Oscillation criteria for neutral second-order half-linear differential equations with applications to Euler type equations

where $\mathrm{\Phi }(t)={|t|}^{\alpha -1}t$ is the power type nonlinearity, $\alpha \ge 1$, which ensures that the function $\mathrm{\Phi }(\cdot )$ is a convex function on $(0,\mathrm{\infty })$. The coefficients r and p are subject of usual conditions $r\in C^1([t_0,\mathrm{\infty }),{\mathbb{R}}^{+})$, $p\in C^1([t_0,\mathrm{\infty }),{\mathbb{R}}_0^{+})$ and the coefficient q is positive $q\in C([t_0,\mathrm{\infty }),{\mathbb{R}}^{+})$. Further we suppose that the deviating arguments are unbounded and sufficiently smooth functions: $\tau \in C^2([t_0,\mathrm{\infty }),\mathbb{R})$, ${\tau }^{\prime }(t)>0$, ${lim}_{t\to \mathrm{\infty }}\tau (t)=\mathrm{\infty }$, $\sigma \in C^1([t_0,\mathrm{\infty }),\mathbb{R})$, ${\sigma }^{\prime }(t)>0$, ${lim}_{t\to \mathrm{\infty }}\sigma (t)=\mathrm{\infty }$ and the deviating arguments from the differential term and potential satisfy either $\sigma (\tau (t))=\tau (\sigma (t))$ or $\sigma (\tau (t))\ge \tau (\sigma (t))$ (in the latter case we use stronger condition on the coefficient $p(t)$ and the conclusion is weaker than in the commutative case).

By the solution of (1) we understand any differentiable function $x(t)$ which does not identically equal zero eventually, such that $r(t)\mathrm{\Phi }(z^{\prime }(t))$ is differentiable and (1) holds for large t.

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 first-order 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 ${\int }^{t}x(\sigma (s))\mathrm{d}s$ and we naturally obtain the second-order neutral differential equation.

with $0\le p(t)\le p_0<\mathrm{\infty }$, $\alpha \ge 1$, $\beta >0$, ${\lambda }_1\in (0,1)$.

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 $\tau (t)$. In contrast to the results of some other authors [3, 5, 6], we do not simply sum up the arising Riccati equations, but we develop an advanced technique based on suitable linear combination and careful comparison of $q(t)$ and $q(\tau (t))$. We show that an application of this technique allows to remove the above mentioned disadvantages of the paper [3] and allows to derive sharper results comparing the results published in the literature. Finally, we also discuss the case when the usual assumption $\sigma \circ \tau =\tau \circ \sigma$ is broken. This step opens applications to the neutral delay differential equations with proportional delay σ and constant delay τ.

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 $l^{\ast }$, the multiplicative factor φ, and a function $Q(t)$.

As far as we know, these ideas have never been used in the context of Riccati technique even in the linear case $\alpha =1$ and in the section with main results we show that these points are crucial points of the paper which allow to derive sharper results than the results published in the literature.

Throughout the paper $A_{+}$ denotes the positive part of A, i.e. $A_{+}=max\{A,0\}$.

Proof The inequality is trivial if $A\le 0$ and a special case of the Young inequality if $A>0$. □

Proof The proof follows immediately from the convexity of $x^{\alpha }$. □

holds for positive mutually conjugate numbers l, $l^{\ast }$, and every t which satisfies $x(\sigma (t))\ge 0$ and $x(\sigma (\tau (t)))\ge 0$.

Proof From the previous lemma using $a=lx(\sigma (t))$ and $b=l^{\ast }p(\sigma (t))x(\sigma (\tau (t)))$ and also from the definition of $z(t)$ and the fact that $x(\sigma (\tau (t)))\ge x(\tau (\sigma (t)))$. □

whenever $x(\sigma (t))$ and $x(\sigma (\tau (t)))$ are nonnegative and ${[r(\tau (t))\mathrm{\Phi }(z^{\prime }(\tau (t)))]}^{\prime }$ is negative.

where $z^{\prime }(\tau (t))=\frac{\mathrm{d}z(s)}{\mathrm{d}s}{|}_{s=\tau (t)}$.

Now (10) follows from the definition of Q and from (9). Inequality (12) follows from (10) and from the fact that ${[r(\tau (t))\mathrm{\Phi }(z^{\prime }(\tau (t)))]}^{\prime }$ is negative. □

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: $\sigma (t)\le \tau (t)$ and $\tau (t)\le \sigma (t)$. Let us start with the first case.

Then (1) is oscillatory.

for every $t>t_1$.

for large t.

Letting $t\to \mathrm{\infty }$ we have a negative upper bound for the function z and large t. However, the positivity of both $x(t)$ and $x(\tau (t))$ implies positivity of z. This contradiction proves that $\mathrm{\Phi }(z^{\prime }(t))>0$ and $z^{\prime }(t)>0$ eventually.

for every $t>t_2$.

Multiplying by −1 and taking into account the fact that both $\omega (t)$ and $\frac{p^{\alpha }(\sigma (t))\phi (t)}{{\tau }^{\prime }(t)}v(t)$ are nonnegative we get a finite upper bound for the integral from (15), which contradicts (15). □

holds.

The following corollary is in fact a variant of Theorem 1 if $p(t)$ is bounded above by a nonnegative number and ${\tau }^{\prime }(t)$ is bounded below by a positive number. Since (15) is not simply monotone with respect to $p(t)$ and $\tau (t)$, we have to include the corresponding estimates in the opening part of the proof.

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 $p(t)$ by $p_0$ and ${\tau }^{\prime }(t)$ by ${\tau }_0$. □

and since for ${\lambda }_1<1$ we have ${\lambda }_1^{\alpha }<1<\frac{1}{{\lambda }_1}$, this oscillation constant is smaller than the oscillation constant from (3).

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.

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 $p(t)\equiv p_0$, (8), (14), $\sigma (t)\le t$ and $\sigma (t)\le \tau (t)$ hold. If (20) holds for some mutually conjugate numbers l, $l^{\ast }$ and positive functions $\rho (t)$, $\phi (t)$, 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 $x^{\prime }(t)$ 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, $p(t)$ constant and $x(t)$ 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 $\sigma (\tau (t))\ne \tau (\sigma (t))$.

guarantees that either every solution or derivative of every solution of the equation is oscillatory.

In the following theorem we drop the condition $\sigma (t)\le \tau (t)$ and use the opposite $\sigma (t)\ge \tau (t)$. In this case we modify the denominator in the Riccati type substitutions (16) and (18).

Then (1) is oscillatory.

for every $t>t_2$.