Nonlinear Analysis and Boundary Value Problems: NABVP 2018, Santiago de Compostela, Spain, September 4-7 (Springer Proceedings in Mathematics & Statistics) PDF
Preview Nonlinear Analysis and Boundary Value Problems: NABVP 2018, Santiago de Compostela, Spain, September 4-7 (Springer Proceedings in Mathematics & Statistics)
Springer Proceedings in Mathematics & Statistics Iván Area Alberto Cabada José Ángel Cid Daniel Franco Eduardo Liz Rodrigo López Pouso Rosana Rodríguez-López Editors Nonlinear Analysis and Boundary Value Problems NABVP 2018, Santiago de Compostela, Spain, September 4–7 Springer Proceedings in Mathematics & Statistics Volume 292 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 á Iv n Area Alberto Cabada (cid:129) (cid:129) é Á Jos ngel Cid Daniel Franco (cid:129) (cid:129) ó Eduardo Liz Rodrigo L pez Pouso (cid:129) (cid:129) í ó Rosana Rodr guez-L pez Editors Nonlinear Analysis and Boundary Value Problems NABVP 2018, Santiago de Compostela, – Spain, September 4 7 123 Editors Iván Area AlbertoCabada Departamento deMatemática AplicadaII Departamento deEstatística, Análise Universidade deVigo Matemática eOptimización Ourense,Spain Universidade deSantiagodeCompostela SantiagodeCompostela, ACoruña,Spain JoséÁngel Cid Departamento deMatemáticas DanielFranco Universidade deVigo E.T.S. Ingenieros Industriales Ourense,Spain Universidad NacionaldeEducación a Distancia (UNED) Eduardo Liz Madrid,Spain Departamento deMatemática AplicadaII Universidade deVigo RodrigoLópezPouso Vigo, Pontevedra,Spain Departamento deEstatística, Análise Matemática eOptimización Rosana Rodríguez-López Universidade deSantiagodeCompostela Departamento deEstatística, Análise SantiagodeCompostela, ACoruña,Spain Matemática eOptimización Universidade deSantiagodeCompostela SantiagodeCompostela, ACoruña,Spain ISSN 2194-1009 ISSN 2194-1017 (electronic) SpringerProceedings in Mathematics& Statistics ISBN978-3-030-26986-9 ISBN978-3-030-26987-6 (eBook) https://doi.org/10.1007/978-3-030-26987-6 MathematicsSubjectClassification(2010): 34A08,34BXX,34K37,34KXX,34LXX,35BXX,37-XX, 47HXX,92BXX ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This book consists of contributions presented at the International Conference on Nonlinear Analysis and Boundary Value Problems, held in Santiago de Compostela, Spain, from September 4 to 7, 2018, and is dedicated to Prof. Juan J. Nieto, on the occasion of his 60th birthday. The conference was organized by the Nonlinear Differential Equations Group at the University of Santiago de Compostela. Thebookcomprises17contributionsthatcoverawidevarietyoftopicslinkedto Prof. Nieto’s scientific work, ranging from differential, difference, and fractional equationstoepidemiologicalmodelsanddynamicalsystemsandtheirapplications. Itisprimarilyintendedforresearchersinvolvedinnonlinearanalysisandboundary value problems in a broad sense. Radu Precup presents a variational analogue of Krasnoselskii’s cone compres- sion–expansionfixed-pointtheorem,basedonEkeland’sprinciple.Healsoincludes a general scheme of applications to semilinear equations, making use of Mikhlin’s variational theory on positive linear operators. AurelianCerneastudiesa certain second-order evolution inclusiondefinedby a family of linear closed operators that is the generator for an evolution system of operators and by a set-valued map with nonconvex values in a separable Banach space. In his work, results are provided concerning the differentiability of mild solutions with respect to the initial conditions of the problem considered. The results may be interpreted as extensions to a special class of second-order differ- ential inclusions of the classical Bendixson-Picard–Lindelöf theorem concerning the differentiability of the maximal flow of a differential equation. AntonioPumariño,JoséA.Rodríguez,andEnriqueVigildescribetheattractors for a two-parameter family of two-dimensional piecewise affine maps using mea- sure theory. These piecewise affine maps arise when studying the unfolding of homoclinic tangencies for a certain class of three-dimensional diffeomorphisms. They also prove the existence, for each natural number n, of an open set of parameters in which the respective transformation exhibits at least 2n two-dimensional strange attractors. v vi Preface Vladimir E. Fedorov, Anna S. Avilovich and Lidiya V. Borel study initial problems for semilinear differential equations in Banach spaces with fractional Caputo derivative. They apply abstract results to the research into initial boundary value problems for a class of time-fractional order partial differential equations. Feliz Minhós and Infeliz Coxe consider a system of nth-order differential equations with full nonlinearities coupled with two-point boundary conditions. They provide the solvability of such systems by using a Nagumo condition, lower and upper solutions, and Leray–Schauder degree theory. Moreover, they present two applications: to a Lorentz-Lagrangian system, for n=2, and to a stationary system of Korteweg-de Vries equations, for n=3. MarinaV.PlekhanovaandGuzelD.BaybulatovadealwiththeCauchyproblem for two types of semilinear fractional differential equation in Banach spaces dependingonthelowerorderfractionalCaputoderivatives.Theyprovidesufficient conditions for the existence of a unique solution in both cases and illustrate their abstracts results with two examples: a modified Oskolkov–Benjamin–Bona– Mahony–Burgers nonlinear equation with time-fractional derivatives and a non- linearsystemofpartialdifferentialequationsnotsolvablewithrespecttothehighest time-fractional derivative. Bouchra Ben Amma, Said Melliani, and Lalla Saadia Chadli consider an intu- itionistic fuzzy partial hyperbolic differential equation with integral boundary conditions. They obtain an existence and uniqueness result by means of Banach fixed-point theorem and present a procedure to solve some kinds of intuitionistic fuzzypartialhyperbolicdifferentialequation.Severalillustrativeexamplesarealso presented. OlgaRozanovaandMarkoTurzynskyconsideramodelusedforthedescription ofthedynamicsoftheatmosphereofarotatingplanet.Theirmainresultprovesthat taking into account a small correction due to centrifugal force, which is usually neglected in the literature, drastically changes the stability properties of a specific class of vortices. Jose S. Cánovas deals with the chaotic properties of the two-periodic Ricker model.Inparticular,hecomputesthetopologicalentropyandshowstheparameter region where the dynamics is chaotic for this model. Alberto Cabada and Kadda Maazouz prove the existence, uniqueness, and location of solutions for implicit fractional differential equations involving the Hadamard fractional derivative in Banach spaces. The linear equation has been solved by means of the invertibility of the differential operator. Such an inverse operatorischaracterizedbythekernelofasuitableintegraloperator.Itsqualitative properties concerning the sign and boundedness properties allow the application of the Banach contraction principle and the deduction of the existence and uniqueness of the solution of the considered problem. FranciscoJ.Fernández,AureaMartínez,andLinoJ.Alvarez-Vázquezformulate asuitablesystemofnonlinearpartialdifferentialequationstomodelatechniqueof artificial circulation for oxygenating eutrophic water bodies subject to quality problems. Then, they use fixed-point theory to prove the existence of at least one solution for the considered problem in an appropriate functional space. Preface vii Jiří Kadlec and Petr Nečesal deal with a boundary value problem for a second-order differential equation with a non-local boundary condition in integral form. Their main results describe the structure of the Fučík spectrum for this problem as a pair of regular curves. Peter Tomiczek considers a second-order differential equation of Duffing type and uses a variational approach to prove the existence of at least one periodic solution.Forit,heintroducesasuitablefunctionthatsatisfiestheso-calledPalais– Smale condition. Sebastián Buedo-Fernández, Daniel Cao Labora, and Rosana Rodríguez-López present improvements in some comparison results for the periodic boundary value problemrelatedtoafirst-orderdifferentialequationperturbedbyafunctionalterm. The comparison results presented cover many cases such as differential equations with delay, differential equations with maxima, and integrodifferential equations. The authors also analyze the interesting case of functional perturbation with piecewise constant arguments. Gaber M. Bahaa and Delfim F. M. Torres investigate optimal control problems (OCP)forfractionalsystemsinvolvingfractional-timederivativesontimescales.In their analysis, the fractional-time derivatives and integrals are those of Riemann– Liouville. They consider a fractional OCP with a performance index given as a delta-integralfunctionofbothstateandcontrolvariables,withtimeevolvingonan arbitrarily given time scale. Interpreting the Euler–Lagrange first-order optimality condition with an adjoint problem, defined by means of right Riemann–Liouville fractional delta derivatives, they obtain an optimality system for the considered fractional OCP. For that, the authors prove new fractional integration by parts formulas on time scales. AlbertoCabadaandLucíaLópez-SomozashowseveralpropertiesoftheGreen’s functions related to various boundary value problems of arbitrary even order. As a consequence, they write the expression of the Green’s functions related to the generaldifferentialoperatoroforder2ncoupledtoNeumann,Dirichlet,andmixed boundaryconditionsasalinearcombinationoftheGreen’sfunctionscorresponding toperiodicconditionsonadifferentinterval.Thisallowstoensuretheconstantsign ofvariousrelatedGreen’sfunctionsandtodescribethespectrumoftheconsidered differential operator with a given boundary condition as the union of several spectrums of the same operator with different boundary conditions. Abdelkader Moulay and Abdelghani Ouahab consider an abstract evolution equation with random parameter. They introduce the notion of stabilization with respect to the random parameter and fractional integral-feedback. More precisely, they study the well-posedness and polynomial stabilization result for a random evolution equation with fractional integral-feedback. Finally, they show some applications to random heat and wave equations with fractional integral-feedback and bounded damping. This volume would not have been possible without the help of various people who contributed in different ways. First of all, we would like to thank the authors themselvesforsubmittingtheirworktothisissue.Specialthanksgotothereferees viii Preface whoagreedtotakepartinthisprocess:theircommentsandsuggestionshaveledto improvements in most of the contributions. We would also like to express our gratitude to Francesca Ferrari from Springer for her attention and constant support at every step in the editorial process. Moreover, we want to express our thanks for the financial support provided by XuntadeGaliciaandDeputacióndeACoruña(Spain).Wearealsogratefultothe Faculty of Mathematics of the University of Santiago de Compostela for their supportwithrespecttotheconferencelocationandthefacilitiesavailableduringthe conference. Ourense, Spain Iván Area Santiago de Compostela, Spain Alberto Cabada Ourense, Spain José Ángel Cid Madrid, Spain Daniel Franco Vigo, Spain Eduardo Liz Santiago de Compostela, Spain Rodrigo López Pouso Santiago de Compostela, Spain Rosana Rodríguez-López May 2019 Contents A Variational Analogue of Krasnoselskii’s Cone Fixed Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Radu Precup Differentiability Properties of Solutions of a Second-Order Evolution Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Aurelian Cernea How to Analytically Prove the Existence of Strange Attractors Using Measure Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Antonio Pumariño, José A. Rodríguez and Enrique Vigil Initial Problems for Semilinear Degenerate Evolution Equations of Fractional Order in the Sectorial Case. . . . . . . . . . . . . . . . . . . . . . . . 41 Vladimir E. Fedorov, Anna S. Avilovich and Lidiya V. Borel Solvability for nth Order Coupled Systems with Full Nonlinearities . . . 63 Feliz Minhós and Infeliz Coxe Semilinear Equations in Banach Spaces with Lower Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Marina V. Plekhanova and Guzel D. Baybulatova Integral Boundary Value Problem for Intuitionistic Fuzzy Partial Hyperbolic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Bouchra Ben Amma, Said Melliani and Lalla Saadia Chadli On the Periodic Ricker Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Jose S. Cánovas The Stability of Vortices in Gas on the l-Plane: The Influence of Centrifugal Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Olga Rozanova and Marko Turzynsky ix
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