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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 75, pp. 1--11.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/75\hfil Some basic theorems]
{Some basic theorems on difference-differential equations}

\author[B. G. Pachpatte\hfil EJDE-2008/75\hfilneg]
{Baburao G. Pachpatte}

\address{Baburao G. Pachpatte \newline
Shri Niketan Colony, Near Abhinay Talkies,
Aurangabad 431 001 (Maharashtra), India}
\email{bgpachpatte@gmail.com}

\thanks{Submitted December 21, 2007. Published May 22, 2008.}
\subjclass[2000]{34K05, 34K10, 34K40}
\keywords{Difference-differential equations; Banach fixed point theorem;
\hfill\break\indent
 Gronwall-Bellman integral inequality; existence and uniqueness;
  continuous dependence}

\begin{abstract}
 In this paper some basic theorems on the existence, uniqueness
 and continuous dependence of solutions of a certain
 difference-differential equation are established.
 The well known Banach fixed point theorem and the Gronwall-Bellman
 integral inequality are used to establish these results.
\end{abstract}

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

\section{Introduction}
Let $ \mathbb{R}^n$ denote the real $n$-dimensional Euclidean
space with the corresponding norm  $ |\cdot|$. Let
$ \mathbb{R}_+= [{0,\infty })$ be a subset of the real numbers.
Consider the difference-differential
equation
\begin{equation}
x'(t)= f({t,x(t),x({t - 1})}), \label{e1.1}
\end{equation}
for $
t \in \mathbb{R}_+
$ under the initial conditions
\begin{equation}
x({t - 1})= \phi (t)\quad ({0 \leq t < 1}), \quad    x(0)= x_0 ,
\label{e1.2}
\end{equation}
with $c_0  = \phi ({1 - 0})$,
where $f \in C({\mathbb{R}_+   \times \mathbb{R}^n  \times \mathbb{R}^n ,
\mathbb{R}^n })$ and
$\phi (t) $ is a continuous function for which
$\lim_{t \to 1 - 0}\phi (t)$ exists.
If we consider the solutions of \eqref{e1.1} for $t \in \mathbb{R}_+$,
 we obtain a function $x(t-1)$ which is unable to define as
a solution for $ 0 \leq t < 1$.
Hence, we have to impose some additional condition, for example
the first condition in \eqref{e1.2}. In this case it is sufficient
to consider the ordinary differential equation
\[
x'(t)= f({t,x(t),\phi (t)}),%\label{e1.3}
\]
for $0 \leq t < 1$, with the second condition in \eqref{e1.2}.
We note that, here it is essential to obtain the solutions of
\eqref{e1.1} for $0 \leq t < \infty$.
It is easy to observe that the integral equations which are equivalent
to  \eqref{e1.1}-\eqref{e1.2} are
\[
x(t)= x_0  + \int_0^t {f({s,x(s),\phi (s)})ds,}
%\label{e1.4}
\]
for $0 \leq t < 1$ and
\[
x(t)= x_0  + \int_0^1 {f({s,x(s),\phi (s)})ds
+ \int_1^t {f({s,x(s),x({s - 1})})ds} ,}
%\label{e1.5}
\]
for $0 \leq t < \infty$.

In 1960, Sugiyama \cite{s1} (see also \cite{s2}) studied
the existence and uniqueness of solutions to \eqref{e1.1} under the
initial conditions \eqref{e1.2} by using Tychonov's fixed point theorem,
the method of successive approximations, and the comparison method.
The main objective of the present paper is to study the existence,
uniqueness and continuous dependence of solutions  to the
initial-value problem \eqref{e1.1}-\eqref{e1.2}.
The main tools are the applications
of the Banach fixed point theorem  \cite[p. 37]{c2}
and the Gronwall-Bellman integral inequality (see  \cite[p.11]{p2}).

\section{Existence and uniqueness}

Let $S$ be the space of functions $z(t)\in \mathbb{R}^n$ which
are continuous for $t \in \mathbb{R}_+$ and fulfill the condition
\begin{equation}
 |{z(t)}| = O({\exp ({\lambda t})}), \label{e2.1}
\end{equation}
for some positive constant ${\lambda  > 0}$. In this space we define
the norm (see \cite{b2,p1})
\begin{equation}
 |z|_S  = \sup_{t \in \mathbb{R}_+} [|z(t)|\exp (- \lambda t)].
\label{e2.2}
\end{equation}
It is easy to see that $S$ with the above norm is a Banach
space. Note that condition \eqref{e2.1} implies the existence
of a nonnegative constant $N$ such that
$ |{z(t)}| \leq N\exp ({\lambda t})$ for
$t \in \mathbb{R}_+$. Using this fact in \eqref{e2.2} we observe that
\begin{equation}
 |z|_S  \leq N. \label{e2.3}
\end{equation}
\paragraph{}
We need the following integral inequality, often referred to as
Gronwall Bellman  inequality  \cite[p.11]{p2}.

\begin{lemma} \label{lemGB}
Let $u$ and $f$ be continuous functions defined on $\mathbb{R}_+$ and
$c$ be a nonnegative constant. If
\[
u(t)\leq c + \int_0^t {f(s)u(s)ds,}
\]
for $ t \in \mathbb{R}_+ $, then
\[
u(t)\leq c\exp \Big( \int_0^t f(s)ds \Big),
\]
for $t \in \mathbb{R}_+$.
\end{lemma}

Now we shall prove the following main result of this section.

\begin{theorem} \label{thm1}
Assume that:
\begin{itemize}
\item[(i)] The function $f$ in  \eqref{e1.1} satisfies the condition
\begin{equation}
 |{f({t,x,y})- f({t,\bar x,\bar y})}| \leq h(t)[ |{x - \bar x}|
 +  |y - \bar{y}|], \label{e2.4}
\end{equation}
for $({t,x,y}),({t,\bar x,\bar y})\in \mathbb{R}_+
\times \mathbb{R}^n  \times \mathbb{R}^n $,
where $h \in C({\mathbb{R}_+  ,\mathbb{R}_+  })$,

\item[(ii)] for $ \lambda $ as in \eqref{e2.1}:\\
(a) there exists a nonnegative constant $\alpha$ such that
$\alpha  < 1$ and
\begin{equation}
\int_0^t {[{h(s)+ h({s + 1})}]\exp ({\lambda s})ds
\leq \alpha \exp ({\lambda t}),}\label{e2.5}
\end{equation}
for $t \in \mathbb{R}_+$,
\\
(b) there exists a nonnegative constant $\beta$ such that
\begin{equation}
 |x_0| + \int_0^1 {h(s) |{\phi (s)}|ds
 + \int_0^t { |{f({s,0,0})|}ds \leq \beta \exp ({\lambda t}),} }\label{e2.6}
\end{equation}
for ${t \in \mathbb{R}_+ }$.
Then the  \eqref{e1.1}-\eqref{e1.2} has a unique solution on
${\mathbb{R}_+  }$ in $S$.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk1} \rm
We note that the linear systems with such a delay with bounded
coefficients on $\mathbb{R}_+$ satisfy \eqref{e2.4}, \eqref{e2.5},
\eqref{e2.6}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm1}]
Let $x(t)\in S $ and define the operator $T$ by (see \cite{s1})
\begin{equation}
Tx(t)= x_0  + \int_0^t {f({s,x(s),\phi (s)})ds,}\label{e2.7}
\end{equation}
for $0 \leq t < 1 $, and
\begin{equation}
Tx(t)= x_0  + \int_0^1 {f({s,x(s),\phi (s)})ds
 + \int_1^t {f({s,x(s),x({s - 1})})ds} ,}
\label{e2.8}
\end{equation}
for $1 \leq t < \infty $.
First we shall show that $Tx$ maps $S$ into itself. Evidently,
 $Tx$ is continuous on ${\mathbb{R}_+  }$ and
$Tx \in \mathbb{R}^n $. To verify that \eqref{e2.1} is fulfilled,
we consider the following two cases.

\noindent\textbf{Case 1: $0 \leq t < 1$.}
 From \eqref{e2.7}, using the hypotheses and \eqref{e2.3}, we have
\begin{equation}
\begin{aligned}
 |Tx(t)| &\leq  |x_0|
+ \int_0^t  |f({s,x(s),\phi (s)})- f({s,0,0})|ds
+ \int_0^t  |f({s,0,0})|ds \\
& \leq  |x_0| + \int_0^t {h(s)[{ |{x(s)|}
+  |{\phi (s)}}|]ds + \int_0^t { |{f({s,0,0})}|ds} }
\\
&\leq  |x_0| + \int_0^1 {h(s) |{\phi (s)}|ds
+ \int_0^t { |{f({s,0,0})}|ds} }  +
|x|_S \int_0^t {h(s)\exp ({\lambda s})ds} \\
& \leq [{\beta  + N\alpha }]\exp ({\lambda t}).
\end{aligned} \label{e2.9}
\end{equation}

\noindent\textbf{Case 2: $1 \leq t < \infty $}
 From \eqref{e2.8}, using the hypotheses and \eqref{e2.3}, we have
\begin{equation}
\begin{aligned}
 |{Tx(t)}|
&\leq  |x_0| + \int_0^1 { |{f({s,x(s),\phi (s)})- f({s,0,0})}|ds
+ \int_0^1 { |{f({s,0,0})}|ds} } \\
&\quad + \int_1^t { |{f({s,x(s),x({s - 1})})- f({s,0,0})}|ds
+ \int_1^t { |{f({s,0,0})}|ds} } \\
& \leq  |x_0| + \int_0^1 {h(s)[{ |{x(s)}| +  |{\phi (s)}}|]ds
 + \int_0^t { |{f({s,0,0})}} }| ds \\
&\quad + \int_1^t {h(s)[{ |{x(s)|} +  |{x({s - 1})}}|]ds} \\
& =  |x_0| + \int_0^1 {h(s) |{\phi (s)}|ds}
 + \int_0^t { |{f({s,0,0})}|ds} \\
&\quad  + \int_0^1 {h(s) |{x(s)}|ds}  + \int_1^t {h(s) |{x(s)}|ds}
+ \int_1^t {h(s) |{x({s - 1})}|ds} \\
&\leq \beta \exp ({\lambda t})+ \int_0^t {h(s) |{x(s)}|ds}  + I_1 ,
\end{aligned} \label{e2.10}
\end{equation}
where
\begin{equation}
I_1  = \int_1^t {h(s) |{x({s - 1})}|ds} . \label{e2.11}
\end{equation}
By making the change of variable,
 we obtain
\begin{equation}
I_1  = \int_0^{t - 1} {h({\sigma  + 1}) |{x(\sigma )}|d\sigma }
 \leq \int_0^t {h({\sigma  + 1}) |{x(\sigma )}|d\sigma } . \label{e2.12}
\end{equation}
Using \eqref{e2.12} in \eqref{e2.10}, we get
\begin{equation}
\begin{aligned}
 |Tx(t)| &\leq \beta \exp ({\lambda t})
+  |x|_S \int_0^t {[{h(s)+ h({s + 1})}]\exp ({\lambda s})ds}\\
 &\leq [{\beta  + N\alpha }]\exp ({\lambda t}).
\end{aligned} \label{e2.13}
\end{equation}
 From this inequality and \eqref{e2.9}, it follows that
$Tx \in S$. This proves that $T$ maps $S$ into itself.

Next, we verify that the operator $T$ is a contraction map. Let
$x(t),y(t)\in S$.
We consider the following two cases.

\noindent\textbf{Case 1: $0 \leq t < 1$.}
 From \eqref{e2.7} and using the hypotheses, we have
\begin{equation}
\begin{aligned}
 |{Tx(t)- Ty(t)} |
&\leq \int_0^t { |{f({s,x(s),\phi (s)})- f({s,y(s),\phi (s)})}}| ds \\
& \leq \int_0^t {h(s) |{x(s)- y(s)}|ds} \\
& \leq  |{x - y}|_S \int_0^t {h(s)\exp ({\lambda s})ds} \\
& \leq  |{x - y}|_S \alpha \exp ({\lambda t}).
\end{aligned} \label{e2.14}
\end{equation}

\noindent\textbf{Case 2:  $1 \leq t < \infty$.}
 From \eqref{e2.8} and using the hypotheses, we have
\begin{equation}
\begin{aligned}
 |{Tx(t)- Ty(t)}|
&\leq \int_0^1 { |{f({s,x(s),\phi (s)})- f({s,y(s),\phi (s)})}}| ds \\
&\quad  + \int_1^t { |{f({s,x(s),x({s - 1})})- f({s,y(s),y({s - 1})})}}| ds \\
&\leq \int_0^1 h(s) |{x(s)- y(s)}|ds \\
&\quad + \int_1^t h(s)[{ |{x(s)- y(s)}|
 +  |{x({s - 1})- y({s - 1})}}|]ds \\
& = \int_0^t h(s) |x(s)- y(s)|ds + I_2 ,
\end{aligned}\label{e2.15}
\end{equation}
where
\begin{equation}
I_2  = \int_1^t h(s){ |{x({s - 1})- y({s - 1})}|ds.}\label{e2.16}
\end{equation}
By making the change of variable, we obtain
\begin{equation}
I_2  \leq \int_0^t {h({s + 1}) |{x(s)- y(s)}|ds.}\label{e2.17}
\end{equation}
Using this inequality and  \eqref{e2.15} we get
\begin{equation}
\begin{aligned}
 |{Tx(t)- Ty(t)}| &\leq  |{x - y}|_S
 \int_0^t {[{h(s)+ h({s + 1})}]\exp ({\lambda s})ds} \\
& \leq  |{x - y}|_S \alpha \exp ({\lambda t}),
\end{aligned} \label{e2.18}
\end{equation}
for all ${x,y \in S}$.
 From \eqref{e2.14} and \eqref{e2.18}, we observe that
\[
 |{Tx - Ty}|_S  \leq \alpha  |{x - y}|_S .
\]
Since $\alpha  < 1$, it follows from Banach fixed point theorem
  \cite[p. 37]{c2} that $T$ has a unique fixed in $S$.
 The fixed point of $T$ is however a solution of \eqref{e1.1}-\eqref{e1.2}.
The proof is complete.
\end{proof}


\begin{remark} \label{rmk2} \rm
 We note that the norm defined in \eqref{e2.2} was first used
by Bielecki \cite{b2} for proving  the existence and uniqueness of global solutions
for ordinary differential equations. For the developments related to
Bielecki's method, see \cite{c1}.
\end{remark}

The following theorem shows the uniqueness of solutions to
 \eqref{e1.1}-\eqref{e1.2} without the existence part.

\begin{theorem} \label{thm2}
 Assume that the function $f$ in equation \eqref{e1.1} satisfies
 \eqref{e2.4}. Then the \eqref{e1.1}-\eqref{e1.2} has at most one
 solution on $\mathbb{R}_+ $.
\end{theorem}

\begin{proof}
Let $x_1 (t)$ and $x_2 (t)$ be two solutions of  \eqref{e1.1}-\eqref{e1.2}
and $u(t)=  |{x_1 (t)- x_2 (t)}|$, $t \in \mathbb{R}_+ $.
 We consider the following two cases.

\noindent\textbf{Case 1: $0 \leq t < 1$.}
 From the hypotheses, we have
\begin{equation}
\begin{aligned}
u(t)&\leq \int_0^t { |{f({s,x_1 (s),\phi (s)})
 - f({s,x_2 (s),\phi (s)})}}| ds \\
& \leq \int_0^t {h(s) |{x_1 (s)- x_2 (s)}} |ds \\
&= \int_0^t {h(s)u(s)} ds.
\end{aligned} \label{e2.19}
\end{equation}
Now a suitable application of Lemma \ref{lemGB} (with $c=0$) yields
\begin{equation}
 |{x_1 (t)- x_2 (t)} |\leq 0. \label{e2.20}
\end{equation}

\noindent\textbf{Case 2: $1 \leq t < \infty $.}
 From the hypotheses, we have
\begin{equation}
\begin{aligned}
u(t)&\leq \int_0^1 { |{f({s,x_1 (s),\phi (s)})- f({s,x_2 (s),\phi (s)})}}| ds\\
&\quad + \int_1^t { |{f({s,x_1 (s),x_1 ({s - 1})})
 - f({s,x_2 (s),x_2 ({s - 1})})}} |ds \\
& \leq \int_0^1 {h(s)}  |{x_1 (s)- x_2 (s)}|ds\\
&\quad  + \int_1^t {h(s)[{ |{x_1 (s)- x_2 (s)}|
+  |{x_1 ({s - 1})- x_2 ({s - 1})}}|]} ds \\
& = \int_0^t {h(s)}  |{x_1 (s)- x_2 (s)}|ds + I_3 ,
\end{aligned} \label{e2.21}
\end{equation}
where
\[
I_3  = \int_1^t {h(s) |{x_1 ({s - 1})- x_2 ({s - 1})}}| ds.
%\label{e2.22}
\]
By making a change of variable, we observe that
\[
I_3  \leq \int_0^t {h({s + 1}) |{x_1 (s)- x_2 (s)}}| ds.
%\label{e2.23}
\]
Using this inequality in \eqref{e2.21}, we obtain
\[
u(t)\leq \int_0^t {[{h(s)+ h({s + 1})}]} u(s)ds.
%\label{e2.24}
\]
Now a suitable application of Lemma \ref{lemGB} (with $c=0$) yields
\[
 |x_1 (t)- x_2 (t)| \leq 0. %\label{e2.25}
\]
From \eqref{e2.20} and this inequality, we have
${x_1 (t)= x_2 (t)}$ for $t \in \mathbb{R}_+ $.
Thus there is at most one solution to \eqref{e1.1}-\eqref{e1.2} on
$\mathbb{R}_+$.
\end{proof}

\section{Continuous dependence}

In this section we study the continuous dependence of solutions to
\eqref{e1.1} on the given initial data, and on the function $f$.
Also we show the continuous dependence of
solutions of equations of the form \eqref{e1.1} on certain parameters.

First, we shall give the following theorem concerning the continuous
dependence of solutions to \eqref{e1.1} on the given initial data.

\begin{theorem} \label{thm3}
 Assume that the function $f$ in  \eqref{e1.1} satisfies the condition
\eqref{e2.4}. Let $x_1 (t)$ and $x_2 (t)$ be the solutions of
\eqref{e1.1} with the initial conditions
\begin{gather}
x_1 ({t - 1})= \phi _1 (t)\quad
({0 \leq t < 1}),\quad x_1 (0)= c_1 , \label{e3.1}
\\
x_2 ({t - 1})= \phi _2 (t)\quad
({0 \leq t < 1}),\quad x_2 (0)= c_2 , \label{e3.2}
\end{gather}
respectively, where $c_1 ,c_2$ are constants. Then
\begin{equation}
 |{x_1 (t)- x_2 (t)}| \leq c\exp \Big({\int_0^t {h(s)ds} }\Big), \label{e3.3}
\end{equation}
for $0 \leq t < 1$ and
\begin{equation}
 |{x_1 (t)- x_2 (t)}| \leq c\exp \Big({\int_0^t {[{h(s)+ h({s + 1})}]ds} }\Big),
\label{e3.4}
\end{equation}
for $1 \leq t < \infty$, where
\begin{equation}
c =  |{c_1  - c_2 }| + \int_0^1 {h(s) |{\phi _1 (s)-\phi _2 (s)}} |ds. \label{e3.5}
\end{equation}
\end{theorem}

\begin{proof}
Let $u(t)=  |{x_1 (t)- x_1 (t)}$ for $t \in \mathbb{R}_+$. We consider
the following two cases.

\noindent\textbf{Case 1: $0 \leq t < 1$.}
 From the hypotheses, it follows that
\begin{equation}
\begin{aligned}
u(t)&\leq  |{c_1  - c_2 }|
+ \int_0^t {h(s) |{f({s,x_1 (s),\phi _1 (s)})
 - f({s,x_2 (s),\phi _2 (s)})}} |ds \\
& \leq  |{c_1  - c_2 }| + \int_0^t {h(s)[|{ |{x_1 (s)- x_2 (s)}|
 +  |{\phi _1 (s)- \phi _2 (s)}}|]} ds \\
& \leq  |{c_1  - c_2 }| + \int_0^1 {h(s) |{\phi _1 (s)- \phi _2 (s)}} |ds
 + \int_0^t {h(s) |{x_1 (s)- x_2 (s)}}| ds \\
&= c + \int_0^t {h(s)u(s)ds.}
\end{aligned} \label{e3.6}
\end{equation}
Now an application of Lemma \ref{lemGB} to \eqref{e3.6}, yields \eqref{e3.2}.

\noindent\textbf{Case 2: $1 \leq t < \infty$.}
 By following a similar arguments as in the proof of Theorem \ref{thm2}
in case 2, from the hypotheses, it follows that
\begin{align}
u(t)&\leq  |{c_1  - c_2 }|
+ \int_0^1 { |{f({s,x_1 (s),\phi _1 (s)})- f({s,x_2 (s),\phi _2 (s)})}}| ds
\notag\\
&\quad  + \int_1^t { |{f({s,x_1 (s),x_1 ({s - 1})})- f({s,x_2 (s),x_2
 ({s - 1})})}} |ds \notag \\
& \leq  |{c_1  - c_2 }| + \int_0^1 {h(s)[{ |{x_1 (s)- x_2 (s)}|
+  |{\phi _1 (s)- \phi _2 (s)}}|]} ds \notag \\
&\quad  + \int_1^t {h(s)[{ |{x_1 (s)- x_2 (s)}|
 +  |{x_1 ({s - 1})- x_2 ({s - 1})}}|]} ds \notag \\
& =  |{c_1  - c_2 }| + \int_0^1 {h(s) |{\phi _1 (s)- \phi _2 (s)}} |ds
 + \int_0^t {h(s) |{x_1 (s)- x_2 (s)}}| ds \notag \\
&\quad  + \int_1^t {h(s) |{x_1 ({s - 1})- x_2 ({s - 1})}}| ds \notag \\
& \leq c + \int_0^t {[{h(s)+ h({s + 1})}]} u(s)ds.
\label{e3.7}
\end{align}
Now an application of Lemma \ref{lemGB} yields \eqref{e3.4}.
From \eqref{e3.3} and \eqref{e3.4}, it follows that the solutions of
equation \eqref{e1.1} depends on the given initial data.
\end{proof}

Now, we consider \eqref{e1.1}-\eqref{e1.2} and the corresponding
initial-value problem
\begin{equation}
y'(t)= F({t,y(t),y({t - 1})}), \label{e3.8}
\end{equation}
for $t \in \mathbb{R}_+$ under the initial conditions
\begin{equation}
y({t - 1})= \psi (t),\quad y(0)= y_0 , \label{e3.9}
\end{equation}
where $F \in C({\mathbb{R}_+   \times \mathbb{R}^n  \times \mathbb{R}^n ,
\mathbb{R}^n })$ and $\psi (t)$ is a continuous function for which
the limit $\lim _ {t \to 1 - 0} \psi (t)$ exists.

The following theorem shows the continuous dependence of solutions to
\eqref{e1.1}-\eqref{e1.2} on the function $f$/

\begin{theorem} \label{thm4}
 Assume that the function $f$ in  \eqref{e1.1} satisfies \eqref{e2.4} and
\begin{gather}
\begin{aligned}
&|{x_0  - y_0 }| + \int_0^1 {h(s) |{\phi (s)- \psi (s)}|ds} \\
& + \int_0^1 { |{f({s,y(s),\phi (s)})- F({s,y(s),\psi (s)})}}| ds
\leq \varepsilon _1 ,
\end{aligned} \label{e3.10}
\\
\int_1^t { |{f({s,y(s),y({s - 1})})- F({s,y(s),y({s - 1})})}}| ds
\leq \varepsilon _2 , \label{e3.11}
\end{gather}
where $x_0 ,\phi ,f$ and $y_{0,} \psi ,F $ are as in
 \eqref{e1.1}-\eqref{e1.2} and  \eqref{e3.8}-\eqref{e3.9},
$\varepsilon _1 ,\varepsilon _2 $ are nonnegative constants and
$y(t)$ is a solution of \eqref{e3.8}-\eqref{e3.9}.
Then the solution $x(t)$ of  \eqref{e1.1}-\eqref{e1.2} depends
continuously on the functions involved therein as given below
by \eqref{e3.13} and \eqref{e3.15}.
\end{theorem}

\begin{proof}
Let $u(t)=  |x(t)- y(t)|$ for $t \in \mathbb{R}_+$. We consider
the following two cases.

\noindent\textbf{Case 1: $0 \leq t < 1$.}
 From the hypotheses, we have
\begin{align}
u(t)&\leq  |{x_0  - y_0 }|
+ \int_0^t { |{f({s,y(s),\phi (s)})- F({s,y(s),\psi (s)})}}| ds \notag \\
& \leq  |{x_0  - y_0 }| + \int_0^t { |{f({s,y(s),\phi (s)})
- f({s,y(s),\psi (s)})}}| ds \notag \\
&\quad  + \int_0^t { |{f({s,y(s),\psi (s)})- F({s,y(s),\psi (s)})}} |ds \notag \\
& \leq  |{x_0  - y_0 }| + \int_0^t {h(s)[{ |{x(s)- y(s)}|
+  |{\phi (s)- \psi (s)}}|]ds} \notag \\
&\quad  + \int_0^1 { |{f({s,y(s),\psi (s)})- F({s,y(s),\psi (s)})}}| ds \notag \\
& \leq  |{x_0  - y_0 }| + \int_0^t {h(s) |{x(s)- y(s)}|ds}
 + \int_0^1 {h(s) |{\phi (s)- \psi (s)}|ds} \notag \\
& + \int_0^1 { |{f({s,y(s),\psi (s)})- F({s,y(s),\psi (s)})}}|ds \notag \\
& \leq \varepsilon _1  + \int_0^t {h(s)u(s)ds.}
\label{e3.12}
\end{align}
Now an application of Lemma \ref{lemGB} yields that for $0 \leq t < 1$,
\begin{equation}
 |{x(t)- y(t)}| \leq \varepsilon _1 \exp \Big({\int_0^t {h(s)ds} }\Big).
\label{e3.13}
\end{equation}

\noindent\textbf{Case 2: $1 \leq t < \infty$.}
 Following an arguments as in the proof of Theorem \ref{thm2} in case 2,
 from the hypotheses, we have
\begin{align}
u(t)
&\leq  |{x_0  - y_0 }| + \int_0^1 { |{f({s,x(s),\phi (s)})
- F({s,y(s),\psi (s)})}|ds} \notag \\
&\quad  + \int_1^t { |{f({s,x(s),x({s - 1})})- F({s,y(s),y({s - 1})})}|ds}\notag \\
& \leq  |{x_0  - y_0 }|
+ \int_0^1 { |{f({s,x(s),\phi (s)})- f({s,y(s),\psi (s)})}|ds} \notag  \\
&\quad + \int_0^1 { |{f({s,y(s),\psi (s)})- F({s,y(s),\psi (s)})}|ds} \notag  \\
&\quad  + \int_1^t { |{f({s,x(s),x({s - 1})})- f({s,y(s),y({s - 1})})}|ds} \notag \\
&\quad  + \int_1^t { |{f({s,y(s),y({s - 1})})- F({s,y(s),y({s - 1})})}|ds}\notag \\
&\leq  |{x_0  - y_0 }|
+ \int_0^1 {h(s)[{ |{x(s)- y(s)}| +  |{\phi (s)- \psi (s)}}|]ds} \notag \\
&\quad  + \int_0^1 { |{f({s,y(s),\psi (s)})- F({s,y(s),\psi (s)})}|ds} \notag \\
&\quad  + \int_1^t {h(s)[{ |{x(s)- y(s)}| +  |{x({s - 1})- y({s - 1})}}|]ds} \notag \\
&\quad  + \int_1^t { |{f({s,y(s),y({s - 1})})- F({s,y(s),y({s - 1})})}|ds}\notag \\
&\leq  |{x_0  - y_0 }| + \int_0^1 {h(s) |{\phi (s)- \psi (s)}|ds}\notag \\
&\quad + \int_0^1 { |{f({s,y(s),\psi (s)})- F({s,y(s),\psi (s)})}|ds}\notag \\
&\quad + \int_0^t {h(s) |{x(s)- y(s)}ds}|
 + \int_0^t {h({s + 1}) |{x(s)- y(s)}|ds} \notag \\
&\quad + \int_1^t { |{f({s,y(s),y({s - 1})})- F({s,y(s),y({s - 1})})}|ds} \notag \\
&\leq ({\varepsilon _1  + \varepsilon _2 })
 + \int_0^t {[{h(s)+ h({s + 1})}]u(s)ds.} \label{e3.14}
\end{align} 
Now an application of Lemma \ref{lemGB} yields that for $1 \leq t < \infty$,
\begin{equation}
 |x(t)- y(t)| \leq ({\varepsilon _1  + \varepsilon _2 })
\exp \Big({\int_0^t {[{h(s)+ h({s + 1})}]ds} }\Big). \label{e3.15}
\end{equation}
From this inequality and \eqref{e3.13}, it follows that
 \eqref{e1.1}-\eqref{e1.2} depends continuously on the functions
involved therein.
\end{proof}

Next, we consider the  difference-differential equations
\begin{gather}
x'(t)= g({t,x(t),x({t - 1}),\mu }), \label{e3.16} \\
x'(t)= g({t,x(t),x({t - 1}),\mu _0 }), \label{e3.17}
\end{gather}
for $t \in \mathbb{R}_+$, where
$g \in C({\mathbb{R}_+\times \mathbb{R}^n  \times \mathbb{R}^n
\times R,\mathbb{R}^n })$, and
with the initial conditions given by \eqref{e1.2},


The following theorem states the continuous dependency of solutions
to \eqref{e3.16}-\eqref{e1.2} and  \eqref{e3.17}-\eqref{e1.2} on parameters.

\begin{theorem} \label{thm5}
Assume that the function $g$ satisfy the conditions
\begin{gather}
 |{g({t,x,y,\mu })- g({t,\bar x,\bar y,\mu })}|
 \leq p(t)[{ |{x - \bar x}| +  |{y - \bar y}}|], \label{e3.18} \\
 |{g({t,x,y,\mu })- g({t,x,y,\mu _0 })}|
 \leq q(t) |{\mu  - \mu _0 }|,     \label{e3.19}
\end{gather}
where $p,q \in C({\mathbb{R}_+  ,\mathbb{R}_+  })$,
Let $x_1 (t)$ and $x_2 (t)$ be the solutions of \eqref{e3.16}-\eqref{e1.2}
and \eqref{e3.17}-\eqref{e1.2} respectively. Then
\begin{equation}
 |{x_1 (t)- x_2 (t)}| \leq \Big({ |{\mu  - \mu _0| }
\int_0^1 {q(s)} }\Big)\exp \Big({\int_0^t {p(s)ds} }\Big), \label{e3.20}
\end{equation}
for $0 \leq t < 1$ and
\begin{equation}
 |{x_1 (t)- x_2 (t)}| \leq \Big({ |{\mu  - \mu _0 }|\int_0^t {q(s)} }\Big)
\exp \Big({\int_0^t {[{p(s)+ p({s + 1})}]ds} }\Big),
\label{e3.21}
\end{equation}
for $1 \leq t < \infty $.
\end{theorem}

\begin{proof}
Let $u(t)=  |{x_1 (t)- x_2 (t)}$ for ${t \in \mathbb{R}_+  }$.
We consider the following two cases.

\noindent\textbf{Case 1: $0 \leq t < 1$.}
 From the hypotheses, we have
\begin{align*}
u(t)&\leq \int_0^t { |{g({s,x_1 (s),\phi (s),\mu })- g({s,x_2 (s),
 \phi (s),\mu _0 })}} |ds \\
&\leq \int_0^t { |{g({s,x_1 (s),\phi (s),\mu })- g({s,x_2 (s),
 \phi (s),\mu })}} |ds \\
&\quad + \int_0^t { |{g({s,x_2 (s),\phi (s),\mu })- g({s,x_2 (s),
 \phi (s),\mu _0 })}} |ds \\
&\leq \int_0^t {p(s) |{x_1 (s)- x_2 (s)}}| ds
 + \int_0^t {q(s) |{\mu  - \mu _0 }}| ds \\
& \leq  |{\mu  - \mu _0 }|\int_0^1 {q(s)ds + \int_0^t {p(s)u(s)ds.} }
%\label{e3.22}
\end{align*}
Now a suitable application of Lemma \ref{lemGB} yields \eqref{e3.20}.

\noindent\textbf{Case 2: $1 \leq t < \infty$.}
 By following the arguments in the proof of Theorem \ref{thm2} in case 2,
 from the hypotheses, we have
\begin{align*}
u(t)&\leq \int_0^1 { |{g({s,x_1 (s),\phi (s),\mu })- g({s,x_2 (s),
  \phi (s),\mu _0 })}} |ds \\
&\quad  + \int_1^t { |{g({s,x_1 (s),x_1 ({s - 1}),\mu })
  - g({s,x_2 (s),x_2 ({s - 1}),\mu _0 })}}| ds \\
& \leq \int_0^1 { |{g({s,x_1 (s),\phi (s),\mu })- g({s,x_2 (s),
  \phi (s),\mu })}} |ds \\
&\quad  + \int_0^1 { |{g({s,x_2 (s),\phi (s),\mu })
 - g({s,x_2 (s),\phi (s),\mu _0 })}} |ds \\
&\quad  + \int_1^t { |{g({s,x_1 (s),x_1 ({s - 1}),\mu })
 - g({s,x_2 (s),x_2 ({s - 1}),\mu })}} |ds \\
&\quad  + \int_1^t { |{g({s,x_2 (s),x_2 ({s - 1}),\mu })
 - g({s,x_2 (s),x_2 ({s - 1}),\mu _0 })}} |ds \\
&\leq \int_0^1 {p(s) |{x_1 (s)- x_2 (s)}} |ds
 + \int_0^1 {q(s) |{\mu  - \mu _0 }} |ds \\
&\quad  + \int_1^t {p(s)[{ |{x_1 (s)- x_2 (s)}|
 +  |{x_1 ({s - 1})- x_2 ({s - 1})}}|]} ds
 + \int_1^t {q(s) |{\mu  - \mu _0 }}| ds \\
&= \int_0^t {q(s) |{\mu  - \mu _0 }}|ds
 + \int_0^t p(s) |x_1 (s)- x_2 (s)|ds\\
&\quad + \int_1^t {p(s) |{x_1 ({s - 1})- x_2 ({s - 1})}}| ds \\
& \leq  |\mu  - \mu _0 |\int_0^t q(s)ds+ \int_0^t [{p(s)+ p({s + 1})}]u(s)ds.
%\label{e3.23}
\end{align*}
Now a suitable application of Lemma \ref{lemGB}  yields \eqref{e3.21}.
 From \eqref{e3.20} and \eqref{e3.21}, it follows that the solutions
 to \eqref{e3.16}-\eqref{e1.2} and \eqref{e3.17}-\eqref{e1.2}
 depend continuously on the parameter $\mu$.
\end{proof}

Note that there are many papers and monographs concerning the existence,
uniqueness and other properties of solutions of \eqref{e1.1};
see \cite{p2,s1},  \cite[p. 342]{b1}, \cite[p.308]{d1},  \cite[p. 18]{k1}
and the references cited therein. We believe that the results given here,
using elementary analysis, present some useful basic results for
future reference.

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\end{document}