Linearized Riccati Technique and (Non )Oscillation Criteria for Half Linear 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 ´y1 _{and Simona Fiˇsnarov ´a}2

1_{Department of Mathematics and Statistics, Masaryk University, Jan´aˇckovo n´am. 2a,} 66295 Brno, Czech Republic

2_{Department 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

ΔrkΦ

Δxk

ckΦxk1 0, Φx:|x|p−2x, p >1, 1.1

where r, care real-valued sequences andrk > 0. This equation can be regarded as a discrete counterpart of the half-lineardifferentialequation

rtΦxctΦx 0 1.2

It is known that oscillatory properties of 1.1are very similar to those of the second-order Sturm-Liouville difference equationwhich is a special case ofp2 in1.1:

ΔrkΔxk

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 oscillaequa-tion 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-lineardiff er-entialequation

Φxγp

tpΦx 2

p−1

p

_p−1_δt

tp Φx 0, γp :

p−1

p

_p

, 1.4

is viewed as a perturbation of the Euler-type half-linear differential equation

Φx γp

tpΦx 0, 1.5

and oscillatory properties of1.4are studied via thelinearequation

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≥2and 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:

ΔrkΦ

Δxk

ckΦ

xk1

0, 1.7

and oscillatory properties of solutions of1.1are related to those of the linear second-order difference equation

ΔRkΔyk

Ckyk10, 1.8

where

Rk 2

qrkhkhk1Δhk

p−2

, Ck

ck−ck

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 ageneralized zeroin 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 bedisconjugatein a discrete interval m, nif the solutionxof1.1given by the initial conditionxm0,xm1/0, has no generalized zero inm, n1. Equation1.1is said to

benonoscillatoryif there existsm∈Nsuch that it is disconjugate on m, nfor everyn > m, and it is said to beoscillatoryin the opposite case.

If x is a solution of1.1 such that xk/0 in some discrete interval m,∞, then wk

rkΦΔxk/xkis a solution of the associated Riccati-type equation

R wk

: Δwkckwk

1− rk

ΦΦ−1_r

k

Φ−1_w

k

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 distinguishedsolutionw which has the property that there exists an interval m,∞such that any other solutionwof2.1for whichΦ−1_r

k 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∞. Ifw is the distinguished solution of2.1, then the associated solution of1.1given by the formula

xk k−1

jm

1 Φ−1 wj

rj

2.2

is said to be therecessive solutionof1.1 see 13. Note that in the linear casep2 a solution

xof1.3is recessive if and only if

∞

₁

rkxkxk1 ∞.

Our first statement presents a comparison theorem for distinguished solutions of2.1 and2.4given below.

Lemma 2.1see 13. Let1.1be nonoscillatory and letck ≥ ck for largek. Further, letwk,vk be

distinguished solutions of the corresponding generalized Riccati equations2.1and

R vk

: Δvkckvk

1− rk

ΦΦ−1_r

k

Φ−1_v

k

0, 2.4

respectively. Then there existsm ∈Zsuch thatwk ≥ vk fork ∈ m,∞. In particular, ifck ≥ 0and _∞

r_k1−q∞, thenwk ≥0for largek.

The next statement relates nonoscillation of1.1to the existence of a certain solution of the Riccati inequality associated with2.1.

Lemma 2.2 see 2, Theorem 8.2.7. Equation1.1is nonoscillatory if and only if there exists a sequencewk satisfyingrkwk >0and

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

hp_k1∞, 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, _∞

r_k−1∞, and∞ck <∞. If

lim inf k→∞

k−1

₁

rj

_∞

jk

cj

> 1

4, 2.7

then1.3is oscillatory. If

lim sup k→∞

k−1

₁

rj

_∞

jk

cj

< 1

4, 2.8

For the remaining part of this section, we suppose that1.7is nonoscillatory and we let

and define the function

Hk, v:vrkhk1Φ

Proof. By a direct computation and using the fact thatwkis a solution of2.4, we obtain

Ifwk is a solution of2.1, thenvk satisfies2.13. We prove the nonnegativity of the function proves the last statement ofLemma 2.5.

Lemma 2.6. LetR,Gbe defined by1.9and2.9, respectively, and suppose thatGk > 0fork ∈ N.

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

in some right neighborhood ofv0. Further, we have

Denote byAvthe expression in brackets in the last expression. By a direct computation, we have

Av q−2Φ−1_{vG qR−GvG}q−2_−2rq−1_hq_{, 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

ΔRkΔyk

Ckyk10, 3.1

whereRandCare given by1.9, that is,

Rk 2

qrkhkhk1

Δhk

_p−2

, Ck

ck−ck

hp_k1. 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 3.1 with R, C given 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

v2

k

Rkvk 0

3.4

such that Rk vk > 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

ΔvkCk H

k, vk

≤0. 3.5

Theorem 3.2. Letp ∈ 1,2,ck ≥ ck for largek, and let hbe 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

wk rk > 0 for largek. Put againv hpw−w, where w is the distinguished solution of 2.4. Thenvsolves the equation

ΔvkCk H

k, vk

0, 3.6

and byLemma 2.1, we havewk ≥ wk for largek, hencevk ≥ 0,and thereforeRk vk > 0 for largek. ByLemma 2.6,

ΔvkCk

v2_k Rkvk ≤

0. 3.7

This means that3.1is nonoscillatory byLemma 2.2.

4. Criteria without restriction onp

Throughout this section, we suppose thatRk,Ck, andGk are given by1.9and2.9, respec-tively, and that1.7is nonoscillatory.

Theorem 4.1. Letck ≥ck for largekand lethk >0be the recessive solution of 1.7such that

∞

ck−ck

hp_k1<∞. 4.1

Further, suppose that condition3.3holds and

lim k→∞rkhkΦ

Δhk

∞. _4.2

If there existsε >0such that the equation

ΔRkΔyk

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, that 1.1 is nonoscillatory, and let xk be its recessive solution. Denote by wk rkΦΔxk/

xkandwk rkΦΔhk/hkthe distinguished solutions of the Riccati equations2.1and2.4, respectively, and putvk : hp_kwk −wk. Sinceck ≥ ck for 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 ofvk such that

0≤ lim

Next, letN ∈Nbe sufficiently large,k > N. Summing4.4fromNtok, we obtain regarded as a parameter. By a direct computation, we have

and hence4.11can be written in the form existsN1such that

q−εrkhk computationsee also the proof ofLemma 2.6, we havekis regarded again as a parameter

This means that there existsN3∈Nsuch that

1−ε

1 1−εvk/Rk

< 1−ε/2

1vk/Rk for k≥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

v2

k

Rk/1−ε vk < H

k, vk

for k≥maxN2, N3, 4.26

and according toLemma 2.5,

ΔvkCk

v_k2

Rk/1−ε vk <

0 fork≥maxN2, N3. 4.27

The last inequality is the Riccati inequality associated with the equation

Δ

Rk 1−εΔyk

Ckyk10, 4.28

that is, with 4.3. SinceRk/1−ε vk > 0, it follows from Lemma 2.2that this equation is nonoscillatory, which is a contradiction.

Theorem 4.2. Letck ≥ck for largekand lethk >0be a solution of 1.7such that conditions3.3, 4.1, and

lim inf k→∞ rkhkΦ

Δhk

>0 _4.29

are satisfied. If there existsε >0such that the equation

ΔRkΔyk

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 solutionvk of the associated Riccati equation

ΔvkCk

v2_k

Rk/1ε vk 0

such thatRk/1ε vk > 0.This means that vk 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−vk1

k

jN

Cj k

jN

v2

j

Rj/1ε vj, 4.33

and hence

vN ≥ k

jN

Cj k

jN

v2_j

Rj/1ε vj. 4.34

Lettingk→ ∞and using4.1, we have

∞

v2_k

Rk/1ε vk <∞.

4.35

This, together with conditions3.3and 4.5, implies thatvk → 0 as k → ∞.Hence by the Taylor formula for the function Fk, v : Rk vHk, vat the center v 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 _v2

k

Rkvk < H

k, vk

<

1 ε

2 _v2

k

Rkvk for large k,

4.36

and we can show similarly as in the proof ofTheorem 4.1note that4.29implies that4.23 holds in view of4.22that

1 ε

2 _v2

k

Rkvk <

v2

k

Rk/1ε vk for large k.

4.37

Consequently, from4.32,

ΔvkCk H

k, vk

<0, 4.38

which according toLemma 2.5means that

wk1ck− _ΦΦ−1 rkwk

rk

Φ−1_w

k

<0, _4.39

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

rtΦxctΦx 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

vct−cthp_{t p−1r}1−q_th−q_{tHt, v 0, 5.2}

where

Ht, v:vGtq−qvΦ−1Gt−Gtq. 5.3

By a direct computation, similar to that given in the proof of Lemma 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−q_t|w|q _{0 5.4}

has a solution on T,∞. These solutions are related by the formulavhpw−wh, wherewh

rΦh/h. The functionH in 5.3is 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

2Gt q−2

v2, p≥2,

Ht, v≥ q

2Gt q−2

v2, p∈1,2

5.5

for every v ∈ R. Moreover see 19, for every M ≥ 0, there exist constants K1 K1M,

K2K2Msuch that

K1Gtq−2v2_{≤Ht, v≤K2Gt}q−2_v2 _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−q_h−q_|G|q−2_v2 K rh2_hp−2v

2_,

5.7

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 by 2.10 is substantially more complicated than its continuous counterpart 5.3; in particular, we were able to formulate inequalities for the function H in Lemma 2.6 only under more restrictive assumptions than in the continuous case. This is also the reason why assumptions of non-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 4.2 of the previous section.

Theorem 5.1. Consider the perturbed Euler-type difference equation

ΔΦΔxk

Then5.9is nonoscillatory provided

ask → ∞; hence

To prove that5.9is nonoscillatory, we applyTheorem 4.2with

ck

γp

k1p dk 5.17

and ck given by 5.16. The term Ok 1−p−1 is of the form akk1−p−1, where ak is a bounded sequence; so we have

ck−ck dk− ak

k1p1 >0 5.18

because of5.10. Condition4.1is also satisfied since from5.11we have that the series

Concerning assumptions3.3and4.29, tion, one findsusing the discrete l’Hospital rule; see, e.g., 20, page 29that

lim

This implies, by Lemma 2.4, that 4.30 is nonoscillatory and the statement follows from Theorem 4.2.

with ck 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 yet note that the results of 24do not apply to our case; so we are not able to prove thathk kp−1/p is 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

ΔΦΔxk

_γ

p k1p

λ

k1plog2k1

Φxk1

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

Δ2_x

k 1 4k2

1 1

log2k · · ·

1 _n−1

j1

log_jk2

λ

_n j1

log_jk2

xk10, _5.30

where log₀k k, log_jk loglog_j−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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