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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 67, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/67\hfil Lyapunov-type inequalities]
{Lyapunov-type inequalities for $n$-dimensional quasilinear systems}

\author[M. F. Akta\c{s} \hfil EJDE-2013/67\hfilneg]
{Mustafa Fahri Akta\c{s}}  % in alphabetical order

\address{Mustafa Fahri Akta\c{s} \newline
Gazi University\\
Faculty of Sciences\\
Department of Mathematics\\
06500 Teknikokullar, Ankara, Turkey}
\email{mfahri@gazi.edu.tr}

\thanks{Submitted January 4, 2013. Published March 5, 2013.}
\subjclass[2000]{34A40}
\keywords{Lyapunov-type inequality; quasilinear system}

\begin{abstract}
 In this article, inspired by the paper of Yang et al
 \cite{YangKimLo2}, we establish new versions of Lyapunov-type
 inequalities for a certain class of Dirichlet quasilinear systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

 In this article, we  prove generalized Lyapunov-type inequalities
for a special case of the  system
\begin{equation}
-\big(r_k(x) \phi _{p_k}(u_k')\big) '=f_k(x)\phi _{\alpha _{kk}}(u_k)
\prod_{ i=1,\, i\neq k}^n| u_{i}|^{\alpha _{ki}},  \label{1}
\end{equation}
where $n\in\mathbb{N}$,  $\phi _{\gamma }(u) =| u| ^{\gamma -2}u$,
$\gamma >1$, $r_k$, $f_k\in C([ a,b] ,\mathbb{R}) $,
$r_k(x)>0$ for $k=1,2,\dots ,n$ and $x\in \mathbb{R}$.
Let $(u_1(x),u_2(x)$, $\dots ,u_{n}(x))$ be a real nontrivial solution of
the system \eqref{1} such that
\begin{equation}
u_k(a) = u_k(b) =0  \quad  k=1,2,\dots ,n,\label{2}
\end{equation}
for $a,b\in\mathbb{R}$ with $a<b$
are consecutive zeros of  $u_k$, and $u_k$  are not
identically zero on $[a,b]$, $1<p_k<\infty $ and $\alpha _{ki}$
are nonnegative constants, for $k,i=1,2,\dots ,n$.

 Lyapunov \cite{lyapunov} proved the following remarkable result, for
 problem \eqref{1}-\eqref{2} with $n=1$, $p_1=2$, and $r_1(x) =1$,
\begin{gather}
-u_1'' = f_1(x) u_1,  \label{6} \\
u_1(a) = u_1(b)=0.  \label{7}
\end{gather}
 \begin{theorem} \label{thmA}
If $f_1\in C([ a,b],[0,\infty )) $ and $u_1(x) $ is a
nontrivial solution on $[ a,b] $ for the problem
\eqref{6}-\eqref{7}, then the so-called Lyapunov inequality holds,
\begin{equation}
\frac{4}{b-a}\leq \int_{a}^{b}f_1(s) ds \,. \label{8}
\end{equation}

\end{theorem}

 Lyapunov-type inequalities have been studied extensively; see for example
the references in this article and their references.
\c{C}akmak and Tiryaki \cite{CakmakTiryaki1} obtained the following
inequality for system \eqref{1} with $n=2$ under the condition
$\sum_{k=1}^{2}\frac{\alpha _{ik}}{p_k}=1$ for $i=1,2$.

\begin{theorem} \label{thmB}
 If $f_k\in C([ a,b],\mathbb{R}) $ for $k=1,2$
 and $(u_1(x) ,u_2(x) ) $ is a nontrivial solution on
$[ a,b] $ for the system \eqref{1} with $n=2$, then the
inequality
\begin{equation}
\begin{aligned}
2^{\alpha _{21}+\alpha _{12}}
&\leq \Big(\int_{a}^{b}f_1^{+}(s) ds\Big) ^{\frac{\alpha _{21}}{p_1}}
\Big(\int_{a}^{b}f_2^{+}(s) ds\Big) ^{\frac{\alpha _{12}}{p_2}}\\
&\times
\Big(\int_{a}^{b}(r_1(s) ) ^{\frac{1}{1-p_1}
}ds\Big) ^{\frac{\alpha _{21}(p_1-1) }{p_1}}\Big(
\int_{a}^{b}(r_2(s) ) ^{\frac{1}{1-p_2}
}ds\Big) ^{\frac{\alpha _{12}(p_2-1) }{p_2}}
\end{aligned}  \label{65}
\end{equation}
holds, where $f_k^{+}(x)=\max \{ 0,f_k(x)\} $
 is the nonnegative part of $f_k(x)$ for $k=1,2$.
\end{theorem}

Recently, \c{C}akmak and Tiryaki \cite{Cakmak-1} obtained the following
inequality for system \eqref{1} with $r_k(x) =1$ and
$\alpha _{ik}=\alpha _{kk}$, $k,i=1,2,\dots ,n$, under the condition
$\sum_{k=1}^n\frac{\alpha _{kk}}{p_k}=1$.

\begin{theorem} \label{thmC}
If $f_k\in C([ a,b],\mathbb{R})$  for
 and $(u_1( x) ,u_2(x) ,\dots ,u_{n}(x) ) $
is a nontrivial solution on $[ a,b] $ for system \eqref{1} with
$r_k(x) =1$  and $\alpha _{ik}=\alpha_{kk} $,
$k,i=1,2,\dots ,n$, then the inequality
\begin{equation}
\prod_{k=1}^n[ (c_k-a) ^{1-p_k}+(
b-c_k) ^{1-p_k}] ^{\alpha _{kk}/p_k}\leq
\prod_{k=1}^n\Big(\int_{a}^{b}f_k^{+}(s)ds\Big) ^{\alpha
_{kk}/p_k}  \label{25}
\end{equation}
holds, where $| u_k(c_k)| =\max_{a<x<b} | u_k(x)| $  and
$f_k^{+}(x)=\max \{ 0,f_k(x)\} $ for $k=1,2,\dots ,n$.
\end{theorem}

Throughout this article, for the sake of brevity, we denote
\begin{gather}
D_k(x) =\frac{[ \xi _k(x)\eta _k(x)] ^{p_k-1}
}{\xi _k^{p_k-1}(x)+\eta _k^{p_k-1}(x)}, \quad
E_k( x) =2^{p_k-2}\Big(\frac{\xi _k(x)\eta _k(x)}{\xi _k(x)+\eta
_k(x)}\Big) ^{p_k-1},  \label{2ca}
\\
F_k=2^{-p_k}(\xi _k(x) +\eta _k(x)) ^{p_k-1}
=2^{-p_k}\Big(\int_{a}^{b}r_k^{1/(1-p_k) }(s)ds\Big) ^{p_k-1},  \label{2cb}
\end{gather}
where
\begin{equation}
\xi _k(x)=\int_{a}^{x}r_k^{1/(1-p_k) }(s)ds, \quad
\eta _k(x)=\int_{x}^{b}r_k^{1/(1-p_k) }(s)ds \label{2a}
\end{equation}
for $k=1,2,\dots ,n$.

Recently, Yang et al. \cite{YangKimLo2} obtained the following
inequality for system \eqref{1}.

\begin{theorem} \label{thmD}
Assume that there exist nontrivial solutions
$(e_1,e_2,\dots ,e_{n}) $ of the linear homogeneous system
\begin{equation}
e_k\big(1-\frac{\alpha _{kk}}{p_k}\big)
-\sum_{i=1,\, i\not=k}^n \frac{\alpha _{ik}}{p_k}e_{i}=0,  \label{100}
\end{equation}
where $e_k\geq 0$ for $k=1,2,\dots ,n$ and
$ \sum_{k=1}^ne_k^{2}>0$. If
$f_k\in C([ a,b] ,\mathbb{R}) $ for $k=1,2,\dots ,n$  and
$(u_1(x),u_2(x) ,\dots $, $u_{n}(x) )$  is
a nontrivial solution on $[ a,b] $ for system \eqref{1}, then the inequality
\begin{equation}
1<\prod_{k=1}^n\Big(F_k\int_{a}^{b}f_k^{+}(s)ds\Big)^{e_k}  \label{103}
\end{equation}
holds, where $f_k^{+}(x)=\max\{ 0,f_k(x)\} $\
 for $k=1,2,\dots n$.
\end{theorem}

Our motivation comes from the recent papers of \c{C}akmak and Tiryaki
\cite{CakmakTiryaki1, Cakmak-1}, Sim and Lee \cite{Sim}, Tang and He \cite{Tang},
and Yang et al. \cite{YangKimLo2}. In this article, we state and prove new
generalized Lyapunov-type inequalities for system \eqref{1} under the
condition $\alpha _{ki}=\alpha _{ik}$ for $k,i=1,2,\dots ,n$.

Since our attention is restricted to the Lyapunov-type inequalities for the
quasilinear systems of differential equations, we shall assume the existence
of the nontrivial solution of the system \eqref{1}. For readers
interested in the existence of solutions of these type systems, we refer
to the paper by Afrouzi and Heidarkhani \cite{Afrouzi}.

Now, we present some inequalities on $D_k(x) $,
$E_k( x) $, and $F_k$ for $k=1,2,\dots ,n$ which are useful in the comparison
of our main results. We know that since the function $h(x)=x^{p_k-1}$ is
concave for $x>0$ and $1<p_k<2$, Jensen's inequality
$h(\frac{\omega +v}{2}) \geq \frac{1}{2}[ h(\omega )+h(v)] $ with
$\omega = 1/\xi _k(x)$ and $v=1/\eta _k(x)$ implies
\begin{equation}
D_k(x) \geq E_k(x)  \label{e16a}
\end{equation}
for $1<p_k<2$, $k=1,2,\dots ,n$. If $p_k>2$ for $k=1,2,\dots ,n$, then the
function $h(x)=x^{p_k-1}$ is convex for $x>0$. Thus, the inequality
\eqref{e16a} is reversed; i.e.,
\begin{equation}
D_k(x) \leq E_k(x)  \label{18c}
\end{equation}
for $p_k>2$, $k=1,2,\dots ,n$. In addition, since the function
$l(x)=x^{1-p_k}$ is convex for $x>0$ and $p_k>1$, Jensen's inequality
 $l(\frac{\omega +v}{2}) \leq \frac{1}{2}[ l(\omega )+l(v)] $ with
$\omega =\xi _k(x)$ and $v=\eta _k(x)$ implies
\begin{equation}
D_k(x) \leq F_k  \label{30}
\end{equation}
for $k=1,2,\dots ,n$. By using inequality
\begin{equation}
4AB\leq (A+B) ^{2}  \label{9a}
\end{equation}
with $A=\xi _k(x) >0$ and $B=\eta _k(x) >0$ for
$k=1,2,\dots ,n$ in $E_k(x) $, we obtain the  inequality
\begin{equation}
E_k(x) \leq F_k  \label{30a}
\end{equation}
for $k=1,2,\dots ,n$.

\section{Main results}

One of the main results of this paper is the following theorem.

\begin{theorem} \label{thm2.1}
Assume that  there exist nontrivial solutions
$(e_1,e_2,\dots ,e_{n}) $ of the  linear homogeneous system
\begin{equation}
e_k\Big(1-\frac{\alpha _{kk}}{p_k}\Big)
-\sum_{\substack{ i=1,\, i\neq k}}^n
\frac{\alpha _{ki}}{p_k}e_{i}=0,  \label{121}
\end{equation}
where $e_k\geq 0$ for $k=1,2,\dots ,n$ and
$\sum_{k=1}^ne_k^{2}>0$. If $f_k\in C([ a,b] ,\mathbb{R}) $
 for $k=1,2,\dots ,n$ and $(u_1(x),u_2(x) ,\dots $, $u_{n}(x) )$
is a nontrivial solution on $[ a,b] $ for system \eqref{1} with
$\alpha_{ki}=\alpha _{ik}$ for $k,i=1,2,\dots ,n$,
then the inequality
\begin{equation}
1<\prod_{k=1}^n\Big[ \int_{a}^{b}f_k^{+}(s)\prod
_{i=1}^nD_{i}^{\alpha _{ki}/p_{i}}(s) ds\Big] ^{e_k}
\label{20a}
\end{equation}
holds, where $f_k^{+}(x)=\max \{ 0,f_k(x)\} $ for $k=1,2,\dots n$.
\end{theorem}

\begin{proof}
Let $u_k(a)=0=u_k(b)$ for $k=1,2,\dots ,n$ where
$n\in \mathbb{N}, $ $a,b\in\mathbb{R}$ with $a<b$ are consecutive
zeros and $u_k$ for $k=1,2,\dots ,n$ are
 not identically zero on $[a,b]$. By using $u_k(a) =0$ and
H\"{o}lder's inequality, we obtain
\begin{align*}
| u_k(x) |
&\leq \int_{a}^{x}|u_k'(s) | ds\\
&\leq \Big(
\int_{a}^{x}r_k^{1/(1-p_k) }(s) ds\Big)^{(p_k-1) /p_k}
\Big(\int_{a}^{x}r_k(s)
| u_k'(s) | ^{p_k}ds\Big)^{1/p_k} \\
&=\xi _k^{^{(p_k-1) /p_k}}(x)\Big(
\int_{a}^{x}r_k(s) | u_k'(s)| ^{p_k}ds\Big) ^{1/p_k}
\end{align*}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. Thus, we have
\begin{equation}
| u_k(x) | ^{p_k}\xi_k^{^{1-p_k}}(x)\leq \int_{a}^{x}r_k(s) |
u_k'(s) | ^{p_k}ds  \label{20b}
\end{equation}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. Similarly, by using
$u_k(b) =0$ and H\"{o}lder's inequality, we obtain
\begin{equation}
| u_k(x) | ^{p_k}\eta_k^{^{1-p_k}}(x)
\leq \int_{x}^{b}r_k(s) | u_k'(s) | ^{p_k}ds  \label{20c}
\end{equation}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. Adding \eqref{20b} and
\eqref{20c}, we have
\begin{equation}
| u_k(x) | ^{p_k}\leq D_k(x) \int_{a}^{b}r_k(s) | u_k'(s) | ^{p_k}ds  \label{20dA}
\end{equation}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. After that by using a
technique similar to the one in \cite[Theorem 3.1]{Tang}, it
can be showed that the equality case in \eqref{20dA} does not hold. Thus, we
have
\begin{equation}
| u_k(x) | ^{p_k}<A_kD_k(x) , \quad x\in (a,b) ,  \label{20d}
\end{equation}
where $A_k=\int_{a}^{b}r_k(s) | u_k'(s) | ^{p_k}ds$ for $k=1,2,\dots ,n$.
If we take the $\frac{\alpha _{ki}}{p_k}$-th power of both side of
 inequality \eqref{20d}, we obtain
\begin{equation}
| u_k(x) | ^{\alpha _{ki}}<A_k^{\alpha
_{ki}/p_k}D_k^{\alpha _{ki}/p_k}(x) ,  \label{20e}
\end{equation}
for $k,i=1,2,\dots ,n$. 

Multiplying both sides of \eqref{20e} with
$i=k$ by
$f_k^{+}(x) \prod_{i=1,\, i\neq k} ^n| u_{i}(x)| ^{\alpha _{ki}}$ for
$k=1,2,\dots ,n$,
integrating from $a$ to $b$, we have
\begin{equation}
\int_{a}^{b}f_k(s) \prod_{i=1}^n|
u_{i}(s) | ^{\alpha _{ki}}ds<A_k^{\alpha
_{kk}/p_k}\int_{a}^{b}f_k^{+}(s) D_k^{\alpha
_{kk}/p_k}(s) \prod_{i=1,\, i\neq k}^n| u_{i}(s)| ^{\alpha _{ki}}ds  \label{20f}
\end{equation}
for $k=1,2,\dots ,n$. On the other hand, multiplying the $k$-th equation of
system \eqref{1} by $u_k$ and integrating from $a$ to $b$, we get
\begin{equation}
A_k=\int_{a}^{b}r_k(s) | u_k'(s)|
^{p_k}ds=\int_{a}^{b}f_k(s)\prod_{i=1}^n|
u_{i}(s)| ^{\alpha _{ki}}ds  \label{20g}
\end{equation}
for $k=1,2,\dots ,n$. By using \eqref{20g} in \eqref{20f}, we have
\begin{equation*}
A_k<A_k^{\alpha _{kk}/p_k}\int_{a}^{b}f_k^{+}(s)D_k^{\alpha
_{kk}/p_k}(s) \prod_{i=1,\, i\neq k}^n| u_{i}(s) | ^{\alpha _{ki}}ds
\end{equation*}
and hence from $\alpha _{ki}=\alpha _{ik}$ for
$k,i=1,2,\dots ,n$
\begin{equation}
A_k<A_k^{\alpha _{kk}/p_k}\int_{a}^{b}f_k^{+}(s)D_k^{\alpha
_{kk}/p_k}(s) \prod_{i=1,\, i\neq k}^n| u_{i}(s) | ^{\alpha _{ik}}ds
\label{20h}
\end{equation}
for $k=1,2,\dots ,n$. Therefore, by using \eqref{20e} in \eqref{20h},
we have
\begin{equation*}
A_k^{1-\alpha _{kk}/p_k}<\int_{a}^{b}f_k^{+}(s)D_k^{\alpha
_{kk}/p_k}(s) \prod_{i=1,\, i\neq k}
^nA_{i}^{\alpha _{ik}/p_{i}}D_{i}^{\alpha _{ik}/p_{i}}(s) ds,
\end{equation*}
and hence
\begin{equation}
A_k^{1-\alpha _{kk}/p_k}<\prod_{i=1,\, i\neq k}
^nA_{i}^{\alpha_{ik}/p_{i}}\int_{a}^{b}f_k^{+}(s)\prod_{i=1}^nD_{i}^{\alpha
_{ik}/p_{i}}(s) ds  \label{20i}
\end{equation}
for $k=1,2,\dots ,n$. Raising the both sides of the inequality \eqref{20i} to
the power $e_k$ for each $k=1,2,\dots ,n$, respectively, and multiplying the
resulting inequalities side by side, we obtain
\begin{equation*}
\prod_{k=1}^nA_k^{e_k(1-\alpha _{kk}/p_k)
}<\prod_{k=1}^n\Big[ \prod_{i=1,\, i\neq k}
^nA_{i}^{\alpha _{ik}/p_{i}}\Big] ^{e_k}\prod_{k=1}^n\Big[
\int_{a}^{b}f_k^{+}(s)\prod_{i=1}^nD_{i}^{\alpha
_{ik}/p_{i}}(s) ds\Big] ^{e_k}
\end{equation*}
and hence
\begin{equation}
\prod_{k=1}^nA_k^{e_k(1-\alpha _{kk}/p_k) }<
\Big[ \prod_{k=1}^nA_k^{\sum_{i=1,\, i\neq k}^n
\frac{\alpha _{ki}}{p_k}e_{i}}\Big] \prod_{k=1}^n[
\int_{a}^{b}f_k^{+}(s)\prod_{i=1}^nD_{i}^{\alpha
_{ik}/p_{i}}(s) ds] ^{e_k}.  \label{40}
\end{equation}
It is easy to see that by using a technique similar to the one in
\cite[Theorem 3.1]{Tang}, we obtain the
inequalities $A_k>0$ for $k=1,2,\dots ,n$. Thus, we have
\begin{equation}
\prod_{k=1}^nA_k^{\theta _k}<\prod_{k=1}^n\Big[
\int_{a}^{b}f_k^{+}(s)\prod_{i=1}^nD_{i}^{\alpha
_{ki}/p_{i}}(s) ds\Big] ^{e_k},  \label{101}
\end{equation}
where $\theta _k=e_k(1-\frac{\alpha _{kk}}{p_k})
-\sum_{i=1,\, i\neq k}^n\frac{\alpha _{ki}}{p_k}
e_{i} $ for $k=1,2,\dots ,n$.
By assumption, system \eqref{121} has nonzero
solutions $(e_1,e_2,\dots ,e_{n}) $ such that $\theta _k=0$
for $k=1,2,\dots ,n$, where $e_k\geq 0$ for $k=1,2,\dots ,n$ and at least one
$e_{j}>0 $ for $j=\left\{ 1,2,\dots ,n\right\} $.
Choosing one of the solutions $(e_1,e_2,\dots e_{n}) $, we obtain
from \eqref{101} the inequality \eqref{20a}.
This completes the proof.
\end{proof}

Another main result of this paper is the following theorem.

\begin{theorem} \label{thm2.2}
Assume that there exist nontrivial solutions
$(e_1,e_2,\dots ,e_{n}) $ of system \eqref{121}. If
$f_k\in C([ a,b] ,\mathbb{R}) $ for $k=1,2,\dots ,n$
and $(u_1(x) ,u_2(x) ,\dots ,u_{n}(x) ) $ is a nontrivial solution on
 $[ a,b] $ for the system \eqref{1} with $\alpha _{ki}=\alpha _{ik}$
for $k,i=1,2,\dots ,n$, then the inequality
\begin{equation}
1<\prod_{k=1}^n\Big[ \int_{a}^{b}f_k^{+}(s)\prod
_{i=1}^nE_{i}^{\alpha _{ki}/p_{i}}(s) ds\Big] ^{e_k}
\label{e3}
\end{equation}
holds, where $f_k^{+}(x)=\max \{ 0,f_k(x)\} $ for
$k=1,2,\dots n$.
\end{theorem}

\begin{proof}
Let $u_k(a)=0=u_k(b)$ for $k=1,2,\dots ,n$ where
 $n\in \mathbb{N}, $ $a,b\in\mathbb{R}$ with $a<b$ are consecutive
zeros and $u_k$ for $k=1,2,\dots ,n$ are not
identically zero on $[a,b]$. As in the proof of Theorem \ref{thm2.1},
we have inequalities \eqref{20b} and \eqref{20c}.
Multiplying the inequalities \eqref{20b} and \eqref{20c} by
$\eta _k^{p_k-1}(x) $ and $\xi_k^{p_k-1}(x) $, $k=1,2,\dots ,n$,
respectively, we obtain
\begin{equation}
\eta _k^{p_k-1}(x)| u_k(x) |
^{p_k}\leq \eta _k^{p_k-1}(x)\xi
_k^{p_k-1}(x)\int_{a}^{x}r_k(s) | u_k'(s) | ^{p_k}ds  \label{401}
\end{equation}
and
\begin{equation}
\xi _k^{p_k-1}(x)| u_k(x) |
^{p_k}\leq \xi _k^{p_k-1}(x)\eta
_k^{p_k-1}(x)\int_{x}^{b}r_k(s) | u_k'(s) | ^{p_k}ds  \label{402}
\end{equation}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. Thus, adding the
inequalities \eqref{401} and \eqref{402}, we have
\begin{equation}
| u_k(x) | ^{p_k}(\xi
_k^{p_k-1}(x)+\eta _k^{p_k-1}(x)) \leq (\xi _k(x)\eta
_k(x)) ^{p_k-1}\int_{a}^{b}r_k(s) |
u_k'(s) | ^{p_k}ds  \label{403}
\end{equation}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. It is easy to see that the
functions $\xi _k^{p_k-1}(x)+\eta _k^{p_k-1}(x)$ take the minimum
values at $c_k\in (a,b) $ such that $\xi _k(
c_k) =\eta _k(c_k) $ for $k=1,2,\dots ,n$. Thus, we obtain
\begin{equation}
| u_k(x) | ^{p_k}(\xi
_k^{p_k-1}(c_k)+\eta _k^{p_k-1}(c_k)) \leq (\xi
_k(x)\eta _k(x)) ^{p_k-1}\int_{a}^{b}r_k(s)
| u_k'(s) | ^{p_k}ds  \label{405}
\end{equation}
for $k=1,2,\dots ,n$. Since $\xi _k(c_k)+\eta _k(c_k)=\xi _k(x)+\eta
_k(x)$, $\forall x,c_k\in (a,b) $, and 
$$
\xi _k(c_k)= \frac{\xi _k(x)+\eta _k(x)}{2}=\frac{1}{2}\int_{a}^{b}r_k^{1/(
1-p_k) }(s)ds,
$$
 we have
\begin{equation}
\begin{aligned}
&| u_k(x) | ^{p_k}[ 2^{2-p_k}(\xi _k(x)+\eta _k(x)) ^{p_k-1}]\\
&=| u_k( x) | ^{p_k}[ 2\xi _k^{p_k-1}(c_k)] \\
&\leq (\xi _k(x)\eta _k(x)) ^{p_k-1}\int_{a}^{b}r_k(
s) | u_k'(s) | ^{p_k}ds
\end{aligned} \label{406}
\end{equation}
and hence
\begin{equation}
| u_k(x) | ^{p_k}\leq E_k(x) \int_{a}^{b}r_k(s) | u_k'(s) | ^{p_k}ds
\label{407}
\end{equation}
for $k=1,2,\dots ,n$ and $x\in [ a,b] $. After that by using a technique
similar to the one in \cite[Theorem 3.1]{Tang}, it
can be showed that the equality case in \eqref{407} does not hold. Thus, we
have
\begin{equation}
| u_k(x) | ^{p_k}<A_kE_k(x), \quad x\in (a,b) ,  \label{408}
\end{equation}
where $A_k=\int_{a}^{b}r_k(s) | u_k'(s) | ^{p_k}ds$ for $k=1,2,\dots ,n$.
The rest of the proof is the same as in the proof of Theorem \ref{thm2.1},
and hence is omitted.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
It is easy to see from the inequality
\eqref{e16a} that if we take $1<p_k<2$  for $k=1,2,\dots ,n$,
then inequality \eqref{e3} is better than \eqref{20a} in the sense
that \eqref{20a} follows from \eqref{e3}, but not conversely. 
Similarly, from the inequality \eqref{18c}, if $p_k>2$  for 
$k=1,2,\dots ,n$, then  inequality \eqref{20a} is better than \eqref{e3} 
in the sense that \eqref{e3} follows from \eqref{20a}, but not conversely.
\end{remark}

By using the inequality \eqref{30} in Theorem \ref{thm2.1} or \eqref{30a} 
in Theorem \ref{thm2.2}, we obtain the following result.

\begin{corollary} \label{coro2.1}
Assume that there exist nontrivial solutions $(e_1,e_2,\dots ,e_{n}) $ 
of system \eqref{121}. If $f_k\in C([ a,b] ,\mathbb{R}) $ for $k=1,2,\dots ,n$
 and $(u_1(x) ,u_2(x) ,\dots ,u_{n}(x) ) $ is a nontrivial solution on 
$[ a,b] $ for system \eqref{1} with $\alpha _{ki}=\alpha _{ik}$
for $k,i=1,2,\dots ,n$, then 
\begin{equation}
1<\prod_{k=1}^n\Big(F_k\int_{a}^{b}f_k^{+}(s)ds\Big)^{e_k}\,.  \label{e21}
\end{equation}
\end{corollary}

\begin{remark} \label{rmk2.2} \rm
Note that Theorem \ref{thm2.1} or \ref{thm2.2} yields a new Lyapunov-type inequality which is
 not covered by Theorem \ref{thmD} given by Yang et al \cite{YangKimLo2}. 
It is easy to see that Corollary \ref{coro2.1}
coincides with Theorem \ref{thmD} under the condition $\alpha _{ki}=\alpha _{ik}$
 for $k,i=1,2,\dots ,n$.
\end{remark}

\begin{remark} \label{rmk2.3}\rm 
Since $| f(x)|\geq f^{+}(x)$, the functions $f_k^{+}(x)$ for 
in the above results can also be replaced by 
$| f_k(x)| $ for $k=1,2,\dots ,n$.
\end{remark}

Now, we give an application of the obtained Lyapunov-type inequalities for
the  eigenvalue problem
\begin{equation}
\begin{gathered}
-(r_k(x) \phi _{p_k}(u_k')) '=\lambda _kh(x)\phi _{\alpha _{kk}}(u_k)
\prod_{i=1,\, i\neq k}^n| u_{i}|
^{\alpha _{ki}}\\
u_k(a) =u_k(b)=0 ,
\end{gathered}\label{51}
\end{equation}
where $h(x)>0$. Thus, if there exist nontrivial solutions 
$(e_1,e_2,\dots ,e_{n}) $ of linear homogeneous system \eqref{121},
then we have
\begin{equation*}
\Big\{ \Big(\prod_{k=1}^{n-1}\lambda _k^{e_k}\Big)
\prod_{k=1}^n\Big[ \int_{a}^{b}h(s)\prod
_{i=1}^nD_{i}^{\alpha _{ki}/p_{i}}(s) ds\Big]
^{e_k}\Big\} ^{-\frac{1}{e_{n}}}<\lambda _{n}
\end{equation*}
or
\begin{equation*}
\Big\{ \Big(\prod_{k=1}^{n-1}\lambda _k^{e_k}\Big)
\prod_{k=1}^n\Big[ \int_{a}^{b}h(s)\prod
_{i=1}^nE_{i}^{\alpha _{ki}/p_{i}}(s) ds\Big]
^{e_k}\Big\} ^{-\frac{1}{e_{n}}}<\lambda _{n}.
\end{equation*}

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\end{document}