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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 128, pp. 1--10.\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-2013/128\hfil Lyapunov-type inequalities]
{Lyapunov-type inequalities for nonlinear systems involving the
 $(p_1,p_2,\dots ,p_n)$-Laplacian}

\author[D. \c{C}akmak, M. F. Akta\c{s},   A. Tiryaki \hfil EJDE-2013/128\hfilneg]
{Devrim \c{C}akmak, Mustafa Fahri Akta\c{s},  Aydin Tiryaki}  % in alphabetical order

\address{Devrim \c{C}akmak \newline
Gazi University, Faculty of Education\\
Department of Mathematics Education\\
06500 Teknikokullar, Ankara, Turkey}
\email{dcakmak@gazi.edu.tr}

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

\address{Aydin Tiryaki \newline
Izmir University, Faculty of Arts and Sciences\\
Department of Mathematics and Computer Sciences\\
35350 Uckuyular, Izmir, Turkey}
\email{aydin.tiryaki@izmir.edu.tr}

\thanks{Submitted August 20, 2012. Published May 27, 2013.}
\subjclass[2000]{26D10, 34A40, 34C10}
\keywords{ Lyapunov-type inequality; lower bound;
$(p_1,p_2,\dots ,p_n)$-Laplacian}

\begin{abstract}
 We  prove some generalized Lyapunov-type inequalities for
 $n$-di\-mensional Dirichlet nonlinear systems. We extend the results
 obtained by \c{C}akmak and Tiryaki \cite{Cakmak-1} for a parameter $1<p_k<2$.
 As an application, we  obtain  lower bounds for the
 eigenvalues of the corresponding system.
\end{abstract}

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

\section{Introduction}

In 1907, Lyapunov \cite{lyapunov} obtained the  remarkable
inequality
\begin{equation}
\int_a^b| f_1(s)| ds\geq \dfrac{4}{b-a},
\label{1a}
\end{equation}
if Hill's equation
\begin{equation}
x_1''+f_1(t)x_1=0  \label{1}
\end{equation}
has a real nontrivial solution $x_1(t)$ such that $x_1(a)=0=x_1(b)$,
where $a,b\in \mathbb{R}$ with $a<b$ are consecutive zeros and $x_1$
is not identically zero on $[a,b]$, where $f_1$ is a real-valued continuous
function defined on $\mathbb{R}$. We know that the constant 4 in the
right hand side of inequality \eqref{1a}
cannot be replaced by a larger number (see \cite[p. 345]{Hartman:1982}).

Since this result has proved to be a useful tool in oscillation theory,
disconjugacy, eigenvalue problems and numerous other applications in the
study of various properties of solutions for differential equations, many
proofs and generalizations or improvements of it have appeared in the
literature. For authors, who contributed to the Lyapunov-type inequalities,
we refer to [1-19].

Here, we give some inequalities 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 $\omega =\frac{1
}{c_k-a}$ and $v=\frac{1}{b-c_k}$ for $k=1,2,\dots ,n$ implies
\begin{equation}
2^{2-p_k}[ \frac{1}{c_k-a}+\frac{1}{b-c_k}] ^{p_k-1}\geq
\frac{1}{(c_k-a)^{p_k-1}}+\frac{1}{(b-c_k)^{p_k-1}}=m_1(
c_k)\label{e18}
\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{e18} is reversed, i.e.
\begin{equation}
\frac{1}{(c_k-a)^{p_k-1}}+\frac{1}{(b-c_k)^{p_k-1}}\geq 2^{2-p_k}
\big[ \frac{1}{c_k-a}+\frac{1}{b-c_k}\big] ^{p_k-1}=m_2(
c_k)\label{18c}
\end{equation}
for $p_k>2$, $k=1,2,\dots ,n$. Moreover, if we obtain the minimum of the right
hand side of inequalities \eqref{e18} and \eqref{18c} for $c_k\in (
a,b)$, $k=1,2,\dots ,n$, then it is easy to see that
\begin{equation}
\min_{a<c_k<b} m_i(c_k)=m_i(\frac{a+b}{2})
=\frac{2^{^{p_k}}}{(b-a)^{p_k-1}}  \label{10}
\end{equation}
for $i=1,2$ and $k=1,2,\dots ,n$.

In 2006, Napoli and Pinasco \cite{napoli} obtained the following inequality
\begin{equation}
\Big(\int_a^bf_1(s)ds\Big)^{\alpha _1/p_1}
\Big(\int_a^bf_2(s)ds\Big)^{\alpha _2/p_2}\geq \frac{
2^{\alpha _1+\alpha _2}}{(b-a)^{\alpha _1+\alpha _2-1}},
\label{d3}
\end{equation}
if the quasilinear system
\begin{equation}
\begin{gathered}
-(\phi _{p_1}(x_1'))'=f_1(t)| x_1| ^{\alpha _1-2}x_1|
x_2| ^{\alpha _2} \\
-(\phi _{p_2}(x_2'))'=f_2(t)| x_1| ^{\alpha _1}|
x_2| ^{\alpha _2-2}x_2
\end{gathered} \label{d1}
\end{equation}
has a real nontrivial solution $(x_1(t),x_2(t))$ such that
$x_1(a)=x_1(b)=0=x_2(a)=x_2(b)$ where $a,b\in\mathbb{R}$
 with $a<b$ consecutive zeros, and $x_k$ for $k=1,2$ are not
identically zero on $[a,b]$, where $\phi _{\alpha }(u)
=| u| ^{\alpha -2}u$, $f_1$ and $f_2$ are real-valued
positive continuous functions defined on $\mathbb{R}$,
$1<p_1,p_2<+\infty $ and the nonnegative parameters
$\alpha_1,\alpha _2$ satisfy
$\frac{\alpha _1}{p_1}+\frac{\alpha _2}{p_2}=1$.

In 2010, \c{C}akmak and Tiryaki \cite{Cakmak-1} obtained the following
inequality
\begin{equation}
\prod_{k=1}^{n}\Big(\int_a^bf_k^{+}(s)ds\Big)
^{\alpha _k/p_k}\geq \prod_{k=1}^{n}\big[ \frac{1}{
(c_k-a)^{p_k-1}}+\frac{1}{(b-c_k)^{p_k-1}}\big] ^{\alpha
_k/p_k},  \label{25}
\end{equation}
where $| x_k(c_k)| =\max_{a<t<b}|x_k(t)| $ and
 $f_k^{+}(t)=\max \left\{ 0,f_k(t)\right\} $
for $k=1, 2,\dots ,n$, if the $n$-dimensional problem
\begin{equation}
\begin{gathered}
-(\phi _{p_1}(x_1'))'=f_1(t)| x_1| ^{\alpha _1-2}x_1|
x_2| ^{\alpha _2}\dots | x_n| ^{\alpha _n}
\\
-(\phi _{p_2}(x_2'))'=f_2(t)| x_1| ^{\alpha _1}|
x_2| ^{\alpha _2-2}x_2\dots | x_n|
^{\alpha _n} \\
\dots  \\
-(\phi _{p_n}(x_n'))'=f_n(t)| x_1| ^{\alpha _1}|
x_2| ^{\alpha _2}\dots | x_n| ^{\alpha _n-2}x_n
\end{gathered}  \label{e1}
\end{equation}
has a real nontrivial solution
 $(x_1(t),x_2(t),\dots ,x_n(t)) $ satisfying the Dirichlet boundary
conditions
\begin{equation}
x_k(a)=0=x_k(b) \label{*}
\end{equation}
where $a,b\in\mathbb{R}$ with $a<b$ consecutive zeros,
$x_k\not\equiv 0$ for $k=1,2,\dots ,n$  on $[a,b]$.
Here, $n\in \mathbb{N}$, $\phi _{\alpha }(u)=| u| ^{\alpha -2}u$,
$f_k$ are real-valued continuous functions defined on
$\mathbb{R}$, $1<p_k<+\infty $ and the nonnegative parameters
$\alpha _k$ satisfy $\sum_{k=1}^{n}\frac{\alpha _k}{p_k}=1$
for $k=1,2,\dots ,n$.
Using \eqref{10} in the inequality \eqref{25}, \c{C}akmak and Tiryaki
\cite{Cakmak-1} also obtained the  inequality
\begin{equation}
\prod_{k=1}^{n}\Big(\int_a^bf_k^{+}(s)ds\big)^{\alpha _k/p_k}
\geq \frac{2^{\sum_{k=1}^{n}\alpha _k}}{(b-a)^{(
\sum_{k=1}^{n}\alpha _k)-1}}.  \label{e4}
\end{equation}

Recently, Yang et al \cite{Yang} obtained the  inequality
\begin{equation}
\int_a^bf_k(s)ds\geq \frac{2^{^{p_k}}}{(b-a)^{p_k-1}}
H_k,  \label{11-}
\end{equation}
where
\begin{equation}
H_k=\frac{M_k^{p_k-1}}{g_k(M_1,M_2,\dots ,M_n)}
\label{11a}
\end{equation}
with $M_k=| x_k(c_k)| =\max_{a<t<b}| x_k(t)| $ for
$k=1,2,\dots ,n$, at least one inequality in \eqref{11-} is also strict,
if the following nonlinear system
involving $(p_1,p_2,\dots ,p_n)$-Laplacian operators
\begin{equation}
\begin{gathered}
(\phi _{p_1}(x_1'))'+F_1(t,x_1,x_2,\dots ,x_n)=0 \\
(\phi _{p_2}(x_2'))'+F_2(t,x_1,x_2,\dots ,x_n)=0 \\
\dots  \\
(\phi _{p_n}(x_n'))'+F_n(t,x_1,x_2,\dots ,x_n)=0
\end{gathered} \label{8}
\end{equation}
has a real nontrivial solution $(x_1(t),x_2(t),\dots ,x_n(t))$
satisfying the boundary condition \eqref{*},
where $n\in\mathbb{N}$, $\phi _{\alpha }(u)=| u| ^{\alpha -2}u$,
$1<p_k<+\infty $ and
$F_k\in C([ a,b] \times\mathbb{R}^{n},\mathbb{R})$ for $k=1,2,\dots ,n$,
under the following hypothesis:
\begin{itemize}
\item[(C1)] There exist the functions $f_k\in
C([ a,b] ,[0,\infty ))$ and
$g_k\in C(\mathbb{R}^{n},[0,\infty ))$ for $k=1,2,\dots ,n$ such that
\begin{equation}
| F_k(t,x_1,x_2,\dots ,x_n)|
\leq f_k(t)g_k(x_1,x_2,\dots ,x_n)\label{9}
\end{equation}
and
\begin{equation}
g_k(x_1,x_2,\dots ,x_n) \text{ is monotonic nondecreasing
in each variable }  \label{8a}
\end{equation}
for $k=1,2,\dots ,n$.
\end{itemize}

Yang et al \cite{Yang} claim that the inequality \eqref{e4} with
$f_k(t)>0 $ for $k=1, 2,\dots ,n$ of \c{C}akmak and Tiryaki \cite{Cakmak-1}
can be obtained by using the inequality \eqref{11-} under the following
conditions
\begin{equation}
F_k(t,x_1,x_2,\dots ,x_n)
=f_k(t)g_k( x_1,x_2,\dots ,x_n),\quad k=1,2,\dots ,n,  \label{11b}
\end{equation}
where $g_k(x_1,x_2,\dots ,x_n)=| z_k(
x_1,x_2,\dots ,x_n)| $ with
\begin{equation}
\begin{gathered}
z_1(x_1,x_2,\dots ,x_n)=| x_1|
^{\alpha _1-2}x_1| x_2| ^{\alpha _2}\dots |x_n| ^{\alpha _n} \\
z_2(x_1,x_2,\dots ,x_n)=| x_1|^{\alpha _1}| x_2| ^{\alpha _2-2}x_2\dots |
x_n| ^{\alpha _n} \\
\dots  \\
z_n(x_1,x_2,\dots ,x_n)=| x_1|
^{\alpha _1}| x_2| ^{\alpha _2}\dots |
x_n| ^{\alpha _n-2}x_n,
\end{gathered}\label{11c}
\end{equation}
where $\alpha _k\geq 0$ for $k=1,2,\dots ,n$ such that $\sum_{k=1}^{n}
\frac{\alpha _k}{p_k}=1$. 
It is easy to see from \eqref{8a} that
the nondecreasing condition on each variable of $g_k$ with \eqref{11c} for
$k=1,2,\dots ,n$ is not satisfied. 
Therefore, \cite[Remarks 1--3,  Corollary 3]{Yang} fail. 
So, \cite[Corollary 3]{Yang} does not apply to this example.

Now, we present the following hypothesis instead of (C1):
\begin{itemize}
\item[(C1*)] There exist the functions $
f_k\in C([ a,b] ,[0,\infty ))$ and
$g_k\in C(\mathbb{R}^{n},[0,\infty ))$ for $k=1,2,\dots ,n$ such that
\begin{equation}
| F_k(t,x_1,x_2,\dots ,x_n)|
\leq f_k(t)g_k(| x_1| ,|x_2| ,\dots ,| x_n| )\label{11e}
\end{equation}
and
$g_k(u_1,u_2,\dots ,u_n)$ is monotonic nondecreasing
in each variable $u_i$, such that
either $g_k(0,0,\dots ,0)=0$  or
$g_k(u_1,u_2,\dots ,u_n)>0$ for at least one $u_i\neq 0$ for
$i=1,2,\dots ,n$,
for $k=1,2,\dots ,n$.
\end{itemize}

It is clear that if the hypothesis (C1) is replaced by (C1*)
for system \eqref{8}, then \eqref{e4} with $f_k(t)>0$ for $
k=1, 2,\dots ,n$ of \c{C}akmak and Tiryaki \cite{Cakmak-1} obtain by using
inequality \eqref{11-} under the condition $\alpha _k\geq 1$ for $k=1,
2,\dots ,n$.

In this article, our purpose is to obtain Lyapunov-type inequalities for
system \eqref{8} similar to the ones given in Yang et al \cite{Yang} by
imposing somewhat different conditions on the function $F_k$ for
$ k=1,2,\dots ,n$, and improve and generalize the results of
\c{C}akmak and Tiryaki \cite{Cakmak-1} when $1<p_k<2$ for $k=1,2,\dots ,n$.
In addition, the positivity conditions on the function $f_k$ for
$k=1,2,\dots ,n$ in hypothesis (C1) are dropped. We also obtain a better
lower bound for the eigenvalues of corresponding system as an application.

We derive some Lyapunov-type inequalities for system \eqref{8}, where all
components of the solution $(x_1(t),x_2(t),\dots ,x_n(t))$
have consecutive zeros at the points $a,b\in\mathbb{R}$ with $a<b$ in
$I=[t_0,\infty )\subset\mathbb{R}$. For system \eqref{8}, we also
derive some Lyapunov-type inequalities
which relate not only points $a$ and $b$ in $I$ at which all components of
the solution $(x_1(t),x_2(t),\dots ,x_n(t))$ have
consecutive zeros but also a point in\ $(a,b)$ where all components of the
solution $(x_1(t),x_2(t),\dots ,x_n(t))$ are maximized.

Since our attention is restricted to the Lyapunov-type inequalities for
system of differential equations, we shall assume the existence of the
nontrivial solution $(x_1(t),x_2(t),\dots ,x_n(t))$ of system \eqref{8}.

\section{Main results}

We give the following hypothesis for system \eqref{8}.
\begin{itemize}
\item[(C2)]  There exist the functions $f_k\in C([ a,b] ,\mathbb{R})$
and $g_k\in C(\mathbb{R}^{n},[0,\infty ))$ such that
\begin{equation}
F_k(t,x_1,x_2,\dots ,x_n)x_k
\leq f_k(t)g_k(| x_1| ,| x_2|,\dots ,| x_n| )\label{H2}
\end{equation}
and
\begin{equation}
\parbox{9cm}{
$g_k(u_1,u_2,\dots ,u_n)$ is monotonic nondecreasing
in each variable $u_i$ such that
either $g_k(0,0,\dots ,0)=0$  or
$g_k( u_1,u_2,\dots ,u_n)>0$ for at least one $u_i\neq 0$,
$i=1,2,\dots ,n$,}
\end{equation}
for $k=1,2,\dots ,n$.
\end{itemize}

One of the main results of this article is the following theorem, whose
proof is different from the that of \cite[Theorem 1]{Yang} and
modified that of \cite[Theorem 2.1]{Sim}.

\begin{theorem} \label{thm2.1}
Assume that hypothesis {\rm (C2)} is satisfied.
If  \eqref{8} has a real nontrivial solution
$(x_1(t),x_2(t),\dots ,x_n(t))$ satisfying the boundary condition
\eqref{*}, then the inequalities
\begin{equation}
\int_a^bf_k^{+}(s)ds\geq 2^{2-p_k}\big[ \frac{1}{
c_k-a}+\frac{1}{b-c_k}\big] ^{p_k-1}M_kH_k  \label{r1}
\end{equation}
hold, where $f_k^{+}(t)=\max \{ 0,f_k(t)\} $, and
$H_k$, $M_k$ for $k=1,2,\dots ,n$  are as in \eqref{11a}.
Moreover, at least one inequality in \eqref{r1} is strict.
\end{theorem}

\begin{proof}
Let the boundary condition \eqref{*} hold; i.e.,
 $x_k(a)=0=x_k(b)$ for $k=1,2,\dots ,n$ where $n\in \mathbb{N}$,
 $a,b\in\mathbb{R}$ with $a<b$ consecutive zeros and $x_k$ for
$k=1,2,\dots ,n$ are not
identically zero on $[a,b]$. Thus, by Rolle's theorem, we can choose
$ c_k\in (a,b)$ such that
\begin{equation*}
M_k=\max_{a<t<b}|x_k(t)|=| x_k(c_k)|\quad \text{and}\quad
 x_k'(c_k)=0
\end{equation*}
for $k=1,2,\dots ,n$. By using $x_k(a)=0$ and H\"{o}lder's inequality,
we obtain
\begin{equation}
| x_k(c_k)|
\leq \int_a^{c_k}| x_k'(s)| ds
\leq (c_k-a)^{(p_k-1)/p_k}\Big(\int_a^{c_k}| x_k'(s)|^{p_k}ds\Big)^{1/p_k}  \label{13}
\end{equation}
and hence
\begin{equation}
| x_k(c_k)| ^{p_k}\leq (c_k-a)^{p_k-1}\int_a^{c_k}| x_k'(s)| ^{p_k}ds  \label{t1}
\end{equation}
for $k=1,2,\dots ,n$ and $c_k\in (a,b)$. Similarly, by using
$x_k(b)=0$ and H\"{o}lder's inequality, we obtain
\begin{equation}
| x_k(c_k)| ^{p_k}\leq (
b-c_k)^{p_k-1}\int_{c_k}^b| x_k'(
s)| ^{p_k}ds  \label{t2}
\end{equation}
for $k=1,2,\dots ,n$ and $c_k\in (a,b)$. Multiplying the
inequalities \eqref{t1} and \eqref{t2} by $(b-c_k)^{p_k-1}$
and $(c_k-a)^{p_k-1}$ for $k=1,2,\dots ,n$, respectively, we
obtain
\begin{equation}
(b-c_k)^{p_k-1}| x_k(c_k)| ^{p_k}
\leq [ (c_k-a)(b-c_k)] ^{p_k-1}\int_a^{c_k}| x_k'(s)| ^{p_k}ds  \label{t7}
\end{equation}
and
\begin{equation}
(c_k-a)^{p_k-1}| x_k(c_k)| ^{p_k}
\leq [ (c_k-a)(b-c_k)] ^{p_k-1}\int_{c_k}^b| x_k'(s)| ^{p_k}ds  \label{t8}
\end{equation}
for $k=1,2,\dots ,n$ and $c_k\in (a,b)$. Thus, adding the
inequalities \eqref{t7} and \eqref{t8}, we have
\begin{equation}
| x_k(c_k)| ^{p_k}[ (b-c_k)^{p_k-1}+(c_k-a)^{p_k-1}]
 \leq [(c_k-a)(b-c_k)] ^{p_k-1}\int_a^b| x_k'(s)| ^{p_k}ds  \label{t9}
\end{equation}
for $k=1,2,\dots ,n $ and $c_k\in (a,b)$. It is easy to see that
the functions $z_k(x)=(b-x)^{p_k-1}+(x-a)^{p_k-1}$ take the minimum
values at $\tfrac{a+b}{2}$; i.e.,
\begin{equation*}
z_k(x)\geq \min_{a<x<b} z_k(x)
=z_k(\frac{ a+b}{2})=2(\dfrac{b-a}{2})^{p_k-1}
\end{equation*}
for $k=1,2,\dots ,n$. Thus, we obtain
\begin{equation}
| x_k(c_k)| ^{p_k}[ 2(\dfrac{b-a}{2})^{p_k-1}]
\leq [ (c_k-a) (b-c_k)] ^{p_k-1}\int_a^b| x_k'(s)| ^{p_k}ds  \label{405}
\end{equation}
and hence
\begin{equation}
2M_k^{p_k}=2| x_k(c_k)| ^{p_k}\leq
[ \dfrac{2}{b-a}(c_k-a)(b-c_k)]
^{p_k-1}\int_a^b| x_k'(s)|
^{p_k}ds  \label{16}
\end{equation}
for $k=1,2,\dots ,n$ and $c_k\in (a,b)$. Multiplying the $k$-th equation of
system \eqref{8} by $x_k(t)$, integrating from $a$ to $b$ by
using integration by parts and taking into account that
$x_k(a)=0=x_k(b)$ and the inequalities \eqref{H2} for
$k=1,2,\dots ,n$, then the monotonicity of $g_k$ yields
\begin{equation}
\begin{aligned}
\int_a^b| x_k'(s)| ^{p_k}ds
&= \int_a^bF_k(s,x_1(s),x_2(s) ,\dots ,x_n(s))x_k(s)ds   \\
&\leq \int_a^bf_k(s)g_k(| x_1(s)| ,| x_2(s)|,\dots ,| x_n(s)| )ds   \\
&\leq \int_a^bf_k^{+}(s)g_k(|x_1(s)| ,| x_2(s)| ,\dots ,| x_n(s)| )ds \\
&= g_k(M_1,M_2,\dots ,M_n)\int_a^bf_k^{+}(s)ds.
\end{aligned}  \label{r2}
\end{equation}
Then, using \eqref{r2} in \eqref{16}, we have
\begin{equation}
\int_a^bf_k^{+}(s)ds
\geq \frac{2M_k^{p_k}}{g_k(M_1,M_2,\dots ,M_n)}\big[ \frac{b-a}{2(
c_k-a)(b-c_k)}\big] ^{p_k-1}  \label{r3}
\end{equation}
for $k=1,2,\dots ,n$. Since $(x_1(t),x_2(t),\dots ,x_n(t))$ is a
nontrivial solution of system \eqref{8}, it is easy to see that at least
 one inequality in \eqref{r3} is strict, which completes the proof.
\end{proof}

Another main result of this paper is the following theorem whose
proof is almost the same to that of \cite[Theorem 1]{Yang};
hence it is omitted.

\begin{theorem} \label{thm2.2}
Let all the assumptions of Theorem \ref{thm2.1} hold. Then the inequality
\begin{equation}
\int_a^bf_k^{+}(s)ds\geq \big[ \frac{1}{
(c_k-a)^{p_k-1}}+\frac{1}{(b-c_k)^{p_k-1}}\big] M_kH_k
\label{r4}
\end{equation}
holds, where $f_k^{+}(t)$, $H_k$ and $M_k$ for
$ k=1,2,\dots ,n$ are as in Theorem \ref{thm2.1}.
Moreover, at least one inequality in \eqref{r4} is strict.
\end{theorem}

\begin{remark} \label{rmk2.1} \rm
The right-hand side of inequalities \eqref{r1} in Theorem \ref{thm2.1}
 or \eqref{r4} in Theorem \ref{thm2.2} shows that $c_k$, for $k=1,2,\dots ,n$,
cannot be too close to $a$ or $b$,
since the exponents satisfy $1<p_k<+\infty $
for $k=1,2,\dots ,n$.
We have $\int_a^bf_k^{+}(s)ds<\infty $  for
$k=1,2,\dots ,n$, but
\begin{gather*}
\lim_{ c_k\to a^{+},\, c_k\to b^{-}}
\big[\frac{1}{c_k-a}+\frac{1}{b-c_k}\big] ^{p_k-1}=\infty, \text{ or}\\
\lim_{c_k\to a^{+},\, c_k\to b^{-}}
\big[ \frac{1}{(c_k-a)^{p_k-1}}+\frac{1}{(b-c_k)^{p_k-1}}\big]
=\infty
\end{gather*}
for $ k=1,2,\dots ,n$.
\end{remark}

Now, according to the value of $p_k$, we compare Theorem \ref{thm2.1}
 with Theorem \ref{thm2.2} as follows.

\begin{remark} \label{rmk2.2}\rm
It is easy to see from  inequality
\eqref{e18} that if we take $1<p_k<2$, for $k=1,2,\dots ,n$,
then  inequality \eqref{r1} is better than \eqref{r4} in the sense
that \eqref{r4} follows from \eqref{r1}, but not conversely.
Similarly, from inequality \eqref{18c}, if $p_k>2$, for $k=1,2,\dots ,n$,
 then  inequality \eqref{r4} is better than \eqref{r1} in the sense
that \eqref{r1} follows from \eqref{r4}, but not conversely.
 Moreover, if $p_k=2$ or $c_k=\frac{a+b}{2}$ for
$k=1,2,\dots ,n$, then Theorem \ref{thm2.1} is exactly the same as 
Theorem \ref{thm2.2}.
\end{remark}

By using \eqref{10} in Theorem \ref{thm2.1} or \ref{thm2.2}, 
we obtain the following result.

\begin{theorem} \label{thm2.3}
Let all the assumptions of Theorem \ref{thm2.1} hold. Then the inequality
\begin{equation}
\int_a^bf_k^{+}(s)ds\geq \frac{2^{^{p_k}}}{(b-a)^{p_k-1}}M_kH_k  \label{22a}
\end{equation}
holds, where $f_k^{+}(t)$, $H_k$ and $M_k$ for
$k=1,2,\dots ,n$ are as in Theorem \ref{thm2.1}.
Moreover, at least one inequality in \eqref{22a} is strict.
\end{theorem}

Now, we present the following hypothesis which gives the importance of our
theorems for system \eqref{e1}.
\begin{itemize}
\item[(C3)] There exist the functions
 $f_k\in C([ a,b] ,\mathbb{R})$ and $g_k\in C(\mathbb{R}^{n},[0,\infty ))$
such that
\begin{equation}
F_k(t,x_1,x_2,\dots ,x_n)x_k
=f_k(t) g_k(| x_1| ,| x_2|,\dots ,| x_n| )\label{22b}
\end{equation}
and
\begin{equation}
\parbox{9cm}{$g_k(u_1,u_2,\dots ,u_n)$ is monotonic nondecreasing
in each variable $u_i$ such that
either $g_k(0,0,\dots ,0)=0$  or
$g_k(u_1,u_2,\dots ,u_n)>0$ for at least one $u_i\neq 0$ for
$i=1,2,\dots ,n$,}
\end{equation}
where $g_k(| x_1| ,| x_2|,\dots ,| x_n| )=x_kz_k(
x_1,x_2,\dots ,x_n)$ with \eqref{11c} for $k=1,2,\dots ,n$ such that
$\alpha _k\geq 0$ and $\sum_{k=1}^{n}\frac{\alpha _k}{p_k}=1$.
\end{itemize}

It is easy to see that system \eqref{8} with hypothesis (C3) reduces to
system \eqref{e1}. Since
\begin{equation}
\prod_{k=1}^{n}(M_kH_k)^{\alpha _k/p_k}=1,
\label{22ba}
\end{equation}
 we have the following results from Theorems \ref{thm2.1} and \ref{thm2.2}, 
respectively.

\begin{theorem} \label{thm2.4}
Assume that hypothesis {\rm (C3)} is satisfied. If\eqref{8}
has a real nontrivial solution $(x_1(t),x_2(t),\dots ,x_n(t))$
satisfying the boundary condition \eqref{*}, then 
\begin{equation}
\prod_{k=1}^{n}\Big(\int_a^bf_k^{+}(s)ds\Big)^{\alpha _k/p_k}
\geq \prod_{k=1}^{n}\big[ 2^{2-p_k}(
\frac{1}{c_k-a}+\frac{1}{b-c_k})^{p_k-1}\big] ^{\alpha_k/p_k},  \label{22c}
\end{equation}
 where $| x_k(c_k)| =\max_{a<t<b} | x_k(t)| $  and
$f_k^{+}(t)=\max\{ 0,f_k(t)\} $ for $k=1, 2,\dots ,n$.
Moreover, at least one inequality in \eqref{22c} is strict.
\end{theorem}

\begin{theorem} \label{thm2.5}
Let all the assumptions of Theorem \ref{thm2.4} hold. Then the inequality
\begin{equation}
\prod_{k=1}^{n}\Big(\int_a^bf_k^{+}(s)ds\Big)
^{\alpha _k/p_k}\geq \prod_{k=1}^{n}\big[ \frac{1}{
(c_k-a)^{p_k-1}}+\frac{1}{(b-c_k)^{p_k-1}}\big] ^{\alpha
_k/p_k}  \label{22d}
\end{equation}
holds, where $c_k$\textit{\ and }$f_k^{+}(t)$ for
$k=1, 2,\dots ,n$ are as in Theorem \ref{thm2.4}.
Moreover, at least one inequality in \eqref{22d} is strict.
\end{theorem}

By using \eqref{10} in Theorem \ref{thm2.4} or \ref{thm2.5} and 
\eqref{22ba} in Theorem \ref{thm2.3},
we have the following result.

\begin{corollary} \label{coro2.1}
Let all the assumptions of Theorem \ref{thm2.4} hold. Then the inequality
\begin{equation}
\prod_{k=1}^{n}\Big(\int_a^bf_k^{+}(s)ds\Big)^{\alpha _k/p_k}
\geq \frac{2^{\sum_{k=1}^{n}\alpha _k}}{(b-a)^{(
\sum_{k=1}^{n}\alpha _k)-1}}  \label{22e}
\end{equation}
holds, where $f_k^{+}(t)$ for $k=1,\ 2,\dots ,n$ is as in
Theorem \ref{thm2.4}.
Moreover, at least one inequality in \eqref{22e} is strict.
\end{corollary}

\begin{remark} \label{rmk2.3}\rm
It is easy to see from \eqref{e18} that if we take
$1<p_k<2$  for $k=1,2,\dots ,n$,
 then \eqref{22c} is better than \eqref{25} in the sense
that \eqref{25} follows from \eqref{22c}, but not conversely.
Similarly, from  \eqref{18c}, if $p_k>2$  for $k=1,2,\dots ,n$,
 then  \eqref{25} is better than \eqref{22c} in the
sense that \eqref{22c} follows from \eqref{25}, but not conversely.
\end{remark}

\begin{remark} \label{rmk2.4}\rm
It is easy to see that  inequality
\eqref{22d} is exactly the same as \eqref{25},
and \eqref{22e} is exactly the same as \eqref{e4}.
\end{remark}

\begin{remark} \label{rmk2.5}\rm
When $\alpha _k=p_k$ for $k=1, 2,\dots ,n$,  and for
$i\neq k$, $\alpha _i=0$ for $i=1, 2,\dots ,n$ in system \eqref{e1},
we obtain the result for the case of a single equation from
Theorems \ref{thm2.4}, \ref{thm2.5} or Corollary \ref{coro2.1}.
\end{remark}

\begin{remark} \label{rmk2.6}\rm
Since $| f(x)|\geq f^{+}(x)$, the integrals of
$\int_a^bf_k^{+}(s)ds$ for $k=1,2,\dots ,n$ in the above results can also be
replaced by $\int_a^b| f_k(s)| ds$ for $k=1, 2,\dots ,n$, respectively.
\end{remark}

\section{Applications}

In this section, we present some applications of the Lyapunov-type
inequalities obtained in Section 2.

Firstly, we give the same example of Yang et al \cite{Yang} which gives the
importance of our results. Note that our Corollary \ref{coro2.1}
 is applicable to the following example, but \cite[Corollary 3]{Yang} 
is not applicable to it,
since the nondecreasing condition on each variable of $g_k$ for 
$k=1,2,\dots ,n$ is not satisfied.

\begin{example}\label{examp3.1} \rm
Consider the  quasilinear system
\begin{equation}
\begin{gathered}
(\phi _{p_1}(x_1'))'+f_1(t)(3+\sin 2x_1)| x_1| ^{\alpha
_1-2}x_1| x_2| ^{\alpha _2-1}x_2=0 \\
(\phi _{p_2}(x_2'))'+f_2(t)(1+\sin ^{2}2x_2)| x_1|
^{\alpha _1-1}x_1| x_2| ^{\alpha _2-2}x_2=0,
\end{gathered}  \label{s1}
\end{equation}
where $\phi _{\alpha }(u)=| u| ^{\alpha-2}u$, $p_1,p_2>1$,
$\alpha _1,\alpha _2\geq 0$ with $\frac{\alpha _1}{p_1}+\frac{\alpha _2}{p_2}=1$,
$f_1$ and $f_2$ are nonnegative continuous functions on $[ a,b] $.
Assume that system \eqref{s1} has a real nontrivial solution
$(x_1(t),x_2(t))$ satisfying the Dirichlet boundary condition
$x_1(a)=x_1(b)=0=x_2(a)=x_2(b)$. Since
\begin{equation}
\begin{gathered}
F_1(t,x_1,x_2)x_1\leq 4f_1(t)|x_1| ^{\alpha _1}| x_2| ^{\alpha _2}\quad
\text{and}\\
F_2(t,x_1,x_2)x_2\leq 2f_2(t)| x_1| ^{\alpha _1}|x_2| ^{\alpha _2},
\end{gathered}  \label{s2}
\end{equation}
where $g_k(u_1,u_2)=u_1^{\alpha _1}u_2^{\alpha _2} $ for $k=1,2$
which are satisfied the nondecreasing condition on each
variable $u_i$ for $i=1,2$, we have the following inequalities
\begin{equation}
4\int_a^bf_1(s)ds>\frac{2^{^{p_1}}}{(b-a)^{p_1-1}}
M_1H_1,\quad
2\int_a^bf_2(s)ds>\frac{ 2^{^{p_2}}}{(b-a)^{p_2-1}}M_2H_2  \label{s4}
\end{equation}
with $M_1H_1=M_1^{p_1-\alpha _1}M_2^{-\alpha _2}$ and
$ M_2H_2=M_1^{-\alpha _1}M_2^{p_2-\alpha _2}$ from Theorem \ref{thm2.4}.
Hence, we have
\begin{equation}
\Big(\int_a^bf_1(s)ds\Big)^{\frac{\alpha _1}{p_1}}
\Big(\int_a^bf_2(s)ds\Big)^{\frac{\alpha _2}{p_2}}
>\frac{2^{\alpha _1+\alpha _2-\frac{\alpha _1}{p_1}-1}}{
(b-a)^{\alpha _1+\alpha _2-1}}  \label{s6}
\end{equation}
from Corollary \ref{coro2.1}.
\end{example}

Secondly, we give another application of the Lyapunov-type inequalities
obtained for system \eqref{e1}. Note that the lower bounds are found by
using inequality \eqref{22d} in Theorem \ref{thm2.5} coincide with 
that of \cite[Theorem 9]{Cakmak-1}. Now, we present new lower bounds
by using inequality \eqref{22c} in Theorem \ref{thm2.4} which give a better lower
bound for the eigenvalues of following system than that of 
\cite[Theorem 9]{Cakmak-1} when $1<p_k<2$ for $k=1,2,\dots ,n$.

Let $\lambda _k$ for $k=1,2,\dots ,n$ be generalized eigenvalues of system
\eqref{e1}, and $r(t)$ be a positive function for all
$t\in\mathbb{R}$. Therefore, system \eqref{e1} with
$f_k(t)=\lambda _k\alpha _kr(t)>0$ for $k=1,2,\dots ,n$ and all
$t\in\mathbb{R}$ reduces to the system
\begin{equation}
\begin{gathered}
-(| x_1'| ^{p_1-2}x_1')^{\prime }=\lambda _1\alpha _1r(t)| x_1|
^{\alpha _1-2}x_1| x_2| ^{\alpha _2}\dots |
x_n| ^{\alpha _n} \\
-(| x_2'| ^{p_2-2}x_2')'=\lambda _2\alpha _2r(t)| x_1|
^{\alpha _1}| x_2| ^{\alpha _2-2}x_2\dots |
x_n| ^{\alpha _n} \\
\dots  \\
-(| x_n'| ^{p_n-2}x_n')'=\lambda _n\alpha _nr(t)| x_1|
^{\alpha _1}| x_2| ^{\alpha _2}\dots |
x_n| ^{\alpha _n-2}x_n\,.
\end{gathered}  \label{u1}
\end{equation}
By using similar techniques to the technique in \cite{Cakmak-1}, we obtain
the following result which gives lower bounds for the $n$-th eigenvalue
$ \lambda _n$. The proof of following theorem is based on above
generalization of the Lyapunov-type inequality, as in that of
\cite[Theorem 9]{Cakmak-1} and hence is omitted.

\begin{theorem} \label{thm3.1}
There exist a function $k_1(\lambda _1,\lambda _2,\dots ,\lambda
_{n-1})$ such that
\begin{equation}
\lambda _n\geq k_1(\lambda _1,\lambda _2,\dots ,\lambda _{n-1})
\label{u1a}
\end{equation}
for every generalized eigenvalue $(\lambda _1,\lambda
_2,\dots ,\lambda _n)$ of system \eqref{u1}, where
$|x_k(c_k)| =\underset{a<t<b}{\max }|x_k(t)| $ for $k=1,2,\dots ,n$
 and
\begin{equation}
\begin{split}
&k_1(\lambda _1,\lambda _2,\dots ,\lambda _{n-1})\\
&=\frac{1}{\alpha _n}\Big\{ \prod_{k=1}^{n}
\big[ 2^{2-p_k}(\tfrac{1}{c_k-a}+
\tfrac{1}{b-c_k})^{p_k-1}\big] ^{\alpha _k/p_k}
\big[\prod_{k=1}^{n-1}(\lambda _k\alpha _k)^{\alpha
_k/p_k}\int_a^br(s)ds\big] ^{-1}\Big\} ^{p_n/\alpha _n}.  \label{u2}
\end{split}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk3.1}\rm 
Let $1<p_k<2$ for $k=1,2,\dots ,n$. 
If we compare Theorem \ref{thm3.1} with \cite[Theorem 9]{Cakmak-1}, 
we obtain $k_1(\lambda _1,\lambda _2,\dots,
\lambda _{n-1})\geq h_1(\lambda _1,\lambda _2,\dots ,\lambda _{n-1})$
 since the inequality \eqref{e18} holds. Thus, Theorem \ref{thm3.1} gives a
better lower bound than \cite[Theorem 9]{Cakmak-1}.
\end{remark}

\begin{remark} \label{rmk3.2}\rm
Since $k_1$ is a continuous function, 
 $k_1(\lambda _1,\lambda _2,\dots ,\lambda
_{n-1})\to +\infty $ as any eigenvalue of 
$\lambda _k\to 0^{+}$ for $k=1,2,\dots ,n-1$. Therefore,
there exists a ball centered in the origin such that the generalized
spectrum is contained in its exterior. Also, by rearranging terms in 
\eqref{u1a} we obtain 
\begin{equation}
\prod_{k=1}^{n}\lambda _k^{\alpha _k/p_k}
\geq \prod_{k=1}^{n}[ 2^{2-p_k}(\frac{1}{c_k-a}+\frac{1}{
b-c_k})^{p_k-1}] ^{\alpha _k/p_k}
\Big[\prod_{k=1}^{n}\alpha _k^{\alpha
_k/p_k}\int_a^br(s)ds\Big] ^{-1}.  \label{u7}
\end{equation}
It is clear that when the interval collapses, right-hand side of 
\eqref{u7} approaches infinity.
\end{remark}

\subsection*{Acknowledgments} 
The authors want to thank the anonymous referee
for his/her valuable suggestions and comments that helped us improve 
this article.

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\end{document}