\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 70, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2015/70\hfil Oscillation of solutions]
{Oscillation of solutions to fourth-order trinomial delay differential
equations}
\author[J. D\v{z}urina, B. Bacul\'ikov\'a, I. Jadlovsk\'a \hfil EJDE-2015/70\hfilneg]
{Jozef D\v{z}urina, Blanka Bacul\'ikov\'a, Irena Jadlovsk\'a}
\address{Department of Mathematics, Faculty of Electrical
Engineering and Informatics, Technical University of Ko\v{s}ice,
Letn\'a 9, 042 00 Ko\v{s}ice, Slovakia}
\email{jozef.dzurina@tuke.sk}
\email{blanka.baculikova@tuke.sk}
\email{irena.jadlovska@student.tuke.sk}
\thanks{Submitted February 5, 2015. Published March 24, 2015.}
\subjclass[2000]{34C10, 34K11}
\keywords{Fourth-order functional differential equation;
oscillation; \hfill\break\indent delay argument; comparison theorem}
\begin{abstract}
The objective of this article is to study the oscillation properties of
the solutions to the fourth-order linear trinomial delay differential
equation
\[
y^{(4)}(t)+p(t)y'(t)+q(t)y(\tau(t))=0.
\]
Applying suitable comparison principles, we present new criteria for oscillation.
In contrast with the existing results, we establish oscillation of all solutions,
and essentially simplify the examination process for oscillation.
An example is included to illustrate the importance of results obtained.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
We consider the trinomial fourth-order differential equation with delay argument
\begin{equation}\label{E}
y^{(4)}(t)+p(t)y'(t)+q(t)y(\tau(t))=0,
\end{equation}
under the assumptions
\begin{itemize}
\item[(H1)] $p(t), q(t)\in C([t_0,\infty))$, $p(t), q(t)$ are positive,
$ \tau(t) \in C^1([t_0,\infty))$, $\tau'(t)>0$, $\tau(t)\leq t$,
$\lim_{t\to\infty}\tau(t)=\infty$.
\end{itemize}
By a solution we mean a function $y(t) \in C^{4}([T_{y},\infty ))$,
$T_{y}\geq t_{0}$, which satisfies \eqref{E} on $[T_{y},\infty)$.
We consider only those solutions $y(t)$ of \eqref{E} which satisfy
$\sup \{|y(t)|:t \geq T\}>0$ for all $T\geq T_{y}$.
We assume that \eqref{E} possesses such a solution.
A solution of \eqref{E} is called oscillatory if it has arbitrarily large zeros
on $[T_{y},\infty)$ and otherwise it is called to be
nonoscillatory. Equation \eqref{E} is said to be oscillatory if all
its solutions are oscillatory.
The investigation of linear fourth-order differential equations,
firstly originated with the vibrating rod problem of mathematical physics
\cite{courant}, is generally of great practical importance.
Such equations form part of an immense collection of higher-order differential
equations and are encountered in various fields of science and engineering
as the more basic mathematical models. For instance, it is well known that
the problem of beam deflection in linear theory of elasticity is described
by classical linear fourth order equation
\[
y^{(4)}+q(t)y=0,
\]
where $y(t)$ approximates the shape of a beam, deflected from the equilibrium
due to some external forces. Another particularly interesting model of
physiological systems represented by fourth order differential equations
with time delay concerns the oscillatory movements of muscles which can
arise due to the interaction of a muscle with its inertial load \cite{stein}.
In view of the above, the study on oscillations of the fourth-order differential
equations has received considerable portion of attention and some profound
results have been obtained.
By establishing comparison theorems of Sturm's type, Leighton and
Nehari \cite{leighton}, Howard \cite{howard} studied extensively the
nature and behavior of solutions of a self-adjoin linear fourth-order
differential equations of the form
\[
(r(t)y'')''\pm q(t)y=0
\]
and their results obtained were fundamental in the next research.
Thereafter, the problem of obtaining sufficient conditions for oscillatory
and non-oscillatory properties of different classes of two-term fourth-order
differential equations, including, for instance, delay and neutral delay
dynamic equations on time scales, partial differential equations and difference
equations has been and still is receiving intensive attention. We refer the
reader to \cite{agarwal2009oscillation} - \cite{thandapani2007asymptotic}
and the references cited therein.
The problem of the oscillation of trinomial differential equations has been
widely studied by many authors who have provided various techniques especially
for lower order. The motivation for this work was twofold: a continuation of
the pioneering work of Hou and Chengmin \cite{hou} and on the other hand,
the thought of a missing analogy with investigation of third order trinomial
differential equations taking advantage of existing results, which will be
briefly stated.
A systematic study of asymptotic behavior of solutions of the third-order
differential equation with damping of the form
\[
y'''(t)+p(t)y'(t)+q(t)y(t)=0,
\]
has been made by \cite{hanan1961oscillation}, followed
by \cite{erbe1976existence, cecchi1999third,lazer1966behavior,jones1973asymptotic},
to mention just a few.
The articles \cite{aktacs2011qualitative, aktacs, Ti} deal with the
third order delay nonlinear differential equation of the form
\begin{equation}\label{eee}
(r_2(t)(r_1(t)y'(t))')'+p(t)y'(t)+q(t)f(y(\tau(t)))=0
\end{equation}
Using a generalized Riccati transformation and integral averaging technique,
authors establish some sufficient conditions which ensure that any solution of
oscillates or converges to zero. The another oscillation criteria have
been obtained by establishing a useful comparison principle with either first
or second order delay differential inequality, given in \cite{agarwal2009oscillation}.
The key assumption in the above papers is the existence of a positive solution
of the auxiliary second-order linear differential equation
\begin{equation}\label{ee}
(r_2(t)v'(t))'+\frac{p(t)}{r_1(t)}v(t)=0.
\end{equation}
Recently, the present authors have established in \cite{baculikova2010comparison}
new comparison theorems that reduce examination of the properties of the partial
case of equation \eqref{eee}; that is,
\begin{equation}\label{dz}
y'''(t)+p(t)y'(t)+q(t)y(\tau(t))=0
\end{equation}
to the study of the properties of associated first order delay differential equations. This is possible by rewriting equation into the binomial form
\[
\Big(v^2(t)\big(\frac{1}{v(t)}y'(t)\big)'\Big)'+v(t)q(t)y(\tau(t))=0,
\]
making use of the solution \eqref{ee}. It is worth to note that another approaches
exist, for example, in \cite{graef2013oscillation} it is generalized Lazer's
result \cite{lazer1966behavior} and established new criteria depending on the
sign of particular functional without requirement of an additional information
of related second-order equation. Contrary to most known results, we stress that
the technique used in the paper \cite{baculikova2010comparison} has established
oscillation of \emph{all} solutions.
The equation \eqref{dz} can be viewed either as the lowest possible prototype of
a higher-order trinomial differential equation
\begin{equation}\label{1}
y^{(n)}(t)+p(t)y^{(n-2)}(t)+q(t)y(\tau(t)) =0,
\end{equation}
or
\begin{equation}\label{2}
y^{(n)}(t)+p(t)y'(t)+q(t)y(\tau(t)) =0
\end{equation}
While for equations of the first type the same approach holds,
there is a few literature concerning the asymptotic and oscillatory properties
of equations of the second type.
The authors in \cite{hou} studied equation \eqref{E} by means of Riccati
transformation and presented conditions under which every nonoscillatory
solution tends to zero as $t\to \infty$. They also indicated an
interesting application in column-beam theory, where the middle term
is incorporated to control the slope of a beam. Their crucial theorem
ensures a constant sign first-derivative $y'(t)$ when an auxiliary third-order
differential equation
\begin{equation}\label{z}
z'''(t)+p(t)z(t)=0
\end{equation}
has increasing solution.
In this article, we are dealing with the oscillation and asymptotic behavior
of the solutions of the fourth-order delay trinomial differential
equation \eqref{E}.
Establishing oscillatory criteria for fourth-order trinomial differential
equations is far from easy, because the presence of the middle term $p(t)y'(t)$
causes the structure of possible nonoscillatory solutions to be unclear.
Our technique permits us to rewrite trinomial equations as binomial
differential equations with quasiderivative.
We offer a new approach, which uses a decreasing solution of an auxiliary
differential equation \eqref{z} (which always exists) and a positive solution
of related second-order differential equation (we provide condition under which
it exists) in order to obtain its associated binomial form. Furthermore,
by comparison with a couple of first-order delay differential equations,
we establish oscillation of all solutions of \eqref{E}.
\section{Preliminary results}
Before giving the main results, we will state Lemmas, which permit us
to rewrite the studied trinomial equation \eqref{E} into the binomial equation.
Consider the operator
$$
L_y=y^{(4)}(t)+p(t)y'(t).
$$
\begin{lemma}\label{L1}
Let $z(t)$ be a positive solution of
\begin{equation}\label{P1}
z'''(t)+p(t)z(t)=0.
\end{equation}
Then
\begin{equation}\label{Ly}
L_{y}=\Big[\frac{1}{z(t)}\Big(z^2(t)\big(\frac{y'(t)}{z(t)}\big)'\Big)'\Big]'
+z''(t)\big(\frac{y'(t)}{z(t)}\big)'.
\end{equation}
\end{lemma}
\begin{proof}
Simple computations show that the right hand side of \eqref{Ly} equals
\begin{equation}\label{1.1}
\begin{aligned}
&\big[\frac{1}{z(t)}\big(y''(t)z(t)-y'(t)z'(t)\big)'\big]'
+z''(t)\big(\frac{y'(t)}{z(t)}\big)' \\
&=\big[y'''(t)-\frac{y'(t)z''(t)}{z(t)}\big]'+z''(t)\big(\frac{y'(t)}{z(t)}\big)' \\
&=y^{(4)}(t)-y'(t)\frac{z'''(t)}{z(t)},
\end{aligned}
\end{equation}
which in view of \eqref{P1} leads to
$$
L_y=y^{(4)}(t)+p(t)y'(t)
= \Big[\frac{1}{z(t)}\Big(z^2(t)\big(\frac{y'(t)}{z(t)}\big)'\Big)'\Big]'
+z''(t)\big(\frac{y'(t)}{z(t)}\big)'.
$$
The proof is complete.
\end{proof}
\begin{remark}\label{R1} \rm
It follows from Chanturia and Kiguradze's result \cite{KCH}, that
\eqref{P1} always possesses a positive decreasing solution,
so-called Kneser solution.
\end{remark}
We recall the another result of Chanturia and Kiguradze \cite{KCH},
that will be useful in the sequel.
\begin{lemma}\label{non}
Assume that
\begin{equation}\label{P2crit}
\limsup_{t\to \infty}t^3 p(t)<\frac{2}{3\sqrt{3}},
\end{equation}
then all solutions of \eqref{P1} are non-oscillatory.
\end{lemma}
It is important to note that the above transformation does not reduce
\eqref{E} into the binomial form, as desired. However, it permits us
to decrement a difference in the derivative order between the first
and the second term of \eqref{E}. In other words, \eqref{E}, which is exactly
a case of \eqref{2}, turns out to be a more general version of the second
higher-order mentioned prototype \eqref{1}.
Now, consider an another operator
$$
M_{y}=\frac{1}{v(t)}\Big[\frac{v^{2}(t)}{z(t)}\Big(\frac{z^{2}(t)}{v(t)}
\big(\frac{1}{z(t)}y'(t)\big)'\Big)'\Big]'.
$$
\begin{lemma}\label{L2}
Let $z(t)$ be a positive decreasing solution of \eqref{P1} and let
the equation
\begin{equation} \label{P2}
(\frac{1}{z(t)}v'(t))'+\frac{z''(t)}{z^{2}(t)}v(t)=0
\end{equation}
possess a positive solution. Then
\begin{equation}\label{Lyy}
L_{y}=\frac{1}{v(t)}\Big[\frac{v^{2}(t)}{z(t)}
\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)'\Big)'\Big]'.
\end{equation}
\end{lemma}
\begin{proof}
We shall show, that operators $M_{y}$ and $L_{y}$ are equivalent.
It is easy to see that:
\begin{equation}\label{2.1}
\begin{split}
M_{y}
&=\frac{1}{v(t)}\Big[-\frac{v'(t)}{z(t)}z^{2}(t)
\big(\frac{1}{z(t)}y'(t)\big)'+v(t)\frac{1}{z(t)}\Big(z^{2}(t)
\big(\frac{1}{z(t)}y'(t)\big)'\Big)'\Big]' \\
&=\frac{1}{v(t)}\Big[-(\frac{v'(t)}{z(t)})'z^{2}(t)\big(\frac{1}{z(t)}y'(t)\big)'
-\frac{v'(t)}{z(t)}\Big(z^{2}(t)\big(\frac{1}{z(t)}y'(t)\big)'\Big)' \\
&\quad +v'(t)\frac{1}{z(t)}(z^{2}(t)\big(\frac{1}{z(t)}y'(t)\big)')'
+v(t)\Big(\frac{1}{z(t)}\Big(z^{2}(t)\big(\frac{1}{z(t)}y'(t)\big)'\Big)'\Big)'\Big] \\
&=\Big(\frac{1}{z(t)}\Big(z^{2}(t)\big(\frac{1}{z(t)}y'(t)\big)'\Big)'\Big)'
-\frac{z^{2}(t)}{v(t)}(\frac{v'(t)}{z(t)})'\big(\frac{1}{z(t)}y'(t)\big)'.
\end{split}
\end{equation}
Applying \eqref{Ly} from Lemma \ref{L1}, we obtain
\begin{align*}
M_{y}
&=y^{(4)}(t)+p(t)y'(t)-z''(t)\big(\frac{1}{z(t)}y'(t)\big)
'-\frac{z^{2}(t)}{v(t)}(\frac{v'(t)}{z(t)})'\big(\frac{1}{z(t)}y'(t)\big)\\
&=y^{(4)}(t)+p(t)y'(t)-\big(\frac{1}{z(t)}y'(t)\big)'\frac{z^{2}(t)}{v(t)}
\big[(\frac{1}{z(t)}v'(t))'+\frac{z''(t)}{z^{2}(t)}v(t)\big].
\end{align*}
Since $v(t)$ is a solution of \eqref{P2}, the previous equality yields
$$
M_{y}=y^{(4)}(t)+p(t)y'(t)=L_{y}.
$$
The proof is complete.
\end{proof}
Lemmas \ref{L1} and \ref{L3} permit us to rewrite studied trinomial equation \eqref{E}
in the binomial equation
\begin{equation}\label{Ec}
\Big[\frac{v^{2}(t)}{z(t)}\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)
\big)'\Big)'\Big]'+v(t)q(t)y(\tau(t))=0.
\end{equation}
As already stated, principal theorems in this paper will relate properties
of solutions of the fourth order delay differential equation \eqref{E}
to those of solutions of a couple of auxiliary linear ordinary differential
equations of the third and the second order. We need to explore conditions
that guarantee existence of positive solutions to the auxiliary equation \eqref{P2}.
Moreover, for our next purposes, it is desirable to have \eqref{Ec} in
a canonical form
\begin{gather}\label{can1}
\int_{t_0}^{\infty}\frac{z(t)}{v^2(t)}\,\mathrm{d}{t}=\infty, \\
\label{can2}
\int_{t_0}^{\infty}\frac{v(t)}{z^2(t)}\,\mathrm{d}{t}=\infty,\\
\label{H2}
\int_{t_0}^{\infty}z(t)\,\mathrm{d}{t}=\infty,
\end{gather}
since properties of canonical equations are generally nicely explored.
We will assume throughout the remainder of the paper that \eqref{H2} holds.
In the next result we crack the problem of the existence of positive solution
for \eqref{P2}.
\begin{lemma}\label{eqiuv}
Assume that all solutions of \eqref{P1} are non-oscillatory, then \eqref{P2}
possesses a positive solution.
\end{lemma}
\begin{proof}
It is clear that all solutions of \eqref{P2} are either oscillatory or
non-oscillatory. We admit that \eqref{P2} has an oscillatory solution $v(t)$.
Then $v(t)$ also satisfies
$$
-z'(t)v'(t)+z(t)v''(t)+z''(t)v(t)=0.
$$
Differentiating the last equality, one can see that
$$
z(t)v'''(t)+z'''(t)v(t)=0.
$$
But \eqref{P1} implies that $z'''(t)/z(t)=-p(t)$. Therefore, $v(t)$ is
the oscillatory solution of the differential equation
\begin{equation}\label{vt}
v'''(t)-p(t)v(t)=0.
\end{equation}
On the other hand, Chanturia and Kiguradze \cite{KCH} have shown that
all solutions of \eqref{P1} are nonoscillatory if and only if all
solutions of \eqref{vt} so does. This contradicts to oscillation of $v(t)$
and we conclude that all solutions of \eqref{P2} are non-oscillatory.
\end{proof}
Combining Lemma \ref{non} and \ref{eqiuv}, we get easily verifiable
criterion for \eqref{P2} to be non-oscillatory.
\begin{corollary}\label{C11}
Assume that \eqref{P2crit} hold. Then
\eqref{P2} possesses a positive solution.
\end{corollary}
Now, we show that under assumption \eqref{P2crit}, the conditions
\eqref{can1} and \eqref{can2} are always satisfied.
\begin{lemma}
Let \eqref{P2crit} hold. Then \eqref{P2} always has a solution
$v(t)$ such that \eqref{can1} and \eqref{can2} are satisfied.
\end{lemma}
\begin{proof}
The existence of a positive solution $v(t)$ of \eqref{P2} follows from
Corollary \ref{C11}. Moreover, the monotonicity properties of $v(t)$
and $z(t)$ implies that $0c_2>0$.
Thus \eqref{can2} is satisfied.
On the other hand,
If $v(t)$ does not satisfy \eqref{can1}; i.e.,
$$
\int^{\infty}\frac{z(s)}{v^{2}(s)}\,\mathrm{d} s < \infty,
$$
then it is easy to see that $v_*(t)$ given by
\begin{equation}\label{sol}
v_*(t) = v(t)\int_t^{\infty}\frac{z(s)}{v^2(s)}\,\mathrm{d} s
\end{equation}
satisfies
\begin{align*}
\big(\frac{1}{z(t)}v_* '(t)\big)'
&=\big(\frac{1}{z(t)}v'(t)\big)'\int_t^{\infty}\frac{z(s)}{v^2(s)}\,\mathrm{d} s \\
& = -\frac{z''(t)}{z^2(t)}v(t)\int_t^{\infty}\frac{z(s)}{v^2(s)}\,\mathrm{d} s
=- \frac{z''(t)}{z^2(t)}v_*(t) .
\end{align*}
Thus $v_*(t)$ is another positive solution of \eqref{P2}. Moreover,
$v_*(t)$ meets \eqref{can1} by now. To see this, let us denote
$$
\mathcal{V}(t) = \int_t^{\infty}\frac{z(s)}{v^2(s)}\,\mathrm{d} s,
$$
then $ \lim_{t\to \infty} \mathcal{V}(t) = 0$ and
\[
\int_{t_0}^{\infty} \frac{z(t)}{v_*^2(t)}\,\mathrm{d} t
= -\int_{t_0}^{\infty}\frac{\mathcal{V}'(t)}{ \mathcal{V}^2(t)}\,\mathrm{d} t
= \lim_{t\to \infty} (\frac{1}{ \mathcal{V}(t)} -\frac{1}{ \mathcal{V}(t_0)})
= \infty.
\]
\end{proof}
An immediate consequence of the above reasoning is the following result.
\begin{corollary}
Let \eqref{H2} hold. Assume that \eqref{P2crit} is fulfilled, then the
trinomial equation \eqref{E} can always be rewritten in its binomial
form \eqref{Ec} and what is more, \eqref{Ec} is in the canonical form.
\end{corollary}
\section{Oscillation of \eqref{E}}
Now, we are ready to study the properties of \eqref{E} with the help
of \eqref{Ec}. Without loss of generality, we can consider only with the
positive solutions of \eqref{Ec}.
The following result is a modification of Kiguradze's lemma \cite{KCH}.
\begin{lemma}\label{L3}
Let {\rm (H2)} hold. Assume that $y(t)$ is an eventually positive solution
of \eqref{Ec}, then
$\Big[\frac{v^{2}(t)}{z(t)}\Big(\frac{z^{2}(t)}{v(t)}
\big(\frac{1}{z(t)}y'(t)\big)'\Big)'\Big]'<0$ and, moreover, either
$$
y(t)\in \mathscr{N}_1 \iff y'(t)>0,\,\big(\frac{1}{z(t)}y'(t)\big)'<0,
\,\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)'\Big)'>0
$$
or
$$
y(t)\in \mathscr{N}_3 \iff y'(t)>0,\,\big(\frac{1}{z(t)}y'(t)\big)'>0,\,
\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)'\Big)'>0.
$$
\end{lemma}
Consequently, assuming (H2), the set $\mathscr{N}$ of all positive solutions
of \eqref{E} has the decomposition
$$
\mathscr{N}=\mathscr{N}_1\cup\mathscr{N}_3.
$$
To obtain oscillation of studied equation \eqref{E}, we need to eliminate
booth cases of possible non-oscillatory solutions.
Let us denote
$$
Q_1(t)=\Big(\frac{v(t)}{z^2(t)}\int_{t_1}^{\tau(t)}z(s)\,\mathrm{d}{s}\Big)
\Big(\int_t^{\infty}\frac{z(u)}{v^2(u)}
\int_{u}^{\infty}v(s)q(s)\,\mathrm{d}{s}\mathrm{d}{u}\Big)
$$
and
$$
Q_2(t)=v(t)q(t) \int_{t_1}^{\tau(t)}z(s_1)\int_{t_1}^{s_1}\frac{v(u)}{z^2(u)}
\int_{t_1}^{u}\frac{z(s)}{v^2(s)}\,\mathrm{d}{s}\mathrm{d}{u}\mathrm{d}{s_1}.
$$
\begin{theorem}\label{T1}
Let (H2) hold.
Assume that both first-order delay differential equations
\begin{equation}\label{E1}
x'(t)+Q_1(t)x(\tau(t))=0
\end{equation}
and
\begin{equation}\label{E2}
x'(t)+Q_2(t)x(\tau(t))=
0.
\end{equation}
are oscillatory. Then \eqref{E} is oscillatory.
\end{theorem}
\begin{proof}
Assume that $y(t)$ is an eventually positive solution of \eqref{E}.
Then $y(t)$ obeys also \eqref{Ec}.
It follows from Lemma \ref{L3} that either $y(t)\in \mathscr{N}_1$ or
$y(t)\in \mathscr{N}_3$. At first, we admit that
$y(t)\in \mathscr{N}_1$. Noting that $\frac{1}{z(t)}y'(t)$
is decreasing, we see that
\begin{equation}\label{1.1b}
y(t)\geq
\int_{t_{1}}^{t}z(u)\,\frac{y'(u)}{z(u)}\,du\geq \frac{y'(t)}{z(t)}
\int_{t_{1}}^{t}z(u)\,du.
\end{equation}
Integrating \eqref{Ec} from $t$ to $\infty$, we have
\begin{equation}\label{1.2}
\frac{v^{2}(t)}{z(t)}\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)
\big)'\Big)'\geq \int_t^{\infty} v(s)q(s)y(\tau(s))\,\mathrm{d}{s}.
\end{equation}
Taking into account that $y(\tau(t))$ is increasing, the last inequality yields
\begin{equation}\label{1.2b}
\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)
\big)'\Big)'
\geq y(\tau(t))\,\frac{z(t)}{v^{2}(t)}\ \int_t^{\infty} v(s)q(s)\,\mathrm{d}{s}.
\end{equation}
Integrating once more, we are led to
\begin{equation}\label{1.2c}
-\big(\frac{1}{z(t)}y'(t)
\big)'\geq y(\tau(t))\,\frac{v(t)}{z^{2}(t)}
\int_t^ {\infty}\frac{z(u)}{v^{2}(u)}
\int_{u}^{\infty} v(s)q(s)\,\mathrm{d}{s}\mathrm{d}{u}.
\end{equation}
Combining the last inequality with \eqref{1.1}, one gets
\begin{equation}
-\big(\frac{1}{z(t)}y'(t)\big)'\geq \frac{1}{z(\tau(t)) }y'(\tau(t))Q_1(t).
\end{equation}
Thus, the function $x(t)=\frac{y'(t)}{ z(t)}$ is positive a solution
of the differential inequality
\[
x'(t)+Q_1(t)x(\tau(t))\leq
0.
\]
Hence, by Philos theorem \cite{Phi}, we
conclude that the corresponding differential equation \eqref{E1}
also has a positive solution, which contradicts to assumptions of the theorem.
Now, we shall assume that $y(t)\in \mathscr{N}_3$.
Since $\frac{v^{2}(t)}{z(t)}(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)')'$
is decreasing, we are led to
\begin{align*} %\label{2.1}
\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)'
&\geq\int_{t_{1}}^{t} \frac{z(s)}{v^{2}(s)}
\frac{v^{2}(s)}{z(s)}\Big(\frac{z^{2}(s)}{v(s)}\big(\frac{1}{z(s)}y'(s)\big)'\Big)'\,du\\
&\geq \frac{v^{2}(t)}{z(t)}\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)
\big)'\Big)'\int_{t_{1}}^{t} \frac{z(s)}{v^{2}(s)} \,\mathrm{d}{s}.
\end{align*}
Integrating the above inequality, one can verify that
$$
y'(t)\geq z(t) \frac{v^{2}(t)}{z(t)}
\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)'\Big)'
\int_{t_{1}}^{t} \frac{v(u)}{z^{2}(u)}\int_{t_{1}}^{t} \frac{z(s)}{v^{2}(s)}
\,\mathrm{d}{s}\mathrm{d}{u}.
$$
Integrating once more, we see that
$x(t)=\frac{v^{2}(t)}{z(t)}
\Big(\frac{z^{2}(t)}{v(t)}\big(\frac{1}{z(t)}y'(t)\big)'\Big)'$
satisfies
$$
y(t)\geq x(t) \int_{t_{1}}^{t}z(s_1) \int_{t_{1}}^{s_1}
\frac{v(u)}{z^{2}(u)}\int_{t_{1}}^{t} \frac{z(s)}{v^{2}(s)}
\,\mathrm{d}{s}\mathrm{d}{u}\mathrm{d}{s_1}.
$$
Setting the last estimate into \eqref{Ec}, we see that $x(t)$ is a
positive solution of the differential inequality
$$
x'(t)+Q_2(t)x(\tau(t))\leq
0,
$$
which in view of Philos theorem in \cite{Phi} guarantees that
the corresponding differential equation \eqref{E2} has also a positive solution.
This is a contradiction and the proof is complete now.
\end{proof}
Applying suitable criteria for oscillation of
\eqref{E1}, \eqref{E2}, we obtain immediately criteria for oscillation of (E). The first one is due to Ladde et al. \cite{Lad}, while the second one pertains to Kusano and Kitamura \cite{KK}.
\begin{corollary}
Let (H2) hold. Assume that for $i=1,2$
\begin{equation}\label{C1}
\liminf_{t\to\infty}\int_{\tau(t)}^{t} Q_i(s)\mathrm{d}{s}>\frac{1}{\mathrm{e}}
\end{equation}
hold.
Then \eqref{E} is oscillatory.
\end{corollary}
\begin{corollary}\label{yyy}
Let {\rm (H2)} hold. Assume that for $i=1,2$
\begin{equation} \label{C2}
\liminf_{t\to\infty}\int_t^{\infty} Q_i(s)\mathrm{d}{s}=\infty
\end{equation}
hold. Then \eqref{E} is oscillatory.
\end{corollary}
\begin{example}\label{Ex1} \rm
Let us consider the fourth order delay differential equation
\begin{equation}\label{Ex}
y^{(4)}(t)+\frac{0.231}{t^3}\,y'(t)+\frac{a}{t^4}\,y(\lambda t)=0,\quad a>0,
\quad\lambda\in(0,1),\quad t\geq1.
\end{equation}
For considered equation \eqref{Ex}, the auxiliary equation \eqref{P1}
takes the form
\[
z'''(t)+\frac{0.231}{t^3}\,z(t)=0
\]
with positive solution $z(t)=t^{-0.1}$.
On the other hand, \eqref{P2} reduces to
\[
(t^{0.1}\,v'(t))'+\frac{0.11}{t^{1.9}}\,v(t)=0
\]
which possesses the positive solution $v(t)=t^{\alpha}$, where
$\alpha=\frac{0.9-\sqrt{0.37}}{2}$. It is easy to verify that (H2) holds.
Moreover, simple computation shows that
$$
Q_1(t)=\frac{a\lambda^{0.9}}{0.9(3-\alpha)(2.1+\alpha)}
\frac{1}{t}-\frac{K}{t^{1.9}},\quad K\in\mathbb{R}.
$$
and
$$
Q_2(t)=\frac{a\lambda^{3-\lambda}}{(3-\alpha)(2.1-\alpha)(0.9-2\alpha)}
\frac{1}{t}-\frac{K_1}{t^{1.9-2\alpha}}
-\frac{K_2}{t^{3.1-\alpha}} -\frac{K_2}{t^{4-\alpha}},\quad K_i\in\mathbb{R}
$$
Criteria \eqref{C1} and \eqref{C2} from Corollary \ref{yyy} yield
\begin{equation}\label{ex1}
a\lambda^{0.9}\ln\frac{1}{\lambda}>\frac{0.9(3-\alpha)(2.1+\alpha)}{\mathrm{e}}
\end{equation}
and
\begin{equation}\label{ex2}
a\lambda^{3-\alpha}\ln\frac{1}{\lambda}
>\frac{(3-\alpha)(2.1-\alpha)(0.9-2\alpha)}{\mathrm{e}}
\end{equation}
respectively. By Corollary \ref{yyy}, we conclude that \eqref{Ex}
is oscillatory if \eqref{ex1} and \eqref{ex2} hold simultaneously.
For e.g. $\lambda=0.6$ it happens provided that $a>10.4991$.
We note that criteria from \cite{hou}, \cite{HL} does not provide
any information about oscillation of\eqref{E}.
\end{example}
In this article we have established a new approach for studying the oscillation
of the fourth order trinomial delay differential equation.
One of the key elements in the method is to study the associated binomial
representation \eqref{Ec} of equation \eqref{E}. Our technique is based on
the existence of positive solutions of a couple of auxiliary differential
equations of the second and third order. Furthermore, we establish some basic
properties of related linear operator to ensure the canonical form of \eqref{Ec}.
As consequence, employing some comparison principles, one can easily deduce
oscillation of all solutions of studied equation. The presented technique is
new and essentially simplifies investigation of oscillation of fourth-order
trinomial differential equations.
\subsection*{Acknowledgements}
This work was supported by Slovak Research and Development Agency under
contracts No. APVV-0404-12 and APVV-0008-10.
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