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\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)] $0\max \{ f_{\text{1}},g_1,1\}$;
\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 $H_10$, with $| f(t,\phi )| 0$ $(x\neq 0)$ 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)] $\delta_1=\frac{d_1h_0a_1M}{c_0m}
+\frac{c_1M+\delta _0}{a_0m}0$. 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