\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 $$X''' + AX'' + BX' + sH(t,X) = P(t) \label{e1.1}$$ subject to the periodic boundary conditions $$X(0) - X(T) = X'(0) - X'(T) = X''(0) - X''(T) = 0 \label{e1.2}$$ 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 $X\in\mathbb{R}^n$, $$\delta_d \|X\|^2 \leq \langle DX, X\rangle \leq \Delta_d \|X\|^2, \label{e1.3}$$ 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{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} \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 $$\min_{1\leq j,k\leq n} \{ {\lambda}_j (D_1 ) {\lambda}_k (D_2 )\} \leq \lambda_i (D_1 D_2 ) \leq \max_{1\leq j,k\leq n} \{ {\lambda}_j (D_1 ) {\lambda}_k (D_2 )\} \label{e1.5}$$ \end{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 $$k^2 {\omega}^2 < {\Delta}_a^{-1} {\delta}_h \leq \frac{\langle sA^{-1} H(t,X) , X \rangle}{{\| X\|}^2} \leq {\delta}_a^{-1} {\Delta}_h < (k+1)^2 {\omega}^2 \label{eH1}$$ 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 $$k^2 \leq \frac{\gamma^- (t)}a \leq \liminf_{|x| \to \infty} \frac{h(t,x)}{ax} \leq \limsup_{|x| \to \infty} \frac{h(t,x )}{ax} \leq \frac{\gamma^+ (t)}a \leq (k+1)^2 \label{eh1}$$ 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{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} 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{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} 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 $$X''' + AX'' + BX' = -A\lambda X \label{e2.1}$$ 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 $$X''' + AX'' + BX' + \lambda AX = P(t) \equiv P(t+T) , P\in L^1 \label{e2.2}$$ 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}) $$-\gamma_D \| X\|^2 \leq \sigma_{\min} \| X\|^2 \leq \langle D_s X, X\rangle = \langle DX, X\rangle \leq \sigma_{\max} \| X\|^2 \leq \Gamma_D \| X\|^2. \label{e2.3}$$ 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 $$0 \le \alpha \le \beta \label{e*}$$ with the strict inequality holding on subsets of $[0,T]$ of positive measure. Let us now consider the linear system $$X''' (t) + AX'' (t) + BX' (t) + C(t)X(t) = 0 \label{e2.4}$$ 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 $$k^2 {\omega}^2 {\| X\|}^2 \leq \Gamma_A^{-1} \gamma_C (t) {\| X\|}^2 \leq \langle A^{-1} C(t)X,X\rangle \leq \gamma_A^{-1} \Gamma_C (t) {\| X\|}^2 \leq (k+1)^2 {\omega}^2 {\| X\|}^2 \label{e2.5}$$ 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{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} 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{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} 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 $$\frac T2 \sum^n_{i=1}\sum_{k =N_0 + 1}^{\infty} (k^2 {\omega}^2 - \rho (t) )^2 ({a_{k,i} }^2 + { b_{k,i} }^2 ) \ge \frac 14 \sum^n_{i=1} \sum_{k =N_0 +1}^{\infty} k^4 {\omega}^4 ({ a_{k,i} }^2 + { b_{k,i} }^2 ). \label{e2.12}$$ But then, we can fix a constant ${\delta}_1$ such that $$\delta_1 \sum^n_{i=1}\sum_{k = 1}^{N_0} (k^2 {\omega}^2 - \rho (t) )^2 ({a_{k,i} }^2 + { b_{k,i} }^2 ) \ge \sum^n_{i=1} \sum_{k = 1}^{N_0} k^4 {\omega}^4 ({ a_{k,i} }^2 + {b_{k,i} }^2 ). \label{e2.13}$$ 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 $$\Lambda_1 (t) \| X\|^2 \leq \langle A^{-1} M(t)X,X \rangle \leq \langle A^{-1} N(t)X,X \rangle \leq \Lambda_2 (t) \| X\|^2 \label{e2.15}$$ 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 $$\langle (A^{-1} M(t) - \epsilon I)X,X \rangle \leq \langle D(t)X,X \rangle \leq \langle (A^{-1} M(t) + \epsilon I)X,X \rangle \label{e2.16}$$ 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 $$\int_0^T \| A^{-1} X''' + X'' + A^{-1} BX' + A^{-1} C(t)X \| \,dt \ge \delta {\| X\|}_{W_T^{3,1}} \label{e2.17}$$ \end{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 $$\langle (A^{-1} M(t) - \frac 1n I)X,X \rangle \leq \langle D_n (t)X,X \rangle \leq \langle (A^{-1} N(t) + \frac 1n I)X,X \rangle , \quad n\in\mathbb{N} , \label{e2.18}$$ 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 $$\int_0^T \| A^{-1} X_n''' (t) + X_n'' (t) + A^{-1} BX_n' (t) + A^{-1} C_n (t)X_n \| \,dt < \frac 1n \label{e2.19}$$ 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 $$X_n \to X \mbox{ in } C^1 ([0,T],\mathbb{R}^n ) , X_n' \to X' , X_n'' \to X'' \mbox { in } L^1 ([0,T],\mathbb{R}^n ) \label{e2.20}$$ Moreover, $D_n \rightharpoonup D$ in $L^1 ([0,T],\mathbb{R}^{n^2} )$, so that by \eqref{e2.18}, $$\langle A^{-1} M(t)X,X \rangle \leq \langle D(t)X,X \rangle \equiv \langle A^{-1} C(t)X,X \rangle \leq \langle A^{-1} N(t)X,X \rangle \label{e2.21}$$ 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{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} 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, $$X''' (t) + AX'' (t) + BX' (t) + C(t)X(t) = 0 ,\label{e2.24}$$ 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{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} 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 $$X''' + AX'' + BX' + (1 - \lambda )C(t)X + \lambda sH(t,X) = \lambda P(t) \label{e3.1}$$ with $\lambda\in [0,1]$, or equivalently $$A^{-1} X''' + X'' + A^{-1} BX' + (1 - \lambda )A^{-1} C(t)X + \lambda sA^{-1} H(t,X) = \lambda A^{-1} P(t) \label{e3.2}$$ 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{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} 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{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} Now, if $\| X\| \ge R_0$, then by \eqref{e2.11} and \eqref{eH4}, \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} 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 $${\| sA^{-1} H(t,X) - A^{-1} C(t)X \|}^2 \leq {\alpha}^2 {\| X\|}^2 + \delta_4 \| X\| + \delta_5 \label{e3.6}$$ 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 $$\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \leq \delta_7 + {\alpha}^2 \int_0^T {\|X \|}^2 \,dt + \delta_8 \int_0^T (\| X'' \| + \| X \|) dt ,\label{e3.7}$$ 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 $$({\beta}^2 - {\alpha}^2 ) \int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \leq \delta_9 + \delta_{10} \int_0^T (\| X'' \| + \| X \|) dt ,\label{e3.8}$$ 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{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} by the Cauchy-Schwartz inequality and the relations \eqref{e2.6} and \eqref{e2.7}. We therefore deduce that $$\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \leq \delta_{14} \label{e3.10}$$ 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 $$\int_0^T {\| X\|}^2 \,dt \leq \delta_{15} , \int_0^T {\| X' \|}^2 \,dt \leq \delta_{15} , \int_0^T {\| X'' \|}^2 \,dt \leq \delta_{15} \label{e3.11}$$ 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 $$\int_0^T {\| X''' \|}^2 \,dt \leq \delta_{16} \label{e3.12}$$ 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{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} 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 $$\langle (A^{-1} M(t) - \epsilon I)X,X \rangle \leq \langle sA^{-1} H(t,X),X \rangle \leq \langle (A^{-1} N(t) + \epsilon I)X,X \rangle \label{e3.13}$$ 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 $$\kappa (t) \equiv \frac 1{\| X\|^2} \Big[\| \langle (A^{-1} M(t) - I)X,X \rangle \| + \| \langle (A^{-1} N(t) + I)X,X \rangle \| \Big] \label{e3.14}$$ so that by construction and \eqref{e3.13}, we deduce that for a.e. $t\in [0,T]$ and $X\in\mathbb{R}^n$ $$\langle A^{-1} M(t) - \epsilon I)X,X \rangle \leq \langle sA^{-1} \mu (t,X),X \rangle \leq \langle A^{-1} N(t) + \epsilon I)X,X \rangle \label{e3.15}$$ 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 $$\|\Theta (t,X)\| \leq \sup_{\| X\|\le\zeta} \| H(t,X) - \tilde{H} (t,X)\| \leq \varphi (t) \label{e3.16}$$ 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 $$X''' (t) + AX'' (t)+ BX' (t) + s\tilde{H} (t,X(t)) + s\Theta (t,X(t)) = P(t) \label{e3.17}$$ 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 $$X''' + AX'' + BX' + (1 - \lambda )N(t)X + s\lambda \tilde{H} (t,X) + s\lambda \Theta (t,X) = \lambda P(t) \label{e3.18}$$ 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 $$LX + (1 - \lambda ) \widetilde{A} X + \lambda (QX + RX) = 0 \label{e3.19}$$ 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{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} Then, from \eqref{e3.15}, for all $X\in \mathop{\rm dom} L$ we have \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} for a.e. $t\in [0,T]$ and $\lambda\in [0,1]$. Thus, we may set $$(1 - \lambda )A^{-1} N(t)X + s\lambda A^{-1} \tilde{H} (t,X) \equiv A^{-1} C(t)X , \label{e3.22}$$ 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 $$\langle (A^{-1} M(t) - \epsilon I)X,X \rangle \leq \langle A^{-1} C(t)X,X \rangle \leq \langle (A^{-1} N(t) + \epsilon I)X,X \rangle \label{e3.23}$$ 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{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} It follows from \eqref{e3.22} that \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} where $\gamma_A > 0$ is as defined in the previous section. That is, by Lemma \ref{lem2.2} and \eqref{e3.16}, we obtain $$0 \ge \delta {\| X\|}_{W^{3,1}} - \gamma_A^{-1} \bigl( \|\varphi \|_{L^1} - {\| P\|} _{L^1} \bigr) , \label{e3.26}$$ 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 $$k^2 {\omega}^2 < \Gamma_A^{-1} \gamma_H (t) \leq \frac{\| sA^{-1} (H(t,X_1 ) - H(t,X_2 ) )\|}{\| X_1 - X_2 \|} \leq \gamma_A^{-1} \Gamma_H (t) < (k+1)^2 {\omega}^2 \label{eH6}$$ or \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} 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 $$X''' (t) + AX'' (t) + BX' (t) + \widetilde{H} (t,X(t)) = 0 \label{e3.27}$$ 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 $$\int_0^T \langle X'' + sA^{-1} \widetilde{H} (t,X) , X'' + A^{-1} C(t)X\rangle \,dt = 0 \label{e3.28}$$ 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 $$\int_0^T {\| X'' + A^{-1} C(t)X \|}^2 \,dt \leq \int_0^T {\| A^{-1} \widetilde{H} (t,X) - A^{-1} C(t)X \|}^2 \,dt \label{e3.30}$$ Now if $X(t) \ne 0$, then by \eqref{eH6}, \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} 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}, $${\beta}^2 \int_0^T {\| X \|}^2 \,dt \leq {\alpha}^2 \int_0^T {\| X \|}^2 \,dt \label{e3.32}$$ 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 $$X''' (t) + AX'' (t) + BX' (t) + \widetilde{C} (t,X(t))X(t) = 0 , \label{e3.33}$$ 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. 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