Functional and impulsive differential equations of fractional order : qualitative analysis and applications PDF
Preview Functional and impulsive differential equations of fractional order : qualitative analysis and applications
Functional and Impulsive Differential Equations of Fractional Order Qualitative Analysis and Applications Functional and Impulsive Fermented Meat Products Differential Equations of Health Aspects Fractional Order Qualitative Analysis and Applications Edited by Nevijo Zdolec University of Zagreb, Faculty of Veterinary Medicine Department of Hygiene Technology and Food Safety Heinzelova 55 10000 Zagreb, Croatia Ivanka M. Stamova and Gani Tr. Stamov The University of Texas at San Antonio Texas, USA CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20160707 International Standard Book Number-13: 978-1-4987-6483-4 (Hardback) This book contains information obtained from authentic and highly regarded sources. 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For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com This book is dedicated to Trayan and Alex Preface This book is an exposition of the most recent results related to the qualitative analysisofavarietyoffractional-orderequations. The mathematical investigations of derivatives of non-integer order mark their beginning with the correspondence between Leibniz and L’Hospital in 1695(see[Leibniz1695],[Podlubny1999]).Althoughfractionalcalculushasa more than three centuries long history, the subject of fractional differential equations has gained considerable popularity and importance during the past few decades, due mainly to its demonstrated applications to many real-world phenomenastudiesinphysics,mechanics,chemistry,engineering,finance,etc. There are several excellent books devoted to equations of fractional order. The book written by I. Podlubny (see [Podlubny 1999]) played an outstanding role in the development of the theory of fractional ordinary differential equations. It is the most cited and used book which is entirely devoted to a systematic presentation of the basic ideas and methods of fractional calculus in the theory and applications of such equations. The monograph of Kilbas, Srivastava and Trujillo (2006) provides uptodate developments on fractional differential and fractional integro-differential equations involving many differentpotentiallyusefuloperatorsoffractionalcalculus.Inadditiontothese, the books of Abbas, Benchohra and N’Gue´re´kata (2012), Baleanu, Diethelm, ScalasandTrujillo(2012),Caponetto,Dongola,FortunaandPetra´s˘ (2010),Das (2011), Diethelm (2010), Hilfer (2000), Kiryakova (1994), Lakshmikantham, Leela and Vasundhara Devi (2009), Magin (2006), Miller and Ross (1993) are good sources for the theory of fractional operators and equations, as well as theirnumerousapplications. On the other hand, relatively recently functional differential equations of fractional order have started to receive an increasing interest [Baleanu, Sadati, Ghaderi, Ranjbar, Abdeljawad and Jarad 2010], [Banas´ and 2011], [Benchohra, Henderson, Ntouyas and Ouahab 2008], [El-Sayed, Gaaraf and vii viii (cid:4) FunctionalandImpulsiveDifferentialEquationsofFractionalOrder Hamadalla 2010], [Henderson and Ouahab 2009], [Lakshmikantham 2008]. Indeed,fractionaloperatorsareaverynaturaltooltomodelmemory-dependent phenomena. Also, fractional calculus has been incorporated into impulsive differential equations [Ahmad and Nieto 2011], [Ahmad and Sivasundaram 2010], [Anguraj and Maheswari 2012], [Cao and Chen 2012], [Chauhan and Dabas 2011], [Fecˇkan, Zhou and Wang 2012]. Many interesting results on the fundamental theory of such equations have been reported. However, due to the lackofabookonthesetopics,manyresearchersremainunawareofthisfield. The questions related to the qualitative theory of different classes of differential equations are the age-old problems of a great importance. The methods used in the qualitative investigation of their solutions, and their wide applications have all advanced to the extent that aspects in each of these areas havedemandedindividualattention. Thesurveypublishedin2011byLiandZhang(see[LiandZhang2011]),is a very good overview on the recent stability results of fractional differential equations without impulses and delays, and the analytical methods used. It is seen that, at the time, a few stability results rely on a restrictive modeling of fractional differential systems: the basic hypothesis deals with commensurability, i.e. the fractional derivative orders have to be an integer multipleofminimalfractionalorder.Inthelastdecades,manyresearchershave moreinterestsinthestabilityoflinearsystemsandsomemethodshaveemerged in succession. For example, there are the frequency domain methods, Linear Matrix Inequalities methods, and conversion methods [Li and Zhang 2011]. By contrast,thedevelopmentofstabilityofnonlinearfractionaldifferentialsystems evenwithoutimpulsiveperturbationswasabitslow. During the last few years the authors’ research in the area of the qualitative theory of different classes of functional and impulsive fractional-order equations have undergone rapid development. A string of extensive results on thestability,boundedness,asymptoticbehaviorandalmostperiodicityforthese classes of equations have been obtained. The primary aim of this book is to gatherunderonecovermanyoftheseresultswhichwillbeofprimeimportance for researchers on the topic. It fills a void by making available a source book whichdescribesexistingliteratureonthetopic,methodsandtheirdevelopment. The second motivation comes from the applicable point of view, since the qualitative properties have significant practical applications in the emerging areas such as optimal control, biology, mechanics, medicine, bio-technologies, electronics, economics, etc. For the applied scientists it is important to have an introduction to the qualitative theory of fractional equations, which could help intheirinitialstepstoadopttheresultsandmethodsintheirresearch. Thebookconsistsoffourchapters.Itpresentsresultsfordifferentclassesof fractional equations, including fractional functional differential equations, fractional impulsive differential equations, fractional impulsive functional differentialequations,whichhavenotbeencoveredbyotherbooks.Itshowsthe Preface (cid:4) ix manifestations of different constructive methods by demonstrating how these effective techniques can be applied to investigate qualitative properties of the solutions of fractional systems. Since many applications are also included, the demonstrated techniques and models can be used in training of students in mathematical modeling and as an instigation in the study and development of fractional-ordermodels. The book is addressed to a wide audience of professionals such as mathematicians,appliedresearchersandpractitioners. The authors are extremely grateful and very much indebted to Dr. Sandy Norman, Chair of the Department of Mathematics at the University of Texas at San Antonio for ensuring the opportunity for successful work on this book. In addition, the authors have the pleasure to express their sincere gratitude to all theirco-authors,theworkwithwhomexpandedtheirexperience.Theyarealso thankful to all friends, colleagues and reviewers for their valuable comments andsuggestionsduringthepreparationofthemanuscript. I.M.StamovaandG.Tr.Stamov Correspondingauthor:[email protected]
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