Abstract
We study the secondorder neutral delay halflinear differential equation
MSC: 34K11, 34K40.
Keywords:
halflinear differential equation; oscillation criteria; Riccati technique; delay equation; neutral equation; Euler type equation1 Introduction
In the paper we study the equation
where
By the solution of (1) we understand any differentiable function
The solution of (1) is said to be oscillatory if it has infinitely many zeros tending to infinity. Equation (1) is said to be oscillatory if all its solutions are oscillatory. In the opposite case, i.e., if there exists an eventually positive solution of (1), (1) is said to be nonoscillatory.
The neutral equations naturally arise in the mathematical models where the rate of
the growth depends not only on the current state and the state in the past, but also
on the rate of change in the past. The paper [1] suggests to imagine a child, which begins to grow more rapidly at the age of about
12 years, growing more and more rapidly until a certain height is approached, at which
time there is a rapid slowing of the growth, stopping at the adult height dictated
by genes. This process can easily be modeled by neutral equation. Similarly, the paper
[2] suggests one to use a logistic neutral differential equation to model a population
of Daphnia magna. If in the system modeled by firstorder neutral differential equation the mature
individuals produce some toxin which inhibits the rate the growth and if the production
of this toxin is constant per capita and unit time, then the rate of the growth is
inhibited by the term like
This research is motivated by the paper [3], where the main results of [3] are illustrated by an example of the equation
with
If
If
Note that if we put
which is well known to be an optimal oscillation constant for the Euler type equation
see [[4], Chapter 1.4.2]. The aim of this paper is to develop sharper results than those of
[3]. The main idea is to use the classical approach based on Riccati type inequality.
Due to the neutral nature of (1) we have to work with (1) and with the same equation
shifted from t to
The paper is organized as follows. In the following section we formulate inequalities
which are used to prove the main results. Section 3 contains main results of the paper
and examples which prove that we provide sharp oscillation constant for Euler type
differential equation. These criteria are expressed in terms of positive mutually
conjugate numbers l and
2 Preliminary results
First we derive some technical lemmas  inequalities which are necessary to reduce secondorder differential equation into a combination of two firstorder Riccati type equations. Note the changes against [3] and other related papers.
(i) We use convex linear combination (6) instead of arithmetic mean in Lemma 2. To
achieve this, we consider two positive mutually conjugate numbers l and
(ii) We relax the condition on the commutativity of the composition of σ and τ if x is an increasing function in Lemma 3.
(iii) We use new multiplicative factor
As far as we know, these ideas have never been used in the context of Riccati technique
even in the linear case
Throughout the paper
Lemma 1The following inequality holds for everyAand every
Proof The inequality is trivial if
Lemma 2The following inequality holds for
Proof The proof follows immediately from the convexity of
Lemma 3Suppose that either
or suppose thatxis an increasing function and
The inequality
holds for positive mutually conjugate numbersl,
Proof From the previous lemma using
Lemma 4Letxbe solution of (1). Suppose that either (7) holds or suppose thatxis an increasing function and (8) holds. The inequality
where
is valid for positive mutually conjugate numbersl,
whenever
Proof To obtain the second term from the definition of z we shift (1) from t to
where
Now we multiply (1) by
Now (10) follows from the definition of Q and from (9). Inequality (12) follows from (10) and from the fact that
3 Main results with applications to Euler type equation
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:
Theorem 1Suppose that (7) and
hold. Further suppose that
Then (1) is oscillatory.
Proof Suppose, by contradiction, that all of the assumptions of the theorem hold and there
exists a solution
for every
Condition (14) ensures that the corresponding function z is eventually increasing. In fact, from (1) we have
for
for large t.
Suppose that there exists
and
for
Letting
Consequently, we will work on the interval
for every
Define
Clearly
From
and combining these computations we get
Further we define
we use the obvious fact
Using the monotonicity of
and hence
Multiplying (17) by
Using the product rule for derivatives we obtain
and Lemma 1 implies
Integrating from
Multiplying by −1 and taking into account the fact that both
Remark 1 Under the conditions
holds.
The following corollary is in fact a variant of Theorem 1 if
Corollary 1Suppose that (7), (14),
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
Example 1 For the Euler type equation (2) with
and (2) is oscillatory if
Note that if
and since for
Further, if
and, since
which is well known to be an optimal and nonimprovable oscillation constant for (5). In this sense we consider our result as reasonably sharp.
Finally, taking into account that
A simple computation shows that the function
satisfies
and has a global minimum at
Example 2 Baculíková et al. [[5], Example 2.1] considered the equation
with
(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
Taking into account that
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.
Figure 1. Bounds for oscillation constantbfrom (23) for
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 2Suppose that
Proof Suppose, by contradiction, that the assumptions are satisfied and x is an eventually positive solution of (1) such that
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,
In the following example we show an application of Corollary 2 to the equation where
Example 3 Consider the equation
with
Using this computation and using the fact that the function
guarantees that either every solution or derivative of every solution of the equation is oscillatory.
In the following theorem we drop the condition
Theorem 2Suppose that (7), (14),
Then (1) is oscillatory.
Proof Suppose, by contradiction, that all the conditions are satisfied and an eventually
positive solution
for every
Define
As in the proof of Theorem 1, we have
From
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
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 3Suppose that (7), (14),
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
Let us compare this result with (4). The inequalities
where f is defined by (22) and
and (27) is sharper than (4).
The following corollary is a variant of Corollary 2 for
Corollary 4Suppose that
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
4 Conclusion
New oscillation theorems for secondorder halflinear differential equations have
been obtained. The novelty is in the point that we employed general linear combination
based on conjugate numbers l and
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The paper has been prepared by both authors. Both authors have made the same contribution and approved the final manuscript.
Acknowledgements
This research was supported by the Grant P201/10/1032 of the Czech Science Foundation.
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