## Functional Differential Equations with Infinite DelayIn the theory of functional differential equations with infinite delay, there are several ways to choose the space of initial functions (phase space); and diverse (duplicated) theories arise, according to the choice of phase space. To unify the theories, an axiomatic approach has been taken since the 1960's. This book is intended as a guide for the axiomatic approach to the theory of equations with infinite delay and a culmination of the results obtained in this way. It can also be used as a textbook for a graduate course. The prerequisite knowledge is foundations of analysis including linear algebra and functional analysis. It is hoped that the book will prepare students for further study of this area, and that will serve as a ready reference to the researchers in applied analysis and engineering sciences. |

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### Contents

Phase spaces 1 | ix |

Fundamental theorems | 35 |

Stieltjes integrals and linear operators on | 53 |

Copyright | |

9 other sections not shown

### Common terms and phrases

a-periodic A(L+f absolutely continuous assume asymptotically stable Axiom C1 Banach space Borel measurable Borel set bounded linear operator bounded set bounded variation compact set compactly consider constant continuous function converges Corollary D-S integrable defined denote eigenspace Equation 1.1 equations with infinite exists a sequence fading memory space fdg is D-S fi(F finite following result function f functional differential equations Furthermore Hence holds implies inequality infinite delay Lebesgue measure Lemma locally bounded Lp(ji matrix measurable function Moreover Naito norm periodic solution phase space proof of Theorem Proposition 1.1 prove R-bounded solution Relation relatively compact resp respect Rudin satisfies Axiom Section seminorm solution of Equation solution of System solution semigroup SQ(t stability properties subset Suppose System G t i a t i s Theorem 2.1 uniform fading memory uniformly bounded uniformly continuous uniformly equicontinuous unique WUAS zero solution