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377

A. M. Fink Iowa State University of Science and Technology, Ames, lA/USA

Almost Periodic Differential Equations

Springer-Verlag Berlin' Heidelberg' New York 1974

AMS Subject Classifications (1970): 34-02, 34-C-25, 42-02, 42-A-84, 34-C-30, 34-0-20 ISBN 3-540-06729-9 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-06729-9 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin' Heidelberg 1974. Library of Congress Catalog Card Number74-391. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

Preface

These lecture notes are written with the hope that the recent advances in the subject of almost periodic differential equations can become more accessible if the basic facts are collected in one place.

It is well known that

in Celestial Mechanics,

almost periodic solutions and stable solutions are intimately related.

In the same way, stable electronic circuits exhibit

almost periodic behavior.

A vast amount of research has been

directed toward studying these phenomena.

A great portion of the

roughly five hundred items in our bibliography are dated after

1955. These lecture notes are about almost periodic solution to ordinary differential equations.

I spend the first four chapters

on the theory of almost periodic functions.

Included in those

chapters is the skeleton of almost periodic theory.

I have

taken the tack of presenting only that material which is germane to the later chapters on differential equations.

I include

essentially no fact about almost periodic functions which is not used to prove something else.

This is no virtue.

It illustrates

the depth of the thoery that is developed here. These notes are self-contained except for the usual preliminary facts about existence and uniqueness of solutions of differential equations.

It should therefore be accessible to a wide audience.

I have taken the periodic case as motivation for much of the material, so an acquaintance with the fourier series theory of periodic functions is helpful.

IV

Much of the material in the first four chapters appears in other books.

Some of the material, however, is only available in

research papers.

This is primarily the case because the theory

of almost periodic differential equations has been a rich source for new developments in almost periodic functions that are not well-known. I have tried to give references to specific results.

The

reader will find an extensive bibliography of the sUbject.

In

addition, the notes at the end of each chapter are helpful in identifying sources of results