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Oscillation criteria for second-order nonlinear neutral difference equations of mixed type Advances in Difference Equations 2012, 2012:4 doi:10.1186/1687-1847-2012-4 Ethiraju Thandapani

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Oscillation criteria for second-order nonlinear neutral difference equations of

mixed type

Advances in Difference Equations 2012, 2012:4 doi:10.1186/1687-1847-2012-4

Ethiraju Thandapani (ethandapani@yahoo.co.in)Nagabhushanam Kavitha (kavitha_snd@hotmail.com)Sandra Pinelas (sandra.pinelas@gmail.com)

ISSN 1687-1847

Article type Research

Submission date 3 October 2011

Acceptance date 27 January 2012

Publication date 27 January 2012

Article URL http://www.advancesindifferenceequations.com/content/2012/1/4

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Oscillation criteria for second-order nonlinear neutral difference equations of mixed type

Ethiraju Thandapani∗1, Nagabhushanam Kavitha1

and Sandra Pinelas2

1Ramanujan Institute for Advanced Study in Mathematics,

University of Madras, Chennai 600 005, India

2Departamento de Matem´atica, Universidade dos A¸cores,

Ponta Delgada, Portugal

∗Corresponding author: ethandapani@yahoo.co.in

Email addresses:

NK: kavitha snd@hotmail.com SP: sandra.pinelas@gmail.com

AbstractSome oscillation criteria are established for the second order nonlinear

neutral difference equations of mixed type.

∆2(x n + ax n−τ1 ± bx n+τ2)α = q n x β n−σ1 + p n x β n+σ2 , n ≥ n0

where α and β are ratio of odd positive integers with β ≥ 1 Results

obtained here generalize some of the results given in the literature.Examples are provided to illustrate the main results

2010 Mathematics Subject classification: 39A10

Keywords: Neutral difference equation; mixed type; comparison orems; oscillation

Let θ = max {τ1, σ1} By a solution of Equation (E ±) we mean a real

sequence {x n } which is defined for n ≥ n0− θ and satisfies Equation (E ±) for

all n ∈ N(n0) A nontrivial solution of Equation (E ±) is said to be oscillatory

if it is neither eventually positive nor eventually negative Otherwise it isknown as nonoscillatory

Equations of this type arise in a number of important applications such

as problems in population dynamics when maturation and gestation are cluded, in cobweb models, in economics where demand depends on the price

in-at an earlier time and in electric networks containing lossless transmissionlines Hence it is important and useful to study the oscillation behavior of

solutions of neutral type difference Equation (E ±)

The oscillation, nonoscillation and asymptotic behavior of solutions of

Equation (E ± ), when b = 0 and p n ≡ 0 or a = 0 and p n ≡ 0 or b = 0 and

q n ≡ 0 have been considered by many authors, see for example [1–4] and

the reference cited therein However, there are few results available in theliterature regarding the oscillatory properties of neutral difference equations

of mixed type, see for example [1–8] Motivated by the above observation, in

this article we establish some new oscillation criteria for the Equation (E ±)which generalize some of the results obtained in [1–3,5–7]

In Section 2, we present conditions for the oscillation of all solutions of

equation (E ±) Examples are provided in Section 3 to illustrate the results

2 Oscillation results

In this section, we obtain sufficient conditions for the oscillation of all

solu-tions of Equation (E ± ) First we consider the Equation (E −), viz,

∆2y n+q n

b β y β/α n−σ1−τ2 + p n

b β y n+σ2−τ2 β/α ≤ 0 (2.5)

has no eventually positive solution.

Then every solution of Equation (E − ) is oscillatory.

Proof Let {x n } be a nonoscillatory solution of Equation (E −) Without loss

of generality, we may assume that there exists n1 ∈ N(n0) such that x n−θ > 0

for all n ≥ n1 Set z n = (x n + ax n−τ1 − bx n+τ2)α

Then

∆2z n = q n x β n−σ1 + p n x β n+σ2 > 0, n ≥ n1,

which implies that {z n } and {∆z n } are of one sign for all n2 ≥ n1 We claim

that z n > 0 eventually To prove it assume that z n < 0 Then we let

Consequently {y n } and {∆y n } are of one sign, eventually Now we shall prove

that y n > 0 If not, then let

We obtain that {v n } is a positive solution of inequality (2.5), a contradiction.

Next we consider the following two cases:

Case 1: Let ∆z n < 0 for n ≥ n3 ≥ n2 We claim that ∆y n < 0 for n ≥ n3.

If not, then we have y n > 0, ∆y n > 0 and ∆2y n ≥ 0 which implies that

lim

n→∞ y n = ∞ On the other hand, z n > 0 , ∆z n < 0 implies that lim

n→∞ z n = c <

∞ Then applying limits on both sides of (2.6) we obtain a contradiction.

Thus ∆y n < 0 for n ≥ n 3. Using the monotonicity of {z n }, we now get

Thus {y n } is a positive decreasing solution of inequality (2.4), a contradiction.

Case 2: Let ∆z n > 0 for n ≥ n 3. Now we consider the following two cases

Case (i): Assume that ∆y n < 0 for n ≥ n 3. Proceeding similarly as above

and using the monotonicity of {z n } we obtain

Case (ii): Assume that ∆y n > 0 for n ≥ n3 Then y n+σ2 ≤ (1 + a β )z n+σ2

which in view of (2.8) implies

of solutions of Equation (E − ) even if not all assumptions of Theorem 2.2

are satisfied If the difference inequality (2.3) has an eventually positive creasing solution then the conclusion of Theorem 2.2 is replaced by “Every solution of Equation (E − ) is either oscillatory or |x n | → ∞ as n → ∞”.

in-Remark 2.2 In [2, Theorem 7.6.26] , the author considered the Equation

(E − ) with α = β = 1, p n ≡ p, and q n ≡ q and obtain oscillation results with

(1 + a − b) > 0 Hence Theorem 2.2 generalize and improve the results of [2,

Theorem 7.6.26].

Remark 2.3 Applying existing conditions sufficient for the inequalities (2.3), (2.4), and (2.5) to have no above mentioned solutions, we immediately obtain various oscillation criteria for Equation (E − ).

Theorem 2.3 Let σ1 > τ1, σ2 ≥ 2, and β = α Assume that

Remark 2.4 Taking into account the result of [2], we see that the absence

of positive solution of (2.5) can be replaced by the assumption that for the corresponding equation

∆2y n+q n

b β y β/α n−σ1−τ2 + p n

b β y n+σ2 β/α −τ2 = 0

every solution of this equation are oscillatory.

Next we consider the difference Equation (E+)

∆2(x n + ax n−τ1 + bx n+τ2)α = q n x β n−σ1 + p n x β n+σ2 (E+)

and present conditions for the oscillation of all solutions of Equation (E+).

Theorem 2.4 Assume that σ1 ≥ τ1, σ2 ≥ τ2+ 2, q ∗

n = min {q n−σ1 , q n , q n+τ2 } and p ∗

n = min {p n−σ1 , p n , p n+τ2 } If

∆2y n − p

∗ n

Proof Let {x n } be a nonoscillatory solution of (E+) Without loss of

gener-ality, we assume that there exists an integer n1 ∈ N(n0) such that x n−θ > 0

for all n ≥ n1 Setting

z n = (x n + ax n−τ1 + bx n+τ2)αand

y n = z n + a β z n−τ1 + b β

2β−1 z n+τ2 (2.13)

Then z n > 0, y n > 0 and

∆2z n = q n x β n−σ1 + p n x β n+σ2 ≥ 0. (2.14)

Then {∆z n } is of one sign, eventually On the other hand

∆2y n =q n x β n−σ1 + p n x β n+σ2 + a β q n−τ1 x β n−σ1−τ1 + a β p n−τ1 x β n+σ2 −τ1

+ b β

2β−1 q n+τ2 x β n−σ1 +τ2 + b β

2β−1 p n+τ2 x β n+σ2 +τ2. (2.15)Using (2.1) in (2.15) we obtain

∆2y n ≥ q

∗ n

2β−1 (x n−σ1 + ax n−σ1−τ1)β +q n+τ2

2β−1 b β x β n−σ1 +τ2

+ p

∗ n

Case 1: Assume that ∆z n > 0 Then ∆y n > 0 In view of (2.16), we have

∆2y n+τ2 ≥ 1

4β−1 p ∗ n+τ2 z n+σ2 β/α +τ2. (2.17)

Applying the monotonicity of z n , we find

∆2y n+τ2 ≥ p

∗ n+τ2

From the monotonicity of {z n } we find

Therefore {y n } is a positive decreasing solution of the difference inequality

(2.12), a contradiction This completes the proof

Theorem 2.5 Assume that σ1 ≥ τ1, σ2 ≥ τ2+2, β = α, q ∗

n = min {q n−σ1 , q n , q n+τ2 } and p ∗

has no eventually positive increasing solution, and

Proof. Conditions (2.21) and (2.22) are sufficient for the inequality (2.11)

to have no increasing positive solution and for (2.12) to have no decreasingpositive solution, respectively (see e.g., [2, Lemma 7.6.15]) The proof thenfollows from Theorem 2.4

Remark 2.5 When α = β = 1 , Theorem 2.5 involves result of Theorem 7.6.6 of [2].

has no eventually negative solution, and

∆v n − ψ n+δ2 v n+δ2 ≥ 0 (2.26)

has no eventually positive solution, then every solution of Equation (E+) is

oscillatory.

Proof Let {x n } be a nonoscillatory solution of Equation (E+) Without loss

of generality, we assume that there exists an integer n1 ∈ N(n0) such that

x n−θ > 0 for all n ≥ n1 Define z n and y nas in Theorem 2.4 Proceeding as inthe proof of Theorem 2.4, we obtain (2.16) Next we consider the followingtwo cases

Case 1: Assume ∆z n > 0 Clearly ∆y n > 0 Then as in case 1 of Theorem

2.4, we find that {y n } is a positive increasing solution of inequality (2.19).

Let B n = ∆y n + ψ n y n+δ2 Then B n > 0 Using (2.24), we have

Define B n = ψ n v n Then {v n } is a positive solution of (2.26), a contradiction.

Case 2: Assume that ∆z n < 0 Clearly ∆y n < 0 Then as in case 2 of

Theorem 2.4, we find that {y n } is a positive decreasing solution of inequality

(2.20) Let A n = ∆y n − φ n y n−δ1 Then A n < 0 Using (2.23), we have

Define A n = φ n v n. Then {v n } is a negative solution of inequality (2.25), a

contradiction This completes the proof

From Theorem 2.6 and the results given in [7] we have the following

oscillation criteria for Equation (E+).

Corollary 2.7 Let β = α, δ1 = σ1− τ1

2 > 0, and δ2 =

σ2− τ2

2 > 0.

Suppose that there exist two positive real sequence {φ n } and {ψ n } with ∆φ n ≥

0 and ∆ψ n ≤ 0 such that (2.23) and (2.24) holds If

then every solution of Equation (E+) is oscillatory.

Proof. It is known (see [7]) that condition (2.27) is sufficient for inequality(2.25) to have no eventually negative solution On the other hand, condition

(2.28) is sufficient for inequality (2.26) to have no eventually positive solution.

Remark 2.6 From the results presented in this section, we observe that when the coefficient p n = 0 or the condition on {p n } is violated the conclusion

of the theorem may be replaced by “Every solution {x n } of equation (E ± ) is

oscillatory or x n → ∞ as n → ∞”.

Once again from the proofs, we see that if q n = 0 or condition on {q n }

is violated then the conclusion of the theorems may be replaced by “Every solution {x n } of Equation (E ± ) is oscillatory or x n → 0 as n → ∞”.

In this section, we provide some examples to illustrate the main results

Example 1 Consider the difference equation

Example 2 Consider the difference equation

By Corollary 2.6, we see that all solutions of Equation (3.3) are oscillatory.

We conclude this article with the following remark

Remark 3.1 It would be interesting to extend the results of this article to the equation

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