Oscillatory properties of half linear difference equations: two term perturbations

R E S E A R C H

Open Access

Oscillatory properties of half-linear difference

equations: two-term perturbations

Simona Fišnarová

*_{Correspondence:}

fi[email protected] Department of Mathematics, Mendel University in Brno, Zem ˇed ˇelská 1, Brno, CZ-613 00, Czech Republic

Abstract

We consider the nonoscillatory half-linear difference equation

(

)

and we study the influence of the perturbations˜r,c˜on the oscillatory properties of the equation

The presented oscillation and nonoscillation criteria are obtained using the variational principle and the so-called modified Riccati technique.

1 Introduction

In this article, we study oscillatory properties of the second-order half-linear difference equation of the form

L[xk] :=

rk(xk)

+ck(xk+) = , (x) :=|x|p–sgnx, p> , ()

wherer,care real-valued sequences,rk= . Ifp= , then () reduces to the linear Sturm-Liouville difference equation

(rkxk) +ckxk+= . ()

The basic qualitative theory of () has been established in the article [] and is summarized in the books [, ]. Many oscillatory properties of () are very similar to that of (), however the absence of the linearity requires sometimes to use different methods in half-linear case. In this article, we deal with the so-called perturbation principle. We suppose that equa-tion () is nonoscillatory and thathis a solution of () and we give conditions under which the perturbed equation

˜

L[xk] :=

(rk+r˜k)(xk)

+ (ck+c˜k)(xk+) = , ()

whererk+r˜k= , is oscillatory or nonoscillatory. Similar problem has been studied in [, ], where the caser˜k=  has been considered. We extend some results of those papers to

the general caser˜k=  and we also show that the assumptionlimk→∞rkhk(hk) =∞ considered in [] can be replaced by alternative conditions. We are motivated also by the results of [, ], where the two-term perturbations of the half-linear differential equation

r(t)x+c(t)(x) = , r(t) > 

are studied.

The article is organized as follows. In the next section, we recall the basic methods of oscillation theory for (), in particular the variational principle and the Riccati technique. Section  is devoted to the so-called modified Riccati technique. In Section , we present the main results of this article, the oscillation and nonoscillation criteria for the perturbed equation () and in the last section, we show how the results can be applied to the per-turbed equation of the Euler type.

2 Preliminaries

Oscillatory properties of () are defined using the concept of the generalized zero. We say that a solutionxof () has ageneralized zeroin an interval (m,m+ ] ifxm=  and rmxmxm+≤. Equation () is said to bedisconjugateon an interval [m,n] if any solution

of () has at most one generalized zero on (m,n+ ] and the solution for whichxm=  has no generalized zero on (m,n+ ]. Consequently, equation () is said to benonoscillatoryif there existsm∈Nsuch that this equation is disconjugate on [m,n] for everyn>m. In the opposite case, () is said to beoscillatory.

One of the basic methods used to investigate (non)oscillation of () is the variational technique which relates nonoscillation of () to a positivity of a certainp-degree func-tional.

Lemma [] Equation () is nonoscillatory if and only if there exists m∈Nsuch that

F(y,m,∞) :=

∞

k=m

rk|yk|p–ck|yk+|p

> 

for every nontrivial sequence y∈U(m), where

U(m) :=y={yk}∞_k=;yk= ,k≤m,∃n>m:yk= ,k≥n

.

The second basic method used is the Riccati technique which is based on the relation-ship between nonoscillation of () and the solvability (in a neighborhood of infinity) of the Riccati-type equation

R[wk] :=wk++ck–

rkwk

(–_(r

k) +–(wk))

= , ()

where–_{is the inverse function of,i.e.}–_{(s) =|s|}q–_sgns,q:= p

p–. Indeed, ifxis a

solution of () such thatxk=  on some discrete interval [m,∞), thenwk=rk(xk/xk) is a solution of () on [m,∞). More precisely, we have the following equivalent statements.

(i) Equation () is nonoscillatory.

(ii) There exists a solutionwof () such thatrk+wk> for largek.

(iii) There exists a solutionwof the Riccati inequalityR[wk]≤such thatrk+wk> for

largek.

If () is nonoscillatory, then there exists a solutionwkof () such thatrk+wk>  for large k. Among all solutionswwith this property, there is the so-calledminimalsolutionw˜ for whichw˜k<wkon some interval [m,∞), whererk+wk>  andrk+w˜k> . The minimal solutionw˜ can be constructed as follows. Let () be disconjugate on [n,∞) and letN>n. Denote byxN _{the solution of () which satisfiesx}N

N= ,xNN+=  and letwN=r(xN/xN)

be the solution of () associated withxN_{. Then}

˜ wk= lim

N→∞w N

k for everyk∈[n+ ,∞). ()

For details of this construction see []. The solutionx˜of () which is associated with the minimal solutionw˜of () by the substitutionw˜=r(x˜/x˜), is called therecessivesolution of (). The recessive solution is defined uniquely up to the multiplication by a real constant. Next, we formulate a comparison statement for minimal solutions of two Riccati equa-tions. The Riccati equation associated with () is

˜

R[wk] :=wk++ck+˜ck–

(rk+r˜k)wk

(–_(r

k+r˜k) +–(wk))

= . ()

Lemma []

(i) Let () be nonoscillatory and letr˜k≤,c˜k≥for largek. Further, letw˜ andw¯ be the

minimal solutions of the corresponding Riccati equations () and (), respectively. Then there existsm∈Zsuch thatw˜k≤ ¯wkfork∈[m,∞).

(ii) Ifck≥and ∞rk–q=∞, then the minimal solution of () satisfiesw˜k≥for

largek.

Note that the statement (ii) of the previous lemma is a special case of the statement (i). Condition ∞r–_k q=∞implies that the recessive solution of the equation(rk(xk)) =  is a constant sequence, hence the minimal solution of the associated Riccati equation is

w=  and this is compared with the minimal solution of ().

3 Modified Riccati equation and related results

In this section, we suppose that equation () is nonoscillatory, byhwe denote a positive so-lution of this equation and suppose that both the coefficientsrk,rk+r˜kare positive for large k. This sign restriction is needed when proving inequalities (), () and estimate () be-low, for details see []. Since these estimates play the crucial role in the (non)oscillation criteria based on the modified Riccati technique presented in this section, we suppose that

rk> ,rk+r˜k>  for largekthroughout the whole Section . Denote

define the function

H(k,v) :=v+ (rk+r˜k)hk+(hk) –

(rk+˜rk)(v+Gk)hpk+

(hq_k–_(r

k+r˜k) +–(v+Gk))

, ()

and consider the so-called modified Riccati equation

Rm[vk] :=vk+

˜rk(hk)

+c˜k(hk+)

hk++H(k,vk) = . ()

We show that there is a relation between operators given in () and () and estimate the functionH(k,v) by a term involvingv, which, in turn, enables us to compare Riccati equation associated with () with the Riccati equation related to a certain linear equation.

Lemma  Let w be a sequence such that rk+˜rk+wk= and suppose that vk=hpkwk–Gk. Then

Rm[vk] =hpk+R˜[wk].

Proof By a direct computation

vk=hpk+wk+–hpkwk–

(rk+r˜k)(hk)

hk+– (rk+˜rk)|hk|p

and

H(k,vk) =hp_kwk+ (rk+˜rk)|hk|p–

(rk+˜rk)hpkh p k+wk

(hq_k–_(r

k+˜rk) +–(hpkwk))

=hp_kwk+ (rk+˜rk)|hk|p–

(rk+˜rk)hpk+wk

(–_(r

k+˜rk) +–(wk)) .

Hence,

vk+

(rk+r˜k)(hk)

hk++H(k,vk) =hp_k+wk+–

(rk+r˜k)hpk+wk

(–_(r

k+r˜k) +–(wk)) .

Adding the termhp_k+(ck+c˜k) to both sides of the last equality, we obtain

vk+hk+L˜[hk] +H(k,vk) =hpk+R˜[wk],

whereL˜ is the operator defined in (). Sincehis a solution of (), we have the required

identity.

Lemma  Let H(k,v)be defined in ().

(i) It holdsH(k,v)≥forv> –(rk+˜rk)hk((hk) +(hk))with the equality if and

only ifv= .

(ii) Suppose thathkhk> fork∈[m,∞)and denoteRk=_q(rk+r˜k)hkhk+|hk|p–.

Then we have the following inequalities forv≥andk∈[m,∞):

(Rk+v)H(k,v)≥v, p∈(, ], ()

(iii) Suppose thatlim infk→∞Gk> , then for largek:

and since the first condition in () holds, we have

–

Consequently, again using () and conditions (),

ask→ ∞. Hence,H(k,v) can be written in the form

_{) follows from the second condition in ().}

This means, by the first condition in (), that ∞H(k,v) =∞.

4 Oscillation and nonoscillation criteria

We start with a statement based on the variational principle. This statement generalizes a result of [] dealing with the case˜rk= . Here, we do not need the sign restriction onrk,

Proof The proof is based on Lemma . LetN∈Nbe arbitrary and let K,L, M,N be

positive integers satisfyingN<K<L<M<N, whereK is such that () is disconjugate

on [K,∞) and(r˜k/rk)≤ holds on this interval, andLis such thathhas no generalized gN = . The fact that such a solution really exists follows from the disconjugacy of () on [K,∞) and from the homogeneity of the solution space of (). Indeed, ifxis a solution of

the corresponding solutions of Riccati equation (). Set

F(y;K,L– ) =

Next, using summation by parts, and sincehis a solution of (), we have

F(y;L,M– ) =

Concerning interval [M,N], sincegN=  andhhas no generalized zero in this interval, it follows from [, Lemma ] thatw[_kg]<w_k[h]fork∈[M,N– ]. This means that

rk(gk)gk<rk

(hk)

(hk) |

At the same time,  < rkgkgk+

Next, again summing by parts and using the boundary conditions and the fact thatgis a solution of (),

= n–

k=M

r˜k(hk)

+˜ck(hk+)

hk+.

Combining the above computations, we have

F(y;M,N– )≤–

 +r˜M

rM

w[_Mg]|hM|p– n–

k=M

˜rk(hk)

+c˜k(hk+)

hk+. ()

Consequently, using (), () and (), we obtain

F(y;N,∞) =

∞

k=N

(rk+r˜k)|yk|p– (ck+˜ck)|yk+|p

=F(y;K,L– ) +F(y;L,M– ) +F(y;M,N– )

≤α+α+

 +˜rM

rM

|hM|p

w[_Mh]–w[_Mg]

– n–

k=L

˜rk(hk)

+c˜k(hk+)

hk+.

Letα>  be arbitrary. It follows from condition () thatMcan be taken so large that

n–

k=L

r˜k(hk)

+c˜k(hk+)

hk+≥α+α+α.

Sincehis the recessive solution of (),i.e. w[h] _{is the minimal solution of (), from its}

construction () we have

w[_Mh]= lim

N→∞w

[g]

M.

Hence,Ncan be taken so large that

 +r˜M

rM

|hM|p

w[_Mh]–w[_Mg]<α.

Consequently, ifM,Nare taken as above, then

F(y;N,∞)≤,

and this means that equation () is oscillatory by Lemma . The proof is complete.

The following results are based on the modified Riccati technique. Here we suppose that

his a positive solution of the nonoscillatory equation () andrk> ,rk+r˜k>  for large k. Denote

Ck:=

˜rk(hk)

+c˜k(hk+)

hk+, Rk:= 

q(rk+˜rk)hkhk+|hk|

First we present a statement, where, using the global inequalities (), (), equation () is compared with the linear equation

(Rkyk) +Ckyk+= . ()

This statement generalizes [, Theorems . and .].

Theorem  Let h be a solution of () such that hk> , hkhk> for large k.

(i) Letp≥, and suppose thatCk≥for largekand ∞_Rk =∞. If the linear

equation () is nonoscillatory, then equation () is also nonoscillatory.

(ii) Letp≤, and suppose thathis the recessive solution of (),˜rk≤,c˜k≥for largek.

If the linear equation () is oscillatory, then equation () is also oscillatory.

Proof (i) Nonoscillation of equation () means that there exists a solutionvof the Riccati equation

vk+Ck+ v_k Rk+vk

= 

such thatRk+vk>  for largek. Assumptions of the theorem imply, in view of Lemma (ii), thatvk≥ for largek. Hence, from Lemma (ii) it follows

(Rk+vk)H(k,vk)≤vk for largek,

i.e. vksolves the inequality

vk+Ck+H(k,vk)≤

for large k. Consequently, by Lemma , we have that the sequence w=h–p_{(v+G) is a} solution of Riccati inequalityR˜[wk]≤. Moreover,

rk+r˜k+wk=rk+˜rk+h–kpvk+ (rk+r˜k)(hk/hk) =h–kpvk+

 +˜rk

rk

rk+w[kh]

,

wherew[_kh] is a solution of () related to a nonoscillatory solutionhof () and hencerk+ w[_kh]>  for largek. Sincevk≥ for largek, we haverk+r˜k+w˜k>  for largek. Hence, equation () is nonoscillatory according to Lemma .

(ii) Suppose, by contradiction, that equation () is nonoscillatory and letwbe a solution of the associated Riccati equation (). Then, by Lemma ,v=hp_{w–Gsolves the modified} Riccati equation (). Sincer˜k≤,c˜k≥, it follows from Lemma  thatwk≥w[kh]for large k, wherew[_kh] =rk(hk/hk) is the minimal solution of (). This means thatvk=hpkwk –Gk=hpk(wk– ( +_rk˜rk)w[kh])≥ for largek. Applying inequality () we have

(Rk+vk)H(k,vk)≥vk

for largekand hencevsolves the inequality

vk+Ck+ v_k Rk+vk≤

which, together with the fact thatRk+vk>  for largek, means that () is nonoscillatory.

The next two statements are based on the estimate () which holds for allp> . Hence we do not have to distinguish between the casesp> ,p< . The idea of the proofs is the same as in Theorem  and we skip the proofs since they are similar to that in [, Theo-rems . and .]. The only difference is that we replacerkbyrk+˜rk.

Theorem  Suppose that h is a solution of () such that hk> , hkhk> for large k and let

∞



Rk =∞,

∞

Ck<∞, Ck≥for large k ()

and

lim inf

k→∞ Gk> . ()

If there existsε> such that the linear equation

(Rkyk) + ( +ε)Ckyk+=  ()

is nonoscillatory, then equation () is also nonoscillatory.

Theorem  Suppose that h is the recessive solution of () such that hk> , hkhk> , ˜

ck≥,˜rk≤for large k, conditions () hold and

lim

k→∞Gk=∞. ()

If there existsε> such that the linear equation

(Rkyk) + ( –ε)Ckyk+=  ()

is oscillatory, then equation () is also oscillatory.

The next statement is a version of Theorem . In the proof of the statement we use Lemma (iv). This enables to replace condition () by alternative conditions. This state-ment is new also in caser˜= .

Theorem  Suppose that h is the recessive solution of () such that hk > , hkhk> , ˜

ck≥,˜rk≤for large k and conditions (), () and () hold. If there existsε> such that the linear equation () is oscillatory, then equation () is also oscillatory.

Proof Suppose, by contradiction, that equation () is nonoscillatory and letwbe a solu-tion of the associated Riccati equasolu-tion () andv=hp_{w–Gthe associated solution of the} modified Riccati equation (). Similarly as in the proof of Theorem (ii), we conclude that

vk≥ for largek. SinceCk≥ for largekandH(k,vk) is nonnegative, we have for largek

This means that there exists a finite limitlimk→∞vk. Summing the modified Riccati equa-tion fromN tok,Nbeing sufficiently large, and using the fact thatvis nonnegative, we obtain

vN≥ k

j=N Cj+

k

j=N H(j,vj).

Lettingk→ ∞and since ∞Ck<∞, we have ∞H(k,vk) <∞. From Lemma (iv) it follows thatlimk→∞vk= . Now we can use the estimate (). There existsNsuch that

H(k,vk) >

 –ε 

v

k Rk+vk

fork>N.

We have

Rk= 

q(rk+˜rk)hkhk+|hk| p–₌

qGk

 + hk

hk

>

qGk.

Hence, by (),lim_k→∞ vk

Rk =  and similarly as in [, Theorem .], we obtain for

suffi-ciently largekthe estimate _Rk

–ε+vk < –ε

Rk+vk. Hence,

H(k,vk) >

 –ε 

v

k Rk+vk

> v



k Rk

–ε+vk

for sufficiently largek. This means thatvksolves the inequality

vk+Ck+ v_k Rk

–ε+vk

< ,

which is the Riccati inequality associated with equation () and since _–Rk_ε +vk> , we

have nonoscillation of (). This is a contradiction.

It is known (see []), that if ck≥,rk> , ∞r–k =∞, ∞

ck <∞, then the linear equation () is nonoscillatory provided

lim sup

k→∞ k–

r–_j ∞

j=k cj<



, ()

and it is oscillatory provided

lim inf

k→∞ k–

r_j– ∞

j=k cj>



. ()

Applying this criterion to equations (), (), we obtain the following result.

(i) If

lim sup

k→∞ k–



(rj+r˜j)hjhj+|hj|p–

∞

j=k Cj<



q, ()

then equation () is nonoscillatory.

(ii) Let moreoverhbe the recessive solution of (),˜ck≥,˜rk≤for largekand let either

condition () or conditions () and () be satisfied. If

lim inf

k→∞ k–



(rj+r˜j)hjhj+|hj|p–

∞

j=k Cj>

 q,

then equation () is oscillatory.

Proof Consider,e.g., the case (i), the case (ii) is analogous. Condition () can be written in the form

lim sup

k→∞ k–

R–_j ∞

j=k Cj<

 .

Consequently, there existsε>  such that

lim sup

k→∞ k–

R–_j ∞

j=k

( +ε)Cj<  ,

hence equation () is nonoscillatory and this implies, by Theorem , nonoscillation of

().

Remark  If˜rk= ,c˜k≥, then the statements of Theorem  reduce to [, Theorems . and .]. More precisely, (non)oscillation of

rk(xk)

+ (ck+˜ck)(xk+) =  ()

is compared with that of the linear equation

R_kyk

+C_kyk+= , Rk:= 

qrkhkhk+|hk| p–_,C

k:=c˜khp_k+. ()

In part (ii) of Theorem  we suppose thatp≤,r˜k≤,c˜k≥. Under these conditions, if equation () is oscillatory, then () is oscillatory by [, Theorem .] and oscillation of () follows then by the Sturm comparison theorem, see,e.g., []. However, Theorem , part (ii) extends [, Theorem .] in case when the perturbation˜ckis ‘not too much positive’ so that equation () and hence also () is nonoscillatory.

Remark  The conditions˜rk≤,˜ck≥ considered in Theorem (ii), Theorem  and Theorem  are used to show thatwk– ( +˜rk/rk)wk[h]≥, wherewandw[h]are the solu-tions of the Riccati type equasolu-tions associated with equasolu-tions () and (), respectively. We conjecture thatwkand (+˜rk/rk)w[_kh]can be compared by another argument than Lemma , so the sign restriction on the perturbation termsr˜k,c˜kcan be relaxed, similarly as in the continuous case [].

5 Application

In this section, we apply the previous results to the perturbed Euler-type difference equa-tion

( +r˜k)(xk)

+

γp (k+ )p +c¯k

(xk+) = , γp:=

p– 

p

p–

. ()

This equation is considered as a perturbation of the nonoscillatory equation

(xk)

+ck(xk+) = , ()

where

ck= –

(hk)

(hk+)

, hk=k p–

p _{. ()}

It is easy to see thathk=k p–

p _{is a solution of () and it was shown in [] that ifp}≥_,

then it is the recessive solution. By a direct computation, as shown in [], the coefficient

ckis of the form

ck=

γp (k+ )p

 +O(k+ )–, ask→ ∞

and also

hk= p– 

p k

–p_{ +Ok}–_,

hk(hk) =

p– 

p

p–

 +Ok–,

hkhk+(hk)p–=

p– 

p

p–

k +Ok–,

(hk) = –

p– 

p

p

(k+ )–+p_{ +O(k+ )}–_,

ask→ ∞. Consequently,

hk hk

=p– 

p k

–_{ +Ok}–_{, ask→ ∞,}

hence conditions () are satisfied and we have

Gk=

p– 

p

p–

( +˜rk)

 +Ok–, Rk= 

p– 

p

p–

( +r˜k)k

and

ask→ ∞. Using these computations, Corollary  applied to () reads as follows.

Corollary  Let Ckbe given in () and suppose that

∞

then equation () is nonoscillatory.

(ii) Suppose moreover thatp≥,limk→∞c¯k(k+ )p+=∞,˜rk≤for largek. If

then equation () is oscillatory.

Competing interests

The author declares that she has no competing interests.

Acknowledgements

Research supported by the Grant P201/10/1032 of the Czech Science Foundation.

Received: 23 January 2012 Accepted: 11 June 2012 Published: 5 July 2012

References

1. ˇRehák, P: Oscillatory properties of second order half-linear difference equations. Czechoslov. Math. J.51, 303-321 (2001)

2. Agarwal, RP, Bohner, M, Grace, SR, O’Regan, D: Discrete Oscillation Theory. Hindawi Publishing Corporation, New York (2005)

3. Došlý, O, ˇRehák, P: Half-Linear Differential Equations. North-Holland Mathematics Studies, vol. 202. Elsevier, Amsterdam (2005)

4. Došlý, O, Fišnarová, S: Linearized Riccati technique and (non)oscillation criteria for half-linear difference equations. Adv. Differ. Equ.2008, Article ID 438130 (2008)

5. Došlý, O, ˇRehák, P: Recessive solution of half-linear second order difference equations. J. Differ. Equ. Appl.9, 49-61 (2003)

6. Došlý, O, Fišnarová, S: Half-linear oscillation criteria: Perturbation in term involving derivative. Nonlinear Anal., Theory Methods Appl.73, 3756-3766 (2010)

7. Došlý, O, Fišnarová, S: Variational technique and principal solution in half-linear oscillation criteria. Appl. Math. Comput.217, 5385-5391 (2011)

8. Došlý, O: Oscillation criteria for higher order Sturm-Liouville difference equations. J. Differ. Equ. Appl.4, 425-450 (1998)

9. Erbe, LH, Zhang, BG: Oscillation of second order linear difference equations. Chin. J. Math.16, 239-252 (1988) 10. Došlý, O, Fišnarová, S: Summation characterization of the recessive solution for half-linear second order difference

doi:10.1186/1687-1847-2012-101

Cite this article as:Fišnarová:Oscillatory properties of half-linear difference equations: two-term perturbations.