. 227-236.