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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 97, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/97\hfil Existence and nonexistence of periodic solutions]
{Existence and nonexistence of periodic solutions of n-vector
differential equations of orders six and seven}

\author[V. F. Payne, H. O. Tejumola \hfil EJDE-2012/97\hfilneg]
{Victor F. Payne, Haroon Oladipo Tejumola}  % in alphabetical order

\address{Victor F. Payne \newline
Department of Mathematics, University of Ibadan <br>
Ibadan Nigeria}
\email{vpayne34@gmail.com}


\address{Haroon Oladipo Tejumola \newline
Department of Mathematical Sciences, Redeemer's University,
Mowe, Ogun State, Nigeria}
\email{haroontej@yahoo.co.uk}


\thanks{Submitted April 12, 2012. Published June 10, 2012.}
\subjclass[2000]{34C25, 34K05, 34K13}
\keywords{A priori bound; Leray-Schauder fixed-point teorem;
\hfill\break\indent  parameter-dependent system; periodic solution}

\begin{abstract}
 In this article, we extend our earlier results and establish new ones
 on the existence and non-existence of periodic solutions for $n$-vector
 non-dissipative, nonlinear ordinary differential equations. Our
 results involve both the homogeneous and non-homogeneous cases.
 The setting for non-existence results of periodic solutions
 involves a suitably defined scalar function endowed with
 appropriate properties relative to each equation. But the
 framework for proving existence results is via the standard
 Leray-Schauder fixed-point technique whose central theme is the
 verification of a-priori bounded periodic solutions for a
 parameter-dependent system of equations.
\end{abstract}

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

\section{Introduction}

The article by Ezeilo \cite{e1} on sixth-order equations marks the
beginning of systematic study of the problem of existence, and
nonexistence in the homogeneous case, of periodic solutions of
non-dissipative, nonlinear ordinary differential equations of
orders six and above. Bereketoglu \cite{b1} extended Ezeilo's work to
equations of order seven, while Tejumola \cite{t1} widened the scope of
these earlier investigations to more general class of equations
and to situations, which hitherto were not considered.  An
$n$-vector analogue of the result of Bereketoglu \cite[Theorem 1]{b1}
was recently obtained by Tun\c{c}  \cite{t2,t3,t4} in the
seventh order homogeneous case
\begin{align*}
& x^{(6)} + a_1 x^{(5)} + a_2 x^{(4)} + a_3 x^{(3)} + f(x')x'' +
g(x)x' + h(x) \\
&= p(t,x,x',x'',x''',x^{(4)},x^{(5)}
\end{align*}
and
\begin{align*}
&x^{(7)} + a_1 x^{(6)} + a_2 x^{(5)} + a_3 x^{(4)} + a_4
x''' + f(x')x'' + g(x)x' + h(x)\\
& = p(t,x,x',x'',x''',x^{(4)},x^{(5)},x^{(6)}),
\end{align*}
the problem of existence in the non-homogeneous case was however
not considered. Our present investigation arose from our desire to
provide results in the nonhomogeneous case and, more importantly,
to extend our earlier results \cite{t1} to $n$-vector equations.  The
results we have been able to obtain, which include \cite{b1} and \cite{t2} as
special cases, are stated in \S2 and \S3.

To end this section, we introduce some notation.  $\mathbb{R}^n$ denotes
the usual $n$-dimensional real Euclidean space with inner product
$ \langle X,Y\rangle = \sum^n_{i=1}x_iy_i$ and norm
$\|X\| = \big(\sum^n_{i=1} x^2_i\big)^{1/2}$, for
$X = (x_1,x_2,\dots, x_n)$, $ Y =(y_1,y_2,\dots, y_n)\in \mathbb{R}^n$.
For a constant $n\times n$ matrix $A$, we define sign of
$A$, $\operatorname{sgn}A$, by $\operatorname{sgn}A=1$ or $-1$ according as $A$ is positive
definite or negative definite and we write $\gamma_A = \operatorname{sgn}A$. $A$
is definite if $A$ is positive or negative definite. For any
function $ h :\mathbb{R}^n \to \mathbb{R}^n$, $ J(h(X))$ denotes
the Jacobian of $h$, if it exists.

\section{Statement of Results}

We start with sixth order nonlinear differential equations of the form
\begin{equation} \label{e2.1}
\begin{split}
&X^{(6)}+A_1X^{(5)}+A_2X^{(4)}+f_3(\dot{X}, \ddot{X})\dddot{X}
+f_{4}(\dot{X})\ddot{X}+f_5(\dot{X}, \ddot{X})\dot{X} + f_6(X)\\
&= P_1(t,X,\dot{X},\dots, X^{(5)})
\end{split}
\end{equation}
where $A_1,A_2$ are constant $n\times n$ symmetric matrices, $f_3, f_5$
are symmetric $n\times n$ continuous matrices, $f_4$ is an $n\times n$
continuous matrix, $ f_6 : \mathbb{R}^n \to \mathbb{R}^n$,
$P_1 : R \times \mathbb{R}^n \times \dots \times \mathbb{R}^n \to \mathbb{R}^n$
are continuous functions and
$P_1(t+\omega, X, \dot{X},\dots, X^{(5)}) = P_1(t, X, \dot{X},\dots, X^{(5)})$
for some $\omega>0$.  It will be assumed further that the Jacobians
$ J(f_4(\dot{X})\dot{X}), J(f_6(X))$ exist and are continuous.

\begin{theorem} \label{thm1}
Let $A_1$ be definite and let
\begin{equation} \label{e2.2}
 f_6(0) =0,\quad f_6(X) \neq 0\quad \text{for } X \neq 0.
\end{equation}
Suppose that $ (\gamma_{A_3}f_3)$ is negative  semi-definite and that
 $(\gamma_{A_1}f_5)$ is positive definite.
Suppose further that
\begin{equation} \label{e2.3}
 \inf_{X_2,X_3}\frac{ \langle (\gamma_{A_1}f_5)X_2,X_2\rangle}{\|X_2\|}
> \frac{1}{2a_1} \frac{\langle f^2_3X_2,.X_3\rangle}{\|X_2\|^2},\quad
 X_2 \neq 0,
\end{equation}
where $a_1>0$ is a constant such that
\begin{equation} \label{e2.4}
 \langle(\gamma_{A_1}A_1)Y,Y\rangle \geq a_1\|Y\|^2\quad\text{for all }
 Y \in \mathbb{R}^n.
\end{equation}
Then \eqref{e2.1} with $P_1 \equiv 0$ has no nontrivial periodic solution
of any period.
\end{theorem}

\begin{theorem} \label{thm2}
Let all the conditions of Theorem \ref{thm1}  hold, except for \eqref{e2.2} and \eqref{e2.3}.
Let
\begin{equation} \label{e2.5}
 f_6(X)\operatorname{sgn}X \to +\infty (-\infty)\quad\text{as } \|X\| \to\infty.
\end{equation}
Suppose further that there exist constants
 $\beta_3>0$, $B_1>0$, $B_2 \geq 0$, with $B_2$ sufficiently small,
such that
\begin{gather} \label{e2.6}
 \inf_{X_2,X_3}  \frac{ \langle(\gamma_{A_1}f_5)X_2,X_2\rangle}{\|X_2\|^2}
> \frac{1}{2a_1}\beta^2_3,\quad _2\neq 0,\\
\label{e2.7}
\|P_1(t,X_1,X_2,\dots,X_6)\| \leq B_1 + B_2(\|X_2\|+\|X_3\|)
\end{gather}
for all $t\in R$ and $X_1,X_2,\dots, X_6 \in \mathbb{R}^n$, where
$\beta_3>0$ is a constant such that
$\|f_3(X_2,X_3)\| \leq \beta_3$.
Then \eqref{e2.1} has at least one
periodic solution of period $\omega$.
\end{theorem}

Note the absence of any restrictions on $A_2$ and $f_6$ in
Theorem \ref{thm1}, similarly in Theorem \ref{thm2} except for the additional condition
\eqref{e2.5} required to ensure uniform boundedness of $X_1$. Our
results, which place restrictions on even-subscript terms (that
is, $f_6 $), concern a slightly different class of sixth order
equations of the form
\begin{equation} \label{e2.8}
\begin{aligned}
&X^{(6)}+A_1X^{(5)}+A_2X^{(4)}+A_3\dddot{X}+g_4(\dot{X}, \ddot{X})\ddot{X}
+g_5(X)\dot{X}+g_6(X,\dot{X},\ddot{X}) \\
&= P_2(t,X,\ddot{X},\dots,X^{(5)}).
\end{aligned}
\end{equation}
Here, $A_1,A_2,A_3$ are constant $n\times n$ symmetric matrices, $g_4, g_5$
 are  symmetric $n\times n$ continuous matrices, $J(g_5(X))$ exists and is
continuous, $g_6$ and $P_2$ are continuous $n$-vector functions of their
respective arguments and
 $ P_2(t+\omega, X_1,X_2,\dots,X_6) = P_2(t,X_1,X_2,\dots, X_6)$ for some
 $\omega>0$.  The results are as follows.

\begin{theorem} \label{thm3}
Let $A_2$ be negative definite and let
\[
g_6(0,X_2,X_3) = 0\quad \text{and} \quad g_6(X_1,X_2,X_3) \neq 0\quad
\text{if } X_1\neq 0.
\]
Suppose that
\begin{equation} \label{e2.9}
\sup_{X_1,X_2,X_3} \frac{\langle g_6(X_1,X_2,X_3),X_1\rangle}{\|X_1\|^2}
< \frac{1}{4a_2} \frac{ \langle g^2_4(X_2.X_3)X_1,X_1\rangle}{\|X_1\|^2},\quad
X_1 \neq 0
\end{equation}
where $a_2<0$ is a constant such that
\begin{equation} \label{e2.10}
\langle A_2Y,Y\rangle < a_2\langle Y,Y\rangle\quad\text{for all }
Y\in \mathbb{R}^n.
\end{equation}
Then  \eqref{e2.8} with $P_2\equiv 0$ has no nontrivial periodic solution
of any period.
\end{theorem}


\begin{theorem} \label{thm4}
Let $A_2$ be negative definite so that \eqref{e2.10} holds, and let $\beta_4>0$ be a
constant such that
\begin{equation} \label{e2.11}
\beta_4 = \inf \|g_4(X_4,X_3)\|.
 \end{equation}
Suppose that
\begin{gather} \label{e2.12}
\sup_{X_1,X_2,X_3} \frac{ \langle g_6(X_1,X_2,X_3),X_1\rangle}{\|X_1\|^2}
< \frac{1}{4a_2} \beta^2_4,\quad X_1 \neq 0, \\
\label{e2.13}  \|P_2(t,X_1,X_2,\dots,X_6\| \leq
B^*_1+B^*_2(\|X_1\|+\|X_2\|+\|X_3\|)
\end{gather}
where $B^*_1>0$, $B^*_2\geq 0$ are constants, with $B^*_2$ sufficiently small.
Then  \eqref{e2.8} has at least one $\omega$-periodic solution.
\end{theorem}

Theorems \ref{thm1}--\ref{thm4} are $n$-dimensional analogue of the results in \cite[\S2]{t1}.
Note also that Theorem \ref{thm3} holds true (as in \cite[Theorem 3]{t1}) with $g_4$
and $g_6$ depending also on $ \dddot{X}, X^{(4)}$ and $X^{(5)}$.

\section{Further Results}

 We now state some parallel results in the seventh order case.
 The equations are of the form
\begin{gather}
\label{e3.1}
\begin{aligned}
&X^{(7)}+\sum^4_{k=1}A_kX^{(7-k)}+\varphi_5(\dot{X}, \ddot{X})\ddot{X}
+ \varphi_6(X)\dot{X} + \varphi_7(X,\dot{X},\ddot{X})\\
& = Q_1(t,X,\dot{X},\dots,X^{(6)})
\end{aligned}\\
\label{e3.2}
\begin{aligned}
& X^{(7)}+\sum^3_{k=1}A_kX^{(7-k)}+\psi_4(\dot{X},\ddot{X}, \dddot{X})\dddot{X}
+ \psi_5(\dot{X})\ddot{X} + \psi_6(\dot{X},\ddot{X},\dddot{X})+\psi_7(X)\\
&=Q_2(t,X,\dot{X},\dots,X^{(6)})
\end{aligned}
\end{gather}
where $A_i, i=1,2,3,4$ are constant $n\times n$ symmetric matrices,
$\varphi_5, \varphi_6, \psi_4$ and $\psi_5$ are symmetric $n\times n$
continuous matrices, $\varphi_7, \psi_7, Q_1$ and $Q_2$ are continuous
$n$-vector functions of their respective arguments,
$$
Q_i(t+\omega,X_1,X_2,\dots,X_7) = Q_i(t,X_1,X_2,\dots,X_7),
$$
$i = 1,2$,  for some $\omega>0$ and $ J(\varphi_6(X))$,
 $ J(\psi_5(\dot{X})\ddot{X})$  exist and are continuous.

Our first result concerns equation \eqref{e3.1} with restrictions on terms
with odd subscripts.

\begin{theorem} \label{thm5}
Let $A_1, A_3$ be definite matrices and let
\begin{equation} \label{e3.3}
 \gamma_{A_1}, \gamma_{A_3} =-1.
\end{equation}
Suppose that $ \varphi_7(0, X_2,X_3) = 0$,
 $\varphi_7(X_1,X_2,X_3) \neq 0$ ($X_1\neq 0)$, and that
\begin{equation} \label{e3.4}
\inf_{X_1,X_2,X_3} \frac{\langle \gamma_{A_3}\varphi_7(X_1,X_2,X_3),X_1\rangle}
{\|X_1\|^2} > \frac{1}{4a_3} \frac{ \langle \varphi_5(X_2,X_3)X_1,X_1\rangle}
{\|X_1\|^2},\; X_1 \neq 0
\end{equation}
where $a_3>0$ is a constant such that
\begin{equation} \label{e3.5}
\langle(\gamma_{A_3}A_3)Y,Y\rangle \geq a_3\langle Y,Y\rangle\quad
\text{for all } Y\in \mathbb{R}^n.
\end{equation}
Then  \eqref{e3.1} with $Q_1\equiv 0$ has no nontrivial periodic solution
of any period.
\end{theorem}

Theorem \ref{thm5} extends the result in \cite{t2}, and is an $n$-dimensional analogue
of the nonexistence result \cite[Theorem 2]{t1}.

\begin{theorem} \label{thm6}  Let $A_1,A_3$ be definite matrices such that
 \eqref{e3.3} holds.  Let $\beta_5>0$ be a constant such that
 $\sup\|\varphi_5(X_2,X_3)\| \leq \beta_5$.
Suppose that
\begin{gather} \label{e3.6}
\inf_{X_1,X_2,X_3}\frac{\langle \gamma_{A_3}\varphi_7(X_1,X_2,X_3),X_1\rangle}
{\|X_1\|^2} > \frac{1}{4a_3} \beta^2_5,\quad X_1\neq 0,
\\
\label{e3.7}  \|Q_1(t,X_1,X_2,\dots, X_7\| \leq C_1+C_2(\|X_1\|+\|X_2\|+\|X_3\|)
 \end{gather}
where $a_3>0$ is a constant satisfying \eqref{e3.5} and
 $C_1>0$, $C_2 \geq 0$ are constants, with $C_2$ sufficiently small.
Then  \eqref{e3.1} has at least one periodic solution with period $\omega$.
\end{theorem}

Our results in the other direction (that is, involving even subscripts)
 concern equation \eqref{e3.2}, and are as follows.

\begin{theorem} \label{thm7}
Let $A_2$ be negative definite and let
\begin{equation} \label{e3.8}
\psi_7(0) = 0,\quad \psi_7(X_1) \neq 0\quad \text{if }
 X_1 \neq 0,\quad \psi_6(0, X_3, X_4) = 0.
\end{equation}
Suppose that
\begin{equation} \label{e3.9}
\sup_{X_2,X_3,X_4}\frac{ \langle \psi_6(X_2,X_3,X_4),X_2\rangle}{\|X_2\|^2}
< \frac{1}{4a_2} \frac{ \langle\psi^2_4(X_2,X_3,X_4)X_2,X_2\rangle}{\|X_2\|^2},
\quad X_2 \neq 0,
\end{equation}
where $a_2<0$ is a constant satisfying \eqref{e2.10}.
Then  \eqref{e3.2}, with $Q_2 \equiv 0$, has no
nontrivial periodic solution of any period.
\end{theorem}

\begin{theorem} \label{thm8}
Let $A_2$ be negative definite so that \eqref{e2.10} holds and let
 $\beta^*_4>0$ be a constant such that
 $ \inf\|\psi_4(X_2,X_3,X_4)\| \leq \beta^*_4$.
Suppose that
\begin{gather} \label{e3.10}
\sup_{X_2,X_3,X_4}\frac{\langle \psi_6(X_2,X_3,X_4),X_2\rangle}{\|X_2\|^2}
 <\frac{1}{4a_2}\beta^{*2}_4,\; X_2\neq 0, \\
\label{e3.11}
\begin{gathered}
 \psi_7(X)\operatorname{sgn}X\to +\infty (-\infty)\quad \text{as}\quad
 \|X\|\to\infty\\
 \|Q_2(t,X_1,X_2,\dots,X_7)\| \leq C^*_1 + C^*_2(\|X_2\|+\|X_3\|+\|X_4\|)
\end{gathered}
\end{gather}
where $C^*_1>0$, $C^*_2\geq 0$ are constants, with $C^*_2$ sufficiently small.
 Then  \eqref{e3.2} has at least one periodic solution of period $\omega$.
\end{theorem}

Theorems \ref{thm6}, \ref{thm7}, \ref{thm8}
 are $n$-dimensional analogue of \cite[Theorems 3, 6, 7]{t1}.

The procedure for the proof the theorems is as in \cite{b1,e1,t1}.
 For nonexistence of periodic solutions, a suitably defined scalar function
 with appropriate properties relative to each equation is required; while
for the existence of periodic solutions, the setting for each proof is the
now standard Leray-Schauder fixed-point technique, the central problem of
 which is the verification of an a-priori bound for all possible $\omega$-periodic
solutions of a suitably defined parameter-dependent system of equations.
We shall outline the salient points in the proof of each theorem in
sections 4 and 5.

\section{Proofs of Theorems \ref{thm1}, \ref{thm2}, \ref{thm3}, \ref{thm4}}

 Consider, instead of equation \eqref{e2.1} with $P_1 \equiv 0$,
the equivalent system
\begin{equation} \label{e4.1}
\begin{gathered}
 \dot{X}_i =  X_{i+1},\quad i = 1,2,3,4,5,\quad X_1\equiv X,\\
 \dot{X}_6 =  -A_1X_6-A_2X_5-f_3(X_2,X_3)X_4-f_4(X_2)X_3-f_5(X_2,X_3)X_2-f_6(X_1),
\end{gathered}
\end{equation}
together with the scalar function $ W = W(X_1,X_2,\dots, X_6)$ defined by
\begin{equation} \label{e4.2}
 W = \gamma_{A_1}V, \quad V = V_0+V_1,
\end{equation}
where
\begin{gather} \label{e4.3}
 V_0 = -\int^1_0 \langle\sigma f_4(\sigma X_2)X_2,X_2\rangle d\sigma
 -\int^1_0\langle f_6(\sigma X_1), X_1\rangle d\sigma,\\
\label{e4.4}
  V_1 = -\langle X_2, X_6+A_1X_5+A_2X_4\rangle + \langle X_3,X_5+A_1X_4\rangle
 + \frac{1}{2}\langle A_2X_3,X_3\rangle - \frac{1}{2}\langle X_4,X_4\rangle.
\end{gather}
Let $ (X_1,X_2,\dots,X_6) \equiv (X_1(t), X_2(t),\dots, X_6(t))$
be an arbitrary nontrivial periodic solution of \eqref{e4.1} of period $\alpha$ say.
 Then since
\[
-\dot{V}_0 = \langle f_4(X_2)X_2,X_3\rangle + \langle f_6(X_1),X_2\rangle,
\]
as can be verified as in \cite[\S2]{t2},
we have from \eqref{e4.2}, \eqref{e4.3}, \eqref{e4.4} and \eqref{e4.1} that
\[
\dot{V} = \langle A_1X_4,X_4\rangle + \langle f_5X_2,X_2\rangle
+ \langle f_3X_2,X_4\rangle
\]
Thus, by \eqref{e2.4},
\begin{equation} \label{e4.5}
\begin{aligned}
\dot{W}
&= \langle(\gamma_{A_1}A_1)X_4,X_4\rangle
 + \langle(\gamma_{A_1}f_5)X_2,X_2\rangle
 + \langle (\gamma_{A_1}f_3)X_2,X_4\rangle\\
& \geq  \frac{1}{2}a_1\langle X_4,X_4\rangle
 +\frac{1}{2}a_1\|X_4+\frac{1}{a_1}(\gamma_{A_1}f_3)X_2\|^2
 +\langle(\gamma_{A_1}f_5)X_2,X_2\rangle\\
&\quad -\frac{1}{2a_1} \langle f^2_3X_2,X_2\rangle\\
& \geq  \frac{1}{2}a_1\langle X_4,X_4\rangle
 + \langle(\gamma_{A_1}f_5)X_2,X_2\rangle
 -\frac{1}{2}a_1\langle f^2_3X_2,X_2\rangle.
\end{aligned}
\end{equation}
The hypothesis \eqref{e2.3} now implies that $\dot{W} \geq 0$,
so that $W$ is monotone increasing.  By \eqref{e4.5} and the periodicity
of $W(t)$, it will follow, as in \cite[\S3]{b1}, that
 $ X_1 = X_2 = X_3 = X_4 = X_5 = X_6 = 0$.

Turning now to Theorem \ref{thm2}, consider the parameter $\lambda$-dependent system
\begin{equation}\label{e4.6}
\begin{gathered}
 \dot{X}_i = X_{i+1},\quad i = 1,2,\dots, 5\\
\begin{aligned}
 \dot{X}_6 &= -A_1X_6-A_2X_5-\lambda f_3(X_2,X_3)X_4-\lambda f_4(X_2)X_3
-(1-\lambda)a_5\gamma_{A_1}X_2\\
&\quad -\lambda f_5(X_2,X_3)X_2
-(1-\lambda)a_6X_1-\lambda f_6(X_1)+\lambda P_1(t,X_1,X_2,\dots, X_6),
\end{aligned}
\end{gathered}
\end{equation}
where $0 \leq \lambda \leq 1$,
$a_6$ is a constant chosen as positive or negative according as
$ f_6(X)\operatorname{sgn}X \to +\infty$ or $-\infty$ as $ \|X\|\to\infty$,
and $a_5$ is a constant chosen, in view \eqref{e2.6}, such that
\begin{equation} \label{e4.7}
\frac{ \langle(\gamma_{A_1}f_5)X_2,X_2\rangle}{\|X\|^2} \geq a_5
> \frac{1}{2a_1}\beta^2_3.
\end{equation}
Clearly the system \eqref{e4.6} with $\lambda = 0$, or equivalently, the equation
\[
X^{(6)}+A_1X^{(5)} + A_2X^{(4)}+(a_5\gamma_{A_1})\dot{X}+a_6X = 0.
\]
has no nontrivial periodic solution of any period.
 Therefore, to prove the theorem, it suffices (by the Lerray-Schauder
technique \cite{b1})
 here to establish an a-priori bound
\begin{equation} \label{e4.8}
 \max_{0\leq t \leq\omega}(\|X_1(t)\|+\|X_2(t)\|+\dots+\|X_6(t)\|) \leq D
\end{equation}
for all possible $\omega$-periodic solutions $(X_1(t), X_2(t),\dots, X_6(t))$
of \eqref{e4.6} with $D>0$ a finite constant independent of $\lambda$
and of solutions.  Indeed, in view of the remark in \cite[\S4]{t2} and the form
of system \eqref{e4.6}, \eqref{e4.8} will follow once an estimate of the form
\[
\max_{0 \leq t \leq\omega}(\|X_1(t)\|
+\|X_2(t)\|+\|X_3(t)\|+\|X_4(t)\|) \leq D.
\]
is obtained, with $D>0$ as in \eqref{e4.8}.

To this end, consider the function $W_{\lambda} = W_{\lambda}(X_1,X_2,\dots, X_6)$
defined by
\begin{equation} \label{e4.9}
 W_{\lambda} = \gamma_{A_1}V_{\lambda},\quad
 V_{\lambda} = \lambda V_0 + V_1
\end{equation}
with $V_0, V_1$ given by \eqref{e4.3} and \eqref{e4.4}.
Let $(X_1, X_2, \dots, X_6)$ be an arbitrary $\omega$-periodic solution
of \eqref{e4.6}.  Then on differentiating $W_{\lambda}$ and using
\eqref{e4.9}, \eqref{e4.3} and \eqref{e4.4} we have that
\begin{align*}
\dot{W}_{\lambda}
&= \langle(\gamma_{A_1}A_1)X_4,X_4\rangle
+ \langle(1-\lambda)a_5X_2+\lambda(\gamma_{A_1}f_5)X_2,X_2\rangle\\
&\quad +\lambda\langle(\gamma_{A_1}f_3)X_4,X_2\rangle
- \lambda\langle P_1, X_2\rangle,
\end{align*}
so that by \eqref{e2.4}, \eqref{e2.7} and \eqref{e4.7}
\begin{equation} \label{e4.10}
\begin{split}
\dot{W}_{\lambda}
&\geq \frac{1}{2}a_1\langle X_4,X_4\rangle+a_4\langle X_2,X_2\rangle
 + \frac{1}{2}a_1\|X_4+\frac{1}{a_1}(\gamma_{A_1}f_3)X_2\|^2\\
&\quad -\frac{1}{2a_1}\langle f^2_3X_2,X_2\rangle
 -\frac{3}{2}B_2(\|X_2\|^2+\|X_3\|^2)-B_2\|X_2\|\\
&\geq  \frac{1}{2}a_1\langle X_4,X_4\rangle
 +(a_5-\frac{1}{2a_1}\beta^2_3)\langle X_2,X_2\rangle
 - B_1\|X_2\|\\
&\quad -\frac{3}{2}B_2(\|X_2\|^2+\|X_3\|^2)\\
&\geq  D_1(\|X_2\|^2+\|X_4\|^2)+(D_1\|X_2\|^2-B_1\|X_2\|)
 + [D_1\|X_4\|^2\\
&\quad -\frac{3}{2}(\|X_2\|^2+\|X_4\|^2)]
\end{split}
\end{equation}
where $ 2D_1 = \min[\frac{1}{2}a_1, (a_5-\frac{1}{2a_1}\beta^2_3)]$.
But
\begin{equation} \label{e4.11}
\int^{\omega}_0\|X_{1+i}\|^2dt
\leq \frac{\omega^2}{4\pi^2}\int^{\omega}_0\|X_{2+i}\|^2dt,\quad i = 1,2,
\end{equation}
for any $\omega$-periodic solution $ (X_1,X_2,\dots, X_6)$ of \eqref{e4.6}.
 Thus, on integrating \eqref{e4.10} and using the $\omega$-periodicity of
$W_{\lambda}$ and \eqref{e4.11}, it will follow that
\[
0 \geq D_1\int^{\omega}_0(\|X_2\|^2+\|X_4\|)^2)dt-D_2\omega
\]
where $D_2>0$ is a constant chosen so that $ D_1\|X_2\|^2 - B_1\|X_2\| \geq -D_2$,
and $B_2$ is fixed such that
\[
B_2 \leq \frac{2}{3}\Big[\frac{\omega^2}{4\pi^2}
\big(1+\frac{\omega^2}{4\pi^2}\big)\Big]^{-1}D_1.
\]
Hence
\begin{equation} \label{e4.12}
 \int^{\omega}_0\|X_2\|^2dt \leq D^{-1}_1D_2\omega,\quad
 \int^{\omega}_0\|X_4\|^2dt \leq D^{-1}_1D_2\omega,
\end{equation}
and by periodicity of the solution,
$\|X_2(t)\| \leq D_3,\;\; \|X_3(t)\| \leq D_3$
for some constant $D_3>0$.  Multiplying \eqref{e4.6} by $\operatorname{sgn}X_1$,
and using the continuity of $f_3, f_4, f_5$ and \eqref{e2.7},
 it can be readily shown, in view of \eqref{e2.5}, that
$ \|X_1(t_0)\|\leq D_4$, $t_0 \in [0, \omega]$, and hence that
 $\|X_1(t)\| \leq D_5$ for some constants $D_4>0$.
 The estimate for $\|X_4(t)\|$ can be obtained as in \cite{e1} and the desired
estimate will follow.

We turn next to the proof of Theorems \ref{thm3} and \ref{thm4}.
Consider  equation \eqref{e2.8}, with $P_2 \equiv 0$, in the equivalent system form
\begin{equation} \label{e4.13}
\begin{gathered}
 \dot{X}_i = X_{i+1},\;\; i = 1,2,\dots, 5\\
 \dot{X}_6 = A_1X_6-A_2X_5-A_3X_4-g_4(X_2,X_3)X_3-g_5(X_1)X_2-g_6(X_1,X_2,X_3),
\end{gathered}
\end{equation}
together with the scalar function $ W = W(X_1,X_2,\dots, X_6)$ defined
 by $W_0+W_1$, where
\begin{gather} \label{e4.14}
 W_0 = \int^1_0\langle \sigma g_5(\sigma X_1)X_1,X_1\rangle d\sigma,\\
 \label{e4.15}\begin{aligned}
W_1&=\langle X_1,X_6+A_1X_5+A_2X_4+A_3X_3\rangle
 - \langle X_2,X_5+A_1X_4+A_2X_3\rangle\\
&\quad + \langle X_3,X_4\rangle
  + \frac{1}{2}\langle A_1X_3,X_3\rangle
 -\frac{1}{2}\langle A_3X_2,X_2\rangle.
\end{aligned}
\end{gather}
For any nontrivial periodic solution $(X_1,X_2,\dots, X_6)$ of \eqref{e4.13}
 of period $\alpha$ say, it is readily verified that
\[
\dot{W} = \langle X_4, X_4\rangle - \langle A_2X_3,X_3\rangle
- \langle g_4X_3,X_1\rangle - \langle g_6(X_1),X_1\rangle,
\]
so that by \eqref{e2.10} and \eqref{e2.9},
\begin{align*}
\dot{W}
 & \geq  \langle X_4,X_4\rangle-a_2\|X_3+\frac{1}{2a_2}g_4X_1\|^2
 -\langle g_6(X_1),X_1\rangle + \frac{1}{4a_2}\langle g^2_4X_1,X_1 \rangle\\
& \geq  \langle X_4,X_4\rangle - \langle g_6(X_1),X_1\rangle
 + \frac{1}{4a_2}\langle g^2_4X_1,X_1\rangle> 0.
\end{align*}
The conclusion of Theorem \ref{thm3} now follows from the arguments in \cite[\S3]{b1}.

For the proof of Theorem \ref{thm4}, consider the parameter $\lambda$-dependent system
\begin{equation} \label{e4.16}
\begin{gathered}
 \dot{X}_i = X_{i+1},\quad i = 1,2,\dots, 5,\quad 0 \leq\lambda \leq 1,\\
\begin{aligned}
 \dot{X}_6 &= -A_1X_6-A_2X_5-A_3X_4-g^{\lambda}_4(X_2,X_3)X_3\\
&\quad -\lambda g_5(X_1)X_2-g_6^{\lambda}(X_1,X_2,X_3)+\lambda P_2,
\end{aligned}
\end{gathered}
\end{equation}
where
\begin{equation} \label{e4.17}
\begin{gathered}
 g^{\lambda}_4(X_2,X_3) = (1-\lambda)\beta_4 I+\lambda g_4(X_2,X_3), \\
 g^{\lambda}_6(X_1,X_2,X_3) = (1-\lambda)a_6X_1+\lambda g_6(X_1,X_2,X_3),
\end{gathered}
\end{equation}
where $I$ is the identity $n\times n$ matrix, $\beta_4>0$ is defined by
\eqref{e2.11} and $a_6<0$ is a constant chosen, in view of \eqref{e2.12}, such that
\begin{equation} \label{e4.18}
 \frac{\langle g_6(X_1,X_2,X_3),X_1\rangle}{\|X_1\|^2}
 < a_6 < \frac{1}{4a_2}\beta^2_4,\;\;X_1 \neq 0
\end{equation}
The scalar function $ W^{\lambda}= W^{\lambda}(X_1,X_2,\dots,X_6)$ is defined by
$ W^{\lambda} = \lambda W_0 +W_1$,
 with $W_0, W_1$ given by \eqref{e4.14} and \eqref{e4.15}.
By \eqref{e4.16}, \eqref{e4.17}, \eqref{e2.10}, \eqref{e2.13} and \eqref{e4.18}
it can be verified that
\begin{equation} \label{e4.19}
\begin{split}
\dot{W}^{\lambda} &= \langle X_4,X_4\rangle-\langle A_2X_3,X_3\rangle
 -\langle g^{\lambda}_4X_3,X_1\rangle
 - \langle g_6^{\lambda}(X_1,X_2,X_3),X_1\rangle + \lambda\langle P_2,X_1\rangle\\
& \geq  \langle X_4,X_4\rangle-a_2\|X_3+\frac{1}{2a_2}g^{\lambda}_4X_1\|^2
 -\langle g^{\lambda}_6(X_1,X_2,X_3),X_1\rangle\\
 &\quad + \frac{1}{4a_2}\langle (g^{\lambda}_4)^2X_1,X_1\rangle
 -|\langle p_2X_1\rangle|\\
& \geq  \|X_4\|^2+D_6\|X_1\|^2-B^*_1\|X_1\| - B^*_2(\|X_1\|^2+\|X_2\|^2+\|X_2\|^2),
\end{split}
\end{equation}
where $ D_6 \equiv \big(\frac{1}{4a_2}\beta^2_4-a_6\big)>0$.
 Now on integrating \eqref{e4.19} and using the $\omega$-periodicity of
$W^{\lambda}$ and \eqref{e4.11}, it will follow readily that
\begin{equation} \label{e4.20}
0 \geq \int^{\omega}_0\big(\frac{1}{2}D_6-B^*_2\big)\|X_1\|^2dt
+ \int^{\omega}_0\big\{1-B^*_2\big(\frac{\omega^2}{4\pi^2}
+ \frac{\omega^4}{16\pi^4}\big)\big\}\|X_4\|^2dt-D_7,
 \end{equation}
for some constant $D_7>0$ such that $\frac{1}{2}D_6\|X_1\|^2-B^*_1\|X_1\|\geq -D_7$.
 Fix $B^*_2$ such that
\[
B^*_2 < \min\big[\frac{1}{2}D_6, \big(\frac{\omega}{2\pi}\big)^{-2}
\big(1+\frac{\omega^2}{4\pi^2}\big)^{-1}\big].
\]
Then from \eqref{e4.20},
\begin{equation} \label{e4.21}
\int^{\omega}_0\|X_1\|^2dt \leq D_7D^{-1}_8,\quad
\int^{\omega}_0 \|X_4\|^2dt \leq D_4D^{-1}_9,
\end{equation}
where $ D_8 = (\frac{1}{2}D_6-B^*_2)>0$,
$ D_9 = [1-B^*_2(\frac{\omega^2}{4\pi^2}+\frac{\omega^4}{16\pi^4})]>0$,
and by \eqref{e4.11}
\begin{equation} \label{e4.22}
\int^{\omega}_0\|X_2\|^2dt \leq D_{10},\quad
\int^{\omega}_0\|X_3(t)\|^2dt \leq D_{10}
\end{equation}
for some $D_{10}>0$.  Since \eqref{e4.21} implies the existence of a
$t_0 \in [0, \omega]$ and a constant $D_{11}>0$ such that
$\|X(t_0)\| \leq D_{11}$, it is clear, by periodicity, from \eqref{e4.21}
and \eqref{e4.22}, that
\[
 \|X_1(t)\| \leq D_{12},\quad
 \|X_2(t)\| \leq D_{12},\quad
 \|X_3(t)\| \leq D_{12}.
\]
for some constant $D_{12}>0$.  Using the arguments in \cite{e1}, the estimate
for $\|X_4(t)\|$ can be easily obtained.

\section{Outline of Proof of Theorems \ref{thm5}, \ref{thm6}, \ref{thm7}, \ref{thm8}}

  Observe first that the results embodied in Theorems \ref{thm5},
\ref{thm6}, \ref{thm7} and \ref{thm8} for
seventh order equations are essentially the same as those in
Theorems \ref{thm1}, \ref{thm2}, \ref{thm3} and \ref{thm4}
 for sixth order equations.
Since the proofs of Theorems \ref{thm5}--\ref{thm8} require the same arguments as
those employed for Theorems \ref{thm1}--\ref{thm4} in \S4, with some obvious modifications,
 we shall merely indicate here the appropriate equivalent system of
equations and the scalar functions required in each case, and corresponding
 modifications in arguments.

We start with Theorem \ref{thm5}.  The appropriate equivalent
(to \eqref{e3.1} with $Q_1=0$) system is
\begin{equation} \label{e5.1}
\begin{gathered}
 \dot{X}_i = X_{i+1},\quad i = 1,2,\dots, 6\\
\begin{aligned}
 \dot{X}_1 &=-A_1X_7-A_2X_6-A_3X_5-A_4X_4-\varphi_5(X_2,X_3)X_3\\
&\quad -\varphi_6(X_1)X_2-\varphi_7(X_1,X_2,X_3),
\end{aligned}
\end{gathered}
\end{equation}
and the scalar function is given by
\begin{equation} \label{e5.2}
 V = \gamma_{A_3}U,\;\; U = U_0+U_1
\end{equation}
where
\begin{gather} \label{e5.3}
\begin{aligned}
U_1 &= -\langle X_1,X_7+\sum^4_{k=1}A_kX_{7-k}\rangle + \langle X_2,X_6
 +\sum^3_{k=1}A_kX_{6-k}\rangle\\
&\quad -\langle X_3,X_5+A_1X_4\rangle
 +\frac{1}{2}\langle X_4,X_4\rangle + \frac{1}{2}\langle A_4X_2,X_2\rangle
- \frac{1}{2}\langle A_2X_3,X_3\rangle
\end{aligned}\\
\label{e5.4}
 U_0 = \int^1_0\langle\sigma\varphi_6(\sigma X_1)X_1,X_1\rangle d\sigma
\end{gather}
From \eqref{e5.2}, \eqref{e5.3}, \eqref{e5.4} and \eqref{e5.1} it will be clear,
on proceeding  as in \S4, that $\dot{V} \geq 0$.

For the proof of Theorem \ref{thm6}, observe first from \eqref{e3.6} that there exists a
constant $a_7>0$ such that
\begin{equation} \label{e5.5}
 \frac{ \langle \gamma_{A_3}\varphi_7(X_1,X_2,X_3),X_1\rangle}{\|X_1\|^2}
\geq a_7 > \frac{1}{4a_3}\beta^2_5,\quad X_1\neq 0.
\end{equation}
Set
\begin{gather*}
\varphi^{\lambda}_7(X_1,X_2,X_3)
= (1-\lambda)\gamma_{A_3}a_7X_1+\lambda\varphi_7(X_1,X_2,X_3),\quad
 0 \leq \lambda \leq 1,\\
\varphi^{\lambda}_5(X_2,X_3) = (1-\lambda)\beta_5I + \lambda\varphi_5(X_2,X_3),
\quad I \text{ the identity $n\times n$ matrix.}
\end{gather*}
Then, by \eqref{e5.5} and the fact that $ \|\varphi_5(X_2,X_3)\| \leq \beta_5$,
it will follow that
\begin{equation} \label{e5.6}
\|\varphi^{\lambda}_5(X_2,X_3)\| \leq \beta_5,\quad
\frac{ \langle \gamma_{A_3}\varphi_7^{\lambda}(X_1,X_2,X_3), X_1\rangle}{\|X_1\|^2}
 \geq a_7,\quad X_1 \neq 0.
\end{equation}
With $\varphi^{\lambda}_5,\;\varphi_7^{\lambda}$ defined as above and
satisfying \eqref{e5.6}, the appropriate equivalent (to \eqref{e3.1})
 system to consider is
\begin{equation} \label{e5.7}
\begin{gathered}
 \dot{X}_i = X_{i+1},\quad i = 1,2,\dots, 6\\
\begin{aligned}
 \dot{X}_7 &=-A_1X_7-A_2X_6-A_3X_5-A_4X_4-\varphi^{\lambda}_5(X_2,X_3)X_3
 -\lambda\varphi_6(X_1)\\
&\quad -\varphi^{\lambda}_7(X_1,X_2,X_3)+\lambda Q_1,\quad 0 \leq \lambda \leq 1,
\end{aligned}
\end{gathered}
\end{equation}
and the scalar function $V^{\lambda}$ is defined by
\begin{equation} \label{e5.8}
 V^{\lambda} = \gamma_{A_3}U,\quad  U = \lambda U_0+U_1,
\end{equation}
with $U_0$, $U_1$ given by \eqref{e5.4} and \eqref{e5.3} respectively.
Now, by proceeding as in \S4, using obvious adaptations of the arguments
in \cite[\S4]{t1}, it can be readily shown that
\[
\int^{\infty}_0 \|X_1\|^2dt \leq D_{13},\quad
\int^{\omega}_0\|X_4\|^2dt \leq D_{13}
\]
for some constant $D_{13}>0$, and the desired a-priori bound will follow
as in \cite{e1}.

Turning next to Theorem \ref{thm7}, the appropriate equivalent system is
\begin{equation} \label{e5.9}
\begin{gathered}
 \dot{X}_i = X_{i+1},\quad i = 1,2,\dots, 6\\
\begin{aligned}
\dot{X}_7 &=-A_1X_7-A_2X_6-A_3X_5-\psi_4(X_2,X_3,X_4)X_4\\
&\quad -\psi_5(X_2)X_3 -\psi_6(X_2,X_3,X_4)-\psi_7(X_1)
\end{aligned}
\end{gathered}
\end{equation}
and the scalar function is defined by
\begin{equation} \label{e5.10}
 V = U_0+U_1,
\end{equation}
where
\begin{gather} \label{e5.11}
 U_0 = \int^1_0\langle\psi_7(\sigma X_1),X_1\rangle d\sigma +
\int^1_0\sigma\langle\psi_5(\sigma X_2)X_2,X_2\rangle d\sigma ,\\
\label{e5.12}
\begin{aligned}
U_1 &= \langle X_2,X_7+\sum^3_{k=1}A_kX_{7-k}\rangle
 - \langle X_3,X_6+\sum^2_{k=1}A_kX_{6-k}\rangle+\langle X_4,X_5\rangle\\
&\quad -\frac{1}{2}\langle A_3X_3,X_3\rangle
 + \frac{1}{2}\langle A_1X_4,X_4\rangle.
\end{aligned}
\end{gather}
It is readily shown that $\dot{V} \geq 0$.

Lastly for Theorem \ref{thm8}.  Let $a_6<0$ be a constant chosen,
in view of \eqref{e3.9}, such that
\begin{equation} \label{e5.13}
 \frac{ \langle\psi_6(X_2,X_3,X_4),X_2\rangle}{\|X_2\|^2} \leq a_6
 < \frac{1}{4a_2}\beta^{2*}_4,\quad X_2 \neq 0,
\end{equation}
and set
\begin{equation} \label{e5.14}
\begin{gathered}
\psi^{\lambda}_4(X_2, X_3, X_4)
= (1-\lambda)\beta^*_4I+\lambda\psi_4(X_2,X_3,X_4)\\
 \psi^{\lambda}_6(X_2,X_3,X_4) = (1-\lambda)a_6I+\lambda\psi_6(X_2,X_3,X_4),
\quad 0 \leq \lambda \leq 1,
\end{gathered}
\end{equation}
where $I$ the identity $n\times n$ matrix.  The equivalent system is
\begin{equation} \label{e5.15}
\begin{gathered}
 X_i = X_{i+1}, \quad i = 1,2,\dots,6\\
\begin{aligned}
 \dot{X}_7 &= -A_1X_7-A_2X_6-A_3X_5-\psi^{\lambda}_4(X_2,X_3,X_4)X_4
 -\lambda\psi_5(X_2)X_3\\
&\quad -\psi^{\lambda}_6(X_2,X_3,X_4)-\lambda[\psi_7(X_1)-Q_2]
\end{aligned}
\end{gathered}
\end{equation}
and the scalar function $V^{\lambda} = V^{\lambda}(X_1,X_2,\dots,X_7)$ is defined by
\begin{equation} \label{e5.16}
 V^{\lambda} =\lambda U_0+U_1,
\end{equation}
with $U_0, U_1$ given by \eqref{e5.11} and \eqref{e5.12} respectively.
 It can be readily shown from \eqref{e5.11} to \eqref{e5.16}, that
\begin{equation} \label{e5.17}
\begin{aligned}
\dot{V}^{\lambda}
& \geq  \langle X_5, X_5\rangle - \langle \psi^{\lambda}_6, X_2\rangle
  - \frac{1}{4a_2}\|\psi^{\lambda}_4X_2\|^2-|\lambda\langle X_2,Q_2\rangle|\\
& \geq  \|X_5\|^2+\big(-a_6-\frac{1}{4a_2}\beta^{*2}_4\big)\|X_2\|^2-C^*_1(\|X_2\|)\\
&\quad  -2C^*_2(\|X_2\|^2+\|X_3\|^2+\|X_4\|^2),
\end{aligned}
\end{equation}
where $D_{12}=(-a_6-\frac{1}{4a_2}\beta^{*2}_4)>0$ by \eqref{e5.13}.
 Direct integration of \eqref{e5.17}, for any $\omega$-periodic solution
$(X_1,X_2,\dots, X_7)$ of \eqref{e5.15}, using the $\omega$-periodicity
of $V^{\lambda}$ and \eqref{e4.11}, will yield, for some constants
$D_{15}>0$, $D_{16}>0$,
\begin{equation} \label{e5.18}
\int^{\omega}_0\|X_2\|^2dt \leq D_{15},\quad
\int^{\omega}_0\|X_5\|^2dt \leq D_{16}
\end{equation}
 provided
\[
C^*_2 < \min[\frac{1}{4}D_{14},\frac{2\pi}{\omega}(1+\frac{\omega^2}{4\pi^2})^{-1}].
\]
Clearly, the condition on $\psi_7$ in \eqref{e3.11} implies the existence
of a $t_0 \in [0, \omega]$ such that $\|X_1(t_0)\| \leq D_{17}$,
for some constant $D_{17}>0$. Thus, from \eqref{e5.18}, $\|X_1(t)\| \leq
D_{16}$ for some constant $D_{18}>0$.  The rest of the proof
follows from \eqref{e5.18}, in view of \eqref{e4.11}.

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\end{document}