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In this paper, we study the oscillatory and asymptotic behavior of second order neutral delay difference equation with “maxima” of the form Examples are given to illustrate the main result.

Consider the oscillatory and asymptotic behavior of second order neutral delay difference equation with “maxima” of the form

where Δ is the forward difference operator defined by

(C_{1})

(C_{2})

(C_{3})

(C_{4})

Let

From the review of literature it is well known that there is a lot of results available on the oscillatory and asymptotic behavior of solutions of neutral difference equations, see [

In Section 2, we obtain some sufficient conditions for the oscillation of all solutions of Equation (1). In Section 3, we present some sufficient conditions for the existence of nonoscillatory solutions for the Equation (1) using contraction mapping principle. In Section 4, we present some examples to illustrate the main results.

In this section, we present some new sufficient conditions for the oscillation of all solutions of Equation (1). Throughout this section we use the following notation without further mention:

and

Lemma 2.1. Let

(I)

(II)

Proof. Let _{3}) we have

Hence

Lemma 2.2. Let

(I)

(II)

Proof. The proof is similar to that of Lemma 2.1.

Lemma 2.3. The sequence

The assertion of Lemma 2.3 can be verified easily.

Lemma 2.4. Let

Proof. From the definition of _{3}), we have

Lemma 2.5. Let

Proof. Since

or

The proof is now complete.

Lemma 2.6. Let

Proof. Since

Theorem 2.1. Assume that

and

then every solution of Equation (1) is oscillatory.

Proof. Assume to the contrary that there exists a nonoscillatory solution

Case(I). From Lemma 2.4 and Equation (1), we have

or

Define

or

Summing the last inequality from

Letting

Case(II). Define

Then

Summing the last inequality from

Since

or

or

Thus

So, by

where

By Mean Value Theorem,

where

Therefore,

Since

From Lemma 2.6,

From (8) and (9), we have

Multiply (10) by

Summation by parts formula yields

Using Mean Value Theorem, we obtain

Since

or

Therefore, from (7) and (11), we have

Letting

Theorem 2.2. Assume that

hold, then every solution of equation (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 2.1, we see that Lemma 2.1 holds for

Case(I). Proceeding as in the proof of Theorem 2.1 (Case(I)) we obtain a contradiction to (12).

Case(II). Proceeding as in the proof of Theorem 2.1 (Case(II)) we obtain (7) and (10). Multiplying (10) by

Using the summation by parts formula in the first term of the last inequality and rearranging, we obtain

Inview of (7), we have

As

Theorem 2.3. Assume that

then every solution of equation (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 2.1, we see that Lemma 2.1 holds and Case(I) is eliminated by the condition (2).

Case(II). Proceeding as in the proof of Theorem 2.1 (Case(II)) we have

where

and

Hence

Summing the last inequality from

Again summing the last inequality from

Letting

a contradiction to (14). This completes the proof.

Next, we obtain sufficient conditions for the oscillation of all solutions of Equation (1) when

Theorem 2.4. Assume that

and

then every solution of equation (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 2.1, we see that Lemma 2.4 holds for

Case(I). Define

Then

Using Lemma 2.5 in (18), we obtain

From the monotoncity of

and hence

for some constant

Letting

Case(II). Define a function

Then

Since

Therefore

Since

Now using (15) in (22), we obtain

for some constant

Multiplying the last inequality by

Using the summation by parts formula in the first term of the above inequality and rearranging we obtain

Using completing the square in the las term of the left hand side of the last inequality, we obtain

or

Letting

In this section, we provide sufficient conditions for the existence of nonoscillatory solutions of Equation (1) in case

Theorem 3.1. Assume that

and

then Equation (1) has a bounded nonoscillatory solution.

Proof. Choose

and

for

and let

Define a mapping

Clearly, T is continuous. Now for every

Also, from (26) we have

Thus, we have that

By the Mean Value Theorem applied to the function

Thus, T is a contraction mapping, so T has a unique fixed point

Theorem 3.2. Assume that

then Equation (1) has a bounded nonoscillatory solution.

Proof. Choose

Let

and let

Define a mapping

It is easy to see that T is continuous,

By the Mean Value Theorem applied to the function

and we see that T is a contraction on S. Hence, T has a unique fixed point which is clearly a positive solution of Equation (1). This completes the proof of the theorem.

In this section we present some examples to illustrate the main results.

Example 4.1. Consider the difference equations

Here

see that

Example 4.2. Consider the difference equations

Here

Example 4.3. Consider the difference equations

Here

Example 4.4. Consider the difference equations

Here