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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 140, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/140\hfil
Integrodifferential equation on time scales]
{Fredholm type integrodifferential equation on time scales}

\author[D. B. Pachpatte\hfil EJDE-2010/140\hfilneg]
{Deepak B. Pachpatte}

\address{Deepak B. Pachpatte \newline
 Department of Mathematics,
 Dr. B.A.M. University, Aurangabad,
 Maharashtra 431004, India}
\email{pachpatte@gmail.com}

\thanks{Submitted May 20, 2010. Published October 5, 2010.}
\subjclass[2000]{39A10, 39A12}
\keywords{Fredholm type; integrodifferential equations;
explicit estimate; \hfill\break\indent
integral inequality; continuous dependence}

\begin{abstract}
 The aim of this article is to study some basic qualitative
 properties of solutions to Fredholm type integrodifferential
 equations on time scales. A new integral inequality with
 explicit estimate on time scales is obtained and used to
 establish the results.
\end{abstract}

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

\section{Introduction}

The theory of time scales was introduced by Stefan Hilger
\cite{Hig} in 1988 which unifies continuous and discrete analysis.
Since then many authors have studied various aspects
of dynamic integral equations on time scales by using various
techniques \cite{Kul,Dbp1,Dbp2,Tis,Tis2}. In this paper we
consider the integrodifferential equation
\begin{equation}
x^\Delta  (t )= f( {t,x(t ),x^\Delta (t ),Hx(t )} ),\quad
x(\alpha)= x_0, \label{e1.1}
\end{equation}
for $t \in [\alpha,\beta] \subset I_\mathbb{T}$, where
\begin{equation}
Hx(t )= \int_\alpha ^\beta h({t,\tau ,x(\tau ),x^\Delta (\tau  )}
)\Delta\tau , \label{e1.2}
\end{equation}
$f, h$ are given functions and $x$ is unknown function to be found,
and $\Delta$ denotes the delta derivative. We assume that
$h:I_\mathbb{T}^2  \times \mathbb{R}^n  \times \mathbb{R}^n  \to
\mathbb{R}^n $, $f:I_\mathbb{T}  \times \mathbb{R}^n  \times
\mathbb{R}^n \times \mathbb{R}^n  \to \mathbb{R}^n$ are
rd-continuous functions, $t$ is from a time scale $\mathbb{T}$,
which is nonempty closed subset of $\mathbb{R}$, the set of real
numbers, $\tau  \le t$ and $I_\mathbb{T}  = I \cap \mathbb{T}$,
$I = [t_0 ,\infty )$ the given subset of $\mathbb{R}$,
$\mathbb{R}^n$ the real n-dimensional Euclidean space with
appropriate norm defined by $|\cdot|$. The integral sign represents
the delta integral. Recently in \cite{Boh1, Boh2, Kul, Dbp1,
Dbp2,Tis, Tis2} the authors have studied the existence, uniqueness
and other qualitative properties of solutions of various dynamic
equations on time scales by using different techniques. In fact
the study of equations of the form \eqref{e1.1} is a challenging
task, because of the occurrence of the $x^\Delta$ on the right
hand side in \eqref{e1.1}. One can formulate existence and
uniqueness result for \eqref{e1.1} by using the idea recently
employed in [8, Theorem 3.1]. Motivated by the results obtained by
the present author in \cite{Dbp2}, in this paper we study some
fundamental qualitative properties of solutions of equation
\eqref{e1.1}. Time scale analogue of a variant of a certain
integral inequality with explicit estimate is obtained and used to
establish the results.

\section{Preliminaries}

  A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset
of the real numbers $\mathbb{R}$. We define the jump operators
$\sigma, \rho$ on $\mathbb{T}$ by the two mapping $\sigma ,\rho
:\mathbb{T} \to \mathbb{R}$ satisfying conditions
$$
\sigma (t) = \inf \{ s \in \mathbb{T}:s > t\}, \quad
\rho (t) = \sup \{ s \in \mathbb{T}:s < t\}.
$$
The jump operators classify the points of time scale $\mathbb{T}$
as left dense, left scattered, right dense and right scattered
according to whether  $\rho (t) = t$ or $\rho (t) < t$,
$\sigma (t)=t$ and $\sigma (t) > t$ respectively for
$t \in \mathbb{T}$. A function $f:\mathbb{T} \to \mathbb{R}$
is said to be rd-continuous if it is continuous at each right
 dense point in $\mathbb{T}$. The set of all rd-continuous
functions is denoted by $C_{rd}$, If $\mathbb{T}$ has left
scattered maximum $m$, then
\begin{equation}
\mathbb{T}^k  =  \begin{cases}
   \mathbb{T} - m &\text{if }\sup \mathbb{T} < \infty   \\
   \mathbb{T} & \text{if }\sup \mathbb{T} = \infty  \\
\end{cases}
\end{equation}
A function $F:\mathbb{T} \to \mathbb{R}$ is said to be an
antiderivative of $f:\mathbb{T} \to \mathbb{R}$ provided
$F^\Delta=f(t)$ holds for all $t \in I_\mathbb{T}$.
We define the integral of $f$ by
\begin{equation}
\int_s^t {f(t )\Delta \tau }  = F(t )- F( s ),
\end{equation}
where $s,t \in \mathbb{T}$. The graininess function  $\mu
:\mathbb{T} \to \mathbb{R}_+ = [0,\infty )$ is defined by $\mu (t)
= \sigma (t) - t$. The function $p:\mathbb{T} \to \mathbb{R}$ is
said to be regressive if $1 + \mu (t )p(t )\ne 0$ for all $t \in
I_\mathbb{T}$. We denote by $\Re$ the set of all regressive and
rd-continuous functions and define the set of all regressive
functions by
 \begin{equation}
 \Re^ +    = \{ {p \in \mathbb{R}:1 + \mu (t )p(t )> 0
\text{ for all }t \in \mathbb{T}} \}.
 \end{equation}
For $p \in \Re^ +$ we define (see \cite{Boh1}) the exponential
function $e_p ( {.,t_0 } )$ on time scale $\mathbb{T}$ as the
unique solution to the scalar initial value problem
\begin{equation}
x^\Delta  (t )= p(t )x(t),\quad x({t_0 } )= 1.
\end{equation}

If $p \in \Re^ +$, then $e_p ( {t,t_0 } ) >0$ for all
$t \in \mathbb{T}$. The exponential function $e_p ( {.,t_0 } )$
is given by
\begin{equation}
e_p ({t,t_0 } )= \begin{cases}
   \exp \big(\int_{t_0 }^t {p(s )\Delta s} \big)
&\text{for } t \in \mathbb{T},\; \mu  = 0;  \\[4pt]
\exp \big(\int_{t_0 }^t \frac{\log(1 + \mu (s)p(s))}
{\mu (s )} \Delta s\big) &\text{for } t \in \mathbb{T},\; \mu  > 0;
\end{cases}
\end{equation}
where log is a principle logarithm function. To allow a comparison
of the results in the paper with the continuous case, we note that,
if $\mathbb{T}=\mathbb{R}$, the exponential function is given by
\begin{equation}
e_p ({t,s} )= \exp \Big(\int_s^t {p(\tau )d\tau } \Big),\quad
e_\alpha({t,s} )= \exp ({\alpha ({t - s} )} ),\quad
e_\alpha ({t,0} )= \exp ({\alpha t} ),
\end{equation}
for $s,t \in \mathbb{R}$, where $\alpha \in \mathbb{R}$ is a constant
and $p:\mathbb{R} \to \mathbb{R}$ is a continuous function.
To compare with the discrete case, if $\mathbb{T}=\mathbb{Z}$
(the set of integers), the exponential function is given by
\begin{equation}
e_p ({t,s} )= \prod_{\tau  = s}^{t - 1} {[ {1 + p(\tau )}]} ,\quad
e_\alpha  ({t,s} )= ( {1 + \alpha } )^{t - s},\quad
e_\alpha ({t,0} )= ({1 + \alpha } )^t ,
\end{equation}
for $s,t \in \mathbb{Z}$ with $s < t$, where $\alpha \neq -1$ is a
constant and $p:\mathbb{Z} \to \mathbb{R}$ is a
sequence satisfying $p(t) \neq -1 $ for all $t \in \mathbb{Z} $.
We use the following fundamental result proved in Bohner and
Peterson \cite{Boh1} (see also [3]).

\begin{lemma}
Suppose $u,b \in C_{rd}$ and $a \in   \Re^ +$. Then
\begin{equation}
u^\Delta  (t )\le a(t )u(t )+ b(t )
\end{equation}
for all $t \in \mathbb{T}$, implies
\begin{equation}
u(t )\le u({t_0 } )e_a ({t,t_0 } )+ \int_{t_0 }^t {e_a ({t,\sigma
(\tau  )} )} b(\tau  )\Delta \tau,
\end{equation}
for all $t \in \mathbb{T}$.
\end{lemma}

\section{Basic integral inequality on time scale}

In this section we establish the time scale analogue of the
variant of the integral inequality given in
 [6,Theorem 1.3.1 part(a2), p.41].

\begin{theorem} \label{thm1}
Let  $u,a,b,c,d,f,g \in C_{rd} ({I_\mathbb{T} ,\mathbb{R}_+  } )$
and suppose that
\begin{equation}
\begin{aligned}
u(t )
&\le a(t )+ b(t )\int_\alpha ^t {f(t )} \Big[ {u(s )+ d(s )
\int_\alpha ^\beta  {g(\tau  )} u(\tau  )\Delta \tau } \Big]\Delta s\\
&\quad + c(t )\int_\alpha ^\beta  {g(\tau  )u} ( \tau )\Delta \tau ,
\end{aligned}\label{e3.1}
\end{equation}
for $t \in I_\mathbb{T}$. If
\begin{equation}
k = \int_\alpha ^\beta  {g(\xi  )} K_2 (\xi )\Delta \xi  < 1,
\label{e3.2}
\end{equation}
then
\begin{equation}
u(t )\le K_1 (t )+ MK_2 (t ), \label{e3.3}
\end{equation}
for $t \in I_\mathbb{T}$, where
\begin{gather}
K_1 (t )= a(t )+ b(t )\int_\alpha ^t {f(\tau )} a(\tau )e_{fb}
({t,\sigma (\tau )} )\Delta \tau, \label{e3.4}
\\
K_2 (t )= c(t )+ b(t )\int_\alpha ^t {f(\tau )\{ {c(\tau  )+
d(\tau )} \}} e_{fb} ({t,\sigma (\tau )} )\Delta \tau,
\label{e3.5}
\\
M = \frac{1}{{1 - k}}\int_\alpha ^\beta  {g(\xi  )K_1 (\xi )\Delta
\xi } . \label{e3.6}
\end{gather}
\end{theorem}

\begin{proof}
Let
\begin{equation}
\lambda  = \int_\alpha ^\beta  {g(\tau  )u(\tau )\Delta \tau ,}
\label{e3.7}
\end{equation}
and
\begin{equation}
z(t )
= \int_\alpha ^t {f(s )[{u(s )+ d(s )\int_\alpha ^\beta  {g(\tau  )
 u(\tau  )\Delta \tau } } ]} \Delta s
= \int_\alpha ^t {f(s )[{u(s )+ d( s )\lambda } ]}
\Delta s, \label{e3.8}
\end{equation}
then $z(\alpha)=0$ and \eqref{e3.1} can be restated as
\begin{equation}
u(t )\le a(t )+ b(t)z(t) + c(t)\lambda. \label{e3.9}
\end{equation}
 From \eqref{e3.8} and \eqref{e3.9}, we have
\begin{equation}
\begin{aligned}
z^\Delta  (t )
&\le f(t )[{a(t )+ b(t)z(t) + c(t)\lambda  + d(t )\lambda } ] \\
& = f(t )b(t )z(t )+ f(t )[{a( t )+ \{ {c(t )+ d(t )}
\}} \lambda ].
\end{aligned} \label{e3.10}
\end{equation}
Applying lemma 2.1 to \eqref{e3.10} yields
\begin{equation}
z(t )\le \int_\alpha ^t {f(\tau  )} [{a(\tau )+ \lambda
\{ {c(\tau )+ d(\tau )} \}} ] e_{fb} ({t,\sigma
(\tau )} )\Delta \tau . \label{e3.11}
\end{equation}
Using \eqref{e3.11} in \eqref{e3.9}, we obtain
\begin{equation}
\begin{aligned}
u(t )
& \le a(t )+ b(t)\Big\{ {\int_\alpha ^t {f(\tau  )[{a(\tau  )
+ \lambda \{ {c(\tau  )+ d(\tau  )} \}} ]}
e_{fb} ({t,\sigma (\tau  )} )\Delta \tau} \Big\}+c(t )\lambda \\
&= a(t) + b(t)\int_\alpha ^t {f(\tau  )} a(\tau  )e_{fb} ({t,\sigma (\tau  )} )\Delta \tau\\
&\quad + \lambda \Big\{ {c(t) + b(t)\int_\alpha ^t {f(\tau  )
\{ {c(\tau ) + d(\tau )} \}} e_{fb} (t,\sigma (t ))\Delta \tau } \Big\}\\
& =K_1(t)+\lambda K_2(t).
\end{aligned} \label{e3.12}
\end{equation}
 From this inequality and \eqref{e3.7}, it is easy to observe
that $\lambda  \le M$.
Using this inequality in \eqref{e3.12}, we obtain \eqref{e3.3}.
\end{proof}

\section{Estimates on the solutions}

In this section we obtain estimates on the solutions of equation
\eqref{e1.1} by applying Theorem 3.1, under some suitable
conditions on the functions involved therein.

First, we shall give the following theorem concerning the estimate
on the solution of equation \eqref{e1.1}.

\begin{theorem}
Suppose that the functions $f,h$ in \eqref{e1.1} satisfy the
conditions
\begin{gather}
| {f({t,u,v,w} )} | \le \gamma [{|u| + |v | + | w |} ], \label{e4.1}\\
| {h({t,u,v,w} )} | \le q(t )r (\tau )[{| u| + | v |} ], \label{e4.2}
\end{gather}
where $0 \le \gamma <1$ is a constant and
$q,r \in C_{rd}(I_\mathbb{T},\mathbb{R}_+)$. Let
\begin{gather}
L_1 (t )= \frac{{| {x_0 } |}}{{1 - \gamma }} +
\frac{1}{{1 - \gamma }}\Big[{\int_\alpha ^t {\frac{\gamma }{{1 -
\gamma }}| {x_0 } |e_{\frac{\gamma }{{1 - \gamma }}}
({t,\tau } )\Delta \tau } } \Big], \label{e4.3}
\\
L_2 (t )= \frac{\gamma }{{1 - \gamma }}q(t )+ \frac{1}{{1 - \gamma
}}\Big[{\int_\alpha ^t {\gamma [{\frac{\gamma }{{1 -
\gamma }} + 1} ]} q(t )e_{\frac{\gamma }{{1 - \gamma }}}
({t,\tau } )\Delta \tau } \Big], \label{e4.4}
\end{gather}
for  $t \in I_\mathbb{T}$ and
\begin{equation}
\lambda  = \int_\alpha ^\beta  {r(\xi  )} L_2 ( \xi )\Delta \xi<1
, \quad % \label{e4.5}
Q = \frac{1}{{1 - \lambda }}\int_\alpha ^\beta  {r(\xi )} L_1 (\xi
)\Delta \xi . \label{e4.6}
\end{equation}
If $x(t)$ is a solution of  \eqref{e1.1} on $I_\mathbb{T}$,
then
\begin{equation}
| {x(t )} | + | {x^\Delta  (t )} | \le L_1
(t )+ QL_2 (t ), \label{e4.7}
\end{equation}
for  $t \in I_\mathbb{T}$.
\end{theorem}

\begin{proof}
Let $m(t) = | {x(t )} | + | {x^\Delta (t )}|$,
$t \in I_\mathbb{T}$. Using the fact that $x(t)$ is a
solution of \eqref{e1.1} and the hypotheses, we have
\begin{align*}
m(t)
&  = \Big| {x_0  + \int_\alpha ^t {f({s,x(s),x^\Delta  (s),Hx(s)} )
 \Delta s} } \Big| + | {f(t ,x(t),x^\Delta  (t),Hx(t))} | \\
& \le | {x_0 } | + \int_\alpha ^t {\gamma
\Big[{m(s) + \int_\alpha ^\beta  {q(s)r(\tau )m(\tau )\Delta \tau } }
\Big]} \Delta s \\
&\quad + \gamma \Big[{m(t )+ \int_\alpha ^\beta {q(t )r(\tau )m(\tau
)\Delta \tau } } \Big]. % \label{e4.8}
\end{align*}
 From this inequality,  we have
\begin{align*}
m(t)
& \le \frac{{| {x_0 } |}}{{1 - \gamma }}
+ \frac{1}{{1 - \gamma }}\int_\alpha ^t {\gamma
\Big[{m(s) + q(s)\int_\alpha ^\beta  {r(\tau )m(\tau )\Delta \tau } }
\Big]} \Delta s \\
&\quad + \frac{\gamma }{{1 - \gamma }}q(t)\int_\alpha ^\beta  {r(\tau
)} m(\tau )\Delta \tau. % \label{e4.9}
\end{align*}
Now an application of theorem 3.1 to the above inequality, we have
\eqref{e4.7}.
\end{proof}

\begin{remark} \label{rmk1} \rm
The estimate obtained in  \eqref{e4.7} yields
bounds on the solution $x(t)$ and its delta derivative. If the
estimate in \eqref{e4.7} is bounded, then the solution $x(t)$ of
\eqref{e1.1} and its delta derivative are also bounded on
$I_\mathbb{T}$.
\end{remark}

 Consider the IVP \eqref{e1.1} with the IVP
\begin{equation}
z^\Delta  (t )= g({t,z(t),z^\Delta  (t),Hz(t)} ),\quad
z(\alpha ) = z_0, \label{e4.10}
\end{equation}
for $t \in I_\mathbb{T}$, where $H$ is given by \eqref{e1.2} and
$g\in C_{rd} ({I_\mathbb{T}  \times \mathbb{R}^n  \times
\mathbb{R}^n \times \mathbb{R}^n , \mathbb{R}^n } )$.


 The next result deals with the closeness of solutions of
\eqref{e1.1} and \eqref{e4.10}.

\begin{theorem}
Suppose that the functions $f,h$ in \eqref{e1.1} satisfy the
conditions
\begin{gather}
| {f({t,u,v,w} )- f({t,\overline u ,\overline v ,\overline w
} )} | \le \gamma [{| {u - \overline u } |
+ | {v - \overline v } | + | {w - \overline w }
|} ], \label{e4.11} \\
| {h({t,\tau ,u,v} )- h({t,\tau ,\overline u ,\overline v }
)} | \le q(t)r(\tau )[{| {u - \overline u }
| + | {v - \overline v } |} ], \label{e4.12}
\end{gather}
where $0 \le \gamma <1$ is a constant and $q,r \in C_{rd}
({I_\mathbb{T} ,\mathbb{R}_+  } )$ and
\begin{gather}
| {f({t,u,v,w} )- g(t,u,v,w)} | \le  \epsilon ,
\label{e4.13} \\
| {x_0  - z_0 } | \le \delta ,
\label{e4.14}
\end{gather}
where $f,x_0$ and $g,z_0$ are as in \eqref{e1.1} and
\eqref{e4.10}. Let
\begin{equation}
w(t) = \delta  +  \epsilon [{1 + t - \alpha } ],
\label{e4.15}
\end{equation}
$\lambda ,L_2 (t)$ be as in \eqref{e4.4}, \eqref{e4.6} and
\begin{equation}
Q_0  = \frac{1}{{1 - \lambda }}\int_\alpha ^\beta  {r(\xi )} A_0
(\xi )\Delta \xi, \label{e4.16}
\end{equation}
in which
\begin{equation}
A_0 (t) = \frac{{w(t )}}{{1 - \gamma }} + \frac{1}{{1 - \gamma
}}\Big[{\int_\alpha ^t {\frac{\gamma }{{1 - \gamma }}w(\tau
)e_{\frac{\gamma }{{1 - \gamma }}} ({t,\tau } )\Delta \tau } }
\Big]. \label{e4.17}
\end{equation}
Let $y(t)$ and $z(t)$ be respectively, solutions of \eqref{e1.1}
and \eqref{e4.10} on $I_\mathbb{T}$, then
\begin{equation}
| {x(t) - z(t)} | + | {x^\Delta  (t) - z^\Delta
(t)} | \le A_0 (t )+ Q_0 L_2 (t ), \label{e4.18}
\end{equation}
for $t \in I_\mathbb{T}$.
\end{theorem}

\begin{proof}
Let $u(t )= | {x(t) - z(t)} | + | {x^\Delta (t) -
z^\Delta  (t)} |$, $t \in I_\mathbb{T}$. Using the hypotheses,
we have
\begin{align*}
u(t)
& \le | {x_0  - z_0 } | + \int_\alpha ^t {| {f({s,x(s ),x^\Delta  (s),Hx(s)} )- f({s,z(s),z^\Delta  (s),Hz(s)} )} |\Delta s} \\
&\quad + \int_\alpha ^t {| {f({s,z(s),z^\Delta  (s),Hz(s)} )- g({s,z(s),z^\Delta  (s),Hz(s)} )} |} \Delta s  \\
&\quad + | {f({t,x(t ),x^\Delta  (t),Hx(t)} )- f({t,z(t),z^\Delta  (t),Hz(t)} )} | \\
&\quad + | {f({t,z(t),z^\Delta  (t),Hz(t)} )- g({t,z(t),z^\Delta  (t),Hz(t)} )} | \\
& \le \delta  + \int_\alpha ^t {\gamma
\Big[{u(s) + q(s)\int_\alpha ^\beta  {r(\tau )u(\tau )} \Delta \tau }
\Big]} \Delta s \\
&\quad + \int_\alpha ^t { \epsilon \Delta s}  + \gamma [{u(t) + g(t)\int_\alpha ^\beta  {r(\tau )u(\tau )} \Delta \tau } ] +  \epsilon \\
&= w(t) + \int_\alpha ^t {\gamma [{u(s) + q(s)\int_\alpha ^\beta  {r(\tau )u(\tau )} \Delta \tau } ]} \Delta s \\
&\quad  + \gamma \Big[{u(t) + q(t)\int_\alpha ^\beta  {r(\tau )u(\tau
)} \Delta \tau } \Big]. %\label{e4.19}
\end{align*}
 Then we obtain
\begin{align*}
u(t)
& \le \frac{{w(t)}}{{1 - \gamma }}
+ \frac{1}{{1 - \gamma }}\int_\alpha ^t {\gamma \Big[{u(s) + q(s)
\int_\alpha ^\beta  {r(\tau )u(\tau )} \Delta \tau }\Big]} \Delta s \\
& +\frac{\gamma }{{1 - \gamma }}q(t)\int_\alpha ^\beta  {r(\tau
)u(\tau )} \Delta \tau . %\label{e4.20}
\end{align*}
Now an application of Theorem 3.1  yields
\eqref{e4.18}.
\end{proof}


\begin{remark} \label{rmk2} \rm
 The result given in theorem 4.2 relates the
solutions of \eqref{e1.1} and \eqref{e4.10} in the sense that if
$f$ is close to $g$ and $x_0$ is close to $z_0$, then the
solutions of \eqref{e1.1} and \eqref{e3.10} are also close to
each other.
\end{remark}

\section{Continuous dependence of Solutions}

 In this section we study continuous dependence of solutions of
 \eqref{e1.1} and its variants.
The following theorem deals with the continuous dependence
of solution of \eqref{e1.1} on given initial values.

\begin{theorem} \label{thm5.1}
Suppose that  $f,h$ in \eqref{e1.1} satisfy
 \eqref{e4.11}, \eqref{e4.12}. Let $x_i(t)$, $(i=1,2)$ be
respectively solutions of equation
\begin{equation}
x^\Delta  (t) = f({t,x(t),x^\Delta  (t),Hx(t)} ), \label{e5.1}
\end{equation}
with the given initial conditions
\begin{equation}
x_i (\alpha  )= c_i, \label{e5.2}
\end{equation}
on $I_\mathbb{T}$, where $f,H$ are as in \eqref{e1.1} and $c_i$
are given constants. Let $\lambda$ and $L_2(t)$ be as in
\eqref{e4.4} and \eqref{e4.6} and
\begin{equation}
Q_1  = \frac{1}{{1 - \lambda }}\int_\alpha ^\beta  {r(\xi )} A_1
(\xi )\Delta \xi , \label{e5.3}
\end{equation}
where $A_1(t)$ is defined by the right hand side of \eqref{e4.17}
by replacing $w(t)$ with the expression $|c_1-c_2|$. Then
\begin{equation}
| {x_1 (t) - x_2 (t)} | + | {x_1^\Delta  (t) -
x_2^\Delta  (t)} | \le A_1 (t )+ Q_1 L_2 (t ), \label{e5.4}
\end{equation}
for $t \in I_\mathbb{T}$.
\end{theorem}

\begin{proof}
Let $v(t )= | {x_1 (t) - x_2 (t)} | + |
{x_1^\Delta  (t) - x_2^\Delta  (t)} |$, $t \in
I_\mathbb{T}$.  From the hypotheses, we have
\begin{align*}
v(t)
&\le | {c_1  - c_2 } | + \int_\alpha ^t {| {f({s,x_1 (s),
 x_1^\Delta  (s),Hx_1 (s)} )} }
 -  {f({s,y_2 (s),y_2^\Delta  (s),Hy_2 (s)} )} |\Delta s \\
&\quad + | {f({t,x_1 (t),x_1^\Delta  (t),Hx_1 (t)} )
 - f({t,x_2 (t),x_2^\Delta  (t),Hx_2 (t)} )} | \\
&\le | {c_1  - c_2 } | + \int_\alpha ^t {\gamma
\Big[{v(s )+ \int_\alpha ^\beta  {q(s)r(\tau )v(\tau )\Delta \tau } }
\Big]} \Delta s \\
&\quad + \gamma \Big[{v(t )+ \int_\alpha ^\beta {q(t )r(\tau )v(\tau
)\Delta \tau } } \Big]. %\label{e5.5}
\end{align*}
 Then
\begin{align*}
v(t )
&\le \frac{{| {c_1  - c_2 } |}}{{1 - \gamma }}
+ \frac{1}{{1 - \gamma }}\int_\alpha ^t {\gamma
\Big[{v(s) + q(s)\int_\alpha ^\beta  {r(\tau )v(\tau )\Delta \tau } }
\Big]} \Delta s \\
&\quad + \frac{\gamma }{{1 - \gamma }}q(t)\int_\alpha ^\beta {r(\tau
)v(\tau  )} \Delta \tau. %\label{e5.6}
\end{align*}
Now applying theorem 3.1 gives \eqref{e5.4}, which
shows the dependency of solution of \eqref{e5.1} on given initial
values.
\end{proof}

\begin{remark} \label{rmk3}\rm
 If we put $c_1=c_2=0$, then we have $A_1(t)=0$,
$Q_1=0$ and the uniqueness of solutions of equation \eqref{e5.1}
follows.
\end{remark}

Now we consider the  integrodifferential equations on time scales
\begin{gather}
z^\Delta  (t) = f({t,z(t),z^\Delta  (t ),Hz(t),\mu } ),z(\alpha )
= z_0 , \label{e5.7} \\
z^\Delta  (t) = f({t,z(t),z^\Delta  (t ),Hz(t),\mu _0 } ),z(\alpha
) = z_0 , \label{e5.8}
\end{gather}
for $t \in I_\mathbb{T}$, where H is given as in \eqref{e1.2},
$f \in C_{rd} ({I_\mathbb{T}  \times \mathbb{R}^n  \times
\mathbb{R}^n \times \mathbb{R}^n  \times \mathbb{R}, \mathbb{R}^n
} )$ and $\mu,\mu _0$ are parameters.

Our next theorem deals with the dependency of solutions of
\eqref{e5.7} and \eqref{e5.8} on parameters.

\begin{theorem}
Suppose that the functions $h$ and $f$ in \eqref{e5.7} and
\eqref{e5.8} satisfy respectively the conditions \eqref{e4.12} and
\begin{gather}
| {f(t,u,v,w,\mu ) - f(t,\overline u ,\overline v ,
\overline w ,\mu )} |
\le \gamma [{| {u - \overline u } | + | {v - \overline v } |
+ | {w - \overline w } |} ],
\label{e5.9} \\
| {f(t,u,v,w,\mu ) - f(t,u,v,w,\mu _0 )} |
\le m(t)| {\mu  - \mu _0 } |,
\label{e5.10}
\end{gather}
where $0 \le \gamma  \le 1$ is a constant and
$m \in C_{rd}({I_\mathbb{T} ,\mathbb{R}_ +  } )$. Let
\begin{equation}
\overline m (t) = m(t) + \int_\alpha ^\beta  {m(s)\Delta s,}
\label{e5.11}
\end{equation}
$\lambda,  L_2(t)$ be as in \eqref{e4.4}, \eqref{e4.6} and
\begin{equation}
Q_2  = \frac{1}{{1 - \lambda }}\int_\alpha ^\beta  {r(\xi )} A_2
(\xi  )\Delta \xi , \label{e5.12}
\end{equation}
where $A_2(t)$ is defined by the right hand side of \eqref{e4.17}
by replacing $w(t)$ with the expression
$| {\mu  - \mu _0 } |\overline m (t)$.

Let $z_1(t)$ and $z_2(t)$ be respectively, the solutions of
\eqref{e5.7} and \eqref{e5.8} on $I_\mathbb{T}$. Then
\begin{equation}
| {z_1 (t )- z_2 (t)} | + | {z_1^\Delta (t) -
z_2^\Delta  (t)} | \le A_2 (t) + Q_2 L_2 (t), \label{e5.13}
\end{equation}
for $t \in I_\mathbb{T}$.
\end{theorem}

\begin{proof}
Let $z(t) = | {z_1 (t) - z_2 (t)} | + | {z_1^\Delta
(t) - z_2^\Delta  (t)} |$, $t \in I_\mathbb{T}$.
Using  that $z_1 (t)$ and $z_2 (t)$ are respectively, the solutions
of \eqref{e5.7} and \eqref{e5.8} and hypotheses, we have
\begin{align*}
z(t)
&\le \int_\alpha ^t {| {f(s,z_1 (s),z_1^\Delta  (s),Hz_1 (s),\mu ) - f(s,z_2 (s),z_2^\Delta  (s),Hz_2 (s),\mu )} |} \Delta s \\
&\quad + \int_\alpha ^t {| {f(s,z_2 (s),z_2^\Delta  (s),Hz_2 (s),\mu ) - f(s,z_2 (s),z_2^\Delta  (s),Hz_2 (s),\mu _0 )} |}\Delta s   \\
&\quad + | {f(t,z_1 (t),z_1^\Delta  (t),Hz_1 (t),\mu ) - f(t,z_2 (t),z_2^\Delta  (t),Hz_2 (t),\mu )} | \\
&\quad + | {f(t,z_2 (t),z_2^\Delta  (t),Hz_2 (t),\mu ) - f(t,z_2 (t),z_2^\Delta  (t),Hz_2 (t),\mu _0 )} | \\
&\le \int_\alpha ^t {\gamma [{z(s) + \int_\alpha ^\beta  {q(s)r(\tau )z(\tau )\Delta \tau } } ]\Delta s + \int_\alpha ^t {m(s)| {\mu  - \mu _0 } |} } \Delta s \\
&\quad + \gamma [{z(t) + \int_\alpha ^\beta  {q(t)r(\tau )z(\tau )\Delta \tau } } ] + m(t)| {\mu  - \mu _0 } | \\
&= | {\mu  - \mu _0 } |\overline m (t) + \int_\alpha ^t {\gamma [{z(s) + q(s)\int_\alpha ^\beta  {r(\tau )z(\tau )\Delta \tau } } ]\Delta s} \\
&\quad + \gamma [{z(t) + q(t)\int_\alpha ^\beta  {r(\tau )z(\tau
)\Delta \tau } } ]. %\label{e5.14}
\end{align*}
 Then we have
\begin{align*}
z(t)
&\le \frac{{| {\mu  - \mu _0 } |\overline m (t)}}{{1 - \gamma }}
+ \frac{1}{{1 - \gamma }}\int_\alpha ^t {\gamma \Big[{z(s )
+ q(s )\int_\alpha ^\beta  {r(\tau )z(\tau )\Delta \tau } } \Big]}
 \Delta s \\
&\quad + \frac{\gamma }{{1 - \gamma }}q(t)\int_\alpha ^\beta  {r(\tau
)z(\tau )\Delta \tau .} %\label{e5.14}
\end{align*}
Now applying Theorem 3.1  yields \eqref{e5.13},
which shows the dependency of solutions of \eqref{e5.7} and
\eqref{e5.8} on parameters.
\end{proof}

\subsection*{Application}

It is often difficult to obtain explicitly the solutions to the
equations of the form \eqref{e1.1} and thus need a new insight
for handling the qualitative properties of its solutions. The
method of integral inequalities with explicit estimates provides a
powerful analytic tool in the study of various dynamic equations.
It enable us to obtain valuable information about solutions
without the need to know in advance the solution explicitly. To
illustrate this fact and the main ideas, we consider the following
special version of equation \eqref{e1.1}.
\begin{equation}
x^\Delta  (t )= F({t,x(t),x^\Delta  (t )} ),\quad
x(\alpha ) = x_0 , \label{e6.1}
\end{equation}
for $t \in I_\mathbb{T}$, where $F:I_\mathbb{T}
\times \mathbb{R}^n  \times \mathbb{R}^n  \to \mathbb{R}^n$ is
rd-continuous function and $x(t)$ is unknown function.

Let $y \in C_{rd}(I_\mathbb{T},\mathbb{R}^n)$ be a function
such that $y^\Delta(t) $ exists for $t \in I_\mathbb{T}$ and
satisfies the inequality
\begin{equation}
| {y^\Delta  (t )- F({t,y(t),y^\Delta  (t )} )} | \le
\epsilon , \label{e6.2}
\end{equation}
for a given $\epsilon > 0$, where it is supposed that the
initial condition
$y(\alpha)=x_0$ is fulfilled. Then we call $y(t)$ an
$\epsilon$-approximate solution with respect to problem
\eqref{e6.1}.

The relation between an $\epsilon$-approximate solution of
\eqref{e6.1} and a solution of \eqref{e6.1} is shown in the
following example.

\subsection*{Example}
 Suppose that the function $F$ in \eqref{e6.1}
satisfies the condition
\begin{equation}
| {F({t,u,v} )- F(t,\overline u ,\overline v )} | \le
\gamma [{| {u - \overline u } | + | {v -
\overline v } |} ], \label{e6.3}
\end{equation}
where $0 \leq \gamma <1$ is a constant. Let $x(t),y(t) \in
C_{rd}(I_\mathbb{T},\mathbb{R}^n)$ are respectively a solution of
\eqref{e6.1} and an $\epsilon$-approximate solution of
\eqref{e6.1}. Then from \eqref{e6.1} and \eqref{e6.2}, we have
\begin{equation}
x(t) = x_0  + \int_\alpha ^t {F({s,x(s),x^\Delta  (s)} )\Delta s,}
\label{e6.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
\epsilon ({t - \alpha } )
& \ge \int_\alpha ^t {| {y^\Delta  (s )- F({s,y(s),y^\Delta  (s )} )} |} \Delta s \\
& \ge \Big| {\int_\alpha ^t {\{ {y^\Delta  (s )- F({s,y(s),
y^\Delta  (s )} )} \}\Delta s} } \Big| \\
& = \Big| {y(t )- x_0  - \int_\alpha ^t {F( {s,y(s),y^\Delta  (s
)} )} } \Big|.
\end{aligned} \label{e6.5}
\end{equation}
 From \eqref{e6.3}--\eqref{e6.5}, we observe that
\begin{equation}
\begin{aligned}
| {y(t) - x(t)} |
&= \Big| {y(t) - x_0  - \int_\alpha ^t {F({s,y(s),y^\Delta  (s)} )} }
  \Delta s \\
&\quad  + \int_\alpha ^t {\{ {F({s,y(s),y^\Delta  (s)} )
 - F({s,x(s),x^\Delta  (s)} )} \}\Delta s}  \Big| \\
& \le \Big| {y(t) - x_0  - \int_\alpha ^t {F({s,y(s),y^\Delta  (s)} )}
 \Delta s} \Big| \\
&\quad + \int_\alpha ^t {| {F({s,y(s ),y^\Delta  (s)} )- F({s,x(s ),x^\Delta  (s)} )} |} \\
& \le \epsilon ({t - \alpha } )+ \int_\alpha ^t {\gamma [
{| {y(s) - x(s)} | + | {y^\Delta (s) - x^\Delta
(s)} |} ]} \Delta s.
\end{aligned} \label{e6.6}
\end{equation}
Also, from \eqref{e6.1}--\eqref{e6.3}, we observe
that
\begin{equation}
\begin{aligned}
&| {y^\Delta  (t) - x^\Delta  (t)} |\\
& = \big| {y^\Delta  (t) - F({t,y(t),y^\Delta  (t)} )}
{ + F({t,y(t),y^\Delta  (t)} )- F({t,x(t),x^\Delta  (t)} )} \big|  \\
&\le | {y^\Delta  (t) - F({t,y(t),y^\Delta  (t)} )} |
  + | {F({t,y(t),y^\Delta  (t)} )- F({t,x(t),x^\Delta  (t)} )} | \\
& \le \epsilon  + \gamma [{| {y(t) - x(t)} |
  + | {y^\Delta  (t) - x^\Delta  (t)} |} ].
\end{aligned} \label{e6.7}
\end{equation}
Let $z(t) = | {y(t) - x(t)} | + | {y^\Delta  (t) -
x^\Delta  (t)} |$  for $t \in I_\mathbb{T}$.  From
\eqref{e6.6} and \eqref{e6.7}, we have
\begin{equation}
z(t )\le \varepsilon ({t - \alpha } )+ \int_\alpha ^t {\gamma z(s
)\Delta s}  + \epsilon  + \gamma z(t). \label{e6.8}
\end{equation}
 From \eqref{e6.8}, we observe that
\begin{equation}
z(t) \le \frac{{\varepsilon [{1 + ({t - \alpha } )}
]}}{{1 - \gamma }} + \frac{1}{{1 - \gamma }}\int_\alpha ^t
{\gamma z(s)\Delta s.} \label{e6.9}
\end{equation}
Now a suitable application of
Theorem 3.1 (when $a(t) = \frac{{\varepsilon [{1 + ({t -
\alpha } )} ]}}{{1 - \gamma }}$, $b(t) = \frac{1}{{1 - \gamma
}}$, $f(t) = \gamma, g(t)=0$) to \eqref{e6.9} yields
\[
z(t) \le \frac{{\epsilon [{1 + ({t - \alpha } )}
]}}{{1 - \gamma }} + \frac{1}{{1 - \gamma }}\int_\alpha ^t
{\gamma \big\{ {\frac{{\epsilon [{1 + ({\tau  - \alpha } )}
]}}{{1 - \gamma }}} \big\}} e_{\gamma ({\frac{1}{{1 -
\gamma }}} )} ({t,\sigma ( \tau )} )\Delta \tau . %\label{e6.10}
\]
 From where, we obtain
\begin{equation}
\begin{aligned}
&| {y(t) - x(t)} | \\
&\le \frac{\epsilon }{{1 - \gamma
}}\Big[{[{1 + ({t - \alpha } )} ] + \int_\alpha ^t
{\frac{{\gamma [{1 + ({\tau  - \alpha } )} ]}}{{1 -
\gamma }}e_{\gamma ({\frac{1}{{1 - \gamma }}} )} ({t,\sigma (\tau
)} )\Delta \tau } } \Big].
\end{aligned} \label{e6.11}
\end{equation}
Clearly, this estimate  provides the
relationship between an
$\epsilon$-approximate solution of \eqref{e6.1} and a solution of
\eqref{e6.1}, without knowing in advance their explicit solutions.
Moreover, from \eqref{e6.11}, it follows that if $\epsilon=0$,
then the uniqueness of solutions of equation \eqref{e6.1} is
established.

\subsection*{Acknowledgements}

The author is grateful to the anonymous referee whose suggestions
helped to improve this article.
This research  is supported by UGC
(New Delhi, India) project F.NO 37-538/2009.

\begin{thebibliography}{00}

\bibitem{Boh1} M. Bohner  and A. Peterson;
Dynamic equations on time scales,
\newblock \emph{Birkhauser Boston/Berlin},  2001.

\bibitem{Boh2} M. Bohner  and A. Peterson;
\newblock Advances in Dynamic equations on time scales,
\newblock \emph{Birkhauser Boston/Berlin},  2003.

\bibitem{Boh3} E. A. Bohner, M. Bohner and F. Akin;
\newblock Pachpatte Inequalities on time scale,
\newblock \emph{J. Inequal. Pure. Appl. Math.}, 6(1)(2005), Art 6.

\bibitem{Hig} S. Hilger;
 \newblock Analysis on Measure chain-A unified approch to continuous
and discrete calculus,
\newblock \emph{Results Math}, 18:18--56, 1990.

\bibitem{Kul} T. Kulik and C. C. Tisdell;
\newblock Volterra integral equations on time scales:
Basic qualitative and quantitative results with applications to initial
value problems on unbounded domains,
\newblock \emph{Int. J. Difference Equ.}, Vol. 3 No. 1 (2008), pp. 103-133.

\bibitem{Bgp1}B. G. Pachpatte;
\newblock Integral and Finite Difference Inequalities and Appliations,
\newblock \emph{North Holland Mathematics Studies 205, Elsevier Science B.V.}, 2006

\bibitem{Dbp1}D. B. Pachpatte;
\newblock Properties of solutions to nonlinear dynamic integral equations on time scales,
\newblock \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 136, pp. 1--8.

\bibitem{Dbp2} D. B. Pachpatte;
\newblock  On Nonstandard Volterra  type dynamic integral equations on time scales,
\newblock \emph{Electronic Journal of Qualitative theory of differential equations}, 2009,No 72,1-14.

\bibitem{Tis} C. C. Tisdell and A. Zaidi;
\newblock  Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling,
\newblock \emph{Nonlinear Anal.}, 68 (2008), No. 11, 3504--3524.

\bibitem{Tis2} C. C. Tisdell, A. H. Zaidi;
Successive approximations to
solutions of dynamic equations on time scales.
Comm.  \emph{Appl. Nonlinear Anal}. 16 (2009), no. 1, 61--87.

\end{thebibliography}

\end{document}