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Oscillatory behavior of solutions of third-order delay and advanced dynamic equations
Journal of Inequalities and Applications volume 2014, Article number: 95 (2014)
Abstract
In this paper, we consider oscillation criteria for certain third-order delay and advanced dynamic equations on unbounded time scales. A time scale is a nonempty closed subset of the real numbers. Examples will be given to illustrate some of the results.
MSC:34N05, 39A10, 39A21.
1 Introduction
In this paper, we are concerned with oscillation criteria for solutions of the third-order delay and advanced dynamic equations
and
on such that and , where α is the ratio of two positive odd integers, with
and are nondecreasing functions such that and . We also assume that such that , , and are nondecreasing for satisfying
and
By a solution of (1.1) (or (1.2)) we mean a function , , which has the property that and satisfies (1.1) (or (1.2)) for all large . A nontrivial solution is said to be nonoscillatory if it is eventually positive or eventually negative and it is oscillatory otherwise. A dynamic equation is said to be oscillatory if all its solutions are oscillatory.
Since we are interested in the oscillatory behavior of solutions of (1.1) and (1.2) near infinity, we assume throughout this paper that our time scale is unbounded above. An excellent introduction of time scales calculus can be found in the books by Bohner and Peterson [1] and [2].
The purpose of this paper is to extend the oscillation results given in [3] and [4]. Oscillation criteria for third-order dynamic equations have recently been studied in [5–8]. Other papers related with oscillation of higher-order dynamic equations can be found in [9, 10]. Well-known books concerning the oscillation theory are [11, 12].
For simplification, we define the following operators:
Thus (1.1) and (1.2) can be written as
and
respectively. In what follows we use the following notation. For
and
2 Oscillation criteria for (1.1)
In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (1.1).
Theorem 2.1 Let (1.3) and (1.4) hold and assume that
for and a constant c. If for
and
where A and B are defined as in (1.6) and (1.7), respectively, then (1.1) is oscillatory.
Proof Let x be a nonoscillatory solution of (1.1) and assume that without loss of generality for and so eventually for . Therefore, there exists a such that and are of one sign for all . We now distinguish the following two cases:
-
(I)
and eventually;
-
(II)
and eventually.
We now start with the first case.
-
(I)
Assume that and for . Then we have
Since is nonincreasing, we have
This implies that
Integrating the above inequality from to t, we have
where A is defined as in (1.6). Hence there exists such that
From (2.4) and (1.4) in (1.1) we obtain
Integrating the above inequality from to t, we obtain
or
Dividing both sides of the above inequality by , taking the lim sup of both sides as and using (2.2), we obtain a contradiction to (2.1).
-
(II)
Assume that , for . Then we have
Since is nonincreasing, we have
or
Integrating the above inequality from s to t we obtain
where B is defined as in (1.7). Then there exists such that
Integrating (1.1) from to t and using (1.4) along with the above inequality we have
Dividing the above inequality by , taking the lim sup of both sides of the above inequality as and using (2.3), we obtain a contradiction to (2.1). □
For the bounded solutions of (1.1) we have the following, which is immediate from Theorem 2.1.
Corollary 2.2 In addition to (1.3) and (1.4), assume that (2.1) and (2.3) hold. Then all bounded solutions of (1.1) are oscillatory.
Now we prove the following result.
Theorem 2.3 Let (1.3) and (1.4) hold and assume that
If
where B is defined as in (1.7), then all bounded solutions of (1.1) are oscillatory.
Proof Let x be a nonoscillatory bounded solution of (1.1). Without loss of generality assume that x is positive. Since x satisfies Case (II) in the proof of Theorem 2.1, we have (2.6). Integrating (1.1) from to t and using (1.4) along with (2.6) yields
By dividing the above inequality by and taking the lim sup of both sides of the resulting inequality as , we obtain a contradiction. The proof is now complete. □
We now consider a special case of (1.1) of the form
where β is a ratio of two odd positive integers, and we obtain some oscillatory criteria.
Theorem 2.4 Let and assume that
and
where A and B are defined as in (1.6) and (1.7), respectively. Then (2.7) is oscillatory.
Proof Let x be a nonoscillatory solution of (2.7). Without loss of generality, assume that x is positive. We consider two cases as we did in the proof of Theorem 2.1.
-
(I)
Assume that and for . Using (2.4) in (2.7), we have
Integrating the above inequality from to u and letting , we obtain
Since and the right hand side of the above inequality is infinity by (2.8), we obtain a contradiction to the facts that is positive and nondecreasing.
-
(II)
Assume that , for . Then we have (2.6). Using (2.6) in (2.7), for , we have
Integrating the above inequality from to u and letting , we obtain
Since and the right hand side of the above inequality is infinity by (2.9), we obtain a contradiction to the facts that is positive and nondecreasing. □
3 Oscillation criteria for (1.2)
In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (1.2).
Theorem 3.1 Let (1.3)-(1.5) and (2.1) hold. Also assume that
for and a constant b. If for
and
where A and B are defined as in (1.6) and (1.7), respectively, then (1.2) is oscillatory.
Proof Let x be an eventually positive solution of (1.2). Then eventually and so and are eventually of one sign. We now distinguish the following two cases:
-
(I)
and eventually;
-
(II)
and eventually.
-
(I)
Assume that and for . Then we have
Since is nondecreasing, we have
This implies that
Integrating both sides of the above inequality from s to t, we have
or
where A is defined as in (1.6). Then
Using the above inequality and (1.5) in (1.2), we have
Integrating both sides of the above inequality from t to , we get
Taking the lim sup of both sides of the above inequality as , we obtain a contradiction to (3.1).
-
(II)
Assume that and for . Then we have
Since is nondecreasing, we have
Integrating both sides of the above inequality from to t, we have
where B is defined as in (1.7). This implies that there exists a such that
Using the above inequality and (1.4) in (1.2), we have
Integrating both sides of the above inequality from to t, we find
Taking the lim sup of both sides of the above inequality as , we obtain a contradiction to (2.1). The proof is complete. □
Theorem 3.2 Assume that
and
for and constants and . If
and
where B is defined as in (1.7), respectively, then (1.2) is oscillatory.
Proof Let x be an eventually positive solution of (1.2). As in the proof of Theorem 3.1 we have two cases to consider.
-
(I)
Assume that and for . Integrating (1.2) from t to yields
Integrating the above inequality from t to gives
Again integrating the above inequality from t to we find
Finally, dividing the above inequality by and taking the lim sup of both sides of the above inequality as , we obtain a contradiction to (3.5).
-
(II)
Assume that and for . Integrating
from t to ∞ we have
Using (3.4) along with the above inequality, we have
Dividing the above inequality by and taking the lim sup of both sides of the resulting inequality as , we obtain a contradiction to (3.6). The proof is complete. □
4 Examples
In this section we give examples to illustrate two of our main results. Recall
Theorem 4.1 ([[1], Theorem 1.75])
If and , then
And
Theorem 4.2 ([[1], Theorem 1.79 (ii)])
If consists of only isolated points and , then
Our first example illustrates Theorem 2.1.
Example 4.3 Consider the third-order delay dynamic equation
where . Here , , , and . Observe that if , then
First we show that (1.3) holds. If and , , we have
It is clear that f belongs to , is nondecreasing for , and satisfies for and (1.4). Also, (2.1) holds since
Observe that if and for , then
Note for . Hence
and since f is nondecreasing and on , we obtain
It follows that if
and so (2.2) holds. It remains to show that (2.3) holds for . This requires that we determine . Using the above representations of u, s, t, we have
The monotonicity of f and the fact that on yield
Therefore
which shows that (2.3) holds. By Theorem 2.1, (4.1) is oscillatory.
Our second example illustrates Theorem 3.2.
Example 4.4 Consider the third-order advanced dynamic equation
where α is the ratio of two positive odd integers and , . Here , , , , , and . Then
and so (3.5) and (3.6) hold. Next we show that (3.7) holds. For , we have
Since for all , we have
This implies
Thus (3.8) holds. By Theorem 3.2, (4.2) is oscillatory.
5 Discussion
While we were able to unify most results for (1.1) given in [3] and [4], the comparison result
Theorem Let (1.3)-(1.4) hold. If the first-order delay dynamic equations
and
are oscillatory, then (1.1) is oscillatory.
cannot be extended since is satisfied for few time scales. While the result holds for and , this condition is not satisfied on . Being aware that time scales are not generally closed under addition, we were able to prove the following.
Theorem 5.1 Let (1.3)-(1.4) hold. Furthermore, assume the delay function is a bijection. If the first-order delay dynamic equations
and
are oscillatory, where , then (1.1) is oscillatory.
In order to prove Theorem 5.1, it is necessary to define the function to be a nondecreasing function with respect to its second argument and with the property that for any function . It is also necessary to assume that is a bijection with for all and , and to use the following definition and theorem.
Definition 5.2 Let . By a solution of the dynamic inequality
on an interval , we mean a rd-continuous function y defined on the interval , which is rd-continuously differentiable on and satisfies (5.5) for all . A solution y of (5.5) is said to be positive if for every .
Theorem 5.3 Let y be a positive solution on an interval , of the delay dynamic inequality (5.5). Moreover, we assume that F is positive on any set of the form , . Then there exists a positive solution x on of the delay dynamic equation
such that
and
Proof Let y be a positive solution (5.5). From (5.5) we obtain for all , with
Hence, letting we get
Let X be the set of all nonnegative continuous functions x on the interval with for every . Then by using (5.8) we can easily verify that for any function x in X the formula
defines an operator . If and for , then we also have for since F is nondecreasing with respect to its second argument. Thus, the operator S is monotone. Next, we put
and observe that is a decreasing sequence of functions in X. Furthermore, define
Then by applying the Lebesgue dominated convergence theorem, we obtain . That is
From (5.9) it follows that
and hence x is a solution of (5.6) on the interval . Also (5.9) yields
Moreover, it is clear that for . It remains to prove that x is positive on the interval . Taking into account that y is positive on the interval and the positivity of F on we have
for each . So, x is positive on an interval . Next we will show that x is also positive on . Assume that is the first zero of x in . Then for and . Then (5.9) yields
and consequently
That is, we can choose a such that
On the other hand, taking into account that for and the positivity of F on we get
This leads to a contradiction and the proof is complete. □
Note that Theorem 5.3 holds for any unbounded time scale . Now we present the proof of Theorem 5.1.
Proof Let x be an eventually positive solution of (1.1). We consider two cases as we did in the proof of Theorem 2.1.
-
(I)
Assume that and for . Then (2.4) holds. Using this and (1.4) in (1.1), we obtain
or
where for . Since for all and for all , by Theorem 5.3, there exists a positive solution z of (5.3) such that , which contradicts the hypothesis that (5.3) is oscillatory.
-
(II)
Assume that and for . As in the proof of Theorem 2.1, we have (2.5). Then for , we have
Using this inequality and (1.4) in (1.1) for yields
or
where for . Similar to Case (I) above, by Theorem 5.3, there exists a positive solution y of (5.4) such that , which contradicts the fact that (5.4) is oscillatory. □
In [3] and [4], the authors prove a comparison result for (1.2) similar to the one given at the beginning of this section. That result involved . Again, since time scales are not generally closed under addition, this result cannot be extended to a general time scale .
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Adıvar, M., Akın, E. & Higgins, R. Oscillatory behavior of solutions of third-order delay and advanced dynamic equations. J Inequal Appl 2014, 95 (2014). https://doi.org/10.1186/1029-242X-2014-95
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DOI: https://doi.org/10.1186/1029-242X-2014-95