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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 05, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/05\hfil A generalization of Osgood's test]
{A generalization of Osgood's test and a comparison criterion for
integral equations with noise}
\author[M. J. Ceballos-Lira, J. E. Mac\'ias-D\'iaz, J. Villa\hfil EJDE-2011/05\hfilneg]
{Marcos J. Ceballos-Lira, Jorge E. Mac\'ias-D\'iaz,
Jos\'e Villa} % in alphabetical order
\address{Marcos Josias Ceballos-Lira \newline
Divisi\'on Acad\'emica de Ciencias B\'asicas,
Universidad Ju\'arez Aut\'onoma de Tabasco,\newline
Km. 1 Carretera Cunduac\'an-Jalpa de M\'endez,
Cunduac\'an, Tab. 86690, Mexico}
\email{marjocel\_81@hotmail.com}
\address{Jorge Eduardo Mac\'ias-D\'iaz \newline
Departamento de Matem\'aticas y F\'{\i}sica,
Universidad Aut\'onoma de Aguascalientes, \newline
Avenida Universidad 940, Ciudad Universitaria,
Aguascalientes, Ags. 20131, Mexico}
\email{jemacias@correo.uaa.mx}
\address{Jos\'e Villa Morales\newline
Departamento de Matem\'aticas y F\'{\i}sica,
Universidad Aut\'onoma de Aguascalientes, \newline
Avenida Universidad 940, Ciudad Universitaria,
Aguascalientes, Ags. 20131, Mexico}
\email{jvilla@correo.uaa.mx}
\thanks{Submitted December 7, 2010. Published January 12, 2011.}
\subjclass[2000]{45G10, 45R05, 92F05, 74R10, 74R15}
\keywords{Osgood's test; comparison criterion; time of explosion;
\hfill\break\indent integral equations with noise; crack failure}
\begin{abstract}
In this article, we prove a generalization of Osgood's test
for the explosion of the solutions of initial-value problems.
We also establish a comparison criterion for the solution of
integral equations with noise, and provide estimations of
the time of explosion of problems arising in the investigation
of crack failures where the noise is the absolute value of the
Brownian motion.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\section{Introduction\label{S:Intro}}
Let $x_0$ be a positive, real number, let $b$ be a positive,
real-valued function defined on $[0,\infty )$, and suppose that
$y$ is an extended real-valued function with the same domain as
$b$. The present work is motivated by a criterion for the
explosion of the solutions of ordinary differential equations of
the form
\begin{equation}
\begin{gathered}
\frac{dy(t)}{dt}=b(y(t)), \quad t>0, \\
y(0)=x_0.
\end{gathered} \label{Eq:ODE}
\end{equation}
More precisely, the time of explosion of the solution of this
initial-value problem is the nonnegative, extended real number
$t_e=\sup \{t\geq 0:y(t)<\infty \}$. The above-mentioned
criterion is called \emph{Osgood's test} after its author
\cite{Osgood}, and it states that $t_e$ is finite if and only if
$\int_{x_0}^{\infty }ds/b(s)<\infty $. In such case,
$t_e=\int_{x_0}^{\infty }ds/b(s)$.
A natural question readily arises about the possibility to extend
Osgood's test to more general, initial-value problems, say, to
problems in which the drift function $b$ in the ordinary
differential equation of \eqref{Eq:ODE} is multiplied by a
suitable, nonnegative function of $t$. Another direction of
investigation would be to investigate conditions under which the
solutions of the integral form of such equation with a noise
function added, explode in a finite time. Evidently the
consideration of these two problems as a single one is an
interesting topic of study \emph{per se}. In fact, the purpose of
this paper is to provide a generalization of Osgood's test to
integral equations with noise, which generalize the problem
presented in \eqref{Eq:ODE}. Important, as it is in the recent
literature \cite{Kafini, L-V}, the problem of establishing
analytical conditions under which the time of explosion of the
problem under investigation is finite, is tackled here. In the way, we
establish a comparison criterion for the solutions of integral
equations with noise, and show some applications to the spread of
cracks in rigid surfaces.
Our manuscript is divided in the following way: Section
\ref{S:Osgood} introduces the integral equation with noise that
motivates this manuscript, along with a convenient simplification
for its study; a generalization of Osgood's test is presented in
this stage for the associated initial-value problem for both
scenarios: noiseless and noisy systems. Section \ref{S:Compar}
establishes a comparison criterion for the solutions of two
noiseless systems with comparable initial conditions. A necessary
condition for the explosion of the solutions of the problem under
investigation is provided in this section, together with an
illustrative counterexample and a partial converse. In Section
\ref{S:Approx}, we give upper and lower bounds for the value of
the time of explosion of our integral equation. Finally, Section
\ref{S:Applic} provides estimates of probabilities associated to
the time of explosion of a system in which the noise is the
absolute value of the Brownian motion.
\section{Osgood's test\label{S:Osgood}}
Let $\overline {\mathbb{R}}$ denote the set of extended real
numbers. Throughout, $a, b : [0 , \infty) \to \mathbb{R}$ will
represent positive, continuous functions, while the function $g :
[0 , \infty) \to \mathbb{R}$ will be continuous and nonnegative.
For physical reasons, the function $g$ is called a \emph{noise}.
In this work, $x _0$ will denote a positive, real number, and
$X :[0 , \infty) \to \overline {\mathbb{R}}$ will be a nonnegative
function whose dependency on $t \geq 0$ is represented by $X _t$.
We are interested in establishing
conditions under which the solutions of the integral equation
\begin{equation} \label{Eq:Model}
X _t = x _0 + \int _0 ^t a (s) b (X _s) d s + g (t), \quad t \geq 0,
\end{equation}
explode in finite time. More precisely, we define the \emph{time
of explosion} of $X$ as the nonnegative, extended real number
$T^X _e = \sup \{t \geq 0 : X _t < \infty \}$. In this manuscript, we
investigate conditions under which the time of explosion of $X$ is
a real number.
Letting $Y _t = X _t - g (t)$, one sees immediately that the
problem under consideration is equivalent to finding the time of
explosion of the solution $Y$ of the equation
\begin{equation} \label{Eq:EIM}
Y _t = x _0 + \int _0 ^t a (s) b (Y _s + g (s)) d s, \quad t \geq 0.
\end{equation}
As a matter of fact, $T _e ^X = T _e ^Y$. From this point on, this
common, extended real number will be denoted simply by $T _e$ for
the sake of briefness.
\begin{remark} \label{rmk1} \rm
It is worth noticing that \eqref{Eq:EIM} can be presented in
differential form as the equivalent, initial-value problem
\begin{equation}
\begin{gathered}
\frac{dY_{t}}{dt}=a(t)b(Y_{t}+g(t)), \quad t>0, \\
Y_0=x_0,
\end{gathered} \label{Eq:EDM}
\end{equation}
a problem for which the existence of solutions is guaranteed, for
instance, when $b$ is locally Lipschitzian and $a$ is regulated
(see \cite[(10.4.6)]{Dieudonne})
\end{remark}
Let $r$ be a real number such that $0 0, \\
y (0) = x _0,
\end{gathered}
\end{equation}
has a unique solution given by $y (t) = B ^{- 1} (A (t))$, for
$t < A ^{- 1} (B (\infty))$. The solution explodes in finite
time if and only if $B (\infty) < A (\infty)$, in which case,
$T _e ^y = A ^{- 1} (B (\infty))$.
\end{lemma}
\begin{proof}
The function $y(t)=B^{-1}(A(t))$ is evidently a solution of
\eqref{Eq:EGO}. Additionally, expressing the differential equation
in \eqref{Eq:EGO} as $y'(s)/b(y(s))=a(s)$, integrating both sides
over $[0,t]$ and performing a suitable substitution, we obtain
that $B(y(t))=A(t)$, whence
the uniqueness follows. Moreover, $y(t)$ is real if and only if
$t0 $ such that $v(\widetilde{T}+s)-u(\widetilde{T}+s)>0$
for every $s\in[ 0,\delta )$, whence it follows that
$\widetilde{T}+\frac{\delta }{2}\in N$, a contradiction.
Consequently, $u(t)\leq v(t)$ for every $t\geq 0$.
Now, in case that $x_0=x_1$, the solution of the equation
\[
u_{r}(t)=x_0-r+\int_0^{t}a(s)b(u_{r}(s))ds,\quad 00$.
Expanding the expression $(Y_{s}+e^{s})^{3}$ in \eqref{Eq:EIM},
we obtain $Y_{t}\geq 1+\frac{1}{4}\int_0^{t}Y_{s}^{2}ds$.
Then $Y_{t}\geq (1-\frac{1}{4}t)^{-1}$, which implies that $Y$
explodes in finite time. However, $B(\infty )=2>1=A(\infty )$.
\end{example}
The following result is a partial converse of Theorem \ref{Thm:2}.
We let $\widehat {g} (t) = \sup \{ g (s) : s \in [0 , t] \}$,
for every $t \geq 0$.
\begin{proposition}\label{Prop:1}
Suppose that $b$ is non-decreasing, and that
\[
\widehat{g}(t)** 0 : | W _t | = r \}$. Evidently,
$| W _s | \leq r$, for every $s \in [0 , T _r]$.
\begin{proposition}\label{Prop:4}
Let $0 \leq t \leq T$. For every $r \geq 0$,
\begin{equation} \label{Eq:PRA2}
P \left( T _e \leq t | T _r < T \right)
\leq \frac {1 - \Phi \big(r /\sqrt {A ^{- 1}
(B (\widetilde {B} ^{- 1} _r (A (t))))}\big)}
{1 - \Phi \big(r /\sqrt {T}\big)}.
\end{equation}
\end{proposition}
\begin{proof}
Notice that $|\widehat{W}_{T}|\geq r$ whenever $T_{r}x_0$, then
\[
P( Y_{L}^{-1}\leq t) \leq 1-\Phi \Big( \frac{\widetilde{B}
_{A(t)}^{-1}(L)}{\sqrt{T}}\Big) .
\]
\end{proposition}
\begin{proof}
Let $\widetilde{Y}$ the solution of $\widetilde{Y}_{t}=x_0+
\int_0^{t}a(s)b(\widetilde{Y}_{s}+|\widehat{W}_{T}|)ds$, for
every $0\leq t1$.
\end{example}
\subsection*{Acknowledgments}
M. J. Ceballos-Lira wishes to acknowledge the financial support of
the Mexican Council for Science and Technology (CONACYT) to pursue
postgraduate studies in Universidad Ju\'arez Aut\'onoma de Tabasco
(UJAT); he also wishes to thank UJAT and the Universidad
Aut\'onoma de Aguascalientes (UAA) for additional, partial,
financial support. J. Villa acknowledges the partial support of
CONACYT grant 118294, and grant PIM08-2 at UAA.
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\end{document}
**