\documentclass[twoside]{article}
\usepackage{amssymb}
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\markboth{\hfil Uniqueness theorem for $p$-biharmonic equations
\hfil EJDE--2002/53}
{EJDE--2002/53\hfil Ji\v r\'\i\ Benedikt \hfil}
\begin{document}

\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2002}(2002), No.~53, pp.~1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Uniqueness theorem for $p$-biharmonic equations
 %
\thanks{ {\em Mathematics Subject Classifications:} 34A12, 34C11, 34L30.
\hfil\break\indent
{\em Key words:} $p$-biharmonic operator, existence and uniqueness of solution,
\hfil\break\indent
continuous dependence on initial conditions, jumping nonlinearity.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted April 15, 2002. Published June 10, 2002.} }
\date{}
%
\author{Ji\v r\'\i\ Benedikt}
\maketitle

\begin{abstract}
  The goal of this paper is to prove existence and uniqueness
  of a solution of the initial value problem for the equation
  $$
  (|u''|^{p-2}u'')''=\lambda |u|^{q-2}u
  $$
  where $\lambda\in{\mathbb{R}}$ and $p,q>1$.
  We prove the existence for $p\geq q$ only, and give a
  counterexample which shows that for $p<q$ there need not exist
  a global solution (blow-up of the solution can occur).
  On the other hand, we prove the uniqueness for $p\leq q$,
  and show that for $p>q$ the uniqueness does not hold true
  (we give a corresponding counterexample again).
  Moreover, we deal with continuous dependence of the solution on
  the initial conditions and parameters.
\end{abstract}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definiton}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11
\@addtoreset{equation}{section}
\catcode`@=12

\section{Introduction}\label{sec_i}
In 2000, Dr\'abek and \^Otani proved \cite{DO} that the initial value
problem
\begin{equation}\label{poc_ul_DO}
\begin{array}{c}
\bigl(|u''(t)|^{p-2}u''(t)\bigr)''=\lambda|u(t)|^{p-2}u(t),\quad
t\in[t_0,t_0+\varepsilon], \\[1mm]
u(t_0)=\alpha,\quad u'(t_0)=\beta,\\
|u''(t_0)|^{p-2}u''(t_0)=\gamma,\quad
\left.\bigl(|u''(t)|^{p-2}u''(t)\bigr)'\right|_{t=t_0}=\delta
\end{array}
\end{equation}
where $\lambda>0$ and $p>1$, has a unique locally defined solution (for some
$\varepsilon>0$). The equation in (\ref{poc_ul_DO}) is a generalization of
the one-dimensional version of the well-known linear clamped plate equation,
which we obtain choosing $p=2$ in (\ref{poc_ul_DO}).

It should be mentioned that the existence and uniqueness problem for
(\ref{poc_ul_DO}) cannot be inferred from the classical theory. Indeed, let
us denote $v:=|u''|^{p-2}u''$ and rewrite (\ref{poc_ul_DO}) as the
equivalent problem
\begin{equation}\label{poc_ul_DO2}
\begin{array}{lll}
u''(t)=|v(t)|^\frac{2-p}{p-1}v(t), & u(t_0)=\alpha, & u'(t_0)=\beta, \\[1mm]
v''(t)=\lambda|u(t)|^{p-2}u(t),\quad & v(t_0)=\gamma, & v'(t_0)=\delta,
\end{array}
\quad t\in[t_0,t_0+\varepsilon].
\end{equation}
Whenever $p\ne2$, at least one of the right-hand sides in
(\ref{poc_ul_DO2}) satisfies neither Lipschitz (see, e.g., \cite{CL}) nor any
other general condition that guarantees existence or uniqueness of a
solution. For example, the very general Kamke's Theorem (or its
corollaries---Nagumo's (Rosenblatt's), Osgood's, Tonelli's Criterion, see,
e.g., \cite[pp.~31--35]{H}) cannot be used to prove the uniqueness here.

In what follows, we show how the situation gets more complicated as we carry
forward to more general problems than (\ref{poc_ul_DO}), namely to problems
with
\begin{itemize}
\item different growth of the nonlinearity depending on $u''$ and on $u$
(non-ho\-mo\-ge\-neous equation),
\item jumping nonlinearity,
\item non-constant coefficients.
\end{itemize}
%
Let us consider the problem with a non-homogeneous equation
\begin{equation}\label{poc_ul_neh}
\begin{array}{c}
\bigl(|u''(t)|^{p-2}u''(t)\bigr)''=\lambda|u(t)|^{q-2}u(t),\quad
t\in{\cal I}, \\[1mm]
u(t_0)=\alpha,\quad u'(t_0)=\beta,\\
|u''(t_0)|^{p-2}u''(t_0)=\gamma,\quad
\bigl(|u''(t)|^{p-2}u''(t)\bigr)'\Big|_{t=t_0}=\delta
\end{array}
\end{equation}
where $\lambda\in{\mathbb{R}}$, $p,q>1$ and ${\cal I}=[t_0,t_1]$, $t_0<t_1$, or
${\cal I}=[t_0,\infty)$. Taking $p=q$ in (\ref{poc_ul_neh}) we obtain
(\ref{poc_ul_DO}), but for $p\ne q$ the situation is more complex: for $p<q$
we lose the existence of a globally defined solution (we call this a
``global existence''), and for $p>q$ we lose the uniqueness of a locally
defined solution (we call this a ``local uniqueness''). In Sections
\ref{sec_e} and \ref{sec_u} we introduce the corresponding counterexamples.

We can further generalize (\ref{poc_ul_neh}) adding the jumping nonlinearity to
the right-hand side:
\begin{equation}\label{poc_ul_sk}
\begin{array}{c}
\bigl(|u''(t)|^{p-2}u''(t)\bigr)''=\mu|u(t)|^{q_1-2}u^+(t)-
\nu|u(t)|^{q_2-2}u^-(t),\quad t\in{\cal I}, \\[1mm]
u(t_0)=\alpha,\quad u'(t_0)=\beta,\\
|u''(t_0)|^{p-2}u''(t_0)=\gamma,\quad
\bigl(|u''(t)|^{p-2}u''(t)\bigr)'\Big|_{t=t_0}=\delta
\end{array}
\end{equation}
where $p,q_1,q_2>1$, $\mu,\nu\in{\mathbb{R}}$, $u^+=\max\{u,0\}$ (positive
part of $u$) and $u^-=\max\{-u,0\}$ (negative part of $u$). Putting
$q:=q_1=q_2$ and $\lambda:=\mu=\nu$ into (\ref{poc_ul_sk}) we arrive at
(\ref{poc_ul_neh}). Now the situation is analogous to the previous case
(\ref{poc_ul_neh}): to prove the global existence we have to assume
$p\geq\max\{q_1,q_2\}$, and to prove the local uniqueness we must have
$p\leq\min\{q_1,q_2\}$.

Taking into account non-constant coefficients in (\ref{poc_ul_neh}) we
obtain:
\begin{equation}\label{poc_ul_nek}
\begin{array}{c}
\bigl(|a(t)u''(t)|^{p-2}u''(t)\bigr)''=b(t)|u(t)|^{q-2}u(t),\quad
t\in{\cal I}, \\[1mm]
u(t_0)=\alpha,\quad u'(t_0)=\beta,\\
|a(t_0)u''(t_0)|^{p-2}u''(t_0)=\gamma,\quad
\bigl(a(t)|u''(t)|^{p-2}u''(t)\bigr)'\Big|_{t=t_0}=\delta
\end{array}
\end{equation}
where $a,b\in C({\cal I})$ and $a>0$. When $p>2$, it is not enough to assume
$p\leq q$ for proving the local uniqueness. We have to add a condition on
$b$. It suffices to assume $b\geq0$ or $b\leq0$ on the whole interval $\cal I$,
i.e., that $b$ does not change its sign on ${\cal I}$. Less restrictive is to
assume that $b$ have the property ${\cal P}$ (stated below) on $\cal I$.

\begin{definition}\label{def_vl_p} \rm
We say that a function $f$ has a property ${\cal P}$ on the interval
${\cal I}=[t_0,t_1]$, or ${\cal I}=[t_0,\infty]$, if
\[
\forall\tilde t\in{\cal I}^*\ \exists\xi>0,\quad
f(t)\geq 0\ \forall t\in[\tilde t,\tilde t+\xi]\quad\mbox{or}\quad
f(t)\leq 0\ \forall t\in[\tilde t,\tilde t+\xi]
\]
where ${\cal I}^*=[t_0,t_1)$, or ${\cal I}^*=[t_0,\infty)$,
respectively. In other words, for every point $\tilde t$ of $\cal I$
(except a contingent right boundary point) there exists some right closed
neighborhood of $\tilde t$ in which $f$ does not change its sign.
\end{definition}

Note that a continuous function that does not have the property ${\cal P}$
is, e.g.,
\[
f(t)=(t-t_0)\sin\frac1{t-t_0}\quad\mbox{for }t>t_0,\quad f(t_0)=0.
\]
It is clear that any constant function has the property ${\cal P}$.

We prove the (both local and global) existence and uniqueness for the most
general non-homoge\-neous problem including the jumping nonlinearity and
non-constant coefficients as well:
\begin{equation}\label{poc_ul_ob}
\begin{array}{c}
\setlength{\arraycolsep}{0pt}
\begin{array}{rl}
\bigl(a(t)|u''(t)|^{p-2}u''(t)\bigr)''= & \hspace{1mm}
b_1(t)|u(t)|^{q_1-2}u^+(t)-b_2(t)|u(t)|^{q_2-2}u^-(t),\quad t\in{\cal I},
\end{array} \\[1mm]
u(t_0)=\alpha,\quad u'(t_0)=\beta, \\
a(t_0)|u''(t_0)|^{p-2}u''(t_0)=\gamma,\quad
\bigl(a(t)|u''(t)|^{p-2}u''(t)\bigr)'\Big|_{t=t_0}=\delta
\end{array}
\end{equation}
where $b_1,b_2\in C({\cal I})$ (the other parameters are as above).

Denoting $u_1:=u$ and $u_3:=a|u''|^{p-2}u''$ we can rewrite
(\ref{poc_ul_ob}) as the equivalent initial value problem for a system of
four equations of the first order
\begin{equation}\label{poc_ul_s}
\begin{array}{rll}
u_1'(t)= & \hspace{-2.5mm} u_2(t), & u_1(t_0)=\alpha, \\[1mm]
u_2'(t)= & \hspace{-2.5mm} a^{-\frac1{p-1}}(t)|u_3(t)|^\frac{2-p}{p-1}u_3(t), &
u_2(t_0)=\beta, \\[1mm]
u_3'(t)= & \hspace{-2.5mm} u_4(t), & u_3(t_0)=\gamma, \\[1mm]
u_4'(t)= & \hspace{-2.5mm} b_1(t)|u_1(t)|^{q_1-2}u_1^+(t)-
b_2(t)|u_1(t)|^{q_2-2}u_1^-(t),\ & u_4(t_0)=\delta,
\end{array}
\quad t\in{\cal I}.
\end{equation}

The main results of this paper are the following.

\begin{proposition}[local existence] \label{prop_le}
There exists\/ $\varepsilon>0$ such that\/ {\rm(\ref{poc_ul_ob})} has a
solution on\/ ${\cal I}=[t_0,t_0+\varepsilon]$.
\end{proposition}

\begin{theorem}[global existence]\label{th_ge}
Let\/ $p\geq\max\{q_1,q_2\}$. Then\/ {\rm(\ref{poc_ul_ob})} has a solution
on\/ ${\cal I}=[t_0,\infty)$.
\end{theorem}

\begin{corollary}\label{cor_ge}
If\/ $p\geq\max\{q_1,q_2\}$, then\/ {\rm(\ref{poc_ul_sk})} has a solution
on\/ ${\cal I}=[t_0,\infty)$.

If\/ $p\geq q$, then\/ {\rm(\ref{poc_ul_nek})} and\/ {\rm(\ref{poc_ul_neh})}
have a solution on\/ ${\cal I}=[t_0,\infty)$.
\end{corollary}

\begin{proposition}[local uniqueness]\label{prop_lu}
Let one of these conditions hold true:

\begin{itemize}
\item $|\alpha|+|\beta|+|\gamma|+|\delta|>0$ or
\item $p\leq\min\{q_1,q_2\}$.
\end{itemize}
Moreover, let at least one of the following conditions hold true:
\begin{itemize}
\item $p\leq2$ or
\item $\alpha=\beta=0$ or
\item $|\gamma|+|\delta|>0$ or
\item there exists some right closed neighborhood of\/ $t_0$ in which
neither\/ $b_1$ nor\/ $b_2$ changes its sign.
\end{itemize}
Then there exists\/ $\varepsilon>0$ such that\/ {\rm(\ref{poc_ul_ob})} has
at most one solution on\/ ${\cal I}=[t_0,t_0+\varepsilon]$.
\end{proposition}

\begin{remark}\label{rem_prop_lu} \rm
For the special cases (\ref{poc_ul_neh}) and (\ref{poc_ul_sk}) of
(\ref{poc_ul_ob}) the last condition of the latter four is trivially
satisfied, and so it remains to satisfy only one of the former two
conditions.
\end{remark}

\begin{theorem}[global uniqueness] \label{th_gu}
Let\/ $p\leq\min\{q_1,q_2\}$. Further, let\/ $p\leq2$ or functions\/ $b_1$,
$b_2$ have the property\/ ${\cal P}$ on\/ ${\cal I}$ (see Definition\/
{\rm\ref{def_vl_p}}). Then\/ {\rm(\ref{poc_ul_ob})} has at most one
solution.
\end{theorem}

\begin{corollary}
If\/ $p\leq\min\{q_1,q_2\}$ and, furthermore, $p\leq2$ or neither\/ $b_1$
nor\/ $b_2$ changes its sign on\/ ${\cal I}$, then\/ {\rm(\ref{poc_ul_ob})}
has at most one solution.

If\/ $p\leq q$ and, furthermore, $p\leq2$ or\/ $b$ has the property\/
${\cal P}$ on\/ ${\cal I}$, then\/ {\rm(\ref{poc_ul_nek})} has at most one
solution.

If\/ $p\leq q$ and, furthermore, $p\leq2$ or\/ $b$ does not change its
sign on\/ ${\cal I}$, then\/ {\rm(\ref{poc_ul_nek})} has at most one
solution.

If\/ $p\leq\min\{q_1,q_2\}$, then\/ {\rm(\ref{poc_ul_sk})} has at most one
solution.

If\/ $p\leq q$, then\/ {\rm(\ref{poc_ul_neh})} has at most one solution.
\end{corollary}

The paper is organized as follows. In Section \ref{sec_p} we define the
solution of (\ref{poc_ul_ob}). In Section \ref{sec_e} we prove Proposition
\ref{prop_le} and Theorem \ref{th_ge}. Section \ref{sec_u} contains a proof
of Proposition \ref{prop_lu} and Theorem \ref{th_gu}. In Section
\ref{sec_op} we introduce some open problems related to this paper.

Tables \ref{tab_ex} and \ref{tab_uni} summarize the cases when the global 
existence, and the local uniqueness, respectively, of a solution of
(\ref{poc_ul_neh}) is guaranteed or foreclosed (there exists a
counterexample).

%%%%%%%%%%%%%%%% Tab. 1 %%%%%%%%%%%%%%%%%%%%%%

\begin{table}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\renewcommand{\arraystretch}{1.4}
\setlength{\tabcolsep}{6pt}
\begin{tabular}{|c|l|l|c|}
\hline
$p\geq q$ & \multicolumn{3}{|c|}{{\bf YES} (Corollary \ref{cor_ge})} \\
\hline
& & $\alpha,\beta,\gamma,\delta\geq0$, $\alpha+\beta+\gamma+\delta>0$ &
{\bf NO} (Example \ref{ex_nonex},  \\[-1.5mm]
& & or & Remark \ref{rem_ex_nonex})---blow-up \\[-1.5mm]
& $\lambda>0$ & $\alpha,\beta,\gamma,\delta\leq0$, $\alpha+\beta+\gamma+\delta<0$ &
to $\infty$ or $-\infty$ \\
\cline{3-4}
& & $\alpha=\beta=\gamma=\delta=0$ & {\bf YES} (trivial) \\
\cline{3-4}
$p<q$ & & $\exists\,\kappa_1,\kappa_2\in\{\alpha,\beta,\gamma,\delta\}:$
$\kappa_1\kappa_2<0$ & {\bf ?} \\
\cline{2-4}
& $\lambda=0$ & \multicolumn{2}{|c|}{{\bf YES} (trivial)} \\
\cline{2-4}
& $\lambda<0$ & \multicolumn{2}{|c|}{\bf ?} \\
\hline
\end{tabular}
\caption{\label{tab_ex} Existence of a solution of\/ {\rm (\ref{poc_ul_neh})} on\/
${\cal I}=[t_0,\infty)$.}
\end{center}
\end{table}

%%%%%%%%%%%%%% Tab. 2 %%%%%%%%%%%%%%%%%%%%%%

%\refstepcounter{countermytable}
\begin{table}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\renewcommand{\arraystretch}{1.4}
\setlength{\tabcolsep}{6pt}
\begin{tabular}{|c|l|l|c|}
\hline
$|\alpha|+|\beta|+|\gamma|+|\delta|>0$ & \multicolumn{3}{|c|}{{\bf YES}
(Proposition \ref{prop_lu}, Remark \ref{rem_prop_lu})} \\
\hline
& $p\leq q$ & \multicolumn{2}{|c|}{{\bf YES} (Proposition \ref{prop_lu},
Remark \ref{rem_prop_lu})} \\
\cline{2-4}
$ \alpha=\beta=\gamma=\delta=0$ & & $\lambda>0$ & {\bf NO} (Example
\ref{pr_nejed}) \\
\cline{3-4}
& $p>q$ & $\lambda=0$ & {\bf YES} (trivial) \\
\cline{3-4}
& & $\lambda<0$ & {\bf ?} \\
\hline
\end{tabular}
\caption{\label{tab_uni} Uniqueness of a solution of {\rm (\ref{poc_ul_neh})} on
${\cal I}=[t_0,t_0+\varepsilon]$ for some $\varepsilon>0$.}
\end{center}
\end{table}

The following two corollaries are consequences of the global existence
guaranteed by Theorem \ref{th_ge} and the global uniqueness guaranteed by
Theorem \ref{th_gu}. The reader is invited to accomplish their proofs
following, e.g., that of \cite[Th.~4.1, p.~59]{CL}.

\begin{corollary}\label{cor_sp_zav}
Let\/ $\tilde p\leq\min\{\tilde q_1,\tilde q_2\}$. Further, let\/
$\tilde p\leq2$ or\/ $\tilde b_1$, $\tilde b_2$ have the property\/
${\cal P}$ on\/ $[a,b]$. Let\/ $\tilde{\mbox{\boldmath{$u$}}}$ be a solution of\/
{\rm(\ref{poc_ul_s})} with\/ $p=\tilde p$, $q_1=\tilde q_1$,
$q_2=\tilde q_2$, $a=\tilde a>0$, $b_1=\tilde b_1$, $b_2=\tilde b_2$,
$\alpha=\tilde\alpha$, $\beta=\tilde\beta$, $\gamma=\tilde\gamma$,
$\delta=\tilde\delta$, $t_0=\tilde\tau$ and\/ ${\cal I}=[a,b]$,
$a<\tilde\tau<b$.

Then there exists\/ $\varepsilon>0$ such that for any\/
$p,q_1,q_2,\alpha,\beta,\gamma,\delta,\tau\in{\mathbb{R}}$ and
$a,b_1,b_2\in C({\cal I})$ satisfying
\[
\begin{array}{c}
|p-\tilde p|+|q_1-\tilde q_1|+|q_2-\tilde q_2|+\|a-\tilde a\|_{C({\cal I})}+
\|b_1-\tilde b_1\|_{C({\cal I})}+\|b_2-\tilde b_2\|_{C({\cal I})}+\\[2mm]
+|\alpha-\tilde\alpha|+|\beta-\tilde\beta|+|\gamma-\tilde\gamma|+
|\delta-\tilde\delta|+|\tau-\tilde\tau|<\varepsilon
\end{array}
\]
all solutions\/ $\mbox{\boldmath{$u$}}=\mbox{\boldmath{$u$}}(t,p,q_1,q_2,a,b_1,
b_2,\alpha,\beta,\gamma,\delta,\tau)$ of\/ {\rm(\ref{poc_ul_s})} with\/
$t_0=\tau$ exist over ${\cal I}$, and, as\/ $(p,q_1,q_2,a,b_1,b_2,\alpha,
\beta,\gamma,\delta,\tau)\to(\tilde p,\tilde q_1,\tilde q_2,\tilde a,\tilde b_1,
\tilde b_2,\tilde\alpha,\tilde\beta,\tilde\gamma,\tilde\delta,\tilde\tau)$,
\[
\mbox{\boldmath{$u$}}(t,p,q_1,q_2,a,b_1,b_2,\alpha,\beta,\gamma,\delta,\tau)\to\tilde{\mbox{\boldmath{$u$}}}(t)=
\mbox{\boldmath{$u$}}(t,\tilde p,\tilde q_1,\tilde q_2,\tilde a,\tilde b_1,\tilde b_2,
\tilde\alpha,\tilde\beta,\tilde\gamma,\tilde\delta,\tilde\tau)
\]
uniformly over\/ $[a,b]$.
\end{corollary}

\begin{corollary}
Let\/ $\tilde p=\tilde q_1=\tilde q_2$. Further, let\/ $\tilde p\leq2$ or\/
$\tilde b_1$, $\tilde b_2$ have the property ${\cal P}$ on\/ $[a,b]$. Let\/
$\tilde p,\tilde q_1,\tilde q_2,\tilde\alpha,\tilde\beta,\tilde\gamma,
\tilde\delta,\tilde\tau\in{\mathbb{R}}$, $a<\tilde\tau<b$, and
$\tilde a,\tilde b_1,\tilde b_2\in C({\cal I})$, $a>0$, be fixed. Then there
exists a solution\/ $\tilde{\mbox{\boldmath{$u$}}}$ of\/ {\rm(\ref{poc_ul_s})} with\/
$p=\tilde p$, $q_1=\tilde q_1$, $q_2=\tilde q_2$, $a=\tilde a$,
$b_1=\tilde b_1$, $b_2=\tilde b_2$, $\alpha=\tilde\alpha$,
$\beta=\tilde\beta$, $\gamma=\tilde\gamma$, $\delta=\tilde\delta$,
$t_0=\tilde\tau$ and\/ ${\cal I}=[a,b]$, and the conclusion of Corollary\/
{\rm\ref{cor_sp_zav}} holds true.
\end{corollary}

This paper is a brief version of the first chapter of the author's diploma
thesis \cite{B} which is available in Czech only.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{sec_p}

Let us define a function $\psi_p\colon{\mathbb{R}}\to{\mathbb{R}}$, $p>1$, by
$\psi_p(s)=|s|^{p-2}s$ for $s\ne0$, and $\psi_p(0)=0$. Now we can rewrite
(\ref{poc_ul_ob}) as
\begin{equation}\label{poc_ul_ob2}
\begin{array}{c}
\bigl(a(t)\psi_p(u''(t))\bigr)''=b_1(t)\psi_{q_1}(u^+(t))-
b_2(t)\psi_{q_2}(u^-(t)),\quad t\in{\cal I}, \\[1mm]
u(t_0)=\alpha,\quad u'(t_0)=\beta, \\
a(t_0)\psi_p(u''(t_0))=\gamma,\quad
\left.\bigl(a(t)\psi_p(u''(t))\bigr)'\right|_{t=t_0}=\delta.
\end{array}
\end{equation}
We denote $p'=\frac{p}{p-1}$. One can simply show that $\psi_p$ and
$\psi_{p'}$ are inverse functions. The problem (\ref{poc_ul_s}) then takes
the form
\begin{equation}\label{poc_ul_s2}
\begin{array}{ll}
u_1'(t)=u_2(t), & u_1(t_0)=\alpha, \\[1mm]
u_2'(t)=c(t)\psi_{p'}(u_3(t)), & u_2(t_0)=\beta, \\[1mm]
u_3'(t)=u_4(t), & u_3(t_0)=\gamma, \\[1mm]
u_4'(t)=b_1(t)\psi_{q_1}(u_1^+(t))-b_2(t)\psi_{q_2}(u_1^-(t)),\quad
& u_4(t_0)=\delta,
\end{array}
\quad t\in{\cal I}
\end{equation}
where $c(t)=\psi_{p'}\bigl(\frac1{a(t)}\bigr)$ ($c\in C({\cal I})$, $c>0$).

\begin{definition}\label{def_sol} \rm
By a solution of {\rm(\ref{poc_ul_s2})} we understand a vector function
$\mbox{\boldmath{$u$}}=(u_1,u_2,u_3,u_4)$ of the class\/ $(C^1({\cal I}))^4$
which satisfy the equations in {\rm(\ref{poc_ul_s2})} at every point of
$\cal I$, and fulfill the initial conditions in {\rm(\ref{poc_ul_s2})}.

By a solution of the problem {\rm(\ref{poc_ul_ob2})} we understand a
function $u$ of the class $C^2({\cal I})$, such that\/
$\bigl(u,u',a\psi_p(u''),(a\psi_p(u''))'\bigr)$ is a solution of the
corresponding problem {\rm(\ref{poc_ul_s2})}.
\end{definition}

\begin{remark}\label{rem_prevod} \rm
We transferred the problem of existence and uniqueness of a solution of
(\ref{poc_ul_ob2}) (i.e.~(\ref{poc_ul_ob})) to the equivalent problem for
(\ref{poc_ul_s2}).
\end{remark}

\section{Existence}\label{sec_e}

\paragraph{Proof of Proposition \ref{prop_le}}
By integration of the equations in (\ref{poc_ul_s2}) we obtain that $\mbox{\boldmath{$u$}}$
is a solution of (\ref{poc_ul_s2}) if and only if $(u_1,u_3)$ is a fixed
point of the operator
$T\colon C({\cal I})\times C({\cal I})\to C({\cal I})\times C({\cal I})$
defined by
\begin{eqnarray*}
T(u,v) & = & \Big(\alpha+\beta t+\int_0^t{(t-\tau)c(\tau)\psi_{p'}(v(\tau))
\,{\rm d}\tau},\\
& & \gamma+\delta t+\int_0^t{(t-\tau)\Bigl(b_1(\tau)\psi_{q_1}
(u^+(\tau))-b_2(\tau)\psi_{q_2}(u^-(\tau))\Bigr)\,{\rm d}\tau}\Big).
\end{eqnarray*}
The reader is invited to prove that there exists $\varepsilon>0$ such that
the Schauder Fixed Point Theorem guarantees the existence of at least one
fixed point of $T$. It completes the proof of Proposition \ref{prop_le} (see
Remark \ref{rem_prevod}).
\hfill$\Box$\smallskip

Now we want to prove that the local solution can be extended to $\infty$,
i.e., that there exists a solution of (\ref{poc_ul_s2}) on
${\cal I}=[t_0,\infty)$. We find that it is not always possible, and we must
add some conditions on the parameters in (\ref{poc_ul_s2}). We begin with
the example which shows that it is necessary.

\begin{example}\label{ex_nonex} \rm
Let in (\ref{poc_ul_neh}) $p<q$ and $\lambda>0$. Let $H>t_0$ be arbitrary
(fixed). Then one can compute that the function $u(t)=K(H-t)^r$ where
\[
r=\frac{2p}{p-q}\quad\mbox{and}\quad
K=\left(\frac{2^p(p-1)q\bigl(p(p+q)\bigr)^{p-1}(2pq-p-q)}
{\lambda(q-p)^{2p}}\right)^{1/(q-p)}
\]
is a solution of (\ref{poc_ul_neh}) with
\[
\begin{array}{c}
\alpha=K(H-t_0)^r,\quad\beta=-Kr(H-t_0)^{r-1}, \\[.5mm]
\gamma=\bigl(Kr(r-1)\bigr)^{p-1}(H-t_0)^{(r-2)(p-1)}, \\[.5mm]
\delta=-\bigl(Kr(r-1)\bigr)^{p-1}(r-2)(p-1)(H-t_0)^{(r-2)(p-1)-1}
\end{array}
\]
on ${\cal I}=[t_0,t_1]$ for any $t_1\in(t_0,H)$. However, this solution
cannot be extended to ${\cal I}=[t_0,H]$ because $u(t)\to\infty$ as
$t\to H$. This situation is called a blow-up of the solution.
\end{example}

\begin{remark}\label{rem_ex_nonex} \rm
Using Example \ref{ex_nonex} one can prove that each of the conditions
\begin{itemize}
\item $\alpha,\beta,\gamma,\delta\geq0$, $\alpha+\beta+\gamma+\delta>0$,
$p<q_1$ and $c,b_1\geq C>0$ on $[t_0,\infty)$, and
\item $\alpha,\beta,\gamma,\delta\leq0$, $\alpha+\beta+\gamma+\delta<0$,
$p<q_2$ and $c,b_2\geq C>0$ on $[t_0,\infty)$
\end{itemize}
is sufficient for existence of $H>t_0$ such that there is no solution of
(\ref{poc_ul_ob}) on ${\cal I}=[t_0,H]$. The idea of the proof is based on
comparison of solutions of the initial value problem (\ref{poc_ul_ob}).
\end{remark}

\begin{remark} \rm
Example \ref{ex_nonex} can be generalized for the initial value problem of
the $(2n)^{\rm th}$-order ($n\in{\mathbb{N}}$)
\begin{equation}\label{poc_ul_2n}
\begin{array}{c}
(-1)^n\bigl(\psi_p(u^{(n)}(t))\bigr)^{(n)}=
\lambda\psi_q(u(t)),\quad t\in{\cal I}, \\[1mm]
u^{(i)}(t_0)=\alpha_i,\hspace{2mm}
\left.\bigl(\psi_p(u^{(n)}(t))\bigr)^{(i)}\right|_{t=t_0}=\beta_i,
\quad i=0,\dots,n-1
\end{array}
\end{equation}
where $p<q$ and $(-1)^n\lambda>0$. The solution is defined similarly as for
(\ref{poc_ul_ob2}). Let $H>t_0$ be arbitrary (fixed). The reader is invited
to justify that the function $u(t)=K(H-t)^r$ where
\[
\begin{array}{c}
\displaystyle
r=\frac{np}{p-q}\quad\mbox{and} \\[4mm]
\displaystyle
K=\left(\vrule height 26pt depth 0pt width 0pt\right.\hspace{-3pt}
\frac{\left(\prod\limits_{k=0}^{n-1}{\bigl((n-k)p+kq\bigr)}\right)^
{p-1}\prod\limits_{k=0}^{n-1}{\bigl(npq-kp-(n-k)q\bigr)}}{(-1)^n\lambda(q-p)^
{np}}\hspace{-3pt}\left.\vrule height 26pt depth 0pt width 0pt\right)^
{1/(q-p)}
\end{array}
\]
is a solution of (\ref{poc_ul_2n}) (with some initial conditions) on
${\cal I}=[t_0,t_1]$ for any $t_1\in(t_0,H)$. As in Example \ref{ex_nonex},
this solution cannot be extended to ${\cal I}=[t_0,H]$ because
$u(t)\to\infty$ as $t\to H$.
\end{remark}

\paragraph{Proof of Theorem \ref{th_ge}}
Now we begin the proof of the existence of a solution of (\ref{poc_ul_ob})
on ${\cal I}=[t_0,\infty)$ assuming $p\geq\max\{q_1,q_2\}$. It suffices to
prove that there exists at least one solution of (\ref{poc_ul_s2}) (see
Remark \ref{rem_prevod}) on ${\cal I}=[t_0,t_1]$ for any $t_1$ satisfying
$t_0<t_1$.

Let us have the auxiliary problem
\begin{equation}\label{poc_ul_s_aux}
\begin{array}{ll}
\hat u_1'(t)=\hat u_2(t), & \hat u_1(t_0)=\hat\alpha, \\[1mm]
\hat u_2'(t)=C\psi_{p'}(\hat u_3(t)), & \hat u_2(t_0)=\hat\beta, \\[1mm]
\hat u_3'(t)=\hat u_4(t), & \hat u_3(t_0)=\hat\gamma, \\[1mm]
\hat u_4'(t)=B\psi_p(\hat u_1(t)),\quad & \hat u_4(t_0)=\hat\delta,
\end{array}
\quad t\in[t_0,t_1]
\end{equation}
where $|b_1(t)|\leq B$, $|b_2(t)|\leq B$ and $|c(t)|\leq C$ on $[t_0,t_1]$.
The vector function $\hat{\mbox{\boldmath{$u$}}}=(\hat u_1,\hat u_2,
\hat u_3,\hat u_4)$ where
\[
\begin{array}{c}
\hat u_1(t)=K{\rm e}^{r(t-t_0)},\quad
\hat u_2(t)=Kr{\rm e}^{r(t-t_0)}, \\[1mm]
\displaystyle
\hat u_3(t)=\left(\frac{Kr^2}C\right)^{p-1}{\rm e}^{r(p-1)(t-t_0)},\quad
\hat u_4(t)=\left(\frac{Kr^2}C\right)^{p-1}r(p-1){\rm e}^{r(p-1)(t-t_0)},
\end{array}
\]
$K>0$ is arbitrary and
\[
r=\left(\frac{(p-1)^2}{BC^{p-1}}\right)^{1/(2p)},
\]
is a solution of (\ref{poc_ul_s_aux}) with
\[
\hat\alpha=K,\quad\hat\beta=Kr,\quad
\hat\gamma=\left(\frac{Kr^2}C\right)^{p-1},\quad
\hat\delta=\left(\frac{Kr^2}C\right)^{p-1}r(p-1).
\]
We choose $K$ big enough to have $|\alpha|<\hat\alpha$, $|\beta|<\hat\beta$,
$|\gamma|<\hat\gamma$ and $|\delta|<\hat\delta$. We shall prove that for any
solution $\mbox{\boldmath{$u$}}=(u_1,u_2,u_3,u_4)$ of (\ref{poc_ul_s2}) on
${\cal I}=[t_0,t_1]$
\begin{equation}\label{ex_srov}
|u_1(t)|\leq\hat u_1(t),\quad|u_2(t)|\leq\hat u_2(t),\quad
|u_3(t)|\leq\hat u_3(t),\quad|u_4(t)|\leq\hat u_4(t)
\end{equation}
for every $t\in[t_0,t_1]$.
We have $|u_1(t_0)|<\hat u_1(t_0)$, and so the set
\[
T=\{\tilde t\in[t_0,t_1]:|u_1(\tilde t)|\leq\hat u_1(\tilde t)\}
\]
is non-empty and closed, and there exists
\[
t_m=\max\{\tilde t\in[t_0,t_1]:[t_0,\tilde t]\subseteq T\}.
\]
We can assume $K\geq 1$. Then for any $t\in[t_0,t_m]$ we have
$\hat u_1(t)\geq 1$ and
\[
|u_4'(t)|\leq B|u_1(t)|^{q-1}\leq B(\hat u_1(t))^{q-1}\leq
B(\hat u_1(t))^{p-1}=\hat u_4'(t)
\]
where $q=q_1$ if $u_1(t)\geq0$, and $q=q_2$ if $u_1(t)<0$.
For any $t\in[t_0,t_m]$ we now have
\[
\begin{array}{c}
\displaystyle
|u_4(t)|\leq|\delta|+\int_{t_0}^t{|u_4'(\tau)|\,{\rm d}\tau}\leq
\hat\delta+\int_{t_0}^t{\hat u_4'(\tau)\,{\rm d}\tau}=\hat u_4(t),
\\[4mm]
\displaystyle
|u_3(t)|\leq|\gamma|+\int_{t_0}^t{|u_4(\tau)|\,{\rm d}\tau}\leq
\hat\gamma+\int_{t_0}^t{\hat u_4(\tau)\,{\rm d}\tau}=\hat u_3(t).
\end{array}
\]
Similarly we can show that for $t\in[t_0,t_m]$
\[
|u_2'(t)|\leq\hat u_2'(t)\quad\mbox{and}\quad|u_2(t)|\leq\hat u_2(t).
\]
Thus
\[
|u_1(t_m)|\leq|\alpha|+\int_{t_0}^{t_m}{|u_1'(\tau)|\,{\rm d}\tau}<\hat\alpha+
\int_{t_0}^{t_m}{\hat u_1'(\tau)\,{\rm d}\tau}=\hat u_1(t_m).
\]
This inequality would for $t_m<t_1$ contradict with the maximality of $t_m$,
and so $t_m=t_1$ and (\ref{ex_srov}) is proved. Using the standard
continuation arguments, the proof of Theorem \ref{th_ge} is completed.
\hfill$\Box$\smallskip

\section{Uniqueness}\label{sec_u}

In this section we prove the local uniqueness (Proposition \ref{prop_lu}).
We distinguish the cases (a) $|\alpha|+|\beta|+|\gamma|+|\delta|>0$ and
(b) $\alpha=\beta=\gamma=\delta=0$.

\noindent{\bf(a)} Here $u_1$ does not change its sign on some right neighborhood
of $t_0$. Hence in this case it suffices to prove the uniqueness for the
problem without the jumping nonlinearity, i.e.~(\ref{poc_ul_nek}). The proof
is divided into four parts: Lemma \ref{lem_lu1} (for $p\leq2$, $q\geq2$),
Lemma \ref{lem_lu2} (for $p\leq2$, $q<2$), Lemma \ref{lem_lu3} (for $p>2$,
$q\geq2$) and Lemma \ref{lem_lu4} (for $p>2$, $q<2$).

\noindent{\bf(b)} We assume $p\leq\min\{q_1,q_2\}$ here (see Proposition
\ref{prop_lu}). Lemma \ref{lem_lu5} deals with this case. Before Lemma
\ref{lem_lu5} we introduce the example of non-uniqueness of a solution of
(\ref{poc_ul_neh}) for $\alpha=\beta=\gamma=\delta=0$ and $p>q$.

In all proofs in this section we denote by $A,B,C>0$ such constants that
$|a(t)|\leq A$ (i.e.~$|c(t)|\geq A^{1-p'}$),
$|b_1(t)|\leq B$, $|b_2(t)|\leq B$ (i.e.~$|b(t)|\leq B$ for $b$ from
(\ref{poc_ul_nek})) and $|c(t)|\leq C$ (i.e.~$|a(t)|\geq C^{1-p}$) for every
$t\in{\cal I}$. We can also assume $t_0=0$. According to Remark
\ref{rem_prevod}, we prove the assertions for (\ref{poc_ul_s2}). For Lemmata
\ref{lem_lu1}--\ref{lem_lu4}, formulated for (\ref{poc_ul_nek}), the fourth
equation in (\ref{poc_ul_s2}) takes the form $u_4'(t)=b(t)\psi_q(u(t))$.

\begin{lemma}\label{lem_lu1}
Let\/ $|\alpha|+|\beta|+|\gamma|+|\delta|>0$, $p\leq2$ and\/ $q\geq2$.
Then there exists\/ $\varepsilon>0$ such that\/ {\rm(\ref{poc_ul_nek})} has
at most one solution on\/ ${\cal I}=[t_0,t_0+\varepsilon]$.
\end{lemma}

\paragraph{Proof}
Let $\mbox{\boldmath{$u$}}$ and $\mbox{\boldmath{$v$}}$ be solutions of the
special case of (\ref{poc_ul_s2}), corresponding to (\ref{poc_ul_nek}). From
the former two equations we conclude
\[
u_3(t)-v_3(t)=a(t)\Bigl(\psi_p(u_1''(t))-\psi_p(v_1''(t))\Bigr),
\]
and from the latter two equations we obtain
\[
u_3''(t)-v_3''(t)=b(t)\Bigl(\psi_q(u_1(t))-\psi_q(v_1(t))\Bigr),
\]
$t\in{\cal I}$. Then
\begin{equation}\label{vychozi1}
a(t)\Bigl(\psi_p(u_1''(t))-\psi_p(v_1''(t))\Bigr)=
\int_0^t(t-\tau)b(\tau)\Bigl(\psi_q(u_1(\tau))-\psi_q(v_1(\tau))\Bigr)
\,{\rm d}\tau.
\end{equation}
There exists a constant $K_1>0$ such that $|u_1''(t)|\leq K_1$ and
$|v_1''(t)|\leq K_1$ on ${\cal I}$. Since $p\leq2$,
$\psi_p'(\tau)\geq\psi_p'(K_1)$ for $|\tau|\leq K_1$, and so
\begin{equation}\label{odh_l1}
\begin{array}{c}
\displaystyle
\left|a(t)\Bigl(\psi_p(u_1''(t))-\psi_p(v_1''(t)\Bigr)\right|\geq
C^{1-p}\Bigl|\int_{v_1''(t)}^{u_1''(t)}{\psi_p'(\tau)\,{\rm d}\tau}\Bigr|
\geq \\[4mm]
\displaystyle
\geq(p-1)K_1^{p-2}C^{1-p}|u_1''(t)-v_1''(t)|.
\end{array}
\end{equation}
There exists a constant $K_2>0$ such that $|u_1(\tau)|\leq K_2$ and
$|v_1(\tau)|\leq K_2$ on ${\cal I}$. Since $q\geq2$,
$\psi_q'(\sigma)\leq\psi_q'(K_2)$ for $|\sigma|\leq K_2$. For
$\tau\in{\cal I}$ it yields
\begin{equation}\label{p1}
\left|\psi_q(u_1(\tau))-\psi_q(v_1(\tau))\right|=
\Big|\int^{u_1(\tau)}_{v_1(\tau)}{\psi_q'(\sigma)\,{\rm d}\sigma}\Big|
\leq(q-1)K_2^{q-2}|u_1(\tau)-v_1(\tau)|.
\end{equation}
Using the estimate
\begin{equation}\label{odh_pp}
|u_1(\tau)-v_1(\tau)|=\Big|\int_0^\tau{(\tau-\sigma)(u_1''(\sigma)-
v_1''(\sigma))\,{\rm d}\sigma}\Big|\leq\tau^2\|u_1''-v_1''\|_{C({\cal I})}
\end{equation}
we conclude
\begin{equation}\label{odh_p1}
\Big|\int_0^t{(t-\tau)b(\tau)\Bigl(\psi_q(u_1(\tau))-\psi_q(v_1(\tau))\Bigr)
\,{\rm d}\tau}\Big|\leq t^4(q-1)K_2^{q-2}B\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}
We combine (\ref{vychozi1}), (\ref{odh_l1}) and (\ref{odh_p1}), take the
maximum over $t\in{\cal I}$, and get
\begin{equation}\label{jed_vysl_1}
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^4
\frac{(q-1)K_2^{q-2}B}{(p-1)K_1^{p-2}C^{1-p}}\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}
For $\varepsilon>0$ small enough this implies $u_1''=v_1''$, and so
$u_3=v_3$. Since $u_1(0)=v_1(0)$ and $u_1'(0)=v_1'(0)$, it is then
$u_1=v_1$. Thus $\mbox{\boldmath{$u$}}=\mbox{\boldmath{$v$}}$. \hfill$\Box$


\begin{lemma}\label{lem_lu2}
Let\/ $|\alpha|+|\beta|+|\gamma|+|\delta|>0$, $p\leq2$ and\/ $q<2$.
Then there exists\/ $\varepsilon>0$ such that\/ {\rm(\ref{poc_ul_nek})} has
at most one solution on\/ ${\cal I}=[t_0,t_0+\varepsilon]$.
\end{lemma}

\paragraph{Proof}
We distinguish the cases (i) $\alpha\ne0$, (ii) $\alpha=0$, $\beta\ne0$,
(iii) $\alpha=\beta=0$, $\gamma\ne0$ and (iv) $\alpha=\beta=\gamma=0$,
$\delta\ne0$.

\noindent{\bf(i)} We proceed as in the proof of Lemma \ref{lem_lu1}. The assumption
$u_1(0)=v_1(0)=\alpha\ne0$ guarantees the existence of a constant $K_2>0$
such that $|u_1(\tau)|\geq K_2$ and $|v_1(\tau)|\geq K_2$ in
$[0,\varepsilon]$ for $\varepsilon>0$ small enough. Since $q<2$,
$\psi_q'(\sigma)\leq\psi_q'(K_2)$ for $|\sigma|\geq K_2$. Hence (\ref{p1})
still holds true for all $\tau\in{\cal I}$, and we arrive again at
(\ref{jed_vysl_1}).

\noindent{\bf(ii)} We modify again the proof of Lemma \ref{lem_lu1}. Due to the
assumptions ($\alpha=0$, $\beta\ne0$), $\frac{u_1(\tau)}\tau\to\beta$ and
$\frac{v_1(\tau)}\tau\to\beta\ne0$ as $\tau\to0_+$. Hence there exists a
constant $K_2>0$ such that $\bigl|\frac{u_1(\tau)}\tau\bigr|\geq K_2$ and
$\bigl|\frac{v_1(\tau)}\tau\bigr|\geq K_2$ for all $\tau\in(0,\varepsilon]$
with $\varepsilon>0$ small enough. Thus
\begin{equation}\label{p2}
\Bigl|\psi_q\Bigl(\frac{u_1(\tau)}\tau\Bigr)-
\psi_q\Bigl(\frac{v_1(\tau)}\tau\Bigr)\Bigr|=
\Bigl|\int_{\frac{v_1(\tau)}\tau}^{\frac{u_1(\tau)}\tau}
{\psi_q'(\sigma)\,{\rm d}\sigma}\Bigr|
\leq\frac{(q-1)K_2^{q-2}}\tau|u_1(\tau)-v_1(\tau)|.
\end{equation}
Using (\ref{p2}) instead of (\ref{p1}) we get
\begin{equation}\label{jed_vysl_3}
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^{q+2}\frac{(q-1)K_2^{q-2}B}
{(p-1)K_1^{p-2}C^{1-p}}\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}

\noindent{\bf(iii)} We follow again the proof of Lemma \ref{lem_lu1}. By the
assumptions ($\alpha=\beta=0$, $\gamma\ne0$),
$\frac{u_1(\tau)}{\tau^2}\to\frac12c(0)\psi_{p'}(\gamma)$ and
$\frac{v_1(\tau)}{\tau^2}\to\frac12c(0)\psi_{p'}(\gamma)\ne0$ as
$\tau\to0_+$. Thus, there exists a constant $K_2>0$ such that
$\bigl|\frac{u_1(\tau)}{\tau^2}\bigr|\geq K_2$ and
$\bigl|\frac{v_1(\tau)}{\tau^2}\bigr|\geq K_2$ for every
$\tau\in(0,\varepsilon]$ with $\varepsilon>0$ small enough. Then
\begin{equation}\label{p3}
\Bigl|\psi_q\Bigl(\frac{u_1(\tau)}{\tau^2}\Bigr)-\psi_q\Bigl(
\frac{v_1(\tau)}{\tau^2}\Bigr)\Bigr|=\Bigl|\int_{\frac{v_1(\tau)}{\tau^2}}
^{\frac{u_1(\tau)}{\tau^2}}{\psi_q'(\sigma)\,{\rm d}\sigma}\Bigr|
\geq\frac{(q-1)K_2^{q-2}}{\tau^2}|u_1(\tau)-v_1(\tau)|.
\end{equation}
Using (\ref{p3}) instead of (\ref{p1}) we get
\begin{equation}\label{jed_vysl_4}
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^{2q}\frac{(q-1)K_2^{q-2}B}
{(p-1)K_1^{p-2}C^{1-p}}\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}

\noindent{\bf(iv)} Here we cannot follow the proof of Lemma \ref{lem_lu1}.
Like (\ref{vychozi1}), we can derive that for every $t\in{\cal I}$
\begin{equation}\label{vychozi2}
\bigl|b(t)(u_1(t)-v_1(t))\bigr|\leq|b(t)|\int_0^t{(t-\tau)|c(\tau)|
\Bigl|\psi_{p'}(u_3(\tau))-\psi_{p'}(v_3(\tau))\Bigr|\,{\rm d}\tau}.
\end{equation}
By the assumptions ($\gamma=0$, $\delta\ne0$),
$\frac{u_1''(\tau)}{\tau^{p'-1}}\to c(0)\psi_{p'}(\delta)\ne0$ as
$\tau\to0_+$. So there exists a constant $K_1>0$ such that
$\bigl|\frac{u_1''(\tau)}{\tau^{p'-1}}\bigr|\geq K_1$ for any
$\tau\in(0,\varepsilon]$ with $\varepsilon>0$ small enough. Thus, for every
$t\in{\cal I}$
\[
|u_1(t)|=\int_0^t{(t-\tau)|u_1''(\tau)|\,{\rm d}\tau}\geq\int_0^t{(t-\tau)K_1
\tau^{p'-1}\,{\rm d}\tau}=\frac{K_1t^{p'+1}}{p'(p'+1)},
\]
i.e.~$\bigl|\psi_q\bigl(\frac{u_1(t)}{t^{p'+1}}\bigr)\bigr|\geq
\tilde K_1:=\psi_q\bigl(\frac{K_1}{p'(p'+1)}\bigr)>0$, and analogously
$\bigl|\psi_q\bigl(\frac{v_1(t)}{t^{p'+1}}\bigr)\bigr|\geq\tilde K_1$ for
all $t\in(0,\varepsilon]$. Since $q<2$,
$\psi_{q'}'(\sigma)\geq\psi_{q'}'(\tilde K_1)$ for $|\sigma|\geq\tilde K_1$.
Thus, for all $t\in(0,\varepsilon]$,
\[
\bigl|b(t)(u_1(t)-v_1(t))\bigr|=|b(t)|t^{p'+1}
\Bigl|\psi_{q'}\Bigl(\psi_q\Bigl(\frac{u_1(t)}{t^{p'+1}}\Bigr)\Bigr)-
\psi_{q'}\Bigl(\psi_q\Bigl(\frac{v_1(t)}{t^{p'+1}}\Bigr)\Bigr)\Bigr|=
\]
\begin{equation}\label{odh_l2}
=|b(t)|t^{p'+1}\Bigl|\int_{\psi_q\left(\frac{v_1(t)}{t^{p'+1}}\right)}^
{\psi_q\left(\frac{u_1(t)}{t^{p'+1}}\right)}{\psi_{q'}'(\sigma)\,{\rm d}\sigma}
\Bigr|\geq t^{(2-q)(p'+1)}(q'-1)\tilde K_1^{q'-2}|u_3''(t)-v_3''(t)|.
\end{equation}
Since $\frac{u_3(\tau)}\tau\to\delta$ and $\frac{v_3(\tau)}\tau\to\delta$ as
$\tau\to0_+$, there exists $K_2>0$ such that
$\bigl|\frac{u_3(\tau)}\tau\bigr|\leq K_2$ and
$\bigl|\frac{v_3(\tau)}\tau\bigr|\leq K_2$ for any
$\tau\in(0,\varepsilon]$. Since $p\leq2$,
$\psi_{p'}'(\sigma)\leq\psi_{p'}'(K_2)$ for $|\sigma|\leq K_2$, and so
\begin{equation}\label{p4}
\Bigl|\psi_{p'}\Bigl(\frac{u_3(\tau)}\tau\Bigr)-\psi_{p'}\Bigl(
\frac{v_3(\tau)}\tau\Bigr)\Bigr|=\Bigl|\int_{\frac{v_3(\tau)}\tau}^
{\frac{u_3(\tau)}\tau}{\psi_{p'}'(\sigma)\,{\rm d}\sigma}\Bigr|
\leq\frac{(p'-1)K_2^{p'-2}}\tau|u_3(\tau)-v_3(\tau)|.
\end{equation}
Since, analogously to~(\ref{odh_pp}),
\begin{equation}\label{odh_pp2}
|u_3(\tau)-v_3(\tau)|\leq\tau^2\|u_3''-v_3''\|_{C({\cal I})}\qquad
\forall\tau\in{\cal I},
\end{equation}
we have the following estimate for all $t\in{\cal I}$:
\begin{equation}\label{odh_p2}
\begin{array}{c}
\displaystyle
|b(t)|\int_0^t(t-\tau){|c(\tau)|\psi_{p'}(\tau)\Bigl|\psi_{p'}\Bigl(
\frac{u_3(\tau)}\tau\Bigr)-\psi_{p'}\Bigl(\frac{v_3(\tau)}\tau\Bigr)
\Bigr|\,{\rm d}\tau}\leq \\[3mm]
\leq t^{p'+2}(p'-1)K_2^{p'-2}BC\|u_3''-v_3''\|_{C({\cal I})}.
\end{array}
\end{equation}

Putting (\ref{odh_l2}), (\ref{vychozi2}) and (\ref{odh_p2}) together and
passing to the maximum over ${\cal I}$ (for $t=0$ the inequality is
trivially satisfied) we obtain
\begin{equation}\label{jed_vysl_5}
\|u_3''-v_3''\|_{C({\cal I})}\leq\varepsilon^{(q-1)(p'+1)+1}
\frac{p'-1}{q'-1}\tilde K_1^{2-q'}K_2^{p'-2}BC\|u_3''-v_3''\|_{C({\cal I})}.
\end{equation}
Since for any $p,q>1$ we have $(q-1)(p'+1)+1>0$, the proof is complete.
\hfill$\Box$

\begin{lemma}\label{lem_lu3}
Let\/ $|\alpha|+|\beta|+|\gamma|+|\delta|>0$, $p>2$ and\/ $q\geq2$.
Moreover, let\/ $|\gamma|+|\delta|>0$ or\/ $b$ not change its sign on some
right closed neighborhood of\/ $t_0$. Then there exists\/ $\varepsilon>0$
such that\/ {\rm(\ref{poc_ul_nek})} has at most one solution on\/
${\cal I}=[t_0,t_0+\varepsilon]$.
\end{lemma}

\paragraph{Proof}
We distinguish the cases (i) $\gamma\ne0$, (ii) $\gamma=0$, $\delta\ne0$,
(iii) $\gamma=\delta=0$, $\alpha\ne0$ and (iv) $\gamma=\delta=\alpha=0$,
$\beta\ne0$. Let again $\mbox{\boldmath{$u$}}$ a $\mbox{\boldmath{$v$}}$ be
solutions of the special case (\ref{poc_ul_s2}), corresponding to
(\ref{poc_ul_nek}).

\noindent{\bf(i)} Since $u_1''(0)=v_1''(0)=c(0)\psi_{p'}(\gamma)\ne0$, there exists a
constant $K_1>0$ such that $|u_1''(t)|\geq K_1$ and $|v_1''(t)|\geq K_1$
for all $t\in[0,\varepsilon]$ with $\varepsilon>0$ small enough. We have
$p>2$, and so $\psi_p'(\tau)\geq\psi_p'(K_1)$ for $|\tau|\geq K_1$. Hence
(\ref{odh_l1}) holds true, and we get (\ref{jed_vysl_1}).

\noindent{\bf(ii)} As in the part (iv) of the proof of Lemma \ref{lem_lu2}, there
exists a constant $K_1>0$ such that
$\bigl|\frac{u_1''(t)}{t^{p'-1}}\bigr|\geq K_1$ and
$\bigl|\frac{v_1''(t)}{t^{p'-1}}\bigr|\geq K_1$ for all
$t\in(0,\varepsilon]$ with $\varepsilon>0$ small enough. Hence
\begin{equation}\label{odh_l3}
\begin{array}{c}
\left|a(t)\Bigl(\psi_p(u_1''(t))-\psi_p(v_1''(t))\Bigr)\right|
=|a(t)|t\Bigl|\psi_p\Bigl(\frac{u_1''(t)}{t^{p'-1}}\Bigr)-
\psi_p\Bigl(\frac{v_1''(t)}{t^{p'-1}}\Bigr)\Bigr|\geq \\[3mm]
\displaystyle
\geq C^{1-p}t\Bigl|\int_{\frac{v_1''(t)}{t^{p'-1}}}^{\frac{u_1''(t)}
{t^{p'-1}}}{\psi_p'(\sigma)\,{\rm d}\sigma}\Bigr|
\geq t^{2-p'}(p-1)K_1^{p-2}C^{1-p}|u_1''(t)-v_1''(t)|.
\end{array}
\end{equation}
Using (\ref{odh_l3}) instead of (\ref{odh_l1}) we obtain
\begin{equation}\label{jed_vysl_2}
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^{p'+2}\frac{(q-1)K_2^{q-2}B}
{(p-1)K_1^{p-2}C^{1-p}}\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}

\noindent{\bf(iii)} We can assume
\[
f(t):=\int_0^t(t-\tau){|b(\tau)|\,{\rm d}\tau}>0\quad\forall t\in(0,\varepsilon]
\]
(otherwise $b(\tau)=0$ for all $\tau\in[0,t_0]$ with some $t_0>0$, and the
uniqueness is then trivial). Since $u_1(0)=\alpha\ne0$, there exists a
constant $K_1>0$ such that $|u_1(\tau)|\geq K_1$, and so
$|u_3''(\tau)|\geq K_1^{q-1}|b(\tau)|$ for all $\tau\in[0,\varepsilon]$ with
$\varepsilon>0$ small enough. We suppose that $b$ and $u_3''$ does not
change its sign on ${\cal I}$. Hence for any $t\in{\cal I}$
\[
|u_3(t)|=\int_0^t(t-\tau){|u_3''(\tau)|\,{\rm d}\tau}\geq
K_1^{q-1}\int_0^t(t-\tau){|b(\tau)|\,{\rm d}\tau}=K_1^{q-1}f(t).
\]
Thus
\[
|u_1''(t)|=|c(t)\psi_{p'}(u_3(t))|\geq K_1^{(q-1)(p'-1)}A^{1-p'}f^{p'-1}(t),
\]
and the same estimate holds for $|v_1''(t)|$, $t\in{\cal I}$. For
$t\in(0,\varepsilon]$ we can write
\begin{eqnarray}
&\left|a(t)\Bigl(\psi_p(u_1''(t))-\psi_p(v_1''(t))\Bigr)\right|
\geq C^{1-p}f(t)\Bigl|\psi_p\Bigl(\frac{u_1''(t)}{f^{p'-1}(t)}\Bigr)-
\psi_p\Bigl(\frac{v_1''(t)}{f^{p'-1}(t)}\Bigr)\Bigr|=&\nonumber \\
&\displaystyle
=C^{1-p}f(t)\Bigl|\int_{\frac{v_1''(t)}{f^{p'-1}(t)}}^{\frac{u_1''(t)}
{f^{p'-1}(t)}}{\psi_p'(\tau)\,{\rm d}\tau}\Bigr|\geq&\label{odh_l5} \\
&\geq(p-1)K_1^{\frac{(q-1)(p-2)}{p-1}}A^{-\frac{p-2}{p-1}}C^{1-p}f^{2-p'}(t)
|u_1''(t)-v_1''(t)|.&\nonumber
\end{eqnarray}
Using (\ref{p1}) and (\ref{odh_pp}) we get for $t\in{\cal I}$
\begin{equation}\label{odh_p3}
\Big|\int_0^t(t-\tau){b(\tau)\Bigl(\psi_q(u_1(\tau))-\psi_q(v_1(\tau))
\Bigr)\,{\rm d}\tau}\Big|\leq
t^2(q-1)K_2^{q-2}f(t)\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}
Obviously $f(t)\leq t^2B$. Putting (\ref{odh_l5}), (\ref{vychozi1}) and
(\ref{odh_p3}) together and passing to the maximum over ${\cal I}$ we arrive
at
\[
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^{2p'}\frac{(q-1)K_2^{q-2}
B^{p'-1}}{(p-1)K_1^{\frac{(q-1)(p-2)}{p-1}}A^{-\frac{p-2}{p-1}}C^{1-p}}
\|u_1''-v_1''\|_{C({\cal I})}.
\]

\noindent{\bf(iv)} We proceed as analogically to (iii). We can assume
\[
f(t):=\int_0^t(t-\tau)\tau^{q-1}{|b(\tau)|\,{\rm d}\tau}>0\quad
\forall t\in(0,\varepsilon].
\]
Since $\frac{u_1(\tau)}\tau\to\beta\ne0$, there exists a constant $K_1>0$
such that $\bigl|\frac{u_1(\tau)}\tau\bigr|\geq K_1$, and also
$|u_3''(\tau)|\geq(K_1\tau)^{q-1}|b(\tau)|$ for $\tau\in[0,\varepsilon]$
with $\varepsilon>0$ small enough. We suppose that $b$ and $u_3''$ does not
change sign on ${\cal I}$. For any $t\in{\cal I}$ we have
\[
|u_3(t)|=\int_0^t(t-\tau){|u_3''(\tau)|\,{\rm d}\tau}\geq
K_1^{q-1}\int_0^t(t-\tau){\tau^{q-1}|b(\tau)|\,{\rm d}\tau}=K_1^{q-1}f(t).
\]
Analogously as (\ref{odh_l5}) we can now for $t\in(0,\varepsilon]$ derive
\begin{equation}\label{odh_l6}
\begin{array}{c}
\displaystyle
\left|a(t)\Bigl(\psi_p(u_1''(t))-\psi_p(v_1''(t))\Bigr)\right|\geq \\[2mm]
\displaystyle
\geq(p-1)K_1^{\frac{(q-1)(p-2)}{p-1}}A^{-\frac{p-2}{p-1}}C^{1-p}f^{2-p'}(t)
|u_1''(t)-v_1''(t)|.
\end{array}
\end{equation}

There exists a constant $K_2>0$ such that
$\bigl|\frac{u_1(\tau)}\tau\bigr|\leq K_2$ and
$\bigl|\frac{v_1(\tau)}\tau\bigr|\leq K_2$ for all
$\tau\in(0,\varepsilon]$. Since $q\geq2$, $\psi_q'(\sigma)\leq\psi_q'(K_2)$
for $|\sigma|\leq K_2$. Thus, (\ref{p2}) holds true for all
$t\in(0,\varepsilon]$. Together with (\ref{odh_pp}) it yields for
$t\in{\cal I}$
\begin{equation}\label{odh_p4}
\Bigl|\int_0^t(t-\tau){b(\tau)\Bigl(\psi_q(u_1(\tau))-\psi_q(v_1(\tau))\Bigr)
\,{\rm d}\tau}\Bigr|\leq t(q-1)K_2^{q-2}f(t)\|u_1''-v_1''\|_{C({\cal I})}.
\end{equation}
Obviously $f(t)\leq t^{q+1}B$. Using (\ref{odh_l6}), (\ref{vychozi1}),
(\ref{odh_p4}) we obtain for the maximum over $\cal I$
\[
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^{(q+1)(p'-1)+1}\frac{(q-1)
K_2^{q-2}B^{p'-1}}{(p-1)K_1^{\frac{(q-1)(p-2)}{p-1}}A^{-\frac{p-2}{p-1}}
C^{1-p}}\|u_1''-v_1''\|_{C({\cal I})}.
\]
Since for $p,q>1$ it is $(q+1)(p'-1)+1>0$, we proved the assertion of this
lemma.
\hfill$\Box$

\begin{lemma}\label{lem_lu4}
Let\/ $|\alpha|+|\beta|+|\gamma|+|\delta|>0$, $p>2$ and\/ $q<2$. Moreover,
let\/ $|\gamma|+|\delta|>0$ or\/ $b$ not change its sign on some right
closed neighborhood of\/ $t_0$. Then there exists\/ $\varepsilon>0$ such
that\/ {\rm(\ref{poc_ul_nek})} has at most one solution on\/
${\cal I}=[t_0,t_0+\varepsilon]$.
\end{lemma}

\paragraph{Proof}
We combine the previous techniques. Consequently, we distinguish the cases
(i) $\alpha\ne0$, $\gamma\ne0$, (ii) $\alpha\ne0$, $\gamma=0$, $\delta\ne0$,
(iii) $\alpha\ne0$, $\gamma=\delta=0$, (iv) $\alpha=0$, $\beta\ne0$,
$\gamma\ne0$, (v) $\alpha=0$, $\beta\ne0$, $\gamma=0$, $\delta\ne0$, (vi)
$\alpha=0$, $\beta\ne0$, $\gamma=\delta=0$, (vii) $\alpha=\beta=0$,
$\gamma\ne0$ and (viii) $\alpha=\beta=\gamma=0$, $\delta\ne0$.

\noindent{\bf(i)} As in the part (i) of the proof of Lemma \ref{lem_lu3}, we can use
(\ref{odh_l1}), and, as in the part (i) of the proof of Lemma \ref{lem_lu2},
we can use (\ref{p1}). Using (\ref{vychozi1}) we arrive at
(\ref{jed_vysl_1}).

\noindent{\bf(ii)} From (\ref{odh_l3}), (\ref{vychozi1}) and (\ref{p1}) we derive
(\ref{jed_vysl_2}).

\noindent{\bf(iii)} As (\ref{p1}) holds true, we follow the part (iii) of the proof
of Lemma \ref{lem_lu3}.

\noindent{\bf(iv)} From (\ref{odh_l1}), (\ref{vychozi1}) and (\ref{p2}) we get
(\ref{jed_vysl_3}).

\noindent{\bf(v)} Here (\ref{odh_l3}), (\ref{vychozi1}) and (\ref{p2}) yield
\[
\|u_1''-v_1''\|_{C({\cal I})}\leq\varepsilon^{p'+q}\frac{(q-1)K_2^{q-2}B}
{(p-1)K_1^{p-2}C^{1-p}}\|u_1''-v_1''\|_{C({\cal I})}.
\]

\noindent{\bf(vi)} Since (\ref{p2}) holds true for some $K_2>0$, we can proceed as in
the part (iv) of the proof of Lemma \ref{lem_lu3}.

\noindent{\bf(vii)} From (\ref{odh_l1}), (\ref{vychozi1}) and (\ref{p3}) we conclude
(\ref{jed_vysl_4}).

\noindent{\bf(viii)} We follow the part (iv) of the proof of Lemma \ref{lem_lu2}. Due
to the assumptions ($\gamma=0$, $\delta\ne0$), there exists a constant
$K_2>0$ such that $\bigl|\frac{u_3(\tau)}\tau\bigr|\geq K_2$ and
$\bigl|\frac{u_3(\tau)}\tau\bigr|\geq K_2$ for all
$\tau\in(0,\varepsilon]$ with $\varepsilon>0$ small enough. Since $p>2$,
$\psi_{p'}'(\sigma)\leq\psi_{p'}'(K_2)$ for $|\sigma|\geq K_2$, and so
(\ref{p4}) holds true. Now we put (\ref{odh_l2}), (\ref{vychozi2}) and
(\ref{p4}) together to obtain (\ref{jed_vysl_5}).
\hfill$\Box$\smallskip

As we promised, we show now that if $p>q$, $\alpha=\beta=\gamma=\delta=0$
and $\lambda>0$, then (\ref{poc_ul_neh}) has a non-trivial solution besides
of the trivial one, and so the uniqueness is broken.

\begin{example}\label{pr_nejed} \rm
Let in (\ref{poc_ul_neh}) $p>q$, $\alpha=\beta=\gamma=\delta=0$, $\lambda>0$
and ${\cal I}=[0,\varepsilon]$ with some $\varepsilon>0$. Then one can
compute that $u(t)=0$ and $u(t)=K(t-t_0)^r$ where
\[
r=\frac{2p}{p-q}\quad\mbox{and}\quad
K=\left(\frac{2^p(p-1)q\bigl(p(p+q)\bigr)^{p-1}(2pq-p-q)}
{\lambda(p-q)^{2p}}\right)^{1/(q-p)}
\]
are solutions of (\ref{poc_ul_neh}).
\end{example}

\begin{remark} \rm
Example \ref{pr_nejed} can be generalized (cf.~Example \ref{ex_nonex}) for 
the initial value problem (\ref{poc_ul_2n}) of the $(2n)^{\rm th}$-order, 
$n\in{\mathbb{N}}$,  where $p>q$, $(-1)^n\lambda>0$ and 
$\alpha_i=\beta_i=0$, $i=0,\dots,n-1$. The reader is invited to justify that 
$u(t)=0$ and $u(t)=K(H-t)^r$ where
\[
\begin{array}{c}
\displaystyle
r=\frac{np}{p-q}\quad\mbox{and}\quad \\[4mm]
\displaystyle
K=\left(\vrule height 26pt depth 0pt width 0pt\right.\hspace{-3pt}
\frac{\left(\prod\limits_{k=0}^{n-1}{\bigl((n-k)p+kq\bigr)}\right)^
{p-1}\prod\limits_{k=0}^{n-1}{\bigl(npq-kp-(n-k)q\bigr)}}{(-1)^n\lambda(p-q)^
{np}}\hspace{-3pt}\left.\vrule height 26pt depth 0pt width 0pt\right)^
{1/(q-p)}
\end{array}
\]
are solutions of (\ref{poc_ul_2n}).
\end{remark}

\begin{lemma}\label{lem_lu5}
Let\/ $\alpha=\beta=\gamma=\delta=0$ and\/ $p\leq\min\{q_1,q_2\}$. Then
there exists\/ $\varepsilon>0$ such that\/ {\rm(\ref{poc_ul_ob})} has at
most one solution on\/ ${\cal I}=[t_0,t_0+\varepsilon]$.
\end{lemma}

\paragraph{Proof}
Let $\mbox{\boldmath{$u$}}$ be a solution of (\ref{poc_ul_ob}). We prove that
$u_1=u_2=u_3=u_4=0$ on $[0,\varepsilon]$ for some $\varepsilon>0$. For
$t\in{\cal I}$
\begin{equation}\label{p5}
|a(t)\psi_p(u_1''(t))|\leq\int_0^t(t-\tau)\Bigl(|b_1(\tau)|\psi_{q_1}
(|u_1^+(\tau)|)+|b_2(\tau)|\psi_{q_2}(|u_1^-(\tau)|)\Bigr)\,{\rm d}\tau
\end{equation}
Since for $\tau\in{\cal I}$ obviously
$|u(\tau)|\leq\tau^2\|u''\|_{C({\cal I})}$, we have
\begin{equation}\label{odh_p6}
\begin{array}{c}
|b_1(\tau)|\psi_{q_1}(|u_1^+(\tau)|)+|b_2(\tau)|\psi_{q_2}(|u_1^-(\tau)|)\leq
\\[2mm]
\leq B\Bigl(\tau^{2q_1-2}\|u_1''\|_{C({\cal I})}^{q_1-1}+\tau^{2q_2-2}
\|u_1''\|_{C({\cal I})}^{q_2-1}\Bigr)\leq \\[2mm]
\leq B\Bigl(\tau^{2q_1-2}\|u_1''\|_{C({\cal I}_0)}^{q_1-p}+\tau^{2q_2-2}
\|u_1''\|_{C({\cal I}_0)}^{q_2-p}\Bigr)\|u_1''\|_{C({\cal I})}^{p-1}
\end{array}
\end{equation}
where ${\cal I}_0=[0,\varepsilon_0]$ with $\varepsilon_0>0$ arbitrary, but
fixed, and $\varepsilon\leq\varepsilon_0$. We used the assumption
$p\leq\min\{q_1,q_2\}$ which implies that
$\|u_1''\|_{C[0,\varepsilon]}^{q_i-p}$, $i=1,2$, are increasing functions
of $\varepsilon$. Using the estimate
$|a(t)\psi_p(u_1''(t))|\geq C^{1-p}|u_1''(t)|^{p-1}$ we can infer from
(\ref{p5}) and (\ref{odh_p6}) that for every $t\in{\cal I}$
\[
C^{1-p}|u_1''(t)|^{p-1}\leq B\Bigl(t^{2q_1}\|u_1''\|_{C({\cal I}_0)}^{q_1-p}+
t^{2q_2}\|u_1''\|_{C({\cal I}_0)}^{q_2-p}\Bigr)\|u_1''\|_{C({\cal I})}^{p-1}.
\]
Now we pass to the maximum for $t\in{\cal I}$. If we suppose that
$\varepsilon\leq1$, we obtain
\[
\|u_1''\|_{C({\cal I})}^{p-1}\leq\varepsilon^{2\min\{q_1,q_2\}}BC^{p-1}
\Bigl(\|u''\|_{C({\cal I}_0)}^{q_1-p}+\|u''\|_{C({\cal I}_0)}^{q_2-p}\Bigr)
\|u_1''\|_{C({\cal I})}^{p-1}.
\]
For $\varepsilon>0$ small enough this inequality guarantees that
$\|u''\|_{C({\cal I})}^{p-1}=0$, and so $u_1''=0$, $u_1=0$, and also
$u_2=u_3=u_4=0$ on ${\cal I}=[0,\varepsilon]$.
\hfill$\Box$\smallskip

Now that we completed the proof of Proposition \ref{prop_lu}.
Theorem \ref{th_gu} is a direct consequence of this proposition.

\section{Open Problems}\label{sec_op}

The main problems we leave open are:
\begin{enumerate}
\item Does the conclusion of Proposition \ref{prop_lu} (local uniqueness)
hold true even without the latter four conditions, i.e.~for $p>2$,
$\gamma=\delta=0$, $\alpha$ or $\beta$ nonzero and $b_1$ or $b_2$ changing
its sign on arbitrarily small right neighborhood of $t_0$? If it did, then
the sufficient condition for the global uniqueness (see Theorem \ref{th_gu})
would be $p\leq\min\{q_1,q_2\}$ only (we showed that this assumption cannot
be left out).

We can simplify this problem: Can there exist two different solutions of
\begin{equation}\label{poc_ul_op}
\begin{array}{lll}
u''(t)=\psi_{p'}(v(t)), & u(t_0)=\alpha, & u'(t_0)=\beta, \\
v''(t)=b(t)\psi_p(u(t)),\quad & v(t_0)=0, & v'(t_0)=0,
\end{array}
\quad t\in{\cal I}
\end{equation}
with ${\cal I}=[t_0,t_1]$, $b\in C({\cal I})$ arbitrary, $p>2$ and
$|\alpha|+|\beta|>0$? Note that the system of equations in (\ref{poc_ul_op})
is homogeneous!

\item We gave Example \ref{ex_nonex} which showed that for $p<q$ and some
initial conditions the solution of (\ref{poc_ul_neh}) did not have to exist
on $[t_0,\infty)$. In this Example we assumed $\lambda>0$, for $\lambda=0$
the global existence is trivial, but for $\lambda<0$ we leave the question
of global existence open (see Table \ref{tab_ex}).

\item Analogously to the previous open problem, for $\lambda<0$,
$\alpha=\beta=\gamma=\delta=0$ and $p>q$ we gave neither the proof of the
local uniqueness of the solution of (\ref{poc_ul_neh}) nor a counterexample
(see Table \ref{tab_uni}), and so we leave it as an open question, too.
\end{enumerate}

\paragraph{Acknowledgements}
The author is partially supported by grant 201/00/0376 from
the Grant Agency of the Czech Republic and by grant F971/2002-G4 from
the Ministry of Education of the Czech Republic.

\begin{thebibliography}{99}
\bibitem{B} Benedikt, J., {\it Sturmova--Liouvilleova \'uloha pro
$p$--biharmonick\'y oper\'ator (Sturm--Liou\-vil\-le Problem for
$p$-Biharmonic Operator)}, diploma thesis, Faculty of Applied Sciences,
University of West Bohemia, Plze\v n 2001 (in Czech).
\bibitem{CL} Coddington, E.\,A., Levinson, N., {\it Theory of Ordinary
Differential Equations}, McGraw--Hill Book Company, Inc., New
York--Toronto--London 1955.
\bibitem{DO} Dr\'abek, P., \^Otani, M., `Global Bifurcation Result for
the $p$-Biharmonic Operator', {\em Electron.~J.~Diff.~Eqns.},
Vol.~2001(2001), No.~48, pp.~1--19.
\bibitem{H} Hartman, P., {\em Ordinary Differential Equations}, John Wiley
\& Sons, Inc., Baltimore 1973.
\end{thebibliography}

\noindent\textsc{Ji\v r\'\i\ Benedikt}\\
Centre of Applied Mathematics \\
University of West Bohemia \\
Univerzitn\'\i\ 22, 306 14 Plze\v n \\
Czech Republic \\
e-mail: benedikt@kma.zcu.cz

\section*{Addendum: July 28, 2003.}

It was brought to my knowledge by a colleague that 
in Remark 4.6, fourth line (page 15) there should be
$$u(t)=K(t-t_0)^r$$
instead of 
$$u(t)=K(H-t)^r\,.$$
Even though I think that this mistake is not misleading 
for the reader, I want to correct it.

\end{document}