Regularly Varying Solutions of Second Order Difference Equations with Arbitrary Sign Coefficient

doi:10.1155/2010/673761

Research Article

Regularly Varying Solutions of

Second-Order Difference Equations with

Arbitrary Sign Coefficient

Serena Matucci

_{and Pavel ˇ}

_{Reh ´ak}

1_{Department of Electronics and Telecommunications, University of Florence, 50139 Florence, Italy} 2_{Institute of Mathematics, Academy of Sciences CR, ˇZiˇzkova 22, 61662 Brno, Czech Republic}

Correspondence should be addressed to Pavel ˇReh´ak,[email protected]

Received 15 June 2010; Accepted 25 October 2010

Academic Editor: E. Thandapani

Copyrightq2010 S. Matucci and P. ˇReh´ak. 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.

Necessary and sufficient conditions for regular or slow variation of all positive solutions of a second-order linear difference equation with arbitrary sign coefficient are established. Relations with the so-calledM-classification are also analyzed and a generalization of the results to the half-linear case completes the paper.

1. Introduction

We consider the second-order linear difference equation

Δ2_y

kpkyk10 1.1

onN, wherepis an arbitrary sequence.

The principal aim of this paper is to study asymptotic behavior of positive solutions to1.1in the framework of discrete regular variation. Our results extend the existing ones for 1.1, see 1, where the additional condition pk < 0 was assumed. We point out that

recessive and dominant solutions. A possible extension to the case of half-linear difference equations is also indicated.

The paper is organized as follows. In the next section we recall the concept of regularly varying sequences and mention some useful properties of1.1which are needed later. In the main section, that is,Section 3, we establish sufficient and necessary conditions guaranteeing that1.1has regularly varying solutions. Relations with theM-classification is analyzed in Section 4. The paper is concluded by the section devoted to the generalization to the half-linear case.

2. Preliminaries

In this section we recall basic properties of regularly and slowly varying sequences and present some useful information concerning1.1.

The theory of regularly varying sequencessometimes called Karamata sequences, initiated by Karamata 3in the thirties, received a fundamental contribution in the seventies with the papers by Seneta et al. see 4,5starting from which quite many papers about regularly varying sequences have appeared, see 6 and the references therein. Here we make use of the following definition, which is a modification of the one given in 5, and is equivalent to the classical one, but it is more suitable for some applications to difference equations, see 6.

Definition 2.1. A positive sequencey {yk},k ∈N, is said to beregularly varying of index, ∈R, if there existsC >0 and a positive sequence{α_k}such that

lim

k yk

αk C, limk k Δαk

αk . 2.1

If 0, then{yk}is said to beslowly varying. Let us denote byRVthe totality of

regularly varying sequences of indexand bySVthe totality of slowly varying sequences. A positive sequence {y_k} is said to be normalized regularly varying of index if it satisfies limkkΔyk/yk. If0, thenyis called anormalized slowly varying sequence. In the sequel,

NRV and NSV will denote, respectively, the set of all normalized regularly varying sequences of index, and the set of all normalized slowly varying sequences. For instance, the sequence{yk}{logk} ∈ NSV, and the sequence{yk}{klogk} ∈ NRV, for every ∈R; on the other hand, the sequence{y_k}{1 −1k/k} ∈ SV \ NSV.

The main properties of regularly varying sequences, useful to the development of the theory in the subsequent sections, are listed in the following proposition. The proofs of the statements can be found in 1, see also 4,5.

Proposition 2.2. Regularly varying sequences have the following properties.

iA sequencey ∈ RVif and only ify_k kϕ_kexp{k_j−₁1ψ_j/j}, where{ϕ_k}tends to a positive constant and{ψk}tends to 0 ask → ∞. Moreover,y ∈ RVif and only if ykkLk, whereL∈ SV.

iiA sequence y ∈ RVif and only ifyk ϕkkj−111δj/j, where{ϕk}tends to a positive constant and{δk}tends toask → ∞.

iv Lety∈ RV. If one of the following conditions holds (a)Δyk ≤0andΔ2yk ≥0, or (b) Δyk≥0andΔ2yk≤0, or (c)Δyk≥0andΔ2yk≥0, theny∈ NRV.

vLety∈ RV. Thenlim_ky_k/k−ε∞andlim_ky_k/kε0for everyε >0.

viLetu∈ RV1andv∈ RV2. Thenuv∈ RV12and1/u∈ RV−1. The same holds ifRVis replaced byNRV.

vii Ify ∈ RV, ∈ R, is strictly convex, that is,Δ2_y

k > 0 for everyk ∈ N, theny is decreasing provided ≤ 0, and it is increasing provided > 0. Ify ∈ RV, ∈ R, is strictly concave for everyk∈N, thenyis increasing and≥0.

viii Ify∈ RV, thenlim_ky_k1/yk1.

Concerning1.1, a nontrivialsolutionyof1.1is callednonoscillatoryif it is eventually of one sign, otherwise it is said to beoscillatory. As a consequence of the Sturm separation theorem, one solution of1.1is oscillatory if and only if every solution of1.1is oscillatory. Hence we can speak about oscillation or nonoscillation of equation1.1. A classification of nonoscillatory solutions in case p is eventually of one sign, will be recalled in Section 4. Nonoscillation of 1.1 can be characterized in terms of solvability of a Riccati difference equation; the methods based on this relation are referred to as theRiccati technique: equation

1.1is nonoscillatory if and only if there isa∈Nand a sequencewsatisfying

Δwkpk w

2

k

1w_k 0 2.2

with 1wk > 0 fork ≥a. Note that, dealing with nonoscillatory solutions of1.1, we may

restrict our considerations just to eventually positive solutions without loss of generality. We end this section recalling the definition of recessive solution of1.1. Assume that

1.1is nonoscillatory. A solutiony of1.1is said to be arecessive solutionif for any other solution x of 1.1, with x /λy, λ ∈ R, it holds limkyk/xk 0. Recessive solutions are

uniquely determined up to a constant factor, and any other linearly independent solution is called adominant solution. Letybe a solution of1.1, positive fork≥a≥0. The following characterization holds:yis recessive if and only if∞_ka1/ykyk1 ∞;yis dominant if and

only if∞_ka1/ykyk1<∞.

3. Regularly Varying Solutions of Linear Difference Equations

In this section we prove conditions guaranteeing that1.1has regularly varying solutions. Hereinafter,xk∼ykmeans limkxk/yk1, wherexandyare arbitrary positive sequences.

LetA ∈ −∞,1/4and denote by1 < 2, therealroots of the quadratic equation 2_{−A0. Note that 1−2}

1 √

1−4A >0, 1−12, sgnAsgn1, and2 >0.

Theorem 3.1. Equation 1.1 is nonoscillatory and has a fundamental system of solutions {y, x}

such thatykk1L

k∈ NRV1andxkk2L

k∈ NRV2if and only if

lim

k k ∞

jk

pjA∈

−∞,1

4

whereL,L∈ NSVwithLk∼1/1−21Lkask → ∞. Moreover,yis a recessive solution,xis a dominant solution, and every eventually positive solutionzof 1.1is normalized regularly varying, withz∈ NRV1∪ NRV2.

Proof . First we show the last part of the statement. Let{x, y}be a fundamental set of solutions of1.1, withy ∈ NRV1,x ∈ NRV2, and letzbe an arbitrary solution of1.1, with

we getc2 >0 because of the positivity ofzkforklarge and the strict inequality between the

indices of regular variation1< 2. Moreover,z∈ NRV2. Indeed, taking into account that yk/xk → 0,kΔyk/yk → 1, andkΔxk/xk → 2, it results

Now we prove the main statement.

in particular we see that∞pjconverges. The discrete L’Hospital rule yields

lim

k _∞

jkw2j/

1wj

1/k limk

kk1w2

k

1w_k

2

1. 3.6

Hence, multiplying3.5bykwe get

k∞ jk

pjkwk−k ∞

jk w2

j

1wj −→1−

2

1A 3.7

ask → ∞, that is,3.1holds. The same approach shows thatx∈ NRV2implies3.1.

Su

ffi

ciency

First we prove the existence of a solutiony ∈ NRV1of1.1. Setψk k

_∞

jkpj−A. We

look for a solution of1.1in the form

yk k−1

ja

11ψjwj

j

, 3.8

k ≥ a, with somea ∈ N. In order thatyis a nonoscillatorysolution of1.1, we need to determinewin3.8in such a way that

uk 1ψ_kkwk 3.9

is a solution of the Riccati difference equation

Δukpk u

2

k

1uk 0

3.10

satisfying 1 uk > 0 for large k. If, moreover, limkwk 0, then y ∈ NRV1 by

Proposition 2.2. Expressing3.10in terms ofw, in view of3.9, we get

Δwk−1w_kk−Ak1

1ψkwk 2

k2_k1ψkwk 0, 3.11

that is,

Δwkwk21−1_k2ψk w

2

kψk221ψk

whereGis defined by

0, hencehk is positively decreasing toward zero, seeProposition 2.2. It will be convenient

to rewrite3.12in terms ofh. Multiplying3.12byhand using the identitiesΔhkwk

Solvability of this equation will be examined by means of the contraction mapping theorem in the Banach space of sequences converging towards zero. The following properties of h will play a crucial role in the proof. The first two are immediate consequences of the discrete L’Hospital rule and of the property of regular variation ofh:

sinceΔhk ∼ 21−1hk/k ∼ 21−1hk1/k1 ∼ Δhk1, in view ofh ∈ NRV21−1. to zero, endowed with the sup norm. LetΩdenote the set

Ω w∈∞

0 :|wk| ≤δ, k≥a

3.28

and define the operatorTby

Before we estimateH2_{, we need some auxiliary computations. The Lagrange mean value}

k≥a. Finally, using summation by parts, we get

H3

Now, thanks to the contraction mapping theorem, there exists a unique elementw∈Ω such that w Tw. Thus w is a solution of 3.16, and hence of 3.11, and is positively decreasing towards zero. Clearly, udefined by3.9is such that limkuk 0 and therefore

Hence,1−21xk∼k/ykk1−1/L

k, that is,xk∼k1−1L

k, whereLk1/1−21Lk. Since

L∈ NSVbyProposition 2.2, we getx∈ RV1−1 RV2byProposition 2.2. It remains

to show thatxis normalized. We have

kΔxk xk

kΔykkj−a11/

yjyj1 kyk1/

ykyk1 xk

kΔy_y k k

k xkyk.

3.40

Thanks to this identity, sincekΔyk/yk∼1andk/xkyk∼1−21, we obtain limkkΔxk/xk

1−12, which impliesx∈ NRV2.

Remark 3.2. iIn the above proof, the contraction mapping theorem was used to construct a recessive solutiony ∈ NRV1. A dominant solutionx ∈ NRV2resulted fromyby

means of the known formula for linearly independent solutions. A suitable modification of the approach used for the recessive solution leads to the direct construction of a dominant solutionx∈ NRV2. This can be useful, for instance, in the half-linear case, where we do

not have a formula for linearly independent solutions, seeSection 5.

ii A closer examination of the proof ofTheorem 3.1shows that we have proved a slightly stronger result. Indeed, it results

y∈ NRV1 ⇐⇒lim

k k ∞

jk

pjA < 1₄ ⇐⇒x∈ NRV2 . 3.41

Theorem 3.1can be seen as an extension of 1, Theorems 1 and 2in whichpis assumed to be a negative sequence, or as a discrete counterpart of 2, Theorems 1.10 and 1.11, see also 7, Theorem 2.3.

As a direct consequence of Theorem 3.1 we have obtained the following new nonoscillation criterion.

Corollary 3.3. If there exists the limit

lim

k k ∞

jk pj∈

−∞,1

4

, 3.42

then1.1is nonoscillatory.

Remark 3.4. In 8it was proved that, if

−3

4 <lim infk k ∞

jk

pj≤lim sup k k

∞

jk

pj< 1₄, 3.43

4. Relations with

M

-Classification

Throughout this section we assume that p is eventually of one sign. In this case, all nonoscillatory solutions of1.1are eventually monotone, together with their first difference, and therefore can be a priori classified according to their monotonicity and to the values of the limits at infinity of themselves and of their first difference. A classification of this kind is sometimes calledM-classification, see, for example, 9–12for a complete treatment including more general equations. The aim of this section is to analyze the relations between the classification of the eventually positive solutions according to their regularly varying behavior, and theM-classification. The relations with the set of recessive solutions and the set of dominant solutions will be also discussed. We point out that all the results in this section could be established also in the continuous case and, as far as we know, have never been derived both in the discrete and in the continuous case.

Because of linearity, without loss of generality, we consider only eventually positive solutions of1.1. Since the situation differs depending on the sign ofpk, we treat separately

the two cases. Note that 1.1, with p negative, has already been investigated in 1, and therefore here we limit ourselves to state the main results, for the sake of completeness.

(I)

pk

>

0

for

k

≥

a

Any nonoscillatory solutionyof1.1, in this case, satisfiesykΔyk >0 for largek, that is, all

eventually positive solutions are increasing and concave. We denote this property by saying thatyis of classM, beingM {y : ysolution of1.1,y_k > 0,Δy_k > 0 for largek}. This class can be divided in the subclasses

M ∞,B

y∈M_{: lim}

k yk∞, limk Δyky, 0< y <∞

,

M ∞,0

y∈M_{: lim}

k yk∞, limk Δyk0

,

M

B,0

y∈M_{: lim}

k yky, limk Δyk0, 0< y<∞

4.1

depending on the possible values of the limits of y and ofΔy. Solutions in M_∞,B,M_∞,0,

M

B,0 are sometimes called, respectively, dominant solutions, intermediate solutions, and

subdominant solutions, since, for largek, it holdsxk> yk> zkfor everyx∈M_∞,B,y∈M_∞,0,

andz ∈M_B,0. The existence of solutions in each subclass, is completely characterized by the convergence or the divergence of the series

I ∞ ka

kpk 4.2

see 11,12. The following relations hold

I <∞ ⇐⇒M_M

B,0∪M∞,B, with MB,0/∅,M∞,B/∅,

Let

P lim

k k ∞

jk

pj. 4.4

Sincek∞_jkpj<∞jkjpj, then the following relations betweenIandPhold: iifP >0 thenI ∞;

iiifI <∞thenP0.

FromTheorem 3.1, it follows that, ifP0, then1.1has a fundamental set of solutions{x, y} withx∈ NSV, andy∈ NRV1; if 0< P <1/4, then1.1has a fundamental set of solutions

{u, v} with u ∈ NRV1, andv ∈ NRV2, 0 < 1 < 2 < 1. Further, all the positive

solutions of1.1belong toNSV ∪ NRV1in the first case, and toNRV1∪ NRV2in

the second one. Set

M

SVM∩ NSV, M

RV

M_{∩ NRV , >0.} 4.5

By means of the above notation, the results proved inTheorem 3.1can be summarized as follows

∅/M_M

SV∪MRV1⇐⇒P0,

∅/M_M RV

1 ∪M_RV

2 ⇐⇒P∈

0,1 4

. 4.6

By observing that every solutionx∈ RV1satisfies limkxk∞and thatM_∞,B ⊆M_RV1, we

get the following result.

Theorem 4.1. For1.1, withp_k>0for largek, the following hold.

iIfP 0andI <∞, thenMM_SV∪M_RV1, withM_SVM_B,0, M_RV1 M_∞,B. iiIfP 0andI ∞, thenMM_SV∪M_RV1 M_∞,0.

iiiIfP ∈0,1/4thenM M_RV1∪M_RV 2 M_∞,0.

R denote the set of all positive recessive solutions of 1.1 and D denote the set of all positive dominant solutions of 1.1. From Theorem 4.1, taking into account Theorem 3.1, the following characterization of recessive and dominant solution holds.

iIfP 0 andI <∞, thenRM_SVM_B,0andDM_RV1 M_∞,B.

iiIfP 0 andI∞, thenRM_SV⊂M_∞,0andDM_RV1⊂M_∞,0.

iiiIfP ∈0,1/4andI∞, thenRM_RV1⊂M_∞,0andDM_RV2⊂M_∞,0.

(II)

pk

<

0

for

k

≥

a

In this case, completely analyzed in 1, any positive solution y is either decreasing or eventually increasing. We say thatyis of classMin the first case, of classM− in the second one. It is easy to verify that everyy ∈ M satisfies lim_ky_k ∞, and everyy ∈ M− satisfies limkΔyk0. Therefore the setsMandM−can be divided into the following subclasses

M ∞,B

y∈M _{: lim}

k yk∞, limk Δyky, 0< y<∞

,

M ∞,∞

y∈M _{: lim}

k yk∞, limk Δyk∞

,

M− B,0

y∈M− _{: lim}

k yky, limk Δyk0, 0< y <∞

,

M−

0,0

y∈M− _{: lim}

k yk0, limk Δyk0

.

4.7

Also in this case, the existence of solutions of1.1in each subclass is completely described by the convergence or divergence of the seriesIgiven by4.2

M_M

∞,∞⇐⇒I−∞ ⇐⇒M−M−0,0,

M_M

∞,B ⇐⇒I >−∞ ⇐⇒M− M−B,0.

4.8

Let

M−

SVM−∩ NSV, M−RV

1 M−∩ NRV

1 , 1<0,

M RV

2 M∩ NRV

2 , 2>0.

4.9

Notice that, beingpknegative for largek, it results1≤0, 2≥1. The following holds.

Theorem 4.2see 1. For1.1, withpk<0for largek, it results in what follows.

Relations between recessive/dominant solutions and regularly varying solutions can be easily derived from the previous theorem, see also 1. We have the following.

iIfP 0 andI >−∞, thenRM−_SVM−_B,0, andDM_RV1 M_∞,B.

iiIfP 0 andI−∞, thenRM−_SVM−_0,0, andDM_RV 1 M_∞,∞.

iiiIfP ∈−∞,0, thenRM−_RV1 M−0,0, andDMRV 2 M∞,∞.

We end this section by remarking that in this case positive solutions are convex and therefore they can exhibit also arapidly varying behavior, unlike the previous case in which positive solutions are concave. We address the reader interested in this subject to the paper 1, in which the properties of rapidly varying sequences are described and the existence of rapidly varying solutions of1.1is completely analyzed for the casepk<0.

5. Regularly Varying Solutions of Half-Linear Difference Equations

In this short section we show how the results of Section 3 can be extended to half-linear difference equations of the form

ΔΦΔyk pkΦyk1 0, 5.1

wherep: N → RandΦu |u|α−1_{sgnu,α > 1, for everyu ∈R. For basic information on}

qualitative theory of5.1see, for example, 13.

Let A ∈ −∞,1/αα−1/αα−1 and denote by 1 < 2, the real roots of the

equation||α/α−1_{−A0. Note that sgnAsgn1andΦ}−1

1<α−1/α <Φ−12.

Theorem 5.1. Equation 5.1 is nonoscillatory and has two solutions y, x such that y ∈

NRVΦ−1

1andx∈ NRVΦ−12if and only if

lim

k k α−1∞

jk

pjA∈

−∞,1 α

_α−

1

α

α−1

. 5.2

Proof. The main idea of the proof is the analogous of the linear case, apart from some additional technical problems. We omit all the details, pointing out only the main differences.

Necessity

Setwk ΦΔyk/yk, thenwsatisfies the generalized Riccati equation

Δwkpkwk

1− 1

Φ1 Φ−1_wk

0, 5.3

and lim_kkα−1_w

k1. The proof can then proceed analogously to the linear case.

Su

ffi

ciency

The existence of both solutions y ∈ NRVΦ−1

1 and x ∈ NRVΦ−12 needs to be

formula for computing a linearly independent solution. For instance, a solutiony can be searched in the form

yk k−1

ja

1 Φ−1

1ψjvj jα−1

, 5.4

compare with3.8, whereψkkα−1∞jkpj−Aandvis such thatuk 1ψkvk/kα−1

is a solution of5.3. All the other details are left to the reader.

Remark 5.2. i Theorem 5.1can be seen as an extension of 6, Theorem 1 in which p is assumed to be a negative sequence, and as a discrete counterpart of 14, Theorem 3.1.

ii A closer examination of the proof ofTheorem 5.1shows that we have proved a slightly stronger result which reads as follows:

y∈ NRVΦ−1

1 ⇐⇒lim

k k α−1∞

jk

pjA < 1_{α α−}

1

α _α−1

⇐⇒x∈ NRVΦ−1 2 .

5.5

Similarly as in the linear case, as a direct consequence ofTheorem 5.1we obtain the following new nonoscillation criterion. Recall that a Sturm type separation theorem holds for equation5.1, see 13, hence one solution of5.1is nonoscillatory if and only if every solution of5.1is nonoscillatory.

Corollary 5.3. If there exists the limit

lim

k k α−1∞

jk pj∈

−∞,1_{α α−}

1

α

α−1

, 5.6

then5.1is nonoscillatory.

Acknowledgments

This work was supported by the grants 201/10/1032 and 201/08/0469 of the Czech Grant Agency and by the Institutional Research Plan No. AV0Z10190503.

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