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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 154, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/154\hfil Asymptotic conditions]
{Asymptotic conditions for solving non-symmetric problems
 of third order nonlinear differential systems}

\author[A. S. Ukpera\hfil EJDE-2008/154\hfilneg]
{Awar Simon Ukpera} 

\address{Awar Simon Ukpera \newline
Department of Mathematics \\
Obafemi Awolowo University \\
Ile-Ife, Nigeria}
\email{aukpera@oauife.edu.ng}

\thanks{Submitted November 24, 2006. Published November 7, 2008.}
\subjclass[2000]{34B15, 34C15, 34C25}
\keywords{Asymptotic conditions;
 third order nonlinear differential systems; \hfill\break\indent
 non-symmetric problems; sharp and nonuniform nonresonance}

\begin{abstract}
 This article presents asymptotic conditions, under various sharp
 and nonuniform nonresonance assumptions, for solving
 the forced third order nonlinear system
 $X''' + A X'' + BX' + sH(t,X) = P(t)$,
 in which $A$, $B$ and other associated matrices are not necessarily
 symmetric.  This work generalizes some results in the literature
 which are hinged basically on symmetry considerations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

We shall establish new solvability criteria for the forced
third-order nonlinear system
\begin{equation}
X''' + AX'' + BX' + sH(t,X) = P(t) \label{e1.1}
\end{equation}
subject to the periodic boundary conditions
\begin{equation}
X(0) - X(T) =  X'(0) - X'(T) = X''(0) - X''(T) = 0 \label{e1.2}
\end{equation}
on an interval $[0,T]$ with $T > 0$.

Here, $X \equiv (x_i )_{1\leq i\leq n} : [0,T] \to \mathbb{R}^n$,
$\mathbb{R}^n$ is the n-dimensional Euclidean space, equipped with
the usual norm $\| \cdot \|$ and scalar product
 $\langle X,Y \rangle = \sum^n_{i=1}x_iy_i$, so that
$\langle X,X\rangle = \|X\|^2$ is the usual Euclidean norm in
$\mathbb{R}^n$. $A$ and $B$ are
constant real $n \times n$ matrices (not necessarily symmetric),
$H \equiv (h_i(t,X) )_{1\leq i\leq n}  : [0,T]\times\mathbb{R}^n \to
\mathbb{R}^n$ and $P \equiv (p_i )_{1\leq i\leq n}  :
[0,T]\to\mathbb{R}^n$ are $n$-vectors, which are $T$-periodic in
$t$. Furthermore, $H$ satisfies the Caratheodory conditions, and
$s \in \{ -1, 1 \}$.

The hypotheses for the present problem will be formulated as a
transition from some known results, in previous studies involving
symmetric matrices, to the recent studies carried out in
our joint papers with Afuwape \cite{a3, u2}, and also in \cite{u1}.

The classical spaces of $k$ times continuously differentiable functions 
shall be denoted by $C^k \, ([0,T],\mathbb{R}^n ), k\ge 0$ an integer, 
with norm ${\Vert X\Vert}_{C^k}$; $L^p = L^p([0, T])$, 
$1\le p \le \infty$, will denote the usual Lebesgue spaces, with their 
respective norms $\| X\|_{L^p}$; while $W_T^{k,p} ([0,T],\mathbb{R}^n )$, 
will denote the Sobolev space of $T$-periodic functions of order $k$, 
defined by
\begin{align*} 
W_T^{k,p} = &\{X:[0,T]\to\mathbb{R}^n : X,X',\dots ,X^{(k-1)} 
\mbox{ are absolutely continuous on } [0,T], \\ 
& X^{(k)} \in L^p  (0,T) \mbox{ and } X^{(i)} (0) - X^{(i)} (T) = 0, 
i=0,1,2,\dots ,k-1,~k\in \mathbb{R} \} 
\end{align*} 
with corresponding norm ${\Vert X\Vert}_{W_T^{k,p}}$.
 
It is standard result that if $D$ is a real $n\times n$ symmetric
matrix, then for any $X\in\mathbb{R}^n$,
\begin{equation}
\delta_d \|X\|^2 \leq \langle DX, X\rangle  \leq \Delta_d \|X\|^2,
\label{e1.3}
\end{equation}
where $\delta_d$ and $\Delta_d $ are real constants which represent
respectively, the least and greatest eigenvalues of $D$.
In general, $\lambda_i (D)$,  $i = 1, \dots ,n$, shall denote the
eigenvalues of any matrix $D$; and $\| D\|$ denote the norm of $D$
thought as a linear operator in $\mathbb{R}^n$ (that is, spectral
norm of $D$ with respect to the inner product
$\langle \cdot ,\cdot \rangle$).

Furthermore, the following algebraic inequalities also hold:
Let $D_1$ and $D_2$ be any two real $n\times n$ commuting symmetric matrices.
Then
\begin{itemize}
\item[(i)] the eigenvalues $\lambda_i (D_1 + D_2 )$,
$i = 1,\dots ,n$, of the sum of matrices $D_1$ and $D_2$ are all real
and satisfy
\begin{equation}
\begin{aligned}
\min_{1\leq j\leq n}  {\lambda}_j (D_1 ) + \min_{1\leq k\leq n}{\lambda}_k (D_2 )
&\leq \lambda_i (D_1 + D_2 ) \\
&\leq \max_{1\leq j\leq n}  {\lambda}_j (D_1 )
 + \max_{1\leq k\leq n}  {\lambda}_k (D_2 )
\end{aligned} \label{e1.4}
\end{equation}

\item[(ii)] the eigenvalues $\lambda_i (D_1 D_2 )$,  $i = 1,\dots ,n$,
 of the product of matrices $D_1$ and $D_2$ are all real and satisfy
\begin{equation}
\min_{1\leq j,k\leq n}  \{ {\lambda}_j (D_1 ) {\lambda}_k (D_2 )\} 
\leq \lambda_i (D_1 D_2 ) 
\leq \max_{1\leq j,k\leq n}  \{ {\lambda}_j (D_1 ) {\lambda}_k (D_2 )\} \label{e1.5}
\end{equation}

\end{itemize}
(See \cite{a1,e2} for proofs). However, note that if $D_1$ and $D_2$ are
 two commuting symmetric matrices, their product $D_1 D_2$ is also
symmetric so that its eigenvalues are real.

Relying significantly on the analysis of an underlying linear differential
operator,  various results at both the resonance and nonresonance situations
were proposed in \cite{a3} without much rigorous verification. Given under
 symmetry assumptions, our main result was built around the sharp
nonresonance relation
\begin{equation}
k^2 {\omega}^2 < {\Delta}_a^{-1} {\delta}_h
\leq \frac{\langle sA^{-1} H(t,X) , X \rangle}{{\| X\|}^2}
\leq {\delta}_a^{-1} {\Delta}_h < (k+1)^2 {\omega}^2
 \label{eH1}
\end{equation}
holding for some $k\in\mathbb{N}$ and $\omega = \frac{2\pi}T$,
where ${\delta}_h$, ${\Delta}_h$ are constants which satisfy
$ 0 < \delta_h \leq \lambda_i ( J_h (t,X)) \leq \Delta_h $ uniformly
in $X$ for a.e. $t\in [0,T]$, $J_h (t,X)$ being the Jacobian matrix
of $H(t,X)$, for the existence of $T$-periodic solutions of \eqref{e1.1}.

We have obtained a further generalisation of \cite{a3}, still subject to
symmetry considerations, by weakening the sharp nonresonance
relation $({\mathcal{H}}_1 )$, and thereby permitting some
interaction between the ratio $\langle sA^{-1} H(t,X) ,
X\rangle\big/ \| X\|^2$ and the spectrum $k^2 \omega^2$ of the
associated eigenvalue problem. This coincided with the problem of
finding vector analogues to Ezeilo and Nkashama's \cite{e1} nonuniform
nonresonance assumptions
\begin{equation}
k^2 \leq \frac{\gamma^- (t)}a \leq
\liminf_{|x| \to \infty} \frac{h(t,x)}{ax}  \leq \limsup_{|x| \to
\infty} \frac{h(t,x )}{ax} \leq \frac{\gamma^+ (t)}a \leq (k+1)^2
\label{eh1}
\end{equation}
holding uniformly in $x \in\mathbb{R}$ for a.e.
$t\in [0,2\pi ]$, where  $\gamma^{\pm} \in L^1 (0,2\pi )$ such
that strict inequalities hold on subsets of $[0,2\pi ]$ of
positive measure; for the existence of $2\pi$-periodic solutions
of a scalar prototype of \eqref{e1.1}. This was achieved by
employing two intermediate symmetric matrices $M(t)$ and $N(t)$
whose eigenvalues interact with the spectrum for many values of
$t\in [0,T]$. Thus, as has been demonstrated in \cite{u1,u2}, we
require that the inequalities
\begin{equation}
\begin{aligned}
k^2 \omega^2
&\leq \frac{\langle A^{-1} M(t)X,X \rangle}{\| X\|^2}\\
& \leq \liminf_{\| X\| \to \infty}
 \frac{\langle sA^{-1} H(t,X),X\rangle}{{\| X\|}^2 } \\
&\leq \limsup_{\| X\| \to \infty} \frac{\langle sA^{-1}
 H(t,X),X\rangle}{{\| X\|}^2 } \\
& \leq \frac{\langle A^{-1} N(t)X,X \rangle}{\| X\|^2}\\
&\leq (k+1)^2 \omega^2
\end{aligned} \label{eH2}
\end{equation} hold uniformly in $X\in \mathbb{R}^n$ for a.e. $t\in [0,T]$,
$k\in\mathbb{N}$, with $M, N \in L^1 ([0,T], \mathbb{R}^{n^2} )$
symmetric matrices such that $k^2 \omega^2 \| X\|^2 < \langle
A^{-1} M(t)X,X \rangle$ and $\langle A^{-1} N(t)X,X \rangle <
(k+1)^2 \omega^2 \| X\|^2$ on subsets of $[0,T]$ of positive
measure; with the corresponding expression for uniqueness given by
\begin{equation}
\begin{aligned}
k^2 \omega^2
&\leq \frac{\langle A^{-1} M(t)(X_1 - X_2 ) ,
 X_1 - X_2 \rangle}{\| X_1 - X_2 \|^2} \\
&\leq \frac{\langle sA^{-1} \bigl( H(t,X_1 ) - H(t,X_2 ) \bigr) ,
 X_1 - X_2 \rangle }{{\| X_1 - X_2 \|}^2 } \\
&\leq  \frac{\langle A^{-1} N(t)(X_1 - X_2 ) , X_1 - X_2 \rangle}
{\| X_1 - X_2 \|^2} \\
&\leq (k+1)^2 \omega^2
\end{aligned} \label{eH3}
\end{equation}
for $X_1 , X_2 \in\mathbb{R}^n$ with $X_1 \ne X_2$ where
$k\in \mathbb{N}$ and $M,N$ are as in \eqref{eH2}.

The relation \eqref{eH3} is in harmony with some abstract results
given in Amann \cite{a4} for the unique solvability of semi-linear
operator equations of the form $ {\bf{L}}U = F(U) $ in a real
Hilbert space ${\mathcal{H}}$, for ${\bf{L}}$ self-adjoint, where
it is required that there exist symmetric operators $B^{\pm} \in
{\mathcal{L}} ({\mathcal{H}})$ such that
\begin{gather*}
B^- \leq F' (U) \leq B^+   \forall U\in {\mathcal{H}} \\
\langle ({\bf{L}} - B^- )U,U \rangle \leq -\gamma \| U\|^2 ,\quad
 \langle  ({\bf{L}} - B^+ )U,U \rangle  \ge \gamma \| U\|^2  , \quad
   \gamma > 0
\end{gather*}
Clearly, \eqref{eH3} is an application of this result to systems
of type \eqref{e1.1}, even though our linearity here is not
self-adjoint, with the intermediate operators $B^-$ and $B^+$
replaced by the generalised intermediate matrices $M$ and $N$ in
$L^1 ([0,T],\mathbb{R}^{n^2} )$  respectively.

The basic challenge of this paper is to investigate the changes to
these solvability conditions arising from the transition to
non-symmetry assumptions. Specifically, we shall find analogous
repesentations to some of the above relations when symmetry
considerations on $A$, $B$ and other associated matrices are
dropped.


\section{Preliminary estimates and results}

Consider the eigenvalue problem \begin{equation} X''' + AX'' + BX' = -A\lambda
X \label{e2.1} \end{equation} together with \eqref{e1.2}, with $A ,B$
nonsingular and not  neccessarily symmetric, and $\lambda$ a real
parameter. The following results are vector analogues and
adaptations of their corresponding scalar based counterparts found
in Ezeilo-Nkashama \cite{e1} and Minh\'{o}s \cite{m3} respectively:

\begin{proposition} \label{prop2.1}
The following statements hold for \eqref{e2.1}:
\begin{itemize}
\item[(i)] $\lambda = 0$ is an eigenvalue;
\item[(ii)] $\lambda = k^2 {\omega}^2$, for some
$k = 1 , 2 ,\dots ,   \omega = \frac{2\pi}T$, is an eigenvalue if
and only if $|\lambda_i (B)| = k^2 {\omega}^2$;
\item[(iii)] any $\lambda \ne k^2 {\omega}^2$, for each
$k = 1 , 2 ,\dots ,   \omega = \frac{2\pi}T$ is not an eigenvalue

\end{itemize}
Furthermore, let ${\mathcal{E}}_k$ be the eigenspace corresponding
to the unique eigenvalue $k^2 \omega^2$, when it exists. Then for
every $X\in W_T^{3,2} (0,2\pi )$, we have
\[
\int_0^T \langle X''' + AX'' + BX' + k^2 \omega^2 AX  ,  X''' + AX'' + BX' +
(k+1)^2 \omega^2 AX \rangle  \,dt   \ge   0  ,
\]
and the equality holds if and only if $X = 0$ or either $k^2
\omega^2$ or $(k+1)^2 \omega^2$ is an eigenvalue of \eqref{e2.1}
and $X \in {\mathcal{E}}_k$ or $X \in {\mathcal{E}}_{k+1}$,
respectively.
\end{proposition}

\begin{proof}
Let the solution $X$ of \eqref{e2.1}-\eqref{e1.2} have the Fourier
series expansion
\[
X(t) \sim \sum_{i=1}^n  \sum_{k=0}^{\infty} (c_{k,i} \cos k\omega
t + d_{k,i} \sin k\omega t) ,
\]
$ d_{0,i} = 0$,  $\omega = \frac{2\pi}T $,  $k \in \mathbb{N}$,
$i=1,\dots ,n$.
Then substituting into \eqref{e2.1} yields
\begin{align*}
 0 &=  I\sum_{i=1}^n \sum_{k=0} ^{\infty} - (k\omega )^3
  \left[ c_{k,i} \sin (k\omega t) + d_{k,i} \cos (k\omega t) \right] \\
&\quad +A\sum_{i=1}^n \sum_{k=0}^{\infty} (k\omega )^2
 \left[ c_{k,i} \cos (k\omega t) + d_{k,i} \sin (k\omega t) \right] \\
&\quad + B\sum_{i=1}^n \sum_{k=0}^{\infty} k\omega
 \left[ c_{k,i} \sin (k\omega t) - d_{k,i} \cos (k\omega t) \right] \\
&\quad - \lambda A\sum_{i=1}^n \sum_{k=0}^{\infty}
 \left[ c_{k,i} \cos (k\omega t) + d_{k,i} \sin (k\omega t) \right] \\
&= - \lambda A\sum_{i=1}^n c_{0,i} - \sum_{i=1}^n \sum_{k=1}^{\infty}
 \left[ A(\lambda - k^2 {\omega}^2 ) c_{k,i} + k\omega
 (B - k^2 {\omega}^2 I)d_{k,i} \right] \cos (k\omega t) \\
&\quad +\quad \sum_{i=1}^n \sum_{k=1}^{\infty}
 \left[ k\omega (B - k^2 {\omega}^2 I)c_{k,i} - A(\lambda
  - k^2 {\omega}^2 )d_{k,i} \right] \sin (k\omega t)
\end{align*}
where $I$ is the identity matrix. Thus, for nontrivial solution to exist,
we conclude that statements (i)--(iii) must hold.

Also, noting from the last equality that
\begin{align*}
&X''' + AX'' + BX'  +  k^2 \omega^2 AX \\
&= k^2 {\omega}^2 A\sum_{i=1}^n c_{0,i}
  + \sum_{i=1}^n \sum_{r=1}^{\infty}
  \left[ A{\omega}^2 (k^2 - r^2 ) c_{r,i}
  + r\omega (B - r^2 {\omega}^2 I)d_{r,i} \right] \cos (r\omega t) \\
&\quad - \sum_{i=1}^n \sum_{r=1}^{\infty}
  \left[ r\omega (B - r^2 {\omega}^2 I)c_{r,i} - A{\omega}^2
  (k^2 - r^2 )d_{r,i} \right] \sin (r\omega t)
\end{align*}
we obtain
\begin{align*}
&\int_0^T \langle  X''' + AX'' + BX' + k^2 \omega^2 AX ,  X''' + AX''
+ BX' + (k+1)^2 \omega^2 AX \rangle  \,dt \\
&\geq \frac T2 \sum_{i=1}^n \sum_{r=1}^{\infty}
  \left[ \| A\|{\omega}^2 (k^2 - r^2 ) c_{r,i} + r\omega (|\lambda_i (B)|
   - r^2 {\omega}^2 ) d_{r,i} \right] \\
&\quad \times \left[ \| A\|{\omega}^2 ((k+1)^2 - r^2 ) c_{r,i}
 + r\omega (|\lambda_i (B)| - r^2 {\omega}^2 ) d_{r,i} \right] \\
&\quad + \frac T2 \sum_{i=1}^n \sum_{r=1}^{\infty}
 \left[ r\omega (|\lambda_i (B)| - r^2 {\omega}^2 ) c_{r,i}
- \| A\|{\omega}^2 (k^2 - r^2 )d_{r,i} \right] \\
&\quad \times \left[ r\omega (|\lambda_i (B)| - r^2 {\omega}^2 ) c_{r,i}
- \| A\|{\omega}^2 ((k+1)^2 - r^2 )d_{r,i} \right] \\
&\geq \frac T2 \sum_{i=1}^n \sum_{r=1}^{\infty}
 \left[ \omega^4 \| A\|^2 (k^2 - r^2 )((k+1)^2 - r^2 ) + r^2 \omega^2
(|\lambda_i (B)| - r^2 {\omega}^2 )\right] \\
&\quad \times \left[ c_{r,i}^2 + d_{r,i}^2 \right]
\ge0
\end{align*}
It follows that the equality holds if and only if $ c_{r,i} =  d_{r,i} = 0$
unless $ r^2 = k^2 $ or $ r^2 = (k+1)^2 $ and
$ |\lambda_i (B)| = k^2 \omega^2 I$. That means, if and only if
$X = 0$ or either $ k^2 {\omega}^2 $ or $ (k+1)^2 {\omega}^2 $
is an eigenvalue of \eqref{e2.1} and $X \in {\mathcal{E}}_k$ or
$X \in {\mathcal{E}}_{k+1}$, respectively.
\end{proof}

Each of the statements (i)--(iii) above has direct implication on the
solvability of the corresponding non-autonomous system
\begin{equation}
X''' + AX'' + BX' + \lambda AX = P(t) \equiv P(t+T) ,
 P\in L^1  \label{e2.2}
\end{equation} It is clear for instance, from (iii) and the Fredholm
alternative,  that a solution for \eqref{e2.2}-\eqref{e1.2}
exists, provided that some control is put on the closeness of
$\lambda$ to $k^2 \omega^2$ and $(k+1)^2 \omega^2$. Furthermore,
if $\lambda = 0 ,  k^2 \omega^2$,  then \eqref{e2.2}-\eqref{e1.2}
does not, in general, have a $T$-periodic solution. This
observation underscores the importance of any existence result for
\eqref{e1.1} in which $B$ is arbitrary, to be such that
$\lim_{\|X\|\to\infty} \frac{\langle sA^{-1} H(t,X) ,
X\rangle}{\langle X ,X\rangle}$ is allowed to 'touch' $k^2
\omega^2$ for some values of $t$.

For any given $n\times n$ matrix $D$ (not necessarily symmetric),
we set $D_s = (D + D^T )/2$ and $D_d = (D - D^T )/2$, where $D^T$
is the transpose of $D$. Moreover we define $\gamma_D = |\min \{
0, \sigma_{\min} \} |$, $\Gamma_D = |\max \{ 0, \sigma_{\max} \}
|$, where $\sigma_{\min}$ and $\sigma_{\max}$ represent
respectively the minimum and the maximum eigenvalues of $D_s$ (the
symmetric part of $D$). Then for any $X\in\mathbb{R}^n$ (see
Omari-Zanolin \cite{o1})
\begin{equation}
-\gamma_D \| X\|^2 \leq \sigma_{\min} \|
X\|^2 \leq \langle D_s X, X\rangle = \langle DX, X\rangle \leq
\sigma_{\max} \| X\|^2 \leq \Gamma_D \| X\|^2. \label{e2.3}
\end{equation}
Moreover,
\[
\max\{ \| D_s \| , \| D_d \| \}  \leq \| D\| ,
\| D_s \| = \max\{ \gamma_D , \Gamma_D \}.
\]
Let $\rho (t)$ be defined by
\[ 
\rho (t) = \frac 12 (\min {\lambda}_i (A^{-1} C(t)) + \max {\lambda}_i (A^{-1} C(t))) 
= \frac 12 (\Gamma_A^{-1} \gamma_C (t) + \gamma_A^{-1} \Gamma_C (t)) \]
so that 
\[ k^2 {\omega}^2 < \Gamma_A^{-1} \gamma_C (t) \le \rho (t) \le \gamma_A^{-1} \Gamma_C (t) < (k+1)^2 {\omega}^2 \]
Thus setting 
\[
\alpha = \frac 12 \max(\gamma_A^{-1} \Gamma_C (t)
 - \Gamma_A^{-1} \gamma_C (t)) \mbox{~and~} \beta = \min\{ (k+1)^2 {\omega}^2 
 - \rho (t) , \rho (t) - k^2 {\omega}^2 \} ,
\]
we easily deduce that 
\begin{equation}
 0 \le \alpha \le \beta \label{e*}
\end{equation}
with the strict inequality holding on subsets of $[0,T]$ of positive measure.

Let us now consider the linear system
\begin{equation}
X''' (t) + AX'' (t) + BX' (t) + C(t)X(t) = 0 \label{e2.4}
\end{equation}
together with \eqref{e1.2}, where $A$ and $B$ are constant,
not necessarily symmetric matrices, and $C(t) \equiv (c_{ij} (t))$,
with $c_{ij} \in L^1 (0,T)$, is an $n\times n$ arbitrary matrix function.
The following result holds for system \eqref{e2.4}-\eqref{e1.2}.

\begin{lemma} \label{lem2.1}
Let $A$ be a nonsingular matrix which commutes with $C(t)$, and suppose that
\begin{equation}
k^2 {\omega}^2 {\| X\|}^2 \leq \Gamma_A^{-1} \gamma_C (t) {\| X\|}^2 
\leq \langle A^{-1} C(t)X,X\rangle \leq \gamma_A^{-1} \Gamma_C (t) {\| X\|}^2
\leq (k+1)^2 {\omega}^2 {\| X\|}^2 \label{e2.5}
\end{equation}
uniformly for $X\in\mathbb{R}^n$ and a.e. $t\in [0,T]$, with the strict
inequality holding on subsets of $[0,T]$ of positive measure, where
$\gamma_A , \Gamma_A , \gamma_C , \Gamma_C$ are associated to $A$ and
$C(t)$ as defined in \eqref{e2.3}.
Then, for any arbitrary matrix $B$, \eqref{e2.4}-\eqref{e1.2}
admits in $W^{3,1} ([0,T], \mathbb{R}^n )$ only the trivial solution.

Moreover, there exist positive constants $\beta$ and $\delta_0$,
with $ \beta = \min\{ (k+1)^2 {\omega}^2 - \rho (t) , \rho (t)
- k^2 {\omega}^2 \} $, for $ \rho (t)
= \frac 12 (\Gamma_A^{-1} \gamma_C (t) + \gamma_A^{-1} \Gamma_C (t))$
for a.e $t\in [0,T] $, such that
\begin{gather}
{\beta}^2 \int_0^T {\| X\|}^2 \,dt
\leq \int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \,,\label{e2.6} \\
\int_0^T {\| X'' \|}^2 \,dt
\leq {\delta}_0 \int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt\,. \label{e2.7}
\end{gather}
\end{lemma}

\begin{proof}
Let the solution be $X(t) = \overline{X} (t) + \widetilde{X} (t)$, where
\begin{gather*}
\overline{X} = \sum_{i=1}^n \Big( c_{0,i}
 + \sum_{k=1}^N (c_{k,i} \cos k\omega t + d_{k,i} \sin k\omega t)\Big),\\
\widetilde{X} = \sum_{i=1}^n \sum_{k=N+1}^{\infty} \bigl( c_{k,i}
\cos k\omega t + d_{k,i} \sin k\omega t\bigr) .
\end{gather*}
Then multiplying \eqref{e2.4} scalarly by
$\overline{X} (t) - \widetilde{X} (t)$ and integrating over $[0,T]$,
setting $D(t) = A^{-1} C(t)$, we obtain (see \cite{u1})
\begin{equation}
\begin{aligned}
0 &=  \int_0^T \bigl(\|\widetilde{X}' (t)\|^2 - \langle D(t)\widetilde{X} (t)
,\widetilde{X} (t) \rangle\bigr) \,dt - \int_0^T \bigl(\|\overline{X}'
(t)\|^2 - \langle D(t)\overline{X} (t) , \overline{X} (t) \rangle \bigr) \,dt
 \\
&\ge  \int_0^T (\|\widetilde{X}' (t)\|^2 - \rho (t)\|\widetilde{X} (t)\|^2 )\,
dt - \int_0^T (\|\overline{X}' (t)\|^2 - \rho (t)\|\overline{X} (t)\|^2 )\, dt
 \end{aligned} \label{e2.8}
\end{equation}
where we fix $\rho (t) = \frac 12 (\Gamma_A^{-1} \gamma_C (t) + \gamma_A^{-1}
\Gamma_C (t))$, for a.e $t\in [0,T]$, so that
\begin{gather*}
k^2 \omega^2 \leq \rho (t) \leq (k+1)^2 \omega^2 , \quad   \mbox{for a.e }
t\in [0,T],\\
k^2 \omega^2 < \rho (t) < (k+1)^2 \omega^2  ,\quad
 \mbox{  on subsets of $[0,T]$ of positive measure.}
\end{gather*}
By Parseval's identity given by
\[
\int_0^T {\| X\|}^2 \,dt = \sum^n_{i=1}
\Big( c_{0,i}^2 T + \frac{T}{2} \sum_{k = 1}^{\infty} (c_{k,i}^2
+  d_{k,i}^2 )\Big)  ,
\]
Equation \eqref{e2.8} becomes
\[
\frac T2 \sum_{i=1}^n \Big[\sum_{k=N+1}^{\infty} (k^2 \omega^2
- \rho (t) ) (c_{k,i}^2 + d_{k,i}^2 ) + 2\rho c_{0,i}^2 T
+ \sum_{k=1}^N (\rho (t) - k^2 \omega^2 )(c_{k,i}^2 + d_{k,i}^2 )\Big]
\leq 0  %\label{e2.9}
\]
It follows that $c_{k,i} = 0  (k=0,1,2,\dots )$ and
$d_{k,i} = 0  (k=1,2,\dots )$, for all $i=1,\dots ,n$. Thus, $X\equiv 0$.

Furthermore, using the Fourier expansion of $X(t)$ given above
\begin{align*}
\int_0^T {\| X''} + {A^{-1} C(t)X \|}^2 \,dt
& \geq \rho (t)  \sum^n_{i=1} ({ a_{0i}}^2 T + \frac T2
 \sum_{k = 1}^{\infty} (k^2 {\omega}^2 - \rho (t) )^2  ({ a_{k,i}}^2
 + { b_{k,i} }^2 )) \\
&\geq  {\beta}^2 \sum^n_{i=1} ({a_{0i} }^2 T + \frac T2
\sum_{k_i = 1}^{\infty} ({a_{k,i}}^2 + {b_{k,i}}^2 ))
\end{align*} %\label{e2.10}
by the definition of $\rho (t)$ and $\beta$ given above.
Thus, \eqref{e2.6} now follows by Parseval's identity. Again
\begin{equation}
\begin{aligned}
\int_0^T {\| X''} + {A^{-1} C(t)X \|}^2 \,dt
& \geq \rho (t)  \sum^n_{i=1} ({ a_{0i}}^2 T + \frac T2
 \sum_{k = 1}^{\infty} (k^2 {\omega}^2 - \rho (t) )^2  ({ a_{k,i} }^2
 + { b_{k,i} }^2 )) \\
&\geq  \frac T2 \sum^n_{i=1}\sum_{k = 1}^{\infty} (k^2 {\omega}^2
- \rho (t) )^2  ({a_{k,i} }^2 + { b_{k,i} }^2 )
\end{aligned} \label{e2.11}
\end{equation}
Now, fix an integer $N_0 = N_0 (\nu )$ such that
$ (k^2 {\omega}^2 - \rho (t) )^2  \ge \frac 14 k^4 {\omega}^4 $,
for all $k \ge N_0$. Then,  we have
\begin{equation}
\frac T2 \sum^n_{i=1}\sum_{k =N_0 + 1}^{\infty} (k^2 {\omega}^2
- \rho (t) )^2  ({a_{k,i} }^2 + { b_{k,i} }^2 )
\ge \frac 14 \sum^n_{i=1} \sum_{k =N_0 +1}^{\infty}
k^4 {\omega}^4 ({ a_{k,i} }^2 + { b_{k,i} }^2 ). \label{e2.12}
\end{equation}
But then, we can fix a constant ${\delta}_1$ such that
\begin{equation}
  \delta_1 \sum^n_{i=1}\sum_{k = 1}^{N_0} (k^2 {\omega}^2
- \rho (t) )^2  ({a_{k,i} }^2 + { b_{k,i} }^2 )
\ge  \sum^n_{i=1} \sum_{k = 1}^{N_0} k^4 {\omega}^4 ({ a_{k,i} }^2
+ {b_{k,i} }^2 ). \label{e2.13}
\end{equation}
Combining \eqref{e2.12} and \eqref{e2.13} now gives
\[
\int_0^T \| X'' \|^2 \,dt = \sum^n_{i=1}
\sum_{k = 1}^{\infty} k^4 {\omega}^4 ({ a_{k,i} }^2 + { b_{k,i} }^2 )
\leq \delta_2 \frac T2 \sum^n_{i=1}\sum_{k = 1}^{\infty} (k^2 {\omega}^2
- \rho (t) )^2  ({a_{k,i} }^2 + { b_{k,i} }^2 ) %\label{e2.14}
\]
where ${\delta}_2 = \max\{ 4,{\delta}_1 \}$,
and the inequality \eqref{e2.7} now follows from \eqref{e2.11}.
\end{proof}

\begin{lemma} \label{lem2.2}
Let $A$ and $C(t)$ be as in Lemma \ref{lem2.1}, and suppose that
$M, N \in L^1 ([0,T],\mathbb{R}^{n^2} )$ are nonsingular matrices
which commute with $A$, and satisfy the following condition
\begin{equation}
\Lambda_1 (t) \| X\|^2 \leq \langle A^{-1} M(t)X,X \rangle
\leq \langle A^{-1} N(t)X,X \rangle
\leq \Lambda_2 (t) \| X\|^2 \label{e2.15}
\end{equation}
uniformly in $X\in\mathbb{R}^n$, for a.e. $t\in [0,T]$, where
$\Lambda_1 (t) = \min\lambda_i (A^{-1} M(t))$,
$\Lambda_2 (t) = \max\lambda_i (A^{-1} N(t))$, are such that
$ k^2 \omega^2 \leq |\Lambda_1 (t)| $ and
$ |\Lambda_2 (t)| \leq (k+1)^2 \omega^2 $, $k \in \mathbb{N}$,
$\omega = \frac{2\pi}T$, with the strict inequality holding
on subsets of $[0,T]$ of positive measure.

Then, there exist constants $\epsilon = \epsilon (M, N, C) > 0$
and $\delta = \delta (M, N, C) > 0$ uniformly a.e. on $[0,T]$,
such that for all $D(t) \equiv A^{-1} C(t) \in L^1 ([0,T],\mathbb{R}^{n^2} )$
satisfying
\begin{equation}
\langle  (A^{-1} M(t) - \epsilon I)X,X \rangle
\leq \langle D(t)X,X \rangle  \leq \langle  (A^{-1} M(t)
+ \epsilon I)X,X \rangle  \label{e2.16}
\end{equation}
uniformly in $X\in\mathbb{R}^n$, a.e. on $[0,T]$, and all
$X \in W_T^{3,1} ([0,T], \mathbb{R}^n )$, one has
\begin{equation}
\int_0^T \| A^{-1} X''' + X'' + A^{-1} BX' + A^{-1} C(t)X \| \,dt
\ge \delta {\| X\|}_{W_T^{3,1}} \label{e2.17}
\end{equation}
\end{lemma}

\begin{proof}
We shall prove by contradiction. Hence, let us assume that the conclusion
of the Lemma does not hold, that is, $\epsilon$ and $\delta$ do not exist.
Then there exists a sequence $(X_n ) \in W^{3,1} ([0,T], \mathbb{R}^n )$
 with ${\| X_n \|}_{W^{3,1}} = 1$, and a sequence
$(C_n ) \in L^1 ([0,T], \mathbb{R}^{n^2} )$ of nonsingular matrices with
\begin{equation}
\langle (A^{-1} M(t) - \frac 1n I)X,X \rangle   \leq \langle D_n (t)X,X \rangle  
\leq \langle (A^{-1} N(t)
+ \frac 1n I)X,X \rangle  ,  \quad   n\in\mathbb{N} , \label{e2.18}
\end{equation}
uniformly in $X\in\mathbb{R}^n$, for a.e. $t\in [0,T]$, where
$D_n (t) \equiv A^{-1} C_n (t)$, such that for all $X\in W^{3,1}$, one has
\begin{equation}
\int_0^T \| A^{-1} X_n''' (t) + X_n'' (t) + A^{-1} BX_n' (t)
+ A^{-1} C_n (t)X_n \| \,dt < \frac 1n \label{e2.19}
\end{equation}
Then, by \eqref{e2.18}, there exists some $\kappa \in L^1 ([0,T],\mathbb{R} )$
such that $\| D_n (t)\| \leq \kappa (t) $, $n = 1,2,\dots$ for a.e.
$t\in [0,T] , n\in\mathbb{N}$. For example, one can take
\[
\kappa (t) \equiv \frac 1{\| X\|^2} \Big[\| \langle (A^{-1} M(t)
- I)X,X \rangle \| + \| \langle (A^{-1} N(t) + I)X,X \rangle \| \Big].
\]
Now, by the compact embedding of $W^{3,1} ([0,T],\mathbb{R}^n )$ into
$W^{2,1} ([0,T],\mathbb{R}^n )$ and the continuous embedding of
$W^{2,1} ([0,T],\mathbb{R}^n )$ into $C^1 ([0,T],\mathbb{R}^n )$
 we infer that by going to subsequences if necessary, we can assume that
\begin{equation}
X_n \to X \mbox{ in } C^1 ([0,T],\mathbb{R}^n ) ,   X_n' \to X'  ,   X_n'' \to X''
 \mbox { in } L^1 ([0,T],\mathbb{R}^n ) \label{e2.20}
\end{equation}
Moreover, $D_n \rightharpoonup D$ in $L^1 ([0,T],\mathbb{R}^{n^2} )$,
 so that by \eqref{e2.18},
\begin{equation}
\langle A^{-1} M(t)X,X \rangle \leq \langle D(t)X,X \rangle 
\equiv \langle A^{-1} C(t)X,X \rangle
\leq \langle A^{-1} N(t)X,X \rangle \label{e2.21}
\end{equation}
for a.e. $t\in [0,T]$. Thus for every
$\Phi \in L^{\infty} ([0,T],\mathbb{R}^n )$, we have by the Schwartz
inequality
\begin{align*}
&\|\int_0^T \bigl\langle  D_n (t)X_n (t) - D(t)X(t) ,
 \Phi (t) \bigr\rangle \,dt \| \\
&\leq \|\int_0^T \bigl\langle  D_n (t) \bigl( X_n (t) - X(t)\bigr) , \Phi (t)
   \bigr\rangle \,dt \|
 + \|\int_0^T \bigl\langle \bigl( D_n (t) - D(t)\bigr) X(t) , \Phi (t)
  \bigr\rangle \,dt \| \\
&\leq {\|\Phi\|}_{\infty} {\|\kappa\|}_{L^1} {\| X_n - X \|}_{\infty}
 + \|\int_0^T \bigl\langle \bigl( D_n (t) - D(t)\bigr) X(t) ,
 \Phi (t)\bigr\rangle \,dt \|
\end{align*} %\label{e2.22}
The right hand side of the above inequality clearly tends to zero, and
we deduce that $ D_n X_n \rightharpoonup DX $ in
$L^1 ([0,T],\mathbb{R}^n )$ It follows that
\begin{equation}
\begin{aligned}
X_n''' &= -AX_n'' - BX_n' - C_n (\cdot )X_n \\
&\rightharpoonup  -AX'' - BX' - C(\cdot )X \quad
\mbox{in } L^1 ([0,T],\mathbb{R}^n )
\end{aligned} \label{e2.23}
\end{equation}
Since the operator $\frac{d^3  }{dt^3} : W^{3,1} ([0,T],\mathbb{R}^n )
 \subset L^1 ([0,T],\mathbb{R}^n ) \to L^1 ([0,T],\mathbb{R}^n )$
is weakly closed, this implies that $X\in W_T^{3,1} ([0,T],\mathbb{R}^n )$,
and $ X''' =  -AX'' - BX' - C(\cdot )X $, that is,
\begin{equation}
X''' (t) + AX'' (t) + BX' (t) + C(t)X(t) = 0  ,\label{e2.24}
\end{equation}
for a.e. $t\in [0,T]$ and $X\in W^{3,1} ([0,T],\mathbb{R}^n )$.
It follows from \eqref{e2.15}, \eqref{e2.21} and Lemma \ref{lem2.1}
that $X \equiv 0$, that is, $X_n \to 0$ in $W^{3,1} ([0,T],\mathbb{R}^n )$
as $n\to \infty$. But this clearly contradicts the initial
assumption that ${\| X_n \|}_{W^{3,1}} = 1$ for all $n$,
and the proof is complete.
\end{proof}

\section{Some sharp and nonuniform nonresonance results}

We shall prove the following sharp nonresonance result.

\begin{theorem} \label{thm3.1}
Let $A$ and $C(t)$ be real $n\times n$ (not necessarily symmetric)
matrices which commute, with $A$ nonsingular, and satisfying the
relation \eqref{e2.5}.
Suppose that the Jacobian matrix
$J_H (t,X) \equiv (\frac{\partial h_i}{\partial x_j} )$ of $H(t,X)$
(not necessarily symmetric) exists, and has eigenvalues
$\lambda_i (J_H (t,X))$ which satisfy
${\delta}_h (t)  \leq {\lambda}_i (J_H (t,X)) \leq {\Delta}_h (t)$,
$i=1,\dots ,n$, uniformly in $X\in\mathbb{R}^n$ for a.e $t\in [0,T]$,
where ${\delta}_h$ and ${\Delta}_h$ are respectively the least and
greatest eigenvalues of $J_H (t,X)$; and further the relation
\begin{equation}
\begin{aligned}
k^2 {\omega}^2 {\| X\|}^2
&< \Gamma_A^{-1} \gamma_H (t)  {\| X\|}^2 \\
&\leq \langle sA^{-1} H(t,X) , X\rangle \\
& \leq \gamma_A^{-1} \Gamma_H (t)  {\| X\|}^2 \\
&< (k+1)^2 {\omega}^2 {\| X\|}^2
\end{aligned} \label{eH4}
\end{equation}
holds for a.e $t\in [0,T]$ and $X\in\mathbb{R}^n$ with $\| X\| \ge R_0 > 0$,
with $k\in\mathbb{N}$,  $\omega = \frac{2\pi}T , s\in\{ -1,1\}$,
where $ \gamma_H (t) = |\min \{ 0, \delta_h (t) \} | $,
$ \Gamma_H (t) = |\max \{ 0, \Delta_h (t) \} | $, and
$\gamma_A , \Gamma_A$ are as defined in Lemma \ref{lem2.1}.

Then for each $s\in \{ -1 ,1\}$ and all arbitrary matrix $B$,
the system \eqref{e1.1}-\eqref{e1.2} has at least one solution for
every $P\in L^1 ([0,T],\mathbb{R}^n )$.
\end{theorem}

\begin{proof}
Let us embed \eqref{e1.1} in the $\lambda$ parameter-dependent system
\begin{equation}
X''' + AX'' + BX' + (1 - \lambda )C(t)X
+ \lambda sH(t,X) = \lambda P(t) \label{e3.1}
\end{equation}
with $\lambda\in [0,1]$, or equivalently
\begin{equation}
A^{-1} X''' + X'' + A^{-1} BX' + (1 - \lambda )A^{-1} C(t)X
+ \lambda sA^{-1} H(t,X)
 = \lambda A^{-1} P(t) \label{e3.2}
\end{equation}
Then proceeding as in \cite{a3}, we multiply both sides of \eqref{e3.2}
scalarly by $X'' + A^{-1} C(t)X$ and integrate over $[0,T]$,
yielding after simplification
\begin{equation}
\begin{aligned}
&\int_0^T \bigl( \langle X'' , X'' \rangle  + \langle X'' ,
A^{-1} C(t)X\rangle + \langle H_{\lambda} (t,X) , X'' \rangle\\
& + \langle H_{\lambda} (t,X) , A^{-1} C(t)X\rangle \bigr) \,dt\\
&= \int_0^T \langle \lambda A^{-1} P(t) , X'' + A^{-1} C(t)X\rangle \,dt
\end{aligned}\label{e3.3}
\end{equation}
where we have set
$H_{\lambda} (t,X) = (1 - \lambda )A^{-1} C(t)X + \lambda sA^{-1} H(t,X)$.
One can easily verify that the integrand on the left-hand-side of
\eqref{e3.3} is identically equal to
\[
\frac 12 \{ {\| X'' + A^{-1} C(t)X \|}^2 + {\| X''
+ H_{\lambda} (t,X) \|}^2 - {\| H_{\lambda} (t,X) - A^{-1} C(t)X \|}^2 \}
\]
Thus, since
$ {\| H_{\lambda} (t,X) - A^{-1} C(t)X \|}^2 = {\lambda}^2 {\| sA^{-1} H(t,X)
-A^{-1} C(t)X \|}^2 $, it follows from \eqref{e3.3} that
\begin{equation}
\begin{aligned}
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt
&\leq  \int_0^T {\| sA^{-1} H(t,X) - A^{-1} C(t)X \|}^2 \,dt \\
&\quad + \int_0^T 2 \| A^{-1} \| \| P(t)\| \| X''
+ A^{-1} C(t)X \| \,dt
\end{aligned} \label{e3.4}
\end{equation}
Now, if $\| X\| \ge R_0$, then by \eqref{e2.11} and \eqref{eH4},
\begin{equation}
\begin{aligned}
\| sA^{-1} H(t,X) - A^{-1} C(t)X \|
&= \Big( \frac{\langle sA^{-1}
H(t,X),X\rangle  - \langle A^{-1} C(t)X,X\rangle }{\langle X,X\rangle } \Big)
\| X\| \\
&\leq  (\gamma_A^{-1} \Gamma_H (t) - \rho (t)) \| X\|
\leq \alpha \| X\|
\end{aligned} \label{e3.5}
\end{equation}
uniformly for a.e. $t\in [0,T]$, where
$\rho (t) = \frac 12 (\Gamma_A^{-1} \gamma_C (t) + \gamma_A^{-1} \Gamma_C (t))$
and $\alpha = \frac 12 \max(\gamma_A^{-1} \Gamma_C (t) 
- \Gamma_A^{-1} \gamma_C (t))$. 
On the other hand, if $\| X\| < R_0$, then there exists a constant 
$\delta_3 > 0$ such that
$ \| sA^{-1} H(t,X) - A^{-1} C(t)X \| \leq \delta_3 $,
uniformly for a.e. $t \in [0, T]$. Hence,
\[
\| sA^{-1} H(t,X) - A^{-1} C(t)X \| \leq \alpha \| X\| + \delta_3
\]
which implies that for all $X\in\mathbb{R}^n$ and a.e. $t\in [0,T]$,
there exist constants $\delta_4 > 0$, $\delta_5 > 0$ such that
\begin{equation}
{\| sA^{-1} H(t,X) - A^{-1} C(t)X \|}^2 \leq {\alpha}^2 {\| X\|}^2
+ \delta_4 \| X\| + \delta_5 \label{e3.6}
\end{equation}
Next, noting that $ 2 \| A^{-1} \| \| P(t)\| \| X''
+ A^{-1} C(t)X \| \leq \delta_6 (\| X'' \| + \| X \|) $,
for some $\delta_6 > 0$, \eqref{e3.4} becomes
\begin{equation}
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \leq \delta_7
 + {\alpha}^2 \int_0^T {\|X \|}^2 \,dt + \delta_8 \int_0^T (\| X'' \|
 + \| X \|) dt ,\label{e3.7}
\end{equation}
where $\delta_7 = \delta_5 T$ and $\delta_8 = \delta_4 + \delta_6$.
Thus, from the identity
\[
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt
= \Big( \frac{{\beta}^2 - {\alpha}^2 }{{\beta}^2}
+ \frac{{\alpha}^2 }{{\beta}^2} \Big) \int_0^T {\| X''
+ A^{-1} C(t)X \|}^2 \,dt
\]
and using relation \eqref{e2.6}, \eqref{e3.7} now becomes
\begin{equation}
({\beta}^2 - {\alpha}^2 ) \int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt
\leq \delta_9
+ \delta_{10} \int_0^T (\| X'' \| + \| X \|) dt  ,\label{e3.8}
\end{equation}
where $\delta_9 = {\beta}^2 \delta_7$, $\delta_{10} = {\beta}^2 \delta_8$, and
$\beta^2 - \alpha^2 > 0$ a.e on $[0,T]$ by relation \eqref{e*}. That is,
\begin{equation}
\begin{aligned}
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt
& \leq \delta_{11} + \delta_{12} \int_0^T (\| X'' \| + \| X \|) dt \\
&\leq \delta_{11} + \delta_{13} \Big(\int_0^T {\| X''
+ A^{-1} C(t)X \|}^2 \,dt \Big)^{\frac 12}
\end{aligned} \label{e3.9}
\end{equation}
by the Cauchy-Schwartz inequality and the relations \eqref{e2.6}
and \eqref{e2.7}. We therefore deduce that
\begin{equation}
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \leq \delta_{14} \label{e3.10}
\end{equation}
for some $\delta_{14} > 0$. Thus, by \eqref{e2.6}, \eqref{e2.7}
and Wirtinger's inequality, there exists a constant $\delta_{15} > 0$
such that
\begin{equation}
\int_0^T {\| X\|}^2 \,dt \leq \delta_{15} ,    \int_0^T {\| X' \|}^2 \,dt \leq
\delta_{15} ,    \int_0^T {\| X'' \|}^2 \,dt
 \leq \delta_{15} \label{e3.11}
\end{equation}
Finally, multiply \eqref{e3.2} scalarly by $X''' (t)$ and integrate
 over $[0,T]$ with respect to $t$, and using the estimates
in \eqref{e3.5}, it is easily verified that
\begin{equation}
\int_0^T {\| X''' \|}^2 \,dt \leq \delta_{16} \label{e3.12}
\end{equation}
for some constant $\delta_{16} > 0$.
The estimates \eqref{e3.5} and \eqref{e3.6} show clearly that there
exist a constant $K > 0$, independent of $\lambda\in (0,1)$ such
that any $T$-periodic solution $X(t)$ of \eqref{e3.2} satisfies
$ {\| X\|}_{C^2} \leq K  $, and the proof is complete.
\end{proof}

The following result is a nonuniform counterpart of Theorem \ref{thm3.1}.

\begin{theorem} \label{thm3.2}
Let $A, C, M$ and $N$ be the matrices defined in Lemma \ref{lem2.2}, and
suppose that $H$ is an $L^1$-Carath\'{e}odory function which satisfies
\begin{equation}
\begin{aligned}
\langle A^{-1} M(t)X,X \rangle
&\leq \liminf_{\| X\| \to \infty} \langle sA^{-1} H(t,X),X\rangle \\
&\leq \limsup_{\| X\| \to \infty} \langle sA^{-1} H(t,X),X\rangle \\
&\leq \langle A^{-1} N(t)X,X \rangle \label{eH5}
\end{aligned}
\end{equation}
uniformly in $X\in \mathbb{R}^n$ for a.e. $t\in [0,T]$, such that
$ \Lambda_1 (t) \| X\|^2 \leq \langle A^{-1} M(t)X,X \rangle $,
$ \langle A^{-1} N(t)X,X \rangle \leq \Lambda_2 (t) \| X\|^2 $ a.e.
on $[0,T]$, where $\Lambda_1$, $\Lambda_2$ are as defined in \eqref{e2.15}.

Then there exists $\epsilon = \epsilon (M, N, C) > 0$ satisfying
 \eqref{e2.16} such that for each $s\in \{ -1 ,1\}$ and all
arbitrary matrix $B$, the system \eqref{e1.1}-\eqref{e1.2}
has at least one solution for every $P\in L^1 ([0,T],\mathbb{R}^n )$.
\end{theorem}

\begin{proof}
Let $\epsilon > 0$ be associated to $M, N$ by Lemma \ref{lem2.2}. Then,
by (${\mathcal{H}}_5$), there exists a constant vector
$\zeta = \zeta (\epsilon )$ with each $\zeta_i > 0$ such that
\begin{equation}
\langle (A^{-1} M(t) - \epsilon I)X,X \rangle
\leq \langle sA^{-1} H(t,X),X \rangle \leq \langle (A^{-1} N(t)
+ \epsilon I)X,X \rangle \label{e3.13}
\end{equation}
for a.e. $t\in [0,T]$ and all $X\in\mathbb{R}^n$ with $|x_i | \ge \zeta_i$.

Let us now define
$\mu (t,X) \equiv \bigl(\mu_i (t,X)\bigr)_{1\leq i\leq n} :
[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ by
\[
{\mu}_i (t,X) =
\begin{cases}
x_i^{-1} h_i (t,X) ,&\mbox{if  } |x_i | \ge \zeta_i ;  \\[3pt]
x_i \zeta_i^{-2} h_i (t,x_1 ,\dots ,x_{i-1} ,\zeta_i ,x_{i+1} ,\dots ,x_n )\\
+  (1 - \frac{x_i} {\zeta_i} ) \kappa (t) ,
&\mbox{if  } 0\leq x_i < \zeta_i ; \\[3pt]
x_i \zeta_i^{-2} h_i (t,x_1 ,\dots ,x_{i-1} ,-\zeta_i ,x_{i+1} ,\dots ,x_n )\\
+  (1 +  \frac{x_i} {\zeta_i} ) \kappa (t) ,
&\mbox{if  } -\zeta_i \leq x_i < 0 .
\end{cases}
 \]
for a.e. $t\in [0,T]$, where
\begin{equation}
\kappa (t) \equiv \frac 1{\| X\|^2} \Big[\| \langle (A^{-1} M(t) - I)X,X \rangle \|
+ \| \langle (A^{-1} N(t) + I)X,X \rangle \| \Big] \label{e3.14}
\end{equation}
so that by construction and \eqref{e3.13}, we deduce that for a.e.
$t\in [0,T]$ and $X\in\mathbb{R}^n$
\begin{equation}
\langle A^{-1} M(t) - \epsilon I)X,X \rangle \leq \langle sA^{-1}
 \mu (t,X),X \rangle \leq \langle A^{-1} N(t) + \epsilon I)X,X
\rangle \label{e3.15}
\end{equation}
The function $\tilde{H} \equiv (\tilde{h}_i (t,X))_{1\leq i\leq n}
: [0,T]\times\mathbb{R}^n \to\mathbb{R}^n$ defined by
$\tilde{h}_i (t,X) = \mu_i (t,X)x_i$ satisfies the Carath\'{e}odory conditions,
 by construction. Hence, setting $\Theta (t,X) = H(t,X) - \tilde{H} (t,X)$,
 then $\Theta (t,X)$ is also $L^1$-Carath\'{e}odory with
\begin{equation}
\|\Theta (t,X)\| \leq \sup_{\| X\|\le\zeta} \| H(t,X) - \tilde{H} (t,X)\|
\leq \varphi (t) \label{e3.16}
\end{equation}
for a.e. $t\in [0,T]$ and $X\in \mathbb{R}^n$, for some
$\varphi \in L^1 ([0,T],\mathbb{R} )$ depending only on $M, N$ and
$\gamma_r$, associated with $H$ by the Carath\'{e}odory
assumption $\| H(t,X)\| \leq \gamma_r$.

Finally, \eqref{e1.1} is equivalent to
\begin{equation}
X''' (t) + AX'' (t)+ BX' (t) + s\tilde{H} (t,X(t)) + s\Theta (t,X(t))
= P(t) \label{e3.17}
\end{equation}
By the Leray-Schauder technique (see Mawhin \cite{m1}), the proof of the
theorem now follows by showing that there is a constant $K > 0$,
independent of $\lambda\in [0,1]$, such that $\| X\|_{C^2} < K$,
for all possible solutions $X$ of the homotopy
\begin{equation}
X''' + AX'' + BX' + (1 - \lambda )N(t)X + s\lambda \tilde{H} (t,X)
+ s\lambda \Theta (t,X) = \lambda P(t) \label{e3.18}
\end{equation}
with $\lambda\in[0,1]$. Let ${\bf X} := C^2 ([0,T], \mathbb{R}^n )$,
${\bf Z} := L^1 ([0,T], \mathbb{R}^n )$ and
$\mathop{\rm dom}L \subset W^{3,1} (0,T)$ and define the mappings
\begin{gather*}
L : \mathop{\rm dom}L \subset {\bf X} \to {\bf Z}  ,
X \mapsto X''' (\cdot ) + AX'' (\cdot )+ BX' (\cdot ),\\
Q : {\bf X} \to {\bf Z}  ,  X \mapsto s\widetilde{H} (\cdot ,X(\cdot ))
= s\langle \mu (\cdot ,X(\cdot )) , X(\cdot )\rangle, \\
R : {\bf X} \to {\bf Z}  ,  X \mapsto s\Theta (\cdot ,X(\cdot ))
 - P (\cdot ),\\
\widetilde{A} : {\bf X} \to {\bf Z}  ,  X \mapsto N(\cdot )X(\cdot )
= \langle \mu (\cdot ,0) , X(\cdot )\rangle
\end{gather*}
Accordingly, the existence of a solution will follow from
\cite[Theorem 4.5]{m1} if we can show that the possible solutions
of the equivalent homotopy
\begin{equation}
LX + (1 - \lambda ) \widetilde{A} X + \lambda (QX + RX) = 0 \label{e3.19}
\end{equation}
in $\mathop{\rm dom}L$ are {\it a-priori} bounded independently of
$\lambda\in (0,1)$ and of the solutions. Let us re-write \eqref{e3.18} as
\begin{equation}
\begin{aligned}
&A^{-1}X''' + X'' + A^{-1} BX' + (1 - \lambda )A^{-1} N(t)X  \\
& + s\lambda A^{-1} \tilde{H} (t,X)
+ s\lambda A^{-1} \Theta (t,X)\\
&= A^{-1} \lambda P(t)
\end{aligned}\label{e3.20}
\end{equation}
Then, from \eqref{e3.15}, for all $X\in \mathop{\rm dom} L$ we have
\begin{equation}
\begin{aligned}
\langle A^{-1} M(t)X,X \rangle - \epsilon\| X\|^2
&\leq \langle  (1 - \lambda )A^{-1} N(t)X  + s\lambda A^{-1}
  \tilde{H} (t,X),X \rangle  \\
&\leq  \langle A^{-1} N(t)X,X \rangle + \epsilon\| X\|^2
\end{aligned}\label{e3.21}
\end{equation}
for a.e. $t\in [0,T]$ and $\lambda\in [0,1]$. Thus, we may set
\begin{equation}
(1 - \lambda )A^{-1} N(t)X + s\lambda A^{-1} \tilde{H} (t,X)
 \equiv A^{-1} C(t)X , \label{e3.22}
\end{equation}
for a.e. $t\in [0,T], X\in dom L$ and $\lambda\in [0,1]$, where,
 by \eqref{e3.21}, $C(t)$ is such that
\begin{equation}
\langle (A^{-1} M(t) - \epsilon I)X,X \rangle
\leq \langle A^{-1} C(t)X,X \rangle \leq \langle (A^{-1} N(t)
+ \epsilon I)X,X \rangle \label{e3.23}
\end{equation}
for a.e. $t\in [0,T], X\in dom L$ and $\lambda\in [0,1]$.
Thus, if $X\in \mathop{\rm dom}L$ is a solution of \eqref{e3.20}, then
\begin{equation}
\begin{aligned}
0 &= \int_0^T \bigm| A^{-1} X''' (t) + X'' (t) + A^{-1} BX' (t)
+ (1   - \lambda ) A^{-1} N(t)X(t) \\
&\quad + s\lambda A^{-1} \tilde{H} (t,X)
 + \lambda A^{-1} \bigl( s\Theta (t,X(t)) - P(t)\bigr) \bigm| \,dt
\end{aligned} \label{e3.24}
\end{equation}
It follows from \eqref{e3.22} that
\begin{equation}
\begin{aligned}
0 &\ge \int_0^T \| A^{-1} X''' + A^{-1} X'' + A^{-1} BX' (t)
    + A^{-1} C(t)X \| \,dt\\
&\quad -  \gamma_A^{-1} \bigl(\int_0^T \|\Theta (t,X) \| \,dt
- \int_0^T \| P(t) \| \,dt \bigr)
\end{aligned}\label{e3.25}
\end{equation}
where $\gamma_A > 0$ is as defined in the previous section. That is,
by Lemma \ref{lem2.2} and \eqref{e3.16}, we obtain
\begin{equation}
0 \ge \delta {\| X\|}_{W^{3,1}} - \gamma_A^{-1}
\bigl( \|\varphi \|_{L^1} - {\| P\|} _{L^1} \bigr) , \label{e3.26}
\end{equation}
which finally yields a constant $K_0 > 0$ such that
$\|X\|_{W^{3,1}} \leq K_0$. Hence, we obtain the required constant
$K > 0$ such that $\| X\|_{C^2} < K$, following a standard procedure
just as in \cite{e1}, completing the proof.
\end{proof}

Lastly, we give some uniqueness results for \eqref{e1.1}-\eqref{e1.2}.

\begin{theorem} \label{thm3.3}
Let $A$ be nonsingular and suppose that $H$ is such that either
\begin{equation}
k^2 {\omega}^2 < \Gamma_A^{-1} \gamma_H (t)
\leq \frac{\| sA^{-1} (H(t,X_1 ) - H(t,X_2 ) )\|}{\| X_1 - X_2 \|}
\leq \gamma_A^{-1} \Gamma_H (t) < (k+1)^2 {\omega}^2 \label{eH6}
\end{equation}
or
\begin{equation}
\begin{aligned}
\Lambda_1 (t)
&\leq \frac{\langle A^{-1} M(t)(X_1 - X_2 ) , X_1 - X_2 \rangle}{\| X_1
 - X_2 \|^2} \\
& \leq \frac{\langle  sA^{-1} \bigl( H(t,X_1 ) - H(t,X_2 ) \bigr) ,
 X_1 - X_2  \rangle}{{\| X_1 - X_2 \|}^2 } \\
&\leq  \frac{\langle A^{-1} N(t)(X_1 - X_2 ) , X_1 - X_2
 \rangle}{\| X_1 - X_2 \|^2} \\
&\leq \Lambda_2 (t)
\end{aligned} \label{eH7}
\end{equation}
uniformly for $X_1 , X_2 \in\mathbb{R}^n$ with $X_1 \ne X_2$, and a.e.
$t\in [0,T]$, $k\in \mathbb{N}$, $\omega = \frac{2\pi}T$, where
$\gamma_A$, $\Gamma_A$, $\gamma_H$, $\Gamma_H$, and
$\Lambda_1$, $\Lambda_2$, $M$, $N$ are as appearing in
Theorems \ref{thm3.1} and \ref{thm3.2} respectively.

Then, for each $s\in \{ -1 ,1\}$ and all arbitrary $n\times n$ matrix $B$,
the system \eqref{e1.1}-\eqref{e1.2} has a unique $T$-periodic solution
for every $P\in L^1 ([0,T],\mathbb{R}^n )$.
\end{theorem}

\begin{proof}
\textbf{Case (i)} ($H$ subject to \eqref{eH6}:
Suppose that $X_1 (t)$ and $X_2 (t)$ are two $T$-periodic solutions
of \eqref{e1.1}-\eqref{e1.2} and set $X(t)  = X_1 (t) - X_2 (t)$.
Then $X(t)$ satisfies
\begin{equation}
X''' (t) + AX'' (t) + BX' (t) + \widetilde{H} (t,X(t)) = 0 \label{e3.27}
\end{equation}
where $\widetilde{H} (t,X) = sH(t,X_1 ) - sH(t,X_2 )$.
Rewrite \eqref{e3.27} in the form \eqref{e3.2}, multiply scalarly by
$X'' + A^{-1} C(t)X$ and integrate over $[0,T]$ yields
\begin{equation}
\int_0^T \langle X'' + sA^{-1} \widetilde{H} (t,X) , X''
+ A^{-1} C(t)X\rangle \,dt = 0 \label{e3.28}
\end{equation}
We observe as before that the integral in \eqref{e3.28} can be reset
in the form
\[
\frac 12 \int_0^T \{ {\| X'' + A^{-1} C(t)X \|}^2 + {\| X''
+ \widetilde{H} (t,X) \|}^2 - {\| \widetilde{H} (t,X) - A^{-1}
C(t)X \|}^2 \} \,dt = 0 %\label{e3.29}
\]
so that
\begin{equation}
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt
\leq \int_0^T {\| A^{-1} \widetilde{H} (t,X) - A^{-1} C(t)X \|}^2 \,dt
\label{e3.30}
\end{equation}
Now if $X(t) \ne 0$, then by \eqref{eH6},
\begin{equation}
\begin{aligned}
&\| A^{-1} \widetilde{H} (t,X) - A^{-1} C(t)X \| \\
&= \big| \frac{\| sA^{-1}
H(t,X) - sA^{-1} H(t,Y) \| - \langle A^{-1} C(t)X,X\rangle ^{\frac 12} }{\|
X - Y\|} \big| \| X\| \\
&\leq  (\gamma_A^{-1} \Gamma_H (t) - \rho (t)) \| X\| \\
&\leq \alpha \| X\| ,
\end{aligned} \label{e3.31}
\end{equation}
Hence \eqref{e3.30} becomes
\[
\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt
\leq {\alpha}^2 \int_0^T {\| X\|}^2 \,dt %\label{e3.31}
\]
Hence, by relation \eqref{e2.6} of Lemma \ref{lem2.1},
\begin{equation}
{\beta}^2 \int_0^T {\| X \|}^2 \,dt \leq {\alpha}^2 \int_0^T {\| X \|}^2 \,dt
\label{e3.32}
\end{equation}
But ${\alpha}^2 < {\beta}^2$ a.e on $[0,T]$ by relation \eqref{e*}, 
so that \eqref{e3.32} holds
only when $ \int_0^T {\| X \|}^2 \,dt = 0 $. Thus $X\equiv 0$;
that is, $X_1 = X_2$, and are done.

\noindent\textbf{Case (ii)} ($H$ subject to \eqref{eH7}:
Again setting $X = X_1 - X_2$, then here $X$ is a solution of the problem
\begin{equation}
X''' (t) + AX'' (t) + BX' (t) + \widetilde{C} (t,X(t))X(t) = 0 ,
\label{e3.33}
\end{equation}
satisfying \eqref{e1.2}, where the matrix $\widetilde{C} (\cdot ,X(\cdot ))$
is defined by
\[
\widetilde{C} (t,X(t))X(t) = \begin{cases}
sH(t,X+X_2 ) - sH(t,X_2 ) &\mbox{if  } X \ne 0   \\
M(t) , &\mbox{if  } X = 0
\end{cases}
\]
and by \eqref{eH7},
\[
\Lambda_1 (t) {\| X\|}^2 \leq \langle A^{-1} M(t)X,X \rangle
\leq \langle A^{-1} \widetilde{C} (t)X,X\rangle
\leq \langle A^{-1} N(t)X,X \rangle \leq \Lambda_2 (t) {\| X\|}^2
% \label{e3.34}
\]
uniformly for $X\in\mathbb{R}^n$ and a.e. $t\in [0,T]$, where
$ k^2 \omega^2 \leq |\Lambda_1 (t)| $ and
$ |\Lambda_2 (t)| \leq (k+1)^2 \omega^2 $, with the strict inequality
holding on subsets of $[0,T]$ of positive measure.

By Lemma \ref{lem2.1}, we deduce that $X = 0$, and the uniqueness result
is thereby established.
\end{proof}

\begin{thebibliography}{99}

\bibitem{a1}  A. U. Afuwape;
On the convergence of solutions of certain systems of nonlinear
third order differential equations, {\it Simon Steven},
{\bf 57} (4), (1983), 255 - 271.

\bibitem{a2} A. U. Afuwape, P. Omari, F. Zanolin;
 Nonlinear perturbations of differential operators with nontrivial
kernel and applications to third-order periodic boundary value problems,
{\it J. Math. Anal. Appl.}, {\bf 143} (1), (1989), 35 - 56.

\bibitem{a3} A. U. Afuwape, A. S. Ukpera;
Existence of solutions of periodic boundary value problems for some
vector third order differential equations, {\it J. Nig. Math. Soc.},
{\bf 20}, (2001), 1 - 17.

\bibitem{a4} H. Amann;
On the unique solvability of semi-linear operator equations in Hilbert spaces,
{\it J. Math. Pures Appl.}, {\bf 61}, (1982), 149 - 175.

\bibitem{e1} J. O. C. Ezeilo,  M. N. Nkashama;
 Resonant and nonresonant oscillations for third-order nonlinear
ordinary differential equations, {\it Nonlinear Analysis, T.M.A.},
 {\bf 12} (10), (1988), 1029 - 1046.

\bibitem{e2} J. O. C. Ezeilo; H. O. Tejumola;
Boundedness and periodicity of solutions of a certain system of
third-order nonlinear differential equations, {\it Ann. Math. Pura Appl.},
 {\bf 74} (4), (1966), 283 - 316.

\bibitem{m1} J. Mawhin;
 {\it Topological Degree Methods in Nonlinear Boundary Value Problems},
 CBMS Regional Conference Series in Mathematics {\bf 40},
 American Math. Soc., Providence, R.I. (1979).

\bibitem{m2} J. Mawhin, J. R. Ward;
 Nonuniform nonresonance conditions at the two first eigenvalues
for periodic solutions of forced Li\'{e}nard and Duffing equations,
{\it Rocky Mount. J. Math.}, {\bf 12} (4), (1982), 643 - 653.

\bibitem{m3} F. Minh\'{o}s;
 Periodic solutions for a third order differential equation under
conditions on the potential, {\it Portugaliae Math.},
{\bf 55} (4) (1998), 475 - 484.

\bibitem{o1} P. Omari, F. Zanolin;
 On forced nonlinear oscillations in $n$-th order differential systems
with geometric conditions, {\it Nonlinear Analysis T.M.A.}
{\bf 8} (7), (1984), 723 - 748.

\bibitem{u1} A. S. Ukpera;
Nonuniform conditions for certain forced third order nonlinear
 periodic systems, {\it Internat.J. Diff. Equ. Appl} {\bf 9} (4), (2004) ,
 289 - 307.

\bibitem{u2} A. S. Ukpera, a. U. Afuwape;
 Resonant oscillations for forced third order nonlinear periodic systems,
{\it Nonlinear Studies}, {\bf 12} (3), (2005), 215 - 231.

\end{thebibliography}

\end{document}