Advances in Difference Equations Volume 2008, Article ID 438130,18pages doi:10.1155/2008/438130
Research Article
Linearized Riccati Technique and (Non-)Oscillation Criteria for Half-Linear Difference Equations
Ondˇrej Do ˇsl ´y1and Simona Fiˇsnarov ´a2
1Department of Mathematics and Statistics, Masaryk University, Jan´aˇckovo n´am. 2a, 66295 Brno, Czech Republic
2Department of Mathematics, Mendel University of Agriculture and Forestry in Brno, Zemˇedˇelsk´a 1, 61300 Brno, Czech Republic
Correspondence should be addressed to Ondˇrej Doˇsl ´y,[email protected] Received 23 August 2007; Accepted 26 November 2007
Recommended by John R. Graef
We consider the half-linear second-order difference equationΔrkΦΔxk ckΦxk1 0,Φx:
|x|p−2x,p >1, wherer,care real-valued sequences. We associate with the above-mentioned equa- tion a linear second-order difference equation and we show that oscillatory properties of the above- mentioned one can be investigated using properties of this associated linear equation. The main tool we use is a linearization technique applied to a certain Riccati-type difference equation correspond- ing to the above-mentioned one.
Copyrightq2008 O. Doˇsl ´y and S. Fiˇsnarov´a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we deal with oscillatory properties of solutions of the half-linear second-order difference equation
Δ rkΦ
Δxk
ckΦxk1 0, Φx:|x|p−2x, p >1, 1.1
wherer,care real-valued sequences andrk > 0. This equation can be regarded as a discrete counterpart of the half-linear differential equation
rtΦ
xctΦx 0 1.2
which attracted considerable attention in the recent years. We refer to the books in 1,2and the references given therein. The basic qualitative theory of1.1has been established in the series of papers in 3–7and it is summarized in the books 8, Chapter 3and 2, Chapter 8.
It is known that oscillatory properties of1.1are very similar to those of the second-order Sturm-Liouville difference equationwhich is a special case ofp2 in1.1:
Δ rkΔxk
ckxk10. 1.3
In particular, the discrete linear Sturmian theory extends verbatim to1.1, and hence this equa- tion can be classified as oscillatory or nonoscillatory. We will recall elements of the oscillation theory of1.1in more detail in the next section.
The basic idea of the discrete linearization technique which we establish in this paper is motivated by the paper of Elbert and Schneider 9, where the second-order half-linear differ- ential equation
Φ
xγp
tpΦx 2 p−1
p
p−1δt
tp Φx 0, γp: p−1
p p
, 1.4
is viewed as a perturbation of the Euler-type half-linear differential equation Φ
xγp
tpΦx 0, 1.5
and oscillatory properties of1.4are studied via the linear equation tyδt
t y0 1.6
under the assumption that∞
t δs/s ds≥0 for larget. In particular, the following statements are presented in 9.
iLetp≥2 and let linear equation1.6be nonoscillatory. Then1.4is also nonoscillatory.
iiLetp∈1,2and let half-linear equation1.4be nonoscillatory. Then linear equation1.6 is also nonoscillatory.
The linearization technique for1.2has been further developed in 10–12; see also references given therein.
In our paper, we introduce a similar linearization technique for the investigation of os- cillatory properties of1.1. This equation is regarded as a perturbation of the nonoscillatory equation of the same form:
Δ rkΦ
Δxk
ckΦ xk1
0, 1.7 and oscillatory properties of solutions of1.1are related to those of the linear second-order difference equation
Δ
RkΔyk
Ckyk10, 1.8 where
Rk 2
qrkhkhk1Δhkp−2, Ck ck−ck
hpk1, 1.9
withqp/p−1being the conjugate number ofp, and with a certain distinguished solution hof1.7. This enables to apply the deeply developed linear oscillation theory when investi- gating oscillations of half-linear equation1.1. As we will see in the next sections, compared to the continuous case, the linearization technique is technically more difficult in the discrete case since a nonlinear function which appears in the so-called modified Riccati equation is considerably more complicated in the discrete case.
The paper is organized as follows. In the next section, we recall basic oscillatory proper- ties of1.1, including a quadratization formula for a certain nonlinear function which plays an important role in subsequent sections of the paper. InSection 3, we present a discrete version of the above-mentioned result of Elbert and Schneider 9. InSection 4, we show that under certain additional restriction on properties of solutions of1.7we do not need to distinguish between the casesp≥2 andp∈1,2. The last section of the paper is devoted to an application of the results of the previous sections of the paper.
2. Preliminaries
Oscillatory properties of1.1are defined using the concept of the generalized zero which is defined in the same way as for1.3 see, e.g., 8, Chapter 3or 2, Chapter 7. A solutionxof 1.1has a generalized zero in an intervalm, m1ifxm/0 andxmxm1rm≤0. Since we suppose thatrk>0oscillation theory of1.1generally requires onlyrk/0, a generalized zero ofxin m, m1is either a “real” zero atkm1 or the sign change betweenmandm1. Equation 1.1is said to be disconjugate in a discrete interval m, nif the solutionxof1.1given by the initial conditionxm0,xm1/0, has no generalized zero inm, n1. Equation1.1is said to be nonoscillatory if there existsm∈Nsuch that it is disconjugate on m, nfor everyn > m, and it is said to be oscillatory in the opposite case.
Ifxis a solution of 1.1such thatxk/0 in some discrete interval m,∞, thenwk rkΦΔxk/xkis a solution of the associated Riccati-type equation
R wk
: Δwkckwk
1− rk
Φ Φ−1
rk
Φ−1 wk
0. 2.1 Moreover, ifxhas no generalized zero in m,∞, thenΦ−1rk Φ−1wk > 0,k ∈ m,∞. If we suppose that1.1is nonoscillatory, among all solutions of2.1there exists the so-called distinguished solutionw which has the property that there exists an interval m,∞such that any other solutionwof2.1for whichΦ−1rk Φ−1wk> 0,k ∈ m,∞, satisfieswk >wk, k∈ m,∞. Therefore, the distinguished solution of2.1is, in a certain sense, minimal solution of this equation near∞. Ifwis the distinguished solution of2.1, then the associated solution of1.1given by the formula
xkk−1
jm
1 Φ−1 wj
rj
2.2
is said to be the recessive solution of1.1 see 13. Note that in the linear casep2 a solution
xof1.3is recessive if and only if
∞ 1
rkxkxk1 ∞. 2.3
Our first statement presents a comparison theorem for distinguished solutions of2.1 and2.4given below.
Lemma 2.1see 13. Let1.1be nonoscillatory and letck ≥ckfor largek. Further, letwk,vkbe distinguished solutions of the corresponding generalized Riccati equations2.1and
R vk
: Δvkckvk
1− rk
Φ Φ−1
rk
Φ−1 vk
0, 2.4
respectively. Then there existsm∈Zsuch thatwk ≥vkfork ∈ m,∞. In particular, ifck ≥0 and ∞
r1−qk ∞, thenwk≥0 for largek.
The next statement relates nonoscillation of1.1to the existence of a certain solution of the Riccati inequality associated with2.1.
Lemma 2.2see 2, Theorem 8.2.7. Equation1.1is nonoscillatory if and only if there exists a sequencewksatisfyingrkwk>0 and
R wk
≤0 2.5
for largek.
The next statement is the discrete version of the generalized Leighton-Wintner oscillation criterion. In this criterion,1.1is viewed as a perturbation of1.7.
Lemma 2.3see 13. Lethbe the positive recessive solution of nonoscillatory equation1.7. If ∞
ck−ck
hpk1∞, 2.6
then1.1is oscillatory.
The last auxiliary oscillation results of this section are Hille-Neharinon-oscillation cri- teria for linear difference equation1.3.
Lemma 2.4see 14. Suppose thatck≥0,rk>0,∞
rk−1∞, and∞
ck<∞. If
lim inf
k→∞
k−11 rj
∞
jk
cj
> 1
4, 2.7
then1.3is oscillatory. If
lim sup
k→∞
k−11 rj
∞
jk
cj
< 1
4, 2.8
then1.3is nonoscillatory.
For the remaining part of this section, we suppose that1.7is nonoscillatory and we let hbe its solution such thathk>0 for largek. Further, put
Gk:rkhkΦ Δhk
2.9
and define the function
Hk, v:vrkhk1Φ Δhk
− rk
vGk
hpk1 Φ
hqkΦ−1 rk
Φ−1 vGk
. 2.10 Lemma 2.5. Put
vk:hpk
wk−wk
, 2.11 wherewkrkΦΔhk/hkis a solution of 2.4andwkis any sequence satisfyingrkwk/0. Then
Δvk ck−ck
hpk1H k, vk
hpk1R wk
. 2.12 In particular, ifwkis a solution of 2.1, then
Δvk ck−ck
hpk1H k, vk
0. 2.13
Moreover,Hk, v≥0 forv >−rkhkΦΔhk hp−1k with the equality if and only ifv0.
Proof. By a direct computation and using the fact thatwkis a solution of2.4, we obtain Δvkhpk1
wk1−wk1
−vk
hpk1
wk1ck− rkwk
Φ Φ−1
rk
Φ−1 wk
−vk
hpk1
wk1ck−rkΦΔhk
hk1
−vk
hpk1
wk1ck
−rkhk1Φ Δhk
−vk.
2.14
Next, sincevkhpkwk−Gk,we have rk
vkGk
Φ
hqkΦ−1 rk
Φ−1
vkGk rkhpkwk
Φ hqkΦ−1
rk Φ−1
hpkwk rkwk
Φ Φ−1
rk
Φ−1 wk
,
2.15
and hence Δvk
ck−ck
hpk1H k, vk
hpk1
wk1ck− rk
vkGk
Φ
hqkΦ−1 rk
Φ−1 vkGk
hpk1R wk
.
2.16
Ifwkis a solution of2.1, thenvksatisfies2.13. We prove the nonnegativity of the function Hk, vforv >−rkhkΦΔhk hp−1k as follows. By a direct computation, we have
Hv
k, v
1− rkqhqkhpk1 hqkΦ−1
rk
Φ−1 vkGk
p,
Hvvk, v qrkqhqkhpk1vkGkq−2 hqkΦ−1
rk
Φ−1 vkGk
p1.
2.17
HenceHvk, v 0 if and only ifv0 and the functionHk, vis convex with respect tovfor vsatisfyingh−qk Φ−1rk Φ−1vGk>0 which is equivalent tov >−rkhkΦΔhk hp−1k . This proves the last statement ofLemma 2.5.
Lemma 2.6. LetR,Gbe defined by1.9and2.9, respectively, and suppose thatGk >0 fork ∈N.
Then we have the following inequalities forv≥0 andk∈N: Rkv
Hk, v≥v2, p∈1,2, Rkv
Hk, v≤v2, p≥2. 2.18
Proof. In this proof, we write explicitly an index by a sequence only if this index is different fromk; that is, no index means the indexk. In addition to2.17, we have
Hvvvk,0 q
r2h2h2k1Δh2p−3 q−2hk1−2q−1Δh
. 2.19
DenoteFk, v: RkvHk, v−v2. Then we haveFvk,0 0Fvvk,0and Fvvvk,0 RHvvvk,0 3Hvvk,0
2
rhhk1Δhp−1 q−2hk1−2q−1Δh
3q rhhk1Δhp−2 1
rhhk1Δhp−1 2q−2h Δh 2−qΔh q−2
rhhk1Δhp−1 hh Δh q−2
rhhk1Δhp−1 hhk1 .
2.20
Consequently,
sgnFk, v sgnq−2 2.21
in some right neighborhood ofv0. Further, we have Fvvk, v 2Hvk, v RvHvvk, v−2
− 2rqhqhpk1
hqΦ−1r Φ−1vGp qrqhqhpk1vGq−2Rv hqΦ−1r Φ−1vGp1
rqhqhpk1
hqΦ−1r Φ−1vGp1 −2rq−1hq−2Φ−1vG qvGq−2Rv . 2.22
Denote byAvthe expression in brackets in the last expression. By a direct computation, we have
Av q−2Φ−1vG qR−GvGq−2−2rq−1hq, 2.23 hence sgnAv sgnFvvk, v sgnq−2for largev, and from the computation ofFvvvk,0, we also haveq−2Av>0 in some right neighborhood ofv0. Since
Av q−2vGq−3 q−1vG qR−G
2.24 has no positive root observe that q − 1v G qR − G 0 if and only if v
−1/q − 1rhΔhp−2hk1 h < 0, this means that q − 2Av and hence also q−2Fvvk, vhave a constant sign forv∈0,∞. Therefore, the functionFk, vis convex for q≥2 and concave forq≤2, and this together with2.21implies the required inequalities.
3. (Non-)oscillation criteria:p≥2versusp∈1,2
In this section, we suppose that1.7is nonoscillatory and possesses a positive increasing so- lutionh. We associate with1.1the linear Sturm-Liouville second-order difference equation
Δ RkΔyk
Ckyk10, 3.1 whereRandCare given by1.9, that is,
Rk 2
qrkhkhk1 Δhk
p−2
, Ck ck−ck
hpk1. 3.2
The results of this section can be regarded as a discrete version of the results given in 9.
Theorem 3.1. Letp≥2,ck≥ckfor largek, ∞ 1
Rk ∞, 3.3
and suppose that linear equation 3.1with R, Cgiven by 1.9 is nonoscillatory. Then half-linear equation1.1is also nonoscillatory.
Proof. The proof is based onLemma 2.2. Nonoscillation of3.1implies the existence of a solu- tionvof the associated Riccati equation
ΔvkCk v2k
Rkvk 0 3.4
such thatRkvk > 0 for large k. Moreover, since 3.3 holds and Ck ≥ 0 for large k, by Lemma 2.1vk ≥0 for largek. ByLemma 2.6, we haveRkvHk, v≤v2; hencevis also a solution of the inequality
ΔvkCkH k, vk
≤0. 3.5 Now, substituting forv hpw−w, where w rΦΔh/h, we see fromLemma 2.5thatwis a solution of Riccati inequality2.5. Moreover,rkwkrkh−pk vkwk>0 sincevk≥0 andh is a nonoscillatory solution of1.7; that is, the corresponding solution of the associated Riccati equationwsatisfiesrkwk>0. Therefore,1.1is nonoscillatory.
Theorem 3.2. Letp ∈1,2,ck ≥ckfor largek, and lethbe the recessive solution of 1.7. If half- linear equation1.1is nonoscillatory, then linear equation3.1is also nonoscillatory.
Proof. We proceed similarly as in the previous proof. Nonoscillation of 1.1implies the ex- istence of the distinguished solution w of the associated Riccati equation 2.1 such that wkrk > 0 for largek. Put againv hpw−w, where w is the distinguished solution of 2.4. Thenvsolves the equation
ΔvkCkH k, vk
0, 3.6 and byLemma 2.1, we havewk ≥wkfor largek, hencevk ≥ 0,and thereforeRkvk >0 for largek. ByLemma 2.6,
ΔvkCk v2k
Rkvk ≤0. 3.7
This means that3.1is nonoscillatory byLemma 2.2.
4. Criteria without restriction onp
Throughout this section, we suppose thatRk,Ck, andGkare given by1.9and2.9, respec- tively, and that1.7is nonoscillatory.
Theorem 4.1. Letck≥ckfor largekand lethk>0 be the recessive solution of1.7such that ∞
ck−ck
hpk1<∞. 4.1
Further, suppose that condition3.3holds and
k→∞limrkhkΦ Δhk
∞. 4.2
If there existsε >0 such that the equation Δ
RkΔyk
1−εCkyk10 4.3
is oscillatory, then1.1is also oscillatory.
Proof. Let ε > 0 be such that 4.3 is oscillatory i.e., ε < 1. Suppose, by contradiction, that1.1 is nonoscillatory, and letxk be its recessive solution. Denote by wk rkΦΔxk/ xkandwkrkΦΔhk/hkthe distinguished solutions of the Riccati equations2.1and2.4, respectively, and putvk : hpkwk−wk. Sinceck ≥ ckfor largek, it follows fromLemma 2.1 thatwk≥wk, and hence alsovk≥0 for largek. According toLemma 2.5, we have
Δvk−Ck−H k, vk
. 4.4
Hencevkis nonnegative and nonincreasing for largek, and this means that there exists a limit ofvksuch that
0≤lim
k→∞vk<∞. 4.5
Next, letN∈Nbe sufficiently large,k > N. Summing4.4fromNtok, we obtain
vN−vk1k
jN
Cjk
jN
H j, vj
, 4.6
and hence
vN≥k
jNCjk
jNH j, vj
. 4.7
Lettingk→ ∞and using condition4.1, we have ∞
H k, vk
<∞. 4.8
Substitutingzkvk/GkintoHk, vk, we obtain
H
k, Gkzk
Gkzkrkhk1Φ Δhk
− rk zk1
hpk1 Φ
hk/Δhk Φ−1
zk1 :H k, zk
. 4.9
Now, it follows from conditions4.2and4.5thatzk→0 ask→ ∞. Hence we may approx- imate the functionHk, z by the second-degree Taylor polynomial at the centerz 0kis regarded as a parameter. By a direct computation, we have
Hk, 0 0, Hzk,0 0, Hzzk,0 qrkhk
Δhk
p
hk1 , 4.10
and hence
Hk, z qrkhk
Δhk
p 2hk1 z2o
z2
asz−→0. 4.11
The termoz2is of the formHzzzk, ξz3for someξ∈0, z. By a direct computation, we have Hzzzk,0 qrkhk
Δhk
p
h2k1 q−2hk1−2q−1Δhk
, 4.12
that is,
Hzzzk,0≤qrkhk
Δhkp
hk1 |q−2| 2q−1
. 4.13
SinceHzzzk, zis continuous with respect toznearz0, there exists a constantM >0 such that
Hzzzk, ξ≤Mrkhk
Δhk
p
hk1 , 4.14
and hence4.11can be written in the form Hk, z qrkhk
Δhkp
2hk1 z2
1o1
asz−→0 4.15
and the convergenceo1 → 0 asz → 0 is uniform with respect tok. This means that there existsN1such that
q−εrkhk Δhkp
2hk1 z2k<H k, zk
< qεrkhk Δhkp
2hk1 z2k fork≥N1, 4.16 and consequently
∞>
∞
H k, vk
∞H k, zk
>q−ε 2
∞ rkhk
Δhk
p
hk1 z2k q−ε
2
r∞ khk
Δhk
p
vk2
hk1G2k q−ε 2
∞ vk2 rkhkhk1
Δhk
p−2.
4.17
Taking into account condition 3.3, it follows that vk → 0 ask → ∞.Thus we can apply Taylor’s formula to the functionFk, v : RkvHk, vat the centerv 0.By a direct computationsee also the proof ofLemma 2.6, we havekis regarded again as a parameter
Fk, 0 0, Fvk,0 0, Fvvk,0 2, 4.18 and hence
Fk, v v2o v2
v2
1o1
asv−→0. 4.19
Similarly as in the case ofHk, z, the convergence o1→0 asv→0 is uniform with respect tokbecause of4.2and2.20. Hence
Hk, v v2 Rkv
1o1
asv−→0. 4.20
Consequently, there existsN2> N1such that
1−ε 2
vk2
Rkvk < H k, vk
<
1ε
2 vk2
Rkvk fork≥N2. 4.21 Since
Rk2 qGkhk1
Δhk 2 qGk
1 hk
Δhk
> 2
qGk, 4.22
from conditions4.2and4.22we have vk
Rk −→0 ask−→ ∞. 4.23
This means that there existsN3∈Nsuch that 1−ε
1 1−ε vk/Rk
< 1−ε/2
1vk/Rk fork≥N3, 4.24
that is,
1−ε
Rk 1−εvk 1
Rk/1−ε vk < 1−ε/2 Rkvk
fork≥N3. 4.25
Consequently, from4.21we obtain v2k
Rk/1−ε vk < H k, vk
fork≥max
N2, N3
, 4.26
and according toLemma 2.5,
ΔvkCk v2k
Rk/1−ε vk <0 fork≥max N2, N3
. 4.27
The last inequality is the Riccati inequality associated with the equation Δ
Rk
1−εΔyk
Ckyk10, 4.28 that is, with4.3. SinceRk/1−ε vk > 0, it follows fromLemma 2.2that this equation is nonoscillatory, which is a contradiction.
Theorem 4.2. Letck ≥ckfor largekand lethk >0 be a solution of 1.7such that conditions3.3, 4.1, and
lim inf
k→∞ rkhkΦ Δhk
>0 4.29
are satisfied. If there existsε >0 such that the equation Δ
RkΔyk
1εCkyk10 4.30
is nonoscillatory, then1.1is also nonoscillatory.
Proof. It follows from nonoscillation of4.30, that is,
Δ Rk
1εΔyk
Ckyk10, 4.31 that there exists a solutionvkof the associated Riccati equation
ΔvkCk v2k
Rk/1ε vk 0 4.32
such thatRk/1ε vk > 0.This means thatvk is nonincreasing sinceCk ≥ 0 for large k.
Moreover, since∞
1/Rk ∞, it follows fromLemma 2.1thatvk≥0.Hence condition4.5 holds. Summing4.32fromNtokletN∈Nbe sufficiently large,k > N, we obtain
vN−vk1k
jN
Cjk
jN
vj2
Rj/1ε vj, 4.33
and hence
vN≥k
jN
Cjk
jN
vj2
Rj/1ε vj. 4.34
Lettingk→ ∞and using4.1, we have ∞ v2k
Rk/1ε vk <∞. 4.35
This, together with conditions3.3and4.5, implies thatvk → 0 ask → ∞.Hence by the Taylor formula for the functionFk, v : RkvHk, vat the centerv 0see the com- putations in the proof ofTheorem 4.1and observe that4.29is still sufficient for the uniform convergenceo1→0 asv→0 in4.19, we have
1−ε
2 v2k
Rkvk < H k, vk
<
1ε
2 v2k
Rkvk
for largek, 4.36
and we can show similarly as in the proof ofTheorem 4.1note that4.29implies that4.23 holds in view of4.22that
1ε
2 v2k
Rkvk < v2k
Rk/1ε vk for largek. 4.37
Consequently, from4.32,
ΔvkCkH k, vk
<0, 4.38
which according toLemma 2.5means that
wk1ck− rkwk
Φ Φ−1
rk Φ−1
wk <0, 4.39
wherewkh−pk vkwkandwkrkΦΔhk/hkis a solution of the Riccati equation associated with1.7, hencerkwk >0. Sincevk ≥0 for largek, we haverkwk rkh−pk vkwk>0 and nonoscillation of1.1follows fromLemma 2.2.
5. Remarks and applications
We start this section with a discussion of the continuous counterparts of the results presented in the previous sections. In 15and the subsequent papers 10,16,17,1.2is viewed as a perturbation of another half-linear differential equation of the same form
rtΦ
xctΦx 0 5.1
and it is supposed that this equation possesses a positive solutionhsuch thatht/0 for large t. DenoteGt:rthtΦhtand consider the differential equation
v
ct−ct
hpt p−1r1−qth−qtHt, v 0, 5.2 where
Ht, v:vGtq−qvΦ−1 Gt
−Gtq. 5.3
By a direct computation, similar to that given in the proof ofLemma 2.5, one can show that this equation has a solution defined on some interval T,∞if and only if the Riccati equation associated with1.2
wct p−1r1−qt|w|q0 5.4
has a solution on T,∞. These solutions are related by the formulavhpw−wh, wherewh rΦh/h. The functionH in5.3is the continuous counterpart of the functionH given by 2.10. We have the following estimates for the functionHt, vwhich are proved, for example, in 18,19 we present here these estimates in a modified form with respect to 18:
Ht, v≤ q
2Gtq−2v2, p≥2, Ht, v≥ q
2Gtq−2v2, p∈1,2
5.5
for everyv ∈ R. Moreoversee 19, for everyM ≥ 0, there exist constantsK1 K1M, K2K2Msuch that
K1Gtq−2v2≤Ht, v≤K2Gtq−2v2 5.6 for |v| ≤ M and any p > 1. These estimates enable to approximate the function p − 1r1−qh−qHt, vin5.2by the function
Kr1−qh−q|G|q−2v2 K
rh2hp−2v2, 5.7
where K is a real constant, and after this approximation,5.2becomes the classical Riccati equation corresponding to a linear Sturm-Liouville differential equation. This linear equation is then used to study oscillatory properties of1.2.
In our paper, we follow a similar idea in the discrete case. The essential difference with respect to the continuous case is that the function H given by2.10 is substantially more complicated than its continuous counterpart 5.3; in particular, we were able to formulate inequalities for the functionH inLemma 2.6 only under more restrictive assumptions than in the continuous case. This is also the reason why assumptions ofnon-oscillation criteria formulated inSection 3are more restrictive than those of oscillation criteria for1.2given in 10,16.
Now we comment in more detail on assumptions of Theorems4.1and4.2of the previous section. Assumption4.1is natural since if the sum of this series is∞,1.1is oscillatory by Lemma 2.3. Assumptions3.3and4.2are technical and we needed them to show that the solutionvof3.6satisfiesvk→0 ask→ ∞. Assumptions3.3and4.2can be replaced by a formally less restrictive assumption that
∞
Hk, α ∞ 5.8
for everyα >0, but it may be difficult to verify this assumption in particular cases. Concerning assumptions ofTheorem 4.2, we also needed them to prove thatvk→0ask→ ∞.
We conclude the paper with a statement illustrating application ofTheorem 4.2of the previous section.
Theorem 5.1. Consider the perturbed Euler-type difference equation ΔΦ
Δxk
γp
k1pdk
Φ
xk1
0, γp: p−1
p p
, 5.9
and suppose that
k→∞limdkk1p1∞ 5.10
and that
∞
dkk1p−1<∞. 5.11
Then5.9is nonoscillatory provided
lim sup
k→∞ logk ∞ jk
djj1p−1< 1 2
p−1 p
p−1
. 5.12
Proof. Consider the sequencehk:kp−1/pand letck:−ΔΦΔhk/Φhk1.We have Δhk k1p−1/p−kp−1/pkp−1/p
k1 k
p−1/p
−1 kp−1/p
1p−1
pk − p−1 2p2k2 O
k−3
−1 p−1
p k−1/p
1− 1 2pk O
k−2
5.13
ask→ ∞; hence
Φ Δhk
p−1
p p−1
k−p−1/p
1−p−1 2pk O
k−2
, 5.14
and using the fact thatk−1 k1−1Ok1−2ask→ ∞, we have
ΔΦ Δhk
p−1
p p−1
k1−p−1/p−k−p−1/p
−p−1
2p k1−21/p−k−21/p ΔO
k−31/p
p−1 p
p−1
k1−p−1/p
1− k1
k
p−1/p
−p−1
2p k1−21/p 1−
k1 k
2−1/p
O
k1−31/p −
p−1 p
p
k1−21/pO
k1−31/p .
5.15
Consequently,
ck−ΔΦ Δhk Φ
hk1 γp
k1pO
k1−p−1
. 5.16
To prove that5.9is nonoscillatory, we applyTheorem 4.2with
ck γp
k1pdk 5.17
and ck given by5.16. The termOk 1−p−1is of the form akk1−p−1, where ak is a bounded sequence; so we have
ck−ckdk− ak
k1p1 >0 5.18
because of5.10. Condition4.1is also satisfied since from5.11we have that the series ∞
ck−ck
hpk1∞
dkk1p−1− ak
k12
5.19
is convergent.
Concerning assumptions3.3and4.29, hkhk1
Δhk
p−2kp−1/pk1p−1/p p−1
p p−2
k−p−2/p 1O
k−1
5.20
p−1
p p−2
k 1O
k−1
5.21
and similarly
GkhkΦ Δhk
p−1
p p−1
1O
k−1 5.22
ask→ ∞. This means that both3.3and4.29are satisfied. Moreover, by a routine computa- tion, one findsusing the discrete l’Hospital rule; see, e.g., 20, page 29that
k→∞lim k−1
j11/j
logk 1. 5.23
Now, if5.12holds, there existsε >0 such that lim sup
k→∞ logk ∞ jk
djj1p−1< 1 21ε
p−1 p
p−1
1 2q1ε
p−1 p
p−2
. 5.24
This means that
lim sup
k→∞ logkq 2
p p−1
p−2∞ jk
1εCj< 1
4, 5.25
whereCis given by1.9withhk kp−1/p. Then using5.21and5.23 withRgiven by 1.9, we have
lim sup
k→∞
k−1 R−1j
∞
jk
1εCj
< 1
4. 5.26
This implies, by Lemma 2.4, that 4.30 is nonoscillatory and the statement follows from Theorem 4.2.
Remark 5.2.iWe conjecture that5.9is oscillatoryunder5.10and5.11provided lim inf
k→∞ logk ∞ jk
djj1p−1> 1 2
p−1 p
p−1
. 5.27
The proof of this conjecture could follow essentially the same line as that of the previous theo- rem, with the only difference thatTheorem 4.1instead ofTheorem 4.2is usedand, of course, 2.7is used instead of2.8. However,Theorem 4.1needshto be the recessive solution of the equation
ΔΦ Δxk
ckΦ xk1
0 5.28
withck given by 5.16, and in contrast to the continuous case see 18,21–23, a suitable summation characterization of the recessive solution is not known yetnote that the results of 24do not apply to our case; so we are not able to prove thathk kp−1/pis really the recessive solution of5.28withcgiven by5.16. Moreover, we conjecture that condition4.2 inTheorem 4.1can be replaced by the weaker condition4.29.
iiA typical equation to which the previous theorem applies is the Riemann-Weber-type difference equation
Δ Φ
Δxk
γp
k1p λ
k1plog2k1
Φ xk1
0. 5.29
ByTheorem 4.2, this equation is nonoscillatory ifλ <1/2p−1/pp−1.
iii The previous statement can be viewed as a partial extension of the non- oscillation criterion given in 25, where it is proved, among others, that the difference equation
Δ2xk 1 4k2
1 1
log2k· · · 1 n−1
j1
logjk2 λ n
j1
logjk2
xk10, 5.30
where log0k k, logjk loglogj−1k,j 1, . . . , n, is oscillatory ifλ > 1 and nonoscillatory if λ <1.
Acknowledgments
The research is supported by Grant no. 201/07/0145 of the Czech Grant Agency of the Czech Republic and the Research Project no. MSM0022162409 of the Czech Ministry of Education.
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