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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 136, pp. 1--9.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/136\hfil Differential equations with $p$-Laplacian]
{Existence of global solutions for systems of second-order
differential equations with  $p$-Laplacian}

\author[M. Medve\v d, E. Pek\'arkov\'a\hfil EJDE-2007/136\hfilneg]
{Milan Medve\v d, Eva Pek\'arkov\'a}  % in alphabetical order

\address{Milan Medve\v d \newline
Department of Mathematical Analysis and Numerical
Mathematics, Faculty of Mathematics, Physics and Informatics,
Comenius University, 842 48 Bratislava, Slovakia}
\email{medved@fmph.uniba.sk}

\address{Eva Pek\' arkov\' a \newline
Department of Mathematics and Statistics, Faculty of Science, 
Masaryk University, Jan\'a\v ckovo n\'am. 2a, CZ-602 00 Brno, Czech Republic}
\email{pekarkov@math.muni.cz}

\thanks{Submitted April 17, 2007. Published October 15, 2007.}
\subjclass[2000]{34C11}
\keywords{Second order differential equation;
$p$-Laplacian; global solution}

\begin{abstract}
 We obtain sufficient conditions for the existence of global
 solutions for the systems of differential equations
 $$
 \big(A(t)\Phi_p(y')\big)' + B(t)g(y') + R(t)f(y) = e(t),
 $$
 where $\Phi_p(y')$ is the  multidimensional $p$-Laplacian.
\end{abstract}

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

\section{Introduction}

 The $p$-Laplace differential equation
\begin{equation}\label{e:1}
\mathop{\rm div} (\|\nabla v\|)^{p-2}\nabla v) = h(\|x\|, v)
\end{equation}
plays an important role in the theory of partial differential
equations (see e. g. \cite{21}), where $\nabla$ is the gradient,
$p> 0$ and $\|x\|$ is the Euclidean norm of $x \in \mathbb{R}^n$, $n > 1$
and $h(y,v)$ is a nonlinear function on $\mathbb{R} \times \mathbb{R}$. Radially
symmetric solutions of the equation \eqref{e:1} depend on the
scalar variable $r =\|x\|$ and they are solutions of the ordinary
differential equation
\begin{equation}\label{e:2}
r^{1 - n}(r^{n - 1}|v'|)' = h(r, v),
\end{equation}
where $v' = \frac{{\rm d} v}{{\rm d} r}$ and $p > 1$. If $p \ne n$ then the
change of variables $r = t^{\frac{p-1}{p-n}}$
 transforms the equation \eqref{e:2} into the equation
\begin{equation}\label{e:3}
(\Psi_p(u'))'  = f(t, u),
\end{equation}
 where $\Psi_p(u') = |u'|^{p-2}u'$ is so called one-dimensional, or scalar
$p$-Laplacian \cite{21}, and
$$
f(t, u) = \big|\frac{p-1}{p-n}\big|^p
t^{\frac{p-n}{p(1-n)}}
h(t^{\frac{p-1}{p-n}}, u)\,.
$$
In \cite{22} the existence of periodic solutions of the system
\begin{equation}\label{e:4}
 (\Phi_p(u'))' + \frac{{\rm d}}{{\rm d} t}\nabla F(u) + \nabla G(u) = e(t)
\end{equation}
is studied, where
$$
 \Phi_p: \mathbb{R}^n \to \mathbb{R}^n, \quad
 \Phi_p(u) = (|u_1|^{p-2}u_1, \dots, |u_n|^{p-2}u_n)^T.
$$
The operator $\Phi_p(u')$ is called multidimensional $p$-Laplacian.
The study of radially symmetric solutions of the system of $p$-Laplace
equations
$$
\mathop{\rm div} (\|\nabla v_i\|^{p-2}\nabla v_i) = h_i(\|x\|, v_1, v_2,
\dots, v_n),\quad i = 1, 2, \dots, n,\quad p > 1
$$
leads to the system of ordinary differential equations
\begin{equation}\label{e:5}
(|u'_i|^{p-2}u'_i)'= f_i(t, u_1, u_2, \dots, u_n),\quad i = 1, 2,
\dots, n,\quad p \ne n
\end{equation}
where
$$
f_i(t, u_1, u_2, \dots, u_n) = |
\frac{p-1}{p-n}|^pt^{\frac{p-n}{p(1-n)}}h_i(t^{\frac{p-1}{p-n}},
u_1, u_2, \dots, u_n).
$$
This system can be written in the form
\begin{equation}\label{e:6}
(\Phi_p(u'))'= f(t, u),
\end{equation}
where $f = (f_1, f_2, \dots, f_n)^T$ and $\Phi_p(u')$ is the
$n$-dimensional $p$-Laplacian. Throughout this paper we consider
the operator $\Phi_{p+1}$ with $p > 0$ and for the simplicity we
denote it as $\Phi_p$, i. e. $\Phi_p(u) = (|u_1|^{p-1}u_1,
|u_2|^{p-1}u_2, \dots, |u_n|^{p-1}u_n)$.

   We shall study the initial value problem
\begin{gather}
(A(t)\Phi_p(y'))' + B(t)g(y') + R(t)f(y) = e(t),\label{e:7} \\
y(0) = y_0,\quad y'(0) = y_1,\label{e:8}
\end{gather}
where $p > 0$,  $y_0, y_1 \in \mathbb{R}^n$, $A(t)$, $B(t)$, $R(t)$ are continuous,
matrix-valued functions on $\mathbb{R}_+ := \langle 0, \infty), A(t)$ is
regular for all $t \in \mathbb{R}_+$ , $e: \mathbb{R}_+ \to \mathbb{R}^n$ and $f, g:\mathbb{R}^n \to \mathbb{R}^n$
are continuous mappings. The equation \eqref{e:7} with $n = 1$ has been
studied by many authors (see e.\,g. references in \cite{21}). Many papers
are devoted to the study of the existence of periodic solutions of
scalar differential equation with $p-$Laplacian and in some of them it
is assumed that $A(0) = 0$. We study the system without this singularity.
>From the recently published papers and books see e.g. \cite{12, 13, 21, 22}.
The problems treated in this paper are close to those studied in
\cite{1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 18, 20, 21, 22}.
   The aim of the paper is to study the problem of the existence of global
solutions to \eqref{e:7} in the sense of
 the following definition.

\begin{definition}\rm
A solution $y(t)$,  $t \in \langle 0, T)$ of the initial value
problem \eqref{e:7}, \eqref{e:8} is called nonextendable to the
right if either $T < \infty$ and $\lim_{t \to T^-} [\|y(t)\| +
\|y'(t)\|]=\infty$, or $T = \infty$, i.\,e. $y(t)$ is defined on
$\mathbb{R}_+ =\langle 0, \infty)$. In the second case the solution $y(t)$
is called global.
\end{definition}

   The main result of this paper is the following theorem.

\begin{theorem}\label{t:1}
Let $p > 0$, $A(t)$, $B(t)$, $R(t)$ be continuous matrix-valued
functions on $\langle 0, \infty)$,  $A(t)$ be regular for all
$t \in \mathbb{R}_+$,  $e:\mathbb{R}_+ \to \mathbb{R}^n$,  $f, g: \mathbb{R}^n \to \mathbb{R}^n$ be
continuous mappings and $y_0, y_1 \in \mathbb{R}^n$. Let
\begin{gather}
R_0 = \int_0^\infty \|R(s)\|s^{m-1}ds < \infty    \label{e:9}
\end{gather}
 and there exist constants $K_1, K_2 > 0$ such that
\begin{gather}
 \|g(u)\| \leq K_1 \|u\|^m,\quad \|f(v)\| \leq K_2 \|v\|^m,\quad
u, v \in \mathbb{R}^n. \label{e:10}
\end{gather}
Then the following assertions hold:
\begin{enumerate}
\item[1.] If  $1 < m \leq p$, then any nonextendable to the right solution
$y(t)$ of the initial value problem \eqref{e:7}, \eqref{e:8} is global.

\item[2.] Let $m > p, m > 1$,
\begin{gather*}
A_\infty := \sup_{0 \leq t < \infty}\|A(t)^{-1}\|^{-1} < \infty, \\
E_\infty :=  \sup_{0 \leq t < \infty}\|\int_0^te(s)\,{\rm d} s\| < \infty
\end{gather*}
and
$$
n^{p/2}\frac{m-p}{p}D^{\frac{m-p}{p}}\sup_{0 \leq t
< \infty}\int_0^t\Big(K_1\|B(s)\| +2^{m-1}K_2
\int_s^\infty\|R(\sigma)\|\sigma^{m-1}\,{\rm d} \sigma\Big)\,{\rm d} s <  1,
$$
for all $t \in \langle 0, \infty)$, where
$$ D = n^{p/2}A_\infty\Big(\|A(0)\Phi_p(y_1)\|
+ 2^{m-1} K_2\|y_0\|^mR_0 + E_\infty\Big).
$$
Then any nonextendable to the right solution $y(t)$ of the initial
value problem \eqref{e:7}, \eqref{e:8} is global.
\end{enumerate}
\end{theorem}

In \cite{5} a solution $u:\langle 0, T) \to \mathbb{R}^n $ with $0 < T < \infty$
of the equation \eqref{e:7} with $n = 1$ is called singular of
the second kind, if $\sup_{0 < t < T}|y'(t)| = \infty$.
By \cite[Theorem 1]{5} if $m = p > 0$ (we need to assume $m > 1$) and
the condition \eqref{e:10} is fulfilled then there exists no singular
solution of the second kind of \eqref{e:7} and all solutions
of \eqref{e:7} are defined on $\mathbb{R}_+$, i. e. they are global.
 The proof of this result is based on the
transformation $ y_1(t) = y(t),\,y_2(t) = A(t)|y'(t)|^{p-1}y'(t)$
transforming the scalar equation \eqref{e:7} into the form
\begin{gather}
y'_1 = A(t)^{- \frac{1}{p}}|y_2|^{1/p}\text{sgn}\,y_2,\quad
y'_2= - B(t)g(A(t)^{- \frac{1}{p}}\text{sgn}\,y_2) - R(t)f(y_1) + e(t).
\label{e:11}
\end{gather}
An estimate of the function $v(t) = \max_{0, \leq s \leq
t}|y_2(s)|$ proves the boundedness of $|y'(t)|$ on any bounded
interval $\langle 0, T)$. By \cite[Theorem 2]{5}, if $n = 1$, $R
\in C^1(\mathbb{R}_+, \mathbb{R})$, $R(t) > 0$, $f(x)x > 0$ for all $t \in \mathbb{R}_+$
and either $|g(x)| \leq |x|^p$ for $|x| \geq M$ for some $M \in
(0, \infty)$ or $g(x)x \geq 0$ or
 $g(x) \geq 0$ for all
$x \in \mathbb{R}_+$ then the equation \eqref{e:7} has no singular
solution of the second kind and all its solutions are defined on
$\mathbb{R}_+$, i. e. they are global. The method of proofs are based on
the study of the boundedness from above of the scalar function
$V(t) = \frac{A(t)}{R(t)}|y'(t)|^{p+1} +
\frac{p+1}{p}\int_0^{y(t)}f(s)\,{\rm d} s$ on any bounded interval
$\langle 0, T)$. We remark that in \cite{5} the case $n = 1, m = p
> 0$ is studied. The method of proofs applied in \cite{5} is not
applicable in the case $n > 1$. Our proof of Theorem \ref{t:1} is
completely different from that applied in \cite{5}. The main tool
of our proof is the discrete and also continuous version of the
Jensen's inequality, Fubini theorem and a generalization of the
Bihari theorem (see Lemma), proved in this paper. The application
of the Jensen's inequality is possible only under the assumption
$m > 1$. Therefore we do not study the case $0 < m < 1$. This
means that the problem is open for $n > 1$ and $0< m < 1$. The
natural problem is to formulate sufficient conditions for the
existence of solutions which are not global, or solutions which
are not of the second kind. This problem is not solved even for
the scalar case and it seems to be not simple. By \cite[Remark
5]{5} the existence of singular solutions of the second kind of
\eqref{e:7} is an open problem even in the scalar case. M. Bartu\v
sek proved (see \cite[Theorem 4]{1}) that if $n = 1$, $0 < p < m$
then there exists a positive function $R(t)$, $t \geq 0$ such that
the scalar equation \eqref{e:7} with $A(t) \equiv 1$, $B(t) \equiv
0$, $e(t) \equiv 0$ and $f(y) = |y|^p$ has a singular solution of
the second kind. The case $0 < p < m, n = 1$, studied by
Bartu\v sek, corresponds to the assertion 2 of our
Theorem~\ref{t:1}, however for the example given by  Baru\v sek
in \cite{5} the assumptions of the assertion 2 are not satisfied.
The function $R(t)$ is constructed using a continuous, piecewise
polynomial function and the integral $R_0$ is not finite. Let us
remark that for the case $p = 1$, i.\,e. for second order
differential equations without $p$-Laplacian and also for higher
order differential equations some sufficient conditions for the
existence of singular solutions of the second kind are proved by
Bartu\v sek in the papers \cite{2, 3, 4}. A result on the
existence of singular solutions of the second kind for systems of
nonlinear differential equations (without the
\hbox{$p$-Laplacian)} are proved by Chanturia
\cite[Theorem~3]{7} and also by Mirzov \cite{18}.


\section{Proof of the main result}\label{s:2}

First we shall prove the following lemma.

\begin{lemma} \label{lem0}
 Let $c > 0$, $m > 0$, $p > 0$, $t_0 \in \mathbb{R}$ be
constants, $F(t)$ be a continuous, nonnegative function on $\mathbb{R}_+$
and $v(t)$ be a continuous, nonnegative function on $\mathbb{R}_+$ satisfying
the inequality
\begin{equation}\label{e:12}
 v(t)^p \leq c + \int_{t_0}^tF(s)v(s)^m\,{\rm d} s,\quad t \geq t_0.
\end{equation}
Then the following assertions hold:
\begin{enumerate}
\item[\rm 1.]  If $0 < m < p$ then
\begin{equation}\label{e:13}
v(t) \leq \Big(c^{\frac{p-m}{p}} + \frac{p-m}{p}\int_{t_0}^tF(s)\,{\rm d} s
\Big)^{\frac{1}{p-m}}, \quad t\geq {t_0}
\end{equation}
\item[\rm 2.] If $m > p$, $m > 1$ and
$$
\frac{m-p}{p}c^{\frac{m-p}{p}}\sup_{t_0 \leq t < \infty
}\int_{t_0}^tF(s)\,{\rm d} s < 1
$$
then
\begin{equation}\label{e:14}
v(t) \leq \frac{c}{\Big(1 - \frac{m-p}{p}c^{\frac{m-p}{p}}
\int_{t_0}^tF(s)\,{\rm d} s\Big)^{\frac{1}{m-p}}},\quad t \geq t_0.
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
    Let $G(t)$ be the right-hand side of the inequality \eqref{e:12}. Then
$v(t)^m \leq   G(t)^{\frac{m}{p}}$
whihc yields
$$
\frac{F(t)v(t)^m}{G(t)^{\frac{m}{p}}} \leq F(t),
$$
i.\,e.
$$
\frac{G'(t)}{G(t)^{\frac{m}{p}}} \leq F(t).
$$
Integrating  this inequality from $t_0$ to $t$ we obtain
\begin{align*}
 \int_{t_0}^t\frac{G'(s)}{G(s)^{\frac{m}{p}}}\,{\rm d} s &=
\int_{G(t_0)}^{G(t)}\frac{{\rm d} \sigma}{\sigma^{\frac{m}{p}}} \\
&=\frac{p}{p-m}\Big(G(t)^{\frac{p-m}{p}}-
G(t_0)^{\frac{p-m}{p}}\Big) \\
&\leq \int_{t_0}^tF(s)\,{\rm d} s.
\end{align*}
Since $G(t_0) = c$ we obtain
$$
 v(t) \leq G(t)^{1/p} \leq \Big(c^{\frac{p-m}{p}}
 + \frac{p-m}{p}\int_{t_0}^tF(s)\,{\rm d} s\Big)^{\frac{1}{p-m}}.
 $$
The assertions \eqref{e:1} and \eqref{e:2} follow from this
inequality.
\end{proof}

\begin{remark}\rm
   If $p = 1$, $m > 0$ then this lemma is a consequence of the well known
Bihari inequality (see \cite{6}). Some results on integral inequalities with
power nonlinearity on their left-hand sides can be found in the
B.\,G.~Pachpatte monograph \cite{19}. The idea of the proof  of this
lemma is based on that used in the proofs of results on integral inequalities
with singular kernels and power nonlinearities on their left-hand sides,
published in the papers \cite{16, 17}.
\end{remark}

    Let $y(t)$ be a solution of the initial value problem
\eqref{e:7}, \eqref{e:8} defined on an interval $\langle 0, T)$,
 $0 < T \leq \infty$. If we denote $u(t) = y'(t)$ then
\begin{equation}\label{e:15}
y(t) = y_0 + \int_0^tu(s)\,{\rm d} s,
\end{equation}
and the equation \eqref{e:7} can be rewritten as the following
integro-differential equation for $u(t)$:
\begin{equation}\label{e:16}
\Big(A(t)\Phi_p(u(t))\Big)' + B(t)g(u(t))
+ R(t)f\Big(y_0 + \int_0^tu(s)\,{\rm d} s\Big)= e(t)
\end{equation}
with
\begin{equation}\label{e:17}
u(0) = y_1.
\end{equation}

\begin{theorem}\label{t:2}
Let $p > 0, A(t)$, $B(t)$, $R(t)$ be continuous matrix-valued
functions on $\mathbb{R}_+$, $A(t)$ regular for all $t \in \mathbb{R}_+$,
$e:\mathbb{R}_+ \to \mathbb{R}^n$,  $f, g: \mathbb{R}^n \to \mathbb{R}^n$ be continuous mappings
on $\mathbb{R}_+$,  $y_0, y_1 \in \mathbb{R}^n$, $R_0 := \int_0^\infty
\|R(s)\|s^{m-1}\,{\rm d} s < \infty$ and $0 < T < \infty$. Let the
condition \eqref{e:10} be satisfied and let $u:\langle 0, T) \to
\mathbb{R}^n$ be a solution of the equation \eqref{e:16} satisfying the
condition \eqref{e:17}. Then the following assertions hold:
\begin{enumerate}
\item[1.] If $m = p > 1$, then
$$
\|u(t)\| \leq d_T\hbox{\rm e}^{\int_0^tF_T(s)\,{\rm d} s},\quad 0 \leq t \leq T
$$
where
\begin{gather*}
F_T(t) := n^{p/2}E_T\Big(K_1\|B(s)\| + 2^{m-1}K_2Q(s)\Big),\\
Q(s) = \int_s^\infty\|R(\sigma)\|\sigma^{m-1}\,{\rm d} \sigma,\\
E_T := \max_{0 \leq t \leq T}\|E(t)\|,\,\,E(t) := \int_0^te(s)\,{\rm d} s,\\
d_T = n^{p/2}A_T\Big(\|A(0)\Phi_p(y_1)\| + 2^{m-1} K_2\|y_0\|^mR_0
+ E_T\Big),\\
A_T = \max_{0 \leq t \leq T}\|A(t)^{-1}\|^{-1}.
\end{gather*}

\item[2.] If $1 < m < p$, then
$$ \|u(t)\| \leq \Big(d_T^{\frac{p-m}{p}} +
\frac{p-m}{p}d_T\,\int_0^tF_T(s)\,{\rm d} s\Big)^{\frac{1}{p-m}}.
$$

\item[3.] Let
$m > p$, $m > 1$,
$A_\infty := \sup_{T \geq 0}A_T < \infty$,
$\sup_{0 \leq t \leq \infty}E(t) < \infty$,
$$
n^{p/2}\frac{m-p}{p}D^{\frac{m-p}{p}}\,\sup_{0 \leq t < \infty}\int_0^t
\Big(K_1\|B(s)\| +2^{m-1}K_2Q(s)\Big)\,{\rm d} s < 1,
$$
where
$$
D = n^{p/2}A_\infty \Big(\|A(0)\Phi_p(y_1)\| + 2^{m-1} K_2\|y_0\|^mR_0 + E_\infty\Big).
$$
then
$$
\|u(t)\| \leq D \Big(1 - n^{p/2}\frac{m-p}{p}D^{\frac{m-p}{p}}
\int_0^t\Big(K_1\|B(s)\| + 2^{m-1}K_2Q(s)\Big)\,{\rm d} s\Big)^{-\frac{1}{m-p}},
$$
where $0 \leq t \leq \infty$.
\end{enumerate}
\end{theorem}

\begin{proof}
   We shall give an explicit upper bound for the solution $u(t)$ of
the equation \eqref{e:16}, defined on the interval $\langle 0, T)$,
satisfying \eqref{e:17}. From the equation \eqref{e:16} and the condition
\eqref{e:17} it follows that
\begin{equation}\label{e:18}
\begin{aligned}
\Phi_p(u(t)) &=  A(t)^{-1}\{A(0)\Phi_p(y_1)
  - \int_0^tB(s)g(u(s))\,{\rm d} s  \\
&\quad  + \int_0^tR(s)f\Big(y_0 +
  \int_0^su(\tau)\,{\rm d}\tau \Big)\,{\rm d} s  + E(t)\},
\end{aligned}
\end{equation}
where $E(t) =  \int_0^te(s)\,{\rm d} s$. This inequality together with the
conditions \eqref{e:10} yield
\begin{equation}\label{e:19}
\begin{aligned}
 \|A(t)^{-1}\|\|\Phi_p(u(t))\| \leq&\
\|A(0)\Phi_p(y_1)\| +
K_1\int_0^t\|B(s)\|\|u(s)\|^m\,{\rm d} s\\
&+K_2\int_0^t\|R(s)\|\Big(\|y_0\| + \int_0^t\|u(\tau)\|d\tau \Big)^m\,{\rm d} s
+ \|E(t)\|.
\end{aligned}
\end{equation}

   We shall use the integral version of the Jensen' s inequality
\begin{equation}\label{e:20}
   \Big(\int_0^tH(s)\,{\rm d} s \Big)^\kappa \leq t^{\kappa -
   1}\int_0^tH(s)^\kappa \,{\rm d} s,\quad \kappa > 1,\; t \geq 0
\end{equation}
for  $h \in C(\mathbb{R}_+, \mathbb{R}_+)$ (For a more general integral Jen\-sen's
inequality, see e.\,g. \cite[Chapter VIII, Theorem 2]{15}).
 Also we shall use its discrete version
\begin{equation}\label{e:21}
(A_1 + A_2 + \dots + A_l)^\kappa \leq l^{\kappa - 1}
(A_1^\kappa + A_2^\kappa + \dots + A_l^\kappa),
\end{equation}
for $A_1, A_2, \dots, A_l \geq 0$, $\kappa > 1$
(see \cite[Chapter VIII, Corollary 4]{15}).

   Let $m > 1$. Then using the inequalities \eqref{e:20}
and \eqref{e:21} we obtain the inequality
\begin{align*}
\Big(\|y_0\| + \int_0^s\|u(\tau)\|\,{\rm d}\tau\Big)^m
&\leq 2^{m-1}\Big(\|y_0\|^m + \Big(\int_0^s\|u(\tau)\|\,{\rm d}\tau \Big)^m\Big)\\
&\leq 2^{m-1}\Big(\|y_0\|^m + s^{m-1}\int_0^s\|u(\tau)\|^m\,{\rm d}\tau\Big).
\end{align*}
Putting this inequality into \eqref{e:19} we obtain
\begin{equation}\label{e:22}
\begin{aligned}
&\|A(t)^{-1}\|\|\Phi_p(u(t))\|\\
&\leq
\|A(0)^{-1}\Phi_p(y_1)\| + K_1\int_0^t\|B(s)\|\|u(s)\|^m\,{\rm d} s
+ 2^{m-1}K_2\|y_0\|^m\int_0^t\|R(s)\|\,{\rm d} s \\
&\quad + 2^{m-1}K_2\int_0^t\|R(s)\|s^{m-1}\int_0^s\|u(\tau)\|^m\,{\rm d}\tau \,{\rm d} s
 \|E(t)\| .
\end{aligned}
\end{equation}
Now we shall apply the following consequence of the Fubini
    theorem (see e.\,g. \cite[Theorem 3.10 and Exercise 3.27]{23}):
If $h:\langle a, b\rangle \times \langle a, b\rangle \to \mathbb{R}$
is an integrable function then
$$
\int_a^b\int_a^yh(x, y)\,{\rm d} x\,{\rm d} y = \int_a^b\int_x^bh(x, y)\,{\rm d} y\,{\rm d} x.
$$
If $h(\tau, s) = \|R(s)\|s^{m-1}\|u(\tau)\|^m$, $a = 0$, $b = t$,
$y = s$, $x = \tau$ then
$$
\int_0^t\int_0^sh(\tau, s)\,{\rm d}\tau \,{\rm d} s
= \int_0^t\int_\tau^th(\tau, s)\,{\rm d} s\,{\rm d}\tau,
$$
i.\,e.
$$
\int_0^t\int_0^s\|R(s)\|s^{m-1}\|u(\tau)\|^m\,{\rm d}\tau \,{\rm d} s =
\int_0^t\Big(\int_\tau^t\|R(s)\|s^{m-1}\,{\rm d} s\Big)\|u(\tau)\|^m\,{\rm d}\tau .
$$
This yields
\begin{equation}\label{e:23}
\int_0^t\|R(s)\|s^{m-1}\int_0^s\|u(\tau)\|^m\,{\rm d}\tau \,{\rm d} s \leq
\int_0^tQ(\tau)\|u(\tau)\|^m\,{\rm d}\tau,
\end{equation}
where
$$
Q(\tau):= \int_\tau^\infty\|R(s)\|s^{m-1}\,{\rm d} s
$$
for $\tau \geq 0$.

Let $0 < T  < \infty$ and $t \in \langle 0, T)$.
 From the inequalities \eqref{e:22} and \eqref{e:23} it follows that
\begin{equation}\label{e:24}
\|A(t)^{-1}\|\|\Phi_p(u(t))\|
\leq c_T + \int_0^tF_0(s)\|u(s)\|^m\,{\rm d} s,
\end{equation}
 where
\begin{gather}
c_T = \|A(0)\Phi_p(y_1)\| + 2^{m-1}K_2\|y_0\|^mR_0 + E_T, \label{e:25}\\
F_0(s) = K_1\|B(s)\| + 2^{m-1}K_2Q(s),\label{e:26}\\
 E_T = \max_{0 \leq t \leq T}\|E(t)\|.\label{e:27}
\end{gather}
If $k \in \{1, 2, \dots, n\}$, then
\begin{align*}
|u_k(t)|^p \leq \|\Phi_p(u(t))\|
&= \big(u_1(t)^{2p} +
u_2(t)^{2p} + \dots + u_n(t)^{2p}\big)^{1/2}\\
&\leq A_Tc_T + \int_0^tA_TF_0(s)\|u(s)\|^m\,{\rm d} s;
\end{align*}
i. e.,
\begin{equation}
|u_k(t)|^p \leq c_{0T} + \int_0^tF_{0T}(s)\|u(s)\|^m\,{\rm d} s,
\end{equation}
where
\begin{equation}\label{e:28}
A_T :=  \max_{0 \leq t \leq T}\|A(t)^{-1}\|^{-1},\quad \text{if }
 T < \infty,
\end{equation}
\begin{equation}\label{e:29}
c_{0T} = A_Tc_T,\quad  F_{0T}(t) = A_TF_0(t).
\end{equation}
This yields
$$
\|u(t)\| \leq n^{p/2}\Big(c_{0T} + \int_0^tF_{0T}(s)\|u(s)\|^m\,{\rm d} s\Big)^{1/p},
$$
and therefore we have obtained the inequality
\begin{equation}\label{e:31}
\|u(t)\|^p \leq d_T + \int_0^tF_T(s)\|u(s)\|^m\,{\rm d} s,
\end{equation}
where
\begin{equation}\label{e:30}
d_T = n^{p/2}c_{0T}, F_T(t) = n^{p/2}F_{0T}(t) .
\end{equation}

Now applying Lemma \ref{lem0}
(the case $m = p$ follows from the Gronwall's lemma)
to the inequality \eqref{e:31} we obtain the assertions 1. and 2. In the
proof of the assertion 3. we use the assumptions
$A_\infty := \sup_{0 \leq t < \infty }\|A(t)^{-1}\|^{-1} < \infty$,
$\sup_{0 \leq t \leq \infty}E(t) < \infty$.
>From the inequality \eqref{e:31} we obtain the inequality,
\begin{equation}
\|u(t)\|^p \leq D + \int_0^tG(s)\|u(s)\|^m\,{\rm d} s,
\end{equation}
where $D$ is defined in Theorem \ref{t:1},
$$
G(s) := K_1\|B(s)\| + 2^{m-1}K_2Q(s),
$$
and $Q(s) = \int_s^t\|R(\sigma )\sigma^{m-1}\|\,{\rm d}\sigma$.
Now if we put in Lemma $t_0 = 0$,  $v(t) = \|u(t)\|$,  $c = D$
and $F(t) = G(t)$ then we obtain the
inequality from the assertion 3.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{t:1}]
Let $y:\langle 0, T)\to \mathbb{R}^n$ be a nonextendable to the right
solution of the initial value problem \eqref{e:16}, \eqref{e:17}
with $T < \infty$. Then $y(t) = y_0 + \int_0^tu(s)\,{\rm d} s$, where
$u(t)$ is a solution of the equation \eqref{e:16} satisfying the
condition \eqref{e:17}. From Theorem \ref{t:2} it follows that
$M= \sup_{0 \leq t \leq T}\|u(t)\| < \infty$ and since \eqref{e:15}
yields $\|y(t)\| \leq \|y_0\| + t\sup_{0 \leq s \leq T}\|u(s)\|$
we obtain $\lim_{t \to  T^-} \|y(s)\| < \infty$. This is a
contradiction with nonextendability of $y(t)$.
\end{proof}

\subsection*{Acknowledgements}
The authors are grateful to all the referees for their helpful critical remarks on the first version of the manuscript. This work was supported by
the Grant No. 1/2001/05 of the Slovak Grant Agency VEGA-SAV-M\v S.

\begin{thebibliography}{00}

\bibitem{1} M. Bartu\v sek,
\textit{Singular solutions for the differential equation with $p$-Laplacian,}
Archivum Math. (Brno), \textbf{41} (2005) 123--128.


\bibitem{2} M. Bartu\v sek,
\textit{On singular solutions of a second order differential  equations},
 Electronic Journal of Qualitaive Theory of Differential Equations, \textbf{8}
 (2006), 1--13.

\bibitem{3} M. Bartu\v sek,
\textit{Existence of noncontinuable solutions}, Electronic Journal of Differential Equations, Conference \textbf{15} (2007), 29--39.

\bibitem{4} M. Bartu\v sek
 \textit{On the existence of unbounded noncontinuable solutions}, Annali di Matematica {185} (2006), 93--107.

\bibitem{5} M. Bartu\v sek and E. Pek\' arkov\' a
 \textit{On existence of proper solutions of quasilinear second order
differential equations}, Electronic Journal of Qualitative
Theory of Differential Equations, \textbf{1} (2007), 1--14.

\bibitem{6} J. A. Bihari, \textit{A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations}, Acta Math.
Acad. Sci. Hungar., \textbf{7} (1956), 81--94.

\bibitem{7} T. A. Chanturia, \textit{A singular solutions of nonlinear systems of ordinary differential equations}, Colloquia Mathematica Society J\' anos
Bolyai \textbf(15)Differential Equations, Keszthely (Hungary) (1975), 107--119.

\bibitem{8} M. Cecchi, Z. Do\v sl\' a and M. Marini,
\textit{On oscillatory solutions of differential equations with
$p$-Laplacian}, Advances in Math. Scien. Appl., \textbf{11}
(2001), 419--436.


\bibitem{9} O. Do\v sl\'y, \textit{Half-linear differential
equations,
 Handbook of Differential Equations},  Elsevier
 Amsterdam, Boston, Heidelberg, (2004) 161--357.


\bibitem{10} O. Do\v sl\'y and Z. P\' atikov\' a,
\textit{Hille-Wintner type comparison criteria for half-linear second
order differential equations}, Archivum Math. (Brno),
\textbf{42} (2006), 185--194.


\bibitem{11} Ph. Hartman, \textit{Half-linear differential
equations}, Ordinary Differential Equations, John-Wiley
 and Sons, New-York, London, Sydney 1964.


\bibitem{12} P. Jebelean and J. Mawhin,
\textit{Periodic solutions of singular nonlinear
perturbations of the ordinary $p$-Laplacian}, Adv.
Nonlinear Stud., \textbf{2} (2002), 299--312.


\bibitem{13} P. Jebelean and J. Mawhin,
\textit{Periodic solutions of forced dissipative $p$-Lienard equations with singularities},
Vietnam J. Math., \textbf{32} (2004), 97--103.


\bibitem{14} I. T. Kiguradze and T. A. Chanturia,
\textit{ Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations},
Kluwer, Dortrecht 1993.


\bibitem{15} M. Kuczma, \textit{An Introduction to the Theory of
Functional Equations and Inequalities, Cauchy's Equation and
Jensen's Inequality}, Panstwowe Wydawnictwo Naukowe,
Warszawa--Krakow--Katowice 1985.


\bibitem{16} M. Medve\v d, \textit{A new approach to an analysis of
Henry type integral inequalities and their Bihari type
versions}, J.~Math. Ana. Appl., \textbf{214} (1997), 349--366.


\bibitem{17} M. Medve\v d, \textit{Integral inequalities and global
solutions of semilinear evolution equations}, J.~Math. Anal.
Appl., \textbf{267} (2002) 643--650.


\bibitem{18} M. Mirzov, \textit{Asymptotic Properties of Solutions
of Systems of Nonlinear Nonautonomous Ordinary Differential
Equations}, Folia Facultatis Scie. Natur. Univ. Masarykianae
Brunensis, Masaryk Univ., Brno, Czech Rep. 2004.

\bibitem{19} B. G. Pachpatte,
\textit{Inequalities for Differential and Integral Equations},
Academic Press, San Diego,
Boston, New York 1998.

\bibitem{20} I. Rachunkov\' a and M. Tvrd\' y,
 \textit{Periodic singular problem with quasilinear differential operator}, Math.
Bohemica, \textbf{131} (2006), 321--336.

\bibitem{21} I. Rachunkov\' a, S. Stan\v ek and M. Tvrd\' y,
\textit{Singularities and Laplacians in Boundary Value Problems for
Nonlinear Ordinary Differential Equations}, Handbook of
Differential Equations. Ordinary Differential Equations  \textbf{3}
606--723, Ed. by A. Canada, P. Dr\' abek, A. Fonde,
Elsevier 2006.

\bibitem{22} Shiguo Peng and Zhiting Xu, \textit{On the existence
of periodic solutins for a class of $p$-Laplacian system}, J.
Math. Anal. Appl., \textbf{326} (2007), 166--174.

\bibitem{23} M. Spivak, \textit{Calculus on Manifolds}, W. A.
Benjamin, Amsterdam 1965.

\end{thebibliography}

\end{document}