1. Introduction
We consider the half-linear second order differential equations with mixed deviating argument of the following form.
Throughout this paper, it is assumed that the following is the case:
- ()
, is the ratio of two positive odd integers;
- ()
; .
By a proper solution of Equation (
1), we mean a function
that satisfies Equation (
1) for all sufficiently large
t and
for all
We make the standing hypothesis that (
1) does possess proper solutions. A proper solution is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. An equation itself is said to be oscillatory if all its proper solutions are oscillatory. There are numerous papers devoted to oscillation theory of differential equations, see, e.g., [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12].
It is known (see, e.g., [
7]) that if
is a nonoscillatory solution of (
1), then eventually either:
and we say that
is of degree 0, and we denote the set of such solutions by
or
and we say that
is of degree two, and we denote the corresponding set by
.
Consequently, the set
of all nonoscillatory solutions of (
1) has the following decomposition.
The first aim of this paper is to establish criteria for
and
. This problem has been solved by many authors, and we mention here the pioneering works of Ladas et al. [
12] and Koplatadze and Chanturija [
9]; in general, authors discuss the condition for
only when the deviating argument is delayed (
) and criteria for
only for advanced arguments (
), (see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12]).
In this paper, we establish the desired criteria when deviating argument
is of a mixed type, which means that its delayed part:
and its advanced part
are both unbounded subsets of
.
The second aim of this paper is to join the criteria obtained for
and
to establish the oscillation of (
1).
Our basic results will be formulated for general Equation (
1), i.e., without additional conditions imposed on function
. Then, we provide significant improvements for two partial cases, namely when (
1) is in either canonical form, that is, when it has the following form.
When this situation occurs we employ the following function:
or in noncanonical form (opposite case) when the following is the case.
In this case, we shall use the auxiliary function of the following form.
Thus, our results are of high generality and, what is more, they hold for all
, and our technique does not require discussing cases
and
separately as it is common, see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10].
3. Basic Results
Our first result is applicable to both canonical and noncanonical equations. In all our results, we employ two sequences
and
such that the following is the case:
and we have the following.
Theorem 1. Assume that there exist two sequences and satisfying (4) and (5), respectively. If the following is the case:andthen, (1) is oscillatory. Proof. Assume on the contrary, that (
1) possesses an eventually positive solution
. Then, either
or
. We shall show that (
6) and (
7) imply
and
, respectively.
At first, we admit that
. We remark that since
is increasing, then
implies
. Let
for some
. Using the fact that
is decreasing, an integration of (
1) from
u to
yields the following.
Extracting the
root and integrating from
to
, we are led to the following:
which contradicts the condition (
6), and we conclude that class
.
Now, we assume that
. Then,
is increasing. It is useful to notice that since
is increasing, it follows from
that
. Let
for some
. By integrating (
1) from
to
u, one obtains the following.
An integration of the last inequality from
to
provides the following:
which contradicts (
7) and so
, and the proof is complete. □
Theorem 1 extends the corresponding result of Kusano [
11] formulated for
For the linear case of (
1), namely when
, we can change the order of integration in (
6) and (
7), which essentially simplifies evaluation of these criteria.
Corollary 1. Let and (2) hold. Assume that there exist two sequences and satisfying (4) and (5). If the following is the case:andthen, (1) is oscillatory. Corollary 2. Let and (3) hold. Assume that there exist two sequences and satisfying (4) and (5). If the following is the case:andthen, (1) is oscillatory. To illustrate the above mentioned criteria, we provide the following couple of examples.
Example 1. We consider the second order linear functional differential equation in the canonical form. Clearly, the deviating argument is of mixed type such that and .
We place , . Then, and . Condition (
8)
takes the following form:which means that for , the class for (
12)
. On the other hand, if we set , , then and, moreover, . Condition (9) reduces to the following:which ensures that provided that . Picking up both criteria, we observe that the following condition:implies oscillation of (12). Example 2. We consider the second order linear functional differential equation in the noncanonical form. It is easy to observe that We again substitute and . Condition (
10)
takes the following form:which means that for , the class . To eliminate class , we set , . Condition (
11)
simplifies to the following:which ensures that provided that . Therefore, the following condition guarantees oscillation of (
13)
. In the next two sections, we essentially improve conditions (
6)–(
11) for eliminations of classes
and
. To achieve our goals, it is necessary to study canonical and nocanonical equations separately.
4. Canonical Equations
We establish new monotonic properties of possible nonoscillatory solutions and then apply them to improve the above mentioned criteria. The progress will be presented via Equation (
12). In the first part, we focus our considerations to eliminate class
.
Lemma 1. Let (
2)
hold. Assume that there exist a sequence satisfying (
4)
and a positive constant such that for , we have the following. If is a positive solution of (
1)
such that , then the following is the case. Proof. Assume that
for some
. Since
is decreasing, an integration of (
1) from
to
s yields the following.
It is easy to see that the last inequality, in view of (
14), implies the following.
Consequently, we have the following:
on
and
, and the proof is complete. □
Theorem 2. Let (
2)
hold. Assume that there exists a sequence satisfying (4) and a positive constant such that (
14)
holds. If the following is the case:then, the class for (1) is the case. Proof. Assume on the contrary that (
1) possesses an eventually positive solution
. Let
for some
. Using the fact that
is decreasing on
, an integration of (
1) from
u to
yields the following.
Extracting the
root and integrating from
to
, one obtains the following.
This is a contradiction, and the proof is now complete. □
For
, condition (
16) can be significantly simplified.
Corollary 3. Let and (
2)
hold. Assume that there exists a sequence satisfying (
4)
and a positive constant such that (
14)
holds. If the following is the case:then the class for (1). Now, we turn our attention to the class .
Lemma 2. Assume that there exists a sequence satisfying (
5)
and a positive constant such that for , the following is the case. If is a positive solution of (
1)
such that , then the following is the case. Proof. Assume that
for some
. Taking into account that
is increasing, an integration of (
1) from
u to
produces the following.
By (
18), the last inequality implies the following.
Therefore, we have the following:
on
and
, and the proof is complete. □
Theorem 3. Let (
2)
hold. Assume that there exists a sequence satisying (
5)
and a positive constant satisfying (
18)
. If the following is the case:then the class for (1). Proof. Let us admit that (
1) possesses an eventually positive solution
. Assume that
for some
. Employing the fact that
is increasing on
and integrating (
1) from
to
u, we obtain the following.
By extracting the
root and integrating from
to
, we observe that the following is the case.
This is a contradiction, and the proof is complete now. □
Corollary 4. Let and (
2)
hold. Assume that there exists a sequence satisying (
5)
and a positive constant satisfying (
18)
. If the following is the case:then the class for (1). By picking up the above results, we are prepared to formulate the improvement of Theorem 1 provided that (
1) is a canonical form.
Theorem 4. Let (
2)
hold. Assume that there exist two sequences , satisfying (
4)
and (
5)
and there exist positive constants and for which (
14)
and (
18)
hold. If (
16)
and (
19)
are satisfied, then (
1)
is oscillatory. Note that if
, Theorem 4 reduces to Theorem 1. In the opposite case, the progress that Theorem 4 yields will be demonstrated by means of Equation (
12).
Example 3. We consider again the following differential equation. At first, we shall show that for . Thus, we set . Substituting again and , we observe that . Consequently, Condition (14) reduces to the following. Since is increasing on , we have the following. On the other hand, Criterion (
17)
takes the following form. Substituting simplifies the above term into the following. We used Matlab for evaluating (with ) the following. Finally, we conclude that the following is the case:which by Corollary 3 guarantees that . We obtain essentially better results for value of a than it has been presented in Example 1. We claim that for . To verify this, we let and and . Then, . Equation (18) implies the following. Consequently, we have the following. Condition (20) reduces to the following. To simplify the last integral, we use the substitution , and we obtain the following. By employing Matlab, we find out that for , the following is the case. Therefore, the following is the case:which by Corollary 4 implies that . Again we obtain better results than in Example 1. By combining both criteria, we bserve that condition implies oscillation of (12), while Theorem 1 requires . 5. Noncanonical Equations
Now, we turn our attention to noncanonical equation. Similarly as in the previous section, we introduce new monotonic properties of nonoscillatory solutions and then apply them to improve criteria concerning noncanonical equations. The progress will be demonstrated via Equation (
13).
Lemma 3. Let (
3)
hold. Assume that there exists a sequence satisfying (
4)
and a positive constant such that for , we have the following. If is a positive solution of (
1)
such that , then the following is the case. Proof. Assume that
for some
and
is a solution of (
1). Then, (
15) implies the following.
Consequently, we have the following:
on
and
, and the proof is complete. □
Theorem 5. Let (
3)
hold. Assume that there exists a sequence satisfying (
4)
and a positive constant such that (21) holds. If the following is the case:then the class for (1). Proof. Assume that (
1) has an eventually positive solution
. Let
for some
. Employing that
is decreasing on
and integrating (
1) from
u to
, one obtains the following.
By extracting the
root and integrating
to
, we obtain the following.
This is a contradiction, and the proof is complete now. □
Corollary 5. Let and (
3)
hold. Assume that there exists a sequence satisfying (
4)
and a positive constant such that (21) holds. If the following is the case:then the class for (1). Now, we turn our attention to the class . Since the proofs of the following results are very similar to those presented for canonical equations, they will be omitted.
Lemma 4. Let (
3)
hold. Assume that there exists a sequence satisfying (
5)
and a positive constant such that, for , we have the following. If is a positive solution of (
1)
such that , then the following is the case. Theorem 6. Let (
3)
hold. Assume that there exists a sequence satisfying (
5)
and a positive constant satisfying (24). If the following is the case:then the class for (1). Corollary 6. Let and (
3)
hold. Assume that there exists a sequence satisfying (
5)
and a positive constant such that (24) holds. If the following is the case:then the class for (1). Picking up the above results, we immediately obtain the following improvement of Theorem 1 for noncanonical (
1).
Theorem 7. Let (
3)
hold. Assume that there exist two sequences and satisfying (
4)
and (
5)
, and there exist positive constants and such that (
21)
and (
24)
hold. If (
22)
are (
25)
are satisfied, then (
1)
is oscillatory. The progress that Theorem 7 yields will be demonstrated via equation (
13).
Example 4. We consider again the following differential equation. At first, we shall show that for . Thus, we set . Substituting again and , we observe that . Condition (21) takes the following form.Taking the monotonicity of the above function into account, we see that the following is the case. Criterion (
17)
in terms of coefficients of (
13)
yields the following. By substituting , one can observe that the above inequality transforms into the following.We employ Matlab for evaluating the following (with ). Finally, we conclude that the following is the case:which by Corollary 5 guarantees that . It is useful to notice that we obtained essentially the better parameter a than in Example 2. We shall verify that for . To show this, we let and and . Then, , and it follows from (24) that the following is the case. Thus, we have the following. Condition (26) yields the following. The substitution results in the following. By employing Matlab, we find out that, for , the following is the case. Therefore, the following is the case:which by Corollary 6 implies that . Again, we obtain better results than in Example 1. By combining both criteria, we observe that condition implies oscillation of (13), while Theorem 1 requires .