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Properties of third-order nonlinear differential equations

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9 pages
English

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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.

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Publié le 01 janvier 2012
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BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103
http://www.advancesindifferenceequations.com/content/2012/1/103
RESEARCH OpenAccess
Propertiesofthird-ordernonlinear
differentialequations
*BlankaBaculíkováandJozefDžurina
*Correspondence:
jozef.dzurina@tuke.sk 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
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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 establish the corresponding result also for delay differential
equations.Letusdenote
–p(τ (t))
p (t)=.(.) –τ (τ (t))
Theorem. Let τ(t) ≤t.Ifforsomec ∈(,)thesecondorderdifferentialinequality
/γ  t/β /γz (t) +c z (t) ≤(E )/β /γr (t)p (t)
BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page5of9
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hasnotanysolutionsatisfying
 /β z(t)>, z (t)<, z(t) <, (P )/β
p (t)
then(E)hasproperty(A).
Proof Assumethecontrary,letx(t)beapositivesolutionofEquation(E),satisfying(C ).
Anintegrationof(E)fromt to ∞,yields
∞ ∞ – p(τ (s))γ β βr(t) x(t) ≥ p(s)x τ(s) ds= x (s)ds
–τ (τ (s))t τ(t)
∞ –p(τ (s)) β≥ x (s)ds.
–τ (τ (s))t
γ Using(.),onecanseethaty(t)=[r(t)[x(t)] ] satisfies

∞ s β/γu /γy(t) ≥c p (s) y (u)du ds.(.)  /γr (u)t t
Letusdenotetherighthandsideof(.)by
z(t).ThensimilarlyasintheproofofTheorem.,wecanverifythatz(t)isapositivesolutionof(E )andmoreover,itsatisfies(P ), 
whichcontradictsourassumption.
Establishing, new criteria for elimination of solutions of (E)satisfying(P), i=,,wei i
immediately obtain sufficient conditions for property (A) of (E). Since (E)and(E )are 
ofthesameform,wepresentjustonegeneralcriterionandthen,weadaptthemforboth
(E ).Soweconsiderthenoncanonicaldifferentialinequalityi
α δ *a(t) z (t) +b(t)z (t) ≤, (E )
where
(H) α, δ aretheratiosoftwopositiveoddintegers,
(H) a(t)>,b(t)>.
Letusdenote

–/α(t)= a (s)ds.
t
Theorem. Assumethat δ> α.Ifforallk>

t αδα δlimsup (s)b(s)– ds>,(.)
α+ /αk(α+) (s)a (s) kt→∞ t
*then(E )hasnotanysolutionsatisfying
α *z(t)>, z (t)<, a(t) z (t) <. (P )BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page6of9
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* *Proof Letz(t)beapositivesolutionof(E ),suchthat(P)holds.Wedefine
αa(t)(z(t))
w(t)= .
δz (t)
Thenw(t)<and,moreover,
α –+δ/α(a(t)(z(t)) ) z (t) z (t) +/αw(t)= – δw(t) ≤–b(t)– δw (t) . (.)
δ /αz (t) z(t) a (t)
α /αOn the other hand, noting that –(a(t)(z(t)) ) is positive and increasing, we see that
α /αthereexistsaconstantk >suchthat–(a(t)(z(t)) ) ≥k and 
∞ ∞ α /α –/αz(t) ≥ –z (s)ds= – a(s) z (s) a (s)ds
t t
α /α≥– a(t) z (t) (t) ≥k (t), (.)
orequivalently
–+δ/α –+δ/αz (t) ≥k (t), (.)
–+δ/αwherek =k .Setting(.)into(.),onegets 
–+δ/α (t) +/αw(t) ≤–b(t)– δk w (t).(.) /αa (t)
Itisusefultonoticethat(.)implies
α–δ αz (t) ≥–w(t) (t),
whichtogetherwith(.)implies
 δ≥–w(t) (t). (.)αk
δMultiplying(.)by (t)andthenintegratingfromt tot,weareleadto
t δ– (s)δ δw(t) (t)–w(t ) (t)+ δ w(s)ds  /αa (s)t
t t δ–+δ/α (s)δ +/α≤– b(s) (s)ds– δk w (s)ds, /αa (s)t t 
whichinviewof(.)yields

t δ– (s) δ δ/α +/αb(s) (s)+ δ w(s)+k (s)w (s) ds ≤ . α/αa (s) kt 
+/αAnelementarycalculationshowsthatforthefunctionf(u)=u+Au
,u<thefollowingestimateholds
αα +/αu+Au ≥– .
α+ α(α+) ABaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page7of9
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Consequently,
αα δ/α +/αw(s)+k (s)w (s) ≥– . αα+ δ(α+) k (s)
Therefore,

t αδα δ (s)b(s)– ds ≤ ,
α+ /αk(α+) (s)a (s) kt
αwith k = k .Taking limsup on both sides, we get a contradiction with (.). This t→∞
finishesourproof.
Now,wetransformcondition(.)tothemorepracticalform.
Corollary. Assumethat δ> α.If
δ+ /αlim (t)b(t)a (t)= ∞,(.)
t→∞
* *then(E )hasnotanysolutionsatisfying(P ).
Proof Itfollowsfrom(.)thatforanyk>
αδα δ+ /α (t)b(t)a (t) ≥ + ,
α+k(α+) k
eventually.Thatis
αδα  δ (t)b(t)– ≥ .
α+ /α /αk(α+) (t)a (t) k (t)a (t)
Integratingtheaboveinequalityfromt tot,onegets
t αδα   δ (s) b(s)– ds= ln –ln .
α+ /αk(α+) (s)a (s) k (t) (t )t 
Letting t→∞,weseethat(.) holds true and the assertion now follows from
Theorem..
WecombineTheorems.and.togetherwithCorollary.,toobtaineasilyverifiable
criteriaforproperty(A)of(E).
Theorem. Let β > γ, τ(t) ≥t.If
+/γ∞ β /γ βR (τ(s)) t R (t)
lim p(s)ds = ∞,(.)
β /γ βt→∞ R (s) r (t)p(t)R (τ(t))t
then(E)hasproperty(A).BaculíkováandDžurinaAdvancesinDifferenceEquations2012,2012:103 Page8of9
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/γ ∞–/β tProof We set α=/β, δ=/γ, a(t)= p (t), and b(t)= c .Then (t)= p (s)ds. /γ tr (t)
Since(.)reducesto(.),Corollary.ensuresthat(E )hasnotanysolutionsatisfying
(P ).TheassertionnowfollowsfromTheorem.. 
Theorem. Let β > γ, τ(t) ≤t.If
+/γ∞ /γ τ (t) τ (t)
lim p(s)ds = ∞,(.)
/γt→∞ r (τ(t)) p(t)t
then(E)hasproperty(A).
/γ–/β ∞tProof We set α=/β, δ=/γ, a(t)=p (t), and b(t)=c .Then (t)= p(s)ds.–/γ τ (t)r (t)
As(.)reducesto(.),Corollary.guaranteesthat(E )hasnotanysolutionsatisfy-
ing(P ).TheassertionnowfollowsfromTheorem.. 
Remark. For τ(t) ≡tbothconditions(.)and(.)simplifiestothesamecondition
+/γ∞ /γt 
lim p(s)ds = ∞,
/γt→∞ r (t)p(t)t
forproperty(A)of(E).
Corollary. Assumethat(E)enjoysproperty(A).Ifmoreover,
∞ ∞ ∞ /γ
p(s)dsdu dv= ∞, (.)
/γr (v)t v u
theneverynonoscillatorysolutionof(E)tendstozeroast→∞.
Proof Since(E)hasproperty(A),everyitsnonoscillatorysolutionsatisfies(C ),andwhat
ismore,(.)ensuresthatsuchsolutiontendstozeroast→∞.
Example. Considerthethirdordernonlineardifferentialequation
a t x(t) + x (λt)=, t ≥, (E )xt
wherea>and λ>.Sincebothconditions(.)and(.)hold,Theorems.and.
imply that (E ) enjoys property (A) and, moreover, Corollary . guarantees that everyx
nonoscillatorysolutionof(E )tendstozeroast→∞.Fora=λ onesuchsolutionisx
x(t)=/t.
3
Summary
Ourresultscanbeappliedtobothdelayandadvancedthirdorderdifferentialequations.
Thecriteriaobtainedareeasyverifiableandhavebeenprecedentedbysuitablejointillustrativeexample.
Ourmethodessentiallysimplifiestheexaminationofthethirdorderequationsandwhat
ismore,itsupportsbackwardtheresearchonthesecondorderdelay/advanceddifferential
equationsandinequalities.