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1473

Yoshiyuki Hino Satoru Murakami Toshiki Naito

Functional Differential Equations with Infinite Delay

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors Yoshiyuki Hino Department of Mathematics Chiba University Yayoicho, Chiba 260, Japan Satoru Murakami Department of Applied Mathematics Okayama University of Science Ridaicho, Okayama 700, Japan Toshiki Naito Department of Mathematics The University of Electro-Communications Chofu, Tokyo 182, Japan

Mathematics Subject Classification (1980): 34Kxx, 34C25, 34C27, 34D05, 34DIO, 34D20, 45105, 45MIO, 47D05, 26A42

ISBN 3-540-54084-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54084-9 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidel berg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

Dedicated to Professor Taro Yoshizawa on his 70th birthday

PREFACE

The aim of this text is to deal with functional differential equations with infinite delay on an abstract phase space characterized by several axioms which are satisfied by different kinds of function spaces.

The standard spaces which we have in mind

are the one of continuous functions on

(-00,0)

that are endowed with

-00, and the one of

some restriction on their asymptotic behavior at measurable functions on

(-00,0)

that are integrable with respect to

some Borel measure equipped with mild conditions.

These spaces are

carefully adopted as phase spaces for equations with infinite delay so as to solve each problem which we encounter in applications, nevertheless many fundamental properties of these equations hold good independently of the choice of phase spaces.

The axiomatic

approach is not only advantageous to summarize these properties, but also supply fruitful ideas and methods to investigate the mathematical structure which reflects the effect of infinite delay. Hence, we now intend to develop a unified theory of this field in terms of functional analysis and dynamical systems.

For the sake

of clear understanding of our ground, many elementary facts and proofs are given as completely as possible.

But, few attempts had

been made to give examples of equations in applications, and only a limited number of references are presented at the end of this text. This text consists of nine chapters.

Chapter 1 contains the

formulation of axioms of the phase space together with many example