State-Dependent Impulses Boundary Value Problems on Compact Interval
This book offers the reader a new approach to the solvability of boundary value problems with state-dependent impulses and provides recently obtained existence results for state dependent impulsive problems with general linear boundary conditions. It cove
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Irena Rachůnková Jan Tomeček
State-Dependent Impulses Boundary Value Problems on Compact Interval
Atlantis Briefs in Differential Equations Volume 6
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
More information about this series at http://www.springer.com/series/13609
Irena Rachůnková Jan Tomeček •
State-Dependent Impulses Boundary Value Problems on Compact Interval
Irena Rachůnková Faculty of Science Palacký University Olomouc Czech Republic
Jan Tomeček Faculty of Science Palacký University Olomouc Czech Republic
ISSN 2405-6405 ISSN 2405-6413 (electronic) Atlantis Briefs in Differential Equations ISBN 978-94-6239-126-0 ISBN 978-94-6239-127-7 (eBook) DOI 10.2991/978-94-6239-127-7 Library of Congress Control Number: 2015948780 © 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
Preface
By writing this book we wanted to present a new approach to impulsive boundary value problems, which is applicable both to the state-dependent and the fixed-time impulses. Existence results for boundary value problems with state-dependent impulses are rather rare in the literature and are given mostly for periodic problems. Our main goal here is to provide general existence principles which can serve as a tool for the investigation of solvability of impulsive boundary value problems, in particular with impulses occurring at moments depending on the state variables. To demonstrate it, we bring new existence theorems and examples for such problems. On the other hand, though there is vast literature dealing with fixed-time impulsive boundary value problems, we decided to include existence results for several important problems of this type, for example, problems with time or space singularities. The book is organized as follows: Part I, which consists of Chaps. 2–5, is devoted to fixed-time impulsive boundary value problems. In Chap. 2 we discuss the solvability of the second order problem with nonlinear boundary conditions and show the application of the lower and upper functions method. Chapters 3 and 4 investigate the
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