9 pages
English
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Properties of third-order nonlinear differential equations

-

Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus
9 pages
English

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

Oscillation

Informations

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

Exrait

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
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.
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
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
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)
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-
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)
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 ), 
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 
(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
* *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
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 ≥– .
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
/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