Asymptotic problems for fourth-order nonlinear differential equations

R E S E A R C H

Open Access

Asymptotic problems for fourth-order

nonlinear differential equations

Miroslav Bartušek and Zuzana Došlá

Dedicated to Jean Mawhin on occasion of his seventieth birthday.

*_{Correspondence:}

dosla@math.muni.cz Faculty of Science, Masaryk University Brno, Kotláˇrská 2, Brno, 611 37, The Czech Republic

Abstract

We study vanishing at infinity solutions of a fourth-order nonlinear differential equation. We state sufficient and/or necessary conditions for the existence of the positive solution on the half-line [0,∞) which is vanishing at infinity and sufficient conditions ensuring that all eventually positive solutions are vanishing at infinity. We also discuss an oscillation problem.

1 Introduction

In this paper we study the fourth-order nonlinear differential equation

x()(t) +q(t)x(t) +r(t)x(t)λsgnx(t) =  (t∈R+), () whereλ≥,q∈C_(R

+),q(t) >  for larget,r∈C(R+) such thatr(t)=  for largetand

R+= [,∞).

Jointly with (), we consider a more general equation

x()(t) +q(t)x(t) +r(t)fx(t)=  (t∈R+), () wheref ∈C_{(R) satisfiesf(u)u>  foru= , and the associated linear second-order} equa-tion

h(t) +q(t)h(t) = . ()

By a solution of () we mean a functionx∈C[Tx,∞),Tx≥, which satisfies () on

[Tx,∞). A solution is said to benonoscillatoryifx(t)=  for larget; otherwise, it is said to

beoscillatory. Observe that ifλ≥, according to [, Theorem .], all nontrivial solutions of () satisfysup{|x(t)|:t≥T}>  forT ≥Tx, on the contrary to the caseλ< , when

nontrivial solutions satisfyingx(t)≡ for largetmay exist.

Fourth-order differential equations have been investigated in detail during the last years. The periodic boundary value problem for the superlinear equationx()_{=g(x) +e(t) has} been studied in []. In [], the fourth-order linear eigenvalue problem, together with the nonlinear boundary value problemx()–f(t,x) = , has been investigated. Oscillatory properties of solutions for self-adjoint linear differential equations can be found in []. Equation () withq(t)≡ can be viewed as a prototype of even-order two-term differen-tial equations, which are the main object of monographs [, , ].

Equation () withq(t)≡ fort∈R+is a special case of higher-order differential equa-tions investigated in []. Equation () withqnear to a nonzero constant ast→ ∞has been considered in [] as a perturbation of the linear equationy()_{(t) +q(t)y(t) = , and} the existence of oscillatory solutions of () has been proved. In [], necessary and suffi-cient conditions for the existence of asymptotically linear solutions of () have been given.

In the recent paper [], the equation

x()(t) +kx(t) +fx(t)=  (t∈R),

wherek∈R,f(u)u>  foru= , andf ∈Lip_loc(R) has been investigated and applications to the biharmonic PDE’s can be found there. In particular, the so calledhomoclinics solu-tions, which are defined as nontrivial solutionsxsuch thatlimt→±∞x(t) = , are studied.

The goal of this paper is to investigate asymptotic problems associated with () and the asymptotic boundary condition

x(t) >  for larget, lim

t→∞x(t) = . ()

A solutionxof () satisfying () is said to bevanishing at infinity.

We start with theKneser problemfor (). The Kneser problem is a problem concerning the existence of solutions of () subject to the boundary conditions on the half-line [,∞)

x() =c> , (–)ix(i)(t) >  fort≥,i= , , . ()

We establish necessary and/or sufficient conditions for the solvability of the boundary value problem (), (), (). In the light of these results, as the second problem, we study when all eventually positive solutionsxof () are vanishing at infinity assuming thatλ>  and () is oscillatory. As a consequence, we give a bound for the set of all nonoscillatory solutions. Finally, we discuss when problem (), () is not solvable and solutions to () are oscillatory.

A systematic analysis of solutions of () satisfying () is made according to whether () is nonoscillatory or oscillatory. If () is nonoscillatory, then the following approach will be used. Equation () can be rewritten as the two-term equation

h(t)

x(t)

h(t)

+h(t)r(t)x(t)λsgnx(t) = , ()

wherehis a positive solution of (). According to [], a solutionhof () is said to be a prin-cipal solutionif∞h–_{(t)dt=∞, and such a solution is determined uniquely up to a} mul-tiple constant. Sinceq(t) > , every eventually positive solution of () is nondecreasing for larget. Hence there exists a principal solutionhof () such thath(t) >  fort≥a≥ and

∞

a



h_(t)dt=∞,

∞

a

h(t)dt=∞. ()

Therefore, we can use the known results [, ] stated for systems of differential equations or in [] for fourth order differential equations.

equa-tion and the use of the estimates for positive soluequa-tions of such an equaequa-tion on a compact interval stated in []. This, together with an energy function associated with (), enables us to state an oscillation theorem. In the final section, some extensions of our results to () are given.

2 The Kneser problem

In this section we present necessary and/or sufficient conditions for solvability of bound-ary value problem (), (), ().

2.1 Caser(t) < 0

Proposition  Letλ≥, ()be disconjugate on[,∞),and r(t) < for t∈R+.Then

bound-ary value problem(), ()is solvable for any c> .

To prove this theorem, we use Chanturia’s result [, Theorem ] for the system of dif-ferential equations

dy

dt=f(t,y), ()

where we restrict to the case thatf = (f,f,f,f) :R+×R→Rare continuous functions,

n= ,m= ,l= ,a=  andr=c. Then this result reads as follows.

Theorem A([]) Let there exist c> such that

fi(t,y,y,y,y)≤ (i= , . . . , )

for t∈R+,y∈[,c),yi∈R+(i= , , ).Suppose

–fi(t,y,y,y,y)≥ϕi(t,yi+) (i= , , )

– 

i=

fi(t,y,y,y,y)≤ψ(t)ω

_

i=

yi

for t∈[, ],y∈[,c],yi∈R+(i= , , ),where functionsϕi: [, ]×R+→R+(i= , , )

are continuous and nondecreasing in the second argument such that

lim inf x→∞

 

ϕ(t,x)dt>c,

lim x→∞

 

ϕi(t,x)dt=∞ (i= , ),

ψ: [, ]→R+is a continuous function andω:R+→(,∞)is a continuous nondecreasing

function such that

∞ 

du ω(u)=∞.

Then,for any x∈[,c],system()has a solution satisfying

Proof of Proposition Assumer(t) <  fort∈R+. Since () is disconjugate, it has a positive solutionhonR+, and () can be written as () whereh(t)r(t) <  onR+. Letxbe a solution of () and denote

y(t) =x(t), y(t) = –y_(t), y(t) = – 

h(t)y

(t), y(t) = –h(t)y(t). ()

Then () is equivalent to the system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y_(t) = –y(t),

y_(t) = –h(t)y(t),

y_(t) = – 

h_(t)y(t),

y_(t) = –h(t)|r(t)||y(t)|λ_{(t)sgny(t) (t∈R+).}

()

Letc>  be from (). We apply Theorem A choosing

ϕ(t,x) =x, ϕi(t,x) =kix, i= , , ψ(t) =k, ω(u) =cλ+u,

wherek=mint∈[,]h(t),k=mint∈[,]/h(t) andk=maxt∈[,][h(t) + /h(t) +h(t)r(t)]. By this result, system () has a solution such that

y() =c> , yi(t)≥, yi(t)≤ fort≥, i= , , , .

Sinceλ≥, system () has no solutions such thatyi≡ for somei= , , ,  and larget;

see [] or [, Lemma , Theorem ]. Thus, for anyc> , equation () has a solutionx

such thatx() =c,x(t) =y(t) > ,x(t) = –y(t) <  andx(t) =h(t)y(t) >  fort≥.

Now we state conditions for the existence of a solution for problem (), (), ().

Theorem  Letλ≥,r(t) < and

q(t)(t+ )≤ 

 ()

onR+.If

∞ 

tr(t)dt=∞, ()

then problem(), (), ()is solvable for any c> .

In addition,if

∞ 

tq(t)dt<∞, ()

then the condition

∞ 

tr(t)dt=∞ ()

For the proof, the following lemma will be needed.

Lemma  Consider system()on[a,∞) (a≥),where h(t) > for t≥a and h is a princi-pal solution of ().Let y= (y,y,y,y)be a solution of ()such that yi(t)≥and yi(t)≤ for i= , , , ,and t≥a.Thenlim_t→∞yi(t) = for i= , , ,and if

∞

a

h(t)r(t)

t a



h_(s)

s a

h(τ)τdτds dt=∞, ()

thenlimt→∞y(t) = ,too.Vice versa,ifλ≥andlimt→∞y(t) = ,then()holds.

Proof In view of the monotonicity ofyi, there existlimt→∞yi(t) =yi(∞)≥,i= , , , .

Sincehis the principal solution, () holds, and integrating the first three equations in () fromatot, we getyi(∞) =  fori= , , . Now integrating () fromtto∞, we have

y(t) =

∞

t

h(s)y(s)ds,

y(t) =

∞

t



h_(t)y(s)ds,

y(t) =

∞

t

h(s)r(s)yλ_(s)ds.

Let () hold and assume, by contradiction, thaty(∞) > . Then

y(a) –y(t)≥yλ_(∞)

t a

_∞

s h(u)

_∞

u



h_(v)

_∞

v

h(τ)r(τ)dτdv du ds. ()

Lettingt→ ∞and using the change of the order of integration, we get a contradiction with the boundedness ofy. This proves thaty(∞) = .

Let the integral in () be convergent and assume, by contradiction, thaty(∞) = . Then we have

y(t) =

∞

t

∞

s h(u)

∞

u



h_(v)

∞

v

h(τ)r(τ)yλ

(τ)dτdv du ds,

so

y(t)≤yλ_(t)

∞

t

∞

s h(u)

∞

u



h_(v)

∞

v

h(τ)r(τ)dτdv du ds.

Sinceλ≥, then using the change of the order of integration, we get a contradiction for

larget. This proves thaty(∞) > .

Proof of Theorem In view of (), () is disconjugate onR+. By Proposition , equation () has a solutionxsatisfying (). Therefore, system () has a solution such thatyi(t) > 

andy_i(t) <  fort∈R+. Choosehin () as a principal solution of (). The Euler equation

˜

h+ 

is the majorant of () onR+and has the principal solutionh˜(t) = √

t+ . By the comparison theorem, for the minimal solution of the Riccati equation related to () and (), we have

 <h _(t)

h(t) ≤ ˜

h(t) ˜

h(t)

for larget; see,e.g., []. Thus there exists>  such that≤h(t)≤√t+  fort≥. Assume (). Then () holds, and by Lemma  a solutionxsatisfies ().

Assume (). Then the principal solutionhof () satisfiesh(t)∼for larget(see,e.g., []). Hence, condition () reads as (), and by Lemma  this condition is equivalent to

the property ().

As a consequence of Lemma , we get the following result.

Corollary  Let()be disconjugate on[,∞),and r(t) < for t∈R+.Then any solution x of ()satisfying

x() =c> , x(t) > , t≥, lim

t→∞x(t) =  ()

is a solution of the Kneser problem,i.e., (–)i_x(i)_{(t) > for t≥and i= , .}

Proof Lethbe a positive solution onR+satisfying (), and letxbe a solution of () satis-fying (). Theny= (y,y,y,y), whereyare defined by (), is a solution of system (). Sincey(t) >  fort≥ and () holds, we have by the Kiguradze lemma (see,e.g., []) that eithery_i(t) >  ory_i(t) <  fori= , ,  and larget, say fort≥a≥. Sincexis positive and tends to zero, we havey_≤ fort≥a, so alsoy_i≤ (i= , ) fort≥a. By Lemma , we gety_i(∞) =  (i= , ) fort≥a. Sincex(t) >  fort≥, we havey_(t) <  fort≥ andy is positive and decreasing onR+. Hence, proceeding by the same argument,yi(i= , ) is

positive and decreasing onR+. Now the conclusion follows from ().

2.2 Caser(t) > 0

First we show that the sign condition posed onris necessary for the solvability of problem (), ().

A functiong, defined in a neighborhood of infinity, is said to change sign if there exists a sequence{tk} → ∞such thatg(tk)g(tk+) < .

Theorem  Let r(t) > for large t.Then problem(), ()has no solution and the following hold:

(a) If ()is nonoscillatory,then every nonoscillatory solutionxof ()satisfies x(t)x(t) > andxis of one sign for larget.

(b) If ()is oscillatory,then every nonoscillatory solutionxof ()satisfies either x(t)x(t)≤,orx(t)changes sign.In addition,if a solutionxsatisfies(),thenx changes sign.

Proof Claim (a). Letxbe a positive solution of () on [a,∞), or, equivalently, of () on [a,∞), wherehsatisfies (). Denote

y(t) =x(t), y(t) =y_(t), y(t) = 

h(t)y

Then () is equivalent to the system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y_(t) =y(t),

y_(t) =h(t)y(t),

y_(t) = _h_(t)y(t),

y_(t) = –h(t)r(t)yλ

(t)sgny(t) (t≥a).

()

We havey(t) >  fort≥a. Assume by contradiction thaty(t) <  fort≥a. Lety(t) >  andy(t) < . Sinceyis nonincreasing,y(t)≤y(a) <  and

y(t) –y(a)≤y(a)

t a

/h(s)ds.

Lettingt→ ∞, we get a contradiction with the positiveness ofy. The remaining case

y(t) <  can be eliminated in a similar way using (). Observe that system () is a special case of the Emden-Fowler system investigated in [], and the proof follows also from [, Lemma .].

Claim (b). Without loss of generality, suppose thatr(t) >  fort≥T and there exists a solutionxof () such thatx(t) >  andx(t)≥ on [T,∞),T ≥. Borůvka [] proved that if () is oscillatory, then there exists a functionα∈C_{[T,∞), called aphase function,} such thatα(t) >  and

(α(t)) (α(t)) –

α(t) (α(t))+

q(t)

α_(t)= . ()

Using this result, we can consider the change of variables

s=α(t), x(t) =√

α(t)X(s)

˙= d

ds

()

fort∈[T,∞),s∈[T∗,∞),T∗=α(T). Thus,t=α–(s) and

x(t) = – 

α(t)– 

_{α(t)X(s) +α(t)}/_X˙_(s),

x()(t) = – 

α(t)– 

_{α(t)X(s) +}  

α(t)– 

_α(t)_{X(s) +α(t)}_X¨_(s). Substituting into (), we obtain the second-order equation

¨

X(s) +X(s)

 

α(t)–α(t)– 

α(t)–α(t) +q(t)α(t)–

+α(t)–_r(t)xλ_{(t) = .} From here and (), we obtain

¨

X(s) +X(s) = –α(t)–_r(t)xλ_{(t) < . ()}

Sincex(t)≥, () yieldsX(s)≥ and soX¨(s) < , that is,X˙ is decreasing. If there exists

˙

X(s)≥ andX(s) is nondecreasing. LetT≥T∗be such thatX(s) >  on [T,∞). Thus, using () we obtain

˙

X(s) –X˙(T) =

s T ¨

X(u)du≤–

s T

X(u)du≤–X(T)(s–T).

Hence,lims→∞X˙(s) = –∞, which contradicts the nonnegativity ofX˙(s). Finally, the case

X(s)≡ on [T∗,∞) cannot occur, because ifx(t)≡ on [T,∞), then from () andr(t) > , we havex≡ on [T,∞), which is a contradiction.

Finally, letxbe a positive solution of () satisfying (). Thenxis either oscillatory or

x(t) <  for larget. Assumex(t) <  on someJ= [T,∞), thenxis decreasing and either

x(t)≥ orx(t) <  for larget. Ifx(t)≥ for larget, then we get a contradiction with (). Ifx(t) < , thenx(t)≤x(T) <  fort≥T≥Tandxbecomes negative for larget. Hence

xmust be oscillatory.

Forλ> , the analogous result to Theorem  is the following oscillation result.

Proposition  Letλ> ,r(t) > for large t.Assume either()for large t, (),or(), ().

Then all the solutions of ()are oscillatory.

Proof Letxbe a solution of () andhbe the principal solution of (). Theny= (y,y,y,y), where yi are given by (), is a solution of system (). Proceeding by the similar way

as in the proof of Theorem , we have that () holds. Using the change of the order of integration in (), we can check that conditions of Theorem . in [] applied to system () are verified. Hence by this result all the solutions of () are oscillatory, which gives

the conclusion.

The following result follows from [, Theorem .] and completes Proposition  in the case when () is oscillatory.

Proposition  Letλ> ,q(t)≡and r(t) > for t∈R+.Then the condition∞tr(t)dt= ∞is necessary and sufficient for every solution of ()to be oscillatory.

In the light of these results, in the sequel, we study asymptotic and oscillation problems to () when () is oscillatory.

3 Vanishing at infinity solutions

In this section we study when all nonoscillatory solutions of () are vanishing at infinity.

Theorem  Letλ> and()be oscillatory.Assume that q(t)≥Kt–for large t and some K> ,the functions

q q/,

q q,

q

q/ are bounded onR+, ()

and

|r(t)|

q_(t)↑ ∞ as t→ ∞. ()

The proof of Theorem  is based on the following auxiliary results. Consider the fourth-order quasi-linear differential equation

y()(s) + 

i=

Qi(s)y(i)(s) +R(s)|y|λ(s)sgny(s) = , ()

whereQiandRare continuous functions on [,∞). In [, Theorem .], the following

uniform estimate for positive solutions of () with a common domain was proved.

Proposition ([, Theorem ., Corollary .]) Assumeλ> .Let y be a positive solution

of ()defined on[,b]and

R(s)≥r∗, Qi(s)≤Q–i, i= , , , ()

on[,b]for some constants r∗> and Q> .Then there exists a positive constant M=M(λ)

such that

y(s)≤Mr–



λ– ∗ δ–



λ–_{(s) for s}∈_{[,b], ()}

where

δ(s) =min{s,α,b–s}, α= –Q–.

Remark  In [, Theorem .] the constantMis explicitly calculated.

Lemma  Letλ> .Assume that()holds on[,∞).Then any positive solution of ()

defined on[,∞)satisfies

y(s)≤Mr–



λ–

∗ α–λ– _{for s}∈_[α,∞), _()

whereαand M are constants from Proposition.

Proof Letb≥α. By Proposition , applied on [,b], we haveδ=δ(s)≡αfors∈[α,b–α] and

y(s)≤Mr–



λ– ∗ α–



λ– _fors∈_[α,b–α].

Lettingb→ ∞, we get ().

The next lemma describes the transformation between solutions of () and a certain quasi-linear equation.

Lemma  Let q(t) > on[a,∞)be such that

_∞

a

and consider the transformation

s=

t a

q(t)dt, x(t) =y(s), ˙= d

ds.

Then x=x(t)is a solution of equation()on[a,∞)if and only if y=y(s)is a solution of the equation

d_y

ds + 

q(t)

q/_(t)

d_y

ds +

q _(t)

q_(t)– (q(t))

q_(t) + 

¨

y

+

q(t) q/_(t)–

 

q(t)q(t)

q/_(t) +  

(q(t))

q/_(t) +

q(t) q/_(t)

˙

y

+ r(t)

q_(t)|y| λ_sgn

y=  on[,∞), ()

where t=t(s)is the inverse function to s=s(t).

Proof We have

x(t) =y˙(s)q(t), x(t) =y¨(s)q(t) +˙y(s)q _(t)

q(t),

x(t) = d _y(s)

ds q

/_{(t) +} y¨(s)q

_{(t) +y˙(s)} q(t) q(t)–

(q(t)) q/_(t)

,

x()(t) = d _y(s)

ds q

_{(t) + }dy

dsq

_{(t)q(t) +y¨(s)q(t) –} (q(t)) q(t)

+y˙(s)

q(t) q/_(t)–

 

q(t)q(t)

q/_(t) +  

(q(t))

q/_(t)

.

Substituting into (), we get the conclusion.

Proof of Theorem Letxbe a positive solution of () onI= [a,∞) (a> ). Suppose, for simplicity, thatq(t)≥Kt–fort≥a. Let

q(t)q–_(t)≤_{C, q(t)q}–_(t)≤_{C, q(t)q}–_(t)≤_{C ()}

on [,∞) for some positive constantsC,C,Cand

Q=max

C,

C+  C  +  / , C  +  CC+

 C   + C  / . () Denote

α= –Q–, a¯=exp(α/K). ()

Define the functiont∗=t∗(t) such that

t t∗

fort≥ ¯a. Then, according toq(t)≥Kt–_{, we have}

t∗≥texp(–α/K)≥. ()

Letr/q _{be nondecreasing on [a˜,∞) and putT≥max{a,a¯,a˜}. Chooset}

∈[T,∞) ar-bitrarily fixed. Sinceq(t)≥Kt–_{, we can consider the transformation from Lemma  with}

a=t∗(t),i.e.,

s=

t t∗(t)

q(τ)dτ, x(t) =y(s), ˙= d

ds. ()

Then equation () is transformed into equation () which is a quasilinear equation of the form (), where

Q(s)≤Q, Q(s)≤Q, Q(s)≤Q, R(s) =

r(t(s))

q_(t(s))

andQis defined by () and (). Chooset≥Tarbitrarily. We apply Lemma  to equation () with

r∗=min

s≥ |r(t(s))| q_(t(s))=

|r(t∗(t))| q_(t∗_(t

)) .

Hence estimate () withs=αreads as

x(t) =y(α)≤C

q(t∗(t)) |r(t∗(t))|



λ–

, ()

whereC=Mα–_λ_{–. Lettingt}

→ ∞, we have by () thatt∗(t)→ ∞and the conclusion

follows from () and ().

From the proof of Theorem , we get the estimate for the set of all nonoscillatory solu-tions of () which will be used in the next section.

Corollary  Letλ> ,limt→∞q(t) =∞, ()and()hold.Then,for anyε> ,there exists

a positive constant C=C(λ,q(t),ε)and T≥such that every nonoscillatory solution x of

()satisfies

x(t)≤C

q(t–ε) |r(t–ε)|



λ–

for t≥T. ()

Proof Letε>  be fixed and letT≥ be such that

q(t)≥α 

ε fort≥T,

whereαis given by (). Lett≥Tbe fixed. Using estimate () witht=t≥T, we have

x(t)≤C

q_(t∗_(t)) |r(t∗(t))|



λ–

wheret∗is given by (),i.e.,

α=

t t∗(t)

q(τ)dτ ≥α ε

t–t∗(t).

Thereforet∗(t)≥t–εand estimate () follows from () and ().

Example  Consider the equation

x()(t) + (t+ )x(t) –(t+ )+ (t+ )x(t) =  (t∈R+). () Thenr/q_{→ ∞and by Theorem  all eventually positive solutions are vanishing at} infin-ity. One can check thatx=_t+ is such a solution of ().

Open problem It is an open problem to find conditions for the solvability of boundary value problem (), (), () in caser(t) <  and () is oscillatory.

In view of Theorem , Corollary  and Proposition , it is a question whether () can have vanishing at infinity solutions in caser(t) >  and () is oscillatory.

In the next section, we show that under certain additional assumptions the answer is negative.

4 Oscillation

Here we consider () in caser(t) >  for larget. When () is nonoscillatory, we have es-tablished the oscillation criterion in Proposition . When () is oscillatory, the following oscillation theorem holds.

Theorem  Letλ> ,r(t) > and assumptions(), ()hold.Assume

lim

t→∞q(t) =∞, q

_{(t)≥, q(t)≥ for large t ()}

and

lim sup t→∞ q

+σ (t)

q_(t–ε)

r(t–ε)



λ–

<∞ ()

for someε> andσ> .Then problem(), ()is not solvable and all the solutions of ()

are oscillatory.

Proof Suppose that () holds on [a,∞). First, observe that the assumption () implies that

∞ 

q(t)

q_(t–)

r(t–)



λ–

dt<∞. ()

Indeed, puttingH(t) = (q_(t)/r(t))λ–_{, we have}

t a

q(s)H(s)ds<H(a)q+σ(t)

t a

q(s)

and thus, in view of (), we get (). Consider a solutionxof () such thatx(t) >  for

t≥t≥. According to Corollary , there exists¯tsuch that

 <x(t)≤C

q_(t–)

r(t–)



λ–

fort≥ ¯t≥t, ()

and in view of () we getlim_t→∞x(t) = . Consider the function

F(t) = –x(t) –q(t)x(t) +q(t)x(t).

Then

F(t) =r(t)xλ(t) +q(t)x(t),

and in view of () the functionFis increasing for larget. Hence, there existst≥tsuch that either

F(t) <  fort≥t, ()

or

F(t)≥F(t) >  fort≥t. ()

According to Theorem (b),xoscillates. Define by{τk}∞k=an increasing sequence of zeros ofxtending to∞withτ≥t.

Define

Z(t) = –x(t) –q(t)x(t) – 

∞

t

q(s)x(s)ds. ()

In view of () and () the functionZis well defined and

Z(t) = –x(t) –q(t)x(t) +q(t)x(t) =F(t) ()

on [t,∞). Moreover, we have from (), () and ()

lim

k→∞Z(τk) = . ()

If () holds, thenZ(t) <  and because

Z(τ) = –q(τ)x(τ) – 

∞

τ

q(s)x(s)ds< ,

we getZ(t)≤Z(τ) <  fort≥τ. This is a contradiction with (), so () is impossible. If () holds, thenZ(t) >  andlimt→∞Z(t) =∞. This is again a contradiction with (),

Example  Consider the equation

x()(t) + c

tx

_{(t) +σ|x|}_{(t)sgnx(t) =  (t∈R} +),

whereσ=±. Ifσ= – andc∈(, /), then by Theorem  this equation has a solution satisfying () and (). Ifσ=± andc> /, then by Theorem  any nonoscillatory solution (if any) satisfies ().

5 Extensions

As it was mentioned in [], a certain nonlinear PDE leads to the fourth-order equation with the exponential nonlinearity. In the sequel, we show that the results of this paper can be extended to the nonlinear equation

x()(t) +q(t)x(t) +r(t)fx(t)=  (t∈R+), () whereq,rare as for () andf(u)u>  foru=  such that

f(u)≥kuλ foru∈R+ ()

for someλ≥ andk> . The prototype of such an extension is the functionf(u) =eu_{– }

foru≥.

Theorems - read for () as follows.

Theorem  Letλ≥,r(t) < and()hold for t∈R+.Assume that either(i) (),or(ii) ()and()hold.Then problem(), (), ()has a solution for any c> .

Proof of Theorem It is analogous to the proofs of Proposition  and Theorem  replacing the nonlinearity|y(t)|λ(t)sgny(t) in system () byf(y(t)). Lemma  remains to hold as

a sufficient condition for ().

Theorem  Theoremremains to hold for()without assuming().

Proof of Theorem In the proof of claim (a) of Theorem , we consider system () where the nonlinearity|y(t)|λ_{(t)sgny(t) is replaced byf(y(t)). The proof of claim (b) of}

Theo-rem  is the same for the nonlinearityf.

Theorem  Theoremremains to hold for().

Proof of Theorem Letxbe a positive solution of () on [a,∞). Thenv=xis a solution of the equation

v()_{(t) +q(t)v}()_{(t) +R(t)v}λ_{(t) = , ()}

where

R(t) =r(t)f(x(t))

xλ_(t) ≥kr(t) fort≥a. ()

Theorem  Let the assumptions of Theoremhold.Then()has no eventually positive solutions.

Proof of Theorem It is similar to the one of Theorem . In view of (), the estimate

() holds and the energy functionFis the same.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final draft.

Acknowledgements

Supported by the grant GAP 201/11/0768 of the Czech Grant Agency.

Received: 13 November 2012 Accepted: 16 March 2013 Published: 12 April 2013

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doi:10.1186/1687-2770-2013-89

Cite this article as:Bartušek and Došlá:Asymptotic problems for fourth-order nonlinear differential equations.