Differential Equations with Involutions
This monograph covers the existing results regarding Green’s functions for differential equations with involutions (DEI).The first part of the book is devoted to the study of the most useful aspects of involutions from an analytical point of view and the
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Alberto Cabada F. Adrián F. Tojo
Differential Equations with Involutions
Atlantis Briefs in Differential Equations Volume 2
Series editors Zuzana Dosla, Brno, Czech Republic Sarka Necasova, Prague 1, Czech Republic Milan Pokorny, Praha 8, Czech Republic
About this Series The aim of the series is rapid dissemination of new results and original methods in the theory of Differential Equations, including topics not yet covered by standard monographs. The series features compact volumes of 75–200 pages, written in a concise, clear way and going directly to the point; the introductory material should be restricted to a minimum or covered by suitable references. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books AMSTERDAM—PARIS—BEIJING ATLANTIS PRESS Atlantis Press 29, avenue Laumière 75019 Paris, France
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Alberto Cabada F. Adrián F. Tojo •
Differential Equations with Involutions
Alberto Cabada Departamento de Análise Matemática Universidade de Santiago de Compostela Santiago de Compostela Spain
F. Adrián F. Tojo Departamento de Análise Matemática Universidade de Santiago de Compostela Santiago de Compostela Spain
ISSN 2405-6405 ISSN 2405-6413 (electronic) Atlantis Briefs in Differential Equations ISBN 978-94-6239-120-8 ISBN 978-94-6239-121-5 (eBook) DOI 10.2991/978-94-6239-121-5 Library of Congress Control Number: 2015955852 © Atlantis Press and the author(s) 2015 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
To my dear mother Dolores In loving memory of my father Eliseo Alberto Cabada To my dear brother Jacobo F. Adrián F. Tojo
Preface
Involutions have been an interesting subject of research at least since Rothe first computed the number of different involutions on finite sets in 1800 [1]. After that, Babbage published in 1815 [2] the foundational paper in which functional equations are first considered, in particular those of the form f ðf ðtÞÞ ¼ t which are called involution equations.1 Despite the progresses on the theory of functional equations, we have to wait for Silberstein, who in 1940 [3] solved the first functional differential equation with an involution. The interest on differential equations with involutions is retaken by Wiener in 1969 [8]. Wiener, together with Watkins, will lead the discoveries in this direction in the following decades [4–11]. Quite a lot of work has been done ever since by several authors. We make a brief review of this in Chap. 2. In 2013 the first Green’s function for a differential equation with an involution was computed [12] and the field rapidly expanded [13–16]. This monograph goes through those discoveries related to Green’s functions. In order to do t
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