Oscillation of Second Order Nonlinear Neutral Advanced Functional Difference Equations

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Oscillation of Second Order Nonlinear Neutral Advanced

Functional Difference Equations

A. Murugesan Department of Mathematics, Government Arts College (Autonomous),

Salem - 636 007, Tamil Nadu,

India.

E-Mail: amurugesan3@gmail.com

K. Ammamuthu Department of Mathematics, Arignar Anna Government Arts College,

Attur, Salem - 636 121, Tamil Nadu,

India.

E-Mail: ammuthu75@gmail.com

Abstract

In this paper, we establish some sufficient conditions for the oscil-lations of all solutions of the second order nonlinear neutral advanced functional difference equations

∆[r(n)∆(x(n) +p(n)x(n+τ))] +q(n)f(x(n+σ)) = 0; n≥n0, (∗) where P∞

n=n0

1

r(n) =∞ or

P∞

n=n0

1

r(n) < ∞, and 0 ≤ p(n) ≤ p0 < ∞,

τ is an integer, and σ is a positive integer. The results proved here improve some known results in the literature.

Mathematics Subject Classification(2010): 39A10, 39A12.

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1 Introduction

In this paper we consider the following second order nonlinear neutral advanced functional difference equations of the form:

∆[r(n)∆(x(n) +p(n)x(n+τ))] +q(n)f(x(n+σ)) = 0; n≥n0, (1) where ∆ is the forward difference operator defined by ∆x(n) =x(n+1)−x(n). The following conditions are assumed to be hold throughout the paper:

(a) {r(n)}∞

n=n0 is a sequence of positive real numbers;

(b){p(n)}∞

n=n0 is a sequence of nonnegative real numbers with the property

that 0≤p(n)≤p0 <∞; (c) {q(n)}∞

n=n0 is a sequence of nonnegative real numbers andq(n) is not

identically zero for large values of n;

(d) f(_uu) ≥k >0 foru6= 0, k is a constant; and

(e)τ is an integer and σ is a positive integer.

We shall consider the following two cases,

∞

X

n=n0

1

r(n) =∞ (2)

and _∞

X

n=n0

1

r(n) <∞. (3)

We note that second order neutral functional difference equations have ap-plications in problems dealing with vibrating masses attached to a elastic ball and in some variational problems. In recent years there has been an increasing interest in obtaining sufficient conditions for the oscillation or nonoscillation of solutions for different classes of difference equations, we refer to the books [1,2,7] and the papers [6,13]. Also the oscillatory of behavior of neutral func-tional difference equations has been the subject of intensive study, see, for example, [3-5,8-12,14].

In [12], Zhang et al. established that every solution of the equation

(3)

is either oscillatory or eventually

|x(n)| ≤p|x(n−τ)| if

∞

X

s=n

q(s)

m

X

i=0

pi =∞, where

f(n, u)

u ≥q(n)>0 f or u6= 0.

In [3], Budincevic established that every solution of the equation

∆[a(n)∆(x(n) +px(n−n0))] +q(n)f(x(n−m0)) = 0 (5) is either oscillatory or else x(n)→0 as n → ∞.

In [7], Murugesan et al. established that every solution of the equation (1) is oscillatory if τ > 0, 0 ≤ p(n) ≤ p0 < ∞ and atleast one of the first order advanced difference inequalities

∆w(n)− 1

1 +p0Q1(n)w(n+σ)≥0; (6)

∆w(n)− 1

1 +p0Q2(n)w(n+σ)≥0 (7) has no positive solution.

In this paper our aim is to obtain sufficient conditions for oscillation of all solutions of the equation (1) under the condition

∞

X

n=n0

1

r(n) =∞ or

∞

X

n=n0

1

r(n) <∞.

Letn0 be a fixed nonnegative integer. By a solution of (1) we mean a nontrival real sequence {x(n)}which is defined forn ≥n0 and satisfies the equation (1) forn≥n0. A solution{x(n)}of (1) is said to be oscillatory if for every positive integer N > 0, there exists an n ≥ N such that x(n)x(n+ 1)≤ 0, otherwise {x(n)}is said to be nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

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2 Main Results

In this section, we establish some new oscillation criteria for (1). For the sake of convenience, we define the following notations

Q(n) : = min{q(n), q(n+τ)}, (∆ρ(n))₊ = max{0,∆ρ(n)},

R(n) =

n−1

X

s=n0

1 r(s), and

δ(n) =

∞

X

s=n

1 r(s).

Theorem 2.1 Assume that (2) holds and τ > 0. Moreover suppose that there exists a positive real valued sequence {ρ(n)}∞

n=n0 such that

lim sup

n→∞

n

X

s=n0 "

kρ(s)Q(s)−((1 +p0)r(s)(∆ρ(s))+) 2 4ρ(s)

#

=∞. (8)

Then every solution of (1) is oscillatory.

Proof. Assume the contrary. Without loss of generality we may assume that {x(n)} is an eventually positive solution of (1). Then there exists an integer n1 ≥n0 such that x(n)>0 for all n≥n1. Define

z(n) =x(n) +p(n)x(n+τ). (9) Then z(n)>0 for all n≥n1. From (1), we have

∆(r(n)∆z(n))≤ −kq(n)x(n+σ)≤0; n ≥n1. (10) Therefore the sequence {r(n)∆z(n)} is nonincreasing. We claim that

∆z(n)>0 f or n ≥n1. (11) If not, there exists an integer n2 ≥ n1 such that ∆z(n2) <0. Then from (10) we obtain

r(n)∆z(n)≤r(n2)∆z(n2), n≥n2. Here

z(n)≤z(n2) +r(n2)∆z(n2)

n−1

X

s=n2

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Lettingn→ ∞, we getz(n)→ −∞. This contradiction proves that ∆z(n)>0 for n ≥ n1. On the otherhand, using the definition of z(n) and applying (1), for all sufficiently large values of n,

∆[r(n)∆z(n)] +kq(n)x(n+σ) +p0∆[r(n+τ)∆z(n+τ)] +p0kq(n+τ)x(n+σ+τ)≤0

∆[r(n)∆z(n)] +p0∆[r(n+τ)∆z(n+τ)] +kQ(n)z(n+σ)≤0. Since ∆z(n)>0, we have z(n+σ)≥z(n) and hence

∆[r(n)∆z(n)] +p0∆[r(n+τ)∆z(n+τ)] +kQ(n)z(n)≤0. (12) Define

w(n) = ρ(n)r(n)∆z(n)

z(n) , n≥n1. (13)

Clearly w(n)>0. From (9), we have

r(n+ 1)∆z(n+ 1)≤r(n)∆z(n). From (13), we have

∆w(n) = ρ(n)∆(r(n)∆z(n)) z(n) −

ρ(n)r(n+ 1)∆z(n+ 1) z(n)z(n+ 1) ∆z(n)

+r(n+ 1)∆z(n+ 1) z(n+ 1) ∆ρ(n)

≤ ρ(n)∆(r(n)∆z(n)) z(n) −

ρ(n)w2(n+ 1) ρ2₍_n_{+ 1)}_r₍_n₎ +

w(n+ 1) ρ(n+ 1)∆ρ(n)

≤ρ(n)∆(r(n)∆z(n)) z(n) −

ρ(n)w2(n+ 1) ρ2₍_n_{+ 1)}_r₍_n₎ +

(∆ρ(n))+

ρ(n+ 1) w(n+ 1). (14) Similarly, we define

v(n) =ρ(n)r(n+τ)∆z(n+τ)

z(n) , n ≥n1. (15)

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∆v(n) ≤ ρ(n)∆(r(n+τ)∆z(n+τ))

z(n) −

ρ(n)v2(n+ 1) ρ2₍_n_{+ 1)}_r₍_n₎ +

∆ρ(n)

ρ(n+ 1)v(n+ 1)

≤ ρ(n)∆(r(n+τ)∆z(n+τ))

z(n) −

ρ(n)v2(n+ 1) ρ2₍_n_{+ 1)}_r₍_n₎ +

(∆ρ(n))+

ρ(n+ 1) v(n+ 1).(16) It follows from (14) and (16) that

∆w(n) +p0∆v(n)≤ρ(n)∆[r(n)∆z(n)]

z(n) +p0ρ(n)

∆[r(n+τ)∆z(n+τ)] z(n)

−ρ(n)w

2₍_n_{+ 1)} ρ2₍_n_{+ 1)}_r₍_n₎ +

(∆ρ(n))+

ρ(n+ 1) w(n+ 1)−p0

ρ(n)v2(n+ 1) ρ2₍_n_{+ 1)}_r₍_n₎ +p0

(∆ρ(n))+

ρ(n+ 1) v(n+ 1). In view of (12)and the above ineqality, we obatin

∆w(n) +p0∆v(n) ≤ −kρ(n)Q(n)−

ρ(n)w2₍_n_{+ 1)} ρ2₍_n_{+ 1)}_r₍_n₎ +

(∆ρ(n))+

ρ(n+ 1) w(n+ 1)

−p0

ρ(n)v2₍_n_{+ 1)} ρ2₍_n_{+ 1)}_r₍_n₎ +p0

(∆ρ(n))+

ρ(n+ 1) v(n+ 1)

≤ −kρ(n)Q(n) +(1 +p0)r(n)(∆(ρ(n)+) 2 4ρ(n) . Summing the above inequality from n, to n−1, we have

w(n)+p0v(n)≤w(n1)+p0v(n1)−

n−1

X

s=n1 "

kρ(s)Q(s)−(1 +p0)r(s)((∆ρ(s))+) 2 4ρ(s)

#

,

which follows that

n−1

X

s=n1 "

kρ(s)Q(s)− (1 +p0)r(s)((∆ρ(s))+) 2 4ρ(s)

#

≤w(n1) +p0v(n1),

which contradicts (8). This completes the proof. Choosing ρ(n) = R(n+σ). By Theorem 2.1 we have the following results.

Corollary 2.2 Assume that (2) holds and τ >0. If lim sup

n→∞

n

X

s=n0 "

kR(s+σ)Q(s)− (1 +p0) 4r(s)R(s+σ)

#

=∞, (17)

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Corollary 2.3 Assume that (2) holds and τ >0. If

lim inf

n→∞

1 lnR(n+σ)

n−₁

X

s=n0

R(s+σ)Q(s)> 1 +p0

4k , (18)

then every solution of (1) is oscillatory.

Proof. From (18), there exists a ǫ >0 such that for all largen, 1

lnR(n+σ)

n−1

X

s=n0

R(s+σ)Q(s)> 1 +p0 4k +ǫ, it follows that

n−1

X

s=n0

R(s+σ)Q(s)−

1 +_p0

4k

lnR(s+σ)≥ǫlnR(n+σ),

that is

n−1

X

s=n0 "

R(s+σ)Q(s)− 1 +p0 4kr(s)R(s+σ)

#

≥ǫlnR(n+σ)+

1 +_p0

4k

lnR(n0+σ). (19) Now, it is obvious that (19) implies (17) and the assertion of Corollary 2.3. follows from Corollary 2.2.

Corollary 2.4 Assume that (2) holds and τ >0. If lim inf

n→∞ [Q(n)R

2₍_n₊_σ₎_r₍_n_)]_> 1 +p0

4k , (20)

Proof. It is easy to verify that (20) yields the existence ǫ > 0 such that for all large n,

Q(n)R2(n+σ)r(n)≥ 1 +p0 4k +ǫ.

Multiplying by _R₍_n₊1_σ₎_r₍_n₎ on both sides of the above inequality, we have

R(s+σ)Q(s)− 1 +p0

4kr(s)R(s+σ) ≥

ǫ R(s+σ)r(s)

which implies that (17) holds. Therefore by Corollary 2.2. every solution of (1) is oscillatory.

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Corollary 2.5 Assume that (2) holds and τ >0. If lim sup

n→∞

n

X

s=n0 "

ksQ(s)−(1 +p0)r(s) 4s

#

=∞, (21)

Theorem 2.6 Assume that (3) holds τ >0, and σ ≤τ. Suppose also that there exists a positive real valued sequence {ρ(n)}∞

n=n0 such that (8) holds, and

lim sup

n→∞

n

X

s=n0 "

kδ(s+ 1)Q(s−τ)− (1 +p0)δ(s) 4r(s)δ2₍_s_{+ 1)}

#

=∞. (22)

Proof. Assume that contrary. Without loss of generality; we may suppose that{x(n)}is an eventually positive solution of (1) Then there exists an integer n1 ≥ n0 such that x(n+σ −τ) > 0 for all n ≥ n1. From (1), {r(n)∆z(n)} is nonincreasing eventually, where z(n) is defined by (9). Consequently, it is easy to conclude that there exists two possible cases of the sign of ∆z(n), that is , ∆z(n) > 0 or ∆z(n) < 0 eventually. If ∆z(n) > 0 eventually, then we are back of the case of Theorem 2.1, and we can get a contradiction to (8). If ∆z(n)<0,n ≥n2 ≥n1 then we define the sequence {v(n)} by

v(n) = r(n)∆z(n)

z(n) , n ≥n3 =n2+τ. (23)

Clearly v(n)<0. Noting that {r(n)∆z(n)} is nonincreasing, we get r(s)∆z(s)≤r(n)∆z(n), s ≥n≥n3.

Dividing the above by r(s) and summing it from n to l−1, we obtain

z(l)≤z(n) +r(n)∆z(n)

l−1

X

s=n

1

r(s), l ≥n ≥n3. Letting l → ∞in the above inequality, we see that

0≤z(n) +r(n)∆z(n)δ(n), n≥n3. Therefore,

r(n)∆z(n)

z(n) δ(n)≥ −1, n≥n3. From (23), we have

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Similarly, we introduce the sequence {w(n)}where

w(n) = r(n−τ)∆z(n−τ)

z(n) , n ≥n3. (25)

Obviously w(n) < 0. Noting that {r(n)∆z(n)} is nonincreasing, we have r(n−τ)∆z(n−τ)≥r(n)∆z(n). Then w(n)≥v(n). From (24), we have

−1≤w(n)δ(n)≤0, n ≥n3. (26) From (24) and (25), we have

∆v(n) = ∆(r(n)∆z(n) z(n) −

v2₍_n₎

r(n), (27)

and

∆w(n) = ∆(r(n−τ)∆z(n−τ))

z(n) −

w2(n)

r(n) . (28)

In view of (27) and (28), we obtain,

∆w(n) +p0∆v(n)≤ ∆(r(n−τ)∆z(n−τ)) z(n)

+p0

∆(r(n)∆z(n)) ∆z(n) −

w2₍_n₎ r(n) −

v2₍_n₎

r(n) . (29) On the otherhand, proceed as in the proof of Theorem 2.1, we have

∆(r(n−τ)∆z(n−τ)) +p0∆(r(n)∆z(n))≤ −kQ(n−τ)z(n). Using the above inequality in (29), we have

∆w(n) +p0∆v(n)≤ −kQ(n−τ)− w 2₍_n₎ r(n) −

v2(n)

r(n) . (30) Multiplying (30) by δ(n+ 1) and summing from n3 ton−1, we have

δ(n)w(n) − δ(n3+ 1)w(n3) +

n−1

X

s=n3

w(s) r(s) +

n−1

X

s=n3

w2₍_s₎

r(s) δ(s+ 1) + p0δ(n)v(n)−p0δ(n3+ 1)v(n3) +p0

n−1

X

s=n3

v(s) r(s) + p0

n−1

X

s=n3

v2(s)

r(s) δ(s+ 1) +k

n−1

X

s=n3

(10)

From the above inequality, we have

δ(n)w(n) − δ(n3+ 1)w(n3) +p0δ(n)v(n)−p0δ(n3+ 1)v(n3) + k

n−₁

X

s=n3

δ(s+ 1)Q(s−τ)− (1 +p0) 4

n−₁

X

s=n3

δ(s)

r(s)δ2₍_s_{+ 1)} ≤0. Thus, it follows from the above inequality that

δ(n)w(n) + p0δ(n)v(n) +

n−1

X

s=n3 "

kδ(s+ 1)Q(s−τ)− (1 +p0)δ(s) 4r(s)δ2₍_s_{+ 1)}

#

≤ δ(n3+ 1)w(n3) +p0δ(n3+ 1)v(n3).

By (24) and (26), we get a contradiction with (22) and this completes the proof.

Corollary 2.7 Assume that (3) holds,τ > 0 andσ ≤τ. Furthermore that one of conditions (17), (18), (20) and (21) holds and one has (22). Then every solution of (1) is oscillatory.

Theorem 2.8 Assume that (3) holds, τ > 0, and σ ≤ τ. Further more suppose that there exists a positive real valued sequence {ρ(n)}∞

n=n0 such that

(8) holds, and

lim sup

n→∞

n

X

s=n0

δ2(s+ 1)Q(s−τ) =∞. (31)

Proof. Assume the contrary. Without loss of generality, we assume that {x(n)} is an eventually positive solution of (1). Then there exists an integer n1 ≥ n0 such that x(n − τ) > 0 for all n ≥ n1. By (1), {r(n)∆z(n)} is nonincreasing, eventually where z(n) is defined by (9). Consequently, it is easy to conclude that there exist two possible cases of the sign of {∆z(n)}, that is ∆z(n)> 0 or ∆z(n) <0 eventually. If ∆z(n)< 0, n ≥n2 ≥ n1, then we define w(n) and v(n) as in Theorem 2.6. Then proceed as in the proof of Theorem 2.6, we obtain (24), (26) and (30). Multiplying (30) by δ2(n+ 1), and summing from n3 ton−1, where n3 ≥n2+τ, we get

δ2(n)w(n) − δ2(n3+ 1)w(n3+ 1) + 2

n−1

X

s=n3

w(s)δ(s+ 1) r(s+ 1) +p0δ

2₍_n₎_v₍_n₎

− p0δ2(n3+ 1)v(n3) + 2

n−1

X

s=n3

v(s)δ(s+ 1) r(s+ 1) +

n−1

X

s=n3

(11)

+ p0

n−1

X

s=n3

v2₍_s₎ r(s) δ

2₍_s_{+ 1) +}_k

n−1

X

s=n3

Q(s−τ)δ2(s+ 1)≤0. (32)

It follows from (3) and (24) that

∞ X

s=n3

w(s)δ(s+ 1) r(s+ 1)

≤ ∞ X

s=n3

|w(s)δ(s+ 1)| r(s+ 1) ≤

∞

X

s=n3

1

r(s+ 1) <∞,

n−1

X

s=n3

w2(s)δ2(s+ 1) r(s) ≤

∞

X

s=n3

1

r(s) <∞. In view of (26), we have

∞ X

s=n3

v(s)δ(s+ 1) r(s+ 1)

≤ ∞ X

s=n3

|v(s)δ(s+ 1)| r(s+ 1) ≤

∞

X

s=n3

1

r(s+ 1) <∞,

∞

X

s=n3

v2(s)δ2(s+ 1) r(s) ≤

∞

X

s=n3

1

r(s) <∞. From (32), we get

lim sup

n→∞

n

X

s=n0

δ2(s+ 1)Q(s−τ)<∞,

which is a contradiction with (31). This completes the proof.

Corollary 2.9 Assume that (3) holds, τ >0 and σ≤τ. Suppose also that one of conditions (17), (18), (20) and (21) holds, and one has (31). Then every solution of (1) is oscillatory.

Theorem 2.10 Assume that (2) holds and τ < 0. Moreover suppose that there exists a positive real valued sequence {ρ(n)}∞

n=n0 such that

lim sup

n→∞

n

X

s=n0 "

kρ(s)Q(s)− (1 +p0)r(s+τ)(∆ρ(s)+) 2 4ρ(s)

#

=∞. (33)

Proof. Assume the contrary. Without loss of generality we may assume that {x(n)} is an eventually positive solution of (1). Then there exists an integer n1 ≥ n0 such that x(n−τ) > 0 for all n ≥ n1. Define z(n) by (9). Similar to the proof of Theorem 2.1, there exists an integer n2 ≥n1 such that ∆z(n+τ)>0 for n≥n2 and

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Define the sequence {w(n)} by

w(n) = ρ(n)r(n)∆z(n)

z(n+τ) , n ≥n2. (35)

Then w(n)>0. By (10) we get

r(n+ 1)∆z(n+ 1)≤r(n+τ)∆z(n+τ), and

∆w(n) ≤ ρ(n)∆(r(n)∆z(n)) z(n+τ) −

ρ(n)w2₍_n_{+ 1)} ρ2₍_n_{+ 1)}_r₍_n₊_τ₎+

w(n+ 1) ρ(n+ 1)∆ρ(n)

≤ρ(n)∆(r(n)∆z(n)) z(n+τ) −

ρ(n)w2(n+ 1) ρ2₍_n_{+ 1)}_r₍_n₊_τ₎ +

w(n+ 1)

ρ(n+ 1)(∆ρ(n))+ (36) Again define the sequence {v(n)} by

v(n) =ρ(n)r(n+τ)∆z(n+τ)

z(n+τ) , n≥n2. (37)

The rest of the proof is similar to that of Theorem 2.1 and so is omitted. This completes the proof.

Choosing ρ(n) = R(n+τ). By Theorem 2.10 we have the following oscil-lation criteria.

Corollary 2.11 Assume that (2) holds and τ <0. If lim sup

n→∞

n

X

s=n0 "

kR(s+τ)Q(s)− 1 +p0 4r(s+τ)R(s+τ)

#

=∞, (38)

Corollary 2.12 Assume that (2) holds and τ <0. If lim inf

n→∞

1 lnR(n+τ)

n−1

X

s=n0

R(s+τ)Q(s)> 1 +p0

4k , (39)

Proof. By Corollary 2.11, the proof is similar to that of Corollary 2.3, we omit the details.

Corollary 2.13 Assume that (2) holds and τ <0. If lim inf

n→∞ [Q(n)R

2₍_n₊_τ₎_r₍_n₊_τ_)]_> 1 +p0

4k , (40)

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Proof. By Corollary 2.11, the proof is similar to that of Corollary 2.4, and so the proof is omitted.

Next, we choosing ρ(n) = n. From Theorem 2.4, we have the following result.

Corollary 2.14 Assume that (2) holds and τ <0. If lim sup

n→∞

n

X

s=n0 "

ksQ(s)− (1 +p0)r(s+τ) 4s

#

=∞, (41)

Next, we will give the following results under the case when (3) andσ≤τ. Theorem 2.15 Assume that (3) holds, τ < 0, and σ ≤ −τ. Further, suppose that there exists a positive real valued sequence {ρ(n)}∞

n=n0 such that

(33) holds. Suppose also that one of the following holds:

lim sup

n→∞

n

X

s=n0 "

kδ(s+ 1)Q(s+τ)− (1 +p0)δ(s) 4r(s)δ2₍_s_{+ 1)}

#

=∞, (42)

lim sup

n→∞

n

X

s=n0

δ2(s+ 1)Q(s+τ) =∞. (43) Then every solution of (1) is oscillatory.

Proof. Assume the contrary. Without loss of generality, we may assume that {x(n)} is an eventually positive solution of (1). Then there exists an integer n1 ≥ n0 such that x(n) > 0 for all n ≥ n1. From (1), {r(n)∆z(n)} is nonincreasing eventually where z(n) is defined by (9). Consequently it is easy to conclude that there exist two possible cases of sign of {∆z(n)}. That is ∆z(n) >0 or ∆z(n) < 0 eventually. If ∆z(n) > 0 eventually, then we are back of the case of Theorem 2.10 and we can get a contradiction to (33). If ∆z(n)<0 eventually, then there exists an integern2 ≥n1such that ∆z(n)<0 for all n≥n2.

Define the sequence {w(n)}and {v(n)}as follows:

w(n) = r(n+τ)∆z(n+τ)

z(n) f or n ≥n3 =n2 + 2τ and

v(n) = r(n+ 2τ)∆z(n+ 2τ)

z(n) f or n ≥n3 =n2+ 2τ.

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3 Examples

Example 3.1 Consider the second order neutral advanced difference equa-tion

∆[(n+τ)∆(x(n) +p(n)x(n+τ))] +λ

nf(x(n+σ)) = 0, n = 1,2,3, .... (44) wherer(n) =n+τ,τ andσare positive integers,ρ(n) =n+τ,f(x) =x(1+x2), 0≤p(n)≤p0 <∞, q(n) = λ

n and λ >0. Let k = 1. Then

lim sup

n→∞

n

X

s=1

kρ(s)Q(s)− (1 +p0)r(s)((∆ρ(s))+) 2 4ρ(s)

= lim sup

n→∞

n

X

s=1

"

λ−(1 +p0) 4

#

=∞,

for λ > 1+p0

4 . Hence, by Theorem 2.1, every solution of (44) is oscillatory for λ > 1+p0

4 .

Example 3.2 Consider the following second order neutral advanced differ-ence equation

∆[(n−τ)∆(x(n) +p(n)x(n+τ))] +q(n)f(x(n+σ)) = 0, n ≥1, (45) where r(n) =n−τ, q(n) = λ

n, 0≤p(n) ≤p0 <∞, τ is a negative integer, σ

is a positive integer, and f(x) =x(1 +x2) and λ >0. It is easy to see that

lim sup

n→∞

n

X

s=1

"

ksQ(s)− (1 +p0)r(s+τ) 4s

#

= lim sup

n→∞

n

X

s=1

λ− 1 +p0 4

=∞,

for λ > 1+p0

4 . Hence, by Corollary 2.14, every solution of (45) is oscillatory. Example 3.3 Consider the following second order neutral advanced differ-ence equation

∆[en(x(n) +p(n)x(n+ 2))] +e2nf(x(n+ 1)) = 0, n≥1, (46) where 0≤p(n)≤p0 <∞, f(x) =x(1 +x2), r(n) =en, q(n) = e2n. It is easy to see that

∞

X

n=1 1

(15)

Choose k= 1. Then

lim sup

n→∞

n

X

s=1

δ2(s+ 1)Q(s−2)

= lim sup

n→∞

n

X

s=1

1 e2n₍_e₋₁₎2e

2(n−2)

= ∞. Also

lim inf

n→∞ [Q(n)R

2₍_n_{+ 1)}_r₍_n_{)] = lim inf}

n→∞



e2n

en₋₁

en₍_e₋₁₎

!2

en



>

1 +p0 4

Then by Corollary 2.9 every solution of (46) is oscillatory.

References

[1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York, 1999.

[2] R.P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equa-tions, Kluwer, Dodrecht, 1997.

[3] R.P. Aggarwal, M. M. S. Manual and E. Thandapani, Oscillatory and non oscillatory behaviour of second - order neutral delay difference equations II, Appl. Math. Lett., 10(1997), 103-109.

[4] R.P. Agarwal, et al., Oscillatory and nonoscillatory behaviour of sec-ond order neutral delay difference equations, Math.Comput.Modelling, 24(1996), 5-11.

[5] M. Budincevic, Oscillations and the asymptotic behaviour of certain sec-ond order neutral difference equations, Univ. u Novom sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 21(1991), 165-172.

[6] Bi Ping and Maoan Han, Oscillation of second order difference equations with advanced argument, proceedings of the fourth international confer-ence on Dynamical Systems and Differential equations, May 24-27, 2002, Wilmington NC, USA, P.P. 108-112.

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[8] W.T.Li and S.S. Cheng, Classifications and existence of positive solutions of second order nonlinear neutral difference equations, Funkcialaj Ekvac., 40(1997), 371-393.

[9] A. Murugesan and K. Ammamuthu, Sufficient conditions for oscillation of second order neutral advanced difference equations, Int. Jour. Appl. Math...., 98(2015), 145-156.

[10] A. Murugesan, Oscillation neutral advanced difference equation. Glob. J. Pure. Appl. Math., 9(2013) 83-92.

[11] R.N. Rath et al., Oscillatory and assymptotic behaviour of a homogenous neutral delay difference equation of second order, Academica Sinica (New series), 3(2008), 453-467.

[12] A. Sternal and B. Szmanda; Assymptotic and oscillatory behaviour of certain difference equations, Le Mathematiche Vol. 51, No. 1(1996), 77-86.

[13] A.K.Tripathy, Oscillatory behaviour of a class of nonlinear second order mixed difference equations, EJQTDE, 48(2010), 1-19.