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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 21, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/21\hfil One-phase Stefan problems]
{Existence and uniqueness for one-phase Stefan problems
of non-classical heat equations with temperature boundary condition
at a fixed face}
\author[A. C. Briozzo, D. A. Tarzia\hfil EJDE-2006/21\hfilneg]
{Adriana C. Briozzo, Domingo A. Tarzia}


\address{Adriana C. Briozzo \hfill\break
Departamento de Matem\'{a}tica \\
FCE, Universidad Austral \\
Paraguay 1950, S2000FZF Rosario, Argentina}
\email{Adriana.Briozzo@fce.austral.edu.ar}

\address{Domingo Alberto Tarzia \hfill\break
Departamento de Matem\'{a}tica - CONICET\\
FCE, Universidad Austral \\
Paraguay 1950, S2000FZF Rosario, Argentina}
\email{Domingo.Tarzia@fce.austral.edu.ar}

\date{}
\thanks{Submitted November 1, 2005. Published February 9, 2006.}
\subjclass[2000]{35R35, 80A22, 35C05, 35K20, 35K55, 45G15, 35C15} 
\keywords{Stefan problem; non-classical heat equation; free boundary problem;
\hfill\break\indent similarity solution; nonlinear heat sources; 
Volterra integral equations}

\begin{abstract}
 We prove the existence and uniqueness, local in time, of a
 solution for a one-phase Stefan problem of a non-classical
 heat equation  for a semi-infinite material with temperature
 boundary condition  at the fixed face.
 We use the Friedman-Rubinstein integral representation method
 and the Banach contraction theorem in order to solve an
 equivalent system of two Volterra integral equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

The one-phase Stefan problem for a semi-infinite material 
for the classical heat equation requires the determination of the
temperature distribution $u$ of the liquid phase (melting problem) or of the
solid phase (solidification problem), and the evolution of the free boundary
$x=s(t).$ Phase-change problems appear frequently in industrial processes
and other problems of technological interest
\cite{AlSo,AMR,Ca,CaJa,ChRa,Cr,DHLV,FaPr,Lu,WrBr}.
A large bibliography on the subject was given in \cite{Ta00}.

Non-classical heat conduction problem for a semi-infinite material was
studied in \cite{BeTaVi,CaYi,KePr,TaVi,Vi}, e.g. problems of the type
\begin{equation}
\begin{gathered}
u_{t}-u_{xx}=-F(u_{x}(0,t)), \quad x>0,\;t>0, \\
u(0,t)=0, \quad t>0 \\
u(x,0)=h(x), \quad x>0
\end{gathered}
\label{primero}
\end{equation}
where $h(x),x>0$, and $F(V),V\in \mathbb{R}$, are continuous functions. The
function $F$, henceforth referred as control function, is assumed to satisfy
the  condition
\begin{itemize}
\item[(H1)] $F(0)=0$.
\end{itemize}

As observed in \cite{TaVi,Vi}, the heat flux $w(x,t)=u_{x}(x,t)$ for
problem \eqref{primero} satisfies a classical heat conduction problem with a
nonlinear convective condition at $x=0$, which can be written in the form
\begin{equation}
\begin{gathered}
w_{t}-w_{xx}=0, \quad x>0,\;t>0, \\
w_{x}(0,t)=F(w(0,t)), \quad t>0, \\
w(x,0)=h'(x)\geq 0, \quad x>0.
\end{gathered}  \label{segundo}
\end{equation}

The literature concerning problem \eqref{segundo} has  increased rapidly
since the publication of the papers \cite{MaWo,OlHa,RoMa}. Related
problems have been also studied; see for example \cite{AfCa,GlSp,HaOl}.
In \cite{Ta01}, a one-phase Stefan problem for a non-classical heat equation
for a semi-infinite material was presented. 
There the free boundary problem consists in determining the 
temperature $u=u(x,t)$ and the free boundary
$x=s(t)$ with a control function $F$ which depends on the evolution of the
heat flux at the extremum $x=0$ is given by the  conditions
\begin{equation}
\begin{gathered}
u_{t}-u_{xx}=-F(u_{x}(0,t)), \quad 0<x<s(t),\;0<t<T, \\
u(0,t)=f(t)\geqslant 0, \quad 0<t<T, \\
u(s(t),t)=0,\quad u_{x}(s(t),t)=-\dot {s}(t), \quad 0<t<T, \\
u(x,0)=h(x)\geq 0, \quad 0\leqslant x\leqslant b=s(0)\quad
(b>0).
\end{gathered} \label{Stefan}
\end{equation}

The goal in this paper is to prove the existence and
uniqueness, local in time, of a solution to the one-phase Stefan problem
\eqref{Stefan} for a non-classical heat equation with temperature boundary
condition at the fixed face $x=0$.
First, we prove that problem \eqref{Stefan} is equivalent to a system
of two Volterra integral equations \eqref{ecintegral}-(\ref{ecintegral2})
following the Friedman-Rubinstein's method given in \cite{Fr,Ru}. 
Then, we prove that the problem \eqref{ecintegral}-(\ref{ecintegral2}) 
has a unique local solution by using the Banach contraction theorem.

\section{Existence and Uniqueness of Solutions}


We have the following equivalence for the existence of solutions to
the non-classical free boundary problem \eqref{Stefan}.

\begin{theorem} \label{thm1}
The solution of the free-boundary problem \eqref{Stefan} is
\begin{gather}
\begin{aligned}
u(x,t)
&=\int_{0}^{b}G(x,t;\xi ,0)h(\xi )d\xi
+\int_{0}^{t}G_{\xi}(x,t;0,\tau )f(\tau )\, d\tau   \\
&\quad+\int_{0}^{t}G(x,t;s(\tau ),\tau )v(\tau )\, d\tau
-\iint_{D(t)}G(x,t;\xi ,\tau )F(V(\tau ))d\xi \, d\tau ,
\end{aligned} \label{u}
\\
s(t)=b-\int_{0}^{t}v(\tau )\, d\tau \,, \label{s}
\end{gather}
where $D(t)=\{ (x,\tau ): 0<x<s(\tau ), 0<\tau <t\} $, with
$f\in C^{1}[0,T)$, $h\in C^{1}[0,b]$,
$h(b)=0$, $h(0)=f(0)$, $F$ is a
Lipschitz function over $C^{0}[0,T]$, and the functions
$v\in C^{0}[0,T]$, $V\in C^{0}[0,T]$ defined by
\begin{equation}
v(t)=u_{x}(s(t),t)\,,\quad
V(t)=u_{x}(0,t)  \label{vyV}
\end{equation}
must satisfy the following system of Volterra integral equations
\begin{gather} \label{ecintegral}
\begin{aligned}
v(t)&=2\int_{0}^{b}N(s(t),t;\xi ,0)h'(\xi )d\xi
 -2\int_{0}^{t}N(s(t),t;0,\tau )\dot {f}(\tau )\, d\tau\\
&\quad +2\int_{0}^{t}G_{x}(s(t),t;s(\tau ),\tau )v(\tau )\, d\tau \\
&\quad +2\int_{0}^{t}[N(s(t),t;s(\tau ),\tau )-N(s(t),t;0,\tau )]
F(V(\tau ))\, d\tau .
\end{aligned}
\\  \label{ecintegral2}
\begin{aligned}
V(t)&=\int_{0}^{b}N(0,t;\xi ,0)h'(\xi )d\xi \\
&\quad -\int_{0}^{t}N(0,t;0,\tau )\dot {f}(\tau )\, d\tau
+\int_{0}^{t}G_{x}(0,t;s(\tau ),\tau )v(\tau )\, d\tau \\
&\quad +\int_{0}^{t}[N(0,t;s(\tau ),\tau )-N(0,t;0,\tau )]F(V(\tau
))\, d\tau ,
\end{aligned}
\end{gather}
where $G$, $N$ are the Green and Neumann functions and $K$ is the
fundamental solution of the heat equation, defined respectively by
\begin{gather*}
G(x,t,\xi ,\tau )=K(x,t,\xi ,\tau )-K(-x,t,\xi ,\tau ), \\
N(x,t,\xi ,\tau )=K(x,t,\xi ,\tau )+K(-x,t,\xi ,\tau ), \\
K(x,t,\xi ,\tau )=\begin{cases}
\frac{1}{2\sqrt{\pi (t-\tau )}}\exp \big(-\frac{(x-\xi
)^{2}}{4(t-\tau )}\big)\, & t>\tau \\
0 & t\leq \tau\,,
\end{cases}
\end{gather*}
where $s(t)$ is given by \eqref{s},
\end{theorem}

\begin{proof}
Let $u(x,t)$ be the solution to \eqref{Stefan}. We integrate,
on the domain
$D_{t,\varepsilon }=\{ (\xi ,\tau ): 0<\xi <s(\tau ),\varepsilon <\tau
<t-\varepsilon \} $,  the Green identity
\begin{equation}
(Gu_{\xi }-uG_{\xi })_{\xi }-(Gu)_{\tau }=GF(u_{\xi }(0,\tau ))\,.\label{Green}
\end{equation}
Now we let $\varepsilon \to 0$, to obtain the following integral
representation for $u(x,t)$,
\begin{align*}
u(x,t) &=\int_{0}^{b}G(x,t;\xi ,0)h(\xi )d\xi +\int_{0}^{t}G_{\xi
}(x,t;0,\tau )f(\tau )\, d\tau   \\
&\quad +\int_{0}^{t}G(x,t;s(\tau ),\tau )\;u_{_{\xi }}(s(\tau ),\tau )\,
d\tau
 -\iint_{D(t)}G(x,t;\xi ,\tau )F(u_{\xi }(0,\tau ))d\xi \,
d\tau \,.
\end{align*} % 13
 From the definition of  $v(t)$ and $V(t)$ by $(\ref{vyV})$, we obtain
\eqref{u} and \eqref{s}. If we differentiate
$u(x,t)$ in variable $x$ and we let $x\to 0^{+}$ and
$x\to s(t)$, by using the jump relations, we obtain the integral
equations for $v$ and $V$ given by \eqref{ecintegral} and
\eqref{ecintegral2}.

Conversely, the function $u(x,t)$ defined by \eqref{u} where $v$ and $V$
are the solutions of \eqref{ecintegral}and \eqref{ecintegral2},
satisfy the conditions \eqref{Stefan} (i),(ii),(iv) and (v).
In order to prove condition \eqref{Stefan} (iii) we define
$\psi (t)=u(s(t),t)$. Taking into account that
$u$ satisfy the conditions \eqref{Stefan} (i),(ii),(iv) and (v),
if we integrate the Green identity \eqref{Green} over the domain
$D_{t,\varepsilon }$, ($\varepsilon >0$) and we let $\varepsilon
\to 0$ we obtain that
\begin{align*}
u(x,t)&=\int_{0}^{b}G(x,t;\xi ,0)h(\xi )d\xi
+\int_{0}^{t}G(x,t;s(\tau ),\tau )v(\tau )\, d\tau
\\
&\quad +\int_{0}^{t}\psi (\tau )[G_{x}(x,t;s(\tau ),\tau
)-G(x,t;s(\tau ),\tau )v(\tau )]\, d\tau
\\
&\quad +\int_{0}^{t}G_{\xi }(x,t;0,\tau )f(\tau )\, d\tau
-\iint_{D(t)}G(x,t;\xi ,\tau )F(V(\tau ))d\xi \, d\tau .
\end{align*}
Then, if we compare this last expression with \eqref{u}, we deduce that
\begin{equation}
M(x,t)=\int_{0}^{t}\psi (\tau )[
G_{x}(x,t;s(\tau ),\tau )-G(x,t;s(\tau ),\tau )v(\tau )]\, d\tau \equiv 0
\label{M}
\end{equation}
for $0<x<s(t)$, $0<t<\sigma $. We let $x\to s(t)$ in (\ref{M}) and by
using the jump relations we have that $\psi $ satisfy the integral equation
\[
\frac{1}{2}\psi (t)+\int_{0}^{t}\psi (\tau )[
G_{x}(s(t),t;s(\tau ),\tau )-G(s(t),t;s(\tau ),\tau )v(\tau )]\, d\tau =0\,.
\]
Then we deduce that
\begin{align*}
|\psi (t)| &\leq C\int_{0}^{t}\frac{|\psi
(\tau )| }{\sqrt{t-\tau }}\, d\tau \\
&\leq C^{2}\int_{0}^{t}
\frac{\, d\tau }{\sqrt{t-\tau }}\int_{0}^{\tau }\frac{|\psi (\eta
)| }{\sqrt{\tau -\eta }}d\eta \\
&= C^{2}\int_{0}^{t}|\psi (\eta )| d\eta \int_{\eta
}^{t}\frac{\, d\tau }{[(t-\tau )(\tau -\eta )
]^{1/2}}\\
&=\pi C^{2}\int_{0}^{t}|\psi (\eta )| d\eta
\end{align*}
where $C=C(t)$; therefore by using the Gronwall inequality we
have that $\psi (t)=0$ over $[0,\sigma ]$.
\end{proof}

Next, we  use the Banach fixed point theorem in order to prove the local
existence and uniqueness of solution
$v,V \in C^{0}[0,\sigma ]$ to the system of two Volterra integral
equations (\ref{ecintegral})-(\ref{ecintegral2}) where
$\sigma$ is a positive small number. Consider the Banach space
\[
C_{M,\sigma }=\big\{ \vec {w}=\begin{pmatrix}v \\ V\end{pmatrix}
: v,V:[0,\sigma ]\to \mathbb{R},\text{ continuous, with }
\| \vec {w}\| _{\sigma }\leq M\big\}
\]
with
\[
\|\vec {w}\|_{\sigma }
:=\|v\|_{\sigma }+\|V\|_{\sigma }
:=\max_{t\in [0,\sigma]} |v(t)| +\max_{t\in [0,\sigma ]}|V(t)|
\] % 15
We define $A:C_{M,\sigma }\longrightarrow C_{M,\sigma },$ such that
\[
\vec {\widetilde{w}}(t)=A(\vec w(t))=\begin{pmatrix} A_{1}(v(t),V(t))
\\ A_{2}(v(t),V(t)) \end{pmatrix}
\] %16
where
\begin{equation}
A_{1}(v(t),V(t))=F_{0}(v(t))+2\int_{0}^{t}[N(s(t),t,s(\tau ),\tau
)-N(s(t),t,0,\tau )]F(V(\tau ))\, d\tau  \label{a1}
\end{equation}
with
\begin{align*}
F_{0}(v(t))
&=2\int_{0}^{b}N(s(t),t,\xi ,0)h'(\xi )d\xi
-2\int_{0}^{t}N(s(t),t,0,\tau )\dot {f}(\tau )\, d\tau \\
&\quad +2\int_{0}^{t}G_{x}(s(t),t,s(\tau ),\tau )v(\tau )\, d\tau
\end{align*}
and
\begin{equation} \label{a2}
\begin{aligned}
A_{2}(v(t),V(t))&=\int_{0}^{b}N(0,t,\xi ,0)h'(\xi )d\xi
-\int_{0}^{t}N(0,t,0,\tau )\dot {f}(\tau )\, d\tau \\
&\quad +\int_{0}^{t}G_{x}(0,t,s(\tau ),\tau )v(\tau )\, d\tau\\
&\quad +\int_{0}^{t}[N(0,t,s(\tau ),\tau )-N(0,t,0,\tau )]F(V(\tau ))\, d\tau .
\end{aligned}
\end{equation}

\begin{lemma} \label{lem2}
If $v\in C^{0}[0,\sigma ]$, $\max_{t\in [0,\sigma]} |v(t)| \leq M$
and $2M\sigma \leq b$ then
$s(t)$ defined by (\ref{s}) satisfies
\[
|s(t)-s(\tau )| \leq M|t-\tau| \,\quad
|s(t)-b| \leq \frac{b}{2}\,,\quad \forall t,\tau \in [0,\sigma ].
\]
\end{lemma}

 To prove the following Lemmas we need the inequality
\begin{equation}
\exp \Big(\frac{-x^{2}}{\alpha (t-\tau )}\Big)
/(t-\tau )^{n/2} \leq \Big(\frac{n\alpha }{2ex^{2}}
\Big)^{n/2}, \quad \alpha , x>0,\; t>\tau ,\; n\in \mathbb{N}.
\label{exp}
\end{equation}

\begin{lemma} \label{lem3}
Let $\sigma \leq 1$, $M\geq 1$, $f\in C^{1}[0,T)$, $h\in C^{1}[0,b]$,
$F$ a Lipschitz function over $C^{0}[0,T]$. Under the hypothesis of
Lemma \ref{lem2}, we have the following properties:
\begin{gather}
\int_{0}^{t}|N(s(t),t,0,\tau )| |\dot {f}(\tau
)| \, d\tau \leq \|\dot {f}\|_{t}C_{1}(b) t  \label{i}
\\
\int_{0}^{t}|G_{x}(s(t),t,s(\tau ),\tau )| |v(\tau
)| \, d\tau \leq M^{2}C_{2}(b)\sqrt{t}  \label{ii}
\\
\int_{0}^{b}|N(s(t),t,\xi ,0)| |h'(\xi )|
d\xi \leq \|h'\|\label{iii}
\\
\int_{0}^{t}|N(s(t),t,s(\tau ),\tau )-N(s(t),t,0,\tau )| |
F(V(\tau ))| \, d\tau \leq C_{4}(L)M\sqrt{t}
\label{iv}
\\
\int_{0}^{b}|N(0,t,\xi ,0)| |h'(\xi )| d\xi
\leq \|h'\|\label{v}
\\
\int_{0}^{t}|N(0,t,0,\tau )| |\dot {f}(\tau
)| \, d\tau \leq \frac{2\|\dot {f}\|_{\sigma }}{\sqrt{
\pi }}\sqrt{t}  \label{vi}
\\
\int_{0}^{t}|G_{x}(0,t,s(\tau ),\tau )| |v(\tau )|
\, d\tau \leq C_{3}(b)Mt  \label{vii}
\\
\int_{0}^{t}|N(0,t,s(\tau ),\tau )-N(0,t,0,\tau )| |
F(V(\tau ))| \, d\tau \leq C_{4}(L)M\sqrt{t}  \label{viii}
\end{gather}
where $L$ is the Lipschitz constant of $F$ and
\begin{equation} \label{constantes}
\begin{gathered}
C_{1}(b)=(\frac{8}{eb^{2}})^{1/2} \frac{1}{\sqrt{\pi }},\quad
C_{2}(b)=\frac{1}{2\sqrt{\pi }}+\frac{3b}{4\sqrt{\pi }}(\frac{2}{3eb^{2}})^{3/2}
 \\
C_{3}(b) =\frac{3b}{8\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2},\quad
C_{4}(L)=\frac{4L}{\sqrt{\pi }}\;.
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
To prove \eqref{i}, we have
\begin{align*}
|N(s(t),t,0,\tau )|
&=|K(s(t),t,0,\tau )+K(-s(t),t,0,\tau )| =2K(s(t),t,0,\tau )\\
&=\exp \big(\frac{-s^{2}(t)}{4(t-\tau )}\big)\frac{(t-\tau )
 ^{-1/2}}{\sqrt{\pi }}\\
&\leq \exp \big(\frac{-b^{2}}{16(t-\tau )}\big)
 \frac{(t-\tau )^{-1/2}}{\sqrt{\pi }} \\
&\leq (\frac{8}{eb^{2}})^{1/2}\frac{1}{\sqrt{\pi }}=C_{1}(b)
\end{align*}
then \eqref{i} holds.
To prove \eqref{ii}, we have
\begin{align*}
|G_{x}(s(t),t,s(\tau ),\tau )|
& =\big|K_{x}(s(t),t,s(\tau ),\tau )+K_{x}(-s(t),t,s(\tau ),\tau )\big| \\
&=\frac{(t-\tau )^{-3/2}}{4\sqrt{\pi }}
\Big|(s(t)-s(\tau ))\exp \big(\frac{-(s(t)-s(\tau ))^{2}}{4(t-\tau)}\big)\\
&\quad -(s(t)+s(\tau ))\exp \big(\frac{-(s(t)+s(\tau ))^{2}}{4(t-\tau )}
\big)\Big|
\\
&\leq \frac{(t-\tau )^{-3/2}}{4\sqrt{\pi }}
\Big(M(t-\tau )+3b\exp \big(\frac{-9b^{2}}{4(t-\tau )}\big)\Big)
\\
&\leq \frac{1}{4\sqrt{\pi }}
\Big(M(t-\tau )^{-1/2}+3b\big(\frac{2}{3eb^{2}}\big)^{3/2}\Big)\,.
\end{align*}
Then
\begin{align*}
\int_{0}^{t}|G_{x}(s(t),t,s(\tau ),\tau )| |v(\tau )| \, d\tau
&\leq \frac{M}{4\sqrt{\pi }}\Big(2M\sqrt{t}+3b(\frac{2}{3eb^{2}})
^{3/2}t\Big)
\\
&\leq M^{2}\sqrt{t}\big(\frac{1}{2\sqrt{\pi }}+\frac{3b}{M4\sqrt{\pi }}
(\frac{2}{3eb^{2}})^{3/2}\Big)\\
&\leq M^{2}C_{2}(b)\sqrt{t},
\end{align*}
 which implies \eqref{ii}. To prove \eqref{iii},  we have
\[
\int_{0}^{b}|N(s(t),t,\xi ,0)| |h'(\xi )|
d\xi \leq \|h'\|\int_{0}^{\infty }|N(s(t),t,\xi
,0)| d\xi \leq \|h'\|
\]
because
\[
\int_{0}^{\infty }|N(s(t),t,\xi ,0)| d\xi \leq 1.
\]
To prove \eqref{iv}, by taking into account that
\[
|N(s(t),t,s(\tau ),\tau )
-N(s(t),t,0,\tau )| \leq \frac{2}{\sqrt{\pi (t-\tau )}}
\]
we obtain
\begin{align*}
\int_{0}^{t}|N(s(t),t,s(\tau ),\tau )-N(s(t),t,0,\tau )| |
F(V(\tau ))| \, d\tau
&\leq LM\int_{0}^{t}\frac{2}{\sqrt{\pi (t-\tau )}}\, d\tau \\
& =C_{4}(L)M\sqrt{t}.
\end{align*}
The inequality \eqref{v} is prove in the same way as \eqref{iii}.
To prove \eqref{vi}, we have
\begin{align*}
\int_{0}^{t}|N(0,t,0,\tau )| |\dot {f}(\tau)| \, d\tau
&\leq \|\dot {f}\|_{\sigma }\int_{0}^{t}|N(0,t,0,\tau )| \, d\tau \\
&=\|\dot {f}\|_{\sigma }\int_{0}^{t}\frac{1}{\sqrt{\pi
(t-\tau )}}\, d\tau \\
&=\frac{\|\dot {f}\|_{\sigma }}{\sqrt{\pi }}2\sqrt{t}.
\end{align*}
To prove \eqref{vii}, we have
\begin{align*}
|G_{x}(0,t,s(\tau ),\tau )|
&=\frac{(t-\tau )^{-3/2}}{4\sqrt{\pi }}\,s(\tau )
\exp \Big(\frac{-(s(\tau ))^{2}}{4(t-\tau )}\Big)\\
&\leq \frac{3b}{8\sqrt{\pi }}(t-\tau )^{-3/2}
\exp \Big(\frac{-b^{2}}{16(t-\tau )}\Big)\\
&\leq \frac{3b}{8\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}\,.
\end{align*}
To prove \eqref{viii}, as in \eqref{iv}, we prove that
\[
|N(0,t,s(\tau ),\tau )-N(0,t,0,\tau )| \leq \frac{2}{\sqrt{\pi
(t-\tau )}}
\]
and therefore \eqref{viii} holds.
\end{proof}

\begin{lemma} \label{lem4}
Let $s_{1}$, $s_{2}$ be the functions corresponding to $v_{1}$, $v_{2}$
in $C^{0}[0,\sigma ]$, respectively, with
$\max_{t\in [0,\sigma]}|v_{i}(t)| \leq M$, $i=1,2$, Then we have
\begin{equation}
\begin{gathered}
|s_{2}(t)-s_{1}(t)| \leq t\| v_{2}-v_{1}\|_{t}, \\
|s_{i}(t)-s_{i}(\tau )| \leq M|t-\tau | ,\quad i=1,2, \\
\frac{b}{2}\leq s_{i}(t)\leq \frac{3b}{2},\quad \forall t\in
[0,\sigma ],\; i=1,2.
\end{gathered}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem5}
Let $f\in C^{1}[0,T)$, $h\in C^{1}[0,b]$, $F$ a Lipschitz function
in $C^{0}[0,T]$. We
have
\begin{gather}
|F_{0}(v_{2}(t))-F_{0}(v_{1}(
t))| \leq E(b,h,f)\sqrt{t}\|
v_{2}-v_{1}\|_{t};  \label{i1}
\\
\begin{aligned}
&\int_{0}^{t}|N(s_{2}(t),t,s_{2}(\tau )
,\tau )-N(s_{2}(t),t,0,\tau )| |
F(V_{2}(\tau ))-F(V_{1}(\tau ))| \, d\tau\\
&\leq C_{4}(L)\sqrt{t}\|V_{2}-V_{1}\|_{t};
\end{aligned} \label{ii1}
\\
\begin{aligned}
&\int_{0}^{t}|N(s_{2}(t),t,0,\tau )-N(
s_{1}(t),t,0,\tau )| |F(V_{1}(\tau ))| \, d\tau\\
&\leq C_{5}(b,L,M)t\|v_{2}-v_{1}\|_{t};
\end{aligned} \label{iii1}
\\
\begin{aligned}
&\int_{0}^{t}|N(s_{2}(t),t,s_{2}(\tau )
,\tau )-N(s_{1}(t),t,s_{1}(\tau )
,\tau )| |F(V_{1}(\tau )) | \, d\tau \\
&\leq [C_{6}(L,M)\sqrt{t}+C_{7}(b,L,M)
t]\|v_{2}-v_{1}\|_{t};
\end{aligned} \label{iv1}
\\
\int_{0}^{t}|G_{x}(0,t,s_{2}(\tau ),\tau )| |v_{2}(\tau
)-v_{1}(\tau )| \, d\tau \leq C_{8}(b)t\|
v_{2}-v_{1}\|_{t};  \label{v1}
\\
\begin{aligned}
&\int_{0}^{t}|G_{x}(0,t,s_{2}(\tau ),\tau )v_{2}(\tau
)-G_{x}(0,t,s_{1}(\tau ),\tau )v_{1}(\tau )| \, d\tau\\
&\leq (C_{8}(b)t+C_{9}(b,M)t^{2}) \|v_{2}-v_{1}\|_{t};
\end{aligned} \label{vi1}
\\
\begin{aligned}
&\int_{0}^{t}\Big|[N(0,t,s_{2}(\tau ),\tau )-N(0,t,0,\tau
)]F(V_{2}(\tau ))\\
&-[N(0,t,s_{1}(\tau ),\tau )-N(0,t,0,\tau )]
F(V_{1}(\tau ))\Big| \, d\tau \\
&\leq C_{4}(L)\sqrt{t}\|V_{2}-V_{1}\|
_{t}+C_{5}(b,L,M)t^{2}\|v_{2}-v_{1}\|_{t},
\end{aligned}  \label{vii1}
\end{gather}
where the constants are defined by
\begin{equation}
\begin{gathered}
C_{4}(L)=\frac{4L}{\sqrt{\pi }}, \quad
C_{5}(b,L,M)=LM\frac{3b}{8\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}\,,
\\
C_{6}(L,M)=\frac{LM^{3}}{\sqrt{\pi }}\,, \quad
C_{7}(b,L,M)=(\frac{6}{eb^{2}})^{3/2}\frac{3bLM^{2}}{2\sqrt{\pi }}\,, \\
C_{8}(b)=\frac{3}{4\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}\,,\quad
C_{9}(b,M)=[(\frac{40}{eb^{2}})^{\frac{5}{2}}\frac{9b^{2}}{16\sqrt{\pi }}+\frac{1}{2\sqrt{
\pi }}(\frac{24}{eb^{2}})^{3/2}]\frac{M}{2}.
\end{gathered}
\label{viii1}
\end{equation}
\end{lemma}

\begin{proof}
The proof of \eqref{i1} can be found in  \cite{Fr}.
To prove \eqref{ii1}, we have
\[
|N(s_{2}(t),t,s_{2}(\tau),\tau )-N(s_{2}(t),t,0,\tau )
| \leq \frac{2}{\sqrt{\pi (t-\tau )}}.
\]
 Then
\begin{align*}
&\int_{0}^{t}|N(s_{2}(t),t,s_{2}(\tau ),\tau )-N(s_{2}(t),t,0,\tau )| |
F(V_{2}(\tau ))-F(V_{1}(\tau ) )| \, d\tau \\
&\leq \frac{4L}{\sqrt{\pi }}\sqrt{t}\|V_{2}-V_{1}\|_{t}
\end{align*}
To prove \eqref{iii1}, we use the mean value theorem: There exists
$c=c(t,\tau )$ between $s_{1}(t) $ and $s_{2}(t)$ such that
\begin{align*}
&|N(s_{2}(t),t,0,\tau )-N(s_{1}(t),t,0,\tau )| |F(V_{1}(\tau ))| \\
&=|N_{x}(c,t,0,\tau )| |s_{2}(\tau )-s_{1}(\tau )| |F(V_{1}(\tau ))|\\
&\leq |c| \exp \big(-\frac{c^2}{4(t-\tau )}\big)
\frac{(t-\tau )^{-3/2}}{2\sqrt{\pi }}LM\tau |v_{2}(\tau )-v_{1}(\tau)|
\\
&\leq \frac{3b}{4\sqrt{\pi }}\exp \big(-\frac{b^{2}}{16(t-\tau )}\big)
(t-\tau )^{-3/2}LM\tau |v_{2}(\tau )-v_{1}(\tau )|
\\
 &\leq \frac{3b}{4\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}
 LM\tau |v_{2}(\tau )-v_{1}(\tau )|\,.
\end{align*}
Then
\begin{align*}
&\int_{0}^{t}|N(s_{2}(t),t,0,\tau )-N(
s_{1}(t),t,0,\tau )| |F(V_{1}(\tau ))| \, d\tau \\
&\leq \frac{3b}{8\sqrt{\pi }}(\frac{24}{eb^{2}})^{\frac{3}{2}
}LMt\|v_{2}-v_{1}\|_{t}=C_{5}(b,L,M)t\|
v_{2}-v_{1}\|_{t}.
\end{align*}
To prove \eqref{iv1}, we  have
\begin{align*}
&N(s_{2}(t),t,s_{2}(\tau ),\tau )
-N(s_{1}(t),t,s_{1}(\tau ),\tau ) \\
&=K(s_{2}(t),t,s_{2}(\tau ),\tau )
-K(s_{1}(t),t,s_{1}(\tau ),\tau ) \\
&\quad +K(-s_{2}(t),t,s_{2}(\tau ),\tau )
-K(-s_{1}(t),t,s_{1}(\tau ),\tau )\,.
\end{align*}
As in \cite{Sh}, for each $(t,\tau )$, $0<\tau <t$, we define
\[
f_{t,\tau }(x)=\exp \big(\frac{-x^{2}}{4(t-\tau )}\big).
\]
Then we have
\begin{align*}
&K(s_{2}(t),t,s_{2}(\tau ),\tau )
-K(s_{1}(t),t,s_{1}(\tau ),\tau )\\
&=\frac{(t-\tau )^{-1/2}}{2\sqrt{\pi }}\big[\exp \big(-\frac{
(s_{2}(t)-s_{2}(\tau ))^{2}}{4(t-\tau )}\big)
-\exp \big(-\frac{(s_{1}(t)-s_{1}(\tau ))^{2}}{4(t-\tau )}\big)\big]
\\
&=\frac{(t-\tau )^{-1/2}}{2\sqrt{\pi }}\big[f_{t,\tau
}(s_{2}(t)-s_{2}(\tau ))-f_{t,\tau }(s_{1}(t)-s_{1}(\tau ))\big]
\end{align*}
and
\begin{align*}
&K(-s_{2}(t),t,s_{2}(\tau ),\tau )
-K(-s_{1}(t),t,s_{1}(\tau ),\tau )\\
&=\frac{(t-\tau )^{-1/2}}{2\sqrt{\pi }}\big[\exp \big(-\frac{
(s_{2}(t)+s_{2}(\tau ))^{2}}{4(t-\tau )}\big)
-\exp \big(-\frac{(s_{1}(t)+s_{1}(\tau ))^{2}}{4(t-\tau )}\big)\big]
\\
&=\frac{(t-\tau )^{-1/2}}{2\sqrt{\pi }}\big[f_{t,\tau
}(s_{2}(t)+s_{2}(\tau ))-f_{t,\tau }(s_{1}(t)+s_{1}(\tau ))\big]
\end{align*}
 By the mean value theorem there exists $c=c(t,\tau )$
between $s_{2}(t)-s_{2}(\tau )$ and $s_{1}(t)-s_{1}(\tau )$ such that
\begin{align*}
&f_{t,\tau }(s_{2}(t)-s_{2}(\tau ))-f_{t,\tau }(s_{1}(t)-s_{1}(\tau ))\\
&=f_{t,\tau }^{\;\prime }(c)(s_{2}(t)-s_{2}(\tau )-s_{1}(t)+s_{1}(\tau
)) \\
&=\frac{-c}{2(t-\tau )}\exp (-\frac{c^{2}}{4(t-\tau )})(
s_{2}(t)-s_{2}(\tau )-s_{1}(t)+s_{1}(\tau ))
\end{align*}
Taking into account that
\[
|c| \leq \max \{ |s_{i}(t)-s_{i}(\tau )|,i=1,2\} \leq M(t-\tau )
\]
it results
\begin{align*}
|f_{t,\tau }(s_{2}(t)-s_{2}(\tau ))
 -f_{t,\tau }(s_{1}(t)-s_{1}(\tau))|
&\leq \frac{M}{2}[|s_{2}(t))-s_{1}(t)|
 +|s_{2}(\tau )-s_{1}(\tau )| ] \\
&\leq M^{2}\|v_{2}-v_{1}\|_{t}\,.
\end{align*}
Then we have
\[
|K(s_{2}(t),t,s_{2}(\tau ),\tau
)-K(s_{1}(t),t,s_{1}(\tau ),\tau
)| \leq \frac{M^{2}}{2\sqrt{\pi (t-\tau )}}
\|v_{2}-v_{1}\|_{t}.
\]
In the same way we have
\begin{align*}
&f_{t,\tau }(s_{2}(t)+s_{2}(\tau ))-f_{t,\tau }(s_{1}(t)+s_{1}(\tau ))\\
&=f_{t,\tau }^{\;\prime }(c^{*})(s_{2}(t)+s_{2}(\tau)-s_{1}(t)-s_{1}(\tau ))\\
&=\frac{-c^{*}}{2(t-\tau )}\exp (-\frac{c^{*2}}{4(t-\tau )})
(s_{2}(t)+s_{2}(\tau )-s_{1}(t)-s_{1}(\tau ))
\end{align*}
where $c^{*}=c^{*}(t,\tau )$ is between
$s_{2}(t)+s_{2}(\tau)$ and $s_{1}(t)+s_{1}(\tau )$.
Since $s_{1}(t)+s_{1}(\tau )\leq c^{*}\leq s_{2}(t)+s_{2}(\tau )$,
(or viceversa), we deduce that
$b\leq c^{*}\leq 3b$, that is $\exp (-c^{*2}/4(t-\tau ))\leq
\exp (-b^{2}/4(t-\tau ))$. Then we obtain
\begin{align*}
&|K(-s_{2}(t),t,s_{2}(\tau ),\tau
)-K(-s_{1}(t),t,s_{1}(\tau ),\tau)|\\
&=\frac{(t-\tau )^{-1/2}}{2\sqrt{\pi }}|f_{t,\tau
}(s_{2}(t)+s_{2}(\tau ))-f_{t,\tau }(s_{1}(t)+s_{1}(\tau ))| \\
&\leq \frac{3b}{4\sqrt{\pi }(t-\tau )^{3/2}}
\exp \big(-\frac{b^{2}}{4(t-\tau )}\big)2M\|v_{2}-v_{1}\|_{t}\\
&\leq (\frac{6}{eb^{2}})^{3/2}\frac{3bM}{2\sqrt{\pi }}\|
v_{2}-v_{1}\|_{t}
\end{align*}
and
\begin{align*}
&|N(s_{2}(t),t,s_{2}(\tau ),\tau
)-N(s_{1}(t),t,s_{1}(\tau ),\tau )| \\
&\leq (\frac{M^{2}}{2\sqrt{\pi (t-\tau )}}+(\frac{6}{
eb^{2}})^{3/2}\frac{3bM}{2\sqrt{\pi }})\|
v_{2}-v_{1}\|_{t}.
\end{align*}
Therefore,
\begin{align*}
&\int_{0}^{t}|N(s_{2}(t),t,s_{2}(\tau )
,\tau )-N(s_{1}(t),t,s_{1}(\tau )
,\tau )| |F(V_{1}(\tau ))| \, d\tau \\
&\leq \int_{0}^{t}\Big(\frac{M^{2}}{2\sqrt{\pi (t-\tau )}}
+(\frac{6}{eb^{2}})^{3/2}\frac{3bM}{2\sqrt{\pi }}
\Big)\|v_{2}-v_{1}\|_{t}|F(V_{1}(\tau ))| \, d\tau \\
&\leq LM\Big(\frac{M^{2}\sqrt{t}}{\sqrt{\pi }}+(\frac{6}{eb^{2}}
)^{3/2}\frac{3bM}{2\sqrt{\pi }}t\Big)\|v_{2}-v_{1}\|_{t} \\
&=(C_{6}(L,M)\sqrt{t}+C_{7}(L,M,b)t)\|v_{2}-v_{1}\|_{t}.
\end{align*}
To prove \eqref{v1}, we take into account \eqref{exp}:
\begin{align*}
G_{x}(0,t,s_{2}(\tau ),\tau )
&=K(0,t,s_{2}(\tau ),\tau )\frac{s_{2}(\tau )}{t-\tau } \\
&=\exp \big(-\frac{s_{2}^{2}(\tau )}{4(t-\tau )}\big)
\frac{(t-\tau )^{-3/2}}{2\sqrt{\pi }}s_{2}(\tau ) \\
&\leq \frac{1}{2\sqrt{\pi }}\big(\frac{24}{eb^{2}}\big)^{3/2}s_{2}
(\tau )\leq \frac{3b}{4\sqrt{\pi }}(\frac{24}{eb^{2}})^{
\frac{3}{2}}=C_{8}(b).
\end{align*}
To prove \eqref{vi1}, we have
\begin{align*}
&|G_{x}(0,t,s_{2}(\tau ),\tau )v_{2}(\tau )-G_{x}(0,t,s_{1}(\tau ),\tau
)v_{1}(\tau )| \\
&\leq |G_{x}(0,t,s_{2}(\tau ),\tau )| |v_{2}(\tau )-v_{1}(\tau )| \\
&\quad +|G_{x}(0,t,s_{2}(\tau ),\tau )-G_{x}(0,t,s_{1}(\tau ),\tau )|
|v_{1}(\tau )| .
\end{align*}
Using the mean value theorem there exists $c=c(\tau )$
between $s_{2}(\tau )$\ and$\;s_{1}(\tau )$ such that
$G_{x}(0,t,s_{2}(\tau ),\tau )-G_{x}(0,t,s_{1}(\tau ),\tau )
=G_{x\xi }(0,t,c,\tau )(s_{2}(\tau )-s_{1}(\tau ))$.
 Taking into account the following
properties
\begin{gather*}
G_{x\xi }(0,t,c,\tau )=\frac{K(0,t,c,\tau )}{t-\tau }
\big(\frac{c^{2}}{2(t-\tau )}+1\big),
\\
\frac{K(0,t,c,\tau )}{t-\tau }=\frac{1}{2\sqrt{\pi }}
\exp \big(-\frac{c^{2}}{4(t-\tau )}\big)(t-\tau )^{-3/2}
\leq \frac{1}{2\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2},
\\
K(0,t,c,\tau )\frac{c^{2}}{2(t-\tau )^{2}}=\frac{1}{4\sqrt{\pi }
}\exp \big(-\frac{c^{2}}{4(t-\tau )}\big)(t-\tau )^{-\frac{5}{2}
}c^{2}\leq \frac{9b^{2}}{16\sqrt{\pi }}(\frac{40}{eb^{2}})^{5/2}
\end{gather*}
we have
\begin{align*}
&|G_{x}(0,t,s_{2}(\tau ),\tau )-G_{x}(0,t,s_{1}(\tau ),\tau )|
|v_{1}(\tau )| \\
&\leq (\frac{1}{2\sqrt{\pi }}(\frac{24}{eb^{2}})^{\frac{3
}{2}}+\frac{9b^{2}}{16\sqrt{\pi }}(\frac{40}{eb^{2}})^{\frac{5
}{2}})|s_{2}(\tau )-s_{1}(\tau )| |v_{1}(\tau)|
\\
&\leq M(\frac{1}{2\sqrt{\pi }}(\frac{24}{eb^{2}})^{\frac{3
}{2}}+\frac{9b^{2}}{16\sqrt{\pi }}(\frac{40}{eb^{2}})^{\frac{5}{
2}})\tau |v_{2}(\tau )-v_{1}(\tau )|\,.
\end{align*}
Then
\begin{align*}
&\int_{0}^{t}|G_{x}(0,t,s_{2}(\tau ),\tau )-G_{x}(0,t,s_{1}(\tau ),\tau
)| |v_{1}(\tau )| \, d\tau \\
&\leq (\frac{1}{2\sqrt{\pi }}(\frac{24}{eb^{2}})^{\frac{3
}{2}}+\frac{9b^{2}}{16\sqrt{\pi }}(\frac{40}{eb^{2}})^{\frac{5
}{2}})\frac{Mt^{2}}{2}\|v_{2}-v_{1}\|_{t}.
\end{align*}
Then \eqref{vi1} holds by using \eqref{v1}.
To prove \eqref{vii1}, we have
\begin{equation} \label{e1}
\begin{aligned}
&[N(0,t,s_{2}(\tau ),\tau )-N(0,t,0,\tau )]F(V_{2}(\tau )) \\
&-[N(0,t,s_{1}(\tau ),\tau )-N(0,t,0,\tau )]F(V_{1}(\tau )) \\
&=[N(0,t,s_{2}(\tau ),\tau )-N(0,t,0,\tau )][
F(V_{2}(\tau ))-F(V_{1}(\tau ))] \\
&\quad +[N(0,t,s_{2}(\tau ),\tau )-N(0,t,s_{1}(\tau ),\tau )]
F(V_{1}(\tau ))
\end{aligned}
\end{equation}
Using $|N(0,t,s_{2}(\tau ),\tau )-N(0,t,0,\tau )
| \leq \frac{2}{\sqrt{\pi (t-\tau )}}$ we get
\[
|N(0,t,s_{2}(\tau ),\tau )-N(0,t,0,\tau )| |
F(V_{2}(\tau ))-F(V_{1}(\tau ))|
 \leq \frac{2}{\sqrt{\pi (t-\tau )}}L|V_{2}(\tau
)-V_{1}(\tau )| ,
\]
and
\begin{equation}
\begin{aligned}
&\int_{0}^{t}|N(0,t,s_{2}(\tau ),\tau )-N(0,t,0,\tau )
| |F(V_{2}(\tau ))-F(V_{1}(\tau ))| \, d\tau \\
&\leq  \frac{4\sqrt{t}}{\sqrt{\pi }}L\|V_{2}-V_{1}\|_{t}
=C_{4}(L) \sqrt{t}\|V_{2}-V_{1}\|_{t}.
\end{aligned} \label{ac1}
\end{equation}
Furthermore,
\[
|N(0,t,s_{2}(\tau ),\tau )-N(0,t,s_{1}(\tau ),\tau )| =|
N_{\xi }(0,t,c,\tau )| |s_{2}(\tau )-s_{1}(\tau )|
\]
where $c=c(\tau )$ is between $s_{2}(\tau )$\ and$\;s_{1}(\tau
)$ and
\begin{align*}
|N_{\xi }(0,t,c,\tau )| |s_{2}(\tau )-s_{1}(\tau )|
&=|-G_{x}(0,t,c,\tau )| |s_{2}(\tau )-s_{1}(\tau )| \\
&\leq \frac{|c| }{2\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}
\tau |v_{2}(\tau )-v_{1}(\tau )| \\
&\leq \frac{3b}{4 \sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}\tau |
v_{2}(\tau )-v_{1}(\tau )| .
\end{align*}
Then
\begin{equation}
\begin{aligned}
&\int_{0}^{t}|N(0,t,s_{2}(\tau ),\tau )-N(0,t,s_{1}(\tau ),\tau
)| |F(V_{1}(\tau ))| \, d\tau \\
&\leq LM\frac{3b}{4\sqrt{\pi }}(\frac{24}{eb^{2}})^{3/2}
\frac{t^{2}}{2}\|v_{2}-v_{1}\|_{t}=C_{5}(L,M,b)t^{2}\|
v_{2}-v_{1}\|_{t}
\end{aligned} \label{ac2}
\end{equation}
Therefore, by \eqref{e1}, \eqref{ac1}, and
\eqref{ac2}), the inequality \eqref{vii1} holds.
\end{proof}

\begin{theorem} \label{thm6}
The map $A:C_{M,\sigma }\to C_{M,\sigma }$ is well defined and
is a contraction map if $\sigma $ satisfies the following
inequalities:
\begin{gather}
\sigma \leq 1\,, 2M\sigma \leq b  \label{nose} \\
(2\|\dot {f}\|_{\sigma }C_{1}(b)
+MC_{3}(b))\sigma +(2M^{2}C_{2}(b)+
\frac{2\|\dot {f}\|_{\sigma }}{\sqrt{\pi }}
+3MC_{4}(L))\sqrt{\sigma }\leq 1  \label{des1sigma} \\
D(b,f,h,L,M)\sqrt{\sigma }<1 , \label{des2sigma}
\end{gather}
where
\begin{equation}
M=1+3\|h'\|\label{eme}
\end{equation}
and
\begin{gather*}
D_{1}(b,f,h,L,M)=E(b,f,h)+2C_{6}(L,M)+3C_{4}(L)\\
D_{2}(b,L,M)=2[C_{5}(b,L,M)+2C_{7}(b,L,M)+C_{8}(b)]\\
D_{3}(b,L,M)=C_{9}(b,M)+C_{5}(b,L,M) \\
D(b,f,h,L,M)= D_{1}(b,f,h,L,M)+D_{2}(b,L,M)+D_{3}(b,L,M).
\end{gather*}
Then there exists a unique solution on $C_{M,\sigma }$ to the system of
integral equations \eqref{ecintegral}, (\ref{ecintegral2}).
\end{theorem}

\begin{proof}
Firstly we demonstrate that $A$ maps $C_{\sigma ,M}$ into itself, that
is
\begin{equation}
\|A(\vec {w})\|_{\sigma }
=\max_{t\in [0,\sigma ]}|A_{1}(v(t),V(t))|
+\max_{t\in [0,\sigma ]}|A_{2}(v(t),V(t))| \leq M  \label{1}
\end{equation}
Using the Lemmas \ref{lem3}, \ref{lem4} and the definitions \eqref{a1}-\eqref{a2}, we have
\begin{gather*}
|A_{1}(v(t),V(t))| \leq 2\|\dot {f}\|
_{\sigma }C_{1}(b)t+2M^{2}C_{2}(b)\sqrt{t}
+2\|h'\|+2C_{4}(L)M\sqrt{t},\\
|A_{2}(v(t),V(t))| \leq \|h'\|+(
\frac{2\|\dot {f}\|_{\sigma }}{\sqrt{\pi }}+C_{4}(
L)M)\sqrt{t}+C_{3}(b)Mt.
\end{gather*}
Then
\begin{align*}
\|A(\vec {w})\|_{\sigma }
&=\max_{t\in [0,\sigma ]}|A_{1}(v(t),V(t))|
 +\max_{t\in [0,\sigma ]}|A_{2}(v(t),V(t))| \\
&\leq 3\|h'\|+(2\|\dot {f}\|_{\sigma }C_{1}(b)+C_{3}(b)M)\sigma \\
&\quad +\Big(2M^{2}C_{2}(b)+\frac{2\|\dot {f}\|
_{\sigma }}{\sqrt{\pi }}+3MC_{4}(L)\Big)\sqrt{\sigma }.
\end{align*}
Selecting $M$ by \eqref{eme} and $\sigma $
such that \eqref{nose} and \eqref{des1sigma} hold, we obtain \eqref{1}.

Now, we prove that
\[
\|A(\vec {w_{2}})-A(\stackrel{
\longrightarrow }{w_{1}})\|_{\sigma }\leq D(
b,h,f,L,M)\sqrt{\sigma }\|\vec {w_{2}}-
\vec {w_{1}}\|_{\sigma }
\]
where $\vec {w_{1}}=\binom{v_{1}}{V_{1}}$,
$\vec{w_{2}}=\binom{v_{2}}{V_{2}}$. By selecting  $\sigma$ such that
\eqref{des2sigma} holds, $A$ becomes a contraction mapping on
$C_{\sigma ,M}$ and therefore it has a unique fixed point.
To prove this assertion we consider
\[
A(\vec {w_{1}})(t)-A(
\vec {w_{2}})(t)=\binom{A_{1}(v_{2}(t),V_{2}(t))-A_{1}(
v_{1}(t),V_{1}(t))}{A_{2}(v_{2}(
t),V_{2}(t))-A_{2}(v_{1}(t)
,V_{1}(t))}
\]
where
\begin{align*}
&A_{1}(v_{2}(t),V_{2}(t))-A_{1}(v_{1}(t),V_{1}(t))\\
&=F_{0}(v_{2}(t))-F_{0}(v_{1}(t))
 +2\int_{0}^{t}[N(s_{2}(t),t,s_{2}(\tau),\tau )-N(s_{2}(t),t,0,\tau )
 ]F(V_{2}(\tau ))\, d\tau
\\
&\quad -2\int_{0}^{t}[N(s_{1}(t),t,s_{1}(\tau
),\tau )-N(s_{1}(t),t,0,\tau )
]F(V_{1}(\tau ))\, d\tau
\end{align*}
and
\begin{align*}
&A_{2}(v_{2}(t),V_{2}(t))-A_{2}(v_{1}(t),V_{1}(t))\\
&=\int_{0}^{t}[G_{x}(0,t,v_{2}(\tau ),\tau )v_{2}(\tau
)-G_{x}(0,t,v_{1}(\tau ),\tau )v_{1}(\tau )]\, d\tau
\\
&\quad +\int_{0}^{t}\big\{ [N(0,t,s_{2}(\tau ),\tau
)-N(0,t,0,\tau )]F(V_{2}(\tau ))
\\
&\quad  -[N(0,t,s_{1}(\tau ),\tau )-N(
0,t,0,\tau )]F(V_{1}(\tau ))\big\} \, d\tau\,.
\end{align*}
Taking into account the Lemmas \ref{lem4} and \ref{lem5} it results
\begin{align*}
&|A_{1}(v_{2}(t),V_{2}(t))-A_{1}(v_{1}(t),V_{1}(t))| \\
&\leq  E(b,h,f)\sqrt{t}\|v_{2}-v_{1}\|
_{t}+2C_{4}(L)\sqrt{t}\|V_{2}-V_{1}\|_{t}\\
&\quad +2C_{5}(b,L,M)t\|v_{2}-v_{1}\|_{t}+2[
C_{6}(L,M)\sqrt{t}+C_{7}(b,L,M)t]\|
v_{2}-v_{1}\|_{t},
\end{align*}
and
\begin{align*}
&|A_{2}(v_{2}(t),V_{2}(t))-A_{2}(v_{1}(t),V_{1}(t))|\\
& \leq(C_{8}(b)t+C_{9}(b,M)t^{2})\| v_{2}-v_{1}\|_{t} \\
&\quad +C_{4}(L)\sqrt{t}\|V_{2}-V_{1}\|_{t}+C_{5}(
b,L,M)t^{2}\|v_{2}-v_{1}\|_{t}\,.
\end{align*}
Therefore,
\begin{align*}
&\|A(\vec {w_{2}})-A(\vec {w_{1}})\|_{\sigma }\\
&\leq \max_{t\in [0,\sigma ]}|A_{1}(v_{2}(t),V_{2}(t))
-A_{1}(v_{1}(t),V_{1}(t))| \\
&\quad +\max_{t\in [0,\sigma ]} |A_{2}(v_{2}(t),V_{2}(t))-A_{2}(v_{1}(t),
V_{1}(t) )| \\
&\leq \{ D_{1}(b,f,h,L,M)\sqrt{\sigma }+D_{2}(
b,L,M)\sigma +D_{3}(b,L,M)\sigma ^{2}\} \|
\vec {w_{2}}-\vec {w_{1}}\|_{\sigma } \\
&\leq D(b,f,h,L,M)\sqrt{\sigma }\|\vec {w_{2}}-\vec {w_{1}}\|_{\sigma}\,.
\end{align*}
By hypothesis (\ref{des2sigma}) we have that $A$ is a contraction.
\end{proof}

\subsection*{Remark}
If $F$ satisfies  the  conditions
\begin{itemize}
\item[(H2)]  $F(V)>0$, for all $V\neq 0$ and  $F(0)=0$,
\end{itemize}
then by the maximum principle \cite{Ber}, $u$ is a sub-solution for
the same problem with $F\equiv 0$, that is
\[
u(x,t)\leq u_{0}(x,t)\,,\quad s(t)\leq s_{0}(t)
\]
where $u_{0}(x,t)$ and $s_{0}(t)$ solve the classical Stefan problem
\begin{gather*}
u_{0t}-u_{0xx}=0, \quad 0<x<s_{0}(t),\;0<t<T, \\
u_{0}(0,t)=f(t)\geqslant 0, \quad 0<t<T, \\
u_{0}(s_{0}(t),t)=0,\quad u_{_{0}x}(s_{0}(t),t)=-\dot {s_{0}}(t).
 0<t<T, \\
u_{0}(x,0)=h(x) \quad  0\leqslant x\leqslant b=s_{0}(0)\,.
\end{gather*}

\subsection*{Acknowledgments}
The authors have been partially sponsored by
Project PIP No. 5379 from CONICET - UA (Rosario, Argentina),
 by Project \#53900/4 from Fundaci\'{o}n Antorchas (Argentina),
and and by ANPCYT PICT \# 03-11165 from Agencia (Argentina).

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\end{document}