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
(C1)
(C2)
(C3)
(C4)
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
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
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