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 and and is a nonnegative integer subject to the following conditions:

(C₁) and are positive integers;

(C₂) is a ratio of odd positive integers;

(C₃) and are nonnegative real sequences with and for all;

(C₄) is a positive real sequence such that.

Let. By a solution of Equation (1), we mean a real sequence satisfying Equation (1) for all. Such a solution is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise.

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 [1] -[5] , and the references cited therein. But very few results are available in the literature dealing with the oscillatory and asymptotic behavior of solutions of neutral difference equations with “maxima”, see [6] -[9] , and the references cited therein. Therefore, in this paper, we investigate the oscillatory and asymptotic behavior of all solutions of Equation (1). The results obtained in this paper extend that in [4] for equation without “maxima”.

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 be an eventually positive solution of Equation (1). Then one of the following holds

(I) and;

(II) and.

Proof. Let be an eventually positive solution of Equation (1). Then we may assume that, for all. Then inview of (C₃) we have for all. From the Equation (1), we obtain

Hence and are of eventually of one sign. This completes the proof.

Lemma 2.2. Let be an eventually negative solution of Equation (1). Then one of the following holds

(I) and;

(II) and.

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

Lemma 2.3. The sequence is an eventually negative solution of Equation (1) if and only if is an eventually positive solution of the equation

The assertion of Lemma 2.3 can be verified easily.

Lemma 2.4. Let be an eventually positive solution of Equation (1) and suppose Case (I) of Lemma 2.1 holds. Then there exists such that

Proof. From the definition of and condition (C₃), we have. Further , since is nondecreasing. This completes the proof.

Lemma 2.5. Let be an eventually positive solution of equation (1) and suppose Case (I) of Lemma 2.1 holds. Then there exists such that

Proof. Since, we see that

or

The proof is now complete.

Lemma 2.6. Let be an eventually positive solution of Equation (1) and suppose Case (II) of Lemma 2.1 holds. Then there exists such that is nonincreasing for all.

Proof. Since and then we have for. This completes the proof.

Theorem 2.1. Assume that, and there exists a positive integer k such that. If for all sufficiently large and for all constants,. One has

and

then every solution of Equation (1) is oscillatory.

Proof. Assume to the contrary that there exists a nonoscillatory solution of Equation (1). Without loss of generality we may assume that for all, where N is chosen so that both the cases of Lemma 2.1 hold for all. We shall show that in each case we are led to a contradiction.

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

or

Define, then we have

or

Summing the last inequality from to, we have

Letting, we get a contradictions to (2).

Case(II). Define

Then for. Since is nonincreasing, we have

Summing the last inequality from to, we obtain

Since and by letting, in the last inequality we obtain

or

or

Thus

So, by and (6), we have

where. From (6), we obtain

By Mean Value Theorem,

where. Since and, we have

Therefore,

Since, we have

From Lemma 2.6, for, we have

From (8) and (9), we have

Multiply (10) by and summing it from to, we have

Summation by parts formula yields

Using Mean Value Theorem, we obtain

Since, we have

or

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

Letting in the last inequality, we obtain a contradiction to (3). This completes the proof.

Theorem 2.2. Assume that, and there exists a positive integer k such that. If for all suffi- ciently large and for every constant, (2) holds, and

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 and summing it from to we have

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

Inview of (7), we have as and

As in the last inequality, we obtain a contradiction to (12). This completes the proof.

Theorem 2.3. Assume that, and there exists a positive integer k such that. If for all suffi- ciently large and for every constant, (2) holds, and

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. From Equation (1), we have

and

Hence

Summing the last inequality from to, we obtain

Again summing the last inequality from to, we have

Letting in the above inequality, we obatin

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 there exists a positive integer k such that. If for all sufficiently large and for every constant, one has

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 by

Then and from Equation (1) and Lemma 2.2, we have

Using Lemma 2.5 in (18), we obtain

From the monotoncity of, we have

and hence

for some constant for all large n. Using (20) in (19) and then summing the resulting inequality from to, we have

Letting in (21), we obtain a contradiction to (16).

Case(II). Define a function by

Then for, we have

Since, and is negative and decreasing we have

Therefore

Since is a positive and decreasing, we have. Combining the last two inequalities, we have

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

for some constant. That is

Multiplying the last inequality by, and then summing it from to, we have

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 the above inequality, we obtain a contradiction to (17). The proof is now complete.

In this section, we provide sufficient conditions for the existence of nonoscillatory solutions of Equation (1) in case or. Note that in this section we do not require.

Theorem 3.1. Assume that. If

and

then Equation (1) has a bounded nonoscillatory solution.

Proof. Choose sufficiently large so that

and

for. Let be the set of all bounded real sequences defined for all with norm

and let

Define a mapping by

Clearly, T is continuous. Now for every and, (25) implies

Also, from (26) we have

Thus, we have that. Since S is bounded, closed and convex subset of, we only need to show that T is contraction mapping on S in order to apply the contraction mapping principle. For and, we have

By the Mean Value Theorem applied to the function, we see that for any, we have for all. Hence

Thus, T is a contraction mapping, so T has a unique fixed point such that. It is easy to see that is a positive solution of Equation (1). This complete the proof of the theorem.

Theorem 3.2. Assume that. If

then Equation (1) has a bounded nonoscillatory solution.

Proof. Choose sufficiently large so that

Let be the set of all bounded real sequences defined for all with norm

and let

Define a mapping by

It is easy to see that T is continuous, , and for any and, we have

By the Mean Value Theorem applied to the function, we see that for any, we have for all. Hence

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 and. Then. Choosing, we

see that. Further it is easy to verify that all other conditions of Theorem 2.1 are satisfied. Therefore every solution of Equation (28) is oscillatory.

Example 4.2. Consider the difference equations

Here and. Then and

. Choosing, we see that. Further it is easy to verify that all other conditions of Theorem 2.4 are satisfied. Therefore every solution of Equation (29) is oscillatory.

Example 4.3. Consider the difference equations

Here and. By talking, we see that all conditions of Theorem 3.1 are satisfied and hence Equation (30) has a bounded nonoscillatory solution.

Example 4.4. Consider the difference equations

Here and. By talking, we see that all conditions of Theorem 3.2 are satisfied and hence Equation (31) has a bounded nonoscillatory solution.