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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 115, pp. 1--18.\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{8mm}}

\begin{document}
\title[\hfilneg EJDE-2006/115\hfil Solution to a semilinear pseudoparabolic
problem]{Solution to a semilinear pseudoparabolic problem with integral
conditions}

\author[A. Bouziani, N. Merazga, \hfil EJDE-2006/115\hfilneg]
{Abdelfatah Bouziani, Nabil Merazga}

\address{Abdelfatah Bouziani \newline
D\'{e}partement de Math\'{e}matiques, Centre Universitaire Larbi Ben M'hidi,
Oum El Bouaghi 04000, Algeria}
\email{aefbouziani@yahoo.fr}

\address{Nabil Merazga \newline
D\'{e}partement de Math\'{e}matiques, Centre Universitaire Larbi Ben M'hidi,
Oum El Bouaghi 04000, Algeria}
\email{nabilmerazga@yahoo.fr}

\date{}
\thanks{Submitted March 7, 2006. Published September 21, 2006.}
\subjclass[2000]{35K70, 35A35, 35B30, 35B45, 35D05}
\keywords{Semilinear pseudoparabolic equation; time-discretization method;
\hfill\break\indent integral condition; a priori estimate; 
generalized solution}

\begin{abstract}
  In this article, we use the Rothe time-discretization method 
  to prove the well-posedness of a mixed problem with integral 
  conditions for a third order semilinear pseudoparabolic equation.
  Also we establish the convergence of the method and an error 
  estimate for a semi-discrete approximation.
\end{abstract}

\maketitle

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

\numberwithin{equation}{section} 
\allowdisplaybreaks

\section{Statement of the problem}

This paper concerns the problem of finding a function $v=v(x,t)$ satisfying,
in a weak sense, the semilinear pseudoparabolic equation 
\begin{equation}
\frac{\partial v}{\partial t}-\frac{\partial ^{2}v}{\partial x^{2}}-\eta 
\frac{\partial ^{3}v}{\partial x^{2}\partial t}=F(x,t,v)\,,\quad (x,t)\in
(0,1)\times [0,T],  \label{e1.1}
\end{equation}
subject to the initial condition 
\begin{equation}
v(x,0)=V_{0}(x),\quad 0\leq x\leq 1,  \label{e1.2}
\end{equation}
and to the integral conditions 
\begin{gather}
\int_{0}^{1}v(x,t)dx=E(t),\quad 0 \leq t\leq T,  \label{e1.3} \\
\int_{0}^{1}xv(x,t)dx=G(t),\quad 0 \leq t\leq T,  \label{e1.4}
\end{gather}
where $F$, $V_{0}$, $E$ and $G$ are given functions which are sufficiently
regular, and $T$ and $\eta $ are positive constants.

This problem has a practical relevance, for instance in the context of soil
thermophysics, \eqref{e1.1} describes the dynamics of moisture transfer in a
subsoil layer $0<x<1$ for $t\in [0,T]$, while \eqref{e1.3}-\eqref{e1.4}
represent the moisture moments (see \cite{B3} and references therein).
Equations of type \eqref{e1.1} (with eventually variable coefficients and
additional nonlinear terms) have also many other applications in various
physical situations, notably in the non-steady flows of second order fluids 
\cite{T1, CN}; in the infiltration of homogeneous fluids through fissured
rocks \cite{BZK}; in the diffusion of imprisoned resonant radiation through
a gas \cite{Mi, MZ, Sob} (which has applications in the analysis of certain
laser systems \cite{PS}); in the theory of the two temperatures in heat
conduction \cite{CG}; in the monodirectional propagation of nonlinear
dispersive long waves \cite{BBM, Kar}, and so forth. This is the main reason
for which the investigation of (classical) mixed problems for such equations
have been the subject of many works for a long time, (see, e.g. \cite{B1,
KP,Me,OA,ST,Show,T2,V}).

Recently, mixed problems with integral condition(s) for some generalizations
of equation \eqref{e1.1} have been treated by the second author in \cite{B3,
B4} using the energy-integral method. Differently to these works, in the
present paper we use a constructive method (Rothe time-discretization
method) to build the solution, which is more suitable for numerical
computations. It is interesting to note that the application of Rothe method
to this nonlocal problem is made possible thanks, essentially, to the use of
a nonclassical function space (see also \cite{MB}).

By the the transformation 
\begin{equation*}
u(x,t):=v(x,t)-r(x,t),\quad (x,t)\in (0,1)\times [0,T],
\end{equation*}
where 
\begin{equation*}
r(x,t)=6({2G(t)-E(t)})x-2({3G(t)-2E(t)}),
\end{equation*}
problem \eqref{e1.1}-\eqref{e1.4} with inhomogeneous integral conditions %
\eqref{e1.3} and \eqref{e1.4} is converted to the following equivalent
problem with homogeneous conditions for the new unknown function $u$: 
\begin{gather}
\frac{\partial u}{\partial t}-\frac{\partial ^{2}u}{\partial x^{2}}-\eta 
\frac{\partial ^{3}u}{\partial x^{2}\partial t}=f(x,t,u)\,,\quad (x,t)\in
(0,1)\times I,  \label{e1.5} \\
u(x,0)=U_{0}(x),\quad 0\leq x\leq 1,  \label{e1.6} \\
\int_{0}^{1}u(x,t)dx=0,\quad t\in I,  \label{e1.7} \\
\int_{0}^{1}xu(x,t)dx=0,\quad t\in I,  \label{e1.8}
\end{gather}
where the notation $I:=[0,T]$ is used and 
\begin{gather*}
f(x,t,u) :=F(x,t,u+r)-\frac{\partial r}{\partial t}(x,t), \\
U_{0}(x) :=V_{0}(x)-r(x,0).
\end{gather*}
Hence, instead of looking for the function $v$, we seek the function $u$.
The solution of problem \eqref{e1.1}-\eqref{e1.4} will be simply given by
the formula $v=u+r$.

This paper is organized as follows: In Section 2, we introduce function
spaces needed in our investigation and recall an auxiliary result. We also
state the assumptions on data and make precise concept of the solution. In
Section 3, approximate solutions of problem \eqref{e1.5}-\eqref{e1.8} are
constructed by solving the corresponding linearized time-discretized
problems. Then, some a priori estimates for the approximations are derived
in Section 4, while the convergence of the method and the well-posedness of
the problem under study are established in Section 5.

\section{Preliminaries and main result}

Let $H^{2}(0,1)$ be the (real) second order Sobolev space on $(0,1)$ with
norm $\| \cdot \| _{H^{2}(0,1)}$ and let $(\cdot,\cdot)$ and $\| \cdot \| $
be the usual inner product and the corresponding norm respectively in 
$L^{2}(0,1)$. The nature of the boundary conditions \eqref{e1.7}-\eqref{e1.8}
suggests to introduce the following space 
\begin{equation}
V:=\big\{ \phi \in L^{2}(0,1):\int_{0}^{1}\phi (x)dx=\int_{0}^{1}x\phi
(x)dx=0\big\}  \label{e2.1}
\end{equation}
which is clearly a Hilbert space for $(\cdot ,\cdot)$.

Our analysis requires the use of the nonclassical function space 
$B_{2}^{1}(0,1)$ defined for example in \cite{B2} as the completion of the
space $C_{0}(0,1)$ of real continuous functions with compact support in 
$(0,1)$, for the inner product 
\begin{equation}
(u,v)_{B_{2}^{1}}=\int_{0}^{1}\Im _{x}u\cdot \Im _{x}v\,dx,  \label{e2.2}
\end{equation}
and the associated norm 
\begin{equation*}
\| v\| _{B_{2}^{1}}=\sqrt{(v,v)_{B_{2}^{1}}},
\end{equation*}
where $\Im _{x}v:=\int_{0}^{x}v(\xi )d\xi $ for $x\in (0,1)$. We recall
that, for $v\in L^{2}(0,1)$, the inequality 
\begin{equation}
\| v\| _{B_{2}^{1}}^{2}\leq \frac{1}{2}\| v\| ^{2}  \label{e2.3}
\end{equation}
holds, implying the continuity of the embedding $L^{2}(0,1)\to
B_{2}^{1}(0,1) $. Moreover, we will work in the standard functional spaces 
$C(I,X)$, $C^{0,1}(I,X)$ and $L^{2}(I,X)$ where $X$ is a Banach space, the
main properties of which can be found in \cite{Kuf}.

The notation $\theta (t)$ is automatically used for the same function 
$\theta (x,t)$ considered as an abstract function of the variable $t\in I$
into some functional space on $(0,1)$. Strong or weak convergence are
denoted by $\to $ or $\rightharpoonup $ respectively.\newline
The Gronwall Lemma in the following continuous and discrete forms will be
very useful to us thereafter.

\begin{lemma} \label{lem2.1}
\begin{itemize}
\item[(i)] Let $x(t)\geq 0$, $h(t),y(t)$ be real integrable functions on the
interval $[a,b]$. If
\begin{equation*}
y(t)\leq h(t)+\int_{a}^{t}x(\tau )\,y(\tau )d\tau ,\quad \forall t\in
[a,b],
\end{equation*}
then
\begin{equation*}
y(t)\leq h(t)+\int_{a}^{t}h(\tau )\,x(\tau )\,\exp \Big(\int_{\tau
}^{t}x(s)ds\Big)d\tau ,\quad \forall t\in [a,b].
\end{equation*}
In particular, if $x(\tau )\equiv C$ is a constant and $h(\tau )$ is
nondecreasing, then
\begin{equation*}
y(t)\leq h(t)e^{C(t-a)},\quad \forall t\in [a,b].
\end{equation*}

\item[(ii)]  Let $\{ a_{i}\} $ be a sequence of real nonnegative
numbers satisfying
\begin{gather*}
a_{1}\leq a, \\
a_{i}\leq a+bh\sum_{k=1}^{i-1}a_{k},\quad \forall i=2,\dots ,
\end{gather*}
where $a$, $b$ and $h$ are positive constants. Then
\begin{equation*}
a_{i}\leq ae^{b(i-1)h},\quad \forall i=1,2,\dots \,.
\end{equation*}
\end{itemize}
\end{lemma}

\begin{proof}
The proof of assertion (i) is the same as in \cite[Lemma 1.3.19]{Kac}. To
establish assertion (ii), we use induction on $i$ giving 
\begin{equation*}
a_{i}\leq a(1+bh)^{i-1},\quad \forall i=1,2,\dots \,.
\end{equation*}
from where, the desired inequality follows thanks to the elementary
inequality $1+t\leq e^{t}$, for all $t\in \mathbb{R}_{+}$.
\end{proof}

We shall work under the following hypotheses:

\begin{itemize}
\item[(H1)] $f(t,w)\in L^{2}(0,1)$ for each pair $(t,w)\in I\times
L^{2}(0,1) $ and the Lipschitz condition 
\begin{equation*}
\| f(t,w)-f(t^{\prime},w^{\prime})\| _{B_{2}^{1}} \leq l\big[ |
t-t^{\prime}| (1+\| w\|_{B_{2}^{1}}+\| w^{\prime}\| _{B_{2}^{1}}) +\|
w-w^{\prime}\| _{B_{2}^{1}} \big] ,
\end{equation*}
is satisfied for all $t,t^{\prime}\in I$ and $w,w^{\prime}\in V$, where $l$
is some positive constant.

\item[(H2)] $U_{0}\in H^{2}(0,1)$

\item[(H3)] the compatibility condition: $U_{0}\in V$, i.e. 
$\int_{0}^{1}U_{0}(x)dx=\int_{0}^{1}xU_{0}(x)dx=0$.
\end{itemize}

We look for a weak solution in the following sense:

\begin{definition} \label{def2.1} \rm
By a weak solution of Problem \eqref{e1.5}-\eqref{e1.8}, we mean a function
$u:I\to L^{2}(0,1)$ such that
\begin{itemize}
\item[(i)] $u\in C^{0,1}(I,V)$;

\item[(ii)] $u$ has (a.e. in $I$) a strong derivative
$\frac{du}{dt}\in L^{\infty }(I,L^{2}(0,1))$;

\item[(iii)] $u(0)=U_{0}$ in $V$;

\item[(iv)] the equality
\begin{equation} \label{e2.4}
\big( \frac{du}{dt}(t),\phi\big)_{B_{2}^{1}}
+\big(u(t),{\phi }\big)+\eta \big(\frac{du}{dt}(t),\phi\big)
=\big(f(t,u(t)), {\phi }\big)_{B_{2}^{1}},
\end{equation}
holds for all $\phi \in V$ and all $t\in I$.
\end{itemize}
\end{definition}

We remark that since $u\in C^{0,1}(I,V)\subset C(I,V)$ the condition (iii)
makes sense, and in view of (i), (ii) and Assumption (H1) each term in 
\eqref{e2.4} is well defined. On the other hand, the fulfillment of the
integral conditions \eqref{e1.7} and \eqref{e1.8} is included in the fact
that $u(t)\in V$, for all $t\in I$.

In this paper, we will demonstrated the following main result.

\begin{theorem} \label{thm2.2}
Assuming (H1)--(H3), problem \eqref{e1.5}-\eqref{e1.8} admits a unique
weak solution $u$ in the sense of Definition \ref{def2.1},
that depends continuously upon the data $f$ and $U_{0}$.
Moreover, $u$ is the limit as $n\to \infty $ of the sequence of Rothe
functions \eqref{e3.13} in the following sense:
\begin{gather*}
u^{(n)}\to u\quad \text{in  }C(I,V),\quad (\text{with
convergence order }O(n^{-1/2})),
\\
\frac{du^{(n)}}{dt}\rightharpoonup \frac{du}{dt}\quad \text{in }
L^{2}(I,L^{2}(0,1)).
\end{gather*}
\end{theorem}

\section{Rothe approximations}

To solve problem \eqref{e1.5}-\eqref{e1.8} by the Rothe method, we divide
the time interval $I$ into $n$ subintervals $[t_{j-1},t_{j}]$, 
$j=1,\dots ,n$, where $t_{j}=jh$ and $h:=T/n$ is the time-step. 
Then, replacing $\frac{\partial u}{\partial t}$, at each point 
$t=t_{j}$, $j=1,\dots ,n$, by the
difference quotient $\delta u_{j}:=\frac{u_{j}-u_{j-1}}{h}$, where $u_{j}$
is destined to be an approximation of $u(\cdot ,t_{j})$, we are conducted to 
\textit{solve successively for } $j=1,\dots ,n$ the linearized problem 
\begin{gather}
\delta u_{j}-\frac{d^{2}u_{j}}{dx^{2}}-\eta \frac{d^{2}\delta u_{j}}{dx^{2}}
=f_{j}\,,\quad x\in (0,1),  \label{e3.1} \\
\int_{0}^{1}u_{j}(x)dx=0,  \label{e3.2} \\
\int_{0}^{1}xu_{j}(x)dx=0,  \label{e3.3}
\end{gather}
where $f_{j}:=f(t_{j},u_{j-1})$, starting from 
\begin{equation}
u_{0}=U_{0}.  \label{e3.4}
\end{equation}

To this purpose, it is astute to introduce the following auxiliary functions 
\begin{equation}
w_{j}=u_{j}+\eta \delta u_{j},\quad j=1,\dots ,n.  \label{e3.5}
\end{equation}
In this case, we have 
\begin{equation*}
u_{j}=\frac{h}{\eta +h}w_{j}+\frac{\eta }{\eta +h}u_{j-1},\quad j=1,\dots ,n,
\end{equation*}
from which, it follows 
\begin{equation}
\delta u_{j}=\frac{1}{\eta +h}(w_{j}-u_{j-1}),\quad j=1,\dots ,n,
\label{e3.6}
\end{equation}
so that, problem \eqref{e3.1}-\eqref{e3.3} is seen to be equivalent to the
problem of \textit{finding the function} $w_{j}:(0,1)\to \mathbb{R}$
satisfying: 
\begin{gather}
-\frac{d^{2}w_{j}}{dx^{2}}+\frac{1}{\eta +h}w_{j}=f_{j}+\frac{1}{\eta +h}
u_{j-1}\,,\quad x\in (0,1),  \label{e3.7} \\
\int_{0}^{1}w_{j}(x)dx=\int_{0}^{1}xw_{j}(x)dx=0,  \label{e3.8}
\end{gather}
with the update 
\begin{equation}
u_{j}=\frac{h}{\eta +h}w_{j}+\frac{\eta }{\eta +h}u_{j-1},\quad j=1,\dots ,n.
\label{e3.9}
\end{equation}
Of course, this coupled problem has to be solved successively for
 $j=1,\dots ,n$ starting from $u_{0}=U_{0}$.

Developing an idea of \cite{SC}, we, first, look for a function 
$w_{j}^{\prime}(x)=w_{j}^{\prime}(x;\lambda ,\mu )$ which solves equation 
\eqref{e3.7} supplemented by the classical Dirichlet boundary conditions 
\begin{equation}
w_{j}^{\prime}(0)=\lambda \quad \text{and}\quad w_{j}^{\prime}(1)=\mu ,
\label{e3.10}
\end{equation}
instead of nonlocal conditions \eqref{e3.8}, where $(\lambda ,\mu )$ is for
the moment an arbitrary fixed ordered pair of real numbers.

Since 
\begin{equation*}
f_{1}+\frac{1}{\eta +h}u_{0}:=f(t_{1},U_{0})+\frac{1}{\eta +h} U_{0}\in
L^{2}(0,1),
\end{equation*}
the Lax-Milgram Lemma guarantees the existence and uniqueness of a strong
solution $w_{1}^{\prime}\in H^{2}(0,1)$ to the elliptic problem \eqref{e3.7}
and \eqref{e3.10} with $j=1$. Let us show that the parameters $\lambda $ and 
$\mu $ can be chosen in a suitable way such that the corresponding function 
$w_{1}^{\prime}(\cdot ;\lambda ,\mu )$ is also a solution of problem 
\eqref{e3.7}-\eqref{e3.8} with $j=1$ provided that $n$ is large enough.

In fact, the function $w_{1}^{\prime}(\cdot ;\lambda ,\mu )$ shall be a
solution to problem \eqref{e3.7}-\eqref{e3.8}, with $j=1$, if and only if
the pair $(\lambda ,\mu )$ satisfies 
\begin{equation}
\begin{gathered} \int_{0}^{1}w_{1}'(x;\lambda ,\mu )dx=0, \\
\int_{0}^{1}xw_{1}'(x;\lambda ,\mu )dx=0, \end{gathered}  \label{e3.11}
\end{equation}
thus, the above equation will provide all the solutions to problem 
\eqref{e3.7}-\eqref{e3.8} with $j=1$. But, 
\begin{equation*}
w_{1}^{\prime}(x;\lambda ,\mu )=w_{1}^{\prime}(x;0,0)+\widetilde{w}
_{1}(x;\lambda ,\mu ),\quad x\in (0,1),
\end{equation*}
where $\widetilde{w}_{1}$ is the solution to the problem: 
\begin{gather*}
-\frac{d^{2}\widetilde{w}_{1}}{dx^{2}}+\frac{1}{\eta +h}\widetilde{w}
_{1}=0,\quad x\in (0,1), \\
\widetilde{w}_{1}(0)=\lambda ,\text{ }\widetilde{w}_{1}(1)=\mu .
\end{gather*}
One can readily find that 
\begin{equation*}
\widetilde{w}_{1}(x)=k_{1}e^{x/\sqrt{\eta +h}}+k_{2}e^{-x/\sqrt{\eta +h}},
\end{equation*}
where 
\begin{equation*}
k_{1}=\frac{\mu -\lambda e^{-1/\sqrt{\eta +h}}}{e^{1/\sqrt{\eta +h}}-e^{-1/ 
\sqrt{\eta +h}}},\quad k_{2}=\frac{\lambda e^{1/\sqrt{\eta +h}}-\mu }{e^{1/ 
\sqrt{\eta +h}}-e^{-1/\sqrt{\eta +h}}}.
\end{equation*}
Then, replacing in \eqref{e3.11} and performing some computations and
elementary simplifications, we finally obtain the following equivalent
linear algebraic system 
\begin{equation}
\begin{gathered} \lambda +\mu =\frac{\sinh (1/\sqrt{\eta +h})}{\sqrt{\eta
+h} (1-\cosh (1/\sqrt{\eta +h}))}\int_{0}^{1}w_{1}'(x;0,0)dx, \\
\begin{aligned} &(1-\sqrt{\eta +h}\sinh \frac{1}{\sqrt{\eta +h}})\lambda +(
\sqrt{\eta +h}\sinh \frac{1}{\sqrt{\eta +h}}-\cosh \frac{1}{\sqrt{\eta +h}}
)\mu \\ &=\frac{\sinh (1/\sqrt{\eta +h })}{\sqrt{\eta
+h}}\int_{0}^{1}xw_{1}'(x;0,0)dx \end{aligned} \end{gathered}  \label{e3.12}
\end{equation}
whose determinant is 
\begin{equation*}
D(h)=2\sqrt{\eta +h}\,\sinh \frac{1}{\sqrt{\eta +h}}
-\cosh \frac{1}{\sqrt{\eta +h}}-1.
\end{equation*}
It can be shown that the real function $\Phi (s):=2\sqrt{s}\sinh \frac{1}{ 
\sqrt{s}}-\cosh \frac{1}{\sqrt{s}}-1$ possesses a unique real root 
$\overline{s}\simeq 3.448\times 10^{15}$. Therefore, if 
$\eta \geq \overline{s}$ then $D(h)\neq 0$ for all $h>0$ and the system \eqref{e3.12} which is
equivalent to \eqref{e3.11} admits a unique solution $(\lambda _{1},\mu
_{1})\in \mathbb{R}^{2}$, hence problem \eqref{e3.7}-\eqref{e3.8}, with $j=1$
, is uniquely solvable. In the case where $\eta <\overline{s}$, then $D(h)$
vanishes only for $h= \overline{s}-\eta $, consequently the system 
\eqref{e3.12} which is equivalent to \eqref{e3.11} has a unique solution for
every $h<\overline{s}-\eta $ and so is problem \eqref{e3.7}-\eqref{e3.8}
with $j=1$. To summarize, let $n_{0}$ be the smallest positive integer
satisfying $T/n_{0}\leq h_{0}$ where 
\begin{equation*}
h_{0}:=\begin{cases}
T, & \text{if }\eta \geq \overline{s}, \\ 
\min \{ \overline{s}-\eta ,T\} , & \text{if } \eta <\overline{s}.
\end{cases}
\end{equation*}
Then we have shown that problem \eqref{e3.7}-\eqref{e3.8}, with $j=1$,
admits a unique solution $w_{1}=w_{1}^{\prime}(\cdot ;\lambda _{1},\mu
_{1})\in H^{2}(0,1)$ and consequently the solution $u_{1}\in H^{2}(0,1)$ of
problem \eqref{e3.1}-\eqref{e3.3}, with $j=1$, is uniquely determined via
the formula \eqref{e3.9} provided that $n>n_{0}$ holds. Replacing in 
\eqref{e3.7} with $j=2$, this latter is seen to admit a unique strong
solution $w_{2}^{\prime}\in H^{2}(0,1)$ satisfying \eqref{e3.10} wiht $j=2$.
Thanks to Lax-Milgram Lemma since $f_{2}+\frac{1}{\eta +h}u_{1}\in
L^{2}(0,1) $. Similarly as above, the function $w_{2}^{\prime}(\cdot;\lambda
,\mu )$ is seen to be the unique solution of problem 
\eqref{e3.7}-\eqref{e3.8} with $j=2 $ for a suitable choice of $(\lambda ,\mu )$ 
if $n>n_{0}$ is true. Accordingly, the solution $u_{2}\in H^{2}(0,1)$
 of problem 
\eqref{e3.1}-\eqref{e3.3} with $j=2$ is uniquely determined due to relation 
\eqref{e3.9}. Proceeding in this way step by step, we will be able to state
the following result:

\begin{theorem} \label{thm3.1}
There exists $n_{0}\in \mathbb{N}$ such that, for all $n>n_{0}$ and for all
$j=1,\dots ,n$, the problems \eqref{e3.7}-\eqref{e3.8} and
\eqref{e3.1}-\eqref{e3.3}
admit a unique solution $w_{j}\in H^{2}(0,1)$ and $u_{j}\in H^{2}(0,1)$
respectively.
\end{theorem}

So, for all $n>n_{0}$, we can define the Rothe approximation $u^{(n)}:I\to
H^{2}(0,1)\cap V$ as a piecewise linear interpolation of the $u_{j}$ 
$(j=1,\dots ,n)$ with respect to time, i.e. 
\begin{equation}  \label{e3.13}
u^{(n)}(t)=u_{j-1}+\delta u_{j}(t-t_{j-1}),\quad t\in [ t_{j-1},t_{j}]
,\quad j=1,\dots ,n,
\end{equation}
as well as the corresponding step function $\overline{u}^{(n)}:I\to
H^{2}(0,1)\cap V$: 
\begin{equation}
\overline{u}^{(n)}(t)=\begin{cases}
u_{j} & \text{for } t\in (t_{j-1},t_{j}] ,\; j=1,\dots ,n. \\ 
U_{0} & \text{for }t\in [\frac{-T}{n},0]%
\end{cases}
\label{e3.14}
\end{equation}

\section{A priori estimates for the approximations}

In this section, we will derive some a priori estimates which are the key
points to establish Theorem \ref{thm2.2}. Note that, in the rest of the
paper, positive constants are denoted by $C$, $C_{i}$ ($i=1,2,\dots $).

\begin{lemma} \label{lem4.1}
There exist $C>0$ such that, for all $n>n_{0}$, the solutions $u_{j}$ of the
time-discretized problems \eqref{e3.1}-\eqref{e3.3}, $j=1,\dots ,n$,
satisfy the estimates
\begin{itemize}
\item[(i)] $\| u_{j}\| \leq C$
\item[(ii)] $\| \delta u_{j}\| \leq C$.
\end{itemize}
\end{lemma}

\begin{proof}
First of all, we write problem \eqref{e3.7}-\eqref{e3.8} in a variational
form. Suppose that $n>n_{0}$ and let $\phi $ be any function from the space 
$V$ defined in \eqref{e2.1}. A standard integration by parts yields 
\begin{equation}  \label{e4.1}
\int_{0}^{x}(x-\xi )\phi (\xi )d\xi =\Im _{x}^{2}\phi ,\quad \text{for all }
x\in (0,1),
\end{equation}
where 
\begin{equation*}
\Im _{x}^{2}\phi :=\Im _{x}(\Im _{\xi }\phi )=\int_{0}^{x}d\xi \int_{0}^{\xi
}\phi (\eta )d\eta .
\end{equation*}
Hence, taking $x=1$ in \eqref{e4.1}, we get 
\begin{equation}  \label{e4.2}
\Im _{1}^{2}\phi =\int_{0}^{1}(1-\xi )\phi (\xi )d\xi =\int_{0}^{1}\phi (\xi
)d\xi -\int_{0}^{1}\xi \phi (\xi )d\xi =0.
\end{equation}
Next, taking for all $j=1,\dots ,n$, the inner product in $L^{2}(0,1)$ of
equation \eqref{e3.7} and $\Im _{x}^{2}\phi $, we get 
\begin{equation}
-\int_{0}^{1}\frac{d^{2}w_{j}}{dx^{2}}(x)\Im _{x}^{2}\phi dx
+\frac{1}{\eta +h}\int_{0}^{1}w_{j}(x)\Im _{x}^{2}\phi dx 
=\int_{0}^{1}(f_{j}(x)+\frac{1}{\eta +h}u_{j-1}(x))\Im _{x}^{2}\phi dx.  \label{e4.3}
\end{equation}
Carrying out some integrations by parts and invoking \eqref{e4.2}, we obtain
for each term in \eqref{e4.3}: 
\begin{align*}
\int_{0}^{1}\frac{d^{2}w_{j}}{dx^{2}}(x)\Im _{x}^{2}\phi dx
 &= \frac{dw_{j}}{%
dx}(x)\Im _{x}^{2}\phi \big\vert _{x=0}^{x=1} -\int_{0}^{1}\frac{dw_{j}}{dx}
(x)\Im _{x}\phi dx \\
&= - w_{j}(x)\Im _{x}\phi \big\vert_{x=0}^{x=1} +\int_{0}^{1}w_{j}(x)\phi
(x)dx \\
&= (w_{j},\phi ),
\end{align*}
\begin{align*}
\int_{0}^{1}w_{j}(x)\Im _{x}^{2}\phi dx &= \int_{0}^{1}\frac{d}{dx}(\Im
_{x}w_{j})\Im _{x}^{2}\phi dx \\
&= \Im _{x}w_{j}\Im _{x}^{2}\phi \big\vert _{x=0}^{x=1}-\int_{0}^{1}\Im
_{x}w_{j}\Im _{x}\phi dx \\
&= -(w_{j},\phi )_{B_{2}^{1}},
\end{align*}
and 
\begin{align*}
&\int_{0}^{1}(f_{j}(x)+\frac{1}{\eta +h}u_{j-1}(x))\Im _{x}^{2}\phi dx \\
&= \int_{0}^{1}\frac{d}{dx}\big[ \Im _{x}\big(f_{j}+\frac{ 1}{\eta +h}u_{j-1}%
\big)\big] \Im _{x}^{2}\phi dx \\
&= \Im _{x}\big(f_{j}+\frac{1}{\eta +h}u_{j-1}\big)\Im _{x}^{2}\phi %
\big\vert _{x=0}^{x=1} -\int_{0}^{1}\Im _{x}\big(f_{j}+\frac{1}{\eta +h}%
u_{j-1}\big) \Im_{x}\phi dx \\
&= -\big(f_{j}+\frac{1}{\eta +h}u_{j-1},\phi\big)_{B_{2}^{1}}\,.
\end{align*}
So that \eqref{e4.3} becomes 
\begin{equation}
(w_{j},\phi )+\frac{1}{\eta +h}(w_{j},\phi )_{B_{2}^{1}} =\big(f_{j}+\frac{1 
}{\eta +h}u_{j-1},\phi\big)_{B_{2}^{1}},\quad \forall j=1,\dots ,n.
\label{e4.4}
\end{equation}
Now, testing this identity with $\phi =w_{j}$ which is in $V$ thanks to 
\eqref{e3.8}, with the help of the Cauchy-Schwarz inequality we obtain 
\begin{equation*}
\| w_{j}\| ^{2}+\frac{1}{\eta +h}\| w_{j}\| _{B_{2}^{1}}^{2}\leq \big[ \|
f_{j}\| _{B_{2}^{1}}+\frac{1}{\eta +h}\| u_{j-1}\| _{B_{2}^{1}}\big] \|
w_{j}\| _{B_{2}^{1}},
\end{equation*}
from where we deduce 
\begin{equation}
\| w_{j}\| \leq \| f_{j}\| _{B_{2}^{1}}+\frac{1}{\eta +h}\| u_{j-1}\|
_{B_{2}^{1}},  \label{e4.5}
\end{equation}
as well as 
\begin{equation}
\| w_{j}\| _{B_{2}^{1}}\leq (\eta +h)\| f_{j}\| _{B_{2}^{1}}+\| u_{j-1}\|
_{B_{2}^{1}}.  \label{e4.6}
\end{equation}
Hence, \eqref{e3.9} gives for all $j=1,\dots ,n$, 
\begin{align*}
\| u_{j}\| _{B_{2}^{1}} &\leq \frac{h}{\eta +h} \| w_{j}\| _{B_{2}^{1}} +
\frac{\eta }{\eta +h}\| u_{j-1}\| _{B_{2}^{1}} \\
&\leq \frac{h}{\eta +h}((\eta +h)\| f_{j}\|_{B_{2}^{1}} +\| u_{j-1}\|
_{B_{2}^{1}})+\frac{\eta }{ \eta +h}\| u_{j-1}\| _{B_{2}^{1}},
\end{align*}
i.e., 
\begin{equation*}
\| u_{j}\| _{B_{2}^{1}}\leq h\| f_{j}\| _{B_{2}^{1}} +\| u_{j-1}\|
_{B_{2}^{1}},
\end{equation*}
then, iterating this last inequality, we may arrive at 
\begin{equation}
\| u_{j}\| _{B_{2}^{1}}\leq h\sum_{i=1}^{i=j}\| f_{i}\| _{B_{2}^{1}}+\|
U_{0}\| _{B_{2}^{1}},\quad \forall j=1,\dots ,n.  \label{e4.7}
\end{equation}
But, for all $i\geq 1$ we have in view of Assumption (H1): 
\begin{equation}
\| f_{i}\| _{B_{2}^{1}} \leq \| f(t_{i},u_{i-1})-f(t_{i},0)\| _{B_{2}^{1}}
+\| f(t_{i},0)\| _{B_{2}^{1}} \leq l\| u_{i-1}\| _{B_{2}^{1}}+M,
\label{e4.8}
\end{equation}
where $M:=\max_{t\in I} \| f(t,0)\| _{B_{2}^{1}}$. So that substituting in 
\eqref{e4.7}, 
\begin{align*}
\| u_{j}\| _{B_{2}^{1}} &\leq h\sum_{i=1}^{i=j}(l\| u_{i-1}\|
_{B_{2}^{1}}+M)+\|U_{0}\| _{B_{2}^{1}} \\
&= jhM+(1+lh)\| U_{0}\|_{B_{2}^{1}}+lh\sum_{i=2}^{i=j} \| u_{i-1}\|
_{B_{2}^{1}} \\
&\leq TM+(1+lh_{0})\| U_{0}\|_{B_{2}^{1}} +lh\sum_{i=1}^{i=j-1}\| u_{i}\|
_{B_{2}^{1}},
\end{align*}
from where it comes due to the discrete Gronwall's Lemma 
\begin{equation*}
\| u_{j}\| _{B_{2}^{1}}\leq ( TM+(1+lh_{0})\| U_{0}\|
_{B_{2}^{1}})e^{l(j-1)h}.
\end{equation*}
Then 
\begin{equation}
\| u_{j}\| _{B_{2}^{1}}\leq C_{1},\quad j=1,\dots ,n,  \label{e4.9}
\end{equation}
with $C_{1}:=(TM+(1+lh_{0})\| U_{0}\|_{B_{2}^{1}})e^{lT}$. Then, From 
\eqref{e3.6}, \eqref{e4.6} and \eqref{e4.8}, we have the estimate 
\begin{align*}
\| \delta u_{j}\| _{B_{2}^{1}} &= \frac{1}{\eta +h} \| w_{j}-u_{j-1}\|
_{B_{2}^{1}} \\
&\leq \frac{1}{\eta }(\| w_{j}\| _{B_{2}^{1}}+\| u_{j-1}\| _{B_{2}^{1}}) \\
&\leq \frac{1}{\eta } \Big((\eta +h)\| f_{j}\| _{B_{2}^{1}}+2\| u_{j-1}\|
_{B_{2}^{1}}\Big) \\
&\leq \frac{1}{\eta } \Big(((\eta +h)l+2)\| u_{j-1}\| _{B_{2}^{1}}+(\eta +h)M
\Big),
\end{align*}
finally, due to \eqref{e4.9}, 
\begin{equation}
\| \delta u_{j}\| _{B_{2}^{1}}\leq C_{2},\quad j=1,\dots ,n,  \label{e4.10}
\end{equation}
where $C_{2}:=\frac{1}{\eta }(\left[ (\eta +h_{0})l+2\right] C_{1}+(\eta
+h_{0})M)$. Now, combining \eqref{e4.5} and \eqref{e4.8}, 
\begin{equation*}
\| w_{j}\| \leq \big(l+\frac{1}{\eta +h}\big) \| u_{j-1}\| _{B_{2}^{1}}+M.
\end{equation*}
Consequently by \eqref{e4.9}, we get 
\begin{equation}
\| w_{j}\| \leq C_{3},\quad j=1,\dots ,n,  \label{e4.11}
\end{equation}
with $C_{3}:=(l+\frac{1}{\eta })C_{1}+M$. On the other hand, iterating 
\eqref{e3.9} we may obtain for $j=1,\dots ,n$ 
\begin{align*}
u_{j} &=\frac{h}{\eta +h}w_{j}+\frac{\eta }{\eta +h}u_{j-1} \\
&=\frac{h}{\eta +h}w_{j}+\frac{\eta }{\eta +h}\Big(\frac{h}{\eta +h} w_{j-1}+
\frac{\eta }{\eta +h}u_{j-2}\Big) \\
&=\frac{h}{\eta +h}\big(w_{j}+\frac{\eta }{\eta +h}w_{j-1}\big)
 +\big(\frac{\eta }{\eta +h}\big)^{2}u_{j-2} \\
&=\dots \\
&= \frac{h}{\eta +h}\Big[ w_{j}+\frac{\eta }{\eta +h}w_{j-1}
+(\frac{ \eta }{\eta +h})^{2}w_{j-2}+\dots +(\frac{\eta }{\eta +h}) ^{j-1}w_{1}\Big]
 +(\frac{\eta }{\eta +h})^{j}U_{0}.
\end{align*}
So that by \eqref{e4.11}, we have 
\begin{align*}
\| u_{j}\| &\leq \frac{h}{\eta +h}\Big[ \|w_{j}\| +\frac{\eta }{\eta +h}\|
w_{j-1}\| +(\frac{\eta }{\eta +h})^{2}\| w_{j-2}\| +\dots 
+(\frac{\eta }{\eta +h})^{j-1}\| w_{1}\| \Big] \\
&\quad +(\frac{\eta }{\eta +h})^{j}\| U_{0}\| \\
&\leq \frac{C_{3}h}{\eta +h}\Big[ 1+\frac{\eta }{\eta +h}
+( \frac{\eta }{\eta +h})^{2}+\dots +(\frac{\eta }{\eta +h}) ^{j-1}\Big] 
+\| U_{0}\| ,
\end{align*}
since $\frac{\eta }{\eta +h}<1$. But 
\begin{align*}
1+\frac{\eta }{\eta +h}+(\frac{\eta }{\eta +h})^{2}+\dots 
+( \frac{\eta }{\eta +h})^{j-1} 
&=\frac{1-(\frac{\eta }{\eta +h})^{j}}{1-\frac{\eta }{\eta +h%
}} \\
&\leq \frac{1}{1-\frac{\eta }{\eta +h}} =\frac{\eta +h}{h},
\end{align*}
then 
\begin{equation}
\| u_{j}\| \leq \frac{C_{3}h}{\eta +h}\frac{\eta +h}{h} +\| U_{0}\|
=C_{3}+\| U_{0}\| ,\quad \text{for } j=1,\dots ,n,  \label{e4.12}
\end{equation}
which proves estimate (i) with $C:=C_{3}+\| U_{0}\| $. Finally, using %
\eqref{e3.5}, \eqref{e4.11} and \eqref{e4.12}, we derive 
\begin{equation*}
\| \delta u_{j}\| \leq \frac{1}{\eta }(\| w_{j}\| +\| u_{j}\|) \leq \frac{1}{%
\eta }(2C_{3}+\| U_{0}\|),\quad \text{for } j=1,\dots ,n.
\end{equation*}
Thus, estimate (ii) is proved with $C:=\frac{1}{\eta }( 2C_{3}+\| U_{0}\|)$,
and so the proof is complete.
\end{proof}

We deduce the following estimates that we shall use later.

\begin{corollary} \label{coro4.2}
For all $n>n_{0}$, the functions $u^{(n)}$ and $\overline{u}^{(n)}$
satisfies the inequalities
\begin{itemize}
\item[(i)] $\| u^{(n)}(t)\| \leq C$,
$\| \overline{u}^{(n)}(t)\| \leq C$, for all $t\in I$,

\item[(ii)] $\| \frac{du^{(n)}}{dt}(t)\| \leq C$, a.e. in $I$,

\item[(iii)] $\| \overline{u}^{(n)}(t)-u^{(n)}(t)\| \leq \frac{C}{n}$,
for all $t\in I$

\item[(iv)] $\| \overline{u}^{(n)}(t)-\overline{u}^{(n)}(t-\frac{T}{n})
\| \leq \frac{C}{n}$, for all $t\in I$.
\end{itemize}
\end{corollary}

\begin{proof}
Inequalities (i) follow immediately from Lemma \ref{lem4.1} (i) with the
same constant, whereas inequality (ii) is an easy consequence of Lemma \ref%
{lem4.1} (ii), also with the same constant, noting that we have 
\begin{equation*}
\frac{du^{(n)}}{dt}(t)=\delta u_{j},\quad \forall t\in (t_{j-1},t_{j}],\;
1\leq j\leq n.
\end{equation*}
As for inequalities (iii) and (iv), since we have 
\begin{equation*}
\overline{u}^{(n)}(t)-u^{(n)}(t)=(t_{j}-t)\delta u_{j},\quad \forall t\in
(t_{j-1},t_{j}],\; 1\leq j\leq n,
\end{equation*}
and 
\begin{equation*}
\overline{u}^{(n)}(t)-\overline{u}^{(n)}(t-\frac{T}{n}) =u_{j}-u_{j-1},\quad
\forall t\in (t_{j-1},t_{j}],\; 1\leq j\leq n,
\end{equation*}
it follows that 
\begin{equation*}
\| \overline{u}^{(n)}(t)-u^{(n)}(t)\| \leq h \max_{1\leq j\leq n} \| \delta
u_{j}\|, \quad \forall \,t\in I,
\end{equation*}
and 
\begin{equation*}
\| \overline{u}^{(n)}(t)-\overline{u}^{(n)}(t-\frac{T}{n}) \| \leq
h\max_{1\leq j\leq n} \| \delta u_{j}\| ,\quad \forall \,t\in I.
\end{equation*}
Hence, applying Lemma \ref{lem4.1} (ii), we get the desired inequalities
(iii) and (iv) with $C:=\frac{T}{\eta }(2C_{3}+\| U_{0}\|)$.
\end{proof}

\section{Convergence and Existence result}

Using relations \eqref{e3.5} and \eqref{e3.6}, we can rearrange the
variational equations \eqref{e4.4} as follows 
\begin{equation*}
(\delta u_{j},\phi )_{B_{2}^{1}}+(u_{j},\phi )+\eta (\delta u_{j},\phi
)=(f_{j},\phi)_{B_{2}^{1}},\quad \forall \phi \in V,\; j=1,\dots ,n.
\end{equation*}
If we define, for all $n>n_{0}$, the abstract step function $\overline{f}
^{(n)}:I\times V\to L^{2}(0,1)$ by 
\begin{equation*}
\overline{f}^{(n)}(t,v)=f(t_{j},v),\quad t\in (t_{j-1},t_{j}],\,\,j=1,\dots
,n,
\end{equation*}
the previous equations may be rewritten as 
\begin{equation}
\big(\frac{du^{(n)}}{dt}(t),\phi\big)_{B_{2}^{1}} +\big({\overline{u}
^{(n)}(t),\phi }\big) +\eta\big(\frac{du^{(n)}}{dt}(t),\phi\big) \\
=\big(\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-\frac{T}{n} )),\phi\big)
_{B_{2}^{1}},  \label{e5.1}
\end{equation}
for all $\phi \in V$, $t\in (0,T]$. Before performing a limiting process in
the approximation scheme \eqref{e5.1}, we have to prove some convergence
assertions.

\begin{theorem} \label{thm5.1}
The sequence $\{ u^{(n)}\}_{n>n_{0}}$ converges under the the norm of
$C(I,V)$ to some function $u\in C(I,V)$ and the error
estimate
\begin{equation}
\| u^{(n)}-u\| _{C(I,V)}\leq \frac{C}{n^{1/2}},  \label{e5.2}
\end{equation}
takes place for all $n>n_{0}$.
\end{theorem}

\begin{proof}
The main idea of the proof is to show that $\{ u^{(n)}\}_{n>n_{0}}$ is a
Cauchy sequence in the Banach space $C(I,V)$. Let $u^{(n)}$ and $u^{(m)}$ be
the Rothe functions \eqref{e3.13} corresponding to the step lengths $h_{n}=%
\frac{T}{n}$ and $h_{m}=\frac{T}{m}$ respectively with $m>n>n_{0}$. Testing
the difference of \eqref{e5.1} for $n$ and $m$, with $\phi
=u^{(n)}(t)-u^{(m)}(t)\,(\in V)$, we get for all $t\in \,(0,T]$: 
\begin{align*}
&\Big(\frac{d}{dt}\big(u^{(n)}(t)-u^{(m)}(t)\big) ,u^{(n)}(t)-u^{(m)}(t)\Big)
_{B_{2}^{1}} \\
&+\big(\overline{u}^{(n)}(t)-\overline{u}^{(m)}(t),u^{(n)}(t)-u^{(m)}(t) 
\big) \\
&+\eta \Big(\frac{d}{dt}\big(u^{(n)}(t)-u^{(m)}(t)\big) 
,u^{(n)}(t)-u^{(m)}(t)\Big) \\
& =\Big(\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-\frac{T}{n }))-\overline{f}
^{(m)}(t,\overline{u}^{(m)}(t- \frac{T}{m})),u^{(n)}(t)-u^{(m)}(t)\Big)
_{B_{2}^{1}},
\end{align*}
or after some rearrangement 
\begin{equation}
\begin{aligned} &\frac{1}{2}\frac{d}{dt}\| u^{(n)}(t)-u^{(m)}(t)\|
_{B_{2}^{1}}^{2}+\frac{\eta }{2}\frac{d}{dt}\| u^{(n)}(t)-u^{(m)}(t)\|
^{2}+\| \overline{u}^{(n)}(t)- \overline{u}^{(m)}(t)\| ^{2} \\
&=\big({\overline{u}^{(n)}(t)-\overline{u}^{(m)}(t),(
\overline{u}^{(n)}(t)-u^{(n)}(t))+(u^{(m)}(t)-\overline{u}
^{(m)}(t))}\big)\\ &\quad+\Big({\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-
\frac{T}{n}))-\overline{f}^{(m)}(t,\overline{u}
^{(m)}(t-\frac{T}{m})),u^{(n)}(t)-u^{(m)}(t)}\Big) _{B_{2}^{1}}.
\end{aligned}  \label{e5.3}
\end{equation}
But, from the fact that 
\begin{equation*}
\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-\frac{T}{n})
)=f(t_{j},u_{j-1}):=f_{j},\quad \forall t\in (t_{j-1},t_{j}],\,\,j=1,\dots
,n,
\end{equation*}
we deduce, in view of \eqref{e4.8}, that 
\begin{align*}
\| \overline{f}^{(n)}\big(t,\overline{u}^{(n)}(t-\frac{T}{n} )\big)\|
_{B_{2}^{1}} &\leq \max_{1\leq j\leq n} \| f_{j}\| _{B_{2}^{1}} \\
&\leq l \max_{1\leq j\leq n}\|u_{j-1}\| _{B_{2}^{1}}+M, \quad \forall t\in I.
\end{align*}
Therefore, due to \eqref{e4.9}, 
\begin{equation}
\| \overline{f}^{(n)}\big(t,\overline{u}^{(n)}(t-\frac{T}{n} )\big)\|
_{B_{2}^{1}}\leq lC_{1}+M,\quad \forall t\in I.  \label{e5.4}
\end{equation}
Thus, from the identity 
\begin{equation*}
(\overline{u}^{(n)}(t),\phi) = \Big(\overline{f}^{(n)}\big(t,\overline{u}
^{(n)} (t-\frac{T}{n})\big)-\frac{du^{(n)}}{dt}(t),\phi\Big)_{B_{2}^{1}}
-\eta \big(\frac{du^{(n)}}{dt}(t),\phi\big),
\end{equation*}
for all $t\in I$, $\phi \in V$, which follows from \eqref{e5.1}, due to 
\eqref{e4.10}, \eqref{e5.4} and Corollary \ref{coro4.2}(ii), we obtain 
\begin{equation}  \label{e5.5}
\begin{aligned} \vert (\overline{u}^{(n)}(t),\phi)\vert &\leq \Big[ \|
\overline{f}^{(n)}\big(t,\overline{u}
^{(n)}(t-\frac{T}{n})\big)\|_{B_{2}^{1}} +\| \frac{du^{(n)}}{dt}(t)\|
_{B_{2}^{1}} +\eta \| \frac{du^{(n)}}{dt}(t)\| \Big] \| \phi\| \\ 
&\leq
C_{4}\| \phi \| ,\quad \forall t\in I,\; \forall \phi \in V, \end{aligned}
\end{equation}
with $C_{4}:=lC_{1}+M+C_{2}+2C_{3}+\| U_{0}\| $. Applying \eqref{e5.5} for 
\begin{equation*}
\phi =(\overline{u}^{(n)}(t)-u^{(n)}(t))+( u^{(m)}(t)-\overline{u}^{(m)}(t)),
\end{equation*}
together with Corollary \ref{coro4.2} (iii), we can dominate the first term
in the right-hand side of \eqref{e5.3} as follows 
\begin{equation}
\begin{aligned} &\big({\overline{u}^{(n)}(t)-\overline{u}^{(m)}(t),(
\overline{u}^{(n)}(t)-u^{(n)}(t))+(u^{(m)}(t)-\overline{u}
^{(m)}(t))}\big)\\ &\leq 2C_{4}\big(\| \overline{u}^{(n)}(t)-u^{(n)}(t) \|
+\| \overline{u}^{(m)}(t)-u^{(m)}(t)\|\big)\\ &\leq
C_{5}(\frac{1}{n}+\frac{1}{m}),\quad \forall t\in \,I, \end{aligned}
\label{e5.6}
\end{equation}
with $C_{5}:=\frac{2C_{4}T}{\eta }(2C_{3}+\| U_{0}\|)$. It remains to
majorize the second term in the right hand side in \eqref{e5.3}. To this
end, we use the Cauchy inequality 
\begin{equation*}
\alpha \beta \leq \frac{\varepsilon }{2}\alpha ^{2}+\frac{1}{ 2\varepsilon }
\beta ^{2},\quad \forall \alpha ,\beta \in \mathbb{R},\quad \forall
\varepsilon \in \mathbb{R}_{+}^{\ast },
\end{equation*}
for every $\varepsilon >0$: 
\begin{equation}
\begin{aligned} &\Big(\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-\frac{T}{n}
))-\overline{f}^{(m)}(t,\overline{u}^{(m)}(t-\frac{
T}{m})),u^{(n)}(t)-u^{(m)}(t)\Big)_{B_{2}^{1}} \\ &\leq \|
\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-
\frac{T}{n}))-\overline{f}^{(m)}(t,\overline{u} ^{(m)}(t-\frac{T}{m}))\|
_{B_{2}^{1}}\| u^{(n)}(t)-u^{(m)}(t)\| _{B_{2}^{1}} \\ &\leq
\frac{\varepsilon }{2}\| \overline{f}^{(n)}(t,
\overline{u}^{(n)}(t-\frac{T}{n}))-\overline{f}
^{(m)}(t,\overline{u}^{(m)}(t-\frac{T}{m})) \| _{B_{2}^{1}}^{2} \\ &\quad
+\frac{1}{2\varepsilon }\| u^{(n)}(t)-u^{(m)}(t)\| _{B_{2}^{1}}^{2},\quad
\forall t\in \,I. \end{aligned}  \label{e5.7}
\end{equation}
Now, let $t$ be arbitrary but fixed in $(0,T]$. Then there exist two
integers $k$ and $i$ corresponding to the subdivision of $I$ into $n$ and $m$
subintervals respectively, such that $t\in (t_{k-1},t_{k}]\cap
(t_{i-1},t_{i}]$. Hence thanks to the assumed Lipschitz continuity of $f$, 
\begin{align*}
&\| \overline{f}^{(n)}\big(t,\overline{u}^{(n)}(t-\frac{T}{n })\big) -
\overline{f}^{(m)}\big(t,\overline{u}^{(m)}(t- \frac{T}{m})\big)\|
_{B_{2}^{1}}^{2} \\
&=\| f\big(t_{k},\overline{u}^{(n)}(t-\frac{T}{n}) \big)-f\big(t_{i},
\overline{u}^{(m)}(t-\frac{T}{m})\big) \| _{B_{2}^{1}}^{2} \\
&\leq l^{2}\Big[ \vert t_{k}-t_{i}\vert \big\{ 1+\| \overline{u}^{(n)}(t-
\frac{T}{n})\| _{B_{2}^{1}}+\| \overline{u}^{(m)}(t-\frac{T}{m}) \|
_{B_{2}^{1}}\big\} \\
&\quad +\| \overline{u}^{(n)}(t-\frac{T}{n})- \overline{u}^{(m)}(t
-\frac{T}{m})\| _{B_{2}^{1}}\Big]^{2} \\
&\leq l^{2}\Big[ (h_{n}+h_{m})(1+\| u_{k-1}\| _{B_{2}^{1}}+\| u_{i-1}\|
_{B_{2}^{1}})+\| \overline{u}^{(n)}(t-\frac{T}{n} )-\overline{u}^{(n)}(t)\|
_{B_{2}^{1}} \\
&\quad +\| \overline{u}^{(n)}(t)-\overline{u} ^{(m)}(t)\| _{B_{2}^{1}}+\| 
\overline{u} ^{(m)}(t)-\overline{u}^{(m)}(t-\frac{T}{m}) \| _{B_{2}^{1}}%
\Big] ^{2}\,.
\end{align*}
Then follows with consideration to \eqref{e4.9} and Corollary \ref{coro4.2}
(iv) that 
\begin{align*}
&\| \overline{f}^{(n)}\big(t,\overline{u}^{(n)}(t-\frac{T}{n })\big)-
\overline{f}^{(m)}\big(t,\overline{u}^{(m)}(t- \frac{T}{m})\big)\|
_{B_{2}^{1}}^{2} \\
&\leq l^{2}\big[ T(\frac{1}{n}+\frac{1}{m})( 1+2C_{1})+\frac{T}{\eta }
(2C_{3}+\| U_{0}\| )(\frac{1}{n}+\frac{1}{m})+\| \overline{u} ^{(n)}(t)-
\overline{u}^{(m)}(t)\| _{B_{2}^{1}}\big] ^{2} \\
&= l^{2}\big[ T(1+2C_{1}+\frac{1}{\eta }(2C_{3}+\| U_{0}\|))(\frac{1}{n}+
\frac{1}{m}) +\| \overline{u}^{(n)}(t)-\overline{u}^{(m)}( t)\| _{B_{2}^{1}}%
\big] ^{2} \\
&\leq l^{2}\Big[ C_{6}^{2}(\frac{1}{n}+\frac{1}{m}) ^{2}+2C_{6}(\frac{1}{n}+
\frac{1}{m})(\| \overline{ u}^{(n)}(t)\| _{B_{2}^{1}}+\| \overline{u}
^{(m)}(t)\| _{B_{2}^{1}}) \\
&\quad +\| \overline{u}^{(n)}(t)-\overline{ u}^{(m)}(t)\| _{B_{2}^{1}}^{2}
\Big] \\
&\leq (lC_{6})^{2}(\frac{1}{n}+\frac{1}{m}) ^{2}+4l^{2}C_{6}C_{1}(\frac{1}{n}
+\frac{1}{m})+l^{2}\| \overline{u}^{(n)}(t)-\overline{u}^{(m)}(t) \|
_{B_{2}^{1}}^{2},\quad \forall t\in I,
\end{align*}
with $C_{6}:=T\big(1+2C_{1}+\frac{1}{\eta }(2C_{3}+\| U_{0}\|)\big)$. Thus,
setting $C_{7}:=(lC_{6})^{2}$ and $C_{8}:=4l^{2}C_{6}C_{1}$, we write 
\begin{equation}
\begin{aligned} &\| \overline{f}^{(n)}(t,\overline{u}^{(n)}(t-\frac{T}{n}
))-\overline{f}^{(m)}(t,\overline{u}^{(m)}(t-\frac{ T}{m}))\|
_{B_{2}^{1}}^{2} \\ &\leq C_{7}(\frac{1}{n}+\frac{1}{m})^{2}+C_{8}(\frac{1
}{n}+\frac{1}{m})+l^{2}\| \overline{u}^{(n)}(t)- \overline{u}^{(m)}(t)\|
_{B_{2}^{1}}^{2},\quad \forall t\in I; \end{aligned}  \label{e5.8}
\end{equation}
therefore, going back to \eqref{e5.7}, we have 
\begin{equation}
\begin{aligned} &\Big(\overline{f}^{(n)}(t,\overline{u}^{(n)}(t-\frac{T}{n}
))-\overline{f}^{(m)}(t,\overline{u}^{(m)}(t-\frac{
T}{m})),u^{(n)}(t)-u^{(m)}(t)\Big)_{B_{2}^{1}} \\ &\leq \frac{\varepsilon
}{2}C_{7}(\frac{1}{n}+\frac{1}{m}) ^{2}+\frac{\varepsilon
}{2}C_{8}(\frac{1}{n}+\frac{1}{m})+\frac{ \varepsilon }{2}l^{2}\|
\overline{u}^{(n)}(t)-\overline{ u}^{(m)}(t)\| _{B_{2}^{1}}^{2} \\ 
&\quad +\frac{1}{2\varepsilon }\| u^{(n)}(t)-u^{(m)}(t)\| _{B_{2}^{1}}^{2},\quad
\forall t\in I. \end{aligned}  \label{e5.9}
\end{equation}
Now, combining \eqref{e5.3}, \eqref{e5.6}, \eqref{e5.9} and \eqref{e2.3}, we
get 
\begin{align*}
&\frac{d}{dt}\Big(\| u^{(n)}(t)-u^{(m)}(t)\| _{B_{2}^{1}}^{2}+\eta \|
u^{(n)}(t)-u^{(m)}(t)\| ^{2}\Big)+2\| \overline{u}^{(n)}(t)-\overline{u}
^{(m)}(t)\| ^{2} \\
&\leq \varepsilon C_{7}(\frac{1}{n}+\frac{1}{m}) ^{2}+(\varepsilon
C_{8}+2C_{5})(\frac{1}{n}+\frac{1}{m} )+\frac{\varepsilon l^{2}}{2}\| 
\overline{u}^{(n)}( t)-\overline{u}^{(m)}(t)\| ^{2} \\
&\quad +\frac{1}{2\varepsilon }\| u^{(n)}(t)-u^{(m)}(t)\| ^{2},\quad \forall
t\in I.
\end{align*}
Hence 
\begin{align*}
&\eta \frac{d}{dt}\| u^{(n)}(t)-u^{(m)}(t)\| ^{2}+(2- \frac{\varepsilon l^{2}%
}{2})\| \overline{u}^{(n)}(t)- \overline{u}^{(m)}(t)\| ^{2} \\
&\leq \varepsilon C_{7}(\frac{1}{n}+\frac{1}{m}) ^{2}+(\varepsilon
C_{8}+2C_{5})(\frac{1}{n}+\frac{1}{m} )+\frac{1}{2\varepsilon }\|
u^{(n)}(t)-u^{(m)}(t)\| ^{2}.
\end{align*}
Choosing $\varepsilon >0$ so that $2-\frac{\varepsilon l^{2}}{2}=0$, i.e. 
$\varepsilon =\frac{4}{l^{2}}$ and integrating the just obtained inequality
between $0$ and $t$ taking into account the fact that $%
u^{(n)}(0)=u^{(m)}(0)=U_{0}$, we get for all $t\in I$: 
\begin{align*}
&\| u^{(n)}(t)-u^{(m)}(t)\| ^{2} \\
&\leq \frac{4C_{7}T}{\eta l^{2}}(\frac{1}{n}+\frac{1}{m})^{2} 
+\frac{2T}{\eta }(\frac{2C_{8}}{l^{2}}+C_{5})(\frac{1}{n}+\frac{1}{m}) 
+\frac{l^{2}}{8\eta }\int_{0}^{t}\| u^{(n)}(\tau )-u^{(m)}(\tau )\| ^{2}d\tau .
\end{align*}
Then, by Gronwall's Lemma, 
\begin{equation*}
\| u^{(n)}(t)-u^{(m)}(t)\| ^{2}\leq \big[ C_{9}( \frac{1}{n}+\frac{1}{m}
)^{2}+C_{10}(\frac{1}{n}+\frac{1}{m} )\big] e^{\frac{l^{2}}{8\eta }t}\quad
\forall t\in I,
\end{equation*}
with $C_{9}:=\frac{4C_{7}T}{\eta l^{2}}$ and 
$C_{10}:=\frac{2T}{\eta }(\frac{2C_{8}}{l^{2}}+C_{5})$. Accordingly 
\begin{equation*}
\| u^{(n)}(t)-u^{(m)}(t)\| \leq \big[ C_{9}( \frac{1}{n}+\frac{1}{m}
)^{2}+C_{10}(\frac{1}{n}+\frac{1}{m} )\big] ^{1/2}e^{\frac{l^{2}T}{16\eta }
},\quad \forall t\in I.
\end{equation*}
Then, taking the upper bound with respect to $t$ in the left-hand side of
this inequality, 
\begin{equation}  \label{e5.10}
\| u^{(n)}-u^{(m)}\| _{C(I,V)}\leq \big[ C_{9}( \frac{1}{n}+\frac{1}{m}
)^{2}+C_{10}(\frac{1}{n}+\frac{1}{m} )\big] ^{1/2}e^{\frac{l^{2}T}{16\eta }},
\end{equation}
which shows that $\{ u^{(n)}\} _{n>n_{0}}$ is a Cauchy sequence in $C(I,V)$.
This implies the existence of a function $u\in C(I,V)$ such that 
$u^{(n)}\to u$ in this space. Moreover, letting $m\to \infty $ in \eqref{e5.10}
 we obtain the error estimate \eqref{e5.2} with 
 $C=\sqrt{C_{9}+C_{10}}e^{\frac{l^{2}T}{16\eta }}$, what completes the proof.
\end{proof}

We write down some results for the limit-function $u$.

\begin{corollary} \label{coro5.2}
The function $u$ possesses the following properties:
\begin{itemize}
\item[(i)] $u\in C^{0,1}(I,V)$;
\item[(ii)] $u$ is strongly differentiable a.e. in $I$ and
$\frac{du}{dt}\in L^{\infty }(I,L^{2}(0,1))$;
\item[(iii)] $\overline{u}^{(n)}(t)\to u(t)$ in $V$ for
all $t\in I$;
\item[(iv)] $\frac{du^{(n)}}{dt}\rightharpoonup \frac{du}{dt}$
 in $L^{2}(I,L^{2}(0,1))$.
\end{itemize}
\end{corollary}

\begin{proof}
On the basis of Corollary \ref{coro4.2} (i) and (ii), uniform convergence
statement from Theorem \ref{thm5.1} and the continuous embedding
 $V\hookrightarrow Y:=L^{2}(0,1)$, \cite[Lemma 1.3.15]{Kac} is valid for our
special situation yielding assertions (i), (ii) and (iv) of the present
Corollary. The remaining assertion (iii) is an immediate consequence of the
combination of Corollary \ref{coro4.2} (iii) with the convergence result
stated in Theorem \ref{thm5.1}.
\end{proof}

Collecting all the obtained results, we can state our existence theorem.

\begin{theorem} \label{thm5.3}
The limit function $u$ from Theorem \ref{thm5.1} is the unique weak solution to
problem \eqref{e1.5}-\eqref{e1.8} in the sense of Definition \ref{def2.1}.
Moreover, $u$ depends continuously upon data $f$ and $U_{0}$,
 namely the inequality
\begin{equation} \label{e5.11}
\max_{0\leq s\leq t} \| u^{\ast }(s)-u^{\ast\ast }(s)\|
\leq C\Big(\| U_{0}^{\ast }-U_{0}^{\ast \ast }\|
+\int_{0}^{t}\| f^{\ast }(s,u^{\ast }(s))-f^{\ast \ast }(s,u^{\ast
\ast }(s))\| _{B_{2}^{1}}ds\Big),
\end{equation}
holds for all $t\in I$, with some positive constant $C$ depending only on
$\eta $.
\end{theorem}

\begin{proof}
\textbf{Existence. }It suffices to check all the points (i)-(iv) of
Definition \ref{def2.1}. Obviously, in light of Corollary \ref{coro5.2}, the
first two points of Definition \ref{def2.1} are already fulfilled. Moreover,
since $u^{(n)}\to u$ in $C(I,V)$ as $n\to \infty $ and, by definition, 
$u^{(n)}(0)=U_{0}$, it follows that $u(0)=U_{0}$ holds in $V$ so the initial
condition \eqref{e1.6} is also fulfilled, that is point (iii) of Definition 
\ref{def2.1} takes place. To show that $u$ obeys the integral equation 
\eqref{e2.4}, we investigate the behaviour as $n\to \infty $ of the integral
relation 
\begin{equation}
\begin{aligned} &\big({u^{(n)}(t)-U_{0},\phi }\big)_{B_{2}^{1}}
+\int_{0}^{t}\big({\overline{u}^{(n)}(\tau ),\phi } \big)d\tau +\eta
\big({u^{(n)}(t)-U_{0},\phi }\big)\\
&=\int_{0}^{t}\Big(\overline{f}^{(n)}\big(\tau ,\overline{u}^{(n)}( \tau
-\frac{T}{n})\big),\phi\Big)_{B_{2}^{1}}d\tau ,\quad \forall \phi \in V,\;
\forall t\in I, \end{aligned}  \label{e5.12}
\end{equation}
which results from \eqref{e5.1} by integration over $(0,t)\subset I$ noting
that $u^{(n)}(0)=U_{0}$. This requires some additional convergence
statements.

Firstly, since $u^{(n)}\rightarrow u$ in $C(I,V)$ and since for all fixed 
$\phi \in V$, the linear functional $v\mapsto (v,\phi )_{B_{2}^{1}}$ is
continuous on $V$, we deduce that 
\begin{gather}
\big({u^{(n)}(t),\phi }\big)\underset{n\rightarrow \infty }{\longrightarrow }
\big({u(t),\phi }\big),\quad \forall \phi \in V,\;\forall t\in I,
\label{e5.13} \\
\big({u^{(n)}(t),\phi }\big)_{B_{2}^{1}}\underset{n\rightarrow \infty }{
\longrightarrow }\big({u(t),\phi }\big)_{B_{2}^{1}},\quad \forall \phi \in
V,\;\forall t\in I.  \label{e5.14}
\end{gather}%
Secondly, by virtue of \eqref{e5.5} the Lebesgue Theorem of dominated
convergence may be applied to the convergence statement (iii) from Corollary 
\ref{coro5.2} giving 
\begin{equation}
\int_{0}^{t}\big({\overline{u}^{(n)}(\tau ),\phi }\big)d\tau
 \underset{n\rightarrow \infty }{\longrightarrow }\int_{0}^{t}
 \big(\overset{}{u(\tau
),\phi }\big)d\tau ,\quad \forall \phi \in V,\;\forall t\in I.  \label{e5.15}
\end{equation}%
Thirdly, in view of Assumption (H1), we have 
\begin{align*}
& \Vert \overline{f}^{(n)}\big(\tau ,\overline{u}^{(n)}(\tau -\frac{T}{n})
\big)-f(\tau ,u(\tau ))\Vert _{B_{2}^{1}} \\
& =\Vert f\big(t_{j},\overline{u}^{(n)}(\tau -\frac{T}{n})\big)-f(\tau
,u(\tau ))\Vert _{B_{2}^{1}} \\
& \leq l\big[|t_{j}-\tau |(1+\Vert u_{j-1}\Vert _{B_{2}^{1}}+\Vert u(\tau
)\Vert _{B_{2}^{1}})+\Vert \overline{u}^{(n)}(\tau -\frac{T}{n})-u(\tau
)\Vert _{B_{2}^{1}}\big],
\end{align*}%
for all $\tau \in (t_{j-1},t_{j}]$, $1\leq j\leq n$; therefore 
\begin{equation*}
\Vert \overline{f}^{(n)}\big(\tau ,\overline{u}^{(n)}(\tau -\frac{T}{n})\big)%
-f(\tau ,u(\tau ))\Vert _{B_{2}^{1}}\leq \frac{C}{n}+l\Vert \overline{u}%
^{(n)}(\tau -\frac{T}{n})-u(\tau )\Vert _{B_{2}^{1}},
\end{equation*}%
for all $\tau \in I$, where $C:=lT(1+C_{1}+\Vert u\Vert _{C(I,V)})$.
However, with consideration to estimates (iii)-(iv) from Corollary 
\ref{coro4.2} and inequality \eqref{e5.2}, we can write 
\begin{align*}
\Vert \overline{u}^{(n)}(\tau -\frac{T}{n})-u(\tau )\Vert _{B_{2}^{1}}& \leq
\Vert \overline{u}^{(n)}(\tau -\frac{T}{n})-\overline{u}^{(n)}(\tau )\Vert 
\\
& \quad +\Vert \overline{u}^{(n)}(\tau )-u^{(n)}(\tau )\Vert +\Vert
u^{(n)}(\tau )-u(\tau )\Vert  \\
& \leq C(\frac{1}{n}+\frac{1}{n^{1/2}}),\quad \forall \tau \in I,
\end{align*}%
whence 
\begin{equation*}
\Vert \overline{f}^{(n)}\big(\tau ,\overline{u}^{(n)}(\tau -\frac{T}{n})\big)
-f(\tau ,u(\tau ))\Vert _{B_{2}^{1}}\leq \frac{C}{n^{1/2}},\quad \forall
\tau \in I,
\end{equation*}%
and then 
\begin{equation}
\overline{f}^{(n)}\big(\tau ,\overline{u}^{(n)}(\tau -\frac{T}{n})\big)
\underset{n\rightarrow \infty }{\longrightarrow }f(\tau ,u(\tau ))\quad 
\text{in }B_{2}^{1}(0,1),\quad \forall \tau \in I.  \label{e5.16}
\end{equation}%
Now, due to \eqref{e5.4} the function $|(\overline{f}^{(n)}\big(\tau ,
\overline{u}^{(n)}(\tau -\frac{T}{n})\big),\phi )_{B_{2}^{1}}|$ is uniformly
bounded with respect to both $\tau $ and $n$. So the Lebesgue Theorem of
dominated convergence may be applied again to \eqref{e5.16} yielding 
\begin{equation}
\int_{0}^{t}(\Big(\overline{f}^{(n)}\big(\tau ,\overline{u}^{(n)}(\tau -
\frac{T}{n})\big),\phi \Big)_{B_{2}^{1}}d\tau \underset{n\rightarrow \infty }
{\longrightarrow }\int_{0}^{t}(f(\tau ,u(\tau )),\phi )_{B_{2}^{1}}d\tau ,
\label{e5.17}
\end{equation}%
for all $\phi \in V$ and all $t\in I$. Then, by \eqref{e5.13}, \eqref{e5.14}
, \eqref{e5.15} and \eqref{e5.17}, a limiting process $n\rightarrow \infty $
in \eqref{e5.12} leads to 
\begin{equation*}
\big({u(t)-U_{0},\phi }\big)_{B_{2}^{1}}+\int_{0}^{t}\big({u(\tau ),\phi }
\big)d\tau +\eta \big({u(t)-U_{0},\phi }\big)=\int_{0}^{t}(f(\tau ,u(\tau
)),\phi )_{B_{2}^{1}}d\tau ,
\end{equation*}%
for all $\phi \in V$ and $t\in I$. Finally, the differentiation of this last
identity with respect to $t$ recalling that $u:I\rightarrow L^{2}(0,1)$ is
strongly differentiable for $a.e$. $t\in I$, leads to the required identity 
\eqref{e2.4} by the aid of the equalities $\frac{d}{dt}({u(t),\phi }
)_{B_{2}^{1}}=(\frac{du}{dt}(t),\phi )_{B_{2}^{1}}$ and 
$\frac{d}{dt}({u(t),\phi })=(\frac{du}{dt}(t),\phi )$ which hold for all 
$t\in I$ and all $\phi \in V$. Thus, $u$ weakly solves problem 
\eqref{e1.5}-\eqref{e1.8}.
\newline
\textbf{Uniqueness and continuous dependence upon data. }Let $u^{\ast }$ and 
$u^{\ast \ast }$ be two weak solutions of problem \eqref{e1.5}-\eqref{e1.8}
corresponding respectively to $(U_{0}^{\ast },f^{\ast })$ and $(U_{0}^{\ast
\ast },f^{\ast \ast })$ instead of $(U_{0},f)$. Subtracting the identity 
\eqref{e2.4} written for $u^{\ast \ast }$ from the same identity written for 
$u^{\ast }$ and inserting $\phi =u^{\ast }(t)-u^{\ast \ast }(t)$ in the
resulting relation, we get by integration over $(0,\tau )$, with $\tau \in I:
$ 
\begin{align*}
& \frac{1}{2}\Vert u^{\ast }(\tau )-u^{\ast \ast }(\tau )\Vert
_{B_{2}^{1}}^{2}-\frac{1}{2}\Vert u^{\ast }(0)-u^{\ast \ast }(0)\Vert
_{B_{2}^{1}}^{2}+\int_{0}^{\tau }\Vert u^{\ast }(t)-u^{\ast \ast }(t)\Vert
^{2}dt \\
& +\frac{\eta }{2}\Vert u^{\ast }(\tau )-u^{\ast \ast }(\tau )\Vert ^{2}-
\frac{\eta }{2}\Vert u^{\ast }(0)-u^{\ast \ast }(0)\Vert ^{2} \\
& =\int_{0}^{\tau }\Big(f^{\ast }(t,u^{\ast }(t))-f^{\ast \ast }(t,u^{\ast
\ast }(t)),u^{\ast }(t)-u^{\ast \ast }(t)\Big)_{B_{2}^{1}}dt,
\end{align*}%
hence, ignoring the first and the third terms in the left hand side, we
obtain 
\begin{align*}
& \Vert u^{\ast }(\tau )-u^{\ast \ast }(\tau )\Vert ^{2} \\
& \leq \frac{1}{\eta }\Vert u^{\ast }(0)-u^{\ast \ast }(0)\Vert
_{B_{2}^{1}}^{2}+\Vert u^{\ast }(0)-u^{\ast \ast }(0)\Vert ^{2} \\
& \quad +\frac{2}{\eta }\int_{0}^{\tau }\Vert f^{\ast }(t,u^{\ast
}(t))-f^{\ast \ast }(t,u^{\ast \ast }(t))\Vert _{B_{2}^{1}}\Vert u^{\ast
}(t)-u^{\ast \ast }(t)\Vert _{B_{2}^{1}}dt \\
& \leq (\frac{1}{2\eta }+1)\Vert u^{\ast }(0)-u^{\ast \ast }(0)\Vert ^{2}+
\frac{\sqrt{2}}{\eta }\max_{0\leq t\leq \tau }\Vert u^{\ast }(t)-u^{\ast
\ast }(t)\Vert  \\
& \times \int_{0}^{\tau }\Vert f^{\ast }(t,u^{\ast }(t))-f^{\ast \ast
}(t,u^{\ast \ast }(t))\Vert _{B_{2}^{1}}dt \\
& \leq (\frac{1}{2\eta }+1)\Vert U_{0}^{\ast }-U_{0}^{\ast \ast }\Vert
\max_{0\leq t\leq \tau }\Vert u^{\ast }(t)-u^{\ast \ast }(t)\Vert +\frac{
\sqrt{2}}{\eta }\max_{0\leq t\leq \tau }\Vert u^{\ast }(t)-u^{\ast \ast
}(t)\Vert  \\
& \quad \times \int_{0}^{\tau }\Vert f^{\ast }(t,u^{\ast }(t))-f^{\ast \ast
}(t,u^{\ast \ast }(t))\Vert _{B_{2}^{1}}dt \\
& \leq \Big[(\frac{1}{2\eta }+1)\Vert U_{0}^{\ast }-U_{0}^{\ast \ast }\Vert +
\frac{\sqrt{2}}{\eta }\int_{0}^{\tau }\Vert f^{\ast }(t,u^{\ast
}(t))-f^{\ast \ast }(t,u^{\ast \ast }(t))\Vert _{B_{2}^{1}}dt\Big] \\
& \quad \times \max_{0\leq t\leq \tau }\Vert u^{\ast }(t)-u^{\ast \ast
}(t)\Vert ,
\end{align*}%
where \eqref{e2.3} has been used. Consequently for all $s\in \lbrack 0,\tau ]
$, we have 
\begin{align*}
& \Vert u^{\ast }(s)-u^{\ast \ast }(s)\Vert ^{2} \\
& \leq \Big[(\frac{1}{2\eta }+1)\Vert U_{0}^{\ast }-U_{0}^{\ast \ast }\Vert +
\frac{\sqrt{2}}{\eta }\int_{0}^{\tau }\Vert f^{\ast }(t,u^{\ast
}(t))-f^{\ast \ast }(t,u^{\ast \ast }(t))\Vert _{B_{2}^{1}}dt\Big] \\
& \quad \times \max {0\leq t\leq \tau }\Vert u^{\ast }(t)-u^{\ast \ast
}(t)\Vert ,
\end{align*}
whence 
\begin{align*}
& \max_{0\leq s\leq \tau }\Vert u^{\ast }(s)-u^{\ast \ast }(s)\Vert ^{2} \\
& \leq \Big[(\frac{1}{2\eta }+1)\Vert U_{0}^{\ast }-U_{0}^{\ast \ast }\Vert +
\frac{\sqrt{2}}{\eta }\int_{0}^{\tau }\Vert f^{\ast }(t,u^{\ast
}(t))-f^{\ast \ast }(t,u^{\ast \ast }(t))\Vert _{B_{2}^{1}}dt\Big] \\
& \quad \times \max_{0\leq t\leq \tau }\Vert u^{\ast }(t)-u^{\ast \ast
}(t)\Vert ,
\end{align*}
from which the estimate \eqref{e5.11} follows with 
$C:=\max (\frac{1}{2\eta }+1,\frac{\sqrt{2}}{\eta })$.
 This implies the uniqueness as well as the
continuous dependence of the solution of \eqref{e1.5}-\eqref{e1.8} upon
data. So the proof is complete.
\end{proof}

\begin{thebibliography}{99}
\bibitem{BZK} G. Barenblbatt, Iu. P. Zheltov, and I. N. Kochina, \emph{Basic
concepts in the theory of seepage of homogeneous liquids in fissured rocks
[Strata]}, J. Appl. Math. Mech. \textbf{24} (1960), 1286-1303.

\bibitem{BBM} T. B. Benjamin, J. L. Bona, and J. J. Mahony, \emph{Model
equations for long waves in non-linear dispersive systems, }Phil. Trans.
Roy. Soc. London Ser. A, \textbf{272} (1972), 47-78.

\bibitem{B1} A. Bouziani, \emph{Strong solution to a mixed problem for
certain pseudoparabolic equation of variable order}, Annales de Math\'{e}
matiques de l'Universit\'{e} de Sidi Bel Abb\`{e}s, \textbf{5 }(1998), 1-10.

\bibitem{B2} A. Bouziani, \emph{On the solvability of parabolic and
hyperbolic problems with a boundary integral condition}, Int. J. Math. Math.
Sci. \textbf{31} (2002), no. 4, 201-213.

\bibitem{B3} A. Bouziani, \emph{Initial-boundary value problems for a class
of pseudoparabolic equations with integral boundary conditions}, J. Math.
Anal. Appl. \textbf{291} (2004) 371--386.

\bibitem{B4} A. Bouziani, \emph{Solvability of a nonlinear pseudoparabolic
equation with a nonlocal boundary condition}, Nonlinear Analysis: Theory,
Methods and Applications, \textbf{55} (2003), 883-904.

\bibitem{CG} P. J. Chen and M. E. Gurtin, \emph{On a theory of heat
conduction involving two temperatures}, Z. Angew. Math. Phys. \textbf{19}
(1968), 614-627.

\bibitem{CN} B. D. Coleman and W. Noll, \emph{Approximation theorem for
functionals, with applications in continuum mechanics}, Arch. Rational Mech.
Anal. \textbf{6} (1960), 355-370.

\bibitem{Kac} J. Ka\v{c}ur, \emph{Method of Rothe in evolution equations}.
Teubner-Texte zur Mathematik, vol. 80, BSB B. G. Teubner
Verlagsgesellschaft, Leipzig, 1985.

\bibitem{Kar} G. Karch, \emph{Asymptotic behavior of solutions to some
pseudoparabolic equations, }Math. Meth. Appl. Sci. \textbf{20} (1997), no.
3, 271-289.

\bibitem{KP} A. G. Kartsatos and M. E. Parrott, \emph{On a class of
nonlinear functional pseudoparabolic problems,} Funkcial. Ekvac. \textbf{25}
(1982), 207-221.

\bibitem{Kuf} A. Kufner, O. John, and S. Fu\v{c}ik, \emph{Function spaces}.
Noordhoff, Leiden, 1977.

\bibitem{Me} N. Merazga, \emph{R\'{e}solution d'un probl\`{e}me mixte
pseudo-parabolique par la m\'{e}thode de discr\'{e}tisation en temps,}
(French) 2i\`{e}me Colloque National en Analyse Fonctionnelle et
Applications (Sidi Bel Abb\`{e}s, 1997). Ann. Math. Univ. Sidi Bel Abb\`{e}s 
\textbf{6 }(1999), 105-118.

\bibitem{MB} N. Merazga and A. Bouziani, \emph{Rothe method for a mixed
problem with an integral condition for the two-dimensional diffusion
equation,} Abstract and Applied Analysis, \textbf{2003}:16 (2003), 899--922.

\bibitem{Mi} E. A. Milne, \emph{The diffusion of imprisoned radiation
through a gas}, J. London Math. Soc. \textbf{1} (1926), 40-51.

\bibitem{MZ} A. C. G. Mitchel and N. W. Zemansky, \emph{Resonance radiation
and excited atoms, }Cambridge University Press, Cambridge, England 1934.

\bibitem{OA} K. Oru\c{c}oglu and S. S. Akhiev, \emph{The Riemann function
for the third-order one dimensional pseudoparabolic equation}, Acta Appl.
Math. \textbf{53} (1998), 353-370.

\bibitem{PS} J. C. Pirkle and V. G. Sigillito, \emph{Analysis of optically
pumped CO}$_{2}$\emph{-Laser}, Applied Optics \textbf{13} (1974), 2799-2807.

\bibitem{SC} M. P. Sapagovas and R. Yu. Chegis, \emph{On some boundary value
problems with a nonlocal condition}, Differential Equations \textbf{23}
(1987), no. 7, 858-863, translation from Differentsial'nye Uravneniya 
\textbf{23} (1987), no. 7, 1268-1274.

\bibitem{ST} R. E. Showalter and T. W. Ting, \emph{Pseudoparabolic partial
differential equations, }SIAM J. Math. Anal. \textbf{1} (1970), 733-763.

\bibitem{Show} R. E. Showalter, \emph{Local regularity, boundary values and
maximum principles for pseudoparabolic equations, }Applicable Anal. 
\textbf{16} (1983), 235-241.

\bibitem{Sob} V. V. Sobolev, \emph{A treatise on radiative transfer}, Van
Nostrand, New York 1963.

\bibitem{T1} T. W. Ting, \emph{Certain non-steady flows of second order
fluids}, Arch. Rational Mech. Anal. \textbf{14} (1963), 1-26.

\bibitem{T2} T. W. Ting, \emph{Parabolic and pseudoparabolic partial
differential equations}, J. Math. Soc. Japan \textbf{21} (1969), 440-453.

\bibitem{V} V. A. Vodakhova, \emph{A boundary- value problem with A. M.
Nakhusheve's nonlocal condition for a pseudoparabolic moisture-transfer
equation}, translated from Differentsial'nye Uravneniya \textbf{18} (1982),
no. 2, 280-285.
\end{thebibliography}

\end{document}