\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 66, pp. 1--14.\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 2005 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2005/66\hfil Integrodifferential inequalities]
{Cone-valued impulsive differential
and integrodifferential inequalities}
\author[S. O. Ale, B. O. Oyelami,  M. S. Sesay\hfil EJDE-2005/66\hfilneg]
{Sam Olatunji Ale, Benjamin Oyediran Oyelami, Maligie S. Sesay}  


\address{Sam Olatunji Ale \hfill\break
National Mathematical Centre, Abuja, Nigeria}
\email{samalenmc@yahoo.com}

\address{Benjamin Oyediran Oyelami \hfill\break
National Mathematical Centre, Abuja, Nigeria}
\email{boyelami2000@yahoo.com}

\address{Maligie S. Sesay \hfill\break
Mathematical Sciences Programme,
Abubakar Tafawa Balewa University,
Bauchi, Nigeria}

\date{}
\thanks{Submitted February 7, 2005. Published June 23,2005.}
\subjclass[2000]{34A37}
\keywords{Strict set contraction; monotone iterations techniques;\hfill\break\indent
 measure of non-compactness; maximal solutions}

\begin{abstract}
 In this paper, we present impulsive analogues of the Gronwall-Bellman
 inequalities. Conditions for the existence of maximal solutions of 
 some integrodifferential equations are obtained by finding upper bounds
 for these inequalities. Using monotone iterative techniques and a
 fixed point theorem, we obtained a priori estimates for the
 inequalities.
\end{abstract}

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

\section{introduction}
Integral inequalities play crucial roles in the
study of qualitative properties of systems particularly in the process 
of obtaining results involving the existence, uniqueness, boundedness and
stability and comparison equations for the solution of systems of 
differential and integral equations. The most widely encountered inequalities are 
the Gronwall-Bellman and Pachpatte families and their varieties;
such inequalities have found applications in ordinary differential equations
(ODEs) (Akinyele \cite{a1}, Akinyele and Akpan \cite{a2}, Hale 
\cite{h1}).

 In the study of impulsive differential equations inequalities play the 
same crucial role just like the traditional ordinary differential
equations (ODEs) ones. Hence, in the last few years, series of
impulsive analogues of the Gronwall-Bellman inequalities  have
been evolved to study quantitative and qualitative properties of
impulsive differential equations (Oyelami \cite{o3,o4}, Oyelami et
al. \cite{o1,o2},   Bainov and Svetla \cite{b4}, Bainov and Stamova 
\cite{b5})

 In this paper, some new inequalities are proposed with potential
applications in impulsive ordinary differential equations (IODEs),
impulsive control systems (ICS) and impulsive partial differential
equations (IPDEs) which are still in cradle of development. The
existing inequalities in (Bainov and Svetla \cite{b4}; Lakshimikantham
et.al. cite{l1}) are special cases of these inequalities.

Furthermore, by means of monotone iterative technique couple with
a fixed point theorem of Guo and Lakshimikantham \cite{g1}, we obtain
results on existence of the maxima solutions of the impulsive
equation for the solution which gives the upper bound for the
inequality .Some special cases of the inequalities were used in
Ale et al. \cite{a4} to obtain some biological policies on
normal-malignant cancer model using the Gronwall-Bellman's kind of
impulsive inequality.

\section{Preliminaries, Notation and Definitions}

Let $C(\mathbb{R}^{+},\mathbb{R}^{n})$ be the space of continuous
functions in $\mathbb{R}^{+}=[0,+\infty )$ and taking values in
$\mathbb{R}^{n}$.
Let $C_{0}^{1}(\mathbb{R}^{+},\mathbb{R}^{n})$ be the space of
continuously differentiable and bounded functions on
$\mathbb{R}^{+}$ taking values in the $n$-dimensional Euclidean space
$\mathbb{R}^{n}$. Let
\begin{align*}
PC(\mathbb{R}^{+},\mathbb{R}^{n})
&=\big\{y(t):y(t)\in C(\mathbb{R}^{+}\setminus \{t_k\},\mathbb{R}^{n}),
\,k =1,2\dots, \\
&\quad\lim_{t\to t_k+0} y(t) \text{exists and equals } y(t_k) \big\}\,.
\end{align*}


\noindent\textbf{Definition (cones).}
Let  $\mathbb{R}^{n}$ be $n$-dimensional Euclidean space. A
non-empty set $E\subset \mathbb{R}^{n}$ is said to be a cone if
and only if it satisfies the Following conditions:
\begin{enumerate}
\item  If there exist sequences $(x_{n},y_{n})\subset E$,
$n\in N=\{1,2,\dots\}$ such that $x_{n}\to  x$, $y_{n}\to  y$ as
$n\to \infty$. Then $\alpha x+\beta y\in E$,
where $\alpha$ and $\beta$ are constants;

\item If $x\in E$ then $\alpha x\in E$ for all $\alpha\geq 0$;

\item If $x,-x\in E$ then $\{x\}\cap\{-x\}=\{\phi\}$,
where $\phi$ is the zero element of the cone $E$.
\end{enumerate}
Let the specializing ordering on $E$ be
$x\leq_E y$ if $x-y$ is in $E$; which reads
$y$ weakly specializes $x$.
Also let $x \leq_0 y$ if $y-x$ is in $\mathop{\rm int}E=E\setminus 
\{\phi\}$;
which reads $y$ strongly specializes $x$.

Let $M_{n}(E)$ be $n x n$ symmetric matrices define on the cone and let
$M_{n}^*(E)$ denote its transpose.
We introduce the generalized inner product on $E$ as follows:
\smallskip

\noindent\textbf{Definition (inner products on cones).}
For $X(t)\in M_{n}'(PC(R,E))$ and $B(t)\in M_{n}'(E)$, let
\[
\langle B(t),X(t)\rangle_{E}=\int_{t_0}^{t} B^{\ast}(s)X(s)ds\,.
\]
For the impulse set
$Q_k=\{t_k\in \mathbb{R}^{+}:t_{0}<t_k<t,\; t\in \mathbb{R}^{+},\;
k=1,2,3,\dots\}$, the inner product is
\[
\langle B(t),X(t_k)\rangle_{E}=
\sum_{t_0<t_k<t} B^{\ast}(t_k)X(t_k)\,
\]
where $M_{n}(PC(\mathbb{R}^{+},E))$ is the set of $n x n$
symmetric matrices whose elements are in $PC(\mathbb{R}^{+},E)$ and
* denotes the transpose of the matrix.

Clearly, $\langle .,.\rangle_{E}$ satisfies the following properties:
\begin{enumerate}
\item $\langle x,y\rangle =\phi$ for any $x,y$ in $E$.
$\langle x,y\rangle =\phi$ if $x=y$, where
$\phi$ is zero element of the cone $E$.

\item $\langle \lambda x+y,z\rangle _{E}=\lambda^{\ast}\langle x , y
\rangle _{E}+\langle y, z \rangle _{E}$

\item $\langle x,\mu y+z\rangle_{E}=\mu\langle x , y \rangle _{E}
+\langle x, z \rangle _{E}$,
\end{enumerate}
where $\lambda$ and $\mu$ are complex numbers and $\lambda^{\ast}$ is 
the
complex conjugate of $\lambda$.
\smallskip

\noindent\textbf{Remark.} % 1
If $x\in E$ then $\langle x , x \rangle _{E}=|x|_{E}$ defines the
generalized norm on the cone $E$.
Where
\[
|x|_{E}=(|x_{1}|,|x_{2}|,|x_{3}|,\dots ,|x_{n}|), \quad
x=(x_{1},x_{2},x_{3,\dots .,}x_{n}).
\]
 It must be emphasize that the
classical norm is a real number, whereas, the generalized norm is a
vector.
\smallskip

\noindent\textbf{Definition (adjoint cone).}\;
A cone $E^{\ast}=\{y\in \mathbb{R}^{n}:\langle y,x\rangle \geq_{E}
\phi,x\in E\}\subset \mathbb{R}^{n}$ is defined to
be adjoint cone relative to the cone $E$.

 The set $E^{A}=\{y\in E:\langle y,x\rangle =\phi,x\in E\}$ is
an annihilator of $E$.
While $\langle .,.\rangle $  is the generalized
inner product on $E$.
\smallskip

\noindent\textbf{Remark.} %2
 A necessary and sufficient condition for a point to be an
annihilator of $E$ is that  $x\in E$ for some
$y\in \mathop{\rm int} E^{A}$, $\mathop{\rm int}E=E\setminus \{\phi\}$.
\smallskip

\noindent\textbf{Definition (normal cone).} %4
A cone $E$ is normal if there exists a constant $m>0$
such that $|f|\leq_{E} m|g|$ for any $f,g\in E$ with
$0\leq_{E}f\leq_{E}g$. \smallskip

\noindent\textbf{Remark.} % 3
It is not difficult to show that $E$ is normal if and only
if \for all $\delta>0$ such that $|f+g|>\delta$ for $f,g\in E$ with
$|f|=|g|=1$ where $|.|$ is the Euclidean norm inherited by the cone 
$E$.

\smallskip

\noindent\textbf{Examples of Cones.}
The set
\[
R_{+}^{n}=\{u\in \mathbb{R}^{n}:u_{i} \geq 0,\quad
i=1,2,\dots\,n ,u=(u_{1,}u_{2},\dots ,u_{n})\}
\]
is a cone. The set of non-negative functions in $L_{p}(0,1)$ with
$p\geq1$ is a cone.  The set of non-negative definite matrices
$M_{n}(R^{+})$ is a cone. The set of monotone operators on any
arbitrary Banach space is a cone.
 For further exposition on concept of
abstract cones (see Huston and Pym \cite{h2}, Guo and
Lakshimikantham \cite{g1}, Akinyele and Akpan \cite{a2}, Guo and Liu
\cite{g2}, Akpan \cite{a3}). \smallskip

\noindent\textbf{Definition (order intervals).} %5
The order interval in the cone $E$ can be define as
\[
[U_{0},V_{0}]=\{U(t)\in E:U_{0}\leq U(t)\leq V_{0}(t),t\in
\mathbb{R}^{+}\}\,.
\]
For the rest of this paper we use the following notation:
$M_{n}^{+}(E)$ is the set of $n\times n$ matrices defined
on the cone $E$. \\
$PC(\mathbb{R}^{+},E)$ is the subclass of $PC(\mathbb{R}^{+},\mathbb{R}^{n})$ 
where
the values of $PC(\mathbb{R}^{+},\mathbb{R}^{n})$ is in the cone
$E\subset \mathbb{R}^{n}$
\\
$M(.)$ Denotes the measure of non-compactness of $(.)$
\\
$L^{1}(\mathbb{R}^{+}X\mathbb{R}^{+}XE,E)$ is the space of
absolutely integrable functions on
$\mathbb{R}^{+}X\mathbb{R}^{+}XE$ and taking values in the cone $E$.
\smallskip

\noindent\textbf{Definition.} % 6
Let $X$ be a Banach space. Denote by
$CO\bar{\Omega}$ the convex hull of the set $\Omega \subset X$,
$\bar{\Omega}$ is the closure of $\Omega$ and
$\partial\Omega$ is the boundary of $\Omega$.
To each  bounded set $Y\subset E\subseteq\Omega$, and associate
the nonnegative number $\Psi(Y)$. The function defined this way
is called a measure of non-compactness of the set $Y$ if  the following
conditions are satisfied:
\begin{itemize}
\item[(a)] $\Psi(\overline{COY})=\Psi(Y)$

\item[(b)] If $Y_{1}\subset Y_{2}\in\Omega$ then
$\Psi(Y)\leq\Psi(Y_{2})$
\end{itemize}
\smallskip

\noindent\textbf{Definition.} % 7
The continuous and bounded operator $A$ define on $\Omega$ is called
$\Psi$-condensing if for a noncompact set $Y\subset\Omega$,
$\Psi(Y_{1})\leq\Psi(Y)$ for every $Y_{1}\subset\Omega$.
\smallskip

\noindent\textbf{Definition (set contractive).} % 8
 A map $A:Dom(A)\to  R(A)$, from its domain $Dom(A)$ to its 
range$R(A)$,
is said to be strict set contractive, if it is bounded, continuous and
there exists a constant $\gamma>0$ such that
$M(A(Q))\leq\gamma M(Q)$,
where $M(.)$ denotes the Kuratowski's measure of
non-compactness.

We introduce the concepts of measure of non-compactness and condensing 
maps due
to Krasnose'skii,  Zabreiko and  Sadovskii (see Bainov
and Kazakova \cite{b1}). Many types of measure of non-compactness have 
been defined
by different academicians and employed to study qualitative properties 
of
solutions of varieties of dynamical systems (see Hu et al\cite{h3};
Deimling \cite{d1}, Rzezuchowski \cite{r1}; Guo and Liu \cite{g2}).
Concepts of measure compactness of operator has a fundamental advantage
of estimating a priori bonds without undergoing laborious estimation.

\section{Statement of the problem}

Consider the impulsive inequality
\begin{equation} \label{e3}
\begin{gathered}
u(t)\leq f(t,u(t))+W(t,\int_{t_{0}}^{t}
G(t,s,u(s))ds),\quad t\neq t_k,\; k=1,2,3,\dots \\
\Delta u(t=t_k)\leq I(u(t_k))\\
u(\phi)\leq v_{0}\,.
\end{gathered}
\end{equation}
For an increasing sequence of times,
$0<t_{0}<t_{2}<t_{3}<\dots <t_k$, with  $\lim_{t\to 
\infty}t_k=+\infty$.
Where
$f:\mathbb{R}^{+}XPC(\mathbb{R}^{+},E)\to  E$,
$W:\mathbb{R}^{+}XL^{1}(\mathbb{R}^{+}X\mathbb{R}^{+}XE,E)\to
E$ and $I:E\to  E$.

Before the stage is set up for carrying out our investigations,
we will assume that the following conditions:
\begin{itemize}
\item[(A1)] $f(t,u(t))$ is continuous on $E$ and
Lipschitzian with respect to the second variable.

\item[(A2)] $W(t,\int_{t_{0}}^{t}G(t,s,u(s))ds)$ is a nonnegative 
definite matrix on $E$ and Lipschizian with respect to the second argument.
The function $ G(t,s,u(t))$ is in $C_{0}(\mathbb{R}^{+}X\mathbb{R}^{+}XE,E)$ and 
there exists a constant $k>0$ such that
\[
|G(t,s,u_{2}(t))-G(t,s,u_{1}(t)))|\leq_{E}k|u_{2}(t)-u_{1}(t)|
\]
for  $u_{2}(t),u_{1}(t)\in E$.

\item[(A3)]  $I(.)$ is continuous on $(.)$ and $I(\phi)$.
\end{itemize}

\section{Main results}

Consider the impulsive analogue of the Gronwall-Bellman
inequalities defined on the cone $\mathbb{R}^{+}$:.

\begin{lemma} \label{lem1}
Let $u(t)\in PC(\mathbb{R}^{+},\mathbb{R}^{+})$,
$\beta_k(t)\in PC(\mathbb{R}^{+},\mathbb{R}^{+})$ and
$\gamma(t)\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ and
$C\geq0$ be a nonnegative constant such that
\begin{equation} \label{e4}
u(t)\leq C+\int_{t_{0}}^{t}\gamma(s)u(s)ds+\sum_{t_{0}<t_k<t}\beta_k(t_k)u(t_k),
\quad k=1,2,\dots
\end{equation}
Then
\begin{equation}\label{e5}
u(t)\leq C\prod_{j=i}^{k-1} (1+\beta_j(t_{j})\exp\big(\int_{t_{j-1}}^{t_{j}}\gamma(s)ds\big)\exp\big(\int_{t_{j}}^{t}\gamma(s)ds\big)
\end{equation}
\end{lemma}

\begin{proof}
If $t\in(t_{j}t_{j+1})$, then the proof reduces to the classical continuous Gronwall-Bellman inequality (Hale 
\cite{h1}).
If $t\notin(t_{j}t_{j+1})$, $j=1,2,\dots $ i.e. $t=t_{j}$,
then
\[
u(t_{1})=u(t_{1}+0)\leq C+\int_{t_{0}+0}^{t_{1}}\gamma(s)u(s)ds
\leq C\exp(\int_{t_{0}}^{t}\gamma(s)ds))
\]
similarly,
\begin{align*}
u(t_{2})&\leq C+\int_{t_{0}+0}^{t_{2}}\gamma(s)u(s)ds+\beta_{1}u(t_{1})\\
&\leq(1+\beta_{1}(t_{1}))C\exp\big(\int_{t_{0}}^{t_{1}}\gamma(s)u(s)ds\big)
\exp\big( \int\nolimits_{t_{1}}^{t_{2}}\gamma(s)ds\big)
\end{align*}
 and
\begin{align*}
u(t_{3})&\leq C+\int_{t_{0}}^{t}\gamma(s)u(s)ds+\beta_{1}(t_{1})u(t_{1})
+\beta_{2}(t_{2})u(t_{2})\\
&\leq (1+\beta_{1}(t_{1}))(1+\beta_{2}(t_{2}))C\exp\big(\int_{t_{0}}^{t}\gamma(s)ds)\\
&\quad\times \exp(\int_{t_{1}}^{t_{2}}\gamma(s)ds\big)
\exp\big(\int_{t_{1}}^{t_{2}}\gamma(s)ds\big)
\end{align*}
Thus, by induction on $k\geq4$,
\begin{equation} \label{e6}
u(t)\leq\prod_{j=1}^{k-1}(1+\beta_k(t_k)C)\exp\big(\int_{t_{j}-1}^{t_{j}}\
gamma(s)ds\big)\exp\big(\int_{t_{j}}^{t}\gamma(s)ds\big)\,.
\end{equation}
\end{proof}

Now we consider another version of the above inequality in a 
generalized form:

\begin{lemma} \label{lem2}
Let the hypothesis in Lemma \ref{lem1} be satisfied and let
$\delta(t)\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ and
$\tau_k(t_k)\in PC(\mathbb{R}^{+},\mathbb{R}^{+})$ be such that
\begin{equation}\label{e7}
u(t)\leq C(t)+\int_{t_{0}}^{t}\gamma(s)u(s)ds+\int_{t}
^{\infty}\delta(s)u(s)ds+\sum_{t_{0}<t_k<t}(\beta_k(t_k)+\tau_k(t_k))u(t_k).
\end{equation}
Then
\begin{equation} \label{e8}
u(t)\leq \prod_{j=1}^{k-1}(1+\alpha_{j}(t_{j}))C(t_{j})\exp\big(\int_{t_{j-1}}^{t_{j}}\gamma(s)ds\big)
\exp\big(\int_{t_{j-1}}^{t_{j}}\gamma(s)ds\big)
\exp\big(\int_{t}^{\infty}\delta(s)ds\big)
\end{equation}
where $\alpha_k(t):=\tau_k(t)+\beta_k(t)$, $k=1,2,\dots $
\end{lemma}

\begin{proof} By Lemma \ref{lem1}
\begin{align*}
 u(t)&\leq C(t)+\int_{t_{0}}^{t}\gamma(s)u(s)ds+\int_{t}^{\infty
}\delta(s)u(s)ds+\sum_{t_{0}<t_k<t}(\alpha_k(t_k))u(t_k)\\
&\leq A(t)+ \int_{t}^{\infty}\delta(s)u(s)ds,
\end{align*}
where
\[
A(t)=\prod_{j=1}^{k-1} (1+\alpha_{j}(t_{j}))C(t_{j})\exp\big(\int_{t_{j-1}}^{t_{j}}
\gamma(s)ds)\exp\big(\int_{t_{j-1}}^{t_{j}}\gamma(s)ds\big)\,.
\]
Hence
\begin{equation}\label{e9}
u(t)\leq A(t)\exp\big(\int_{t}^{\infty} \delta(s)ds\big).
\end{equation}
\end{proof}

\noindent\textbf{Remark.} % 4
Lemma \ref{lem2} is a particular case of the inequality in
(Lashimikamtham, et. al.,cite{l1})
where $\delta_{1}=C(t)=P(t)=1$, $\gamma(t)=P(t)V(t)$,
$\delta(t)=0,\alpha(t)=\beta$,
in which the inequality was stated without proof,
whereas, Lemma \ref{lem2} is a generalization of the same inequality when 
$\delta_{1},=C(t)=P(t)=1$, $\gamma(t)=P(t)V(t)$, $\delta(t)=0$, 
$\alpha(t)=\beta=\mathrm{constant}$.

\noindent\textbf{Remark.} % 5
If $i(t_{0},t)$ is the number of points of $t_k$ present in the
interval $(t_{0},t_{k+1})$, $k=0,1,2,\dots$. Then Lemma \ref{lem1} is 
the generalization a of the Bainov-Svetla's inequality \cite[Lemma 2]{b4} 
whenever $\gamma=\gamma(t)$is a constant, $\beta=\beta(t)$ is also a constant.

For the next theorem, we set the following:
\begin{itemize}
\item[(H1)] $(\alpha L_{1}+L_{2})
\sum_{i=1}^{n} [\mathop{\rm diam}(\beta(t_{i})a_{i})<1$,
$\alpha:=\max|t-t_{0}|,t\in \mathbb{R}^{+},t\geq t_{0}$

\item[(H2)] There exist constants $0<L$such that
$\langle B(t_k),\eta(t_{i})>\geq_{E}-L\eta(t_{i})$
for some $\eta(t)\in E$.

\item[(H3)] $G(t,s,.)$ is a nonnegative definite and monotonic
nondecreasing function with respect to second variable such that exists 
a
constant $p>0$ such that
\[ %(11)
W(t,\int_{t_{0}}^{t}G(s,t,\eta_{1}(s))ds)-W(t,\int_{t_{0}
}^{t}G(s,t,\eta_{2}(s))ds)
\geq-p(\eta_{1}(t)-\eta_{2}(t))
\]

\item[(H4)]
\[
u(t)\leq H(t)+W(t,\int_{t_{0}}^{t}G(s,t,u(s))ds)+<B(t),X(t)>_{t=t_k}
\]
\end{itemize}

\begin{theorem} \label{thm1}
Assume $u(t)\in PC(\mathbb{R}^{+},E)$,
$H(t)\in M_{n}'(E),\beta(t)\in M_{n}'(E)$ and
$W\in C^{1}(\mathbb{R}^{+}XE,E)$ and that
(H1)--(H4) are satisfied. Then
\begin{equation} \label{e12}
u(t)\leq H(t)+W(t,u^{\ast}(t))
\end{equation}
where $u^{\ast}(t)$ is the maximal solution of the
impulsive integral equation
\begin{equation} \label{e13}
u(t)=\int_{t_{0}}^{t}G(t,s,u(s))ds+\langle B(t),u(t)\rangle_{t=t_k},\quad k=1,2,\dots
\end{equation}
\end{theorem}

\begin{proof}
The strategy is to show that the solution of the integral
equation in \eqref{e13} exists in a normal cone in an order
interval containing the cone $E$. Moreover, $u^{\ast}(t)$ is the
maximal solution of \eqref{e13} and satisfies the inequality 
\eqref{e12}.
Now define
\begin{gather*}%(14)
A_{1}u(t)=\int_{t_{0}}^{t} G(t,s,u(s))ds\\
A_{2}u(t)=\langle \beta(t_{i}),u(t_{i})\rangle_{i=1,2,3,\dots }\,.
\end{gather*}
Let $A=A_{1}+A_{2}$ be such that
\[
A:\mathop{\rm Dom}(A_{1})\cup \mathop{\rm Dom}(A_{2})\supset
[U_{0},V_{0}]\to  PC(\mathbb{R}^{+},E)
\]
and $M(A_{1}(Q))=\sup_{t\in \mathbb{R}^{+}}M(A_{1}(Q(t))$,
where
\[
Q(t)\in \mathop{\rm Dom}(A_{1})
=\{u(t)\in PC(\mathbb{R}^{+},E):\big|\int_{t_{0}}^{t}
G(t,s,u(s))ds\big|<\infty\}
\]
similarly
\[ %(15)
Q(t_{i})\in \mathop{\rm Dom}
(A)=\{u(t_{i}):|\langle B(t_{i}),u(t_{i})\rangle 
|<+\infty,\;i=1,2,\dots \}\,.
\]
By \cite[Lemma 1]{g2} it follows easily that
there exist constants $L_{1}$ and $L_{2}$ such that
\begin{gather*}
M(A_{1}(Q(t_{i})))\leq\alpha L_{1}M(Q(t)),\\
M(A_{2}(Q(t_{i})))\leq L_{2}M(Q(t))+\epsilon
\end{gather*}
For some arbitrary small positive number $\epsilon$ and since
$t_{0}<t_{i}<t$ for $i=1,2,\dots$, it implies that
\[
M(A_{1}(Q(t_{i}))\leq(\alpha L_{1}+L_{2})M(Q(t))+L_{2}\epsilon
\]
But
\[
M(Q(t_{i}))\leq M(B(t_{i}))M(u(t_{i}))
\leq \sum_{i=1}^{n}
\mathop{\rm diam}B(t_{i})M(u(t_{i}))a_{i}
\]
For some constants $a_{i}$, $i=1,2,\dots $.
Hence
$M(A(u(t_{i}))\leq\gamma M(u(t_{i}))$.
Since $\epsilon$ is arbitrarily small, $A$ is strictly set contractive 
on $[U_{0},V_{0}]$. \smallskip

\noindent\textbf{Step II}
Next it will be shown that $A$ has a fixed point in $[U_{0},V_{0}]$
which is in fact the maximal solution of equation \eqref{e16} below.
Let $u_{n}(t)\to  u_{+}(t)$ as $n\to \infty$.
Now define
\begin{equation} \label{e16}
Au_{n-1}=\int_{t_{0}}^{t}G(t,s,u_{n}(s))ds+\langle 
B(t_{i}),u(t_{i})\rangle\,.
\end{equation}
Suppose that $\eta$ is the solution of equation \eqref{e16}.
Then $A\eta=\eta$ which is a fixed point of $A$. For
$u_{0}\leq\eta_{1}\leq\eta _{2}\leq V_{0}$, we have
\begin{align*}
\eta&=\eta_{2}(t)-\eta_{1}(t)\\
&\geq W(t,\int_{t_{0}}^{t}G(t,s,\eta_{2}(s)ds)
-W(t,\int_{t_{0}}^{t}G(t,s,\eta_{1}(s)ds)
+\langle B(t_{i}),\eta_{2}(t_{i})-\eta_{2}(t_{i})\rangle \\
&\geq L\eta-L\eta=\phi.
\end{align*}
Therefore, $A\eta_{2}-A\eta_{1} \geq_E \phi$ i.e.
$A\eta_{2} \geq_E A\eta_{1}$ for
$\eta_{2}(t_{i})\geq_E \eta_{1}(t_{i})$
Hence $A$ is nondecreasing and strictly set contractive from
$[U_{0},V_{0}]\to PC(\mathbb{R}^{+},E)$.
Since $u_{0}\leq u_{0}(t)$ and
\[
Au_{0}\leq Au_{0}(t)=\int_{t_{0}}^{t}G(t,s,u_{0}(s))ds+\langle B(t_{i}),u_{0}(t_{i})\rangle
=u_{0}(t).
\]
By \cite[Theorem 1.2.1]{g2} there exists, a maximal fixed
point $u(t)$ of $A$ in $[U_{0},V_{0}]$ such that $u_{n}(t)=Au_{n-1}(t)$
and satisfies the condition
$u_{0}\leq u_{1}\leq\dots \leq u_{n}\leq u^{\ast}$,
where
\begin{equation} \label{e17}
Au_{n-1}(t)=\int_{t_{0}}^{t}G(t,s,u_{n}(t)ds+\langle
B(t_k),u_{n}(t_k)\rangle,
\end{equation}
$u^{\ast}(t)$ is the maximal solution of the integral equation
\[ %(18)
u(t)=\int_{t_{0}}^{t}G(t,s,u(t)ds+\langle B(t_k)u(t_k)\rangle
\]
existing in $[U_{0},V_{0}]$.
Hence $u(t)\leq H(t)+W(t,u^{\ast}(t))$.
\end{proof}

\begin{corollary} \label{coro1}
Under the conditions of Theorem \ref{thm1},
replace \eqref{e12} by
\[
 u(t)\leq H(t)+W(t,\int_{t_{0}}^{t}G(t,s,u(s))ds)+\langle B(t_k),u(t_k)\rangle\,.
\]
Then
\begin{equation} \label{e19}
u(t)\leq H(t)+W(t,u^{\ast}(t))+
\langle B(t_k),u^{\ast}(t_k)\rangle ,
\end{equation}
where
$u^{\ast}(t)$ is the maximal solution of the impulsive integral 
equation
\begin{equation} \label{e20}
 u(t)=\int_{t_{0}}^{t}G(t,s,u(s)ds)+\langle B(t_k),u(t_k)\rangle
\end{equation}
existing in $[U_{0},V_{0}]$.
\end{corollary}

For the proof of this corollary, just set $\gamma=\alpha\_{1}<1$ and 
$P_{1}=0$,
as in Theorem \ref{thm1}.

\noindent\textbf{Remark.} % 6
If $W=\int_{t_{0}}^{t}\gamma(s)u(s)ds$, $H$ is constant,
$B(t)=B_k(t)$ and $n=1$,
then Corollary \ref{coro2} is a generalization of our Lemma
\ref{lem1} and
Corollary \ref{coro1} is a generalization of Lemma \ref{lem2}.
On the other hand, if $\langle B(t_k),u(t_k)\rangle=\phi$.
Then theorem \ref{thm1} and Corollary \ref{coro1} are  generalizations
of \cite[Theorem 1]{a1}.

\begin{theorem} \label{thm2}
Under the conditions of Theorem \ref{thm1} assume that
\begin{equation} \label{e22}
\frac{du(t)}{dt}\leq u(t)H(t)+W(t,\int_{t_{0}}^{t}G(t,s,u(s)ds))+\langle B(t_k),u(t_k)\rangle\,.
\end{equation}
Then $u(t)\leq A_{\ast}^{-1}(t)[u_{0}+u^{\ast}(t)]$, % (23)
where
\[
A(t)=\exp\big(\int_{t_{0}}^{t}H(s)ds\big),\quad t\geq t_{0},\; H(t)\in E
\]
and $u^{\ast}(t)$ is the maximal solution of the integral equation in
equation \eqref{e22},
\[ %(24)
u(t)=\int_{t_{0}}^{t}A_{\ast}^{-1}(\tau)[W(\tau,\int_{t_{0}}^{t}
G(\tau,s,u(s))ds)+Y(\tau,t_k)]dt
\]
The constant $\gamma$ is replace by
\[
\gamma:=A_{0}u_{0^{^{\prime}}}+A_{0}\alpha L_{2}+\sum_{i-1}^{n}k_{i}
\mathop{\rm  diam} B(t_{i})<1
\]
where $k_{i}$ are arbitrary constants which are assumed
to exist, and $u_{0'}=\max|u_{0}|_{E}$.
\end{theorem}

of Theorem \ref{thm2}, assume that
$F(t,u(t))\in PC(\mathbb{R}^{+}XE,E)$ and is
measurable, begin{corollary} \label{coro2}
Under the condition monotonically nondecreasing and Lipschitz
with respect to the second variable.
	
Also assume that
$W_{2}:\mathbb{R}^{+}XC(\mathbb{R}^{+},E)XL^{1}(\mathbb{R}^{+}
X\mathbb{R}^{+}XE,E)\to E$ is measurable, monotonically nondecreasing 
and Lipshitz with respect to the third variable such that
\begin{align*}
\frac{du(t)}{dt}&\leq A(t)H(t)u(t)+F(t,u(t))\\
&\quad +\int_{t_{0}}^{t}d\tau H(\tau)A(\tau)W(t,F(\tau,u(\tau)),\int_{t_{0}}^{t}
G(t,s,H(s)A(s)u(s))ds)
\\
&\quad +\int_{t_{0}}^{t} d\tau H(\tau)A(\tau)W_{2}(t,F(\tau,u(\tau)),
\int_{t_{0}}^{t} G(t,s,H(s)A(s)u(s))ds),
\end{align*}
$u(t=t_k)\leq \langle B(t_k),u(t_k)\rangle_{k=1,2,3\dots}$.
Then
\[
u(t)\leq D(t)[u_{0}+u^{\ast}(t)+F(t,u^{\ast}(t)]
+\langle D(t_k)B(t_k ),u^{\ast}(t_k)\rangle _{k=1,2,3\dots},
\]
where
\begin{equation} \label{e26}
\begin{aligned}
 u(t)&=\int_{t_{0}}^{t}d\tau Z(t,\tau)W(t,\int_{t_{0}}^{t}G(t,s,H(s)A(s)u(s))ds)\\
&\quad +\int_{t_{0}}^{t}d\tau Z(\tau)W_{2}(t,F(\tau,u(\tau)),
\int_{t_{0}}^{t}G(t,s,H(s)A(s)u(s))ds)\,.
\end{aligned}
\end{equation}
The function $Z(t,\tau)$ is define as
\[ %(27)
Z(t,\tau)=\begin{cases} D(t)H(t)A(\tau ) &t\geq\tau\geq 0\\
\phi & \tau<t<0,\; t,\tau\in \mathbb{R}^{+}
\end{cases}
\]
and has the properties that $Z(0,\tau)=H(\tau)A(\tau),Z(0,0)=I$,
\[
D(t)=\exp\big(\int_{t_{0}}^{t}H(\tau)A(\tau)d\tau\big), \quad
D(t)\in M_{n'}(E),
\]
and $I$ is identity matrix.

\begin{proof}
By contradiction, let $(u_{n})_{n\in N}$ be a monotonically
nondecreasing sequence in $E$ such that $u_{n}\to  u\ast$ as $n\to 
\infty$.
Let $u^{\ast}(t)$ be the maximal solution of  \eqref{e26} such that
\begin{equation} \label{e28}
u(t)>D(t)[u_{0}+u^{\ast}(t)+F(t,u^{\ast}(t))]+
\langle D(t_k)B(t_k),u^{\ast}(t_k)\rangle ]\,.
\end{equation}
We show that there does not exist a function $u(t)$ in $[U_{0},V_{0}]$ 
such
that $u(t)>u^{\ast}(t)\geq u_{\ast}(t)$, otherwise $u^{\ast}(t)$ would 
cease to be maximal.
Let
\begin{equation} \label{e29}
\begin{aligned}
Au_{n}(t)&=\int_{t_{0}}^{t}d\tau Z(t,\tau)W(t,\int_{t_{0}}^{t}G(t,s,H(s)A_{\ast}(s)u_{n}(s))ds)\\
&\quad +\int_{t_{0}}^{t}d\tau
Z(t,\tau)W_{2}(t,F(t,u_{n}(t),\int_{t_{0}}^{t}G(t,s,H(s)A_{\ast}(s)u_{n}(s))ds).
\end{aligned}
\end{equation}
Then $Au_{0}\geq u_{0}\geq u_{0}$, $Av_{0}\leq v_{0}$ and the operator $A$
is a set contraction if
\begin{gather*}
\gamma=\alpha Z_{0}(h_{0}A_{0}L_{1}+\alpha A_{0}h_{0}L_{2})<1, \quad
Z_{0}=\max_{\tau ,t}\in \mathbb{R}^{+}|Z(t,\tau)|,\\
A_{0}=\max_{\tau} |A_{\ast}(t)|, \quad
h_{0}=\max_{\tau}|H(\tau)|,
\end{gather*}
and $L_{1}, L_{2}$ are constants.

Hence by Theorem \ref{thm1}, there exists a maximal solution 
$u^{\ast}(t)$ of
\eqref{e29} in $[U_{0},V_{0}]$ which is in fact the fixed point of $A$.

Now let
\[
\delta_{\ast}=u_{0}+F(t,u^{\ast}(t))+\langle D(t_k)B(t_k)u^{\ast}(t_k)\rangle
\]
then $\delta_{\ast}(t)\in E$ such that
\begin{gather*}
u(t)>D(t)[u^{\ast}(t)+\delta_{\ast}(t)],\\
\psi(u(t)_{\ast}\delta_{\ast}(t))>u^{\ast}(t),
\end{gather*}
where
$\psi(u(t)_{\ast}\delta_{\ast}(t))-(D^{-1}(t)u(t)-\delta_{\ast}(t))\in[
u_{0}-\delta_{(\ast},v_{0}-\delta_{\ast}]\subseteq\lbrack 
U_{0},V_{0}]$.
But $u^{\ast}(t)$ is maximal.  Hence there does not exist an element
$\psi(u(t)_{\ast}\delta_{\ast}(t))$ in the other 
interval$[U_{0},V_{0}]$ such that equation \eqref{e28} is satisfied which is a contradiction.
Hence,
\[ %(30)
u(t)\leq D(t)[u_{0}+F(t,u^{\ast}(t))+u^{\ast}(t)]
\]
\end{proof}

\begin{theorem} \label{thm3}
Under the conditions of Theorem \ref{thm2}, assume following
conditions:
\begin{align*}
\frac{du(t)}{dt}
&\leq A_{\ast}(t)H(t)u(t)+F(t,u(t))\\
&\quad +\int_{t_{0}}^{t}A_{\ast}(\tau)H(\tau)W(t,\int_{t_{0}}^{t}G(t,s,A(s)H(s)u(s)ds)d\tau )\\
&\quad+ \int_{t_{0}}^{t}A_{\ast}(\tau)H(\tau
)W_{2}(t,F(\tau,u(t)) \int_{t_{0}}^{t}G(t,s,u(s)ds)d\tau)
\end{align*}
$\triangle u(t=t_k)\leq \langle B(t_k),u(t_k)\rangle_{k=1,2,3\dots }$;
and the commutant satisfies
\begin{gather*}
[A(t),H(t)]=A_{\ast}^{\ast}(t)H.(t)-H^{\ast }(t)A\ast(t)=\phi,\\
\det(H(t)A_{\ast}^{\ast}(t)A_{\ast}(t)H(t)>0\,.
\end{gather*}
Also assume that
\begin{gather}
F(t,u(t))=O(|u(t)|_{E}), \label{cond2}\\
\lim_{|x(t)|_{E}\to \phi} 
\frac{|W_{2}(.,.,x(t))|_{E}}{|x(t)|_{E}}=\phi,
\label{cond3}\\
\lim_{|y(t)|_{E}\to \phi} \frac{|W_{{}}(.,Y(t))|_{E}}{|Y(t)|_{E}}=\phi
\label{cond4}
\end{gather}
For $y(t)\in L^{^{\prime}}(E)$.
Then
\begin{equation} \label{e31}
u(t)<D(t)[u_{0}+F(t,u^{\ast}(t)+u^{\ast }(t)]
+ \langle D(t_k)B(t_k),u^{\ast}(t_k)\rangle ,
\end{equation}
where
$u^{\ast}(t)$ is the maximal solution of the integral solution
\begin{equation}
\begin{aligned} \label{e32}
u(t)&=\int_{t_{0}}^{t} D(\tau)A_{\ast}(\tau)W(t,\int_{t_{0}}^{t}
G(t,s,A_{\ast}(s))H(s)u(s)ds)d\tau \\
&\quad +\int_{t_{0}}^{t}D(\tau)A_{\ast}(\tau)
W(t,F(\tau,u(\tau)),\int_{t_{0}}^{t}G(t,s,u(s))ds)
\end{aligned}
\end{equation}
and
\[ %(33)
D(\tau)=\exp\big(\int_{t_{0}}^{t}A_{\ast}(s)H(s)ds\big),\quad
A_{\ast}(t)=\exp\big(\int_{t_{0}}^{t}H(s)ds\big)
\]
\end{theorem}

The proof of the above theorem follows from Corollary \ref{coro2} and 
Theorem \ref{thm2}.
We will like to emphasize that the conditions 
\eqref{cond2}--\eqref{cond4}
do not allow the quantity to be unbounded below and as a consequence of Theorem \ref{thm2}; we 
have the existence of a maximal solution to \eqref{e32} in
a normal cone $K\subseteq[U_{0},V_{0}]$.

\section{Applications}

\subsection*{Example 5.1}

Consider a nonlinear impulsive control system (NLICS)
\begin{gather*}
\frac{dx(t)}{dt}=-xe^{-xy}+f(t,x(t),y(t),u_{1}(t)),\quad t\neq t_k,\;k=0,1,2,\dots, \\
\frac{dy(t)}{dt}=y\sin (xy) +g(t,x(t),y(t),u_{2}(t)),\quad t\neq t_k,\; k=0,1,2,\dots ,\\
\Delta x=\frac{\beta_k^{1}x^{2}(t_k)}{1+x(t_k)},\quad t=t_k,\; k=01,2,\dots ,\\
\Delta y=\frac{\beta_k^{1}y_k^{2}(t_k)}{1+y(t_k)},\quad t=t_k,\;k=01,2,\dots,
\end{gather*}
where $0<t_{0}<t_{2}<\dots <t_k$, $\lim u_{k\to \infty}t_k=+\infty$,
$x(0)=x_{0}$, $y(0)=y_{0}$, $0\leq x\leq\frac{\pi}{2}$ and
$0\leq y\leq\frac{\pi}{2}$.
Also $f:\mathbb{R}^{+}X\mathbb{R}^{+}XC\to  \mathbb{R}^{+}$,
$g:\mathbb{R}^{+}X\mathbb{R}^{+}X\mathbb{R}^{+}XC\to \mathbb{R}^{+}$,
$u:\mathbb{R}^{+}\to  C$,
$u(t)$ is the control variable belonging to the control space
\[
C=\{u(t)=(u_{1}(t),u_{2}(t):0\leq1,t\in \mathbb{R}^{+}\}\subset mathbb{R}^{+}.
\]
To investigate the boundedness or stability property of the above
nonlinear control inequality,
 we often use the comparison equation. In this particular problem, $e^{-xy}\leq1$ for all $x,y\in \mathbb{R}^{+}$ and
$\sin(xy)\leq1$ for the given of $x$ and $y\frac{z}{1+z}\leq1$
for every $z\geq0$. Then the  nonlinear control inequality
\begin{gather*}
\frac{dx}{dt}\leq
-x+f(t,x(t),y(t),u_{1}(t)),\quad t\neq t_k,\; k=0,1,2,\dots \\
\frac{dx}{dt}\leq -y+g(t,x(t),y(t),u_{2}(t)),\quad t\neq t_k,\;
k=0,1,2,\dots \\
\Delta x\leq\beta_k^{1}x(t_k)\\
\Delta x\leq\beta_k^{2}y(t_k)\\
0<t_{0}<t_{1}<t_{2}<\dots <t_k,\quad \lim_{k\to \infty}t_k=+\infty
\end{gather*}
serves as a basic comparison inequality for investigating
(NLICE).  The maxima solution (upper bound for the inequality)
to  NLICE can be found using standard results,
see Lakshimikantham et al \cite{l1}.
Thus
\begin{align*}
x(t)&\leq\prod_{t_{0}<t_k<t}(1+\beta_k^{1})e^{-(t-t_k)}x_{0}\\
&\quad +\int_{t_{0}}^{t}\prod_{t_{0}<t_k
<t}(1+\beta_k^{1})e^{-(t-s)}f(s,x(s),y(s),u_{1}(s))ds,
\\
y(t)&\leq\prod_{t_{0}<t_k<t}(1+\beta_k^{2})e^{-(t-t_k)}y_{0}\\
&+\int_{t_{0}}^{t}\prod_{t_{0}<t_k<t}(1+\beta_k^{2})e^{-(t-s)}
g(s,x(s),y(s),u_{2}(s))ds
\end{align*}
If
\begin{gather*}
f(t,x(t),y(t),u_{1}(t))\leq k_{1}(t)u_{1}(t)x(t)+\sum
_{t_{0}<t_k<t}h^{(1)}(t_k)u_{1}(t_k+0)x(t_k)
\\
 g(t,x(t),y(t),u_{2}(t))\leq k_{2}(t)u_{2}(t)y(t)+\sum
_{t_{0}<t_k<t}h^{(2)}(t_k)u_{2}(t_k+0)y(t_k),
\end{gather*}
where $k_{1}(t),h^{(i)}(t_k)\in \mathbb{R}^{+}$, $i=1,2$, 
$k=0,1,2,\dots $,
\begin{align*}
x(t)&\leq\prod_{t_{0}<t_k<t}(1+\beta_k^{1})e^{-(t-t_k)}x_{0}\\
&\quad +\int_{t_{0}}^{t}\prod_{t_{0}<t_k<t}(1+\beta_k
^{1})e^{-(t-s)}k_{1}(s)u_{1}(s)x(s))ds+\sum_{t_{0}<t_k<t}\phi
_{1}(t_k,t)x(t_k)
\\
y(t)&\leq\prod_{t_{0}<t_k<t}(1+\beta_k^{2})e^{-(t-t_k)}y_{0}\\
&\quad +\int_{t_{0}}^{t}\prod_{t_{0}<t_k<t}(1+\beta_k%
^{2})e^{-(t-s)}k_{2}(s)u_{1}(s)y(s))ds+\sum_{t_{0}<t_k<t}\phi
_{2}(t_k,t)y(t_k)
\end{align*}
where
\begin{gather*}
\phi_{1}(t_k,t)=h^{(1)}(t_k)u_{1}(t_k)\prod_{t_{0}%
<t_k<t}(1+\beta_k^{1})e^{(t-t_k)}\int_{t_{0}}^{t}e^{-(t-s)}ds,
\\
\phi_{2}(t_k,t)=h^{(1)}(t_k)u_{2}(t_k)\prod_{t_{0}%
<t_k<t}(1+\beta_k^{2})e^{(t-t_k)}\int_{t_{0}}^{t}e^{-(t-s)}ds
\end{gather*}
Now let $z(t)=x(t)e^{t}$. Then
\begin{align*}
 z(t)&\leq\prod_{t_{0}<t_k<t}(1+\beta_k
^{1})e^{-(t-t_k)}z_{0}\\
&\quad +\int_{t_{0}}^{t}\prod_{t_{0}<t_k
<t}(1+\beta_k^{1})e^{-(t-s)}k_{1}(s)u_{1}(s)z(s))ds
+\sum_{t_{0}<t_k<t}\phi_{1}(t_k,t)z(t_k).
\end{align*}
Applying the lemma and substituting the value of $x(t)$, we get
\begin{align*}
x(t)&\leq\Big(  \prod_{t_{0}<t_k<t}(1+\beta_k^{1})\Big)
\Big(\prod_{t_{0}<t_k<t}(1+\phi_{1}(t_k,t)\Big) \\
&\quad\times \exp \big(\int_{t_k=t}(1+\beta_k)k_{1}(s)u_{1}(s)ds\big)
\exp\big(-(t-t_k)x_{0}\big).
\end{align*}
By a similar manipulation we obtain
\begin{align*}
y(t)&\leq\Big(  \prod_{t_{0}<t_k<t}(1+\beta_k^{2})\Big)
\Big(\prod_{t_{0}<t_k<t}(1+\phi_{2}(t_k,t)\Big)  \\
&\quad\times 
\exp\big(\int_{t_k=t}(1+\beta_k^{2})k_{2}(s)u_{1}(s)ds\big)
\exp\big(-(t-t_k)y_{0}\big)\,.
\end{align*}
We have obtained the bounds for $x(t)$ and $y(t)$ under the
conditions imposed on $k_{i}(t),h^{(i)}(t_k)\in \mathbb{R}^{+}$,
$i=1,2$, $k=0,1,2,\dots $,
when $u_{i}(t)$, $i=1,2$ are bounded; that is, using control language, 
bounded
input will give rise to bounded output.
Many biological and physical control systems are of bounded 
input-bounded output
types.  Bounds on $x(t)$ and $y(t)$ can be used
to make qualitative deductions about the control system.

\subsection*{Example 5.2}

Consider the impulsive integrodifferential system (IIS)
\begin{gather*}
\frac{du(t)}{dt}\leq \mathop{\rm diag}[ae^{\alpha t}\; be^{\beta t}]u(t)
+F(t,u(t))\\
+\int _{0}^{t}d\tau z(t,\tau)w(t,\int_{0}^{t}
G(t,s,H(s)A(s)u(s)ds),\quad t\neq t_k,\; k=0,1,2,\dots \\
u(t=t_k)\sum_{t_{0}<t_k<t}\beta(t_k)u(t_k)\\
0<t_{0}<t_{1}<\dots <t_k,\lim_{k\to \infty}t_k=\infty\,,
\end{gather*}
where
$H(t)=\mathop{\rm diag}[a\; b]$,
$A(t)=\mathop{\rm diag}[e^{\alpha t}e^{\beta t}]$,
$\alpha>0,\beta>0$, $u(t)=(u_{1}(t),u_{2}(t))$,
\[
G(t,s,H(s)A(s)u(s))=\begin{cases}
\frac{t-s}{h}\mathop{\rm diag}[ae^{\alpha t}\;be^{\beta t}] u(t)
&t\geq s,\\
0 &t<s\,,
\end{cases}
\]
$z(t,\tau)=D(t)H(t)A(t)$, and $w(\phi,\phi)=\phi$.

Assuming $\lim_{|v|\to 0} w(t,v)/|v|=\phi$,
$\phi=(0,0)\in\mathbb{R}^{+}$,
let $v=\int_{0}^{t}G(t,s,\dots )ds$ and $t-s=\theta$.
Therefore,
\[
\begin{pmatrix}
\frac{-ae^{\alpha t}}{h}\int_{\theta}^{\theta+s}e^{-\alpha t}%
u_{1}(-\tau)d\tau\\
\frac{-be^{\alpha t}}{h}\int_{\theta}^{\theta+s}e^{-\beta t}u_{2}%
(-\tau)d\tau
\end{pmatrix}\,.
\]
Also
$z(t,\tau)=\mathop{\rm diag}
[\exp\frac{a}{\alpha}(e^{\alpha t}-1)\;\exp\frac{b}{\beta
}(e^{\beta t}-1)]\mathop{\rm diag}[ae^{\alpha t}be^{\beta t}]$.
Hence
\[
w(t,v)=\begin{pmatrix}
w_{1}(t,v_{1})\\
w_{2}(t,v_{2})
\end{pmatrix}
=\begin{pmatrix}
w_{1}(t,\frac{-ae^{\alpha t}}{h}\int_{\theta}^{\theta+s}e^{-\alpha
t}u_{1}(-\tau)d\tau\\
w_{2}(t,\frac{-be^{\alpha t}}{h}\int_{\theta}^{\theta+s}e^{-\beta
t}u_{2}(-\tau)d\tau
\end{pmatrix}
\]
It can  be shown easily that the commutant of $A(t)$ and $H$
satisfy
$[A(t),H(t))]=\phi$, $\phi=(0,0)\in\mathbb{R}^{+}$
and
\[
\det(H^{ast}(t)A^{ast}(t)A(t)H(t))=a^{2}b^{2}e^{2(\alpha+\beta)t}>0,\quad
\alpha+\beta>0\,.
\]
Therefore, $v_{i}(t)$ are estimated as
\begin{gather*}
 v_{1}(t)\leq\frac{a^{2}}{h}\int_{0}
^{t}\int_{\theta+s}^{\theta}u_{1}(-\tau)d\tau ds,\\
v_{2}(t)\leq\frac{b^{2}}{h}\int_{0}
^{t}\int_{\theta+s}^{\theta}u_{2}(-\tau)d\tau ds, \\
v_{^{\ast}1}(t)\leq\frac{a^{2}}{h}
\int_{0}^{t}\int_{\theta+s}^{\theta}\max_{\tau\in
[0,\theta+s]}u_{1}(-\tau)d\tau ds
=\frac{a^{2}}{h}t^{\ast}|u_{1}(-\tau)|_{R_{0}} \\
v_{^{\ast}2}(t)\leq\frac{b^{2}}{h}
\int_{0}^{t}\int_{\theta+s}^{\theta}\max_{\tau\in [0,\theta+s]}
u_{2}(-\tau)d\tau ds=\frac{b^{2}}{h}t^{\ast}|u_{1}(-\tau)|_{R_{0}}
\end{gather*}
Here $t^{\ast}$ is the threshold value of $t$ taken across the interval
$[0,\theta+s]$. Therefore, applying theorem \ref{thm3} to (llS) yields.
where
\begin{align*}
u^{\ast}(t)&=\int_{t_{0}}^{t}D(\tau)A^{\ast}(\tau)
w(tau,\int_{t_{0}}^{t}G(t,s,A_{1}(s)H(s)u^{\ast}(s)ds)d\tau\\
&\quad + \int_{t_{0}}^{t}d\tau D(\tau)A^{\ast}(\tau)
\begin{pmatrix}
w_{1}(t,\frac{a^{2}t^{\ast}}{h}|u_{1}(-\tau)|R_{0})\\
w_{2}(t,\frac{b^{2}t^{\ast}}{h}|u_{2}(-\tau)|_{R_{0}}
\end{pmatrix}
\end{align*}
Therefore, the right-hand side provides the upper bound for $u(t)$.


\subsection*{Acknowledgements}
The authors hereby acknowledge support from  Abubakar Tafawa Balewa
University, Bauchi Nigeria and from the National
Mathematical Centre, Abuja, Nigeria.  The authors are also grateful for
suggestions from the anonymous referee.

\begin{thebibliography}{00}

\bibitem{a1} Akinyele, Olusola;
\emph{On some fundamental matrix integral inequalities and their
applications}.  Analele stintifice Ale Universitii al,
1-cuza, Din lasi Tomul XX1-75 la Mathematical (1985).

\bibitem{a2}  Akinyele, O. and Akpan, P. Edet;
\emph{On the $\phi$ -stability of comparison differential systems}.
J. Math. Anal Appli 164 (1992), No. 2 , 307--324..

\bibitem{a3} Akpan, Edet P.;
\emph{On the $\phi\sb 0$-stability of perturbed nonlinear differential 
systems}.  Dynam. Systems Appl.  4  (1995),  no. 1, 57--78. 

\bibitem{a4} Ale, S. O.; Deshliev, A.; Oyelami, B. O.;
\emph{On chemotherapy of impulsive model involving malignant
cancer cells}. Abacus J. Math. Asso. Nig. Vol. 24(1996), No. 2, 2-10.

\bibitem{b1} Bainov, D. D. and Kazakova, N. G.;
\emph{Finite difference method for solving the periodic problem
for autonomous differential equation with maxima mathematics}. Reports 
from
Tayoma University 15 (1992) No. 5,1-13.

\bibitem{b5} Bainov, D. D. and Stamova, I. M.;
\emph{Uniform asymptotic stability of impulsive differential
-difference equations of neutral type using lyapunov direct method}.
 J. Comp. Math \& Applic. 62 (1995) No. 5, 359-369.

\bibitem{b4} Bainov, D. D. and Svetla, D. D.;
\emph{Averaging method for neutral type impulsive differential
equations with supremums}. Annales de la aculte des science de Toulause
Vol. Xii (1991), No. 3.

\bibitem{d1} Deimling, K. and Lakshimikantham, V.;
\emph{Quasi-sutions and their role in the qualitative theory
of differential equations}. Nonlinear Analysis 4(1980) No. 1, 657-663.

\bibitem{g1} Guo, Dajun  and Lashimikatham, V.;
\emph{Nonlinear problems in abstract cones}.
Academic Press New York, (1988).

\bibitem{g2} Guo, Dajun and Liu, Xinzhi;
\emph{Extremal solutions of nonlinear impulsive
integrodifferential equations in banach space}.
 J. Math. Anal. Appli., 177(1993), 538-552.

\bibitem{h1} Hale, Jack K.;
\emph{Ordinary differential equations}. 
Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience, 
New York-London-Sydney, (1969). 


\bibitem{h3} Hu, Shouchuan; Lashimikantham, V. and Papageorgions, N.;
\emph{Extremal solutions of functional differential
equations}. J. Math. Anal . Appli., 173  (1993), 430-435.

\bibitem{h2} Huston, V. C. L. and Pym, J. S.;
\emph{Application of functional analysis and operator
theory}. Academic Press London, New York/Toronto/Sydney (1980).

\bibitem{l1} Lakshimikantham, V.; Bainov, D. D. and Simeonov, P. P.;
\emph{Theory of impulsive differential equations}. World
Scientific Publishing Company, Singapore, New Jersey, Hong Kong, 
(1989).

\bibitem{o1} Oyelami, B. O.; Ale, S. O. and Sasey, M. S.;
\emph{On existence of solutions of impulsive differential
difference equations}. Proceeding of Conference on Ordinary 
Differential
Equations July 28-29 at National Mathematical Centre,
Abuja, Vol. 1 (2000), 101-117.

\bibitem{o2} Oyelami, B. O.; Ale, S. O. and Sasey, M. S.;
\emph{Topological degree approach to study of the
existence of solution of impulsive initial-boundary value problems}.
J. Nig. math. Soc.,Vol. 21 (2002), 13-26.

\bibitem{o3} Oyelami, B. O.;
\emph{Asymptotic behavior of impulsive systems}. Unpublished
M. sc. Dissertation University of Ibadan , Ibadan, Nigeria  (1991).

\bibitem{o4} Oyelami, B. O.;
\emph{Instability theorems of certain testable impulsive
systems}. Abacus J. Math. Asso. Nig. vol. 30 (2003) No. 2A, 95-104.

\bibitem{o5} Oyelami, B. O.;
\emph{Boundedness of solution of a delay impulsively
perturbed systems}. Int. J. Nonlinear Diff. Equ. (To appear)

\bibitem{o6} Oyelami, B. O.;
\emph{On military model for impulsive reinforcement
functions using exclusion and marginalization technique.}
Nonlinear Analysis 35 (1999), 947-958.

\bibitem{r1} Rzezuchouski, T.;
\emph{Scorza-Dragoni theorem for upper semi continuous
multi valued function}. Bulletin Polish Academy of Science 28  (1980),
661-666.

\end{thebibliography}


\end{document}