Oscillation of Third-order Nonlinear Delay Difference Equation

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ISSN: 2347-1557

Available Online: http://ijmaa.in/

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International Journal of Mathematics And its Applications

Oscillation of Third-order Nonlinear Delay Difference Equation

Yadaiah Arupula^1,∗ and V. Dharmaiah¹

1 Department of Mathematics, Osmania University, Hyderabad, Telangana, India.

Abstract: Third-order nonlinear difference equation of the form ∆ (cn∆ (dn∆xn)) + pn∆xn+1+ qnf (xn−σ) = 0, n ≥ n0 are considered. Here, {cn}, {d_n}, {p_n}, and {a_n} are sequence of positive real number for n₀∈ N, f is a continuous function such that f (u) /u ≥ k > 0 for u 6= 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.

Keywords: Difference equation, Delay, Third order, Oscillation. Nonoscillation, Riccati transformation.

c

JS Publication. Accepted on: 09.04.2018

1. Introduction

Consider the nonlinear delay difference equation

∆ (cn∆ (dn∆xn)) + pn∆xn+1+ qnf (xn−σ) = 0, n ≥ n0, (1)

where n0∈ N is fixed integer, ∆ denotes the forward difference operator ∆xn= xn+1− xn, and σ is a nonnegative integer.

The real sequence {cn}^∞_n=n

0, {dn}^∞_n=n

0, {pn}^∞_n=n

0, {qn}^∞_n=n

0, and the function f satisfy the following conditions:

(h1) {dn}^∞_n=n0is positive, lim

n→∞R1(n, s) = ∞, where R1(n, s) =

n

P

k=s 1

d_k for n > s ≥ n0; (h2) {cn}^∞_n=n₀ is positive, lim

n→∞R2(n, s) = ∞, where R2(n, s) =

n

P

k=s 1

c_k for n > s ≥ n0; (h3) pn≥ 0, qn≥ 0 and qn6= 0 for infinitely many values of n ∈ N (n0);

(h4) f ∈ C (R, R) , f (u) /u ≥ K for some k > 0 and for all u 6= 0.

By a solution of Equation (1) we mean a nontrivial real sequence {xn} that is defined for n ≥ n0− σ and satisfies Equation (1) for all n ≥ n0. clearly if xn= Anfor n = n0− σ, n0− σ + 1, . . . , n0− 1 are given, then Equation (1) has a unique solution satisfying the above initial conditions. A solution {xn} of Equation (1) is said to be oscillatory if is neither eventually positive nor eventually negative, and nonoscillatory otherwise. Equation (1) is called nonoscillatory if all its solutions are nonoscillatory.

The oscillation problem for difference equations has been investigated in recent years; for first-order, second-order, and higher-order equations, respectively; see [15,20,8,9,11,19,21]. For general theory of oscillation of difference equations, we refer to [1-3,14], and over 500 refer encescited therein. Compared to the second-order difference equations, the study of

∗ E-mail: [email protected]

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third-order difference equations has received considerably less attention in the literature even though such equations arises in economics, mathematical biology, and other areas of Mathematics (5). Some recent results on third-order difference equations can be found [10,24-31]. However, it seems that there is much less known regarding the oscillation of Equation (1).

There are many papers dealing with the oscillatory and asymptotic behaviour of solution of difference and differential equations, see for instance Jiang [12], Jiang and Li [13], Li [17], Li and Yeh [16], Li [17], Luo [18], Philos [22], saker [26, 27].

The oscillatory behaviour of solution of difference equations and that of their discrete analogs may be quite different. For instance, the differential equation

y^m+ 8y = 0

admits a nonoscillatory solution y1(t) = e^−2t and a pair of oscillatory solutions y2(t) = e^tcos√

3t and y3(t) = e^tsin√ 3t, but the difference equation

∆³xn+ 8xn= 0

Which is a discrete analog of the above difference equation, has there oscillatory solutions x¹n = (−1)ⁿ, x²n =

√7n

cosh n

arctanp3/2i

, and x³_n = √ 7n

sinh n

arctanp3/2i

. We note that Equation (1) may be considered as a discrete analog of the delay differential equation

c (t) d (t) x⁰⁰⁰

+ p (t) x⁰+ q (t) f (x (t − σ)) = 0 (2)

For some work regarding the oscillation of Equation (2), we refer to Saker [27] (p (t) ≡ 0) and Tiryaki and Aktas [32] and the references cited therein. A number of dynamical behaviours of solutions of difference equations are possible; here we will only be concerned with conditions which are sufficient for every solution of Equation (1) to be either oscillatory or convergent to zero as n → ∞. Recently, Saker [26] has established some new conditions which are sufficient for all solution of

∆ (cn∆ (dn∆xn)^γ) + qnf (xn−σ) = 0, n ≥ n0, (3)

Where γ ≥ 1 is quotient of odd positive integers, to be either oscillatory or tend to zero as n → ∞. Our aim in this paper is to present some new oscillation criteria for Equation (1) by making use of a Riccati type transformation and arguments developed for differential equations in [32]. It should be noted that the results obtained in this paper extend and improve the related ones in [26]. The paper is organized as follows. In section 2, we will present some lemmas which are useful in establishing our main results. In Section 3 we will state and prove the main results and give examples to illustrate them.

2. Preparatory Lemmas

We begin with the following useful lemma.

Lemma 2.1. Suppose (h5) ∆ (cn∆zn) +_d^pⁿ

n+1zn+1= 0 is nonoscillatory.

If {xn} is a nonoscillatory solution of Equation (1) for n ≥ n0, then there exists a n1≥ n0 such that either xn(dn∆xn) > 0 or xn(dn∆xn) < 0 for all n ≥ n1.

Proof. Suppose that {xn} is a nonoscillatory solution of Equation (1) for n ≥ n0. Without loss of generality, we may take xn> 0 and xn−σ> 0, n ≥ n1 ≥ n0. We see that yn= −dn∆xnis a solution of the order non homogeneous difference

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equation

∆ (cn∆yn) + pn

dn+1

yn+1= qnf (xn−σ) , n ≥ n1 (4)

Indeed, since xnis a oscillatory solution of Equation (1) we have

∆ (cn∆ (−dn∆xn)) + pn

dn+1

(−dn+1∆xn+1) = qnf (xn−σ)

and hence

∆ (cn∆ (dn∆xn)) + pn∆xn+1+ qnf (xn−σ) = 0

We claim that solution of (4) are nonoscillatory. We may assume that zn> 0 for n ≥ n1. Note that {−zn} is also a solution.

Let yn be a oscillatory solution of (4). There exist n3 > n2 > n1 such that yn3 ≥ 0, yn3+1≤ 0, yn2 ≤ 0 and yn2+1≥ 0.

Summing

∆ (cn(yn+1zn− ynzn+1)) = zn+1qnf (xn−σ)

From n2 to n3− 1, we have

cn3(yn3+1zn3− zn3+1yn3) − cn2(yn2+1zn2− zn2+1yn2) =

n3−1

X

k=n2

zk+1qkf (xk−σ) ,

a contradiction. The proof is complete.

Definition 2.2. Let {xn} be a solution of Equation (1). we say that the solution {xn} has property v2 if there exists n∗≥ n0

such that

xn∆xn> 0, xn∆ (dn∆xn) > 0, xn∆ (cn∆ (dn∆xn)) ≤ 0

For every n ≥ n∗.

Lemma 2.3. Let the assumption (h2) hold and {xn} be a nonoscillatory solution of Equation (1) such that xn(dn∆xn) ≥ 0 for every n ≥ n1≥ n0. Then {xn} has property V2.

Proof. Let {xn} be a eventually positive solution of Equation (1) Then there exists an n1 ≥ n0 such that xn−σ > 0 for n ≥ n1. Since xn(dn∆xn) > 0 for every n ≥ n1 ≥ n0, we have ∆xn > 0 for n ≥ n1. From Equation (1) we have

∆ (cn∆ (dn∆xn)) ≤ 0 for n ≥ n1. Then ∆ (dn∆xn) is monotone and eventually of one sign. We claim that there is a n2 ≥ n1 such that for n ≥ n2, ∆ (dn∆xn) > 0. Suppose to the contrary that ∆ (dn∆xn) ≤ 0 for n ≥ n2. Since cn > 0 and cn∆ (dn∆xn) > 0 is nonincreasing there exists a negative constant C and an n3 ≥ n2 such that cn∆ (dn∆xn) ≤ C for n ≥ n3. Dividing both sides by cnand summing from n3 to n − 1 , we obtain

dn∆xn≤ dn3∆xn3+ C

n−1

X

k=n3

1 ck

.

Letting n → ∞, we see that dn∆xn → −∞ by a contradiction with the fact that ∆xn > 0. Then ∆ (dn∆xn) > 0. The proof is complete.

Lemma 2.4. Let {xn} be a solution of Equation (1) and {g_n^∗}^∞_n=n

0 be a sequence of integers which satisfies (h6) g^∗_n≤ n − σ − 1 and lim

n→∞g^∗_n= ∞.

If {xn} be a property V2, then there exists an n1≥ n0 such that dn−σ∆xn−σ≥ R2(n − σ − 1, g^∗n) cn∆ (dn∆xn) for n ≥ n1.

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Proof. Let {xn} be a solution of Equation (1) which has property v2. Without loss of generality, we may also assume that xn> 0 and xn−σ> 0, n ≥ n1 ≥ n0. Since lim

n→∞g^∗n= ∞ and ∆xn> 0, ∆ (dn∆xn) > 0, and ∆ (cn∆ (dn∆xn)) ≤ 0 for every n ≥ n2≥ n1,

dn−σ∆xn−σ= dg_n^∗∆xg^∗_n+

n−σ−1

X

k=g^∗_n

ck∆ (dk∆xk) ck

≥ R2(n − σ − 1, g^∗_n) cn∆ (dn∆xn)

and then we have

dn−σ∆xn−σ≥ R2(n − σ − 1, gn^∗) cn∆ (dn∆xn) .

The proof is complete.

Lemma 2.5. Let µn be a positive sequence defined for n ≥ n0 and set

φn= dn+2∆ (cn+1∆µn) + µnpn.

Furthermore assume that the following conditions are satisfied:

(h7) ∆µn≥ 0, φn≥ 0, ∆ (dn+2∆ (cn+1∆µn)) ≥ 0 (or∆ (µnpn) ≤ 0) for n ≥ n0; (h8)

∞

P

n=n0

(Kµnqn− ∆φn) = ∞, where Kµnqn− ∆φn≥ 0for n ≥ n0.

If (h1) holds and {xn} is a nonoscillatory solution of Equation (1) which satisfies xn(cn∆xn) ≤ 0 for n sufficiently large, then lim

n→∞xn= 0.

Proof. Let {xn} be a nonoscillatory solution of Equation (1). Without loss of generality, we may assume that xn > 0 and xn−σ> 0 for n ≥ n1≥ n0 for some n1 sufficiently large. The proof when {xn} is eventually negative is similar, as the substitution yn= −xn transforms Equation (1) into an equation of the same form. Since xn(cn∆xn) ≤ 0 for n sufficiently large, ∆xnbecomes nonpositive for all n ≥ n2 for some n2≥ n1. Let lim

n→∞xn= λ ≥ 0. Assume that λ 6= 0. There exists an n3≥ n2 such that xn≥ λ for n ≥ n3. Summing Equation (1) from n3 to n − 1, we obtain from that

µncn∆ (dn∆xn) ≤ C1− λ

n−1

X

k=n3

(Kµkqk− ∆φk) ,

where C1is a constant. Employing we see from (2) that µncn∆ (dn∆xn) must take on negative values for n sufficiently large.

By using (h1) we see that xnmust be eventually negative, a contradiction. Hence λ = 0. This complete the proof.

3. Oscillation Criteria

In this section we gave the main of our paper.

Theorem 3.1. Assume that (h1) − (h8) hold, and that there exists a positive sequence {ρn}^∞_n=n₀ such that s

lim sup

n→∞

n

X

k=n₀

Kρkqk−dk−σ(∆ρkdk+1− ρkpkR2(k − σ − 1, g_k^∗))² 4ρkR2(k − σ − 1, g^∗_k) d²_k+1

= ∞. (5)

Then every solution {xn} of Equation (1) is either oscillatory or satisfies xn→ 0 as n → ∞.

Proof. Let {xn} be a nonoscillatory solution of Equation (1) Without loss of generality, we may assume that xn> 0 and xn−σ> 0 eventually. From Lemma 2.1 it following that ∆xn> 0 or ∆xn< 0 for n ≥ n1≥ n0. If ∆xn> 0 for n ≥ N ≥ n1

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then {xn} has property V2 by Lemma 2.2. We define wn= ρncn∆(dn∆xn)

xn−σ , n ≥ N . Then, wn> 0 and in view of Equation (1) by employing Lemma 2.3 we have

∆wn= −f (xn−σ) xn−σ

ρnqn− ρn

xn−σ

pn∆xn+1+cn+1∆ (dn+1∆xn+1) xn−σ∆ρn

xn−σxn−σ+1

−cn+1∆ (dn+1∆xn+1) ρn∆xn−σ

xn−σxn−σ+1

≤ −Kρnqn−

w²n+1

ρnR2(n − σ − 1, g^∗n) (ρn+1)²dn−σ

− wn+1

∆ρn

ρn+1

−pnρnR2(n − σ − 1, gn^∗) dn+1ρn+1

= −Kρnqn− Anw²n+1+ wn+1Bn, (6)

Where

An= ρnR2(n − q − 1, gn^∗) ρ²_n+1dn−σ

, Bn= ∆ρn

ρn+1

−pnqnR2(n − q − 1, g^∗n) dn+1ρn+1

.

Completing the square in (3.2) we obtain

∆wn< −

Kρnqn− B_n² 4An

. (7)

Summing (7) from N to n, we obtain

−wN< wn+1− wN< −

n

X

k=N

Kρkqk− B²_k 4Ak

which yields

n

X

k=N

Kρkqk− B_k² 4An

< wN

For all large n and this is contrary to (5). If ∆xn < 0 for n ≥ N , then by Lemma 2.4 we have lim

n→∞xn= 0. The proof is complete.

Example 3.2. Consider the third order delay difference equation

∆³xn+ 1

5n²∆xn+1+

1 − 1

5n²

xn−3= 0, n ≥ 1 (8)

Note that ∆²zn+_5n¹₂zn+1= 0 is nonoscillatory by [1]. Taking µn= ρn= 1 and g^∗_n= n − 4, we have

lim sup

n→∞

n

X

k=n0

Kρkqk−dk−σ(∆ρkdk+1− ρkpkR2(k − σ − 1, g_k^∗))² 4ρkR2(k − σ − 1, g^∗_k) d²_k+1

=

∞

X

k=1

1 − 1

5k² − 1 100k⁴

= ∞.

Thus, condition (5) is satisfied. The other conditions of Theorem 3.1 are also satisfied. Hence every solution {xn} of Equation (8) is either oscillatory or satisfies xn → 0 as n → ∞. We note that cos^nπ₃

is an oscillatory solution of Equation (8).

Example 3.3. consider the their order difference equation

∆³xn+ 1

2ⁿ⁺¹∆xn+1+1 8

1 + 1

2ⁿ

xn= 0, n ≥ 1. (9)

Note that ∆³zn+ ₂_n+1¹ zn+1 = 0 is nonoscillatory [1]. Taking µn = ρn = 1 and gn^∗ = n − 1, condition (5) is satisfied.

The other conditions of Theorem 3.1 are also satisfied. Hence, every solution {xn} of Equation (9) is either oscillatory or satisfies xn→ 0 as n → ∞. Indeed, the sequence2⁻ⁿ is such a solution of Equation (9).

Theorem 3.4. Assume that (h1) − (h8) hold. Let {ρn}^∞_n=n

0 be a positive sequence and {Hm,n}, m ≥ n ≥ n0, be a double sequence such that

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(1). Hm,m= 0 for m ≥ n0;

(2). Hm,n> 0 for m > n ≥ n0;

(3). ∆2Hm,n= Hm,n+1− Hm,n≤ 0 and −∆2Hm,n= hm,npHm,n for m ≥ n ≥ n0.

If

lim sup

m→∞

1 Hm,m0

m−1

X

n=m₀

KHm,nρnqn−Q²_m,n 4An

= ∞ f or every m0≥ n0, (10)

where

Qm,n= hm,n− ∆ρn

ρn+1

−pnρnR2(n − σ − 1, g_n^∗) dn+1ρn+1

pHm,n, An= ρnR2(n − σ − 1, g_n^∗) ρ²_n+1dn−σ

,

Then every solution {xn} of Equation (1) is either oscillatory or lim

n→∞xn= 0.

Proof. Let {xn} be a nonoscillatory solution of Equation (1) which may assume to be eventually positive. Proceeding as in the proof of Theorem 3.1 we arrive at the inequality 3.2 Then we see that

m−1

X

n=N

KHm,nρnqn≤

m−1

X

n=N

Hm,n −∆wn+ wn+1Bn− Anw²_n+1

= Hm,NwN+

m−1

X

n=N

wn+1∆2Hm,n Bnwn+1− Anw²n+1

= Hm,NwN−

m−1

X

n=N

n

wn+1² AnHm,n+ wn+1

hm,n

pHm,n− Hm,nBn

o

≤ Hm,NwN+

m−1

X

n=N

(hm,n− BnpHm,n)² 4An

, (11)

where Bnis as defined in the proof of Theorem 3.1. Thus we obtain

1 Hm,N

m−1

X

n=N

≤ wN,

Where clearly contradicts (10). If ∆xn< 0 for n ≥ N , then by Lemma 2.4, we have lim

As an immediate consequence of Theorem 3.2 we get the following corollary.

Corollary 3.5. Assume that all the assumption of Theorem 3.2 holds, except that the condition (10) is replaced by

(1). lim sup

m→∞

1 H_m,m0

m−1

P

n=m₀

Hm,nρnqn= ∞ for every m0≥ n0,

(2). lim sup

m→∞

1 H_m,m0

m−1

P

n=m₀ Q²_m,n

A_n < 0 for every m0≥ n0.

Then, every solution {xn} of Equation (1) is either oscillatory or lim

n→∞xn= 0.

∆³xn+ 9

2ⁿ⁺¹∆xn+1+27 32

1 − 1

2ⁿ

xn−2= 0 (12)

Taking µn = ρn = 1, gn^∗ = n − 3 and Hm,n = m − n condition (10) is satisfied. The other condition of Theorem 3.2 are also satisfied. Hence every solution {xn} of Equation (12) is either oscillatory or satisfies xn→ 0 as n → ∞. The sequence {(−1/2)ⁿ} is such a solution of Equation (12).

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∆³xn+27

32xn−2= 0, n ≥ 1 (13)

Taking µn= ρn= 1, g_n^∗= n − 3 and Hm,n= m − n, we have

lim

m→∞sup 1 Hm,n₀

m−1

X

n=n₀

KHm,nρnqn−Q²m,n

4An

= lim sup

m→∞

1 m − 1

m−1

X

n=1

27

32(m − n) − 1 4 (m − n)

= ∞.

Thus, condition (10) is satisfied. The other condition of theorem are also satisfied. Hence every solution {xn} of Equation is either oscillatory or satisfies x → 0 as n → ∞. The sequence2⁻ⁿ is a solution of Equation (13).

Remark 3.8. One may choose {Hm,n} in appropriate manners, to derive several special oscillation criteria for Equation (1) Some choices are

Hm,n= (m − n)^λ, λ ≥ 1, m ≥ n ≥ n0,

Hm,n=

logm + 1 n + 1

λ

, λ ≥ 1, m ≥ n ≥ n0,

Hm,n= (m − n)^(λ), λ > 2, m ≥ n ≥ n0,

Where (m − n)^(λ)= (m − n) (m − n + 1) . . . (m − n + λ − 1).

Theorem 3.9. Let {Hm,n} and {hm,n} be as in Theorem 3.2 and let

0 < inf

n≥n₀

lim inf

m→∞

Hm,n

Hm,n₀

≤ ∞ (14)

lim sup

m→∞

1 Hm,n₀

m−1

X

n=n

Q²m,n

An

< ∞ (15)

If there is a sequence {ΨN} such that

lim sup

m→∞

1 Hm,N

m−1

X

n=N

≥ ψN f or every N ≥ n0 (16)

and

∞

X

n=n₀

Anψ_n+1⁺ 2

= ∞, where ψ⁺_n+1= max {ψn+1, 0} (17)

n→∞xn= 0.

Proof. As in the proof of Theorem 3.2, we have (11), Then, we have

m−1

X

n=N

KHm,nρnqn≤ Hm,NwN+

m−1

X

n=N

Q²m,n

4An

−

m−1

X

n=N

wn+1

pAnHm,n+ Qm,n

2√ An

2

(18)

The remainder of the proof of this case is similar to ones given in [16,18] and hence is omitted.

Example 3.10. Consider the third order difference equation

∆² 1 n³∆xn

+ 2n + 1

[n (n + 1)]² xn+ x³n = 0, n ≥ 1. (19)

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We take µn= n², ρn= 1, g^∗n= n − 1, Hm,n= m − n and ψn=_n¹2. In view of Qm,n=^√_m−n¹ and An= n³, we see that

lim sup

m→∞

1 Hm,n₀

m−1

X

n=n₀

Q²_m,n An

= lim sup

m→∞

1 m − 1

m−1

X

n=1

1

(m − n) n³ = 0 < ∞,

∞

X

n=n₀

Anψ⁺_n+12

=

∞

X

n=1

n³

(n + 1)⁴ = ∞ and

lim sup

m→∞

1 Hm,n

m−1

X

n=N

KHm,nρnqn−Q²m,n

4An

= lim sup

m→∞

1 m − N

m−1

X

n=N

(m − n) 1

n² − 1

(n + 1)² − 1 4 (m − n) n³

= 1 N² = ψN.

Since the conditions of Theorem 3.3 hold, every solution {xn} of Equation (19) is either oscillatory or satisfies xn→ 0 as n → ∞.

Theorem 3.11. Let {Hm,n} and {hm,n} be as in Theorem 3.2 and let (14) hold. Suppose that

lim inf

m→∞

1 Hm,n₀

m−1

X

n=n₀

KHm,nρnqn< ∞, (20)

and there is a sequence {ψN} satisfying (17) and

lim inf

m→∞

1 Hm,N

m−1

X

n=N

≥ ψN f or every N ≥ n0 (21)

Then every solution {xn} of Equation (1) either oscillatory or lim

n→∞xn= 0.

Proof. The proof of Theorem 3.4 is similar to that of Theorem 3.3 and hence is omitted.

Theorem 3.12. Assume that (h1)-(h8) hold. Suppose there exists a positive sequence {ρn}^∞_n=n

0 and a sequence {Fm,n}^∞_m,n=n

0 such that 1 +^F_ρ^m,n

n+1 +_d^pⁿ^ρⁿ^R²

n+1ρ_n+1 −_ρ^∆ρⁿ

n+1 ≥ 0 and

lim sup

m→∞

m

X

n=n₀





n−1

Y

k−n₀

1 +Fm,k

ρk+1

− Bk



 Kρnqn− 1 4An

Fm,n

ρn+1

2!

= ∞, (22)

n→∞xn= 0.

Proof. Let {xn} be a nonoscillatory solution of Equation (1) Without loss of generality, we may assume that xn> 0 and xn−σ> 0 for some N ≥ n0. Proceeding as in the proof of Theorem 3.1 we arrive at

∆wn≤ −Kρnqn+ wn+1Bn− w²_n+1

ρnR2

ρ²_n+1dn−σ

, n ≥ N (23)

From (23) and Young’s inequality, we have

∆wn≤ −Kρnqn+ wn+1Bn− w_n+1² ρnR2

ρ²_n+1dn−σ

− 1

4An

Fm,n

ρn+1

2

+ 1

4AN

Fm,n

ρn+1

2

, n ≥ N

or

wn+1− wn≤ −Kρnqn+ wn+1

Bn−Fm,n

ρn+1

+ 1

4An

Fm,n

ρn+1

2

, n ≥ N It follows that

m

X

n=N

"_n−1 Y

k=N

1 +Fm,k

ρk+1

− Bk

#

Kρnqn− 1 4An

Fm,n

ρn+1

2!

≤ wN.

Hence

lim sup

m→∞

m

X

n=N

"_n−1 Y

k=N

1 +Fm,k

ρk+1

− Bk

#

Kρnqn− 1 4An

Fm,n

ρn+1

2!

≤ wN,

Which contradict (22). If ∆xn< 0 for n ≥ N , then by Lemma 2.4 we have lim

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∆² 1 n⁵∆xn

+ 1

4n³xn(β + e^xⁿ) = 0, n ≥ 1, (24)

where β > 1 and f (u) = u (β + e^u) with K = β. Taking µn= n², ρn= n, g^∗n= n − 1, and Fm,n= n², we have

lim sup

m→∞

m

X

n=1

"_n−1 Y

k=1

1 +Fm,k

ρk+1

− Bk

#

Kρnqn− 1 4An

Fm,n

ρn+1

2!

= β − 1 4 lim sup

m→∞

∞

X

n=1

(n − 1)!

n² = ∞

Thus, condition (22) is satisfied. The other conditions of Theorem 3.6 are also satisfied. Hence every solution {xn} of Equation (24) is either oscillatory or satisfies xn→ 0 as n → ∞.

In the proof of following theorem we use a generalized Riccati transformation technique.

Theorem 3.14. Assume that (h1)-(h8) holds. Let {ρn}^∞_n=n

0 be a positive sequence. Furthermore, we assume that there exists a double sequence − {Hm,n: m ≥ n ≥ n0} such that (i)-(iii). If

lim sup

m→∞

1 Hm,m₀

m−1

X

n=m₀

Hm,nψ − h²_m,n 4An

= ∞ f or every m0≥ n0 (25)

where

ψn= ρn

Kqn−p²ndn−σR2

4d²_n+1 − ∆ (cn−σαn−1)

, αn= −(∆ρndn+1− pnρnR2) dn−σ

2dn+1ρnR2cn−σ+1

. Then every solution {xn} of Equation (1) is either oscillatory or lim

n→∞xn= 0.

Proof. We proceed as in Theorem 3.1, take xn−σ> 0 for all n ≥ N for some N sufficiently large. Define

wn= ρn

cn∆ (dn∆xn) xn−σ

+ cn−σαn−1

, n ≥ N.

Then follows the proof of Theorem 3.1, we obtain

∆wn≤ −Kρnqn+wn+1

ρn+1

∆ρn−pnρnR2

dn+1

− ρnR2

dn−σ

wn+1

ρn+1

− cn−q+1αn

2

+ ρn∆ (cn−σαn−1) +pnρnR2

dn+1

(cn−σ−1αn)

= −ψn− Anw²_n+1, n ≥ N

The remainder of proof is similar to that of the Theorem 3.2 and hence is omitted. If ∆xn< 0 for n ≥ N, then by Lemma 2.4, we have lim

∆³xn+ 2c¹⁻1_/ 2ⁿ √

c − 13

xn(4 − cos xn) = 0, c > 1, n ≥ 1. (26)

Taking µn= ρn= c1_/

2ⁿ, g_n^∗= n − 1 and Hm,n= m − n, we have

lim sup

m→∞

1 Hm,m₀

m−1

X

n=m₀

Hm,nψn−h²_m,n 4An

= lim sup

m→∞

1 m − m₀

m−1

X

n=m₀

6c √

c − 13

(m − n) − 1 4 (m − n)

= ∞.

Thus, condition (25) is satisfied. The other conditions of Theorem 3.6 are also satisfied. Hence every solution {xn} of Equation (26) is either oscillatory or lim

n→∞xn= 0.

Corollary 3.16. Assume that all the assumptions of Theorem 3.5 hold, except that the condition (25) is replaced (1). lim sup

m→∞

1 H_m,m0

m−1

P

n=m0

Hm,nρn

Kqn−^p²ⁿ^ρ_4dⁿ^d2ⁿ^−σR² n+1

− ∆ (cn− σαn−1)

= ∞ for every m0≥ n0,

(2). lim

m→∞

1 H_m,m0

m−1

P

n=m₀ h²_m,n

A_n < ∞ for every m0≥ n0.

Then, every solution {xn} of Equation (1) is either oscillatory or lim

n→∞xn= 0.

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Acknowledgment

The authors thank the referee his/her valuable suggestions.

References

[1] R.P. Agarwal, M. Bohner, S.R. Grace and D. O’Regen, Discrete Oscillation Theory, Hindawi Publishing corporation, New York, (2005).

[2] R.P. Agarwal, Difference Equations and Inqualities, Theory, Methods and Applications, Second Edition, Revised and Expanded, New York, Marcel Dekker, (2000).

[3] R.P. Agarwal and P.J.Y. Wong, Advaned Topics in Difference Equations, Kluwer Academic Publishers, Drodrecht, (1997).

[4] A. Andruch-Sobilo and M. Midga, On the oscillation of solutions of third order linear difference equations of neutral type, Math. Bohem., 130(2005), 19-33.

[5] M. Artzroumi, Generelized stable population theory, J. Math. Biology., 21(1985), 363-381.

[6] Z. Dosla and A. Kobza, On third-order difference euations, Comput. Math. Appl., 48(2004), 191-200.

[7] Z. Dosla and A. Kobza, On third-order linear difference equations involving quasi-differences, Adv. Difference Equ., 113(2006).

[8] S.R. Grace and B.S. Llli, Oscillation theores for forced neutral difference equations, J. Maths. Anal. Appl., 187(1994), 91-106.

[9] S.R. Grace, Oscillation theorems for second order delay and neutral difference euations, Utilities Math., 45(1994), 197-211.

[10] J. Graef and E. Thandapani, Oscillatory and asymptotic behavior of solutions of third order dealy difference equations, Funk. Ekvac., 42(1999), 355-369.

[11] J.W. Hooker and W.T. Patula, A second-order nalinear difference eation: Oscillation and asymptotic behaviour, J.

Math. Anal. Appl., 91(1983), 9-29.

[12] J. Jiang, Oscillation of second order nonlinear neutral differential equations, Appl. Math. Comput., 146(2003), 791-801.

[13] J. Jiang and X. Li, Oscillation of second order nonlinear neutral differential equations, Appl. Math. Comput., 135(2003), 531-540.

[14] Kelley, W.G., Peterson, A.C.: Difference Equations; An Introduction with Applications. New York. Acdemic Press 1991.

[15] G. Ladas, Explicit conditions for oscillations of difference equations, J. Math. Anal. Appl., 153(1990), 276-287.

[16] H.-J. Li and C.-C. Yeh, Oscillation criteria for second-order neutral delay difference equations, Advances in difference equations, II. Compt. Math. Appl., 36(1998), 123-132.

[17] H.-J. Li, Oscillation criteria for second order linear differential equations, J. Math. Anl. Appl., 194(1995), 217-234.

[18] J. Luo, Oscillation criteria for second-order quasi-linear neutral difference equations, Comput. Mat. Appl., 43(2002), 1549-1557.

[19] Ch.G. Philos, Iscillations in a class of difference equations, Appl. Math. Comput., 48(1992), 45-57.

[20] Ch.G. Philos, On oscillations of some difference equations. Funk, Ekvac. 34, 157-172 (1991).

[21] Ch.G. Philos and Y.G. Sficas, Positive solutions of some difference equations, Proc. Amer. Math. Soc., 108(1990), 107-115.

[22] Ch.G. Philos, Oscillation theorems for linear differential equations of second oreder, Arch. Math., 53(1989), 482-492.

(11)

[23] J. Popenda and E. Chmeidel, Nonoscillatory solutions of third order difference equations, Portugal. Math., 49(1992), 233-239.

[24] S.H. Saker, Oscillation and asymptotic behaviour of third-order nonlinear neutral delay difference equations, Dynam.

Systems Appl., 15(2006), 549-567.

[25] S.H. Saker, Oscillation of third-order difference equations, Port. Math., 61(2004), 249-257.

[26] S.H. Saker, Oscillation and asymptotic behaviour of third-order nonlinear neutral delay difference equations, J. Contemp.

Math. to appear.

[27] S.H. Saker, Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca., 56(2006), 433-450.

[28] B. Smith, Linear thire-order difference equations: Oscillatory and asymptotic behaviour, Rocky Mountain J. Math., 22(1992), 1559-1564.

[29] B. Smith and Jr.W.E. Taylor, Nonlinear third order difference equations: Oscillatory and asymptotic behaviour, Tamakang J. Math., 19(1988), 91-95.

[30] B. Simth, Oscillation and nonoscillation theorems for third order quasi adjoint difference equations, Port. Math., 45(1988), 229-243.

[31] B. Smith and Jr.W.E. Taylor, Asymptotic behaviour of solutions of a third order difference equation, Port. Math., 44(1987), 113-11.

[32] A. Tiryaki and M.F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Ansal. Appl., 325(2007), 54-68.