\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 47, pp. 1--13.\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-2017/47\hfil
Fourth-order differential equations]
{Boundedness and square integrability of solutions of nonlinear fourth-order
differential equations with bounded delay}

\author[E. Korkmaz, C. Tun\c{c} \hfil EJDE-2017/47\hfilneg]
{Erdal Korkmaz, Cemil Tun\c{c}}

\address{Erdal Korkmaz \newline
Department of Mathematics,
Faculty of Arts and Sciences,
Mus Alparslan University, 49100, Mu\c{s}, Turkey}
\email{korkmazerdal36@hotmail.com}

\address{Cemil Tun\c{c} \newline
Department of Mathematics, Faculty of Sciences\\
Y\"uz\"unc\"u Yil University, 65080, Van, Turkey}
\email{cemtunc@yahoo.com}

\dedicatory{Communicated by  Mokhtar Kirane}

\thanks{Submitted January 3, 2017. Published February 16, 2017.}
\subjclass[2010]{34D20, 34C11}
\keywords{Stability; boundedness; Lyapunov functional; fourth order;
\hfill\break\indent delay differential equations;  square integrability}

\begin{abstract}
 In this article, we give sufficient conditions for the boundedness, uniformly
 asymptotic stability and square integrability of the solutions to a
 fourth-order non-autonomous differential equation with bounded
 delay by using Lyapunov's second method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

Ordinary differential equations have been
studied for more than 300 years since the seventeenth century after the
concepts of differentiation and integration were formulated by Newton and
Leibniz. By means of ordinary differential equations, researchers can
explain many natural phenomena like gravity, projectiles, wave, vibration,
nuclear physics, and so on. In addition, in Newtonian mechanics, the
system's state variable changes over time, and the law that governs the
change of the system's state is normally described by an ordinary
differential equation. The question concerning the stability of ordinary
differential equations has been originally raised by the general problem of
the stability of motion (Lyapunov \cite{22}).

However, thereafter along with the development of technology, it is seen
that the ordinary differential equations cannot respond to the needs arising
in sciences and engineering. For example, in many applications, it can be
seen that physical or biological background of modeling system shows that
the change rate of the system's current status often depends not only on the
current state but also on the history of the system. This usually leads to
so-called retarded functional differential equations (Smith \cite{33}).

To the best of our knowledge, the study of qualitative properties of
functional differential equations of higher order has been developed at a
high rate in the last four decades. Functional differential equations of
higher order can serve as excellent tools for description of mathematical
modeling of systems and processes in economy, stochastic processes,
biomathematics, population dynamics, medicine, information theory, physics,
chemistry, aerodynamics and many fields of engineering like atomic energy,
control theory, mechanics, etc., Therefore, the investigation of the
qualitative properties of solutions of functional differential equations of
higher order, stability, boundedness, oscillation, integrability etc. of
solutions play an important role in many real world phenomena related to the
sciences and engineering technique fields. In fact, we would not like to
give here the details of the applications related to functional differential
equations of higher order here.

In particular, for more results on the stability, boundedness, convergence,
etc. of ordinary or functional equations differential equations of fourth
order, see the book of Reissig et al.\ \cite{30} as a good survey for the works
done by 1974 and the papers of Burton \cite{6},
Cartwright \cite{7}, Ezeilo \cite{11,12,13,14},
Harrow \cite{15,16}, Tun\c{c} \cite{36,37,38,39,40,41,42},
Remili et al.\ \cite{25,26,27,28,29}, Wu \cite{44} and
others and theirs references. These information indicate the importance of
investigating the qualitative properties, of solutions of retarded
functional differential equations of fourth order.

In this article, we study the uniformly asymptotic stability of the
solutions for $p(t,x,x',x'',x''')\equiv 0$ and also square integrable
 and boundedness of solutions to the
fourth order nonlinear differential equation with delay
\begin{equation}
\begin{aligned}
&x^{(4)}+a(t)( g(x(t))x''(t)) '+b(t)(
q(x(t))x'(t)) ' \\
&+c(t)f(x(t))x'(t) +d(t)h(x(t-r(t)))=p(t,x,x',x'',x''').
\end{aligned} \label{eq1.1}
\end{equation}

For convenience, we let
\[
\theta_1(t)=g'(x(t))x'(t),\quad
\theta_2(t)=q'(x(t))x'(t),\quad
\theta_3(t)=f'(x(t))x'(t).
\]
We write \eqref{eq1.1} in the system form
\begin{equation} \label{eq1.2}
\begin{aligned}
x' &= y,   \\
y' &= z,   \\
z' &= w,   \\
w' &=-a(t)g(x)w-( b(t)q(x)+a(t)\theta_1) z-(
b(t)\theta_2+c(t)f(x)) y\\
&\quad -d(t)h(x) +d(t)\int_{t-r(t)}^{t}h'(x)yd\eta +p(t,x,y,z,w),
\end{aligned}
\end{equation}
where $r$ is a bounded delay, $0\leq r(t)\leq \psi $,
$r'(t)\leq \xi $, $0<\xi <1$, $\xi $ and $\psi $ some positive constants,
$\psi $ which will be determined later, the functions $a,b,c,d$ are continuously
differentiable functions and the functions $f,h,g,q,p$ are continuous
functions depending only on the arguments shown.
Also derivatives $g'(x),q'(x),f'(x)$ and $h'(x)$ exist and are
continuous. The continuity of the functions $a,b,c,d,p,g,g',
q,q',f,h$ guarantees the existence of the solutions of
equation \eqref{eq1.1}. If the right hand side of the system \eqref{eq1.2}
 satisfies a
Lipchitz condition in $x(t),y(t),z(t),w(t)$ and $x(t-r)$ ,and exists of
solutions of system \eqref{eq1.2} , then it is unique solution of system
\eqref{eq1.2}.

Assume that there are positive constants
$a_0$, $b_0$, $c_0$, $d_0$, $f_0$, $g_0$, $q_0$, $a_1$, $b_1$, $c_1$, $d_1$,
$f_1$, $g_1$, $q_1$, $m$, $M$, $\delta$, $\eta_1$
such that the following assumptions hold:

\begin{itemize}
\item[(A1)] $0<a_0\leq a(t)\leq a_1$,
$0<b_0\leq b(t)\leq b_1$, $0<c_0\leq c(t)\leq c_1$,
 $0<d_0\leq d(t)\leq d_1$ for $t\geq 0$;

\item[(A2)] $0<f_0\leq f(x)\leq f_1$, $0<g_0\leq g(x)\leq g_1$,
$0<q_0\leq q(x)\leq q_1$  for $x\in R$ and
$0<m<\min \{ f_{\text{0}},g_0,1\}$,
$M>\max \{ f_{\text{1}},g_1,1\}$;

\item[(A3)] $\frac{h(x)}{x}\geq \delta >0$  for $x\neq 0$, $h(0)=0$;

\item[(A4)] $\int_0^{\infty }( | a'(t)| +| b'(t)| +| c'(t)| +| d'(t)|) dt<\eta_1$;

\item[(A5)] $| p(t,x,y,z,w)| \leq |e(t)|$.
\end{itemize}

Motivated by the results of references, we obtain some new results on the
uniformly asymptotic stability and boundedness of the solutions by means
of the Lyapunov's functional approach. Our results differ from that obtained
in the literature (see, the references in this article and the references therein).
By this way,
we mean that this paper has a contribution to the subject in the literature,
and it may be useful for researchers working on the qualitative behaviors of
solutions of functional differential equations of higher order. In view of
all the mentioned information, it can be checked the novelty and originality
of the current paper.

\section{Preliminaries}

We also consider the functional differential equation
\begin{equation}
\dot{x}=f(t,x_t),\quad x_t(\theta )=x(t+\theta ),\quad
-r\leq \theta \leq 0,\quad t\geq 0.  \label{eq2.1}
\end{equation}
where $f:I\times C_{H}\to\mathbb{R}^n$ is a continuous mapping,
$f(t,0)=0$, $C_{H}:=\{\phi \in (C[-r,0],\mathbb{R}^n):\| \phi \|t \leq H\}$,
and for $H_1<H$, there exists
$L(H_1)>0$, with $| f(t,\phi )| <L(H_1)$ when $\| \phi \|t <H_1$.

\begin{lemma}[\cite{6}]\label{lem1}
 Let $V(t,\phi ):I\times C_{H}\to \mathbb{R}$ be a continuous functional
satisfying a local Lipchitz condition, $V(t,0)=0 $, and wedges $W_i$ such that:
\begin{itemize}
\item[(i)] $W_1(\| \phi \|t )\leq V(t,\phi )\leq W_2(\| \phi \|t )$;

\item[(ii)] $V_{\eqref{eq2.1}}'(t,\phi )\leq -W_3(\|\phi \|t )$.
\end{itemize}
Then, the zero solution of  \eqref{eq2.1} is
uniformly asymptotically stable.
\end{lemma}

\section{Main results}

\begin{lemma}[\cite{19}] \label{lem2}
 Let $h(0)=0$, $xh(x)>0$ $(x\neq 0)$ and $\delta (t)-h'(x)\geq 0$
$(\delta (t)>0)$, then $2\delta (t)H(x)\geq h^2(x)$, where
$H(x)=\int_0^{x}h(s)ds$.
\end{lemma}

\begin{theorem} \label{thm1}
In addition to the basic assumptions imposed on the functions
$a$, $b$, $c$, $d$, $p$, $f$, $h$, $g$, $q$ suppose that there are positive constants
$h_0$, $h_1$, $\delta_0$, $\delta_1$, $\eta_2$, $\eta_3$ such
that the following conditions are satisfied:
\begin{itemize}
\item[(i)] $h_0-\frac{a_0m\delta_0}{d_1}\leq h'(x)\leq
\frac{h_0}{2}$ for $x\in R$;

\item[(ii)] $\delta_1=\frac{d_1h_0a_1M}{c_0m}
+\frac{c_1M+\delta _0}{a_0m}<b_0q_0$;

\item[(iii)] $\int_{-\infty }^{+\infty }( | g'(s)| +| q'(s)| +|
f'(s)| ) ds<\eta_2$;

\item[(iv)] $\int_0^{\infty }| e(t)| dt<\eta_3$.
\end{itemize}
Then any solution $x(t)$  of  \eqref{eq1.1} and its
derivatives $x'(t),x''(t)$ $x'''(t)$ are bounded and satisfy
\[
\int_0^{\infty }( x'^2(s)+x''^2(s) +x'''^ 2(s)) ds<\infty ,
\]
provided that
\[
\psi <\frac{(1-\xi )}{d_1h_1}\min \Big\{ \frac{\varepsilon c_0f_0}{
\alpha +\beta (2-\xi )+1},\frac{2[ b_0q_0-\delta_1-\varepsilon
M(a_1+c_1)] }{(1-\xi )},\frac{2\varepsilon }{\alpha (1-\xi )}
\Big\} .
\]
\end{theorem}

\begin{proof}
We define a Lyapunov functional
\begin{equation}
W=W(t,x,y,z,w)=e^{\frac{-1}{\eta }\int_0^{t}\gamma (s)ds}V,  \label{eq3.1}
\end{equation}
where
\[
\gamma (t)=| a'(t)| +| b'(t)| +| c'(t)| +| d'(t)| +| \theta_1(t)| +| \theta
_2(t)| +| \theta_3(t)| ,
\]
and
\begin{align*}
2V &=2\beta d(t)H(x)+c(t)f(x)y^2+\alpha
b(t)q(x)z^2+a(t)g(x)z^2+2\beta a(t)g(x)yz \\
&\quad +[ \beta b(t)q(x)-\alpha h_0d(t)] y^2-\beta z^2+\alpha
w^2+2d(t)h(x)y+2\alpha d(t)h(x)z \\
&\quad +2\alpha c(t)f(x)yz+2\beta yw+2zw+\sigma
\int_{-r(t)}^{0}\int_{t+s}^{t}y^2(\gamma )d\gamma ds
\end{align*}
with $H(x)=\int_0^{x}h(s)ds$, $\alpha =\frac{1}{a_0m}+\varepsilon $,
$\beta =\frac{d_1h_0}{c_0m}+\varepsilon $, $\varepsilon $ and $\eta $
are positive constants to be determined later in the proof. We can rearrange
$2V$ as
\begin{align*}
2V &=a(t)g(x)\Big[ \frac{w}{a(t)g(x)}+z+\beta y\Big] ^2
+c(t)f(x)\Big[\frac{d(t)h(x)}{c(t)f(x)}+y+\alpha z\Big] ^2 \\
&\quad +\frac{d^2(t)h^2(x)}{c(t)f(x)}
 +2\varepsilon d(t)H(x)
 +\sigma \int_{-r(t)}^{0}\int_{t+s}^{t}y^2(\gamma )d\gamma ds+V_1+V_2+V_3\,,
\end{align*}
where
\begin{gather*}
V_1  = 2d(t)\int_0^{x}h(s)\Big[ \frac{d_1h_0}{c_0m}-2\frac{d(t)}{
c(t)f(x)}h'(s)\Big] ds, \\
V_2 =\big[ \alpha b(t)q(x)-\beta -\alpha ^2c(t)f(x)\big] z^2, \\
V_3 =\big[ \beta b(t)q(x)-\alpha h_0d(t)-\beta ^2a(t)g(x)\big]
y^2+\big[ \alpha -\frac{1}{a(t)g(x)}\big] w^2.
\end{gather*}
Let
\begin{equation}
\varepsilon <\min \Big\{ \frac{1}{a_0m},\frac{d_1h_0}{c_0m},\frac{
b_0q_0-\delta_1}{M(a_1+c_1)}\Big\}   \label{eq3.2}
\end{equation}
then
\begin{equation}
\frac{1}{a_0m}<\alpha <\frac{2}{a_0m},\quad
\frac{d_1h_0}{c_0m}<\beta <2\frac{d_1h_0}{c_0m}.  \label{eq3.3}
\end{equation}
By using conditions (A1)--(A3), (i)--(ii) and inequalities
\eqref{eq3.2}, \eqref{eq3.3} we obtain
\begin{align*}
V_1 &\geq 4d(t)\frac{d_1}{c_0m}\int_0^{x}h(s)[ \frac{h_0}{2}
-h'(s)] ds\geq 0, \\
V_2 &= ( \alpha ( b(t)q(x)-\beta a(t)-\alpha c(t)f(x))
+\beta (\alpha a(t)-1)) z^2 \\
&\geq \alpha \Big( b_0q_0-\frac{d_1h_0a_1}{c_0m}-\frac{c_1M}{
a_0m}-\varepsilon (a_1+c_1M)\Big) z^2+\beta ( \frac{1}{m}-1) z^2 \\
&\geq \alpha ( b_0q_0-\delta_1-\varepsilon M(
a_1+c_1) ) z^2\geq 0,
\end{align*}
and
\begin{align*}
V_3 &\geq \beta \Big( b_0q_0-\frac{\alpha }{\beta }h_0d_1-\beta
a_1M\Big) y^2+( \alpha -\frac{1}{a_0m}) w^2 \\
&\geq \beta \Big( b_0q_0-\frac{c_0}{a_0}-a_1\frac{d_1h_0M}{
c_0m}-\varepsilon (c_0m+a_1M)\Big) y^2+\varepsilon w^2 \\
&\geq \beta ( b_0q_0-\delta_1-\varepsilon M(c_1+a_1))
y^2+\varepsilon w^2\geq 0.
\end{align*}
Thus, it is clear  from the above inequalities that there exists positive
constant $D_0$ such that
\begin{equation}
2V\geq D_0(y^2+z^2+w^2+H(x)).  \label{eq3.4}
\end{equation}
From Lemma \ref{lem2},  (A3) and (i), it follows that there is a positive constant
$D_1$ such that
\begin{equation}
2V\geq D_1(x^2+y^2+z^2+w^2)  \label{eq3.5}
\end{equation}
In this way $V$ is positive definite. From (A1)--(A3), it is clear that there
is a positive constant $U_1$ such that
\begin{equation}
V\leq U_1(x^2+y^2+z^2+w^2).  \label{eq3.6}
\end{equation}
From (iii), we have
\begin{equation}
\begin{aligned}
&\int_0^{t}( | \theta_1(s)| +| \theta_2(s)| +| \theta_3(s)| ) ds\\
&=\int_{\alpha_1(t)}^{\alpha_2(t)}( | g'(u)| +| q'(u)| +| f'(u)| ) du \\
&\leq \int_{-\infty }^{+\infty }( | g'(u)|+| q'(u)| +| f'(u)|) du
&<\eta_2<\infty
\end{aligned} \label{eq3.7}
\end{equation}
where $\alpha_1(t)=\min \{ x(0),x(t)\} $ and
$\alpha_2(t)=\max \{ x(0),x(t)\}$. From inequalities \eqref{eq3.2},
\eqref{eq3.6} and \eqref{eq3.7}, it follows that
\begin{equation}
W\geq D_2(x^2+y^2+z^2+w^2)  \label{eq3.8}
\end{equation}
where $D_2=\frac{D_1}{2}e^{-\frac{\eta_1+\eta_2}{\eta }}.$ Also,
it is easy to see that there is a positive constant $U_2$ such that
\begin{equation}
W\leq U_2(x^2+y^2+z^2+w^2)  \label{eq3.9}
\end{equation}
for all $x,y,z,w$ and all $t\geq 0.$

Now, we show that $\dot{W}$ is negative definite function. The
derivative of the function $V$ along any solution
$(x(t),y(t),z(t),w(t)) $ of system \eqref{eq1.2}, with respect to $t$ is after
simplifying
\[
2\dot{V}_{\eqref{eq1.2}}=-2\varepsilon
c(t)f(x)y^2+V_4+V_5+V_6+V_7+V_8+V_9+2(\beta y+z+\alpha
w)p(t,x,y,z,w)
\]
where
\begin{gather*}
V_4 = -2\Big( \frac{d_1h_0}{c_0m}c(t)f(x)-d(t)h'(x)\Big) y^2
 -2\alpha d(t)( h_0-h'(x)) yz, \\
V_5 = -2( b(t)q(x)-\alpha c(t)f(x)-\beta a(t)g(x)) z^2, \\
V_6 = -2( \alpha a(t)g(x)-1) w^2, \\
\begin{aligned}
V_7 &= 2\alpha d(t)w\int_{t-r(t)}^{t}h'(x(\eta ))x'(\eta
)d\eta +2\beta d(t)y(t)\int_{t-r(t)}^{t}h'(x(\eta ))x'(\eta )d\eta \\
&\quad +2d(t)z(t)\int_{t-r(t)}^{t}h'(x(\eta ))x'(\eta )d\eta
+\sigma r(t)y^2(t)-\sigma (1-r'(t))\int_{t-r(t)}^{t}y^2(\eta
)d\eta ,
\end{aligned}\\
\begin{aligned}
V_8 &= -a(t)\theta_1( z^2+2\alpha zw) -b(t)\theta
_2( \alpha z^2+2\alpha zw+\beta y^2+2yz) \\
&\quad +c(t)\theta_3( y^2+2\alpha yz) ,
\end{aligned} \\
\begin{aligned}
V_9 &= d'(t)[ 2\beta H(x)-\alpha h_0y^2+2h(x)y+2\alpha h(x)z] \\
&\quad +c'(t)[ f(x)y^2+2\alpha f(x)yz] +b'(t)[
\alpha q(x)z^2+\beta q(x)y^2] \\
&\quad +a'(t)[ g(x)z^2+2\beta g(x)yz] .
\end{aligned}
\end{gather*}
By regarding conditions (A1), (A2), (i), (ii) and inequalities \eqref{eq3.3},
\eqref{eq3.4}, we
have
\begin{align*}
V_4 &\leq -2[ d(t)h_0-d(t)h'(x)] y^2-2\alpha d(t)
[ h_0-h'(x)] yz \\
&\leq -2d(t)[ h_0-h'(x)] y^2-2\alpha d(t)[
h_0-h'(x)] yz \\
&\leq 2d(t)[ h_0-h'(x)] [ ( y+\frac{\alpha
}{2}z) ^2-( \frac{\alpha }{2}z) ^2] \\
&\leq \frac{\alpha ^2}{2}d(t)[ h_0-h'(x)] z^2.
\end{align*}
In this case,
\begin{align*}
V_4+V_5
&\leq -2\big[ b(t)q(x)-\alpha c(t)f(x)-\beta a(t)g(x)-\frac{
\alpha ^2}{4}d(t)[ h_0-h'(x)] \big] z^2 \\
&\leq -2\big[ b_0q_0-( \frac{1}{a_0m}+\varepsilon )
c_1M-( \frac{d_1h_0}{c_0m}+\varepsilon ) a_1M-\frac{
\alpha ^2}{4}( a_0m\delta_0) \big] z^2 \\
&\leq -2\big[ b_0q_0-\frac{M}{a_0m}c_1-\frac{d_1h_0a_1M}{
c_0m}-\frac{\delta_0}{a_0m}-\varepsilon M( a_1+c_1)
\big] z^2 \\
&\leq -2[ b_0q_0-\delta_1-\varepsilon M(a_1+c_1) ] z^2\leq 0,
\end{align*}
and
\[
V_6\leq -2[ \alpha a_0m-1] w^2=-2\varepsilon w^2\leq 0.
\]
By taking $h_1=\max \{ | \frac{d_1h_0-a_0m\delta_0}{d_1}| ,| \frac{h_0}{2}|\} $,
 we have
\[
V_7\leq d_1h_1r(t)(\alpha w^2+\beta y^2+z^2)+\sigma r(t)y^2+
[ d_1h_1(\alpha +\beta +1)-\sigma (1-\xi )]
\int_{t-r(t)}^{t}y^2(s)ds
\]
If we choose $\sigma =\frac{d_1h_1(\alpha +\beta +1)}{(1-\xi )}$, we obtain
\[
V_7\leq \frac{d_1h_1}{(1-\xi )}r(t)[ \alpha (1-\xi )w^2+(\alpha
+\beta (2-\xi )+1)y^2+(1-\xi )z^2] .
\]

Thus, there exists a positive constant $D_3$ such that
\[
-\varepsilon c(t)f(x)y^2+V_4+V_5+V_6+V_7\leq
-2D_3(y^2+z^2+w^2).
\]
From \eqref{eq3.4}, and the Cauchy Schwartz inequality, we obtain
\begin{align*}
V_8
&\leq a(t)| \theta_1| ( z^2+\alpha(z^2+w^2))
 +b(t)| \theta_2| ( \alpha z^2+\alpha (z^2+w^2)+\beta y^2+y^2+z^2) \\
&\quad +c(t)| \theta_3| ( y^2+\alpha (
y^2+z^2) ) \\
&\leq \lambda_1( | \theta_1| +|\theta_2| +| \theta_3| ) (y^2+z^2+w^2+H(x)) \\
&\leq 2\frac{\lambda_1}{D_0}( | \theta_1|+| \theta_2| +| \theta_3|) V,
\end{align*}
where $\lambda_1=\max \{ a_1( 1+\alpha ) ,b_1(
1+2\alpha +\beta ) ,c_1( 1+\alpha )\}$.
 Using condition (iii) and Lemma \ref{lem2}, we can write
\[
h^2(x)\leq h_0H(x),
\]
hereby,
\begin{align*}
| V_9|
&\leq | d'(t)|[ 2\beta H(x)+\alpha h_0y^2+h^2(x)+y^2+\alpha (
h^2(x)+z^2) ] \\
&\quad +| c'(t)| [ y^2+\alpha (
y^2+z^2) ] +| b'(t)| [\alpha z^2+\beta y^2] \\
&\quad +| a'(t)| [ z^2+2\beta (y^2+z^2) ] \\
&\leq \lambda_2[ | a'(t)| +|b'(t)| +| c'(t)| +|
d'(t)| ] ( y^2+z^2+w^2+H(x)) \\
&\leq 2\frac{\lambda_2}{D_0}[ | a'(t)| +| b'(t)| +| c'(t)| +| d'(t)| ] V,
\end{align*}
such that $\lambda_2=\max \{ 2\beta +(\alpha +1)h_0,\alpha
h_0+1,\alpha +1\}$. By taking
$\frac{1}{\eta }=\frac{1}{D_0}\max \{ \lambda_1,\lambda_2\} $, we obtain
\begin{equation}
\begin{aligned}
\dot{V}_{\eqref{eq1.2}}
&\leq -D_3( y^2+z^2+w^2) +(\beta y+z+\alpha w) p(t,x,y,z,w)   \\
&\quad +\frac{1}{\eta }( | a'(t)| +|b'(t)| +| c'(t)|
 +|d'(t)| +| \theta_1| +|\theta_2| +| \theta_3| ) V.
\end{aligned}\label{eq3.10}
\end{equation}
From (A4), (A5),(iii),  \eqref{eq3.7},  \eqref{eq3.8},  \eqref{eq3.10} and
the Cauchy Schwartz inequality, we obtain
\begin{align}
\dot{W}_{\eqref{eq1.2}}
&= \Big( \dot{V}_{\eqref{eq1.2}}-\frac{1}{\eta }
\gamma (t)V\Big) e^{-\frac{1}{\eta }\int_0^{t}\gamma (s)ds}  \nonumber \\
&\leq ( -D_3( y^2+z^2+w^2)
  +( \beta y+z+\alpha w) p(t,x,y,z,w)) e^{-\frac{1}{
\eta }\int_0^{t}\gamma (s)ds}  \label{eq3.11} \\
&\leq ( \beta | y| +| z| +\alpha
| w| ) | p(t,x,y,z,w)|  \nonumber \\
&\leq D_4( | y| +| z| +| w| ) | e(t)|  \nonumber \\
&\leq D_4( 3+y^2+z^2+w^2) | e(t)| \nonumber \\
&\leq D_4\big( 3+\frac{1}{D_2}W\big) | e(t)| \nonumber\\
&\leq 3D_4| e(t)| +\frac{D_4}{D_2}W|e(t)| , \label{eq3.12}
\end{align}  
where $D_4=\max \{ \alpha ,\beta ,1\}$.
 Integrating \eqref{eq3.12}
from $0$ to $t$ and using the condition (iv) and the Gronwall inequality, we
have
\begin{equation}
\begin{aligned}
W
&\leq W( 0,x(0),y(0),z(0),w(0)) +3D_4\eta_3   \\
&\quad +\frac{D_4}{D_2}\int_0^{t}W( s,x(s),y(s),z(s),w(s))
| e(s)| ds   \\
&\leq ( W( 0,x(0),y(0),z(0),w(0)) +3D_4\eta_3)
e^{\frac{D_4}{D_2}\int_0^{t}| e(s)| ds}   \\
&\leq ( W( 0,x(0),y(0),z(0),w(0)) +3D_4\eta_3)
e^{\frac{D_4}{D_2}\eta_3}=K_1<\infty
\end{aligned} \label{eq3.13}
\end{equation}
Because of inequalities \eqref{eq3.8} and \eqref{eq3.13}, we write
\begin{equation}
( x^2+y^2+z^2+w^2) \leq \frac{1}{D_2}W\leq K_2, \label{eq3.14}
\end{equation}
where $K_2=\frac{K_1}{D_2}$. Clearly \eqref{eq3.14} implies
\[
| x(t)| \leq \sqrt{K_2},\quad
|y(t)| \leq \sqrt{K_2},\quad
| z(t)| \leq \sqrt{K_2},\quad
| w(t)| \leq \sqrt{K_2}\quad \text{for }t\geq 0.
\]
Hence
\begin{equation}
| x(t)| \leq \sqrt{K_2},\quad | x'(t)| \leq \sqrt{K_2},\quad
| x''(t)| \leq \sqrt{K_2},| x'''(t)| \leq \sqrt{K_2}\label{eq3.15}
\end{equation}
\quad
for $t\geq 0$.
Now, we proof the square integrability of solutions and their derivatives.
We define
\[
F_t=F(t,x(t),y(t),z(t),w(t))
=W+\rho \int_0^{t}( y^2(s)+z^2(s)+w^2(s)) ds,
\]
where $\rho >0$. It is easy to see that $F_t$ is positive definite, since
$W=W(t,x,y,z,w)$ is already positive definite. Using the following estimate
\[
e^{-\frac{\eta_1+\eta_2}{\eta }}\leq e^{-\frac{1}{\eta }
\int_0^{t}\gamma (s)ds}\leq 1
\]
by \eqref{eq3.12} we have
\begin{equation} \label{eq3.16}
\begin{aligned}
\dot{F_t}_{\eqref{eq1.2}}
&\leq -D_3(y^2(t)+z^2(t)+w^2(t)) e^{-\frac{\eta_1+\eta_2}{\eta }}\\
&\quad +D_4( | y(t)| +| z(t)| +| w(t)| ) | e(t)| \\
&\quad +\rho ( y^2(t)+z^2(t)+w^2(t))
\end{aligned}
\end{equation}
By choosing $\rho =D_3e^{-\frac{\eta_1+\eta_2}{\eta }}$ we obtain
\begin{equation}
\begin{aligned}
\dot{F_t}_{\eqref{eq1.2}}
&\leq D_4(3+y^2(t)+z^2(t)+w^2(t)) | e(t)|   \\
&\leq D_4( 3+\frac{1}{D_2}W) | e(t)| \\
&\leq 3D_4| e(t)| +\frac{D_4}{D_2}F_t| e(t)| .
\end{aligned}  \label{eq3.17}
\end{equation}

Integrating  from $0$ to $t$ and using again the
Gronwall inequality and the condition (iv), we obtain
\begin{equation}
\begin{aligned}
F_t
&\leq F_0+3D_4\eta_3+\frac{D_4}{D_2}\int_0^{t}F_{s}| e(s)| ds   \\
&\leq ( F_0+3D_4\eta_3) e^{\frac{D_4}{D_2}\int_0^{t}| e(s)| ds}   \\
&\leq ( F_0+3D_4\eta_3) e^{\frac{D_4}{D_2}\eta_3}=K_3<\infty
\end{aligned} \label{eq3.18}
\end{equation}
Therefore,
\[
\int_0^{\infty }y^2(s)ds<K_3,\quad
\int_0^{\infty }z^2(s)ds<K_3,\quad
\int_0^{\infty }w^2(s)ds<K_3,
\]
which implies
\begin{equation}
\int_0^{\infty }[ x'(s)] ^2ds<K_3,\quad
\int_0^{\infty }[ x''(s)] ^2ds<K_3,\quad
\int_0^{\infty }[ x'''(s)] ^2ds<K_3.
\label{eq3.19}
\end{equation}
which completes the proof.
\end{proof}

\begin{remark} \label{rmk3.3} \rm
If $p(t,x,y,z,w)\equiv 0$, similarly to the above proof, the
inequality \eqref{eq3.11} becomes
\begin{align*}
\dot{W_{\eqref{eq1.2}}}
&= \Big(\dot{V}_{\eqref{eq1.2}}-\frac{1}{\eta
}\gamma(t)V\Big) e^{-\frac{1}{\eta }\int_0^{t}\gamma (s)ds} \\
&\leq -D_3(y^2+z^2+w^2)e^{-\frac{1}{\eta}\int_0^{t}\gamma (s)ds}\\
&\leq -\mu (y^2+z^2+w^2),
\end{align*}
where $\mu =D_3e^{-\frac{\eta_1+\eta_2}{\eta }}$.
 It can also be observed that the only solution of system \eqref{eq1.2}
for which $\dot{W_{\eqref{eq1.2}}}(t,x,y,z,w)=0$ is the solution $x=y=z=w=0$.
 The above discussion guarantees that the trivial solution of equation
\eqref{eq1.1} is uniformly asymptotically stable, and the same conclusion
as in the proof of theorem can be drawn for square integrability of solutions of
equation \eqref{eq1.1}.
\end{remark}


\begin{example} \label{examp3.4} \rm
We consider the  fourth-order nonlinear differential
equation with delay
\begin{equation} \label{eq3.20}
\begin{aligned}
&x^{(4)}+(e^{-2t}\sin 3t+2)\Big( \big(\frac{5x+2e^{x}+2e^{-x}}{e^{x}+e^{-x}}
\big)x''\Big) ' \\
&+\Big(\frac{\sin 2t+11t^2+11}{t^2+1}\Big)\Big( (\frac{\sin x+9e^{x}+9e^{-x}}{
e^{x}+e^{-x}})x'\Big) ' \\
&+\big(e^{-t}\sin t+3\big)\big(\frac{x\cos x+x^{4}+1}{x^{4}+1}\big)x' \\
&+\big(\frac{\sin ^2t+t^2+1}{5t^2+5}\big)\Big( \frac{x(t-\frac{1}{e^{t}+15})}{
x^2(t-\frac{1}{e^{t}+15})+1}\Big) \\
&=\frac{2\sin t}{t^2+1+(xx'x'')^2+(x''')^2}
\end{aligned}
\end{equation}
by taking
\begin{gather*}
g(x)=\frac{5x+2e^{x}+2e^{-x}}{e^{x}+e^{-x}},\quad
q(x)=\frac{\sin x+9e^{x}+9e^{-x}}{e^{x}+e^{-x}},\quad
f(x)=\frac{x\cos x+x^{4}+1}{x^{4}+1}, \\
h(x)=\frac{x}{x^2+1},a(t)=e^{-2t}\sin 3t+2, \quad
b(t)=\frac{\sin 2t+11t^2+11}{t^2+1}, \\
c(t)=e^{-t}\sin t+3, \quad
d(t)=\frac{\sin ^2t+t^2+1}{5t^2+5}, \quad
r(t)=\frac{1}{e^{t}+15}, \\
p(t,x,x'x'',x''')=\frac{2\sin t}{t^2+1+(xx'x'')^2+(x''')^2}.
\end{gather*}
 We obtain
$g_0=0.33$, $g_1=3.7$, $f_0=0.5$, $f_1=1.5$, $q_0=8.5$,
$q_1=9.5$, $a_0=1$, $a_1=3$, $b_0=10$, $b_1=12$, $c_0=2$, $c_1=4$,
$d_0=0.2$, $d_1=0.3$, $m=0.3$, $M=3.8$, $h_0=2$,
$\alpha =\frac{23}{6}$, $\beta =\frac{3}{2}$, $\delta_0=\frac{17}{8}$ and
$\delta_1=69.15$. 
Also we have
\begin{align*}
&\int _{-\infty }^{\infty }| g'(x)| dx\\
&= 5\int_{-\infty }^{\infty }\big| \frac{1}{e^{x}+e^{-x}}+x
\frac{e^{-x}-e^{x}}{( e^{x}+e^{-x}) ^2}\big| dx \\
&\leq 5\int_{-\infty }^{0}\big| \frac{1}{e^{x}+e^{-x}}-x\frac{
e^{-x}-e^{x}}{( e^{x}+e^{-x}) ^2}\big| dx
+5\int_0^{\infty }\big| \frac{1}{e^{x}+e^{-x}}-x\frac{
e^{-x}-e^{x}}{( e^{x}+e^{-x}) ^2}\big| dx \\
&= 5\pi ,
\end{align*}
\begin{align*}
\int_{-\infty }^{\infty }| q'(x)|dx
&= \int_{-\infty }^{\infty }\big| \frac{(
e^{x}+e^{-x}) \cos x-( e^{x}-e^{-x}) \sin x}{(
e^{x}+e^{-x}) ^2}\big| dx \\
&\leq \int_{-\infty }^{\infty }\big| \frac{1}{e^{x}+e^{-x}}+x
\frac{e^{x}-e^{-x}}{( e^{x}+e^{-x}) ^2}\big| dx
= \pi ,
\end{align*}
\begin{align*}
\int_{-\infty }^{\infty }| f'(x)|dx
&= \int_{-\infty }^{\infty }\big| \frac{\cos x}{x^{4}+1}-4x^{4}
\frac{\cos x}{( x^{4}+1) ^2}+-x\frac{\sin x}{x^{4}+1}\big| dx \\
&\leq \int_{-\infty }^{\infty }\big| \frac{5}{x^{4}+1}+\frac{
x^2}{x^{4}+1}\big| dx
= 6\sqrt{2}\pi ,
\end{align*}
\begin{align*}
\int_0^{\infty }| p(t,x,x',x'',x''')| dt
&= \int_0^{\infty }\big| \frac{2\sin t
}{t^2+1+(xx'x'')^2+(x''')^2}\big| dt \\
&\leq \int_0^{\infty }\big| \frac{2\sin t}{t^2+1}\big| dt \\
&\leq \int_0^{\infty }\frac{2}{t^2+1}dt
= \pi ,
\end{align*}
\begin{align*}
\int_0^{\infty }| a'(t)| dt
&= \int_0^{\infty }\big| -2e^{-2t}\sin 3t+3e^{-2t}\cos 3t\big|dt \\
&\leq \int_0^{\infty }5e^{-2t}dt
= \frac{5}{2},
\end{align*}
\begin{align*}
\int_0^{\infty }| b'(t)| dt
&= \int_0^{\infty }\big| \frac{2\cos 2t}{t^2+1}-2t\frac{\sin 2t}{( t^2+1) ^2}\big|
dt \\
&\leq \int_0^{\infty }\frac{3}{t^2+1}dt
= \frac{3\pi }{2},
\end{align*}
\begin{align*}
\int_0^{\infty }| c'(t)| dt
&= \int_0^{\infty }| -e^{-t}\sin t+e^{-t}\cos t| dt \\
&\leq \int_0^{\infty }2e^{-t}dt
= 2,
\end{align*}
\begin{align*}
\int_0^{\infty }| d'(t)| dt
&= \int_0^{\infty }\big| \frac{2\sin t\cos
t}{5t^2+5}-2t\frac{\sin
^2t}{( 5t^2+5) ^2}\big| dt \\
&\leq \frac{11}{25}\int_0^{\infty }\frac{1}{t^2+1}dt
= \frac{11\pi }{50}.
\end{align*}
Consequently
\begin{gather*}
\int_{-\infty }^{+\infty }(
| g'(s)| +| q'(s)| +| f'(s)| )ds<\infty , \\
\int_0^{\infty }( | a'(t)| +| b'(t)| +| c'(t)| +| d'(t)| ) dt<\infty .
\end{gather*}
Thus all the assumptions of Theorem \ref{thm1} hold, this shows that every
solutions of equation \eqref{eq3.20} are bounded and derivatives of
solutions are square integrable.
\end{example}

\subsection*{Conclusion}

A class of nonlinear retarded functional differential equations of fourth
order is considered. Sufficient conditions are established guaranteeing the
uniformly asymptotic stability of the solutions for $p(t,x,x',x'',x''')\equiv 0$
and also square integrable and boundedness of solutions of equation \eqref{eq1.1}
with delay. In the proofs of the main results, we benefit from the Lyapunov functional
approach. The results obtained essentially improve, include and complement
the results in the literature. An example is furnished to illustrate the
hypotheses by MATLAB-Simulink.


\begin{figure*}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1} 
\end{center}
\end{figure*}

\newpage

\begin{thebibliography}{99}

\bibitem{1} Abou-El-Ela, A. M. A.; Sadek, A. I.;
\emph{A stability result for certain fourth order differentialequations}.
 Ann. Differential Equations 6 (1) (1990), 1--9.

\bibitem{2} Adesina, O. A.;  Ogundare, B. S.;
\emph{Some new stability and boundedness results on a certainfourth order
nonlinear differential equation}. Nonlinear Studies 19 (3) (2012), 359--369.

\bibitem{3} Andres, J.; Vlcek, V.;
\emph{On the existence of square integrable
solutions and their derivatives to fourth and fifth order differential
equations}. Acta Universitatis
Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 28 (1)
(1989), 65--86.

\bibitem{4} Barbashin, E. A.;
\emph{The construction of Liapunov function. Differ. Uravn.}, 4 (1968), 2127--2158.

\bibitem{5} Bereketoglu, H.;
\emph{Asymptotic stability in a fourth order delay
differential equation}. Dynam. Systems Appl., 7 (1) (1998) ,105--115.

\bibitem{6} Burton, T. A.;
\emph{Stability and periodic solutions of ordinary and
functional differential equations}. Mathematics in science and engineering,
Volume 178, Academic Press, Inc, (1985).

\bibitem{7} Cartwright, M. L.;
\emph{On the stability of solutions of certain
differential equations of the fourth order}. Quart. J. Mech. Appl. Math.,
 9 (1956), 185--194.

\bibitem{8} Chin, P. S. M.;
\emph{Stability results for the solutions of certain
fourth-order autonomous differential equations}. Internat. J. Control.,
 49 (4) (1989), 1163--1173.

\bibitem{9} Chukwu, E. N.;
\emph{On the stability of a nonhomogeneous differential
equation of the fourth order}. Ann. Mat. Pura Appl., 92 (4) (1972), 1--11.

\bibitem{10} Elsgolts, L.;
\emph{Introduction to the Theory of Differential
Equations with Deviating Arguments}. Translated from the Russian by Robert J.
McLaughlin Holden-Day, Inc. San Francisco,
Calif.-London-Amsterdam, 1966.

\bibitem{11} Ezeilo, J. O. C.;
\emph{A stability result for solutions of a certain
fourth order differential equation}. J. London Math. Soc. 37 (1962), 28--32.

\bibitem{12} Ezeilo, J. O. C.;
\emph{On the boundedness and the stability of
solutions of some differential equations of the fourth order}. J. Math. Anal.
Appl., 5 (1962), 136--146.

\bibitem{13} Ezeilo, J. O. C.;
\emph{Stability results for the solutions of some
third and fourth order differential equations}. Ann. Mat. Pura Appl., 66 (4)
(1964), 233--249.

\bibitem{14} Ezeilo, J. O. C.; Tejumola, H. O.;
\emph{On the boundedness and the
stability properties of solutions of certain fourth order differential
equations}. Ann. Mat. Pura Appl., 95 (4) (1973), 131--145.

\bibitem{15} Harrow, M.;
\emph{A Stability result for Solutions of a Certain
Fourth Order Homogeneous Differential Equations}, J. London Math. Soc. 42,
pp. 51-56, (1967).

\bibitem{16} Harrow, M.;
\emph{Further results on the boundedness and the stability
of solutions of some differential equations of the fourth order}. SIAM J.
Math. Anal., 1 (1970), 189--194.

\bibitem{17} Hu, Chao Yang;
\emph{The stability in the large for certain fourth
order differential equations}. Ann. Differential Equations, 8 (4) (1992),
422--428.

\bibitem{18} Hara, T.;
\emph{On the asymptotic behavior of the solutions of some
third and fourth order nonautonomous differential equations}. Publ. RIMS,
Kyoto Univ., 9 (1974), 649--673.

\bibitem{19} Hara, T.;
\emph{On the asymptotic behavior of solutions of some third
order ordinary differential equations}. Proc. Japan Acad., 47 (1971).

\bibitem{20} Korkmaz, E.; Tun\c{c}, C.;
\emph{On some qualitative behaviors of
certain differential equations of fourth order with multiple retardations}.
Journal of Applied Analysis and Computation, Volume 6, Number 2, (2016),
336--349.

\bibitem{21} Krasovskii, N. N.;
\emph{On the stability in the large of the solution
of a nonlinear system of differential equations} (in Russian). Prikl. Mat. Meh.
18 (1954) 735--737.

\bibitem{22} Lyapunov, A. M.;
\emph{The general problem of the stability of motion}. Translated from
Edouard Davaux's French translation (1907) of the
1892 Russian original and edited by A. T. Fuller. Taylor \& Francis, Ltd.,
London, 1992.

\bibitem{23} Omeike, P. S. M.;
\emph{Boundedness Of Solutions To Fourth-Order
Differential Equation With Oscillatory Restoring And Forcing Terms}.
Electronic Journal of Differential Equations, 2007, (104) (2007), 1--8.

\bibitem{24} Rauch, L. L.;
\emph{Oscillations of a third order nonlinear autonomous system}.
Contributions to the Theory of Nonlinear Oscillations. Ann. Math.
Stud., 20 (1950), 39--88.

\bibitem{25} Remili, M.; Oudjedi, D. L.;
\emph{Uniform stability and boundedness
of a kind of third order delay differential equations}. Bull. Comput. Appl.
Math., 2 (1) (2014), 25--35.

\bibitem{26} Remili, M.; Oudjedi, D. L.;
\emph{Stability and boundedness of the
solutions of non autonomous third order differential equations with delay}.
Cta Univ. Palacki. Olomuc., Fac. Rer. Nat.,
Mathematica 53 (2) (2014), 139--147.

\bibitem{27} Remili, M.; Beldjerd, D.;
\emph{On the asymptotic behavior of the
solutions of third order delay differential equations}. Rend. Circ. Mat.
Palermo, 63 (3) (2014), 447--455.

\bibitem{28} Remili, M.; Rahmane, M.;
\emph{Sufficient conditions for the
boundedness and square integrability of solutions of fourth-order
differential equations}. Proyecciones Journal of Mathematics., Vol 35, No 1,
(2016) pp. 41-61 .

\bibitem{29} Remili, M.; Rahmane, M.;
\emph{Boundedness and square integrability of solutions of nonlinear
fourth-order differential equations}.
Nonlinear Dynamics and Systems Theory, 16 (2) (2016), 192-205.

\bibitem{30} Reissig, R.; Sansone, G.; Conti, R.;
\emph{Non-linear Differential Equations of Higher Order.
Translated from the German}. Noordhoff
International Publishing, Leyden, 1974.

\bibitem{31} Shair, A.;
\emph{Asymptotic properties of linear fourth order
differential equations}. American mathematical society, 59 (1) (1976), 45--51.

\bibitem{32} Sinha, A. S. C.;
\emph{On stability of solutions of some third and
fourth order delay-differential equations}, Information and Control, 23
(1973), 165-172.

\bibitem{33} Smith, H.;
\emph{An introduction to delay differential equations with
applications to the life sciences}. Texts in Applied Mathematics, 57.
Springer, New York, 2011

\bibitem{34} Tejumola, H. O.;
\emph{Further results on the boundedness and the
stability of certain fourth order differential equations}. Atti Accad. Naz.
Lincei Rend. Cl. Sci. Fis. Mat. Natur., 52 (8) (1972), 16--23.

\bibitem{35} Tiryaki, A.; Tun\c{c}, C.;
\emph{Constructing Liapunov functions for some fourth-order autonomous
differential equations}. Indian J. Pure Applied Math,
 26 (3) (1995), 225--232.

\bibitem{36} Tun\c{c}, C.;
\emph{Some stability results for the solutions of
certain fourth order delay differential equations}. Differential Equations
and Applications, 4 (2004), 165--174.

\bibitem{37} Tun\c{c}, C.;
\emph{Stability and boundedness of solutions to certain
fourth order differential equations}. Electronic Journal of Differential
Equations, (35) (2006), 1--10.

\bibitem{38} Tun\c{c}, C.;
\emph{Some remarks on the stability and boundedness of
solutions of certain differential equations of fourth-order}. Computational
and Applied Mathematics, 26 (1) (2007), 1--17.

\bibitem{39} Tun\c{c}, C.;
\emph{On the stability and the boundedness properties of
solutions of certain fourth order differential equations}. Istanbul U niv.
Fen Fak. Mat. Derg., 54 (1997), 161--173.

\bibitem{40} Tun\c{c}, C.;
\emph{A note on the stability and boundedness results of
solutions of certain fourth order differential equations}. Appl. Math.
Comput., 155 (3) (2004), 837--843.

\bibitem{41} Tun\c{c}, C.;
\emph{Some stability and boundedness results for the
solutions of certain fourth order differential equations}. Acta Univ. Palack.
Olomuc. Fac. Rerum Natur. Math., 44 (2005), 161--171.

\bibitem{42} Tun\c{c}, C.;
\emph{Stability and boundedness of solutions to certain
fourth order differential equations}. Electron. J. Differential Equations,
(35) (2006), 10 pp.

\bibitem{43} Yoshizawa, T.;
\emph{Stability Theory by Liapunovs Second Method}, The
Mathematical Society of Japan, Tokyo, (1966).

\bibitem{44} Wu, X.; Xiong, K.;
\emph{Remarks on stability results for the
solutions of certain fourth-order autonomous differential equations}.
Internat. J. Control., 69 (2) (1998), 353--360.

\end{thebibliography}

\end{document}