Oscillation and Convengence Properties of Second Order Nonlinear Neutral Delay Difference Equations

International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 5 Issue 3 || March. 2017 || PP-19-24

Oscillation and Convengence Properties of Second Order Nonlinear Neutral Delay Difference Equations A. Murugesan1 and K. Venkataramanan2 1

Department of Mathematics, Government Arts College (Autonomous), Salem – 636 007, Tamil Nadu, India.

2 Department of Mathematics, Vysya College, Salem – 636 103, Tamil Nadu, India.

Abstract. In this paper, we consider the second order nonlinear neutral delay difference equations of the form

We establish sufficient conditions which ensures that every solution of as

is either oscillatory or tends to zero

. We also gives examples to illustrate our results.

Mathematics Subject Classification 2010 : 39A10, 39A12 Keywords and Phrases: Oscillation, nonoscillation, second order, neutral, nonlinear, delay, difference equation.

I. INTRODUCTION In this paper, we study the oscillation and asymptotic behaviors of the second order nonlinear neutral delay difference equation

where

is the forward difference operator defined by

sequence of nonnegative real numbers, identically zero for large values of n, integers,

is a

is a sequence of nonnegative real numbers and is a sequence of positive real numbers

is the ratio of two odd integers and

and

is not

are nonnegative

is a continuous real valued function.

Throughout the paper the following conditions are assumed to be hold (i) There exist a constant such that ; (ii) u

for all

and there exists a positive constant k such that

for all

In addition to the above, we assume that

If

is an eventually positive solution of (1.1), then its associated sequence

By

a

solution

(1.1),

we

mean

a

real

and satisfies (1.1) for nontrivial for all large n. A solution

sequence .

is defined by

which

is

defined

for

We consider only such solution which are

of (1.1) is said to be non oscillatory if the terms x(n) of the

sequence are eventually positive or eventually negative. Otherwise it is called oscillatory.

Recently, there has been a lot of interest in the study of oscillatory and asymptotic behaviors of second order difference equations see, for example [1,4-10] and the references cited therein. For the general background of difference equations one can refer to [2,3] Sternal et al. [8] established sufficient conditions under which every bounded solution of (1.1) is either oscillatory or tends to zero as for the cases Rath et al. [6] established sufficient conditions which ensures that every solution of or tends to zero as

is oscillatory

for the case

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Oscillation And Convengence Properties Of Second Order Nonlinear Neutral Delay Difference‌ Thandapani et al. [11] consider the second order neutral difference equation

and established sufficient conditions under which every bounded solution of (1.4) is oscillatory and determined sufficient conditions for the existence of positive solution which tends to zero as .

Li et al. [5] consider the following second order nonlinear neutral delay differential equation

and established sufficient conditions which ensures that every solution of (1.5) is either oscillatory or tends to zero.

In this paper, our aim is to determine sufficient conditions under which every solution of (1.1) is either oscillatory or tends to zero. Our established results are discrete analogues of some well-known result due to [5].

II. LEMMAS In this section, we give two lemmas that will be useful for establishing our results.

Lemma 2.1. [4] If X and Y are positive real numbers and then

or There is obviously equality when Lemma 2.2. If

or

.

is an eventually positive solution of the equation (1.1) then z(n) satisfies the following

two possible cases:

for

where

Proof: Since

is sufficiently large. is an eventually positive solution of (1.1) there exist an integer

such that

It follows from (1.1) and (1.3) that

This shows that that

is nonincreasing and of one sign. That is, these exists an integer or

If

for all for

for

, then we have

or

. We prove that

for

is not possible. If

,

. Then, we conclude that

By virtue of the condition (1.2), Case 1: If

.

, then

where

such

as

. We consider now the following two cases separately.

is unbounded, then there exists a sequence

such that

and

where Now from (1.3) we have

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Oscillation And Convengence Properties Of Second Order Nonlinear Neutral Delay Difference‌

which contradicts the fact that Case 2: If

is

bounded,

Hence Lemma 2.3. If

Proof: by

then

is

also

bounded,

which

contradicts

satisfies one of the cases ( ) and ( ). This completes the proof.

is an eventually positive solution of (1.1) and z(n) satisfies the case ( ) then

and

, we have

where is a finite constant. That is, bounded. Using the fact that

is bounded. As in the proof of Case1 in Lemma 2.2,

is also

is bounded, we obtain .

We claim that a=0. If

, then there exists a sequence

of integers such that

and

as

.

Let

. Then for all large k,

and so

which is a contradiction. Thus, a=0 and

This completes the proof.

III. OSCILLATION AND CONVERGENCE OF SOLUTIONS TO (1.1) In what follows we are the following notations:

where

is sufficiently large and where

is a sequence of real numbers.

Theorem 3.1. Suppose that there exists a sequence

of positive real numbers such that for all

sufficiently large

Then every solution of (1.1) is either oscillatory or tends to zero as

.

Proof : Assume the contrary. Without loss of generality, we may suppose that solution of (1.1). Then there exists an integer all

such that

Then from (1.1) and (1.3), we have (2.2). From Lemma 2.2,

is an eventually positive and

for

satisfies one of the cases

or

. We consider each of two cases separately.

Suppose first that

holds. By the definition of z(n).

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Oscillation And Convengence Properties Of Second Order Nonlinear Neutral Delay Difference‌

By the nonincreasing nature of

we have

Also by (2.2), we obtain

Let us define a sequence { (n)} by

Then

for

and

Using (2.2), (3.2) and (3.4) in (3.6), we have

Summing the above inequality (3.7) from

to n-1,

we obtain

which contradicts (3.1).

If z(n) satisfies ( ), then Let

due to Lemma 2.3. The proof is complete.

. We can obtain the following criterion for (1.1) using Theorem 3.1.

Corollary 3.2: If

then every solution of (1.1) is either oscillatory or tends to zero as Theorem 3.2. Assume that

.

Suppose that there exists a sequence

of positive real numbers

such that for sufficiently large

Then every solution of (1.1) is either oscillatory or tends to zero as Proof. Assume the contrary. Without loss of generality, we may suppose that solution of (1.1). Then there exists an integer all

such that

is an eventually positive and

for

.

By virtue of Lemma 2.2. z(n) satisfies one of ( ) and ( ). We discuss each of the two cases separately.

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Oscillation And Convengence Properties Of Second Order Nonlinear Neutral Delay Difference‌ Assume first that {z(n)} satisfies ( ) we obtain (3.3) and (3.4). Define the sequence for

by (3.5).Then

and we obtain the equation (3.6)

By Lemma 2.1, we have Using (3.10) in (3.6), we obtain

Using (2.2) and (3.2) in (3.11), we have

On the other hand, by the decreasing nature of

and (3.4), we have

Using (3.13) and (3.12), we obtain

Summing the above inequality from

to n-1, we obtain

which contradicts (3.9).

If z(n) satisfies ( ), then by Lemma 2.3,

and this completes the proof.

IV. EXAMPLES Example 4.1.

Consider the following second order neutral delay difference equation

where

,

and k=1.

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Oscillation And Convengence Properties Of Second Order Nonlinear Neutral Delay Difference‌ It follows from Corollary 3.2 that every solution of (4.1) is either oscillatory or tends to zero as instance,

For

is an oscillatory solution of the equation (4.1)

Example 4.2.

Consider the following second order neutral delay difference equation

we have Choose

Then

We can easily show that

Then

.

It follows from Theorem 3.1, every solution of (1.1) is either oscillatory or tend to zero

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