Now we are ready to prove the main results of the paper. The function Q which appears in these criteria is a function defined by (11).
We will distinguish two cases: and . Let us start with the first case.
Theorem 1 Suppose that (7) and
(14)
hold. Further suppose that , and there exist positive mutually conjugate numbers l, and positive functions , such that
(15)
Then (1) is oscillatory.
Proof Suppose, by contradiction, that all of the assumptions of the theorem hold and there exists a solution of (1) and a number which satisfies
for every .
Condition (14) ensures that the corresponding function z is eventually increasing. In fact, from (1) we have
for . Hence is decreasing and either
for large t.
Suppose that there exists such that for . There exists a positive constant M such that
and
for . Integrating this inequality over the interval we get
Letting we have a negative upper bound for the function z and large t. However, the positivity of both and implies positivity of z. This contradiction proves that and eventually.
Consequently, we will work on the interval where is such that
for every .
Define
(16)
Clearly and
From and from the monotonicity of we have
and combining these computations we get
(17)
Further we define
(18)
we use the obvious fact and differentiate
Using the monotonicity of and we have
and hence
(19)
Multiplying (17) by , (19) by , adding the resulting inequalities and using (10), we get
Using the product rule for derivatives we obtain
and Lemma 1 implies
Integrating from to t
Multiplying by −1 and taking into account the fact that both and are nonnegative we get a finite upper bound for the integral from (15), which contradicts (15). □
Remark 1 Under the conditions , we can obtain [[3], Theorem 3.1] as a corollary of Theorem 1, since the inequality
holds.
The following corollary is in fact a variant of Theorem 1 if is bounded above by a nonnegative number and is bounded below by a positive number. Since (15) is not simply monotone with respect to and , we have to include the corresponding estimates in the opening part of the proof.
Corollary 1 Suppose that (7), (14), and are satisfied and there exist constants and such that and . If there exist positive mutually conjugate numbers l, , and positive functions , such that
(20)
then (1) is oscillatory.
Proof The proof is the same as the proof of Theorem 1, we just use (12) instead of (10) and in the remaining part of the proof we replace by and by . □
Example 1 For the Euler type equation (2) with we have , , , , , , . Denote and . With this setting we have and hence . Further , , and (20) becomes
and (2) is oscillatory if
(21)
Note that if , then this condition becomes
and since for we have , this oscillation constant is smaller than the oscillation constant from (3).
Further, if and , then (1) becomes (5). Condition (21) becomes
and, since is arbitrary, we get
which is well known to be an optimal and non-improvable oscillation constant for (5). In this sense we consider our result as reasonably sharp.
Finally, taking into account that , condition (21) becomes
A simple computation shows that the function
(22)
satisfies
and has a global minimum at . Thus the choice in (21) produces the smallest oscillation constant
Example 2 Baculíková et al. [[5], Example 2.1] considered the equation
(23)
with , , and . They proved that under the condition (23) is oscillatory if
(24)
(note that this condition is misprinted in [5]). This condition naturally produces poor oscillation constant if β is close to ω. In our notation we have , , , , , , . We choose and . Thus (20) takes the form
Taking into account that and that the function has a local minimum at the point , we find that (23) is oscillatory if
(25)
This condition completes condition (24). It is possible to find constants ω and β for which (25) is better than (24), as well as constants where the opposite is true. The fact that both estimates depend heavily on the parameters is illustrated by Figure 1.
The following corollary suggests another modification of the proof of Theorem 1: we replace condition (7) by weaker condition (8) and add conditions which ensure that x possesses the same type of monotonicity as z.
Corollary 2 Suppose that , (8), (14), and hold. If (20) holds for some mutually conjugate numbers l, and positive functions , , then every solution of (1) is either oscillatory, or the first derivative of this solution is oscillatory.
Proof Suppose, by contradiction, that the assumptions are satisfied and x is an eventually positive solution of (1) such that is not oscillatory.
We proceed as in Theorem 1 with modifications mentioned in the proof of Corollary 1. To ensure that Lemma 3 can be applied even though (7) need not to hold note that from the fact that z is eventually increasing, constant and not oscillatory we conclude easily that x is also eventually increasing. □
In the following example we show an application of Corollary 2 to the equation where .
Example 3 Consider the equation
with , , , , . We have , , , , , , for large t and . We choose and . With this setting the condition (20) takes the form
Using this computation and using the fact that the function takes global minimum on for and we see that the condition
guarantees that either every solution or derivative of every solution of the equation is oscillatory.
In the following theorem we drop the condition and use the opposite . In this case we modify the denominator in the Riccati type substitutions (16) and (18).
Theorem 2 Suppose that (7), (14), and hold. Further suppose that there exist positive mutually conjugate numbers l, and positive functions , such that
Then (1) is oscillatory.
Proof Suppose, by contradiction, that all the conditions are satisfied and an eventually positive solution of (1) exists. As in the proof of Theorem 1, we can show that is decreasing eventually and increasing eventually. Let us work on the interval where is such that
for every .
Define
As in the proof of Theorem 1, we have and
From and from the monotonicity of we have
and combining these computations we get
Further we define
differentiate
and conclude
Similarly as in the proof of Theorem 1 and using the fact that monotonicity of and inequality imply , we get
The remaining part of the proof is the same as in Theorem 1. □
Remark 2 Similarly as in Remark 1, [[3], Theorem 3.3] is a corollary of Theorem 2.
Corollary 3 Suppose that (7), (14), and hold. Furthermore, suppose that there exist constants and such that and . If there exist positive mutually conjugate numbers l, , and positive functions , such that
(26)
then (1) is oscillatory.
Proof The proof is he same as the proof of Corollary 1. We just use Theorem 2 instead of Theorem 1. □
Example 4 Consider (2) with . We choose the functions ρ and φ as in Example 1 and find that (1) is oscillatory if
(27)
Let us compare this result with (4). The inequalities , , and imply
where f is defined by (22) and is a global minimum of f on . Hence
and (27) is sharper than (4).
The following corollary is a variant of Corollary 2 for .
Corollary 4 Suppose that , (8), (14), and hold. If (26) holds for some mutually conjugate numbers l, and positive functions , , then every solution of (1) is either oscillatory, or the first derivative of this solution is oscillatory.
Proof The proof is the same as the proof of Corollary 2; we only replace Theorem 1 by Theorem 2 and Corollary 1 by Corollary 3. □
Remark 3 There are two main approaches how to handle Riccati type transformation in the oscillation theory of neutral differential equations. The first applies if and the shift in the differential term is handled by utilizing the estimate ; see e.g. [7–9]. Thus the results of this type depend on term . Another frequent approach which has been used in [3, 10] and also in this paper is summing up the equation at t and and working with the resulting sum. Since it is necessary to take out common factor, the oscillation criteria usually contain term . Since both and may differ significantly, we developed in this paper a method which replaces this term with the term , where the function is in some sense arbitrary and may have influence on the final oscillation criterion. We also showed on examples in previous section that this idea produces nonempty extension of known results. We conjecture that a similar idea can be used to obtain new results also in the case of a series of papers by Baculíková and Džurina [5, 11, 12], where a sum of two equations (in the original variable and in the shifted variable) is used to derive a certain first-order delay differential equation and the oscillation criteria are formulated in terms of this first-order equation. However, this idea exceeds the scope of this paper and will be examined in other research.