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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 40, 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  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2008/40\hfil Differential equations with $p$-Laplacian]
{Existence of global solutions for systems of second-order
functional-differential equations with  $p$-Laplacian}

\author[M. Bartu\v sek, M. Medve\v d\hfil EJDE-2008/40\hfilneg]
{Miroslav Bartu\v sek, Milan Medve\v d}  % in alphabetical order

\address{Miroslav Bartu\v sek \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{bartusek@math.muni.cz}

\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}

\thanks{Submitted January 29, 2008. Published March 20, 2008.}
\subjclass[2000]{34C11, 34K10}
\keywords{Second order functional-differential equation;
$p$-Laplacian; \hfill\break\indent global solution}

\begin{abstract}
 We find sufficient conditions for the existence of global
 solutions for the systems of functional-differential equations
 $$
 \big(A(t)\Phi_p(y')\big)' + B(t)g(y', y'_t) + R(t)f(y, y_t) = e(t),
 $$
 where $\Phi_p(u) = (|u_1|^{p-1}u_1, \dots, |u_n|^{p-1}u_n)^T$
 which is the  multidimensional $p$-Laplacian.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}\label{s:1}

There are many papers concerning various problems for ordinary
differential equations with $p$-Laplacian. From the recently
published papers and books see e.g. \cite{14, 15, 24, 25, 26}. The
problems treated in this paper are close to those studied in
\cite{1}-\cite{6}, \cite{8}-\cite{26}.
 The recently published paper \cite{10} contains
some results on the existence of positive solutions of a boundary
value problem for a $p$-Laplacian functional- differential
equations. This paper motivated us to study the problem of the
existence of global solutions for such type of equations. This
problem for functional-differential equations of the first order
on the Banach space has been recently studied in the paper \cite{20}. A
survey of papers on this problems concerning systems of ordinary
differential equations and also scalar differential equations with
$p$-Laplacian and  some remarks on results close to the results
proved in \cite{21} can be found in the introduction of this paper.

  In this paper, we are concerned with the  initial value problem
\begin{gather}\label{e:1}
\big(A(t)\Phi_p(y')\big)' + B(t)g(y', y'_t) + R(t)f(y, y_t) = e(t),
\quad t \geq 0, \\
\label{e:2}
y(t) = \varphi_0(t), y'(t) = \varphi_1(t),\quad - r \leq t \leq 0,
\end{gather}
where $n\in \{1,2,\dots\}$,
$\Phi_p(u) = (|u_1|^{p-1}u_1, \dots, |u_n|^{p-1}u_n)^T$,
$u \in \mathbb{R}^n$,
$y_t \in C^1 := C^1(\langle - r, 0\rangle, \mathbb{R}^n)$,
$y_t(\Theta)= y(t + \Theta)$,
$y'_t \in C = C(\langle - r, 0\rangle, \mathbb{R}^n)$,
$y'_t(\Theta) = y'(t + \Theta)$, $A(t)$, $B(t)$, $R(t)$
are continuous, matrix-valued functions on
$\mathbb{R}_+ := \langle 0, \infty), A(t)$ is regular for all
$t \in \mathbb{R}_+$, $e: \mathbb{R}_+ \to \mathbb{R}^n$,
$ \varphi_0:\langle -r, 0 \rangle \to {\mathbb R}^n$,
$\varphi_1 :\langle -r, 0 \rangle \to {\mathbb R}^n$ and $f:\mathbb{R}^n
\times C^1 \to R^n, g:\mathbb{R}^n \times C \to \mathbb{R}^n$
are continuous mappings.

  The aim of the paper is to study the problem of the existence of
global solutions of the equation \eqref{e:1}
in the sense of the following definition.

\begin{definition} \label{def1.1} \rm
A solution $y(t)$,  $t \in \langle - r, T)$ of the initial value
problem \eqref{e:1}, \eqref{e:2} is called non-extendable 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 $\langle - r, \infty)$. In the
second case the solution $y(t)$ is called global.
\end{definition}

   We shall use in the sequel the norm
$\|z\|= \max_{0 \leq i \leq n}|z_i|$ of $ z = (z_1, z_2,\dots, z_n) \in R^n$.
The main results of this paper are formulated in the following theorems.

\begin{theorem}\label{t:1}
Let $m > p, m \geq 1, A(t)$, $B(t)$, $R(t)$ 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 $\varphi_0 \in C^1$, $\varphi_1 \in C$,
$\varphi_0(0) = y_0$, $\varphi_1(0) = y_1$. Let
\begin{equation}
 \int_0^\infty \|R(s)\|s^{m-1}{\rm d}s < \infty    \label{e:3}
\end{equation}
 and there exist constants $K_1, K_2 > 0$ such that
\begin{equation}
 \|g(u, v)\| \leq K_1 (\|u\|^m + \|v\|_C^m),\quad
\|f(u, v)\| \leq K_2 (\|u\|^m + \|v\|_C^m)\,,\label{e:4}
\end{equation}
for all $(u, v) \in \mathbb{R}^n \times C$.
Let $A_\infty =\sup_{0\leq t<\infty} \Vert A(t)^{-1}\Vert$,
$R_\infty =\int_0^\infty \|R(s)\|\,{\rm d} s$,
\begin{gather*}
B_\infty := \sup_{0 \leq t < \infty }\int_0^t\Vert B(\tau)\Vert {\rm d}\tau < \infty,
\quad  E_\infty := \sup_{0 \leq t < \infty}\int_0^t \Vert
e(s)\Vert {\rm d} s < \infty
\end{gather*}
and
\begin{equation}
 \frac{m - p}{p}c^{\frac{m - p}{p}}
 \sup_{0 \leq t < \infty}\int_0^tF(s){\rm d} s<  1, \label{e:5}
\end{equation}
where
\begin{gather*}
\begin{aligned}
c &:= A_\infty \{\Vert A(0)\Phi_p(y_1)\Vert
+ 2^{m-1}K_1\Vert \varphi_1\Vert_C^m  B_\infty\\
&\quad +  2^{m-1} K_2\big(\Vert y_0\Vert ^m + \big(\Vert
\varphi_0\Vert _C + \Vert y_0\Vert\big)^m\big)R_\infty \},
\end{aligned}
\\
F(t) = 2^mK_2A_\infty \int_t^\infty\Vert R(s)\Vert s^{m-1}{\rm d} s
 + (2^{m-1}+1)K_1A_\infty\Vert B(t)\Vert\,.
\end{gather*}
 Then any nonextendable to the right solution $y(t)$ of the initial
value problem \eqref{e:1}, \eqref{e:2} is global.
 \end{theorem}


 Due to the continuous Jensen's inequality, Theorem \ref{t:1} is
 valid for  $m\geq 1$ only.  A similar result is stated in the
 following theorem in case $m<1$ under stronger assumptions.

\begin{theorem}\label{t:3}
Let $m >p>0$, $0<m<1$, $A(t)$, $B(t)$, $R(t)$ be continuous
matrix-valued functions on $\mathbb R_+$, $A$ 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 $\varphi_0\in C^1$, $\varphi_1\in C$,
 $\varphi_0(0)=y_0$, $\varphi_1(0)=y_1$. Let constants $K_1$,
$K_2>0$ exist such that
$$
\Vert  g(u,v)\Vert  \leq K_1 (\Vert u\Vert ^m + \Vert v\Vert
_C^m), \quad  \Vert  f(u,v)\Vert  \leq K_2 (\Vert u\Vert ^m +
\Vert v\Vert _C^m)
$$
for $(u,v)\in {\mathbb R^n \times C}$. Let
$$
\frac{m-p}{p} C_1^{\frac{m-p}{p}} \sup_{0\leq t <\infty } \int_0^t F_1 (s)\,{\rm d} s <1\,,
$$
where $B_\infty $ and $E_\infty $ are given in Theorem
\ref{t:1}.2 and
\begin{gather*}
\begin{aligned}
C_1&= A_\infty  \big\{ \Vert  A(0)\Phi_p(y_1)\Vert
+ 2^m K_1 \Vert  \varphi_1\Vert _C^m B_\infty \\
&\quad + 2^m K_2 R_\infty   \big(\Vert y_0\Vert ^m + (\Vert \varphi_0\Vert +
\Vert y_0\Vert )^m\big)\big\}\,,
\end{aligned}\\
F_1(t)= (2^m+1)A_\infty K_1 \Vert  B(t)\Vert  + 2^{m+1}A_\infty K_2 \Vert
R(t)\Vert  t^m\,.
\end{gather*}
Then any nonextendable to the right solution $y(t)$ of the
initial value problem \eqref{e:1}, \eqref{e:2} is global.
\end{theorem}

The above theorem  solves the problem in case $m\leq p$.

\begin{theorem}\label{t:4}
Let $p>0$, $0<m\leq p$,
$A(t)$, $B(t)$, $R(t)$ be continuous  matrix-valued functions
on $\mathbb R_+$, $A$ 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 $\varphi_0\in C^1$, $\varphi_1\in C$.
Let constants $K_1$, $K_2$, $K_3$, $K_4$, $K_5$ and $K_6$ exist
such that
$$
\Vert g(u,v)\Vert  \leq K_1 \big( \Vert u\Vert ^m + \Vert  v\Vert
_C^m\big)\,, \quad  \Vert f(u,v)\Vert  \leq K_2 \big(\Vert u\Vert
^m + \Vert v\Vert _C^m\big)
$$
for $\Vert u\Vert \geq 1$, $\Vert  v\Vert _C \geq 1$,
$$
\Vert  g(u,v)\Vert  \leq K_3 \Vert  u\Vert ^m\,, \quad  \Vert  f
(u,v)\Vert  \leq K_4 \Vert u\Vert ^m \quad  \mbox{for $\Vert
u\Vert \geq 1$, $0\leq \Vert v\Vert _C<1$}
$$
and
$$
\Vert g(u,v)\Vert  \leq K_5 \Vert v\Vert _C^m\,, \quad
\Vert f(u,v)\Vert  \leq K_6 \Vert  v\Vert _C^m \quad \mbox{for
$0\leq \Vert u\Vert <1$, $\Vert v\Vert _C \geq 1$}.
$$
Then any nonextendable to the right solution $y(t)$ of the
initial value problem \eqref{e:1}, \eqref{e:2} is global.
\end{theorem}

A special case of the equation \eqref{e:1} with $g, f$
independent of $y'_t$ and $y_t$, respectively, i.e.\ the equation
\begin{equation}\label{e:6}
A(t)\Phi_p(y')' + B(t)g(y') + R(t)f(y) = e(t), \quad t \geq 0,
\end{equation}
 and with the initial conditions
\begin{equation}\label{e:7}
y(0) = y_0,\quad y'(0) = y_1
\end{equation}
has been studied in the paper \cite{21}. A similar theorem to
Theorem \ref{t:1} on the existence of a global solution of the initial
value problem
\eqref{e:6}, \eqref{e:7} is proved there.
It is assumed there that there exist positive constants $K_1, K_2$
such that
\begin{equation}\label{e:8}
\|g(u)\| \leq K_1\|u\|^m, \quad  \|f(u)\| \leq K_2\|u\|^m,
u \in \mathbb{R}^n,
\end{equation}
where the constant $c$ and the function $F(t)$ are defined
in \cite[Theorem 1.1]{21} as follows:
\begin{gather}\label{e:9}
c := n^{\frac{p}{2}}A_\infty \big \{\|A(0)\Phi_p(y_1)\|
+ 2^{m-1}K_2\|y_0\|R_\infty + E_\infty \big \}, \\
\label{e:10}
F(t) := K_1\|B(t)\| + 2^{m-1}K_2\int_t^\infty\|R(s)\|s^{m-1}{\rm d} s,
\end{gather}
$\|w\|$ is the Euclidean norm of $w \in \mathbb{R}^n$.
If the condition \eqref{e:8} and one of the assumptions 1., 2.
of \cite[Theorem 1.1]{21}
(with $c, F$ defined by \eqref{e:9} and \eqref{e:10}) is satisfied,
then a solution of the initial value problem
\eqref{e:6}, \eqref{e:7}\quad is
global.

   We remark that in \cite[Theorem 1.1]{21} there is a misprint.
 There must be
$A_\infty = \sup_{0 \leq t < \infty}\|A(t)^{-1}\|$
instead of $A_\infty = \sup_{0 \leq t < \infty}\|A(t)^{-1}\|^{-1}$.

\begin{corollary}\label{coro1.5}
Consider the differential equation
\begin{equation}\label{e:c5}
y^{\prime\prime} =t^\alpha |y|^m \mathop{\rm sgn} y
\end{equation}
with $m>1$. Then  $\varepsilon >0$ exists such that  a solution of the problem
\eqref{e:c5}, $|y(0)|<\varepsilon$, $|y'(0)|<\varepsilon$
is  defined on ${\mathbb R}_+$ if and only if
\begin{equation}\label{e:c6}
\alpha < - m-1\,.
\end{equation}
\end{corollary}


 Corollary \ref{coro1.5} shows that condition \eqref{e:3} cannot be weaken,
the integral cannot be infinite.

\section{Proofs of the main results}\label{s:2}

\subsection*{Proof of Theorem \ref{t:1}}

Let $y:\langle - r, T)\to \mathbb{R}^n$ be a nonextendable to the
right solution of the initial value problem \eqref{e:1}, \eqref{e:2}
with $0 <T < \infty$. If we denote $u(t) = y'(t)$ for $t \geq 0$,
then $y(t) = y_0 + \int_0^tu(\tau)d\tau$ and we can write
 \eqref{e:1} as
\begin{align*}
\Phi_p(u(t)) &= A(t)^{-1}\big \{A(0)\varphi(y_1) -
 \int_0^tB(s)g(u(s), y'_s){\rm d}s \\
&\quad -  \int_0^tR(s) f(y_0 + \int_0^su(\tau){\rm d}
\tau, y_s){\rm d} s + \int_0^te(s){\rm d} s \big \}, t \geq 0\,.
\end{align*}
We need to estimate $\|y_s\|_C$ and $\|y'_s\|_C$.
From the definition of the shift operators we have
\[
\|y_s\|_C  = \max_{- r \leq \Theta \leq 0}\|y(s + \Theta)\| = \max \{\rho_1(s), \rho_2(s)\} \leq \rho_1(s)+ \rho_2(s)\,,
\]
where
\begin{gather*}
\rho_1(s)= \max_{- r \leq s + \Theta \leq 0} \|y(s + \Theta)\|
 \leq \|\phi_0\|_C,
\\
\begin{aligned}
\rho_2(s)&= \max_{ s + \Theta \geq 0} \|y(s + \Theta)\|
 \leq \max_{s + \Theta \geq 0}\{\|y_0\|
 + \int_0^{s + \Theta} \|u(\tau)\|d\tau\} \leq \|y_0\|\\
&\quad  + \int_0^s \|u(\tau)\|{\rm d} \tau
\end{aligned}
\end{gather*}
and this yields
\begin{equation}\label{e:x1}
\|y_s\|_C   \leq  \|\varphi_0\|_C + \|y_0\|
+ \int_0^s \|u(\tau)\|{\rm d} \tau \,.
\end{equation}
We can estimate analogously $\|y'_s\|$:
\begin{gather*}
\|y'_s\|_C  = \max_{- r \leq \Theta \leq 0}\|y'(s + \Theta)\| = \max \{\sigma_1(s), \sigma_2(s)\} \leq
 \sigma_1(s)+ \sigma_2(s)\,,
\end{gather*}
where
\begin{gather*}
\sigma_1(s)= \max_{- r \leq s + \Theta \leq 0}\|y'(s + \Theta)\|
= \max_{- r \leq s + \Theta \leq 0}\|\varphi_1(s + \Theta)\| \leq
\|\varphi_1\|_C,
\\
\sigma_2(s)= \max_{s + \Theta \geq 0}\|y'(s + \Theta)\|
= \max_{s + \Theta \geq 0} \|u(s + \Theta)\|
\leq \max_{0 \leq \tau \leq s}\|u(\tau )\|.
\end{gather*}
Thus we have
\begin{equation}\label{e:x2}
\|y'_s\|_C \leq \|\varphi_1\|_C + \max_{0 \leq \tau \leq s}\|u(\tau)\|\,.
\end{equation}
 From  \eqref{e:1}, the inequalites \eqref{e:x1}, \eqref{e:x2} and
the assumptions of the theorem we obtain
\begin{equation}
\begin{aligned}
\|\Phi_p(u(t)\|
&\leq \|A(t)^{-1}\|\big \{\|A(0)\Phi_p(y_1)\| +
K_1\int_0^t\|B(s)\|\, \|u(s)\|^m {\rm d} s  \\
&\quad +K_1\int_0^t\|B(s)\|\Big( (\|\varphi_1\|_C +
 \max_{0 \leq \tau \leq s}\|u(\tau)\|\big )^m{\rm d}s \\
&\quad + K_2\int_0^t\|R(s)\|\|y_0
 + \int_0^su(\tau){\rm d}\tau\|^m{\rm d}s
+K_2\int_0^t\|R(s)\|\big[\|\varphi_0\|_C \\
&\quad + \|y_0\| + \int_0^s\|u(\tau)\|{\rm d}\tau \big]^m{\rm d}s\Big)\}\,.
\end{aligned}\label{e:x3}
\end{equation}


   Now applying the continuous and discrete versions of the Jensen's
inequality (see \cite[Theorem 2, Chapter VIII]{17} and the Fubini
theorem in a similar way as in the proof of \cite[Theorem 1.2]{21}
we obtain the inequality
\[
v(t)^p \leq c + \int_0^tF_1(\tau)v(\tau)^m{\rm d} \tau
+ \int_0^tF_2(\tau)[\sup_{0 \leq s \leq \tau }v(\tau)]^m{\rm d} \tau,
\quad 0 \leq t < T,
\]
where $c$ is given  in the theorem and $v(t) = \|u(t)\|$.
If we denote by $G(t)$ the right-hand side of this inequality
then $v^p(s) \leq G(t)$
for $s \leq t$ and therefore we obtain the following inequality
for $w(t) := \sup_{0 \leq \sigma \leq t} v(\sigma)$:
\begin{gather*}
w(t)^p \leq c + \int_0^tF(\tau)w(\tau)^m{\rm d} \tau,\quad  0 \leq t < T,
\end{gather*}
where $F = F_1 + F_2$ is the function from the theorem.
 From \cite[Lemma]{21} it follows that
$M = \sup_{0 \leq t < T}\|u(t)\| < \infty$ and since
$w(t) : =  \sup_{0 \leq \sigma \leq t} \|u(\sigma)\|$
we obtain that for the solution $y(t) = y_0 + \int_0^tu(s)ds$ of the
initial value problem (1), (2) we have
$\lim_{t \to T^-}\|y(t)\| \leq \lim_{t \to T^-}\big (\|y_0\| + t\sup_{0 \leq s < T}\|u(s)\|\big ) <
\infty$. Thus we have proved that
$\lim_{t \to T^-}[\|y(t)\| + \|y'(t)\|] < \infty$, i. e. the solution $y(t)$
is global.


\subsection*{Proof of Theorem \ref{t:3}}

Let $y:\langle -r, T)\to {\mathbb R^n}$ be a nonextendable to the
right solution of the initial value problem \eqref{e:1},
\eqref{e:2} with $0<T<\infty $ and  $u(t)=y'(t)$ for $t\geq 0$.
Then \eqref{e:x3} holds. Denote $w(t)=\max_{0\leq s\leq t} \Vert
u(s)\Vert$ $0\leq t<T$. Then \eqref{e:x3} and the
inequality
\begin{equation}\label{e:x4}
(A_1+\dots + A_l)^k \leq l^k (A_1^k +\dots A_l^k)
\end{equation}
for $A_1,A_1,\dots , A_l\geq 0$, $k>0$ yield
\begin{align*}
\Vert \Phi_p(u(t))\Vert
& \leq\Vert  A(t)^{-1}\Vert  \big\{
\Vert A(0)\Phi_p(y_1)\Vert  + K_1 \int_0^t \Vert  B(s)\Vert
w^m(s)\,{\rm d} s\\
&\quad + 2^m K_1 \Vert  \varphi_1\Vert _C^m B_\infty  + 2^m K_1
\int_0^t \Vert  B(s)\Vert  w^m (s)\,{\rm de} s\\
&\quad + 2^m K_2 \Vert  y_0\Vert ^m R_\infty  + 2^m K_2 \int_0^t \Vert
R(s)\Vert  s^m w^m(s)\,{\rm d} s\\
&\quad + 2^m K_2 (\Vert \varphi_0\Vert_C
 + \Vert y_0\Vert )^m R_\infty + 2^m K_2
 \int_0^t \Vert  R(s)\Vert  s^m w^m (s)\,{\rm d} s\big\}\,.
\end{align*}
Hence,
$$
w^p(t)\leq C_1 +\int_0^t F_1 (\tau) w^m (\tau)\,{\rm d} \tau\,,
$$
where $C_1$ and $F_1$ are given in Theorem \ref{t:3}, and the
rest of the proof is the same as in  the end of the proof of Theorem
\ref{t:1}.



\subsection*{Proof of Theorem \ref{t:4}}

Let $y:\langle -r, T)\to {\mathbb R^n}$ be a nonextendable solution
of the initial value problem \eqref{e:1}, \eqref{e:2} with
$0<T<\infty $.
If we denote $u(t)=y'(t)$ for $t\geq 0$ and $\varphi_0(0)=y_0$,
$\varphi_1(0)=y_1$, then the estimations \eqref{e:x1} and
\eqref{e:x2}  are valid.
Let $w(t)=\max\big(1, \max\limits_{0\leq s\leq t} \Vert  u(s)\Vert
\big)$. Furthermore,
\begin{equation}
\begin{aligned}
\Vert  g(u,v)\Vert  &\leq K_1 \Vert  u \Vert ^m + K_1 \Vert
v\Vert _C^m + K_3 \Vert u\Vert ^m + K_5\Vert v\Vert _C^m\\
&\quad + \max_{\Vert u\Vert \leq 1, \Vert v\Vert _C\leq 1} \Vert
g(u,v)\Vert  = K_7 (\Vert u\Vert ^m + \Vert v\Vert _C^m +1)
\end{aligned}\label{e:x5}
\end{equation}
 on  $u,v\in  {\mathbb R}^n\times C$ with
 $$
 K_7 =\max\big\{ K_1+K_3, K_1+K_5 , \max\limits_{\Vert u\Vert
 \leq 1, \Vert v\Vert _C \leq 1} \Vert g(u,v)\Vert \big\}\,.
 $$
 Similarly,
 \begin{equation}\label{e:x6}
\Vert f(u,v)\Vert \leq K_8 (\Vert u\Vert ^m + \Vert v\Vert _C^m
+1 )
\end{equation}
 on  $u,v\in  {\mathbb R}^n\times C$ with
 $$
 K_8 =\max\big\{ K_2+K_4, K_2+K_6, \max\limits_{\Vert u\Vert \leq
 1, \Vert v\Vert _C\leq 1} \Vert f(u,v)\Vert \big\}\,.
 $$
 Then \eqref{e:x1}, \eqref{e:x2}, \eqref{e:x5}, \eqref{e:x6},
the equation \eqref{e:1} and  the assumptions  of the theorem yield
\begin{equation}
 \begin{aligned}
 \Vert \Phi_p(u(t))\Vert
&\leq \Vert  A(t)^{-1}\Vert \Big\{
 \Vert  A(0) \Phi_p (y_1)\Vert  + K_7 \int_0^t \Vert  B(s)\Vert
 w^m(s)\,{\rm d} s\\
 &\quad + K_7 \int_0^t \Vert B(s)\Vert  \big(\Vert \varphi_1\Vert _C +
 w(s)\big)^m + K_7 \int_0^t \Vert  B(s)\Vert \,{\rm d} s\\
 &\quad + K_8 \int_0^t \Vert R(s)\Vert  \big( \Vert  y_0\Vert  +
 sw(s)\big)^m {\rm d} s + K_8 \int_0^t \Vert  R(s)\Vert  \big[\Vert
 \varphi_0 \Vert_C  + \Vert y_0\Vert \\
&\quad + sw(s)\big]^m {\rm d} s + K_8 \int_0^t \Vert R(s)\Vert  \,{\rm d} s\Big\}\,. \label{e:x7}
\end{aligned}
\end{equation}
 From this, the inequalities \eqref{e:x4} and $w(t)\geq 1$, we have
\begin{equation}\label{e:x8}
w^p(t)\leq 1+H+\int_0^t F_2 (s)w^m (s)\,{\rm d} s \leq H+1 +\int_0^t F_2 (s) w^p (s)\,{\rm d} s
\end{equation}
for $t\in [0, T)$, where
\begin{gather*}
\begin{aligned}
H&=\ \max_{0\leq t\leq T} \Vert  A(t)^{-1}\Vert
 \big\{ \Vert A(0)\Phi_p(y_1)\Vert  + (2^m \Vert \varphi_1\Vert _C^m +1 )
K_7 \int_0^T \Vert  B(s)\Vert \,{\rm d} s\\
&\quad +   K_8 \big(2^m \Vert y_0\Vert^m  +2^m (\Vert \varphi_0\Vert_C
 + \Vert y_0\Vert )^m +1\big) \int_0^T \Vert  R(s)\Vert
\,{\rm d} s\big\}\,,
\end{aligned} \\
F_2(t)= \max_{0\leq s\leq T} \Vert A(t)^{-1}\Vert \big \{(2^m +1) K_7
\Vert  B(t)\Vert  + 2^{m +1}K_8 t^m \Vert  R(t)\Vert \big\}\,.
\end{gather*}
Hence, \eqref{e:x8} and Gronwall's inequality yield $w(t)$ and
$y'(t)$ are bounded on $\langle 0,T)$. As according to
$y(t)=y_0+\int_0^t u(\tau)\, {\rm d}\tau$, $y$ is bounded on
$\langle 0, T)$, too, $y$ cannot be nonextendable.
The contradiction proves the statement.

\subsection*{Proof of Corollary \ref{coro1.5}}
The sufficiency of \eqref{e:c6} follows from Theorem \ref{t:1}
and the necessity  of \eqref{e:c6} follows from
\cite[Theorem 17.3]{22}.

\subsection*{Acknowledgements}
The work of the first author was supported by the Ministry of
Education of the Czech Republic  under the project MSM
0021622409 and by the Grant No. 201/08/0469 of the Grant
Agency of the Czech Republic.
 The work  of the second author was supported by the
Grant No. 1/0098/08 of the Slovak Grant Agency VEGA-SAV-M\v S.

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\end{document}