Blow-up collocation solutions of some Volterra integral equations
R. Ben\'{\i}tez$^{1,*}$
V. J. Bol\'os$^{2,*}$
{$^1$ Dpto. Matem\'aticas, Centro Universitario de Plasencia, Universidad de Extremadura.}
{Avda. Virgen del Puerto 2, 10600 Plasencia, Spain.}
{$^2$ Dpto. Matem\'aticas para la Econom\'{\i}a y la Empresa
Facultad de Econom\'{\i}a,}
{Universidad de Valencia. Avda. Tarongers s/n, 46022 Valencia, Spain.}
In this paper we use collocation methods for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, first we generalize the results on existence and uniqueness given in a previous paper in order to consider also non locally bounded kernels, which are present in many cases of blow-up problems. Next, we introduce the concept of ``blow-up collocation solution'' and analyze numerically some blow-up time estimates using collocation methods in particular examples where the previous results about existence and uniqueness can be applied. Finally, we discuss the relationships between necessary conditions for blow-up of collocation solutions and exact solutions.
Integral_Equations_4_2012_JCAM_Neutro.tex
\documentclass[a4paper]{article}
\usepackage{pdfsync}
\usepackage[latin1]{inputenc}
%\usepackage[spanish]{babel}
\usepackage{graphicx}
\usepackage{amssymb,amsfonts,amsmath}
\usepackage{theorem}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{multirow}
\usepackage{verbatim}

\usepackage{color}

\addtolength{\voffset}{-1cm} \addtolength{\textheight}{1.5cm}
\addtolength{\hoffset}{-1cm} \addtolength{\textwidth}{2cm}

\let\realgeq\geq
\let\realleq\leq
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
{\theorembodyfont{\rmfamily}
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\newline \hspace*{\textwidth}\hspace*{-0,4cm} \rule{0.5em}{0.5em} \vspace{0,2cm}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\begin{document}

\title{\textbf{Blow-up collocation solutions of some Volterra integral equations}}

\author{R. Ben\'{\i}tez$^{1,*}$, V. J. Bol\'os$^{2,*}$ \\
\\
{\small $^1$ Dpto. Matem\'aticas, Centro Universitario de Plasencia, Universidad de Extremadura.}\\
{\small Avda. Virgen del Puerto 2, 10600 Plasencia, Spain.}\\
{\small e-mail\textup{: \texttt{rbenitez@unex.es}}} \\
\\
{\small $^2$ Dpto. Matem\'aticas para la Econom\'{\i}a y la Empresa, Facultad de Econom\'{\i}a,}\\
{\small Universidad de Valencia. Avda. Tarongers s/n, 46022 Valencia, Spain.}\\
{\small e-mail\textup{: \texttt{vicente.bolos@uv.es}}} \\
}
\footnotetext{$^*$Work supported by project MTM2008-05460, Spain.}
\date{May 2011}
\maketitle


\begin{abstract}
  In this paper we use collocation methods for detecting blow-up
  solutions of nonlinear homogeneous Volterra-Hammerstein integral
  equations. To do this, first we generalize the results on existence
  and uniqueness given in a previous paper in order to consider also
  non locally bounded kernels, which are present in many cases of
  blow-up problems. Next, we introduce the concept of
  ``blow-up collocation solution'' and analyze numerically
  some blow-up time estimates using collocation methods in
  particular examples where the previous results about existence and
  uniqueness can be applied. Finally, we discuss the relationships
  between necessary conditions for blow-up of collocation solutions
  and exact solutions.
\end{abstract}

\section{Introduction}

Some engineering and industrial problems are described by explosive
phenomena which are modeled by nonlinear integral equations whose
solutions exhibit blow-up at finite time (see \cite{Kirk09,CalCapo09}
and references therein).  Many authors have studied necessary and
sufficient conditions for the existence of such blow-up time. In
particular, in \cite{Myd94,Myd99,Okra07,Okra08,Okra10} equation
\begin{equation}
\label{eq0}
y\left( t\right) =\int _0 ^t K\left( t,s\right) G\left( y\left( s\right) \right) \,
\mathrm{d}s , \qquad t\in I:=\left[ 0,T\right] .
\end{equation}
was considered. In these works, conditions for the existence of a
finite blow-up time, as well as upper and lower estimates of it, were
given, although, in some cases, they were not very accurate.

A way of improving these estimations is to study numerical approximations
of the solution; in this aspect, collocation methods have
proven to be a very suitable technique for approximating nonlinear
integral equations, because of its stability and accuracy (see
\cite{Brun04}). Hence, the aim of this paper is to check the usefulness of collocation
methods for detecting blow-up solutions of the nonlinear HVHIE
(homogeneous Volterra-Hammerstein integral equation) given by equation (\ref{eq0}).

In a previous work \cite{BB11}, we used collocation methods for
approximating the nontrivial solutions of (\ref{eq0}). In
particular we studied the collocation solutions of the implicitly
linear HVIE (homogeneous Volterra integral equation)

\begin{equation}
\label{eq2}
z(t) =G\left( \left( \mathcal{V}z\right)
  \left( t\right) \right) =G\left( \int _0 ^t K\left( t,s\right)
  z(s) \, \mathrm{d}s \right) , \qquad t\in I,
\end{equation}
being $\mathcal{V}$ the linear \textit{Volterra operator}, and
assuming that some \textit{general conditions} were held. In
particular, it was required that the kernel $K$ was locally bounded;
nevertheless, non locally bounded kernels appear in many cases of
blow-up solutions (see \cite{Okra10}) and so, in the present work, we
shall first replace this condition with another that does not exclude
this kind of kernels. Hence, the new \textit{general conditions} that
we are going to impose (even if it is not explicitly mentioned) are:
\begin{itemize}
\item\textbf{Over $K$.} The \textit{kernel} $K:\mathbb{R}^2\to \left[
    0,+\infty \right[ $ has its support in $\left\{ \left( t,s\right)
    \in \mathbb{R}^2\, :\, 0\leq s\leq t\right\} $.

  For every $t>0$, the map $s\mapsto K\left( t,s\right) $ is locally
  integrable, and $\mathcal{K}(t):=\int _0 ^t K\left( t,s\right)
  \mathrm{d}s$ is a strictly increasing function.
% (i.e. $K\left( t,s\right) >0$ with $0<s<t$, almost everywhere).

  \textbf{New
    condition:} %$\mathcal{K}(t)$ continuous (for $t>0$) and $\lim
                %_{t\rightarrow 0^+}\mathcal{K}(t)=0$.
  $\lim _{t\rightarrow a^+} \int _a^t K\left( t,s\right) \,
  \mathrm{d}s=0$ for all $a\geq 0$.

  Note that for convolution kernels (i.e.  $K(t,s)=k(t-s)$) this new condition
  always holds, because obviously $k$ is locally integrable and then
  \[
  \lim _{t\rightarrow a^+}\int _a^t k(t-s)\, \mathrm{d}s = \lim
  _{t\rightarrow a^+}\int _0^{t-a} k(s)\, \mathrm{d}s = \lim
  _{t\rightarrow 0^+}\int _0^t k(s)\, \mathrm{d}s=0.
  \]
\item \textbf{Over $G$.} The \textit{nonlinearity} $G:\left[ 0,+\infty
  \right[ \to \left[ 0,+\infty \right[ $ is a continuous, strictly
  increasing function, and $G\left( 0\right) =0$.
\end{itemize}

There is a one-to-one correspondence between solutions of (\ref{eq0})
and (\ref{eq2}) (see \cite{Brun04,Kras84}). Particularly, if $z$
is a solution of (\ref{eq2}), then $y:=\mathcal{V}z$ is a solution of
(\ref{eq0}). Moreover, $y=G^{-1}\circ z$, since $G$ is injective.

The paper is structured as follows: in Section \ref{sec2} we introduce
briefly the basic definitions and state the notation. Next, we devote
Section \ref{sec3} to generalize the results about existence and
uniqueness of nontrivial collocation solutions obtained in \cite{BB11}
to the new \textit{general conditions}. In Section \ref{sec4}, we
introduce the concept of ``blow-up collocation solution'' and
analyse numerically some blow-up time estimates using collocation
methods in particular examples where the previous results about
existence and uniqueness can be applied. Finally, in Section
\ref{sec5}, we discuss the relationships between necessary conditions
for blow-up of collocation solutions and exact solutions.

\section{Collocation problems for implicitly linear HVIEs}
\label{sec2}

Following the notation of \cite{Brun04}, a \textit{collocation problem}
for equation (\ref{eq2}) is given by a mesh $I_h:=\left\{ t_n\, :\, \,
  0=t_0<t_1<\ldots <t_N=T\right\} $ and a set of $m$
\textit{collocation parameters} $0\leq c_1<\ldots <c_m\leq 1$.  We
denote $h_n:=t_{n+1}-t_n$ $(n=0,\ldots ,N-1)$ and the quantity
$h:=\mathrm{max}\left\{ h_n\, :\, \, 0\leq n\leq N-1\right\} $ is
called the \textit{stepsize}. Moreover, the \textit{collocation
  points} are given by $t_{n,i} :=t_n+c_ih_n$ $(i=1,\ldots ,m)$, and
the set of collocation points is denoted by $X_h$ (see
\cite{Brun04,Brun92}). A \textit{collocation solution} $z_h$ is then
given by the \textit{collocation equation}
%\begin{equation}
%\label{eqcolloc}
\[
z_h\left( t\right) = G\left( \int _0 ^t K\left( t,s\right) z_h\left( s\right)
  \, \mathrm{d}s \right) , \qquad t\in X_h\, ,
\]
%\end{equation}
where $z_h$ is in the space of piecewise polynomials of degree less
than $m$.

\begin{remark}
\label{remnlHVIE}
From now on, a ``collocation problem'' or a ``collocation solution''
will be always referred to the implicitly linear HVIE (\ref{eq2}). So,
if we want to obtain an estimation of a solution of the nonlinear
HVHIE (\ref{eq0}), then we have to consider $y_h:=\mathcal{V}z_h$.
\end{remark}

\begin{remark}
  At the beginning of Section \ref{sec4}, we define the concepts
  of ``blow-up collocation problems'' and ``blow-up collocation
  solutions''.  To do this, we extend the concept of ``collocation
  solution'' to meshes $I_h$ with infinite points. Taking this into
  account, if $z_h$ is a blow-up collocation solution, then $y_h$ also
  blows up at the same blow-up time.
\end{remark}

As it is stated in \cite{Brun04}, a collocation solution $z_h$ is
completely determined by the coefficients $Z_{n,i}:=z_h\left(
  t_{n,i}\right) $ $(n=0,\ldots ,N-1)$ $(i=1,\ldots ,m)$, since
$z_h\left( t_n+vh_n\right) =\sum _{j=1}^m L_j\left( v\right) Z_{n,j}$
for all $v\in \left] 0,1\right] $, where $L_1\left( v\right) :=1$,
$L_j\left( v\right) :=\prod _{k\neq j}^m \frac{v-c_k}{c_j-c_k}$
$(j=2,\ldots ,m)$ are the Lagrange fundamental polynomials with
respect to the collocation parameters.  The values of $Z_{n,i}$ are
given by the system
\begin{equation}
\label{eqznigeneral}
Z_{n,i}=G\left( F_n\left( t_{n,i}\right) +h_n\sum _{j=1}^m
B_n\left( i,j\right) Z_{n,j}\right) ,
\end{equation}
where
%\begin{equation*}
%\label{eqbnijgeneral}
\[
B_n\left( i,j\right) := \int _0^{c_i}K\left( t_{n,i},t_n+sh_n\right)
L_j\left( s\right) \, \mathrm{d}s
\]
%\end{equation*}
and
\begin{equation}
\label{eqlag0}
F_n\left( t\right) :=\int _0^{t_n} K\left( t,s\right)
z_h\left( s\right) \, \mathrm{d}s.
\end{equation}
The term $F_n\left( t_{n,i}\right) $ is called the \textit{lag term}.

\section{Existence and uniqueness of nontrivial collocation solutions}
\label{sec3}

Given a kernel $K$, a nonlinearity $G$ and some collocation parameters
$\left\{ c_1,\ldots ,c_m\right\} $, in \cite{BB11} there were defined
three kinds of existence of nontrivial collocation solutions (of the
corresponding collocation problem) in an interval $I=\left[ 0,T\right]
$ using a mesh $I_h$: \textbf{existence near zero} (collocation
solutions can always be extended a bit more, but it is not ensured the
existence of nontrivial collocation solutions for arbitrarily large
$T$), \textbf{existence for fine meshes} (there exists a fine enough mesh
for which there is a nontrivial collocation solution), and \textbf{unconditional
existence} (given any mesh, there always exists a nontrivial collocation solution). For these two last kinds of existence, it is ensured the
existence of nontrivial collocation solutions in any interval, and
hence, there is no blow-up.

In \cite{BB11}, two particular cases of collocation problems were considered:
\begin{itemize}
\item Case 1: $m=1$ with $c_1>0$.
\item Case 2: $m=2$ with $c_1=0$.
\end{itemize}
In these cases, the system (\ref{eqznigeneral}) is reduced to a single
nonlinear equation, whose solution is given by the fixed points of
$G\left( \alpha +\beta y\right) $ for some $\alpha, \beta $.

The proofs of the results obtained in \cite{BB11} rely on the locally
boundedness of the kernel $K$. Therefore, we will revise them,
imposing the new \textit{general conditions} and avoiding other
hypotheses that turn out to be unnecessary. For example, we extend the
results to decreasing convolution kernels, among others.

\newpage
As in the previous work \cite{BB11}, we will need a lemma on
nonlinearities $G$ satisfying the \textit{general conditions}:
\begin{lemma}
  \label{lema1} The following statements are equivalent to the
  statement that $\frac{G\left( y\right) }{y}$ is unbounded (in
  $\left] 0,+\infty \right[ $):
\begin{itemize}
\item[(i)] There exists $\beta _0>0$ such that $G\left( \beta y\right)
  $ has nonzero fixed points for all $0<\beta \leq \beta _0$.
\item[(ii)] Given $A\geq 0$, there exists $\beta _A>0$ such that
  $G\left( \alpha +\beta y\right) $ has nonzero fixed points for all
  $0\leq \alpha \leq A$ and for all $0<\beta \leq \beta _A$.
\end{itemize}
\end{lemma}

\subsection{Case 1: $m=1$ with $c_1>0$}

First, we shall consider $m=1$ with $c_1>0$.  Equations
(\ref{eqznigeneral}) are then reduced to
\begin{equation}
\label{eqzn11}
Z_{n,1}=G\left( F_n\left( t_{n,1}\right) +h_n B_n Z_{n,1}\right)
\qquad (n=0,\ldots ,N-1),
\end{equation}
where
\begin{equation}
\label{eqbnc1}
B_n:=B_n\left( 1,1\right) = \int _0^{c_1}K\left( t_{n,1},t_n+sh_n\right) \, \mathrm{d}s
\end{equation}
and the lag terms $F_n\left( t_{n,1}\right) $ are given by
(\ref{eqlag0}) with $i=1$. Note that $B_n>0$ because the integrand in
(\ref{eqbnc1}) is strictly positive almost everywhere in $\left]
  0,c_1\right[ $.

\begin{remark}
\label{notahnbn}
In \cite{BB11}, it is ensured that $\lim _{h_n \rightarrow
  0^+}h_nB_n=0$ taking into account (\ref{eqbnc1}) and the old
\textit{general conditions} imposed over $K$. Thus, we must ensure
that the limit also vanishes considering the new \textit{general
  conditions}.  In fact, with a change of variable, we can express
(\ref{eqbnc1}) as
\begin{equation}
\label{newbn}
B_n=\frac{1}{h_n}\int _{t_n}^{t_{n,1}}K\left( t_{n,1},s\right) \, \mathrm{d}s.
\end{equation}
Hence, by (\ref{newbn}) and the new condition, we have
\[
\lim _{h_n \rightarrow 0^+}h_nB_n=\lim _{t\rightarrow t_n^+}\int
_{t_n}^t K\left( t,s\right) \, \mathrm{d}s=0,
\]
since $t_{n,1}=t_n+c_1h_n$.
\end{remark}

Now, we are in position to give a characterization of the existence
near zero of nontrivial collocation solutions:
\begin{proposition}
\label{prop1}
There is existence near zero
if and only if $\frac{G\left( y\right) }{y}$ is unbounded.
\end{proposition}
\begin{proof}

  ($\Leftarrow $) Let us prove that if $\frac{G\left( y\right) }{y}$
  is unbounded, then there is existence near zero. So, we are going to
  prove by induction over $n$ that there exist $H_n>0$ $\left(
    n=0,\ldots ,N-1\right) $ such that if $0<h_n\leq H_n$ then there
  exist solutions of the system (\ref{eqzn11}) with $Z_{0,1}>0$:

\begin{itemize}

\item For $n=0$, taking into account Remark \ref{notahnbn} and Lemma
  \ref{lema1}-\textit{(i)} (see below), we choose a small enough
  $H_0>0$ such that $0<h_0B_0\leq \beta _0$ for all $0<h_0\leq
  H_0$. So, since the lag term is $0$, we can apply Lemma
  \ref{lema1}-\textit{(i)} to the equation (\ref{eqzn11}), concluding
  that there exist strictly positive solutions for $Z_{0,1}$.

\item Let us suppose that, choosing one of those $Z_{0,1}$ and given
  $n>0$, there exist $H_1,\ldots ,$ $H_{n-1}>0$ such that if
  $0<h_i\leq H_i$ ($i=1,\ldots ,n-1$) then there exist coefficients
  $Z_{1,1},\ldots ,Z_{n-1,1}$ fulfilling the equation
  (\ref{eqzn11}). Note that these coefficients are strictly positive,
  and hence, it is guaranteed that the corresponding collocation
  solution $z_h$ is (strictly) positive in $\left] 0,t_n\right] $.

\item Finally, we are going to prove that there exists $H_n>0$ such
  that if $0<h_n\leq H_n$ then there exists $Z_{n,1}>0$ fulfilling the
  equation (\ref{eqzn11}) with the previous coefficients
  $Z_{0,1},\ldots Z_{n-1,1}$:

  Let us define
  \[
  A:=\max \left\{ F_n\left( t_n+c_1\epsilon \right) \, :\, 0\leq
    \epsilon \leq 1\right\} .
  \]
  Note that $A$ exists because $s\mapsto K\left( t,s\right) $ is
  locally integrable for all $t>0$, and $z_h$ is bounded in $\left[
    0,t_n\right] $; hence, $F_n$ (see (\ref{eqlag0})) is bounded in
  $\left[ t_n,t_n+c_1\right] $.  Moreover, $A\geq 0$ because $K$ and
  $z_h$ are positive functions.
  % In fact, $A>0$ because $K\left( t,s\right) >0$ with $0<s<t$ almost
  % everywhere, and $z_h$ is strictly positive in $\left] 0,t_n\right[
  % $.

  So, applying Lemma \ref{lema1}-\textit{(ii)} (see below), there
  exists $\beta _A>0$ such that $G\left( \alpha +\beta y\right) $ has
  nonzero (strictly positive) fixed points for all $0\leq \alpha \leq
  A$ and for all $0<\beta \leq \beta _A$. On one hand, taking into
  account Remark \ref{notahnbn}, we choose a small enough $0<H_n\leq
  1$ such that $0<h_nB_n\leq \beta _A$ for all $0<h_n\leq H_n$; on the
  other hand, choosing one of those $h_n$, we have $0\leq F\left(
    t_{n,1}\right) \leq A$. Hence, we obtain the existence of
  $Z_{n,1}$ as the strictly positive fixed point of $G\left( F\left(
      t_{n,1}\right) +h_nB_ny\right) $.

\end{itemize}

($\Rightarrow $) For proving the other condition, we use Lemma
\ref{lema1}-\textit{(i)}, taking into account Remark \ref{notahnbn}.
\end{proof}

\begin{remark}
  A similar result was proved in \cite{BB11} with the additional
  hypothesis ``$K\left( t,s\right) \leq K\left( t',s\right) $ for all
  $0\leq s\leq t<t'$'' (or ``$k$ is increasing'' for the particular
  case of convolution kernels $K(t,s)=k(t-s)$). We have shown that
  this hypothesis is not needed and the difference between the proofs
  is the choice of $A$.
\end{remark}


Taking into account Remark \ref{notahnbn}, we can adapt the results
given in \cite{BB11} about sufficient conditions on existence for fine
meshes and unconditional existence to our new  \textit{general
  conditions}. This will help us to identify collocation problems
without blow-up (see Proposition \ref{prop10} below). But
first we need to recall the following definition.

\begin{definition}
  We say that a property $\mathcal{P}$ holds \textit{near zero} if
  there exists $\epsilon >0$ such that $\mathcal{P}$ holds on $\left]
    0,\delta \right[ $ for all $0<\delta <\epsilon $.  On the other
  hand, we say that $\mathcal{P}$ holds \textit{away from zero} if
  there exists $\tau >0$ such that $\mathcal{P}$ holds on $\left]
    t,+\infty \right[ $ for all $t>\tau $.

  The concept ``away from zero'' was originally defined in a more
  restrictive way in \cite{BB11}; nevertheless both definitions are
  appropriate for our purposes.
\end{definition}

\begin{proposition}
\label{prop10}
Let $\frac{G(y)}{y}$ be unbounded near zero.

\begin{itemize}
\item If $K$ is a convolution kernel and $\frac{G(y)}{y}$ is bounded
  away from zero, then there is existence for fine meshes.
\item If $\frac{y}{G(y)}$ is unbounded away from zero, then there is
  unconditional existence.
\end{itemize}

If, in addition, $\frac{G(y)}{y}$ is a strictly decreasing function,
then there is at most one nontrivial collocation solution.
\end{proposition}

\begin{remark}
  The proof is analogous to the one given in \cite{BB11}. Note that
  the property ``$\frac{y}{G(y)}$ is unbounded away from zero''
  appears there as ``there exists a sequence $\left\{ y_n\right\}
  _{n=1}^{+\infty }$ of positive real numbers and divergent to
  $+\infty $ such that $\lim _{n\rightarrow +\infty} \frac{G\left(
      y_n\right) }{y_n}=0$'', but both are equivalent.
\end{remark}


\subsection{Case 2: $m=2$ with $c_1=0$}

Considering $m=2$ with $c_1=0$, we have to
solve the following equations:
\begin{eqnarray}
\label{eqzn12}
Z_{n,1}&=&G\left( F_n\left( t_{n,1}\right) \right) \\
\label{eqzn2}
Z_{n,2}&=&G\left( F_n\left( t_{n,2}\right) +h_nB_n\left( 2,1\right)
  Z_{n,1}+h_nB_n\left( 2,2\right) Z_{n,2}\right) ,
\end{eqnarray}
for $n=0,\ldots ,N-1$, where
\begin{equation}
\label{eqbn2j}
B_n\left( 2,j\right) = \int _0^{c_2}K\left( t_{n,2},t_n+sh_n\right)
L_j\left( s\right) \, \mathrm{d}s,\qquad (j=1,2)
\end{equation}
and $F_n\left( t_{n,i}\right)$ $(i=1,2)$ are given by
(\ref{eqlag0}). Note that $B_n\left( 2,j\right) >0$ for $j=1,2$,
because the integrand in (\ref{eqbn2j}) is strictly positive almost
everywhere in $\left] 0,c_2\right[ $.

\begin{remark}
\label{notahnbn2}
As in the first case, we have to ensure
that $\lim _{h_n\rightarrow 0^+}h_nB_n\left( 2,j\right) =0$ for
$j=1,2$, taking into account the new \textit{general
  conditions} (see Remark \ref{notahnbn}). Actually, since $0<L_j\left( s\right)<1$ for $s\in
\left] 0,c_2\right[$, we have
\begin{equation}
\label{hnbn2j}
0 \leq h_n B_n\left( 2,j\right) \leq h_n\int _0^{c_2}K\left( t_{n,2},t_n+sh_n\right) \,
\mathrm{d}s=\int _{t_n}^{t_{n,2}}K\left( t_{n,2},s \right) \, \mathrm{d}s.
\end{equation}
Hence, by (\ref{hnbn2j}) and the new condition, the following inequalities are fulfilled:
\[
0\leq \lim _{h_n \rightarrow 0^+}h_nB_n\left( 2,j\right) \leq \lim
_{t\rightarrow t_n^+}\int _{t_n}^t K\left( t,s\right) \,
\mathrm{d}s=0,
\]
\end{remark}

Analogously to the previous case, we present a characterization of the
existence near zero of nontrivial collocation solutions:

\begin{proposition}
\label{newprop4}
Let the map $t\mapsto K\left( t,s\right) $ be continuous in $\left]
  s,\epsilon \right[ $ for some $\epsilon >0$ and for all $0\leq
s<\epsilon $ (this hypothesis can be removed if $c_2=1$).  Then there
is existence near zero if and only if $\frac{G\left( y\right) }{y}$ is
unbounded.
\end{proposition}

\begin{proof}

  ($\Leftarrow $) Let us prove that if $\frac{G\left( y\right) }{y}$
  is unbounded, then there is existence near zero. So, we are going to
  prove by induction over $n$ that there exist $H_n>0$ $\left(
    n=0,\ldots ,N-1\right) $ such that if $0<h_n\leq H_n$ then there
  exist solutions of the system (\ref{eqzn2}) with $Z_{0,2}>0$:

\begin{itemize}

\item For $n=0$, taking into account Remark \ref{notahnbn2} and Lemma
  \ref{lema1}-\textit{(i)}, we choose a small enough $H_0>0$ such that
  $0<h_0B_0\left( 2,2\right) \leq \beta _0$ for all $0<h_0\leq
  H_0$. So, since the lag terms are $0$ and $Z_{0,1}=G\left( 0\right)
  =0$, we can apply Lemma \ref{lema1}-\textit{(i)} to the equation
  (\ref{eqzn2}), concluding that there exist strictly positive
  solutions for $Z_{0,2}$.

\item Given $n>0$, let us suppose that, choosing one of those $Z_{0,2}$, there exist $H_1,\ldots ,H_{n-1}>0$ such that if $0<h_i\leq
  H_i$ ($i=1,\ldots ,n-1$) then there exist coefficients
  $Z_{1,2},\ldots ,Z_{n-1,2}$ fulfilling the equation
  (\ref{eqzn2}). Moreover, let us suppose that $z_h$ is positive in
  $\left[ 0,t_n\right] $, i.e. these coefficients satisfy $Z_{l,2}\geq
  \left( 1-c_2\right) Z_{l,1}\geq 0$ for $l=1,\ldots ,n-1$, where
  $Z_{l,1}$ is given by (\ref{eqzn12}).

\item Finally, we are going to prove that there exists $H_n>0$ such
  that if $0<h_n\leq H_n$ then there exists $Z_{n,2}>0$ fulfilling the
  equation (\ref{eqzn2}) with the previous coefficients, and $z_h$ is
  positive in $\left[ 0,t_{n+1}\right] $, i.e.  $Z_{n,2}\geq \left(
    1-c_2\right) Z_{n,1}\geq 0$:

    Let us define
    \begin{eqnarray}
    \label{coefa2}
    A &:=& \max \left\{ F_n\left( t_n+c_2\epsilon \right) +\right. \\
    \nonumber
    & & \left. \epsilon  G\left( F_n\left( t_n+c_1\epsilon \right) \right) \int _0^{c_2}K\left( t_{n,2},t_n+sh_n\right)
      L_1\left( s\right) \, \mathrm{d}s \,
      :\, 0\leq \epsilon \leq 1\right\} .
    \end{eqnarray}
    Note that $A$ exists and $A\geq 0$ because $s\mapsto K\left(
      t,s\right) $ is locally integrable for all $t>0$, $z_h$ is
    bounded and positive in $\left[ 0,t_n\right] $, the nonlinearity
    $G$ is bounded and positive, and the polynomial $L_1$ is bounded
    and positive in $\left[ 0,c_2\right] $.
    % In fact, $A>0$ because $K\left( t,s\right) >0$ with $0<s<t$
    % almost everywhere, and the rest of the functions are in fact
    % strictly positive...

    So, applying Lemma \ref{lema1}-\textit{(ii)}, there exists $\beta
    _A>0$ such that $G\left( \alpha +\beta y\right) $ has nonzero
    (strictly positive) fixed points for all $0\leq \alpha \leq A$ and
    for all $0<\beta \leq \beta _A$. On one hand, taking into account
    Remark \ref{notahnbn2}, we choose a small enough $0<H_n\leq 1$
    such that $0<h_nB_n(2,2)\leq \beta _A$ for all $0<h_n\leq H_n$; on
    the other hand, taking into account (\ref{eqlag0}), the lag terms
    $F_n\left( t_{n,i}\right) $ are positive for $i=1,2$, because $K$
    and $z_h$ are positive. Therefore, by (\ref{eqzn12}),
    $Z_{n,1}=G\left( F_n\left( t_{n,1}\right) \right) $ is positive,
    because $G$ is positive.  Moreover, $h_nB_n\left( 2,1\right) $ is
    positive. So,
    \[
    0\leq F_n\left( t_{n,2}\right) +h_nB_n\left( 2,1\right) Z_{n,1}\leq A.
    \]
    Hence, we obtain the existence of $Z_{n,2}$ as the strictly
    positive fixed point of the function $G\left( \alpha +\beta y\right) $ where
    $\alpha :=F_n\left( t_{n,2}\right) +h_nB_n\left( 2,1\right)
    Z_{n,1}$ and $\beta :=h_nB_n(2,2)$.

    Concluding, we have to check that $Z_{n,2}\geq \left( 1-c_2\right)
    Z_{n,1}$. On one hand,
    \[
    Z_{n,2}=G\left( F_n\left( t_{n,2}\right) +h_n\sum_{j=1}^2B_n\left(
        2,j\right) Z_{n,j}\right) \geq G\left( F_n\left(
        t_{n,2}\right) \right)
    \]
    because $B_n\left( 2,j\right) Z_{n,j}\geq 0$ for $j=1,2$.  On the
    other hand, since $t\mapsto K\left( t,s\right) $ is continuous in
    $\left] s,\epsilon \right[ $ for some $\epsilon >0$ and for all
    $0\leq s<\epsilon $, we can suppose that the stepsize is small
    enough for $t_{n,1}<\epsilon $, and then $F_n(t)$ is continuous in
    $t_{n,1}$ (see \ref{eqlag0}). Hence, since $G$ is continuous,
    $G\left( F_n(t)\right) $ is also continuous in $t_{n,1}$,
    i.e. $\lim _{h_n\rightarrow 0^+}G\left( F_n\left( t_{n,2}\right)
    \right) = G\left( F_n\left( t_{n,1}\right) \right) $. Therefore,
    choosing a small enough $h_n$, we have
    \[
    G\left( F_n\left( t_{n,2}\right) \right) \geq \left( 1-c_2\right)
    G\left( F_n\left( t_{n,1}\right) \right) =\left( 1-c_2\right)
    Z_{n,1}.
    \]

\end{itemize}

($\Rightarrow $) For proving the other condition, we use Lemma
\ref{lema1}-\textit{(i)}, taking into account Remark \ref{notahnbn2}.
\end{proof}


\begin{remark}
  A similar result was proved in \cite{BB11}, but the hypothesis on
  the kernel was ``$K\left( t,s\right) \leq K\left( t',s\right) $ for
  all $0\leq s\leq t<t'$'', and so, unlike Proposition \ref{newprop4},
  it could not be applied to decreasing convolution kernels (for
  example). Again, the main difference between both proofs lies in the
  choice of $A$.
\end{remark}

Note that for convolution kernels $K(t,s)=k(t-s)$, the hypothesis on
$K$ is equivalent to say that $k$ is continuous near zero, which is a
fairly weak hypothesis. On the other hand, for general kernels, this
hypothesis is only needed for ensuring that the mapping $t\mapsto K\left( t,s\right)
$ is continuous in $t_{n,1}$. Therefore, if it is not continuous, it
is important to choose a step $h_i$ such that $t\mapsto K\left(
  t,s\right)$ is continuous at the collocation points $t_{i,1}$
$(i=0,\ldots N-1)$.  More specifically, the hypothesis on $K$ is only
needed for checking that $Z_{n,2}\geq \left( 1-c_2\right) Z_{n,1}$, and
hence, for ensuring that $z_h$ is positive in $\left[ 0,t_n\right]
$. Nevertheless, this hypothesis is not needed for verifying that
$Z_{0,2}>Z_{0,1}=0$, and so, it is always ensured the existence of
$Z_{1,2}$ (and obviously $Z_{1,1}$), even if the hypothesis does not
hold.  Hence, if we use another method for checking that $Z_{1,2}\geq
\left( 1-c_2\right) Z_{1,1}$ (for example, numerically), then it is ensured
the existence of $Z_{2,2}$. Repeating this reasoning we can guarantee
the existence of $Z_{n,2}$, removing the hypothesis on $K$.

Following this idea,  in the next proposition we
state a result analogous to Proposition \ref{newprop4} without
hypothesis on the kernel.

\begin{proposition}
\label{prop4b}
$\frac{G\left( y\right) }{y}$ is unbounded if and only if there exists
$H_0>0$ such that there are nontrivial collocation solutions in
$\left[ 0,t_1\right] $ for $0<h_0\leq H_0$.  In this case, there
always exists $H_1>0$ such that there are nontrivial collocation
solutions in $\left[ 0,t_2\right] $ for $0<h_1\leq H_1$.

Moreover, if $\frac{G\left( y\right) }{y}$ is unbounded and there is a
\textbf{positive} nontrivial collocation solution in $\left[
  0,t_n\right] $, then there exists $H_n>0$ such that there are
nontrivial collocation solutions in $\left[ 0,t_{n+1}\right] $ for
$0<h_n\leq H_n$.
\end{proposition}

Considering Remark \ref{notahnbn2} and taking the coefficient $A$
given in (\ref{coefa2}), we can adapt some results of \cite{BB11}
about sufficient conditions on existence for fine meshes and
unconditional existence to our new \textit{general conditions}. These
results are useful for identifying collocation problems without
blow-up.

\begin{proposition}
\label{prop11}
Let $\frac{G(y)}{y}$ be unbounded near zero, and let the map $t\mapsto
K\left( t,s\right) $ be continuous in $\left] s,\epsilon \right[ $ for
some $\epsilon >0$ and for all $0\leq s<\epsilon $ (this hypothesis
can be removed if $c_2=1$).

\begin{itemize}
\item If $K$ is a convolution kernel and $\frac{G(y)}{y}$ is bounded
  away from zero, then there is existence for fine meshes.
\item If $\frac{y}{G(y)}$ is unbounded away from zero, then there is
  unconditional existence.
\end{itemize}

If, in addition, $\frac{G(y)}{y}$ is a strictly decreasing function,
then there is at most one nontrivial collocation solution.
\end{proposition}

Moreover, as in Proposition \ref{prop4b}, we can state a result
about unconditional existence without hypothesis on the kernel:

\begin{proposition}
\label{prop6}
Let $\frac{G\left( y\right) }{y}$ be
unbounded near zero.

If $\frac{y}{G(y)}$ is unbounded away from zero and there is a
\textbf{positive} nontrivial collocation solution in $\left[
  0,t_n\right] $, then there is a nontrivial collocation solution in
$\left[ 0,t_{n+1}\right] $ \textbf{for any} $h_n>0$.

If, in addition, $\frac{G\left( y\right) }{y}$ is a strictly
decreasing function, then there is at most one nontrivial collocation
solution.
\end{proposition}

\subsection{Nondivergent existence and uniqueness}

Our interest is the study of existence of nontrivial collocation
solutions using meshes $I_h$ with arbitrarily small $h>0$. So, we are
not interested in collocation problems whose collocation solutions
``escape'' to $+\infty $ when a certain $h_n\rightarrow 0^+$, since
this is a divergence symptom.  Following this criterion, we define the
concept of ``nondivergent existence'':

Let $0=t_0<\ldots <t_n$ be a mesh such that there exist nontrivial
collocation solutions, and let $S_{h_n}$ be the index set of the
nontrivial collocation solutions of the corresponding collocation
problem with mesh $t_0<\ldots <t_n<t_n+h_n$. Given $s\in S_{h_n}$, we
denote by $Z_{s;n,i}$ the coefficients of the corresponding nontrivial
collocation solution verifying equations (\ref{eqznigeneral}). Then,
we say that there is \textit{nondivergent existence in $t_n^+$} if
\[
\inf _{s\in S_{h_n}} \left\{ \max _{i=1,\ldots
    ,m}\left\{ Z_{s;n,i}\right\} \right\}
\]
exists for small enough $h_n>0$ and it does not diverge to $+\infty $
when $h_n\rightarrow 0^+$.  Given a mesh $I_h= \left\{
  0=t_0<\ldots<t_N\right\} $ such that there exist nontrivial
collocation solutions, we say that there is \textit{nondivergent
  existence} if there is nondivergent existence in $t_n^+$ for
$n=0,\ldots, N$.  See \cite{BB11} for a more detailed analysis, where
we also define the concept of \textit{nondivergent uniqueness}.

In \cite{BB11} it is proved the next result.

\begin{proposition}
\label{prop7}
In cases 1 and 2 with existence of nontrivial collocation solutions,
there is nondivergent existence if and only if $\frac{G\left( y\right)
}{y}$ is unbounded near zero.

If, in addition, $G$ is ``well-behaved'', then there is nondivergent
uniqueness.
\end{proposition}

We say that $G$ is ``well-behaved'' if $\frac{G\left( \alpha +y\right)
}{y}$ is strictly decreasing near zero for all $\alpha >0$. Note that
this condition is very weak (see \cite{BB11}).

So, taking into account Propositions \ref{prop1}, \ref{newprop4} and
\ref{prop7}, the main result of this paper is:

\begin{theorem}
\label{teo1}
(Hypothesis only for case 2 with $c_2\neq 1$: the map $t\mapsto
K\left( t,s\right) $ is continuous in $\left] s,\epsilon \right[ $ for
some $\epsilon >0$ and for all $0\leq s<\epsilon $.)

There is nondivergent existence near zero if and only if
$\frac{G\left( y\right) }{y}$ is unbounded near zero.

If, in addition, $G$ is ``well-behaved'' then, there is nondivergent
uniqueness near zero.
\end{theorem}

In the same way, we can combine Propositions \ref{prop4b} and
\ref{prop7} obtaining a result about nondivergent existence and
uniqueness without hypothesis on the kernel, and, as Theorem
\ref{teo1}, it can be useful to study numerically problems with
decreasing convolution kernels in case 2 with $c_2\neq 1$.

We can also combine Propositions \ref{prop10}, \ref{prop11} and
\ref{prop6} with Proposition \ref{prop7}, obtaining results about
nondivergent existence and uniqueness (for fine meshes and
unconditional) that are useful for identifying collocation
problems without blow-up. Some of these results can be reformulated as
necessary conditions for the existence of blow-up (see Section
\ref{sec5}).

\section{Blow-up collocation solutions}
\label{sec4}

In this section we will extend the concept of collocation problem and
collocation solution in order to consider the case of ``blow-up collocation
solutions''.

\begin{definition}
  We say that a collocation problem is a \textit{blow-up collocation
    problem} (or has a blow-up) if the following conditions are held:
  \begin{itemize}
  \item there exists $T>0$ such that there is no collocation solution
    in $I=\left[ 0,T\right] $ for any mesh $I_h$, and
  \item given $M>0$ there exists $0<\tau<T$, and a
    collocation solution $z_h$ defined on $\left[ 0,\tau \right]$ such
    that $|z_h(t)|> M$ for some $t\in [0,\tau]$.
  \end{itemize}
\end{definition}
We can not speak about ``blow-up collocation solutions'' in the
classic sense, since ``collocation solutions'' are defined in compact
intervals and obviously they are bounded; so, we have to extend first
the concept of ``collocation solution'' to open intervals $I=\left[
  0,T\right[$ before we are in position to define the notion of ``blow-up
collocation solution''.

\begin{definition}
  Let $I:=[0,T[$ and $I_h$ be an infinite mesh given by a strictly
  increasing sequence $\left\{ t_n\right\} _{n=0}^{+\infty }$ with
  $t_0=0$ and convergent to $T$.
  \begin{itemize}
  \item A \textit{collocation solution on $I$} using the mesh $I_h$ is
    a function defined on $I$ such that it is a collocation solution
    (in the classic sense) for any finite submesh $\left\{
      t_n\right\}_{n=0}^N$ with $N\in \mathbb{N}$.
  \item A collocation solution on $I$ is a \textit{blow-up
      collocation solution} (or has a blow-up) with \textit{blow-up
      time} $T$ if it is not bounded.
  \end{itemize}
\end{definition}


Given a collocation problem with nondivergent uniqueness near zero, a
necessary condition for the nondivergent collocation solution to blow-up
is that there is neither existence for fine meshes nor unconditional
existence. So, for example, given a convolution kernel
$K(t,s)=k(t-s)$, in cases 1 and 2 we must require that $\frac{G\left(
    y\right) }{y}$ is unbounded away from zero (moreover, in case
2 with $c_2\neq 1$, we must demand that there exists $\epsilon >0$
such that $k$ is continuous in $\left] 0,\epsilon \right[ $).

For instance, in \cite{Okra07} it is studied equation (\ref{eq0}) with
convolution kernel $k(t-s)=(t-s)^{\beta }$, $\beta \geq 0$, and
nonlinearity $G(y)=t^{\alpha }$, $0<\alpha <1$, concluding that the
nontrivial (exact) solution does not have blow-up. If we consider a
collocation problem with the above kernel and nonlinearity in cases 1
and 2, then we can ensure unconditional nondivergent uniqueness (by
Propositions \ref{prop10}, \ref{prop11} and \ref{prop7}); thus, we can
also conclude that the nondivergent collocation solution has no
blow-up. Actually, we reach the same conclusion considering any kernel
satisfying the general conditions (in case 1 and case 2 with $c_2=1$)
and such that $t\mapsto K\left( t,s\right) $ is continuous in $\left]
  s,\epsilon \right[ $ for some $\epsilon >0$ and for all $0\leq
s<\epsilon $ (in case 2 with $c_2\neq 1$). In Section \ref{sec5} we
discuss in more detail the relationships between necessary conditions
for blow-up of collocation solutions and exact solutions.

\subsection{Numerical algorithms}
\label{sec:algoritmos}
First we need to recall the definition of existence near
zero, mentioned at the beginning of Section \ref{sec3}, and given in
\cite{BB11}.
\begin{definition}
  We say that there is \textbf{existence near zero} if there exists
  $H_0>0$ such that if $0<h_0\leq H_0$ then there are nontrivial
  collocation solutions in $\left[ 0,t_1\right] $; moreover, there
  exists $H_n>0$ such that if $0<h_n\leq H_n$ then there are
  nontrivial collocation solutions in $\left[ 0,t_{n+1}\right] $ (for
  $n=1,\ldots ,N-1$ and given $h_0,\ldots ,h_{n-1}>0$ such that there
  are nontrivial collocation solutions in $\left[ 0,t_n\right]
  $). Note that, in general, $H_n$ depends on $h_0,\ldots ,h_{n-1}$.
\end{definition}

Given a blow-up collocation problem with nondivergent uniqueness near
zero (always in cases 1 and 2), we are going to describe a general
algorithm to compute the nontrivial collocation solution and estimate
the blow-up time. In each iteration, the stepsize $h_n$ is adapted to
$H_n$, since only if $h_n \leq H_n$ we can ensure the existence of the
nontrivial collocation solution.


As long as $h_n\leq H_n$, the solution on $t_n$ for each $n$ is
obtained from the attracting fixed point of a certain function
$y\mapsto G(\alpha+\beta y)$, given by equation (\ref{eqzn11}) in case
1, or (\ref{eqzn2}) in case 2. Therefore, the key point in the
algorithm is to decide whether there is a fixed point or not and, if
so, calculate it.

In order to check that there is no fixed point, the most general
technique consists on iterate and, if a certain bound (which may
depend on the fixed points found in the previous steps) is overcomed,
then it is assumed that there is no fixed point.

So, if for a certain $n$ there is no fixed point, then a smaller $h_n$
should be taken (e.g.  $h_n/2$ in the examples below), and this
procedure is repeated with the new $\alpha$, $\beta $ corresponding to
the new $h_n$. If $h_n$ becomes smaller than a given tolerance
(e.g. $10^{-12}$ in our examples), the algorithm stops and
$t_{n+1}=h_0+\ldots +h_n$ is the estimation of the blow-up time.

Nevertheless, in particular cases, specific algorithms can also be
used, as in the case of Example 1, where the fixed points can be found
analytically:
\begin{itemize}
\item If $0\leq \alpha <1$:
  \begin{itemize}
  \item If $1-\alpha <\beta \leq \frac{1}{4\alpha }$ the fixed point
    is $\frac{1-2\alpha \beta -\sqrt{1-4\alpha \beta }}{2\beta ^2}$.
  \item If $\beta \leq 1-\alpha $ then the fixed point is
    $\frac{1}{2}\left( \beta +\sqrt{\beta ^2+4\alpha }\right) $.
  \end{itemize}
\item If $\alpha \geq 1$ and $\beta \leq \frac{1}{4\alpha }$, then the
  fixed point is $\frac{1-2\alpha \beta -\sqrt{1-4\alpha \beta
    }}{2\beta ^2}$.
\end{itemize}


If $\beta >\frac{1}{4\alpha }$ then there is no solution for the
corresponding $h_n$. Hence a smaller $h_n$ is taken as described in
the general case.

However, both the computational cost and accuracy of the
general and specific algorithms are quite similar.

\subsection{Examples}

We will consider the following examples:

\begin{enumerate}

\item $K\left( t,s\right) =1$ and $G\left( y\right) =
\begin{cases}
\sqrt{y}\, & \textrm{if } y\in \left[ 0,1\right] \\
y^2\, & \textrm{if } y\in \left] 1,+\infty \right[
\end{cases}
$

\item $K\left( t,s\right) =1$ and $G\left( y\right) =
\begin{cases}
\sqrt{y}\, & \textrm{if } y\in \left[ 0,1\right] \\
e^{y-1}\, & \textrm{if } y\in \left] 1,+\infty \right[
\end{cases}
$

\item $K\left( t,s\right) =k(t-s)=t-s$ and $G\left( y\right) $ of Example 1.

\item $K\left( t,s\right) =k(t-s)=\left(\pi \left( t-s\right) \right)^{-1/2}$ and $G\left( y\right) $ of Example 1.

\end{enumerate}

In the first two examples, the blow-up time of the corresponding
(exact) solution is $\hat{t}=3$, while in the other two the blow-up
time is unknown. In \cite{Okra10} it was studied a family of
equations to which Example 4 belongs.

In all the examples, both $K$ and $G$ fulfill the \textit{general
  conditions} and the hypotheses of Theorem \ref{teo1}, and thus it is
ensured the nondivergent existence and uniqueness near zero (note that
since the kernel in Example 4 is decreasing and unbounded near zero,
this case can not be studied in \cite{BB11}).
Moreover, $G(y)/y$ is unbounded away
from zero and $y/G(y)$ is bounded away from zero, and hence
there can exist a blow-up (see Propositions \ref{prop10} and
\ref{prop11}).

He have found the numerical nondivergent collocation solutions for a
given stepsize $h$ ($=h_0$), using the algorithms described in Section
\ref{sec:algoritmos} (the specific for the Example 1 and the general
for the rest). In Figures \ref{figblowup}, \ref{figblowup2},
\ref{figblowup3} and \ref{figblowup4} the estimations of the blow-up
times of the collocation solutions for a given stepsize $h$ are
depicted, varying $c_1$ in case 1 or $c_2$ in case 2. Note that the
different graphs for different $h$ intersect each other in a fairly
good approximation of the blow-up time, as it is shown in Tables
\ref{tabla1c1}, \ref{tabla1c2}, \ref{tabla2c1} and \ref{tabla2c2}.




In Examples 3 and 4 we do not know the exact value of the blow-up
time. However, in order to make a study of the relative error
analogous to the previous examples, we have taken as blow-up time for
Example 3 the approximation for $h=0.001$ in case 2 with $c_2=0.5$:
$t=5.78482$; and for Example 4 the
approximation for $h = 0.0001$ in case 2 with $c_2=2/3$ (\textit{Radau
  I} collocation points): $t=1.645842$. Results are shown in Tables
\ref{tabla3c1}, \ref{tabla3c2}, \ref{tabla4c1} and \ref{tabla4c2}.


The relative error varying $c_1$ (case 1) or $c_2$ (case 2) is the
``relative vertical size'' of the graph, and it decreases at the
same rate as $h$. On the other hand, the relative error of the
intersection decreases faster, sometimes at the same rate as
$h^2$.
Moreover, in case 1, the best approximations are reached with
$c_1\approx 0.5$, and in case 2 with $c_2\approx 2/3$ (approximately
\textit{Radau I}) for Examples 1, 2 and 4 (see Tables
\ref{tabla1c105radau}, \ref{tabla2c105radau} and
\ref{tabla4c105radau}), while for Example 3 the best approximations
are reached with $c_2\approx 0.5$; however, in Table
\ref{tabla3c105radau} are also shown the approximations and their
corresponding errors for the \textit{Radau I} collocation points. On
the other hand, the intersections method offers better results, but
at a greater computational cost.



\section{Discussion and comments}
\label{sec5}

The main necessary conditions for a collocation problem to have a
blow-up are obtained from Propositions \ref{prop10} (case 1) and
\ref{prop11} (case 2), and are mostly related to the nonlinearity,
since the assumption on the kernel ``the map $t\mapsto K\left(
  t,s\right) $ is continuous in $\left] s,\epsilon \right[ $ for some
$\epsilon >0$ and for all $0\leq s<\epsilon $'' is only required in
case 2 with $c_2\neq 1$ and it is a very weak hypothesis. Hence,
assuming that the kernel satisfies this hypothesis, the main necessary
condition for the existence of a blow-up is:
\begin{itemize}
\item[1. ] $\frac{y}{G(y)}$ is bounded away from zero.
\end{itemize}
\newpage
\noindent In addition, for convolution kernels, there is another necessary condition:
\begin{itemize}
\item[2. ] $\frac{G(y)}{y}$ is unbounded away from zero, i.e. there
  exists a sequence $\left\{ y_n\right\} _{n=1}^{+\infty }$ of
  positive real numbers and divergent to $+\infty $ such that $\lim
  _{n\rightarrow +\infty} \frac{y_n}{G\left( y_n\right) }=0$.
\end{itemize}



In \cite{Myd99} it is given a necessary and sufficient condition for
the existence of blow-up (exact) solutions for equation (\ref{eq0}) with a
kernel of the form $K(t,s)=(t-s)^{\alpha -1}r(s)$ with $\alpha >0$,
$r$ nondecreasing and continuous for $s\neq 0$, $r(s)=0$ for $s\leq
0$, and $r(s)>0$ for $s>0$:
\begin{equation}
\label{condnecsuf}
\int _{\delta }^{+\infty } \left( \frac{s}{G(s)}\right) ^{1/\alpha }\frac{\mathrm{d}s}{s}<+\infty ,\qquad \delta >0.
\end{equation}
This also holds for convolution kernels of Abel type
$K(t,s)=(t-s)^{\alpha -1}$ with $\alpha >0$, generalizing some results
given in \cite{Myd94}. Next we will show that necessary conditions 1
and 2 above mentioned are also necessary conditions for the integral
given in (\ref{condnecsuf}) to be convergent, and thus for the
existence of a blow-up (exact) solution.

\begin{proposition}
\label{propfin1}
If (\ref{condnecsuf}) holds, then $\frac{G(y)}{y}$ is unbounded away
from zero.
\end{proposition}
\begin{proof}
  Let us suppose that $\frac{G(y)}{y}$ is bounded away from zero. So,
  there exists $M>0$ such that $\frac{G(y)}{y}<M$ for all $y>\delta
  $. Hence
\[
\int _{\delta }^{+\infty } \left( \frac{s}{G(s)}\right) ^{1/\alpha
}\frac{\mathrm{d}s}{s} > \left( \frac{1}{M}\right) ^{1/\alpha }\int
_{\delta }^{+\infty } \frac{\mathrm{d}s}{s}=+\infty .
\]
\end{proof}

\begin{proposition}
  If (\ref{condnecsuf}) holds, then $\frac{y}{G(y)}$ is bounded away
  from zero.
\end{proposition}
\begin{proof}
  By Proposition \ref{propfin1}, $\frac{G(y)}{y}$ is unbounded away
  from zero and hence, there exists a strictly increasing sequence
  $\left\{ y_n\right\} _{n=1}^{+\infty }$ with $y_1>\delta $ and
  divergent to $+\infty $ such that $\frac{y_n}{G\left( y_n\right)
  }<\frac{1}{2^{\alpha }}$ for all $n$.

Let us suppose that $\frac{y}{G(y)}$ is unbounded away from zero; so,
we can choose $\left\{ y_n\right\} _{n=1}^{+\infty }$ such that there
exist $y'_n\in \left] y_n,y_{n+1}\right[ $ with $\frac{y'_n}{G\left(
    y'_n\right) }=1$ for each $n$. Moreover, since the nonlinearity $G$ is positive and
strictly increasing, we have that $\frac{y'_n}{G\left( y_n\right)
}>\frac{y'_n}{G\left( y'_n\right) }=1$, and then $y'_n>G\left(
  y_n\right) $. Hence, we have
\begin{eqnarray*}
  \int _{y_n}^{y_{n+1}} \left( \frac{s}{G(s)}\right) ^{1/\alpha }\frac{\mathrm{d}s}{s} &>& \int _{y_n}^{y'_n} \left( \frac{s}{G(s)}\right) ^{1/\alpha }\frac{\mathrm{d}s}{s} >\int _{y_n}^{y'_n} \left( \frac{s}{G\left( y'_n\right) }\right) ^{1/\alpha }\frac{\mathrm{d}s}{s} \\
  &=& \alpha \left( 1-\left( \frac{y_n}{y'_n}\right) ^{1/\alpha }\right) >\alpha \left( 1-\left( \frac{y_n}{G\left( y_n\right) }\right) ^{1/\alpha }\right) >\frac{\alpha }{2} .
\end{eqnarray*}
Therefore, (\ref{condnecsuf}) does not hold.
\end{proof}

An interesting problem which is not fully resolved is to continue
clarifying the relationship between the existence of blow-up in exact
solutions and in collocation solutions.

\begin{thebibliography}{00}
\bibitem{Kirk09} C. M. Kirk. Numerical and asymtotic analysis of a
  localized heat source undergoing periodic motion. Nonlinear
  Anal. \textbf{71} (2009), e2168-e2172.

\bibitem{CalCapo09} F. Calabr�, G. Capobianco, Blowing up behavior for
  a class of nonlinear VIEs connected with parabolic
  PDEs. J. Comput. Appl. Math. \textbf{228} (2009), 580-588.

\bibitem{Myd94} W. Mydlarczyk, A condition for finite blow-up time for
  a Volterra integral equation. J. Math. Anal. Appl. \textbf{181}
  (1994), no. 1, 248--253.

\bibitem{Myd99} W. Mydlarczyk, The blow-up solutions of integral
  equations. Colloq. Math. \textbf{79} (1999), no. 1, 147--156.

\bibitem{Okra07} T. Ma\l olepszy, W. Okrasi\'nski, Conditions for
  blow-up of solutions of some nonlinear Volterra integral
  equations. J. Comput. Appl. Math. \textbf{205} (2007), 744--750.

\bibitem{Okra08} T. Ma\l olepszy, W. Okrasi\'nski, Blow-up conditions
  for nonlinear Volterra integral equations with power
  nonlinearity. Appl. Math. Letters. \textbf{21} (2008), 307--312.

\bibitem{Okra10} T. Ma\l olepszy, W. Okrasi\'nski, Blow-up time for
  solutions for some nonlinear Volterra integral
  equations. J. Math. Anal. Appl. \textbf{366} (2010), no. 1,
  372--384.

\bibitem{Brun04} H. Brunner, Collocation Methods for Volterra Integral
  and Related Functional Differential Equations, Cambridge University
  Press, Cambridge, 2004.


\bibitem{BB11} R. Ben�tez, V. J. Bol�s, Existence and uniqueness of
  nontrivial collocation solutions of implicitly linear homogeneous
  Volterra integral equations. J. Comput. Appl. Math. \textbf{235}
  (2011), no. 12, 3661--3672.


\bibitem{Kras84} M. A. Krasnosel'skii, P. P. Zabreiko, Geometric
  Methods of Nonlinear Analysis, Springer Verlag, New York, 1984.


\bibitem{Brun92} H. Brunner, Implicitly linear collocation methods for
  nonlinear Volterra integral equations, Appl. Numer. Math. \textbf{9}
  (1992), 235--247.

\end{thebibliography}





\newpage


\appendix

\section{Figures and tables}

\vspace{3cm}


\begin{figure}[ht]
\begin{center}
  \includegraphics[width=1\textwidth]{blowup_all}
\end{center}
\caption{Example 1. Numerical estimation of the blow-up time of
  collocation solutions varying $c_1$ (case 1) or $c_2$ (case 2), for
  different stepsizes $h$. The blow-up time of the corresponding
  (exact) solution is $\hat{t}=3$.
  \vspace{4.5cm}}
\label{figblowup}
\end{figure}


\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
  \hline
  Case 1 & \multicolumn{3}{|c|}{Varying $c_1$} & \multicolumn{2}{|c|}{Intersection} \\
  \cline{2-6}
  ($m=1$, $c_1>0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
  \hline
  \multirow{2}{*}{$h=0.1$} & $2.60$ & $3.66$ & \multirow{2}{*}{$4\cdot 10^{-1}$} & & \multirow{4}{*}{$10^{-2}$} \\
  & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & $3.03$ & \\
  \cline{1-4}
  \multirow{2}{*}{$h=0.05$} & $2.78$ & $3.40$ & \multirow{2}{*}{$2\cdot 10^{-1}$} & \small{($c_1=0.39$)} & \\
  & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & & \\
  \hline
  \multirow{2}{*}{$h=0.01$} & $2.92$ & $3.11$ & \multirow{2}{*}{$6\cdot 10^{-2}$} & & \multirow{4}{*}{$7\cdot 10^{-4}$} \\
  & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & $3.002$ & \\
  \cline{1-4}
  \multirow{2}{*}{$h=0.005$} & $2.96$ & $3.06$ & \multirow{2}{*}{$3\cdot 10^{-2}$} & \small{($c_1=0.44$)} & \\
  & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & & \\
  \hline
\end{tabular}
\end{center}
\caption{Example 1. Numerical data of Figure \ref{figblowup} (case 1).}
\label{tabla1c1}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
  \hline
  Case 2 & \multicolumn{3}{|c|}{Varying $c_2$} & \multicolumn{2}{|c|}{Intersection} \\
  \cline{2-6}
  ($m=2$, $c_1=0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
  \hline
  \multirow{2}{*}{$h=0.1$} & $2.95$ & $3.20$ & \multirow{2}{*}{$8\cdot 10^{-2}$} & & \multirow{4}{*}{$10^{-4}$} \\
  & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & $3.0003$ & \\
  \cline{1-4}
  \multirow{2}{*}{$h=0.05$} & $2.98$ & $3.10$ & \multirow{2}{*}{$4\cdot 10^{-2}$} & \small{($c_2=0.6178$)} & \\
  & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & & \\
  \hline
  \multirow{2}{*}{$h=0.01$} & $2.995$ & $3.020$ & \multirow{2}{*}{$8\cdot 10^{-3}$} & & \multirow{4}{*}{$10^{-6}$} \\
  & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & $3.000003$ & \\
  \cline{1-4}
  \multirow{2}{*}{$h=0.005$} & $2.998$ & $3.010$ & \multirow{2}{*}{$4\cdot 10^{-3}$} & \small{($c_2=0.6232$)} & \\
  & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & & \\
  \hline
\end{tabular}
\end{center}
\caption{Example 1. Numerical data of Figure \ref{figblowup} (case 2).}
\label{tabla1c2}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
  \hline
   & \multicolumn{2}{|c|}{Case 1 - $c_1=0.5$} & \multicolumn{2}{|c|}{Case 2 - Radau I} \\
  \cline{2-5}
   & Blow-up & Rel. error & Blow-up & Rel. error \\
  \hline
  $h=0.1$ & $2.933883$ & $2.2\cdot 10^{-2}$ & $2.995253$ & $1.6\cdot 10^{-3}$ \\
  \hline
  $h=0.05$ & $2.965002$ & $1.2\cdot 10^{-2}$ & $2.997602$ & $8\cdot 10^{-4}$ \\
  \hline
  $h=0.01$ & $2.992885$ & $2.4q\cdot 10^{-3}$ & $2.999519$ & $1.6\cdot 10^{-4}$ \\
  \hline
  $h=0.005$ & $2.996434$ & $1.2\cdot 10^{-3}$ & $2.999759$ & $8\cdot 10^{-5}$ \\
  \hline
\end{tabular}
\end{center}
\caption{Example 1. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with $c_1=0.5$, and case 2 with \textit{Radau I} collocation points, $c_1=0$, $c_2=2/3$, for different stepsizes $h$.
\vspace{2cm}}
\label{tabla1c105radau}
\end{table}



\begin{figure}[h]
\begin{center}
  \includegraphics[width=1\textwidth]{blowup_all_ej2}
\end{center}
\caption{Example 2. Numerical estimation of the blow-up time of
  collocation solutions varying $c_1$ (case 1) or $c_2$ (case 2), for
  different stepsizes $h$. The blow-up time of the corresponding
  (exact) solution is $\hat{t}=3$.
  \vspace{5.5cm}}
\label{figblowup2}
\end{figure}



\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Case 1 & \multicolumn{3}{|c|}{Varying $c_1$} & \multicolumn{2}{|c|}{Intersection} \\
\cline{2-6}
($m=1$, $c_1>0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
\hline
\multirow{2}{*}{$h=0.1$} & $2.68$ & $3.50$ & \multirow{2}{*}{$3\cdot 10^{-1}$} & & \multirow{4}{*}{$5\cdot 10^{-3}$} \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & $3.015$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.05$} & $2.82$ & $3.30$ & \multirow{2}{*}{$1.5\cdot 10^{-1}$} & \small{($c_1=0.399$)} & \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & & \\
\hline
\multirow{2}{*}{$h=0.01$} & $2.94$ & $3.08$ & \multirow{2}{*}{$4.6\cdot 10^{-2}$} & & \multirow{4}{*}{$2.6\cdot 10^{-4}$} \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & $3.0008$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.005$} & $2.98$ & $3.04$ & \multirow{2}{*}{$2.3\cdot 10^{-2}$} & \small{($c_1=0.444$)} & \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & & \\
\hline
\end{tabular}
\end{center}
\caption{Example 2. Numerical data of Figure \ref{figblowup2} (case 1).}
\label{tabla2c1}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Case 2 & \multicolumn{3}{|c|}{Varying $c_2$} & \multicolumn{2}{|c|}{Intersection} \\
\cline{2-6}
($m=2$, $c_1=0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
\hline
\multirow{2}{*}{$h=0.1$} & $2.97$ & $3.10$ & \multirow{2}{*}{$4.3\cdot 10^{-2}$} & & \multirow{4}{*}{$4\cdot 10^{-5}$} \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & $3.00012$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.05$} & $2.99$ & $3.05$ & \multirow{2}{*}{$2\cdot 10^{-2}$} & \small{($c_2=0.6466$)} & \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & & \\
\hline
\multirow{2}{*}{$h=0.01$} & $2.997$ & $3.010$ & \multirow{2}{*}{$4.3\cdot 10^{-3}$} & & \multirow{4}{*}{$3.6\cdot 10^{-6}$} \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & $3.000011$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.005$} & $2.9985$ & $3.005$ & \multirow{2}{*}{$2.15\cdot 10^{-3}$} & \small{($c_2=0.65114$)} & \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & & \\
\hline
\end{tabular}
\end{center}
\caption{Example 2. Numerical data of Figure \ref{figblowup2} (case 2).}
\label{tabla2c2}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
  \hline
   & \multicolumn{2}{|c|}{Case 1 - $c_1=0.5$} & \multicolumn{2}{|c|}{Case 2 - Radau I} \\
  \cline{2-5}
   & Blow-up & Rel. error & Blow-up & Rel. error \\
  \hline
  $h=0.1$ & $2.941795$ & $2\cdot 10^{-2}$ & $2.999559$ & $1.5\cdot 10^{-4}$ \\
  \hline
  $h=0.05$ & $2.970341$ & $10^{-2}$ & $2.999778$ & $7.4\cdot 10^{-5}$ \\
  \hline
  $h=0.01$ & $2.993979$ & $2\cdot 10^{-3}$ & $2.999955$ & $1.5\cdot 10^{-5}$ \\
  \hline
  $h=0.005$ & $2.996983$ & $10^{-3}$ & $2.999978$ & $7.3\cdot 10^{-6}$ \\
  \hline
\end{tabular}
\end{center}
\caption{Example 2. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with $c_1=0.5$, and case 2 with \textit{Radau I} collocation points, $c_1=0$, $c_2=2/3$, for different stepsizes $h$.
\vspace{2cm}}
\label{tabla2c105radau}
\end{table}

\begin{figure}[h]
\begin{center}
  \includegraphics[width=1\textwidth]{blowup_all_ej3}
\end{center}
\caption{Example 3. Numerical estimation of the blow-up time of
  collocation solutions varying $c_1$ (case 1) or $c_2$ (case 2), for
  different stepsizes $h$.
  \vspace{5cm}}
\label{figblowup3}
\end{figure}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Case 1 & \multicolumn{3}{|c|}{Varying $c_1$} & \multicolumn{2}{|c|}{Intersection} \\
\cline{2-6}
($m=1$, $c_1>0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
\hline
\multirow{2}{*}{$h=0.1$} & $5.20$ & $6.55$ & \multirow{2}{*}{$2.3\cdot 10^{-1}$} & & \multirow{4}{*}{$3.6\cdot 10^{-3}$} \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & $5.8055$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.05$} & $5.45$ & $6.20$ & \multirow{2}{*}{$1.3\cdot 10^{-1}$} & \small{($c_1=0.3961$)} & \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & & \\
\hline
\multirow{2}{*}{$h=0.01$} & $5.71$ & $5.89$ & \multirow{2}{*}{$3.1\cdot 10^{-2}$} & & \multirow{4}{*}{$1.4\cdot 10^{-4}$} \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & $5.7856$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.005$} & $5.735$ & $5.840$ & \multirow{2}{*}{$1.8\cdot 10^{-2}$} & \small{($c_1=0.441$)} & \\
 & \small{($c_1=1$)} & \small{($c_1=0.01$)} & & & \\
\hline
\end{tabular}
\end{center}
\caption{Example 3. Numerical data of Figure \ref{figblowup3} (case 1). The blow-up time of the corresponding (exact) solution is assumed to be $\hat{t}\approx 5.78482$.}
\label{tabla3c1}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Case 2 & \multicolumn{3}{|c|}{Varying $c_2$} & \multicolumn{2}{|c|}{Intersection} \\
\cline{2-6}
($m=2$, $c_1=0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
\hline
\multirow{2}{*}{$h=0.1$} & $5.67$ & $6.10$ & \multirow{2}{*}{$7.4\cdot 10^{-2}$} & & \multirow{4}{*}{$3\cdot 10^{-4}$} \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & $5.78304$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.05$} & $5.73$ & $5.95$ & \multirow{2}{*}{$3.8\cdot 10^{-2}$} & \small{($c_2=0.5286$)} & \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & & \\
\hline
\multirow{2}{*}{$h=0.01$} & $5.775$ & $5.810$ & \multirow{2}{*}{$6\cdot 10^{-3}$} & & \multirow{4}{*}{$3.5\cdot 10^{-6}$} \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & $5.7848$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.005$} & $5.779$ & $5.800$ & \multirow{2}{*}{$3.6\cdot 10^{-3}$} & \small{($c_2=0.514$)} & \\
 & \small{($c_2=1$)} & \small{($c_2=0.01$)} & & & \\
\hline
\end{tabular}
\end{center}
\caption{Example 3. Numerical data of Figure \ref{figblowup3} (case 2). The blow-up time of the corresponding (exact) solution is assumed to be $\hat{t}\approx 5.78482$.}
\label{tabla3c2}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
  \hline
   & \multicolumn{2}{|c|}{Case 1 - $c_1=0.5$} & \multicolumn{2}{|c|}{Case 2 - Radau I} & \multicolumn{2}{|c|}{Case 2 - $c_2=0.5$} \\
  \cline{2-7}
   & Blow-up & Rel. error & Blow-up & Rel. error & Blow-up & Rel. error \\
  \hline
  $h=0.1$ & $5.694865$ & $1.6\cdot 10^{-2}$ & $5.751797$ & $5.7\cdot 10^{-3}$ & $5.788215$ & $5.9\cdot 10^{-4}$ \\
  \hline
  $h=0.05$ & $5.739117$ & $7.9\cdot 10^{-3}$ & $5.767963$ & $2.9\cdot 10^{-3}$ & $5.786612$ & $3.1\cdot 10^{-4}$ \\
  \hline
  $h=0.01$ & $5.775121$ & $1.7\cdot 10^{-3}$ & $5.781037$ & $6.5\cdot 10^{-4}$ & $5.784995$ & $3\cdot 10^{-5}$ \\
  \hline
  $h=0.005$ & $5.780132$ & $8.1\cdot 10^{-4}$ & $5.783066$ & $3\cdot 10^{-4}$ & $5.784875$ & $9.5\cdot 10^{-6}$ \\
  \hline
\end{tabular}
\end{center}
\caption{Example 3. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with $c_1=0.5$, and case 2 with \textit{Radau I} collocation points, $c_1=0$, $c_2=2/3$, and $c_1=0$, $c_2=0.5$, for different stepsizes $h$. The blow-up time of the corresponding (exact) solution is assumed to be $\hat{t}\approx 5.78482$.}
\label{tabla3c105radau}
\end{table}

\begin{figure}[hp]
\begin{center}
  \includegraphics[width=1\textwidth]{blowup_all_ej4}
\end{center}
\caption{Example 4. Numerical estimation of the blow-up time of
  collocation solutions varying $c_1$ (case 1) or $c_2$ (case 2), for
  different stepsizes $h$.
  \vspace{5cm}}
\label{figblowup4}
\end{figure}

\begin{table}[hp]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Case 1 & \multicolumn{3}{|c|}{Varying $c_1$} & \multicolumn{2}{|c|}{Intersection} \\
\cline{2-6}
($m=1$, $c_1>0$) & min. $t$ & max. $t$ & Rel. error & $t$ & Rel. error \\
\hline
\multirow{2}{*}{$h=0.01$} & $1.593$ & $1.722$ & \multirow{2}{*}{$7.8\cdot 10^{-2}$} & & \multirow{4}{*}{$5.2\cdot 10^{-4}$} \\
 & \small{($c_1=1$)} & \small{($c_1=0.05$)} & & $1.6467$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.005$} & $1.613$ & $1.691$ & \multirow{2}{*}{$4.7\cdot 10^{-2}$} & \small{($c_1=0.444$)} & \\
 & \small{($c_1=1$)} & \small{($c_1=0.05$)} & & & \\
\hline
\multirow{2}{*}{$h=0.001$} & $1.6367$ & $1.6571$ & \multirow{2}{*}{$1.2\cdot 10^{-2}$} & & \multirow{4}{*}{$6.6\cdot 10^{-5}$} \\
 & \small{($c_1=1$)} & \small{($c_1=0.05$)} & & $1.64595$ & \\
\cline{1-4}
\multirow{2}{*}{$h=0.0005$} & $1.6406$ & $1.6521$ & \multirow{2}{*}{$7\cdot 10^{-3}$} & \small{($c_1=0.461$)} & \\
 & \small{($c_1=1$)} & \small{($c_1=0.05$)} & & & \\
\hline
\end{tabular}
\end{center}
\caption{Example 4. Numerical data of Figure \ref{figblowup4} (case 1). The blow-up time of the corresponding (exact) solution is assumed to be $\hat{t}\approx 1.645842$.}
\label{tabla4c1}
\end{table}

\begin{table}[hc]
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
Case 2 & \multicolumn{3}{|c|}{Varying $c_2$} \\
\cline{2-4}
($m=2$, $c_1=0$) & min. $t$ & max. $t$ & Rel. error \\
\hline
\multirow{2}{*}{$h=0.01$} & $1.6445$ & $1.6490$ & \multirow{2}{*}{$2.7\cdot 10^{-3}$} \\
 & \small{($c_2=0.95$)} & \small{($c_2=0.33$)} & \\
\cline{1-4}
\multirow{2}{*}{$h=0.005$} & $1.6456$ & $1.6476$ & \multirow{2}{*}{$1.2\cdot 10^{-3}$} \\
 & \small{($c_2=0.9$)} & \small{($c_2=0.3$)} & \\
\hline
\multirow{2}{*}{$h=0.001$} & $1.64577$ & $1.64622$ & \multirow{2}{*}{$2.7\cdot 10^{-4}$} \\
 & \small{($c_2=0.9$)} & \small{($c_2=0.3$)} & \\
\cline{1-4}
\multirow{2}{*}{$h=0.0005$} & $1.64581$ & $1.64607$ & \multirow{2}{*}{$1.6\cdot 10^{-4}$} \\
 & \small{($c_2=0.85$)} & \small{($c_2=0.28$)} & \\
\hline
\end{tabular}
\end{center}
\caption{Example 4. Numerical data of Figure \ref{figblowup4} (case 1). The blow-up time of the corresponding (exact) solution is assumed to be $\hat{t}\approx 1.645842$.}
\label{tabla4c2}
\end{table}

\begin{table}[hb]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
  \hline
   & \multicolumn{2}{|c|}{Case 1 - $c_1=0.5$} & \multicolumn{2}{|c|}{Case 2 - Radau I} \\
  \cline{2-5}
   & Blow-up & Rel. error & Blow-up & Rel. error \\
  \hline
  $h=0.01$ & $1.638700$ & $4.3\cdot 10^{-3}$ & $1.646991$ & $7\cdot 10^{-4}$ \\
  \hline
  $h=0.005$ & $1.642843$ & $1.8\cdot 10^{-3}$ & $1.646540$ & $4.2\cdot 10^{-4}$ \\
  \hline
  $h=0.001$ & $1.645172$ & $4.1\cdot 10^{-4}$ & $1.645985$ & $8.7\cdot 10^{-5}$ \\
  \hline
  $h=0.0005$ & $1.645491$ & $2.1\cdot 10^{-4}$ & $1.645919$ & $4.7\cdot 10^{-5}$ \\
  \hline
\end{tabular}
\end{center}
\caption{Example 4. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with $c_1=0.5$, and case 2 with \textit{Radau I} collocation points, $c_1=0$, $c_2=2/3$, for different stepsizes $h$.
The blow-up time of the corresponding (exact) solution is assumed to be $\hat{t}\approx 1.645842$.}
\label{tabla4c105radau}
\end{table}

\end{document} 
1112.4658.extracted_bib
\begin{thebibliography}{00}
\bibitem{Kirk09} C. M. Kirk. Numerical and asymtotic analysis of a
  localized heat source undergoing periodic motion. Nonlinear
  Anal. \textbf{71} (2009), e2168-e2172.

\bibitem{CalCapo09} F. Calabr�, G. Capobianco, Blowing up behavior for
  a class of nonlinear VIEs connected with parabolic
  PDEs. J. Comput. Appl. Math. \textbf{228} (2009), 580-588.

\bibitem{Myd94} W. Mydlarczyk, A condition for finite blow-up time for
  a Volterra integral equation. J. Math. Anal. Appl. \textbf{181}
  (1994), no. 1, 248--253.

\bibitem{Myd99} W. Mydlarczyk, The blow-up solutions of integral
  equations. Colloq. Math. \textbf{79} (1999), no. 1, 147--156.

\bibitem{Okra07} T. Ma\l olepszy, W. Okrasi\'nski, Conditions for
  blow-up of solutions of some nonlinear Volterra integral
  equations. J. Comput. Appl. Math. \textbf{205} (2007), 744--750.

\bibitem{Okra08} T. Ma\l olepszy, W. Okrasi\'nski, Blow-up conditions
  for nonlinear Volterra integral equations with power
  nonlinearity. Appl. Math. Letters. \textbf{21} (2008), 307--312.

\bibitem{Okra10} T. Ma\l olepszy, W. Okrasi\'nski, Blow-up time for
  solutions for some nonlinear Volterra integral
  equations. J. Math. Anal. Appl. \textbf{366} (2010), no. 1,
  372--384.

\bibitem{Brun04} H. Brunner, Collocation Methods for Volterra Integral
  and Related Functional Differential Equations, Cambridge University
  Press, Cambridge, 2004.


\bibitem{BB11} R. Ben�tez, V. J. Bol�s, Existence and uniqueness of
  nontrivial collocation solutions of implicitly linear homogeneous
  Volterra integral equations. J. Comput. Appl. Math. \textbf{235}
  (2011), no. 12, 3661--3672.


\bibitem{Kras84} M. A. Krasnosel'skii, P. P. Zabreiko, Geometric
  Methods of Nonlinear Analysis, Springer Verlag, New York, 1984.


\bibitem{Brun92} H. Brunner, Implicitly linear collocation methods for
  nonlinear Volterra integral equations, Appl. Numer. Math. \textbf{9}
  (1992), 235--247.

\end{thebibliography}