Oscillation of Solutions of a Linear Second Order Discrete Delayed Equation

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doi:10.1155/2010/693867

Research Article

Oscillation of Solutions of a Linear Second-Order

Discrete-Delayed Equation

J. Ba ˇstinec,

1

_{J. Dibl´ık,}

1, 2

_{and Z. ˇSmarda}

1

1_{Department of Mathematics, Faculty of Electrical Engineering and Communication,}

Brno University of Technology, 61600 Brno, Czech Republic 2_{Brno University of Technology, Brno, Czech Republic}

Correspondence should be addressed to J. Dibl´ık,[email protected]

Received 5 January 2010; Accepted 31 March 2010

Academic Editor: Leonid Berezansky

Copyrightq2010 J. Baˇstinec et al. 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.

A linear second-order discrete-delayed equationΔxn −pnxn−1with a positive coeﬃcient pis considered forn → ∞. This equation is known to have a positive solution ifpfulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality forp, all solutions of the equation considered are oscillating forn → ∞.

1. Introduction

The existence of a positive solution of diﬀerence equations is often encountered when analysing mathematical models describing various processes. This is a motivation for an intensive study of the conditions for the existence of positive solutions of discrete or continuous equations. Such analysis is related to an investigation of the case of all solutions being oscillatingfor relevant investigation in both directions, we refer, e.g., to1–15 and to the references therein. In this paper, sharp conditions are derived for all the solutions being oscillating for a class of linear second-order delayed-discrete equations.

We consider the delayed second-order linear discrete equation

Δxn −pnxn−1, 1.1

where n ∈ Z∞_a : {a, a1, . . .},a ∈ Nis fixed,Δxn xn1−xn, andp : Z∞_a →

R _: _0,_∞_{. A solution}_x_x_n_:_Z∞

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Definition 1.1. Let us define the expression lnqt,q ≥ 1, by lnqt lnlnq−1t, ln0t ≡ t where

t >exp_q−₂1 and exp_stexpexp_s−₁t,s≥1, exp₀t≡tand exp₋₁t≡0 instead of ln0t, ln1t, we

will only writetand lnt.

In 2 a delayed linear diﬀerence equation of higher order is considered and the following result related to1.1on the existence of a positive solution is proved.

Theorem 1.2. Leta∈Nbe suﬃciently large andq∈N. If the functionp:Z∞_a → Rsatisfies

pn≤ 1 4

1 16n2

1 16nlnn2

1 16nlnnln22

1

16nlnnln2nln3n2

· · · 1

16nlnnln2n· · ·lnqn2

1.2

for everyn∈Z∞_a, then there exist a positive integera1≥aand a solutionxxn,n∈Z∞a1of 1.1

such thatxn>0holds for everyn∈Z∞_a₁.

Our goal is to answer the open question whether all solutions of1.1are oscillating if inequality1.2is replaced by the opposite inequality

pn≥ 1 4

1 16n2

1 16nlnn2

1 16nlnnln2n2

1

16nlnnln2nln3n2

· · · 1

16nlnnln2n· · ·lnq−1n

2

κ

16nlnnln2n· · ·lnqn2

1.3

assumingκ >1 andnis suﬃciently large. Below we prove that if1.3holds andκ >1, then all solutions of1.1are oscillatory. The proof of our main result will use a consequence of one of Domshlak’s results8, Corollary 4.2, page 69 .

Lemma 1.3. Letqand r be fixed natural numbers such thatr−q >1. Let{ϕn}∞₁ be a given sequence of positive numbers andν0a positive number such that there exists a numberν∈0, ν0satisfying

r

q1

ϕn≤ π_ν, π_ν ≤r1 q1

ϕn≤ 2π_ν . 1.4

Then, ifpq1≥0and forn∈Zr_q₂

pn≥ sinνϕn−1·sinνϕn1

sinνϕn−1 ϕn·sinνϕn ϕn1 1.5

holds, then any solution of the equation

xn1−xn pnxn−1 0 1.6

has at least one change of sign onZr1

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Moreover, we will use an auxiliary result giving the asymptotic decomposition of the iterative logarithm 7 . The symbols “o” and “O” used below stand for the Landau order symbols.

Lemma 1.4. For fixedr, σ ∈R\ {0}and fixed integers≥1, the asymptotic representation

lnσ_sn−r lnσ_sn 1−

rσ nlnn· · ·lnsn−

r2_σ

2n2_ln_n_{· · ·}_ln_sn

− r2σ

2nlnn2ln2n· · ·lnsn

− · · · − r2σ

2nlnn· · ·lns−1n2lnsn

r2σσ−1 2nlnn· · ·ln_sn2 −

r3_σ₁_o₁

3n3_ln_n_{· · ·}_ln_sn

1.7

holds forn → ∞.

2. Main Result

In this part, we give suﬃcient conditions for all solutions of1.1to be oscillatory asn → ∞.

Theorem 2.1. Let a ∈ N be suﬃciently large, q ∈ N, and κ > 1. Assuming that the function p : Z∞_a → R satisfies inequality1.3for everyn ∈ Z∞_a, all solutions of 1.1are oscillating as n → ∞.

Proof. We set

ϕn: 1

nlnnln2nln3n· · ·lnqn 2.1

and consider the asymptotic decomposition ofϕn−1whennis suﬃciently large. Applying Lemma1.4forσ−1,r1, ands1,2, . . . , q, we get

ϕn−1 1

n−1lnn−1ln2n−1ln3n−1· · ·lnqn−1

_n 1

1−1/nlnn−1ln2n−1ln3n−1· · ·lnqn−1

ϕn· 1 1−1/n·

lnn lnn−1·

ln2n

ln2n−1·

ln3n

ln3n−1· · ·

lnqn lnqn−1

ϕn

1 1

n

1 n2 O

1 n3

× 1 1

nlnn 1 2n2_lnn

1

nlnn2 O

1 n3

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Using the previous decompositions, we have

ϕn−1ϕn1 ϕ2_n ₁ 1

n2

1 n2_ln_n

1 n2_ln_nln

2n

· · · 1

n2_ln_nln

2n· · ·lnqn

1

nlnn2 1 nlnn2ln2n

· · · 1

nlnn2ln2n· · ·lnqn

1

nlnnln2n2

1

nlnnln2n2ln3n

· · · 1

nlnnln2n2ln3n· · ·lnqn

· · · 1 nlnn· · ·ln_q−1n

2

1

nlnn· · ·ln_q−1n

2

ln_q

1

nlnnln2n· · ·lnqn2 O

1 n3

.

2.5

Recalling the asymptotical decomposition of sinxwhenx → 0: sinx xOx3_{, we get}

since limn→ ∞ϕn limn→ ∞ϕn−1 limn→ ∞ϕn1 0

sinνϕn−1 νϕn−1 Oν3ϕ3n−1,

sinνϕn1 νϕn1 Oν3ϕ3n1,

sinνϕn−1 ϕnνϕn−1 ϕnOν3ϕn−1 ϕn3,

sinνϕn ϕn1νϕn ϕn1Oν3ϕn ϕn13

2.6

asn → ∞. Due to2.3and2.4we haveϕn1 Oϕnandϕn−1 Oϕnas n → ∞. Then it is easy to see that, for the right-hand side of the inequality1.5, we have

R: sinνϕn−1·sinνϕn1

sinνϕn−1 ϕn·sinνϕn ϕn1 R1·

1Oν2ϕ2n, n−→ ∞,

2.7

where

R1: ϕn−

1ϕn1

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Moreover, forR1, we will get an asymptotical decomposition asn → ∞. We representR1in

the form

R1

ϕn−1ϕn1/ϕ2_n

1ϕn−1/ϕnϕn1/ϕn ϕn−1ϕn1/ϕ2n. 2.9

As the asymptotical decompositions for

ϕn−1ϕn1 ϕ2_n ,

ϕn−1 ϕn ,

ϕn1

ϕn 2.10

have been derived abovesee2.3–2.5, after some computation, we obtain

R1 1_n1₂ _n₂1

lnn 1

n2_lnnln₂n· · ·

1

n2_lnnln₂n· · ·_ln_qn

1

nlnn2 1 nlnn2ln2n

· · · 1

nlnn2ln2n· · ·lnqn

1

nlnnln2n2

1

nlnnln2n2ln3n

· · · 1

nlnnln2n2ln3n· · ·lnqn

· · · 1 nlnn· · ·lnq−1n

2

1

nlnn· · ·lnq−1n

2

lnq

1

nlnnln2n· · ·lnqn2 O

1 n3

×

1 1 1

n

1 nlnn

1 nlnnln2n

1

nlnnln2nln3n· · ·

1

nlnnln2n· · ·lnqn

1 n2

3 2n2_ln_n

3

2n2_ln_nln₂_n· · ·

3

2n2_ln_nln₂_n_{· · ·}_ln_qn

1

nlnn2

3 2nlnn2ln2n

3

2nlnn2ln3n

· · · 3

2nlnn2ln3n· · ·lnqn

1

nlnnln2n2

3

2nlnnln2n2ln3n

· · · 3

2nlnnln2n2ln3n· · ·lnqn

1

nlnnln2nln3n2

3

2nlnnln2nln3n2ln4n

· · · 3

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Finalizing our decompositions, we see that

It is easy to see that inequality1.5becomes

pn≥ 1

ν ∈0, ν0 withν0 fixed becauseκ >1. Consequently,2.14is satisfied and the assumption

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be so large that inequalities1.4hold. This is always possible since the series∞_n_q₁ϕnis divergent. Then Lemma1.3holds and any solution of1.1has at least one change of sign on

Zr1

q−1. Obviously, inequalities1.4can be satisfied for another couple ofp, r, sayp1, r1with

p1 > randr1 > q11 suﬃciently large, and by Lemma1.3any solution of1.1has at least

one change of sign onZr11

q1−1. Continuing this process, we get a sequence of intervalspn, rn

with limn→ ∞pn ∞such that any solution of1.1has at least one change of sign onZr_qn_n₋11.

This fact concludes the proof.

Acknowledgments

The first author was supported by Grants 201/07/0145 and 201/10/1032 of the Czech Grant AgencyPragueand by the Council of Czech Government MSM 0021630529. The second author was supported by Grants 201/07/0145 and 201/10/1032 of the Czech Grant Agency Pragueand by the Council of Czech Government MSM 00216 30519. The third author was supported by the Council of Czech Government MSM 00216 30503 and MSM 00216 30529.

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