\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 193, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/193\hfil Third-order differential equations]
{Qualitative properties of a third-order differential equation with a
piecewise \\ constant argument}
\author[H. Bereketoglu, M. Lafci, G. S. Oztepe \hfil EJDE-2017/193\hfilneg]
{Huseyin Bereketoglu, Mehtap Lafci, Gizem S. Oztepe}
\address{Huseyin Bereketoglu \newline
Department of Mathematics,
Faculty of Sciences,
Ankara University,
06100, Ankara, Turkey}
\email{bereket@science.ankara.edu.tr}
\address{Mehtap Lafci \newline
Department of Mathematics,
Faculty of Sciences,
Ankara University,
06100, Ankara, Turkey}
\email{mlafci@ankara.edu.tr}
\address{Gizem S. Oztepe (corresponding author) \newline
Department of Mathematics,
Faculty of Sciences,
Ankara University,
06100, Ankara, Turkey}
\email{gseyhan@ankara.edu.tr}
\thanks{Submitted July 7, 2016. Published August 4, 2017.}
\subjclass[2010]{34K11}
\keywords{Third order differential equation;
piecewise constant argument;
\hfill\break\indent oscillation; periodicity}
\begin{abstract}
We consider a third order differential equation with piecewise constant
argument and investigate oscillation, nonoscillation and periodicity
properties of its solutions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
For many years, oscillation, non-oscillation and periodicity of third order
differential equations have been investigated. Kim \cite{kim} studied
oscillation properties of the equation
$$
y'''+py''+qy'+ry=0,
$$
where $p,q$ and $r$ are continuous on an interval.
Tryhuk \cite{tryhuk} established sufficient conditions for the existence
of two linearly independent oscillatory solutions of the third order differential
equation
$$
y'''+p(t)y'+q(t)y=0.
$$
Cecchi \cite{cechi} investigated the oscillatory behavior of the linear third-order
differential equation of the form
$$
y'''+p(x)y'+q(x)y=0,
$$
where the function $p(x)$ changes sign on the positive $x$-axis.
Parhi and Das \cite{parhi1} considered the equation
$$
(r(t)y'')'+q(t)y'+p(t)y=F(t),
$$
and gave necessary and sufficient conditions for the existence of nonoscillatory
or oscillatory solutions of this equation. In \cite{parhi},
they also investigated oscillatory and asymptotic properties of solutions
of the equation
$$
y'''+a(t)y''+b(t)y'+c(t)y=0,
$$
where $a\in C^2$, $b\in C^1$, $c\in C^0$, $a(t), b(t), c(t)\leq0 $
eventually and $b(t)\neq0,c(t)\neq0$ on any
interval of positive measure. The oscillation of the solutions of
$$
(b(t)(a(t) y'(t))')'
+(q_{1}(t)y(t))' +q_{2}(t)y'(t)=0
$$
and
$$
(b(t)(a(t) y'(t))')' +q_{1}(t)y(t) +q_{2}(t))y(\tau(t))=0
$$
was studied in \cite{dahiya} by Dahiya. Adamets and Lomtatidze \cite{adam}
analyzed oscillatory properties of solutions of the third-order
differential equation
$u'''+p(t)u=0$, where $p$ is a locally integrable function on
$[0,\infty )$ which is eventually of one sign. Han, Sun
and Zhang \cite{han} deduced new sufficient conditions which guarantee that every
solution $x$ of the delayed third order differential equation
$$
(x(t)-a(t)x(\tau(t))) '''+p(t)x(\delta(t))=0
$$
is either oscillatory or tends to zero. In \cite{das}, the authors
stated necessary and sufficient conditions for the oscillation of the third-order
nonhomogenous differential equation
$$
y'''+a(t)y''+b(t)y'+c(t)y=f(t),
$$
under certain conditions given in terms of differentiability,
continuity and signs of the
coefficient functions and their derivatives. \cite{bacu} was dedicated to
studying the nonoscillatory solutions of the equation with mixed arguments
$$
(a(t)(x'(t))^\gamma)'' =q(t)f(x(\tau(t)))+p(t)g(x(\sigma(t))),
$$
where $\tau(t)t$.
In 2015, Bartu\v{s}ek and Do\v{s}l\'{a} \cite{bartu} gave conditions under
which every solution of the equation
\begin{equation*}
x'''(t) +q(t) x'(t) +r(t) | x| ^{\lambda }(t) \operatorname{sgn}x(t) =0,\quad t\geq 0,
\end{equation*}
is either oscillatory or tends to zero.
They also studied Kneser solutions vanishing at infinity and the existence of
oscillatory solutions. Shoukaku \cite{shou} considered
$$
y'''(t)-a(t)y''(t)-b(t)y'(t)-\sum_{i=1}^{m}c_{i}(t)y(\sigma_{i}(t))=0
$$
using Riccati inequality.
Ezeilo \cite {eze} studied the equation
$$
x'''+a x''+bx'+h(x)=p(t),
$$
where $a, b$ are constants, $p(t)$ is continuous and periodic with least
period $w$. Using the Leray-Schauder technique, under certain conditions on
$a, b, h, p$, the author guaranteed the existence of one solution of this
equation with least period $w$.
Tabueva In \cite{tabue} studied the existence of a periodic solution of
$$
x'''+\alpha x''+\beta x'+ \sin x =e(t).
$$
Ezeilo \cite{eze1} showed that the equation
$$
x'''+\psi(x')x''+\phi(x)x'+\theta(x)=p(t)+q(t,x,x')
$$
has an $w$-periodic solution, where $\psi, \phi, \theta, p$ and $q $ are
continuous in their respective arguments and $p, q$ have a given period
$w$, $w>0$, in $t$.
In 1979, Tejumola \cite{teju} proved the existence of at least one $w$ periodic
solution of the third order differential equation
$$
x'''+f(x')x''+g(x)x'+h(x)=p(t,x,x',x'' ),
$$
where $p$ is $w$-periodic in its first argument.
In \cite{wang}, the author gave a theorem on the existence of $2\pi$-periodic
solutions of the nonlinear third order differential equation with multiple
deviating arguments
\begin{equation*}
c(t)x'''(t)+\sum_{i=0}^2[a_{i}(x^{(i)})^{2k-1}+b_{i}(x^{(i)})^{2k-1}(t-\tau_{i})]
+g(t,x(t-\tau(t)),x'(t-\tau_{3}) )=p(t),
\end{equation*}
where $a_{i}, b_{i} (i=0,1,2)$ and $\tau_{i} (i=0,1,2,3)$ are constants, $k$
is a positive integer.
Chen and Pan \cite {chen} proved sufficient conditions for the existence of
periodic solutions of third order differential equations with deviating arguments
of the type
$$
x'''(t)+ax''(t-\tau_{2}(t))+bx'(t-\tau_{1}(t))+cx(t)+f(t,x(t-\tau(t))=p(t).
$$
As far as we know, there are some papers on the third-order differential equations
with piecewise constant arguments. The oldest one was published in 1994
by Papaschinopoulos and Schinas \cite{3rdpiecegary94}.
They considered the equation
\begin{equation*}
(y(t)+py(t-1))''' =-qy(2 [\frac{t+1}{2}] )
\end{equation*}
and proved existence, uniqueness and asymptotic stability of the solutions.
Here $t\in [ 0,\infty ) $, $p,q$ are real constants and $[\cdot] $ denotes
the greatest integer function. Liang and Wang \cite{3rdpieceliang2012}
stated several sufficient conditions which insure that any solution of the equation
\begin{equation*}
(r_{2}(t)(r_{1}(t)x'(t))')'+p(t)x'(t)+f(t,x([ t] ) )=0,\quad
t\geq 0
\end{equation*}
oscillates or converges to zero. Shao and Liang \cite{3rdpieceshao2013}
established sufficient conditions for the oscillation and asymptotic behaviour
of the equation
\begin{equation*}
(r(t) x''(t) ) '+f(t,x([ t] )) =0.
\end{equation*}
In \cite{bereket2017}, the authors showed that every solution $x(t)$ of
a third-order nonlinear differential equation with piecewise constant
arguments of the type
\begin{equation*}
(r_{2}(t)(r_{1}(t)x'(t))')'+p(t)x'(t)+f(t,x([ t-1] ) )+g(t,x([ t] ) )=0
\end{equation*}
oscillates or converges to zero, where $t\geq 0,\ r_{1}(t)$, $r_{2}(t)$
are continuous on $[0,\infty ) $ with $r_{1}(t)$, $r_{2}(t)>0$ and
$r_{1}'(t)\geq 0$,
$p(t)$ is continuously differentiable on $[ 0,\infty ) $ with $p(t)\geq 0$.
On the other hand, the first and third authors considered an impulsive
first order delay differential equation with piecewise constant argument
in \cite{fatma}. They investigated its oscillatory and periodic solutions.
Then in \cite{bereket2010}, the same authors studied the oscillation,
nonoscillation, periodicity and global asymptotic stability of an advanced
type impulsive first order nonhomogeneous differential equation with piecewise
constant arguments. Also, in 2011, oscillation, nonoscillation and
periodicity of a second order
\begin{equation*}
x''(t)-a^{2} x(t)=bx([t-1])+cx([t]+dx([t+1]
\end{equation*}
differential equation with mixed type piecewise constant arguments were
investigated \cite{bereket2011}.
In this paper, we extend our results on oscillation, nonoscillation and periodicity
of solutions to first and second order linear differential
equations with piecewise constant arguments to a third order linear differential
equation with piecewise constant argument. For this purpose,
we consider the following third order linear
differential equation with a piecewise constant argument
\begin{equation}
x'''(t)-a^{2} x'(t)=bx([t-1]) \label{1}
\end{equation}
with the initial conditions
\begin{equation}
x(-1) =\alpha_{-1}, x(0) =\alpha_{0}, x'(0) =\alpha_{1},
x''(0) =\alpha_{2}, \label{2}
\end{equation}
where $a\neq 0$ and $a, b, \alpha_{-1}, \alpha_{0}, \alpha_{1},
\alpha_{2} \in \mathbb{R}$.
\section{Existence and uniqueness}
First we give the definition of a solution to \eqref{1}.
Then we use the technique in \cite{wiener} to investigate the solution
of this equation.
A function $x(t) $ defined on $[ 0,\infty ) $ is said
to be a solution of the initial value problem \eqref{1}--\eqref{2}
if it satisfies the following conditions:
\begin{itemize}
\item[(i)] $x$ is continuous on $[0,\infty )$,
\item[(ii)] $x''$ exists and continuous on $[0,\infty )$,
\item[(iii)] $x'''$ exists on $[0,\infty )$ with the possible exception
of the points $[t ]\in [0,\infty )$, where one-sided derivatives exist,
\item[(iv)] $x$ satisfies \eqref{1} on each interval $[n, n+1 )$ with $n\in N$.
\end{itemize}
\begin{theorem} \label{thm1}
Equation \eqref{1} has a solution on $[ 0,\infty ) $.
\end{theorem}
\begin{proof}
Let $x_{n}(t)$ be a solution of \eqref{1} on the interval $[n,n+1 )$
with the conditions
\begin{equation*}
x(n)= c_{n},\ x(n-1)= c_{n-1}, \ x'(n)=d_{n},\ x''(n)=e_{n}.
\end{equation*}
Then \eqref{1} reduces to
\begin{equation*}
x'''(t)-a^{2} x'(t)=bx(n-1). \label{3}
\end{equation*}
The solution of the above equation is found as
\begin{equation}
x_{n}(t)=K_{n}+L_{n}\cosh a(t-n)+M_{n}\sinh a(t-n)
-\frac{b}{a^{2}}tx_{n}(n-1) \label{4}
\end{equation}
with arbitrary constants $K_{n},\ L_{n}$ and $M_{n}$.
Writing $t=n$ in \eqref{4}, we obtain
\begin{equation}
c_{n}=K_{n}+L_{n}-\frac{b}{a^{2}}nc_{n-1}.\label{5}
\end{equation}
If we take $t=n$ in the first and second derivatives of \eqref{4},
respectively, we find
\begin{equation}
M_{n}=\frac{d_{n}}{a}+\frac{b}{a^{3}}c_{n-1}, \quad
L_{n}=\frac{e_{n}}{a^{2}} \label{7}.
\end{equation}
From \eqref{5} and \eqref{7},
\begin{equation}
K_{n}=c_{n}-\frac{e_{n}}{a^{2}}+\frac{b}{a^{2}}nc_{n-1} \label{8}
\end{equation}
is obtained. Substituting \eqref{7} and \eqref{8} in \eqref{4}, we have
\begin{equation}
\begin{aligned}
x_{n}(t)
&=\frac{-1+\cosh a(t-n)}{a^{2}}e_{n}+\frac{\sinh a(t-n)}{a}d_{n}+c_{n}
\\
&\quad +[ \frac{b}{a^{2}}n-\frac{b}{a^{2}}t
+\frac{b}{a^{3}}\sinh a(t-n)] c_{n-1}.\label{8'}
\end{aligned}
\end{equation}
First and second derivatives of \eqref{8'} are found as
\begin{gather}
x'_{n}(t)=\frac{\sinh a(t-n)}{a}e_{n}+\cosh a(t-n)d_{n}
+[ \frac{b}{a^{2}}\cosh a(t-n)-\frac{b}{a^{2}}] c_{n-1}\label{8''},\\
x''_{n}(t)=\cosh a(t-n)e_{n}+a\sinh a(t-n)d_{n}
+ \frac{b}{a}\sinh a(t-n)c_{n-1}\label{8'''}.
\end{gather}
Writing $t=n+1$ in \eqref{8'}, \eqref{8''} and \eqref{8'''}, it follows that
\begin{gather}
c_{n+1}=c_{n}+\frac{\sinh a}{a}d_{n}
+\big(\frac{\cosh a}{a^{2}}-\frac{1}{a^{2}}\big) e_{n}
+\big(\frac{b\sinh a}{a^{3}}-\frac{b}{a^{2}}\big)
c_{n-1}\label{12},\\
d_{n+1}= (\cosh a) d_{n}+\frac{\sinh a}{a}e_{n}
+(\Big(\frac{b\cosh a}{a^{2}}-
\frac{b}{a^{2}}\Big) c_{n-1}\label{13},\\
e_{n+1}=a(\sinh a) d_{n}+(\cosh a) e_{n}+\frac{b\sinh a}{a}c_{n-1}\label{14}.
\end{gather}
Now, let us introduce the vector
$v_{n}=\operatorname{col}(c_{n},d_{n},e_{n})$ and the matrices
\begin{equation*}
A=\begin{pmatrix}
1 & \frac{\sinh a}{a} & \frac{\cosh a}{a^{2}}-\frac{1}{a^{2}} \\
0 & \cosh a & \frac{\sinh a}{a} \\
0 & a\sinh a & \cosh a
\end{pmatrix},\quad
B=\begin{pmatrix}
\frac{b\sinh a}{a^{3}}-\frac{b}{a^{2}} & 0 & 0 \\
\frac{b\cosh a}{a^{2}}-\frac{b}{a^{2}} & 0 & 0 \\
\frac{b\sinh a}{a} & 0 & 0
\end{pmatrix}
\end{equation*}
so we can rewrite the system \eqref{12}-\eqref{14} as
\begin{equation}
v_{n+1}=Av_{n}+Bv_{n-1}.\label{15}
\end{equation}
Looking for a nonzero solution of this difference equation system
in the form of $v_{n}=k\lambda^{n}$, with a constant vector k, leads us to
\begin{equation*}
\det (\lambda ^{2}I-\lambda A-B)=0,
\end{equation*}
and characteristic equation
\begin{equation}
\begin{aligned}
&\lambda^{4}+(-1-2 \cosh a)\lambda^{3}+(1+\frac{b}{a^{2}}-
\frac{b}{a^{3}} \sinh a+2\cosh a )\lambda^{2} \\
&+(-1-\frac{2b}{a^{2}}\cosh a+\frac{2b}{a^{3}}\sinh a )\lambda+(\frac{b}{a^{2}}-
\frac{b}{a^{3}}\sinh a)=0. \label{16}
\end{aligned}
\end{equation}
Assuming that these roots are simple, we write the general solution of \eqref{16},
\begin{equation}
v_{n}=\lambda_{1}^{n}k_{1}+\lambda_{2}^{n}k_{2}+\lambda_{3}^{n}k_{3}
+\lambda_{4}^{n}k_{4}, \label{17}
\end{equation}
where $v_{n}=col(c_{n},d_{n},e_{n})$ and $k_{j}=col(k_{ij})$, $i=1,2,3,4$
which can be found from adequate initial or boundary conditions.
If some $\lambda_{j}$ is a multiple zero of \eqref{16}, then the expression
for $v_{n}$ also includes products of $\lambda_{j}^{n}$ by $n$, $n^{2}$ or $n^{3}$.
Finally, the solution $x_{n}(t)$ is obtained by substituting the appropriate
components of the vectors $v_{n}$ and $v_{n-1}$ in \eqref{8'}.
\end{proof}
\begin{remark} \rm
From \eqref{12}, \eqref{13} and \eqref{14}, we obtain
\begin{gather}
\begin{aligned}
d_{n}
&=\frac{-a}{2\sinh a}c_{n+2}+\frac{a(1+\cosh a ) }{\sinh a}c_{n+1} \\
&\quad +\frac{(-a^{3}-2a^{3}\cosh a-ab+b\sinh a ) }{2a^{2}\sinh a}c_{n}\\
&\quad +\frac{ab+2ab\cosh a-3b\sinh a}{2a^{2}\sinh a}c_{n-1},
\end{aligned} \label{18} \\
\begin{aligned}
e_{n}
&=\frac{-a^{2}}{2(1-\cosh a) }c_{n+2}+\frac{a^{2}\cosh a}{ 1-\cosh a }c_{n+1} \\
&\quad -\frac{(-a^{3}+2a^{3}\cosh a+ab-b\sinh a ) }{2a(1-\cosh a ) }c_{n} \\
&\quad -\frac{ab-2ab\cosh a+b\sinh a}{2a(1-\cosh a ) }c_{n-1}.
\end{aligned} \label{19}
\end{gather}
Substituting \eqref{18} and \eqref{19} in \eqref{12}, gives us the
difference equation
\begin{equation}
\begin{aligned}
&c_{n+3}+(-1-2\cosh a)c_{n+2}+(1+\frac{b}{a^{2}}-
\frac{b}{a^{3}}\sinh a+2\cosh a )c_{n+1}\\
&+(-1-\frac{2b}{a^{2}}\cosh a+\frac{2b}{a^{3}}\sinh a )c_{n}+(\frac{b}{a^{2}}-
\frac{b}{a^{3}}\sinh a)c_{n-1}=0 \label{19*}
\end{aligned}
\end{equation}
whose characteristic equation is the same as \eqref{16}.
\end{remark}
\begin{theorem} \label{thm2}
The boundary-value problem for \eqref{1} with the conditions
\begin{equation}
x(-1)=c_{-1}, x(0)=c_{0}, x(1)=c_{1}, x(N-1)=c_{N-1} \label{20}
\end{equation}
has a unique solution on $ 0\leq t<\infty$ if $N>2$ is an integer and both of the following hypotheses are satisfied:
\begin{itemize}
\item[(i)] The roots of \eqref{16} $\lambda_{j}$ (characteristic roots)
are nontrivial and distinct,
\item[(ii)] $\lambda_{1}^{N}\lambda_{2}\lambda_{3}\lambda_{4}
(-\lambda_{4}^{2}\lambda_{1}\lambda_{2} +\lambda_{4}\lambda_{1}\lambda_{2}^{2}
+\lambda_{4}^{2}\lambda_{1}\lambda_{3} -\lambda_{1}\lambda_{2}^{2}\lambda_{3}
-\lambda_{4}\lambda_{1}\lambda_{3}^{2}+\lambda_{1}\lambda_{2}\lambda_{3}^{2})\\
+\lambda_{1}\lambda_{2}\lambda_{3}^{N}\lambda_{4}(-\lambda_{4}^{2}
\lambda_{1}\lambda_{3} +\lambda_{4}\lambda_{1}^{2}\lambda_{3}
+\lambda_{4}^{2}\lambda_{2}\lambda_{3} -\lambda_{1}^{2}\lambda_{2}\lambda_{3}
-\lambda_{4}\lambda_{2}^{2}\lambda_{3}+\lambda_{1}\lambda_{2}^{2}\lambda_{3})\\
\neq \lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}^{N}
(-\lambda_{4}\lambda_{1}^{2}\lambda_{2} +\lambda_{4}\lambda_{1}\lambda_{2}^{2}
+\lambda_{4}\lambda_{1}^{2}\lambda_{3}
-\lambda_{4}\lambda_{2}^{2}\lambda_{3}-\lambda_{4}\lambda_{1}\lambda_{3}^{2}
+\lambda_{4}\lambda_{2}\lambda_{3}^{2})\\
+\lambda_{1}\lambda_{2}^{N}\lambda_{3}
\lambda_{4}(-\lambda_{4}^{2}\lambda_{1}\lambda_{2}
+\lambda_{4}\lambda_{1}^{2}\lambda_{3}
+\lambda_{4}^{2}\lambda_{2}\lambda_{3} -\lambda_{1}^{2}\lambda_{2}\lambda_{3}
-\lambda_{4}\lambda_{2}\lambda_{3}^{2}+\lambda_{1}\lambda_{2}\lambda_{3}^{2})$.
\end{itemize}
\end{theorem}
\begin{proof}
The first row of the vector equation \eqref{17} gives us
\begin{equation}
c_{n}=\lambda_{1}^{n}k_{11}+\lambda_{2}^{n}k_{21}+\lambda_{3}^{n}k_{31}
+\lambda_{4}^{n}k_{41}. \label{21}
\end{equation}
We get following system by applying the boundary conditions \eqref{20}
to \eqref{21}, respectively.
\begin{gather}
\lambda_{1}^{-1}k_{11}+\lambda_{2}^{-1}k_{21}+\lambda_{3}^{-1}k_{31}
+\lambda_{4}^{-1}k_{41}=c_{-1} \label{22},\\
k_{11}+k_{21}+k_{31}+k_{41}=c_{0} \label{23},\\
\lambda_{1}k_{11}+\lambda_{2}k_{21}+\lambda_{3}k_{31}+\lambda_{4}k_{41}
=c_{1} \label{24},\\
\lambda_{1}^{N-1}k_{11}+\lambda_{2}^{N-1}k_{21}+\lambda_{3}^{N-1}k_{31}
+\lambda_{4}^{N-1}k_{41}=c_{N-1}. \label{25}
\end{gather}
From hypothesis (ii), the determination of the coefficients of this system
is different from zero. Hence, we can find $k_{ij}$ and also $c_{n}$ uniquely.
Furthermore, once the values $c_{n}$ have been found, we calculate $d_{n}$
and $e_{n}$ from \eqref{18} and \eqref{19}, respectively. Substituting $c_{n}$,
$d_{n}$, $e_{n}$ in \eqref{8'}, the unique solution $x_{n}(t)$ is obtained.
\end{proof}
The following four theorems depend on the characteristic roots. Their proofs are
omitted because they are very similar to the proof of Theorem \ref{thm2}.
\begin{theorem} \label{thm3}
Let us assume that all characteristic roots are nontrivial and two of them
are equal $(\lambda_{1}=\lambda_{2}) $, others are different from each other
$(\lambda_{3}\neq\lambda_{4}) $. If
\begin{align*}
&\lambda_{1}^{N}\Bigl[(1-N)\lambda_{1}\lambda_{3}\lambda_{4}^{3}
+(N-2)\lambda_{1}^{2}\lambda_{3}\lambda_{4}^{2}+N\lambda_{4}^{3}\lambda_{3}^{2}
+(2-N)\lambda_{1}^{2}\lambda_{3}^{2}\lambda_{4}\\
&-N\lambda_{4}^{2}\lambda_{3}^{3}+(N-1)\lambda_{1}\lambda_{3}^{3}\lambda_{4} \Bigr] \\
&\neq\lambda_{3}^{N} \bigl[\lambda_{1}\lambda_{3}\lambda_{4}^{3}-2\lambda_{1}^{2}
\lambda_{3}\lambda_{4}^{2}+\lambda_{1}^{3}\lambda_{3}\lambda_{4} \bigr]
+\lambda_{4}^{N}\bigl[2\lambda_{1}^{2}\lambda_{3}^{2}\lambda_{4}
-\lambda_{1}\lambda_{3}^{3}\lambda_{4}-\lambda_{1}^{3}\lambda_{3}\lambda_{4} \bigr],
\end{align*}
then the boundary-value problem for \eqref{1} with the conditions \eqref{20}
has a unique solution on $[0,\infty )$.
\end{theorem}
\begin{theorem} \label{thm4}
If the characteristic roots $\lambda_{j}$ are nontrivial,
$\lambda_{1}=\lambda_{2}, \lambda_{3}=\lambda_{4}$ and
\begin{align*}
&\lambda_{1}^{N}\Bigl[(N-2)\lambda_{1}^{2}\lambda_{2}
+2(1-N)\lambda_{1}\lambda_{2}^{2}+N\lambda_{2}^{3} \Bigr] \\
&\neq\lambda_{2}^{N}\Bigl[(2-N)\lambda_{1}\lambda_{2}^{2}+2(N-1)
\lambda_{1}^{2}\lambda_{2}+N\lambda_{1}^{3}\Bigr],
\end{align*}
then the boundary-value problem for \eqref{1} with the conditions \eqref{20}
has a unique solution on $[0,\infty)$.
\end{theorem}
\begin{theorem} \label{thm5}
If the characteristic roots $\lambda_{j}$ are nontrivial,
$\lambda_{1}=\lambda_{2}=\lambda_{3}$ and
\begin{align*}
&\lambda_{1}^{N}\Bigl[(-N^{2}+3N-2)\lambda_{1}^{3}\lambda_{2}+(2N^{2}-5N+2)
\lambda_{1}^{2}\lambda_{2}^{2}+(2-N)\lambda_{1}^{4}\lambda_{2}^{2}
\\
&+(-N^{2}+2N-1)\lambda_{1}\lambda_{2}^{3}+(N-1)\lambda_{1}^{3}
\lambda_{2}^{3}\Bigr]\neq\lambda_{2}^{N}\Bigl[\lambda_{1}^{5}
\lambda_{2}-\lambda_{1}^{3}\lambda_{2}\Bigr],
\end{align*}
then the boundary-value problem for \eqref{1} with the conditions
\eqref{20} has a unique solution on $[0,\infty)$.
\end{theorem}
\begin{theorem} \label{thm6}
If $\lambda_{1}=\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda$ and
\begin{equation*}
2N-3N^{2}+N^{3}+(-2N+3N^{2}-N^{3})\lambda^{2}\neq 0,
\end{equation*}
then the boundary-value problem for \eqref{1} with the conditions \eqref{20}
has a unique solution on $[0,\infty)$.
\end{theorem}
\section{Main results}
This section deals with the oscillation, nonoscillation and the periodicity
of the solutions of \eqref{1}. Also, we give an example to illustrate our results.
\begin{theorem} \label{thm7}
If
\[
0< \frac{b}{a^{2}}<\frac{1+4\cosh a}{2\cosh a-2},
\]
then there exist oscillatory solutions of \eqref{1}.
\end{theorem}
\begin{proof}
Equation \eqref{16} can be written as a polynomial of $\lambda$,
\begin{equation}
f(\lambda)=\lambda^4+\beta_{1}\lambda^3+\beta_{2}\lambda^2
+\beta_{3}\lambda+\beta_{4}, \label{25'}
\end{equation}
where
\begin{equation}
\begin{aligned}
\beta_{1}= -1-2 \cosh a, \\
\beta_{2}= 1+\frac{b}{a^{2}}-\frac{b}{a^{3}}\sinh a+2\cosh a,\label{25''}\\
\beta_{3}= -1-\frac{2b}{a^{2}}\cosh a+\frac{2b}{a^{3}}\sinh a, \\
\beta_{4}= \frac{b}{a^{2}}-\frac{b}{a^{3}}\sinh a .
\end{aligned}
\end{equation}
To prove the oscillation of solutions, we need to show that there exists a unique
negative root of the characteristic equation \eqref{16}.
For this reason let us take the polynomial
$$
f(-\lambda)=\lambda^4-\beta_{1}\lambda^3+\beta_{2}\lambda^2
-\beta_{3}\lambda+\beta_{4}.
$$
Now, if hypothesis is true, then we find that
$$
\beta_{1}<0, \beta_{2}>0, \beta_{3}<0, \beta_{4}<0.
$$
By using Descartes' rule of signs, we conclude that there exists a unique
negative root of \eqref{16}. Let us take $\lambda_{1}$ as this root.
Now, consider the following boundary conditions
\begin{equation*}
x(0)=c_{0}, x(-1)=c_{-1}=c_{0}\lambda_{1}^{-1}, x(1)
=c_{1}=c_{0}\lambda_{1}, x(2)=c_{2}=c_{0}\lambda_{1}^{2}.
\end{equation*}
Applying these conditions to \eqref{21}, the coefficients $k_{i1}$, $i=1,2,3,4$
are found as
\begin{equation*}
k_{11}=c_{0}, \quad k_{21}=k_{31}=k_{41}=0
\end{equation*}
and therefore, \eqref{21} becomes
$c_{n}=x(n)=c_{0}\lambda_{1}^{n}$.
Since $\lambda_{1}<0$, we see that
\begin{equation*}
x(n)x(n+1)=\lambda_{1}c_{0}^{2}\lambda_{1}^{2n}<0, c_{0}\neq0
\end{equation*}
and so the solution $x(t)$ of \eqref{1} has a zero in each interval $(n,n+1 )$.
So there exist oscillatory solutions.
\end{proof}
\begin{theorem} \label{thm8}
If
\begin{equation}
0**0, \beta_{3}<0, \beta_{4}<0$.
Also, we obtain $\beta_{1}<0, \beta_{2}>0, \beta_{3}>0, \beta_{4}<0$
by using \eqref{g2} where $\beta_{1}, \beta_{2}, \beta_{3}, \beta_{4} $
are given by \eqref{25''}. So, from Descartes' rule of sign, if any of
the above conditions is satisfied, then we obtain that the characteristic
equation \eqref{16} has at least one positive root.
Therefore, there are nonoscillatory solutions of \eqref{1}.
\end{proof}
\begin{theorem} \label{thm9}
If
\begin{equation}
b>\frac{a^{3}(1+2\cosh a)}{\sinh a-a},\label{29}
\end{equation}
then there exist both oscillatory and nonoscillatory solutions of \eqref{1}.
\end{theorem}
\begin{proof}
Condition \eqref{29} implies that $\beta_{1}<0$, $\beta_{2}<0$,
$\beta_{3}<0$, $\beta_{4}<0$, where $\beta_{1}$, $\beta_{2}$,
$\beta_{3}$, $\beta_{4} $ are given by \eqref{25''}.
Hence, from Descartes' rule of sign, we conclude that there exists a
single positive root of \eqref{16}. So, the other roots are negative
or complex. Positive root generates nonoscillatory solutions, and
others give us the oscillatory solutions of \eqref{1}.
\end{proof}
\begin{theorem} \label{thm10}
A necessary and sufficient condition for the solution of
problem \eqref{1}-\eqref{2} to be $k$ periodic,
$k\in N-\{0 \} $, is
\begin{equation}
c(k)=c(0), \quad c(k-1)=c(-1), \quad d(k)=d(0), \quad e(k)=e(0)\label{29'}.
\end{equation}
Here $\{ c(n)\}_{n\geq-1}$ is the solution
of \eqref{19*} with the initial conditions
$$
c(-1) =\alpha_{-1},\quad c(0) =\alpha_{0}, \quad
d(0) =\alpha_{1}, \quad e(0) =\alpha_{2}.
$$
\end{theorem}
\begin{proof}
In this proof, we use technique in \cite{fatma}.
If $x(t)$ is periodic with period $k$, then $x(t+k)=x(t)$ for $t\in[0,\infty)$.
This implies that the equalities \eqref{29'} is true.
For the proof of sufficiency case, suppose that \eqref{29'} is satisfied.
From \eqref{8'},
\begin{gather}
\begin{aligned}
x_{k}(t)&=\frac{-1+\cosh a(t-k)}{a^{2}}e_{k}+\frac{\sinh a(t-k)}{a}d_{k}+c_{k} \\
&\quad +\Big(\frac{b}{a^{2}}k-\frac{b}{a^{2}}t+\frac{b}{a^{3}}\sinh a(t-k)\Big)
c_{k-1}, \quad k\leq t**