Properties of third-order nonlinear differential equations

Baculíková Blanka, Då¾Urina, Då¾Urina Jozef

Description

We establish new comparison theorems, to offer criteria for all nonoscillatory solutions of the third-order functional differential equation [ r ( t ) [ x ′ ( t ) ] γ ] ″ + p ( t ) x β ( τ ( t ) ) = 0 tend to zero. We consider both delay and advanced case of studied equation. The results obtained essentially improve and complement earlier ones. MSC: 34K11, 34C10.

Sujets

• Comparison theorem
• Oscillation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103
http://www.advancesindifferenceequations.com/content/2012/1/103
RESEARCH OpenAccess
Propertiesofthird-ordernonlinear
differentialequations
*BlankaBaculíkováandJozefDžurina
*Correspondence:
[e-mail] Abstract
DepartmentofMathematics,
Weestablishnewcomparisontheorems,tooffercriteriaforallnonoscillatoryFacultyofElectricalEngineeringand
Informatics,TechnicalUniversityof solutionsofthethird-orderfunctionaldifferentialequation
Košice,Letná9,04200Košice,
Slovakia γ βr(t) x (t) +p(t)x (τ(t))=0
tendtozero.Weconsiderbothdelayandadvancedcaseofstudiedequation.The
resultsobtainedessentiallyimproveandcomplementearlierones.
MSC: 34K11;34C10
Keywords: third-orderfunctionaldifferentialequations;comparisontheorem;
oscillation
1 Introduction
We are concerned with the asymptotic behavior of all solutions of the third-order
functionaldifferentialequations
γ βr(t) x(t) +p(t)x τ(t) =. (E)
Throughoutthearticle,wewillassumer,p ∈C([t ,∞)), τ ∈C ([t ,∞))and 
(H) γ, β aretheratiosoftwopositiveoddintegers,
(H) r(t)>,p(t)>, τ (t)>, lim τ(t)= ∞.t→∞
Inthesequel,itisassumedthat(E)isinacanonicalform,i.e.,
t
–/γR(t)= r (s)ds→∞ ast→∞.
t
ByasolutionofEquation(E)wemeanafunctionx(t) ∈C ([T ,∞)),T ≥t ,whichhasx x 
γ theproperty r(t)(x(t)) ∈C ([T ,∞)) andsatisfiesEquation(E)on[T ,∞).Weconsiderx x
only those solutions x(t)of(E) which satisfy sup{|x(t)| : t ≥ T}>forall T ≥ T.Wex
assume that (E) possesses such a solution. A solution of (E) is called oscillatory if it has
arbitrarilylargezeroson[T ,∞)andotherwiseitiscalledtobenonoscillatory.x
Various techniques were established for examination of (E) and its particular cases.
In
thearticles[–],theauthorshaveintroducedcomparisontheoremsforcomparingstudied equation with a set of the first order delay/advanced equation, in the sense that
oscillation of these first order equations yields desired properties of third order
equation.
©2012BaculíkováandDžurina;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and
reproductioninanymedium,providedtheoriginalworkisproperlycited.BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page2of9
http://www.advancesindifferenceequations.com/content/2012/1/103
Propertiesof(E)havebeenusuallystudiedundercondition β = γ or β < γ.Inthisarticle
weestablishresults,where β ≥
γ.Moreover,inthecitedarticlestheauthorsingenerally
considereitherdelayoradvancedequations,butourtechniquepermitstostudybothadvanced and delayed cases. On the other hand, in the existing comparison results of
this
kind,therearestudiedequationsalwayscomparedwithcanonicalsecondorderdifferential equation, but in this article we were able to establish comparison with noncanonical
differentialequation.
We offer new comparison principles, in which we compare our third order equation
with the second order differential inequality and this reduction essentially simplifies the
investigationofthepropertiesofthirdorderdifferentialequations.Ourresultsgeneralize
thosepresentedin[–].
Remark . All functional inequalities considered in this article are assumed to hold
eventually,thatis,theyaresatisfiedforallt largeenough.
2 Mainresults
In the following lemma, we present the classification of the possible nonoscillatory
solutionsof(E).
Lemma . Let x(t) be a nonoscillatory solution of (E). Then x(t) satisfies, one of the
followingconditions:
γ γ (C ) x(t)x(t)<, x(t) r(t) x(t) >, x(t) r(t) x(t) <,
γ γ (C ) x(t)x(t)>, x(t) r(t) x(t) >, x(t) r(t) x(t) <,
eventually.
Proof The proof follows immediately from the canonical form of (E) and details are left
toareader.
Tosimplifyourformulationsofthemainresults,werecallthefollowingdefinition:
Definition . We say that (E) enjoys property (A) if every its nonoscillatory solution
satisfies(C ).
Remark. Itiseasytoverifythatcondition
∞
p(s)ds=
∞,(.)
t
guaranteesproperty(A)of(E).Consequently,inthesequel,wemayassumethattheintegralonthelefthandsideof(.)isconvergent.
Property (A) of (E) has been studied by various authors (see enclosed references). We
offer new technique for investigation property (A) of (E) based on the comparison
theorems,inwhichwereduceproperty(A)of(E)totheabsenceofcertainpositivesolutionof
thesuitablesecondorderdifferentialinequality.Atfirst,weestablishcriteriaforproperty
(A)ofadvanceddifferentialequation.Westartwiththefollowingauxiliaryresult.BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page3of9
http://www.advancesindifferenceequations.com/content/2012/1/103
Lemma. Let τ(t) ≥t.Assumethatx(t)satisfies(C ).Thenforanyk ∈(,)
R(τ(t))
x τ(t) ≥k x(t) , (.)
R(t)
eventually.
γProof Assumethatx(t)>.Themonotonicityofw(t)=r(t)[x(t)] impliesthat
τ(t) τ(t)
/γ –/γx τ(t) –x(t)= x(s)ds= w (s)r (s)ds
t t
τ(t)
/γ –/γ /γ≥w (t) r (s)ds=w (t) R τ(t) –R(t) .
t
Thatis,
/γ x(τ(t)) w (t)
≥+ R τ(t) –R(t).(.)
x(t) x(t)
Ontheotherhand,sincex(t)→∞ast→∞,thenforanyk ∈(,)thereexistsat large
enough,suchthat
t t
/γ –/γ /γ –/γ /γ
kx(t) ≤x(t)–x(t)= w (s)r (s)ds ≤w (t) r (s)ds ≤w (t)R(t)
t t 
orequivalently,
/γw (t) k
≥.(.)
x(t) R(t)
Using(.)in(.),weget
x(τ(t)) k R(τ(t))
≥+ R τ(t) –R(t) ≥k .
x(t) R(t) R(t)
Thiscompletestheproof.
Letusdenote
βR (τ(t))
p (t)= p(t). (.) βR (t)
Theorem. Let τ(t) ≥t.Ifforsomec ∈(,)thesecondorderdifferentialinequality
/γ  t/β /γz (t) +c z (t) ≤(E )/β /γr (t)p (t)
hasnotanysolutionsatisfying
 /β z(t)>, z (t)<, z(t) <, (P )/β
p (t)
then(E)hasproperty(A).BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page4of9
http://www.advancesindifferenceequations.com/content/2012/1/103
Proof Assumethecontrary,letx(t)beanonoscillatorysolutionofEquation(E),satisfying
(C ).Wemayassumethatx(t)>,fort ≥t .Setting(.)into(E),weobtain 
β R (τ(t))γ β βr(t) x(t) +k p(t) x (t) ≤. (.)
βR (t)
γ Ontheotherhand,itfollowsfromthemonotonicityofy(t)=[r(t)[x(t)] ],that
t γ γ/βr(t) x(t) ≥ y(s)ds ≥y(t)(t–t ) ≥c ty(t), (.) 
t
eventually, where c ∈(,) is an arbitrary chosen constant. Evaluating x(t)andthenin-
tegratingfromt (≥t )tot,weareleadto 
t /γs/β /γx(t) ≥c y (s)ds.(.) /γr (s)t
Settingto(.),wehave

βt /γ s γ /γ β y(t)+c k p (t) r(s) x(s) ds ≤.  /γr (s)t
Integratingfromt to ∞,onegets

β∞ s /γu /γy(t) ≥c p (s) y (u)du ds,(.) /γr (u)t t
βwhere c=c k . Let usdenote theright handside of(.)by z(t).Then y(t) ≥z(t)>and
z(t)satisfies(P)andmoreover,
/γ  t (t)/β /γ= z (t) +c y (t)
/β /γr (t)p (t)
/γ  t (t)/β /γ≥ z (t) +c z (t)./β /γr (t)p (t)
Consequently, z(t) is a solution of the differential inequality (E ), which contradicts our
assumption.
We are prepared to establ