Boundary value problems for systems of differential, difference and fractional equations : positive solutions

313 Pages·2016·2.653 MB·English

by  Henderson, Johnny, Luca, Rodica

Boundary Value Problems for Systems of
Differential, Difference and Fractional Equations
Boundary Value Problems
for Systems of
Differential, Difference
and Fractional Equations
Positive Solutions
Johnny Henderson and Rodica Luca
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Johnny Henderson dedicates this book to his siblings, Monty, Madonna, Jana, and
Chrissie, and to the memory of his parents, Ernest and Madora. Rodica Luca ded-
icates this book to her husband, Mihai Tudorache, and her son, Alexandru-Gabriel
Tudorache,andtothememoryofherparents,VioricaandConstantinLuca.
Preface
In recent decades, nonlocal boundary value problems for ordinary differential
equations, difference equations, or fractional differential equations have become a
rapidlygrowingareaofresearch. Thestudyofthesetypesofproblemsisdrivennot
only by a theoretical interest, but also by the fact that several phenomena in engi-
neering, physics, and the life sciences can be modeled in this way. Boundaryvalue
problems with positive solutions describe many phenomena in the applied sciences
such as the nonlinear diffusion generated by nonlinear sources, thermal ignition of
gases,andconcentrationinchemicalorbiologicalproblems. Variousproblemsaris-
inginheatconduction,undergroundwaterflow,thermoelasticity,andplasmaphysics
canbereducedtononlineardifferentialproblemswithintegralboundaryconditions.
Fractionaldifferentialequationsdescribe manyphenomenainseveralfieldsof engi-
neeringandscientificdisciplinessuchasphysics,biophysics,chemistry,biology(such
as bloodflow phenomena),economics, controltheory, signal and imageprocessing,
aerodynamics,viscoelasticity,andelectromagnetics.
Hundredsofresearchersareworkingonboundaryvalueproblemsfordifferential
equations,differenceequations,andfractionalequations,andattheheartofthecom-
munity are researchers whose interest is in positive solutions. The authors of this
monographoccupyanicheinthecenterofthatgroup.Themonographcontainsmany
oftheirresultsrelatedtothesetopicsobtainedinrecentyears.
InChapter1, questionsare addressedontheexistence, multiplicity, andnonexis-
tenceofpositivesolutionsforsomeclassesofsystemsofnonlinearsecond-orderordi-
nary differential equations with parameters or without parameters, subject to
Riemann–Stieltjesboundaryconditions,andforwhichthenonlinearitiesarenonsin-
gular or singular functions. Chapter 2 is devoted to the existence, multiplicity, and
nonexistence of positive solutions for some classes of systems of nonlinear higher-
orderordinarydifferentialequations with parameters or withoutparameters, subject
tomultipointboundaryconditions,andforwhichthenonlinearitiesarenonsingularor
singularfunctions.Asystemofhigher-orderdifferentialequationswithsign-changing
nonlinearitiesandRiemann–Stieltjesintegralboundaryconditionsisalsoinvestigated.
Chapter3dealswiththeexistence,multiplicity,andnonexistenceofpositivesolutions
forsomeclassesofsystemsofnonlinearsecond-orderdifferenceequations,alsowith
orwithoutparameters,subjecttomultipointboundaryconditions.
Chapter4isconcernedwiththeexistence,multiplicity,andnonexistenceofpositive
solutionsforsomeclassesofsystemsofnonlinearRiemann–Liouvillefractionaldif-
ferential equations with parameters or without parameters, subject to uncoupled
Riemann–Stieltjesintegralboundaryconditions,andforwhichthenonlinearitiesare
nonsingularorsingularfunctions.Asystemoffractionalequationswithsign-changing
nonlinearities and integral boundary conditions is also investigated. Chapter 5 is
x Preface
focusedontheexistence,multiplicity,andnonexistenceofpositivesolutionsforsome
classes of systems of nonlinear Riemann–Liouville fractional differential equations
with parameters or without parameters, subject to coupled Riemann–Stieltjes inte-
gralboundaryconditions,andforwhichthenonlinearitiesarenonsingularorsingular
functions. A system of fractional equations with sign-changing nonsingular or sin-
gular nonlinearities and integral boundary conditions is also investigated. In each
chapter,variousexamplesarepresentedwhichsupportthemainresults.
Central to the results of each chapter are applications of the Guo–Krasnosel’skii
fixed point theorem for nonexpansiveand noncontractiveoperators on a cone (The-
orem 1.1.1). Unique to applications of the fixed point theorem is the novel rep-
resentation of the Green’s functions, which ultimately provide almost a checklist
for determining conditions for which positive solutions exist relative to given non-
linearities. In the proof of many of the main results, applications are also made
of the Schauder fixed-point theorem (Theorem 1.6.1), the nonlinear alternative of
Leray–Schaudertype(Theorem2.5.1),andsometheoremsfromthefixedpointindex
theory(Theorems1.3.1–1.3.3).
Therehavebeen otherbooksin thepast onpositivesolutionsforboundaryvalue
problems, but in spite of the area receiving much attention, there have been no new
booksrecently. Thismonographprovidesaspringboardforotherresearcherstoem-
ulatetheauthors’methods. Theaudienceforthisbookincludesthefamilyofmath-
ematical and scientific researchers in boundary value problems for which positive
solutions are important, and in addition, the monographcan serve as a great source
fortopicstobestudiedingraduateseminars.
JohnnyHenderson
RodicaLuca
About the authors
JohnnyHenderson isadistinguishedprofessorofMathematicsattheBaylorUniver-
sity,Waco,Texas, USA.He hasalsoheldfacultypositionsattheAuburnUniversity
and the Missouri University of Science and Technology. His published research is
primarilyintheareasofboundaryvalueproblemsforordinarydifferentialequations,
finite difference equations, functional differential equations, and dynamic equations
ontimescales. HeisanInauguralFellowoftheAmericanMathematicalSociety.
Rodica Luca is a professor of Mathematics at the “Gheorghe Asachi” Technical
University of Iasi, Romania. She obtained her PhD degree in mathematics from
“AlexandruIoanCuza”UniversityofIasi. Herresearchinterestsareboundaryvalue
problems for nonlinear systems of ordinary differential equations, finite difference
equations, and fractionaldifferentialequations, and initial-boundaryvalueproblems
fornonlinearhyperbolicsystemsofpartialdifferentialequations.
Acknowledgments
We aregratefulto all theanonymousrefereesforcarefullyreviewingearly draftsof
themanuscript. WealsoexpressourthankstoGlynJones,theMathematicspublisher
atElsevier,andtoStevenMathews,theEditorialprojectmanagerforMathematicsand
StatisticsatElsevier,fortheirsupportandencouragementofusduringthepreparation
of this book; and to Poulouse Joseph, the project manager for book production at
Elsevier,forallofhisworkonourbook.
The work of Rodica Luca was supported by a grant of the Romanian National
AuthorityforScientificResearch,CNCS-UEFISCDI,projectnumberPN-II-ID-PCE-
2011-3-0557.
Systems of second-order ordinary 1
differential equations with
integral boundary conditions
1.1 Existence of positive solutions for systems
with parameters
Boundaryvalueproblemswithpositivesolutionsdescribemanyphenomenaintheap-
pliedsciences,suchasthenonlineardiffusiongeneratedbynonlinearsources,thermal
ignitionofgases,andconcentrationinchemicalorbiologicalproblems(seeBoucherif
and Henderson, 2006; Cac et al., 1997; de Figueiredo et al., 1982; Guo and Laksh-
mikantham, 1988a,b;Joseph and Sparrow, 1970; Keller and Cohen, 1967). Integral
boundaryconditionsarise in thermalconduction,semiconductor,andhydrodynamic
problems(e.g., Cannon,1964;Chegis,1984;Ionkin,1977;Samarskii,1980). In re-
cent decades, many authors have investigated scalar problems with integral bound-
aryconditions(e.g., Ahmadet al., 2008;Boucherif,2009;Jankowski, 2013;Jia and
Wang,2012;Karakostas and Tsamatos, 2002;Ma and An, 2009;Webband Infante,
2008;Yang,2006).Wealsomentionreferences(CuiandSun,2012;Goodrich,2012;
Haoetal.,2012;Infanteetal.,2012;InfanteandPietramala,2009a,b;KangandWei,
2009;Lan,2011;SongandGao,2011;Yang,2005;YangandO’Regan,2005;Yang
and Zhang, 2012), where the authors studied the existence of positive solutions for
somesystemsofdifferentialequationswithintegralboundaryconditions.
1.1.1 Presentation of the problem
Inthissection,weconsiderthesystemofnonlinearsecond-orderordinarydifferential
equations
(cid:2)
(a(t)u(cid:2)(t))(cid:2)−b(t)u(t)+λp(t)f(t,u(t),v(t))=0, 0<t<1,
(c(t)v(cid:2)(t))(cid:2)−d(t)v(t)+μq(t)g(t,u(t),v(t))=0, 0<t<1, (S)
withtheintegralboundaryconditions
⎧ (cid:7) (cid:7)
⎪⎪⎨αu(0)−βa(0)u(cid:2)(0)= 1u(s)dH (s), γu(1)+δa(1)u(cid:2)(1)= 1u(s)dH (s),
1 2
(cid:7)0 (cid:7)0
⎪⎪⎩α˜v(0)−β˜c(0)v(cid:2)(0)= 1v(s)dK (s), γ˜v(1)+δ˜c(1)v(cid:2)(1)= 1v(s)dK (s),
1 2
0 0
(BC)
BoundaryValueProblemsforSystemsofDifferential,DifferenceandFractionalEquations.
http://dx.doi.org/10.1016/B978-0-12-803652-5.00001-6
Copyright©2016J.HendersonandRodicaL.Tudorache.PublishedbyElsevierLtd.Allrightsreserved.
2 BoundaryValueProblemsforSystemsofDifferential,DifferenceandFractionalEquations
wheretheaboveintegralsareRiemann–Stieltjesintegrals. Theboundaryconditions
aboveincludemultipointandintegralboundaryconditionsandthesumoftheseina
singleframework.
Wegivesufficientconditionsonf andgandontheparametersλandμsuchthat
positivesolutionsofproblem(S)–(BC)exist. Byapositivesolutionofproblem(S)–
(BC)wemeanapairoffunctions(u,v) ∈ C2([0,1])×C2([0,1])satisfying(S)and
(BC)withu(t) ≥ 0,v(t) ≥ 0forallt ∈ [0,1]and(u,v) (cid:5)= (0,0). Thecaseinwhich
thefunctionsH ,H ,K ,andK arestepfunctions—thatis,theboundaryconditions
1 2 1 2
(BC)becomemultipointboundaryconditions
⎧
⎪⎪⎪⎪⎨ αu(0)−βa(0)u(cid:2)(0)=(cid:8)m aiu(ξi), γu(1)+δa(1)u(cid:2)(1)=(cid:8)n biu(ηi),
i=1 i=1
⎪⎪⎪⎪⎩ α˜v(0)−β˜c(0)v(cid:2)(0)=(cid:8)r civ(ζi), γ˜v(1)+δ˜c(1)v(cid:2)(1)=(cid:8)l div(ρi),
i=1 i=1
(BC )
1
wherem,n,r,l ∈ N,(N = {1,2,...})—wasstudiedinHendersonandLuca(2013g).
System (S) with a(t) = 1, c(t) = 1, b(t) = 0, and d(t) = 0 for all t ∈ [0,1],
f(t,u,v) = f˜(u,v), and g(t,u,v) = g˜(u,v)(denotedby (S )) and (BC )was inves-
1 1
tigated in Henderson and Luca (2014e). Some particular cases of the problem from
HendersonandLuca(2014e)werestudiedinHendersonandLuca(2012e)(wherein
(BC ), a = 0 for all i = 1,...,m, c = 0 for all i = 1,...,r, γ = γ˜ = 1, and
1 i i
δ = δ˜ = 0—denotedby(BC )), in Luca (2011)(where in (S ), f˜(u,v) = f˜(v)and
2 1
g˜(u,v)=g˜(u)—denotedby(S )—andin(BC )wehaven =l,b =d,andη = ρ
2 2 i i i i
for all i = 1,...,n, α = α˜, and β = β˜—denoted by (BC )), in Henderson et al.
3
(2008b)(problem(S )–(BC )with α = α˜ = 1 andβ = β˜ = 0), andin Henderson
2 3
and Ntouyas (2008b) and Henderson et al. (2008a) (system (S ) with the boundary
2
conditions u(0) = 0, u(1) = αu(η), v(0) = 0, v(1) = αv(η), η ∈ (0,1), and
0<α <1/η,oru(0)=βu(η),u(1)=αu(η),v(0)=βv(η),andv(1)=αv(η)). In
HendersonandNtouyas(2008a),theauthorsinvestigatedsystem(S )withthebound-
2
ary conditionsαu(0)−βu(cid:2)(0) = 0, γu(1)+δu(cid:2)(1) = 0, αv(0)−βv(cid:2)(0) = 0, and
γv(1)+δv(cid:2)(1)=0,whereα,β,γ,δ ≥0andα+β+γ +δ >0.
In the proofof our main results, we shall use the Guo–Krasnosel’skii fixed point
theorem(seeGuoandLakshmikantham,1988a),whichwepresentnow:
Theorem1.1.1. LetX beaBanachspaceandletC ⊂ X beaconeinX. Assume
(cid:11) and (cid:11) are bounded open subsets of X with 0 ∈ (cid:11) ⊂ (cid:11) ⊂ (cid:11) , and let A :
1 2 1 1 2
C∩((cid:11) \(cid:11) )→Cbeacompletelycontinuousoperator(continuous,andcompact—
2 1
thatis,itmapsboundedsetsintorelativelycompactsets)suchthateither
(1) (cid:9)Au(cid:9)≤(cid:9)u(cid:9), u∈C∩∂(cid:11) ,and(cid:9)Au(cid:9)≥(cid:9)u(cid:9), u∈C∩∂(cid:11) ,or
1 2
(2) (cid:9)Au(cid:9)≥(cid:9)u(cid:9), u∈C∩∂(cid:11) ,and(cid:9)Au(cid:9)≤(cid:9)u(cid:9), u∈C∩∂(cid:11) .
1 2
Then,AhasafixedpointinC∩((cid:11) \(cid:11) ).
2 1