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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 50, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/50\hfil  Dynamic equations on time scales]
{Integral inequalitys for partial dynamic equations on time scales}

\author[D. B. Pachpatte\hfil EJDE-2012/50\hfilneg]
{Deepak B. Pachpatte}

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

\thanks{Submitted January 16, 2012. Published March 27, 2012.}
\subjclass[2000]{26E70, 34N05}
\keywords{Dynamic equations; time scales; qualitative properties;\hfill\break\indent
inequalities with explicit estimates}

\begin{abstract}
 The aim of the present paper is to study some basic qualitative properties
 of solutions of some partial dynamic equations on time scales.
 A variant of certain fundamental integral inequality with explicit estimates
 is used to establish our results.
\end{abstract}

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


\section{Introduction}
During past few years many authors have established the time scale analogue
of well known dynamic equations used in the development of theory of
differential and integral equation see \cite{Boh3,D1,D2,D3,D4,W2,S1,Z2}.
In \cite{Boh4,Boh5,H1,J1,A1} authors have obtained some results on multiple
integration and partial dynamic equations on time scales. Recently in \cite{O1,A2,A3,W1}
authors have obtained inequalities on two independent variables on time scales.
In the present paper we establish some basic qualitative properties of solutions
of some partial dynamic equation on time scales. We use certain fundamental
integral inequality with explicit estimates to establish our results.
We assume understanding of time scales and its notation. Excellent information
about introduction to time scales can be found in \cite{Boh1,Boh2}.

  In what follows $\mathbb{R}  $ denotes the set of real numbers,
$ \mathbb{Z}$ the set of integers and $ \mathbb{T}$ denotes arbitrary time scales.
Let  $C_{rd}$ be the set of all rd continuous function. We assume $\mathbb{T}_1$
and $\mathbb{T}_2$ be two time scales  and $\Omega=\mathbb{T}_1 \times \mathbb{T}_2$.

 In this article, we consider  partial dynamic equation of the type
\begin{equation}
u^{\Delta{t}}  (t,x) = f({t,x,u(t,x)}) + \int_{s_0}^s {g({t,x,y,u({t,y})})\Delta y
+ h(t,x)} ,\label{e1.1}
\end{equation}
which satisfies the initial condition
\begin{equation}
u(t_0,x)=u_0(x), \label{e1.2}
\end{equation}
for $({t_0 ,x}) \in \Omega $, where $u_0 \in C(I,\mathbb{R})$,
$I=[a,b]$ $(a<b)$, $f \in C_{rd}( \Omega \times \mathbb{R} ,\mathbb{R}^+)$,
$g \in C_{rd}( \Omega \times I \times \mathbb{R} ,\mathbb{R}^+)$ and
 $u$ is unknown function to be found.

\section{Basic Inequality}

We will use the following integral inequality.

\begin{lemma} \label{lem2.1}
Let $w,p \in C_{rd}( \Omega,\mathbb{R}_+)$ and let $c \ge  0$ be a constant
\begin{equation}
w(t,x) \le c + \int_{t_0 }^t {[ {p(s,x)w(s,x) + \int_{a}^b {g({s,x,y})w(s,y)\Delta y} } ]} \Delta  s,
\label{e2.1}
\end{equation}
for $(t,x) \in \Omega$, then
\begin{equation}
w(t,x) \le cP(t,x)e_{\int_{a}^b {q({s,x,y})p(s,y)\Delta y } } ({t ,t_0 }),
\label{e2.2}
\end{equation}
for $(t,x) \in \Omega$, where
\begin{equation}
P(t,x) = e_{p(s,x)} ({t_0 ,t}).
\label{e2.3}
\end{equation}
\end{lemma}

\begin{proof}
 Define a function
\begin{equation}
m(t,x) = c + \int_{t_0 }^t {\int_{a}^b {q(s,x,y)w(s,y)\Delta  } } y\Delta  s.
\label{e2.4}
\end{equation}
Then \eqref{e2.1} can be restated as
\begin{equation}
w(t,x) \le m(t,x) + \int_{t_0 }^t {p(s,x)} w(s,x)\Delta  s,
\label{e2.5}
\end{equation}
$m(t,x)$ is non negative for $(t,x) \in \Omega$ and nondecreasing for $t$.
Now considering \eqref{e2.5} as a one dimensional integral inequalities in
$t \in \mathbb{T}$ for every $x \in \mathbb{T}$ and a suitable application of
inequality given in  \cite[Theorem 3.5]{D1}, yields
\begin{equation}
m(t,x) \le c + \int_{t_0 }^t {\int_{a}^b {q(s,x,y)p(s,y)m(s,y)\Delta  } } y\Delta  s.
\label{e2.6}
\end{equation}
Let
\begin{equation}
k(s) = \int_{a}^b {q(s,x,y)p(s,y)m(s,y)\Delta  } y,
 \label{e2.7}
\end{equation}
for every $x \in {\mathbb{T}}$, the inequality \eqref{e2.6} becomes
\begin{equation}
m(t,x) \le c + \int_{a}^b {k(s)\Delta s}.\label{e2.8}
\end{equation}
Let
\begin{equation}
z(t) = c + \int_{a}^b {k(s)\Delta s},\label{e2.9}
\end{equation}
then $z(t_0)=c$ and
\begin{equation}
      m(t,x)\leq z(t),  \label{e2.10}
\end{equation}
for $(t,s) \in \Omega$. From \eqref{e2.9}, \eqref{e2.7} and \eqref{e2.10}, we have
\begin{equation}
\begin{aligned}
z^\Delta  (t) &= k(t)
 = \int_{a}^b {q(t,x,y)p(t,y)m(t,y)\Delta  } y \\
& \le z(t)\int_{a}^b q (t,x,y)p(t,y)\Delta  y.
\end{aligned} \label{e2.11}
\end{equation}
This inequality  implies
\begin{equation}
z(t) \le ce_{\int_{a}^b {q({s,x,y})p(s,y)\Delta  s} } ({t ,t_0}).
\label{e2.12}
\end{equation}
The required inequality \eqref{e2.2} follows from \eqref{e2.12}, \eqref{e2.9}
 and \eqref{e2.6}.
\end{proof}

\section{Main Results}

The following theorem provides some estimates on the solution.

\begin{theorem} \label{thm3.1}
Suppose that the functions $f,g,h,u_0$ in \eqref{e1.1} and \eqref{e1.2}
satisfy the conditions
\begin{gather}
| {f(t,x,u) - f(t,x,\overline u )} | \le c(t,x)| {u - \overline u } |,
\label{e3.1}\\
| {g(t,x,y,u) - g(t,x,y,\overline u )} | \le k(t,x,y)| {u - \overline u } |,
\label{e3.2}\\
d = \sup \big| {\phi (t,x) + \int_{t_0 }^t {\Big[ {f({s,x,t_0 })
+ \int_{a}^b {g(s,x,y,t_0 } )\Delta y} \Big]} \Delta s} \big| < \infty,
\label{e3.3}
\end{gather}
where $c \in \Omega$, $k \in (\Omega \times {\mathbb{R}^n},\mathbb{R}_+)$ and
\begin{equation}
\phi (t,x) = u_0 (x) + \int_{t_0 }^t {h(s,x)\Delta s}. \label{e3.4}
\end{equation}
If $u(t,x)$ is any solution of \eqref{e1.1}-\eqref{e1.2}  then
\begin{equation}
| {u(t,x)} | \le dC(t,x)e_{\int_{a}^b {k(s,x,y)C(s,y)\Delta y} } ({t, t_0}),
\label{e3.5}
\end{equation}
where
\begin{equation}
C(t,x) = e_{c(s,x)} (t, t_0). \label{e3.6}
\end{equation}
\end{theorem}

\begin{proof}
 Since $u(t,x)$ is a solution of \eqref{e1.1}-\eqref{e1.2}
 and hypotheses, we observe that
\begin{equation}
\begin{aligned}
| {u(t,x)} |
& = \Big| \Big\{\phi (t,x) + \int_{t_0 }^t \Big[ \{ {f({s,x,u(s,x)}) - f(s,x,t_0 )
 + f(s,x,t_0 )} \}    \\
&\quad    + \int_{a}^b {\{ {g({s,x,y,u(s,y)}) - g({s,x,y,t_0 }) + g({s,x,y,t_0 })} \}
 \Delta  y}  \Big]\Delta  s\Big\} \Big|
\\
& \le \Big| \phi (t,x)\int_{t_0 }^t \Big[ f(s,x,t_0 )
+ \int_{a}^b g(s,x,y,t_0) \Delta  y \Big] \Delta  s \Big| \\
&\quad + \int_{t_0 }^t \Big[ | {f({s,x,u(s,x)}) - f(s,x,t_0 )} |  \\
&\quad  + \int_{a}^b {| {g({s,x,y,u(s,y)}) - g({s,x,y,t_0 })} |} \Delta  y
 \Big]\Delta  s \\
&\le d + \int_{t_0 }^t {\Big[ {c(s,x)| {u(s,x)} | + \int_{a}^b {k({s,x,y})| {u(s,y)}
|\Delta  y} }\Big]} \Delta  s.
\end{aligned} \label{e3.7}
\end{equation}
Now an application of Lemma \ref{lem2.1} to \eqref{e3.7} yields \eqref{e3.5}.
\end{proof}

Now we give approximation of solutions  to \eqref{e1.1}-\eqref{e1.2}.
We obtain conditions under which we estimate errors between true solution and
approximate solutions.

Let $u(t,x) \in \Omega $, $u^{\Delta{t}}  (t,x)$ exist on $\mathbb{T}$ and satisfy
the inequality
\begin{equation}
\Big| {u^{\Delta{t}}  (t,x) - f({t,x,u(t,x)}) - \int_{a}^b {g({t,x,y,u({t,y})})\Delta  y
- h(t,x)} } \Big| \le  \epsilon
\label{e3.8}
\end{equation}
for a given constant $\epsilon \geq 0$ where we suppose that \eqref{e1.2} holds.
Then we say that $u(t,x)$ has $\epsilon$-approximate solutions with respect
 to \eqref{e1.1}.

\begin{theorem} \label{thm3.2}
Suppose the functions $f,g$ in \eqref{e1.1} satisfy the conditions
\begin{gather}
| {f(t,x,y) - f(t,x,\overline u )} | \le C(t,x)| {u - \overline u } |,
\label{e3.9}\\
| {g(t,x,y,u) - g(t,x,y,\overline u )} | \le K({t,x,y})| {u - \overline u } |,
\label{e3.10}
\end{gather}
Let $u_i(t,x) (i=1,2)$, $(t,x) \in \Omega$ be respectively $\epsilon_i$ approximate
solution of \eqref{e1.1} with
\begin{equation}
u_{i}(t_o,x)=\overline{u}_i(x), \label{e3.11}
\end{equation}
and let
\begin{equation}
\phi _i (t,x) = \overline u _i (x) + \int_{t_0 }^t {h(s,x)\Delta s}. \label{e3.12}
\end{equation}
Suppose that
\begin{equation}
| {\phi _1 (t,x) - \phi _2 (t,x)} | \le \delta , \label{e3.13}
\end{equation}
where $\delta \ge 0$ is a constant and
\begin{equation}
M = \sup_ {t \in T}
[ {({\varepsilon _1  + \varepsilon _2 })t + \delta } ] < \infty, \label{e3.14}
\end{equation}
then
\begin{equation}
| {u_1 (t,x) - u_2 (t,x)} | \le MC(t,x)e_{\int_a^b {k({s,x,y})C(s,y)\Delta s} }
({t,t_0}),
\label{e3.15}
\end{equation}
where
\begin{equation}
C(t,x) = e_{c(s,x)} ({t,t_0}). \label{e3.16}
\end{equation}
\end{theorem}

\begin{proof}
Since $u_i(t,x)$  $(i=1,2) $, $(t,x) \in \Omega$ are respectively $\epsilon_i$-approximate
solutions of \eqref{e1.1} with \eqref{e3.8}, we have
\begin{equation}
\big| {u_i^{\Delta  t} (t,x) - f({t,x,u_i (t,x)})
- \int_a^b {g({t,x,y,u_i ({t,y})})\Delta y - } h(t,x)} \big| \le \varepsilon _i.
\label{e3.17}
\end{equation}
By taking $t=s$ in the above inequality and integrating both sides with respect to
$s$ from $t_0$ to $t$ for $t \in \mathbb{T}$, we obtain
\begin{equation}
\begin{aligned}
\varepsilon _i ({t - t_0})
&\ge \int_{t_0}^t \Big| u_i^{\Delta s} (s,x) - f(s,x,u_i (s,x))
- \int_a^b g(s,x,y,u_i (s,y))\Delta y -  h(s,x) \Big| \Delta s
\\
&\ge \Big| \int_{t_0 }^t \{ u_i^{\Delta s} (s,x) - f(s,x,u_i (s,x))    \\
&   - \int_a^b g(s,x,y,u_i (s,y))\Delta y -  h(s,x) \}\Delta s \Big| \\
&= | u_i (t,x) - \phi _i (t,x)  \\
& \quad  - \int_{t_0 }^t \Big[ f(s,x,u_i (s,x))
+ \int_a^b g(s,x,y,u_i (s,y))\Delta y  \Big]  |\Delta s.
\end{aligned}\label{e3.18}
\end{equation}
From \eqref{e3.18} and using elementary inequalities
\begin{equation}
| {v - z} | \le | v | + | z |,\quad | {v - z} | \le | {v - z} |,
\label{e3.19}
\end{equation}
for $v,z \in \mathbb{R}_+$, we have
\begin{align*}
&(\varepsilon _1  + \varepsilon _2 )(t - t_0 )\\
&\ge \Big| u_1 (t,x) - \phi _1 (t,x)
 - \int_{t_0 }^t {\Big[ {f({s,x,u_1 (s,x)})
 + \int_a^b {g({s,x,y,u_1 (s,y)})\Delta y} }\Big]}  \Big|\Delta s
\\
&\quad+\Big| {u_2 (t,x) - \phi _2 (t,x)}
 - \int_{t_0 }^t {\Big[ {f({s,x,u_2 (s,x)})
 + \int_a^b {g({s,x,y,u_2 (s,y)})\Delta y} } \Big]}  \Big|\Delta s
\\
&\ge \Big| {\Big\{ {\Big| {u_1 (t,x) - \phi _1 (t,x)} } }
 - \int_{t_0 }^t \Big[ {f({s,x,u_1 (s,x)})
  + \int_a^b {g({s,x,y,u_1 (s,y)})\Delta y} } \Big]  \Big|\Delta s \Big\} \\
&\quad - \Big\{ \Big|u_2 (t,x) - \phi _2 (t,x)
  - \int_{t_0 }^t \Big[ f({s,x,u_2 (s,x)})
 + \int_a^b g({s,x,y,u_2 (s,y)})\Delta y  \Big]  \Big|\Delta s \}\Big\}\Big|
 \\
& \ge | {u_1 (t,x) - u_2 (t,x)} | - | {\phi _1 (t,x) - \phi _2 (t,x)} | \\
&\quad - \Big| { {\int_{t_0 }^t {\Big[ {f({s,x,u_1 (s,x)})
 + \int_a^b {g({s,x,y,u_1 (s,y)})\Delta y} } \Big]} } \Big|\Delta s}  \\
& \quad   - \int_{t_0 }^t \Big[ f({s,x,u_2 (s,x)})
  + \int_a^b g({s,x,y,u_2 (s,y)})\Delta y  \Big] \Delta s \Big|.
\end{align*} %\label{e3.20}
Let $u(t,x)=|u_1(t,x)-u_2(t,x)|$, $(t,x) \in \Omega$ from the above inequality
 and using the hypothesis we obtain
\begin{align*}
u(t,x)
&\le ({\varepsilon _1  + \varepsilon _2 })(t - t_0 ) + \delta 
 + \int_{t_0 }^t {\Big[ {c(s,x)u(s,x)
 + \int_a^b {k({s,x,y})u(s,y)\Delta y} } \Big]} \Delta s \\
& \le M   + \int_{t_0 }^t {\Big[ {c(s,x)u(s,x)
 + \int_a^b {k({s,x,y})u(s,y)\Delta y} } \Big]} \Delta s.
\end{align*} %\label{e3.21}
Now an application of Lemma \ref{lem2.1} to the above inequality yields \eqref{e3.15}.
\end{proof}

\begin{remark} \label{rmk3.3}
When $u_1(t,x)$ is a solution of \eqref{e1.1} with $u_1(0,x)=\overline{u}_1(x)$
we obtain $\epsilon_1=0$ and from \eqref{e3.15}, we see that
 $u_2(t,x) \to u_1(t,x)$ as $\epsilon_2 \to \epsilon_1$ and $\delta \to 0$.
Furthermore, if we put $\epsilon_1=\epsilon_2 =0$,
$\overline{u}_1(x)=\overline{u}_2(x)$ in \eqref{e3.15}, then we get the bound which
shows the dependency of solutions of \eqref{e1.1} on given initial values.
\end{remark}

Consider \eqref{e1.1}-\eqref{e1.2} together with following partial dynamic equation
 on time scales
\begin{equation}
v^{\Delta t} (t,x) = \overline{f}({t,x,v(t,x)}) + \int_a^b {\overline{g}({t,x,y,v(t,y)})} \Delta y + h(t,x)
\label{e3.22}
\end{equation}
with given initial condition
\begin{equation}
v(t_0,x) = v_0 (x),
\label{e3.23}
\end{equation}
for $(t,x) \in \Omega$ where $\overline{f} \in C_{rd}( \Omega,\mathbb{R}_+)$,
$\overline{g} \in C_{rd}( \Omega \times \mathbb{R}^n ,\mathbb{R}_+)$,
$h \in C_{rd}( \Omega,\mathbb{R}_+)$.

The following theorem is concerned with the closeness of solutions
 of \eqref{e1.1}-\eqref{e1.2} and \eqref{e3.22}-\eqref{e3.23}.

\begin{theorem} \label{thm 3.4}
Suppose that the functions $f, g$ in \eqref{e1.1}-\eqref{e1.2}
 satisfy the conditions \eqref{e3.9}-\eqref{e3.10} and that there exists constants
$\overline \varepsilon  _i  \ge 0$, $\overline \delta  _i  \ge 0$ $(i = 1,2)$ such that
\begin{gather}
| {f({t,x,u}) - \overline f ({t,x,u})} | \le \overline \varepsilon  _1 , \label{e3.24}\\
| {g({t,x,y,u}) - \overline g ({t,x,y,u})} | \le \overline \varepsilon  _2 ,
\label{e3.25}\\
| {h(t,x) - \overline h (t,x)} | \le \overline \delta  _1 , \label{e3.26}\\
| {u_0 (x) - v_0 (x)} | \le \overline \delta  _2 , \label{e3.27}
\end{gather}
where $f,g,h,u_0$ and $\overline f, \overline g, \overline h,v_0$ are the functions
in \eqref{e1.1}-\eqref{e1.2} and \eqref{e3.22}-\eqref{e3.23} and
\begin{equation}
\overline M  = \sup_ {t \in T}
\big[ {\overline \delta  _2  + [ {\overline \delta  _1  + \overline \varepsilon  _1
 + \overline \varepsilon  _2 ({b - a})} ]t} \big] < \infty.
\label{e3.28}
\end{equation}
Let $u(t,x)$ and $v(t,x)$ be respectively the solutions of \eqref{e1.1}-\eqref{e1.2}
 and \eqref{e3.22}-\eqref{e3.23} for $(t,x) \in \Omega$. Then
\begin{equation}
| {u(t,x) - v(t,x)} | \le \overline M C(t,x)e_{\int_a^b
{k({s,x,y})C(s,y)\Delta y} } ({t,t_0}),
\label{e3.29}
\end{equation}
for $(t,x) \in \Omega$ where $C(t,x)$ is given by \eqref{e3.6}.
\end{theorem}

\begin{proof}
Let $z(t,x)=|u(t,x)-v(t,x)|$, $(t,x) \in \Omega$. Since $u(t,x), v(t,x)$ are
respectively the solutions of  \eqref{e1.1}-\eqref{e1.2} and \eqref{e3.22}-\eqref{e3.23}.
 We have
\begin{align*}
z(t,x)
& \le \Big| {u_0 (x) + \int_{t_0 }^t {h(s,x)\Delta s}  - v_0 (x)
- \int_{t_0 }^t {\overline{h}(s,x)\Delta s} } \Big| \\
&\quad +\int_{t_0 }^t {\Big[ {| {f({s,x,u(s,x)}) - f({s,x,v(s,x)})} |} }
 + | {f({s,x,v(s,x)}) - \overline f ({s,x,v(s,x)})} \Big| \\
&\quad + \int_a^b {\{ {| {g({s,x,y,u(s,y)}) - g({s,x,y,v(s,y)})} |} } \\
&\quad +  { {| {g({s,x,y,v(s,y)}) - \overline g ({s,x,y,v(s,y)})} |} \}\Delta y}\Big]\Delta s
\\
&\le | {u_0 (x) - v_0 (x)} | + \int_{t_0 }^t {| {h(s,x) - \overline h (s,x)} |}
\Delta s \\
&\quad + \int_{t_0 }^t {\Big[ {c(s,x)z(s,x) + \overline \varepsilon  _1
+ \int_a^b {\{ {k({s,x,y})z(s,y) + \overline \varepsilon  _2 } \}\Delta y} } \Big]}
 \Delta s \\
&\le [ {\overline \delta  _2  + \overline \delta  _1 t + \overline \varepsilon  _1 t + \overline \varepsilon  _2 ({b - a})t} ] \\
&\quad + \int_{t_0 }^t {\Big[{c(s,x)z(s,x)
 + \int_a^b {k({s,x,y})z(s,y)\Delta y} } \Big]} \Delta s \\
& \le M + \int_{t_0 }^t {\Big[ {c(s,x)z(s,x)
 + \int_a^b {k({s,x,y})z(s,y)\Delta y} }\Big ]} \Delta s.
\end{align*} %\label{e3.30}
Now an application of Lemma \ref{lem2.1}  to the above inequality yields \eqref{e3.29}.
\end{proof}

\begin{remark} \label{rmk3.5}\rm
 We note that the result given in Theorem \ref{thm3.2} relates the solutions of
\eqref{e1.1}-\eqref{e1.2} and \eqref{e3.22}-\eqref{e3.23} in the sense that if
$f,g,h,u_0$ are respectively close to $f,g,h,v_0$ then the solutions
of \eqref{e1.1}-\eqref{e1.2} and \eqref{e3.22}-\eqref{e3.23} are close together.
\end{remark}

\subsection*{Acknowledgments}
The author is grateful to the anonymous referee and to Professor Julio G. Dix
whose suggestions helped to improve this article.
\frenchspacing

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