Table Of ContentSpringer 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
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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
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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
