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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 105, 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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/105\hfil Existence and uniqueness of solutions]
{Existence and uniqueness of solutions to first--order systems of
nonlinear impulsive boundary--value problems with sub--, super--linear or linear growth}

\author[J. J. Nieto, C. C. Tisdell\hfil EJDE-2007/105\hfilneg]
{Juan J. Nieto, Christopher C. Tisdell}  % in alphabetical order

\address{Juan J. Nieto \newline
Departamento de An\'{a}lisis Matem\'{a}tico \\
        Facultad de Matem\'{a}ticas \\
         Universidad de Santiago de Compostela \\
         Santiago de Compostela 15782, Spain}
\email{amnieto@usc.es}

\address{Christopher C. Tisdell \newline
School of Mathematics and Statistics \\
        The University of New South Wales \\
         Sydney 2052, Australia}
\email{cct@unsw.edu.au}

\thanks{Submitted March 14, 2007. Published July 30, 2007.}
\subjclass[2000]{34A37, 34B15}
\keywords{Existence and uniqueness of solutions; boundary value
 problems; \hfill\break\indent impulsive equations; fixed--point theory; system of equations}

\begin{abstract}
 In this work we present some new results concerning the existence and
 uniqueness of solutions to an impulsive first--order, nonlinear ordinary
 differential equation with ``non--periodic'' boundary conditions. These
 boundary conditions include, as a special case, so--called ``anti--periodic''
 boundary conditions.
 Our methods to prove the existence and uniqueness of
 solutions involve new differential inequalities,
 the classical fixed--point theorem of Schaefer,
 and the Nonlinear Alternative.
 Our new results apply to systems of impulsive differential
 equations where the right-hand side of the equation may grow
 linearly, or sub-- or super--linearly in its second argument.
\end{abstract}

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

\section{Introduction}

At certain points in time, many dynamic phenomena experience sudden,
instantaneous, rapid change exhibited by a jump in their states.
Such behaviour is seen in
a range of physical problems from: mechanics; chemotherapy; population
dynamics; optimal control; ecology; industrial robotics; biotechnology;
spread of
disease; harvesting; and physics.  The reader is referred to
\cite{ GCNT, LBS, LH, N02NA, N97, NR, dO, Saker, SP, TC, YZN, ZS, ZF, ZSW}
 and references therein for some nice examples and applications to the above areas.

The branch of modern, applied analysis known as ``impulsive'' differential
equations furnishes a natural framework to mathematically describe the
aforementioned jumping processes.  Consequently, the area of impulsive
differential equations  has been developing at a rapid rate, with the wide
applications significantly motivating a deeper theoretical study of the
subject.

This paper considers the existence and uniqueness of solutions to the
following first--order differential system:
\begin{gather}
x' = f(t,x),\quad  t \in [0,N], \; t \neq t_1;\label{i1.1} \\
Ax(0) + Bx(N) = \alpha,\quad 0 < N \in \mathbb{R};  \label{i1.2}
\end{gather}
where: $f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ is continuous on $(t,p) \in ([0,N] \setminus \{t_1\})  \times \mathbb{R}^n$; $n\ge 1$;  $A$
and
$B$ are $n \times n$ matrices with real-valued elements; $\alpha$ is a
constant vector in $\mathbb{R}^n$; and the impulse at $t=t_1$ is given by a
continuous function
$I_1: \mathbb{R}^n \to \mathbb{R}^n$ with
\begin{equation} \label{i1.3}
x(t_1^+) = x(t_1^-) + I_1(x(t_1)), \quad t_1 \in (0,1), \; t_1
\text{ fixed};
\end{equation}
using the notation $x(t_1^-):= \lim_{t \to t_1^-}x(t)$ and
$x(t_1^+):= \lim_{t \to t_1^+}x(t)$.

Equations \eqref{i1.1}--\eqref{i1.3} are collectively known as an
impulsive boundary value problem (BVP). Besides the natural physical
applications of impulsive differential
equations, significance of the study of the system \eqref{i1.1}--\eqref{i1.3}
lies in the fact that most types of impulsive BVPs with linear boundary
conditions can be written in the form \eqref{i1.1}--\eqref{i1.3}.
For example, through a simple substitution $x_i:=x^{(i)}$, $i=1,\dots,k$,
impulsive BVPs with second-- or higher--order derivatives may be reduced
to the system \eqref{i1.1}--\eqref{i1.3}.  Thus the study of
\eqref{i1.1}--\eqref{i1.3} can lead to a deeper understanding of a range
of impulsive BVPs, including those of a higher-order.



In the case when: $A=B$ equals the identity
matrix; and $\alpha = 0$ in \eqref{i1.2}, that is,
\eqref{i1.2}
becomes the periodic boundary conditions
$$ x(0) = x(N),$$
some recent and
influential papers examining existence of solutions to
\eqref{i1.1}--\eqref{i1.3} include
\cite{CTY, FN, FNp2, LNS, N02NA, N02JMAA, N97, NR, QL}.

Throughout this paper the condition $\det (A+B) \neq 0$ is assumed
to hold, so in this sense, the boundary conditions \eqref{i1.2} do
not include the periodic conditions $x(0) - x(N) = 0$.  Few
papers, apart from \cite{FN00, LSN, Skora, Skora2}, have examined
the existence and uniqueness of solutions to
\eqref{i1.1}--\eqref{i1.3} under this ``non--periodic'' boundary
condition, even though these types of boundary conditions appear
in many applications, especially the case of ``anti--periodic''
boundary conditions
$$x(0) = -x(N),$$
for example, see \cite{AAB, AR, AMR, Chen, FN00, FNO, Kleinhert, Nakao,
PT, Soup,Yin}.


This article is organised as follows.  Section \ref{sec2} presents some
preliminary ideas associated with the impulsive BVP \eqref{i1.1}--\eqref{i1.3}.  Sections \ref{sec3} and \ref{sec4} contain the main results
of the paper and are devoted to the existence and uniqueness of solutions to
\eqref{i1.1}--\eqref{i1.3}.  There, new differential inequalities in the
impulsive--setting are introduced, developed and applied, in
conjunction with Schaefer's theorem \cite[Theorem 4.4.12]{Lloyd} and the
Nonlinear Alternative \cite[Theorem 5.1, p.61]{DG}, to prove the existence
and uniqueness of solutions to \eqref{i1.1}--\eqref{i1.3}.  The main ideas
rely on: novel differential inequalities; and {\em a priori} bounds on solutions
to a certain family of integral operator equations, with the operator being
compact.


The new results compliment and extend those of \cite{CTY, FN00,
FNp2, LNS, LSN, N02NA, N02JMAA, N97, QL, Skora, Skora2} in the
sense that:  our ideas permit super--linear growth of $\|f(t,p)\|$
in $\|p\|$ in \eqref{i1.1}, whereas the theorems in \cite{LSN} do
not; our investigation tackles a wider range or a different class
of boundary conditions than those in \cite{LNS, LSN, N02NA,
N02JMAA, N97, QL, Skora, Skora2}; and our results apply to systems
of  impulsive BVPs,  unlike the papers \cite{FN00, FNp2} which
have concentrated on scalar--valued equations. This last point is
of particular significance when dealing with large systems of
equations, as traditional methods, like the method of upper and
lower solutions, are rather cumbersome to apply to
\eqref{i1.1}--\eqref{i1.3} when $n$ is large.


Section \ref{sec5} presents an example to illustrate how to apply
some of the newly developed theoretical results. A particular
example is constructed so that the theorems in  \cite{CTY, FN00,
FNp2,LNS, LSN, N02NA, N02JMAA, N97, QL, Skora, Skora2} do not
directly apply.

One could consider impulsive BVPs with a finite number of impulses $I_i$,
so that \eqref{i1.3} could take the form, for $i=1,\dots,p$
$$
x(t_i^+) = x(t_i^-) + I_i(x(t_i)), \quad\text{each $t_i$ in
$(0,1)$ and fixed}.
$$
However, for clarity and brevity, attention is restricted to BVPs with one
impulse.  In addition, the difference between the theory of one or an
arbitrary number of impulses is quite minimal.

Our new results were particularly motivated by the recent works  \cite{DT}, \cite{LNS},
\cite{T1}, \cite{T2} and \cite{WW}.

To understand the notation used above and the ideas in the remainder of the
paper,  some appropriate concepts connected with impulsive differential
equations are now introduced.  The following notation comes from \cite{LSN}
and further information can be found in the references therein.

Assume that
$$
f(t_1^+,x):=\lim_{t \to t_1^+}f(t,x) \quad   \text{and} \quad
f(t_1^-,x):=\lim_{t \to t_1^-}f(t,x)
$$
both exist with
$$ f(t_1^-,x)= f(t_1,x).
$$
Introduce and denote the Banach space $PC([0,N];\mathbb{R}^n)$ by
\begin{align*}
PC([0,N];\mathbb{R}^n):=& \big\{u:[0,N]\to\mathbb{R}^n,  u \in C([0,N] \setminus
\{t_1\};\mathbb{R}^n),   u  \text{ is left continuous}\\
&\text{at $t=t_1$, the right hand limit $u(t_1^+)$ exists} \big\}
\end{align*}
with the norm
$$
\|u\|_{PC} := \sup_{t \in [0,N]}\|u(t)\|
$$
where $\|\cdot\|$ is the usual Euclidean norm and
$\langle\cdot,\cdot\rangle$ will be the Euclidean inner product.

Let $t_0 = 0$ and $t_2=N$.
In a similar fashion to the above, define and denote the Banach space
$PC^1([0,N];\mathbb{R}^n)$ by
\begin{align*}
PC^1([0,N];\mathbb{R}^n):=&\big\{u \in PC([0,N];\mathbb{R}^n), \
u|_{(t_k,t_{k+1})} \in C^1((t_k,t_{k+1});\mathbb{R}^n) \\
&\text{for $k=0,1$, and the limits $u'(t_1^+)$,  $u'(t_1^-)$ exist} \big\}
\end{align*}
with the norm
$$
\|u\|_{PC^1} := \max \{\|u(t)\|_{PC},\|u'(t)\|_{PC}\}.
$$
For an $n \times n$ matrix $A$ with real-valued elements $a_{ij}$,
$\|A\|$ will denote the norm of matrix $A$ given by
$$
\|A\| := \Big(\sum_{j=1}^n \left[ a_{1j}^2 \right] + \dots +
\sum_{j=1}^n \left[ a_{nj}^2 \right]\Big)^{1/2}.
$$
A solution to the impulsive BVP \eqref{i1.1}--\eqref{i1.3} is a
function $x \in PC^1([0,N];\mathbb{R}^n)$ that satisfies
\eqref{i1.1}--\eqref{i1.3} for each $t\in [0,N]$.

\section{Operator Formulation} \label{sec2}

In this section  the impulsive BVP \eqref{i1.1}--\eqref{i1.3} is
reformulated as an appropriate integral equation so that potential
solutions to the integral equation will be solutions to the impulsive BVP
\eqref{i1.1}--\eqref{i1.3}.  The motivation behind this approach is to
define a suitable integral operator, with fixed--points of the operator
corresponding to solutions of the BVP \eqref{i1.1}--\eqref{i1.3}.

The following results are included to keep the paper self-contained for the
benefit of the reader.
Recall that the Heaviside function is defined as $H(s) = 0 $ if $s \leq 0$,
and $H(s) = 1 $ if $s > 0$,

\begin{lemma} \label{lem2.1}
Consider the impulsive BVP \eqref{i1.1}--\eqref{i1.3} with
$\det(A+B) \neq 0$. Let $f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n
\to \mathbb{R}^n$ both be continuous.
\begin{itemize}
\item[(i)] If $x \in PC^1([0,N];\mathbb{R}^n)$ is a solution of
 \eqref{i1.1}--\eqref{i1.3} then
\begin{equation} \label{fp0}
\begin{aligned}
x(t) &= (A+B)^{-1}\Big[\alpha - B\Big( \int_0^N f(s,x(s)) \,ds +
I_1(x(t_1)) \Big) \Big]  \\
&\quad + \int_0^tf(s,x(s)) \,ds + H(t-t_1) \cdot I_1(x(t_1)), \quad
t \in [0,N];
\end{aligned}
\end{equation}
\item[(ii)] If $x \in PC([0,N];\mathbb{R}^n)$ satisfies \eqref{fp0} then
$x \in PC^1([0,N];\mathbb{R}^n)$ and $x$ is a solution of
\eqref{i1.1}--\eqref{i1.3}.
\end{itemize}
\end{lemma}

\begin{proof}
{\bf (i)} Let $x \in PC^1([0,N];\mathbb{R}^n)$  be a solution to
\eqref{i1.1}--\eqref{i1.3}.
Integrating \eqref{i1.1} from $0$ to $t < t_1$ we have
$$
x(t) = x(0) + \int_{0}^{t} f(s,x(s)) \,ds,
$$
and from $t_1$ to $t$ with $t \in (t_1,N]$ we get
$$
x(t) =  x(t_1^+)  + \int_{t_1}^t f(s,x(s)) \,ds.
$$
A similar integration of \eqref{i1.1} from $0$ to $t_1$ shows that
$$
x(t_1^-) = x(0) + \int_{0}^{t_1} f(s,x(s)) \,ds.
$$
Hence combining the previous expressions we  have for $t \in [0,t_1]$
$$
x(t) = x(0) + \int_{0}^{t}{f(s,x(s)) \,ds} = x(0) + H(t-t_1) \cdot
I_1(x(t_1)) + \int_{0}^{t}{f(s,x(s)) \,ds},
$$
and for each $t \in (t_1,N]$
\begin{equation}
\begin{aligned}
x(t) &=  x(0) + x(t_1^+) - x(t_1^-) + \int_0^t  f(s,x(s)) \,ds   \\
 &=  x(0) + I_1(x(t_1)) + \int_0^t f(s,x(s)) \,ds.
\end{aligned}\label{jub}
\end{equation}
Letting $t=N$ in \eqref{jub} and using the boundary conditions \eqref{i1.2}
we obtain
\begin{align*}
B x(N) &=  B \Big[x(0) +  I_1(x(t_1)) + \int_0^N  f(t,x(t)) \,dt \Big] \\
&=  \alpha - Ax(0).
\end{align*}
A rearrangement in the previous expression then gives
$$
x(0) = (A+B)^{-1}\Big[\alpha - B\Big( \int_0^N f(t,x(t)) \,dt +
I_1(x(t_1)) \Big) \Big]
$$
which is substituted into \eqref{jub} and a rearrangement leads to
\eqref{fp0}.

{\bf (ii)}  Let $x \in PC([0,N];\mathbb{R}^n)$ be a solution to \eqref{fp0}.  Since
$f$ is continuous it is easy to see that $x \in PC^1([0,N];\mathbb{R}^n)$. To
verify that $x$ also satisfies the impulsive BVP \eqref{i1.1}--\eqref{i1.3}
just differentiate \eqref{fp0} to obtain \eqref{i1.1} and
also show that \eqref{i1.2} and \eqref{i1.3} hold by direct substitution.
\end{proof}


In view of Lemma \ref{lem2.1} a useful operator will now be introduced so
that fixed--points of the operator will be solutions of the impulsive BVP
\eqref{i1.1}--\eqref{i1.3}.

\begin{lemma} \label{lem2.2}
Consider the impulsive BVP \eqref{i1.1}--\eqref{i1.3} with $\det
(A+B) \neq 0$. Let $f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n
\to \mathbb{R}^n$ both be continuous.  Consider the mapping
$T:PC([0,N];\mathbb{R}^n) \to PC([0,N];\mathbb{R}^n)$  defined by
\begin{equation}
\begin{aligned}
[Tx] (t) &:= (A+B)^{-1}\Big[\alpha - B\Big( \int_0^N f(s,x(s)) \,ds +
I_1(x(t_1)) \Big) \Big]   \\
&\quad + \int_0^t f(s,x(s)) \,ds + H(t-t_1) \cdot I_1(x(t_1)), \quad t \in [0,N].
\end{aligned}\label{TB}
\end{equation}
If $T$ has a fixed--point $q$, that is $Tq = q$ for some $q \in
PC([0,N];\mathbb{R}^n)$, then this fixed--point $q$ is also a solution to the
impulsive BVP \eqref{i1.1}--\eqref{i1.3}.
\end{lemma}

  The above lemma  follows from Lemma \ref{lem2.1}.

The topologically--inspired fixed point theorems that will be used to
guarantee the existence of at least one fixed--point of $T$ requires that
$T$ be a ``compact'' map \cite[pp.54-55]{Lloyd}.

Recall that a mapping between normed spaces is compact if it is continuous
and carries bounded sets into relatively compact sets.

\begin{lemma} \label{lem2.3}
Consider \eqref{TB} with $\det (A+B) \neq 0$. Let $f:[0,N] \times
\mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \to \mathbb{R}^n$ both be continuous.  Then
$T:PC([0,N];\mathbb{R}^n) \to PC([0,N];\mathbb{R}^n)$ is a compact map.
\end{lemma}

\begin{proof} This follows in a standard step--by--step process and
so is omitted. \end{proof}


 The following two well--known fixed--point theorems will be of use in the
sections to follow. In particular, the Nonlinear Alternative
\cite[Theorem 5.1, p.61]{DG} and  Schaefer's Theorem \cite[Theorem 4.4.12]{Lloyd} will be
employed.


\begin{theorem}[Nonlinear Alternative]
Let $X$ be a normed space with $C$ a convex subset of $X$.  Let $U$ be an
open subset of $C$ with $0 \in C$ and consider a compact map $H:{\overline
U} \to C$.  If
$$
u \neq \lambda Hu \quad \text{for all $ u \in \partial U$
 and for all $\lambda \in [0,1]$}
$$
then $H$ has at least one fixed--point.
\end{theorem}



\begin{theorem}[Schaefer]
Let $X$ be a normed space with $H:X \to X$ a compact mapping.  If
the set
$$
S := \{ u \in X: \ u = \lambda Hu  \text{ for some } \lambda \in
[0,1)\}
$$
is bounded then $H$ has at least one fixed--point.
\end{theorem}

\section{Existence: Homogeneous Case} \label{sec3}

This section presents some new existence results for solutions to the
following ``homogenous'' problem ($\alpha = 0$)
\begin{gather}
x' = f(t,x),\quad  t \in [0,N], \; t \neq t_1;\label{h1} \\
Ax(0) + Bx(N) = 0,\quad  0 < N \in \mathbb{R};  \label{ih2} \\
x(t_1^+) = x(t_1^-) + I_1(x(t_1)),\quad  t_1 \in (0,1), \; t_1
\text{ fixed}. \label{h3}
\end{gather}

 The ideas use novel differential inequalities in the impulsive equation
setting  and standard fixed--point methods of integral operators.  In
particular, the Nonlinear Alternative \cite[Theorem 5.1, p.61]{DG} and
Schaefer's Theorem \cite[Theorem 4.4.12]{Lloyd} will be employed.


 The following existence result involves sublinear growth of $\|f(t,p)\|$
in $\|p\|$.

\begin{theorem} \label{thm3.1}
Consider the impulsive BVP \eqref{h1}--\eqref{h3} with $f:[0,N]
\times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \to \mathbb{R}^n$  both being
continuous and $\det (A + B) \neq 0$.  Let $\rho$ and $\sigma$ be
non--negative constants and let $\psi:[0,\infty) \to (0,\infty)$ be
a continuous, non--decreasing function such that
\begin{gather}
\|f(t,p)\| \le \rho \psi(\|p\|),\quad \text{for all }
(t,p) \in [0,N] \setminus \{t_1\}  \times \mathbb{R}^n; \label{WOO} \\
\|I_1(q)\| \le \sigma \|q\|,\quad  \text{for all }  q \in \mathbb{R}^n; \label{ERR} \\
\sup_{c \ge 0} \frac{c}{\psi(c)} > K_1 := \frac{\rho N
\big(\|(A+B)^{-1}B\| + 1\big) }{1 - \sigma\big(\|(A+B)^{-1}B\| +
1\big)};  \label{SANN} \\
\sigma \big(\|(A+B)^{-1}B\| + 1\big) < 1;  \label{SII}
\end{gather}
 then the impulsive BVP \eqref{h1}--\eqref{h3} has at least one solution.
\end{theorem}

\begin{proof} We will use the Nonlinear Alternative.  From
\eqref{SANN} there exists a constant $Q > 0$ such that
\begin{equation} \label{Kay}
\frac{Q}{\psi(Q)} > K_1.
\end{equation}
Consider the mapping $T_1: PC([0,N];\mathbb{R}^n) \to PC([0,N];\mathbb{R}^n)$
\begin{align*}
T_1x(t) &:= (A+B)^{-1}\Big[- B\Big( \int_0^N f(s,x(s)) \,ds + I_1(x(t_1))
\Big) \Big]   \\
&\quad + \int_0^t f(s,x(s)) \,ds + H(t-t_1) \cdot I_1(x(t_1)), \quad
t \in [0,N].
\end{align*}
By Lemma \ref{lem2.3}, $T_1$ is a compact mapping.
Let
$$
{\bar \Omega} := \{x \in PC([0,N];\mathbb{R}^n): \|x\|_{PC} < Q\}.
$$
We consider $T_1: {\bar \Omega} \to PC([0,N];\mathbb{R}^n)$ and the family
of problems
\begin{equation} \label{l1}
x = \lambda T_1x, \quad \lambda \in [0,1].
\end{equation}
Let $x$ be a solution to \eqref{l1} with $x \in {\bar \Omega}$.  We show
that $x \not\in \partial \Omega$.
 From \eqref{l1} and \eqref{TB} we have, for each $t \in [0,N]$,
\begin{align*}
\|x(t)\| &=  \|\lambda T_1x(t)\| \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \int_0^N\|f(t,x(t))\| dt +
\|I_1(x(t_1))\| \Big] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \int_0^N \rho \psi(\|x(t)\|)
dt + \sigma \|x(t_1)\| \Big] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \rho N \psi(\sup_{t \in
[0,N]}\|x(t)\|) + \sigma \sup_{t \in [0,N]} \|x(t)\| \Big].
\end{align*}
Hence we have
$$
\sup_{t \in [0,N]} \|x(t)\| \le \left( 1 + \|(A+B)^{-1}B\| \right)\Big[
\rho N \psi (\sup_{t \in [0,N]} \|x(t)\|) + \sigma \sup_{t \in [0,N]}
\|x(t)\|\Big]
$$
so that a rearrangement in the previous line gives
$$
\sup_{t \in [0,N]} \|x(t)\| \le K_1 \psi (\sup_{t \in [0,N]} \|x(t)\|)
$$
where $K_1$ is defined in \eqref{SANN}.  Hence, by \eqref{Kay} we must have
$\sup_{t \in [0,N]} \|x(t)\| \neq Q$, that is
$\|x\|_{PC} \neq Q$.
The Nonlinear Alternative is applicable and thus the existence of at least
one solution follows.
\end{proof}

The following result allows $\|f(t,p)\|$ to grow more than linearly
in $\|p\|$.

\begin{theorem} \label{npthm3.2}
Consider the impulsive BVP \eqref{h1}--\eqref{h3} with $f:[0,N]
\times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \to \mathbb{R}^n$  both being
continuous and $\det (A+B) \neq 0$.  If there exist non--negative
constants $a$, $b$, $\beta$, $L$ such that:
\begin{gather}
\|f(t,p)\| \le 2 a \langle p,f(t,p)\rangle + b,\quad
\text{for all } (t,p) \in [0,N] \setminus \{t_1\}  \times \mathbb{R}^n; \label{WO} \\
\|I_1(q)\| \le \beta \|q\| + L,\quad  \text{for all } q \in \mathbb{R}^n; \label{ER} \\
\|B^{-1}A\| \le 1;  \label{SAN} \\
\beta(\|(A+B)^{-1}B\| + 1)< 1; \label{SHI}
\end{gather}
 then the impulsive BVP \eqref{h1}--\eqref{h3} has at least one solution.
\end{theorem}


\begin{proof} Consider the mapping $T_1: PC([0,N];\mathbb{R}^n) \to
PC([0,N];\mathbb{R}^n)$
\begin{align*}
T_1x(t) &:= (A+B)^{-1}\Big[- B\Big( \int_0^N f(s,x(s)) \,ds +  I_1(x(t_1))
\Big) \Big]   \\
&\quad + \int_0^t f(s,x(s)) \,ds + H(t-t_1) \cdot I_1(x(t_1)), \quad
t \in [0,N].
\end{align*}
By Lemma \ref{lem2.3}, $T_1$ is a compact mapping.
Consider the equation
\begin{equation} \label{T}
x = T_1x.
\end{equation}
To show that $T_1$ has at least one fixed point, we apply
Schaefer's Theorem by showing that all potential solutions to
\begin{equation} \label{fam}
x = \lambda T_1x, \quad \lambda \in [0,1];
\end{equation}
are bounded {\em a priori}, with the bound being independent of $\lambda$.
With this in mind, let $x$ be a solution to \eqref{fam}.  Note that $x$ is
also a solution to
 \begin{gather*}
x' =  \lambda f(t,x),\quad t \in [0,N], \quad t \neq t_1;\label{fm1.1} \\
A x(0) + Bx(N) =  0;  \label{fm1.2} \\
x(t_1^+) =  x(t_1^-) + \lambda I_1(x(t_1)). \label{fm1.3}
\end{gather*}
Note that \eqref{SAN} and \eqref{i1.2} imply
$$
\|x(N)\| \le \|B^{-1}Ax(0)\| \le \|B^{-1}A\| \|x(0)\| \le \|x(0)\|.
$$
We also  have, for each $t\in[0,N]$,
\begin{align*}
\|x(t)\| &= \lambda \|Tx(t)\| \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \int_0^N\|\lambda
f(t,x(t))\| dt + \|\lambda I_1(x(t_1))\| \Big] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \int_0^N 2 a \langle x(s),
\lambda f(s,x(s)) \rangle + \lambda b \,ds + \beta \|x(t_1)\| + L \Big] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \int_0^N 2 a \langle
x(s),x'(s) \rangle + b \,ds + \beta \|x\|_{PC} + L \Big] \\
&=   \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ \int_0^N  a \frac{d}{ds}
\left(\|x(s)\|^2\right) + b \,ds + \beta \|x\|_{PC} + L \Big] \\
&=   \left(1 + \|(A+B)^{-1}B\| \right) [a (\|x(N)\|^2 - \|x(0)\|^2) +
bN + \beta \|x\|_{PC} + L ] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)[bN + \beta \|x\|_{PC} + L] \\
\end{align*}
Thus, taking the supremum above and rearranging we obtain
$$
\|x\|_{PC} = \sup_{t \in [0,N]} \|x(t)\| \le \frac{[bN+L][1 +
\|(A+B)^{-1}B\|]}{1 - (1 + \|(A+B)^{-1}B\|)\beta}.
$$
Thus we see that the bound on all possible solutions to \eqref{fam} is
independent of $\lambda$ and Schaefer's Theorem applies, yielding the
existence of at least one fixed--point  to $T_1$ and thus
\eqref{h1}--\eqref{h3} has at least one solution.
\end{proof}

Theorem \ref{npthm3.2} may be suitably modified to include an alternate
class of $f$ as follows.

\begin{theorem} \label{npthm3.3}
Consider the impulsive BVP \eqref{i1.1}--\eqref{i1.3} with
$f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \to \mathbb{R}^n$  both being
continuous.  Let the conditions of Theorem \ref{npthm3.2} hold with
\eqref{WO} and \eqref{SAN} respectively replaced by
\begin{gather}
\|f(t,p)\| \le -2 a \langle p,f(t,p)\rangle + b,\quad \text{for all }
 (t,p)
\in [0,N] \setminus \{t_1\}  \times \mathbb{R}^n. \label{WO2} \\
\|A^{-1}B\| \le 1;
\end{gather}
 Then the impulsive BVP \eqref{i1.1}--\eqref{i1.3} has at least one solution.
\end{theorem}

\begin{proof}  The proof is a minor variation to that of Theorem
\ref{npthm3.2} and so is not discussed.
\end{proof}


Although the proofs of Theorems \ref{npthm3.2} and
\ref{npthm3.3} are similar, the two results differ in sense
that Theorem \ref{npthm3.2} may apply to certain problems,
whereas Theorem \ref{npthm3.3} may not apply, and vice--versa.
For example, in the scalar case,
$$
f(t,p) := -p^3 - t, \quad t \in [0,1];
$$
satisfies \eqref{WO2} for the choices $a = 1/2$ and $b= 100$, but
the above $f$ cannot satisfy \eqref{WO} for any choice of non--negative
$a$ and $b$.

\section{Existence and Uniqueness: Inhomogeneous Case} \label{sec4}

This section presents  existence and uniqueness results for solutions to
the general impulsive BVP \eqref{i1.1}--\eqref{i1.3} where $\alpha$ may be
non--zero.

The following general existence result allows linear growth of $\|f(t,p)\|$
in $\|p\|$.


\begin{theorem} \label{inthm1}
Consider the impulsive BVP \eqref{i1.1}--\eqref{i1.3} with
$f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \to \mathbb{R}^n$  both being
continuous and $\det (A + B) \neq 0$.  Let $u,v,w,z$ be
non--negative constants such that
\begin{gather}
\|f(t,p)\| \le u\|p\| + v,\quad \text{for all }  (t,p)  \in [0,N]
\setminus \{t_1\}  \times \mathbb{R}^n; \label{WOOO} \\
\|I_1(q)\| \le w\|q\| + z,\quad \text{for all }  q \in \mathbb{R}^n; \label{ERRR} \\
(\|(A+B)^{-1}B\| + 1)[Nu+w] < 1.  \label{SANNN}
\end{gather}
 Then the impulsive BVP \eqref{i1.1}--\eqref{i1.3} has at least one solution.
\end{theorem}

\begin{proof} We  use Schaefer's Theorem.
Consider the mapping $T: PC([0,N];\mathbb{R}^n) \to PC([0,N];\mathbb{R}^n)$,
\begin{align*}
Tx(t) &:= (A+B)^{-1}\Big[\alpha - B\left( \int_0^N f(s,x(s)) \,ds +
I_1(x(t_1)) \right) \Big]   \\
&\quad + \int_0^t f(s,x(s)) \,ds + H(t-t_1) \cdot I_1(x(t_1)), \quad
t \in [0,N].
\end{align*}
By Lemma \ref{lem2.3}, $T$ is a compact mapping.
Consider the equation
\begin{equation} \label{T2}
x = Tx.
\end{equation}
In order to show that $T$ has at least one fixed point, we apply Schaefer's
Theorem by showing that all potential solutions to
\begin{equation} \label{fam2}
x = \lambda Tx, \quad \lambda \in [0,1];
\end{equation}
are bounded {\em a priori}, with the bound being independent of $\lambda$.
With this in mind, let $x$ be a solution to \eqref{fam2}.  Note that $x$ is
also a solution to
 \begin{gather}
x' =  \lambda f(t,x),\quad t \in [0,N], \quad t \neq t_1,\label{2fm1.1} \\
A x(0) + Bx(N) =  \lambda \alpha;  \label{2fm1.2} \\
x(t_1^+) =  x(t_1^-) + \lambda I_1(x(t_1)). \label{2fm1.3}
\end{gather}
We then have, for each $t\in[0,N]$,
\begin{align*}
&\|x(t)\|\\
 &= \| \lambda Tx(t)\| \\
&\leq  \|(A+B)^{-1}\alpha\| + \left( 1 + \|(A+B)^{-1}B\| \right)\Big[
\int_0^N\|\lambda f(t,x(t))\| \,dt + \|\lambda I_1(x(t_1))\| \Big] \\
&\leq  \|(A+B)^{-1}\alpha\| +  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[
\int_0^N u\|x(t)\| + v \,dt + w \|x(t_1)\| + z \Big] \\
&\leq \|(A+B)^{-1}\alpha\| +  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[ N (
u\|x\|_{PC} + v ) + w \|x\|_{PC} + z \Big] \\
\end{align*}
Thus, taking the supremum and rearranging we obtain
$$
\|x\|_{PC} = \sup_{t \in [0,N]} \|x(t)\| \le \frac{\|(A+B)^{-1}\alpha\|
+ [1 + \|(A+B)^{-1}B\|](Nv + z)}{1 - (\|1 + (A+B)^{-1}B\|)[Nu + w]}.
$$
Thus we see that the bound on all possible solutions to \eqref{fam} is
independent of $\lambda$ and Schaefer's Theorem applies, yielding the
existence of at least one fixed--point  to $T$ and thus
\eqref{i1.1}--\eqref{i1.3} has at least one solution.
\end{proof}

The following uniqueness result for solutions \eqref{i1.1}--\eqref{i1.3}
is now obtained with the help of Theorem \ref{inthm1}.


\begin{theorem} \label{inthm2}
Consider the impulsive BVP \eqref{i1.1}--\eqref{i1.3} with
$f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \to \mathbb{R}^n$  both being
continuous and $\det (A + B) \neq 0$.  Let $u_1$ and $w_1$ be
non--negative constants such that
\begin{gather*}
\|f(t,p) -f(t,q)\| \le u_1\|p-q\|,\quad \text{for all }  (t,p,q)
\in [0,N] \setminus \{t_1\}  \times \mathbb{R}^{2n}; \label{WOOOO} \\
\|I_1(p) - I_1(q)\| \le w_1\|p-q\|,\quad \text{for all }  (p,q) \in
\mathbb{R}^{2n}; \label{ERRRR} \\
(\|(A+B)^{-1}B\| + 1)[Nu_1+w_1] < 1.  \label{SANNNN}
\end{gather*}
 Then the impulsive BVP \eqref{i1.1}--\eqref{i1.3} has a unique solution.
\end{theorem}

\begin{proof}
The conditions of the theorem imply that
\begin{gather*}
\|f(t,p) -f(t,0)\| \le u_1\|p-0\|,\quad \text{for all }  (t,p)  \in
[0,N] \setminus \{t_1\}  \times \mathbb{R}^{n}; \label{WOOOOO} \\
\|I_1(p) - I_1(0)\| \le w_1\|p-0\|,\quad  \text{for all }  p \in
\mathbb{R}^{n}. \label{ERRRRR} \\
\end{gather*}
A rearrangement of the above two inequalities leads to \eqref{WOOO} and
\eqref{ERRR} for
$$
v = \sup_{t \in [0,N]} \|f(t,0)\|,
$$
$u=u_1$, $w = w_1$ and $z = \|I_1(0)\|$.  Thus, since \eqref{SANNN} holds,
all of the conditions of Theorem \ref{inthm1} are satisfied and the
existence of at least one solution to  \eqref{i1.1}--\eqref{i1.3} follows.

Now let $x$ and $y$ be two solutions to  \eqref{i1.1}--\eqref{i1.3}.  We
have, for each $t \in [0,N]$
\begin{align*}
&\|x(t) - y(t)\| \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right)\Big[\int_0^N \|f(t,x(t) -
f(t,y(t))\| dt + \|I_1(x(t_1)) - I_1(y(t_1))\| \Big] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right) \Big[u_1\int_0^N \|x(t) - y(t))\|
dt + w_1\|x(t_1) - y(t_1)\| \Big] \\
&\leq  \left( 1 + \|(A+B)^{-1}B\| \right) [u_1N \|x-y\|_{PC}  + w_1\|x
- y\|_{PC} ].
\end{align*}
Hence rearranging and taking the supremum above we have
$$
\left[ 1 -  [1 + \|(A+B)^{-1}B\|][Nu_1 + w_1] \right] \|x-y\|_{PC} \le 0
$$
and \eqref{SANNNN} ensures $x = y$.  Thus, the solutions are unique.
\end{proof}

The following corollary is a special case of Theorem \ref{inthm1} and
involves global bounds on the functions $f$ and $I_1$.

\begin{corollary}
Consider the impulsive BVP \eqref{i1.1}--\eqref{i1.3} with
$f:[0,N] \times \mathbb{R}^n \to \mathbb{R}^n$ and $I_1:\mathbb{R}^n \times \mathbb{R}^n$  both
being continuous and $\det (A + B) \neq 0$.  Let $v$ and $z$ be
non--negative constants such that
\begin{gather*}
\|f(t,p)\| \le v,\quad \text{for all }  (t,p)  \in [0,N] \setminus
\{t_1\} \times \mathbb{R}^{n};  \\
\|I_1(q)\| \le z,\quad \text{for all }  q \in \mathbb{R}^{n}.
\end{gather*}
 Then the impulsive BVP \eqref{i1.1}--\eqref{i1.3} has at least one solution.
\end{corollary}

\begin{proof}
The proof involves taking $u=0 = w$ so that all of the conditions of
Theorem \ref{inthm1} are satisfied.
\end{proof}

\section{Examples} \label{sec5}
In this section some examples are presented to highlight the theory.
We firstly consider the following scalar--valued differential equation case.

\begin{example} \label{eg2.5} \rm
Consider the impulsive BVP given by
\begin{gather}
x' =  x^3 + x + t, \quad t \in [0,1], \; t \neq t_1; \label{2.10} \\
x(0) =  2x(1);  \label{2.11} \\
x(t_1^+) =  x(t_1^-) + x(t_1)/2 \label{2.12}
\end{gather}
where $x$ is scalar-valued ($n=1$). The above impulsive BVP has at least
one solution.
\end{example}

\begin{proof}
Let $f(t,p) = p^3 + p + t$ and see that $|f(t,p)| \le
|p^3| + |p| + 1$ for $(t,p) \in [0,1] \times \mathbb{R}$.
For $a$ and $b$ to be chosen below, see that
\begin{align*}
 2 a pf(t,p) + b
&=   2 a(p^4+p^2 +pt) + b \\
&=  (p^4 + 1) + [p^2 +pt + 40.25], \quad \text{for the choices }
a = 1/2, b = 41.25 \\
&=   (p^4 + 1) + [ (p+t/2)^2 + 40.25 - t^2/4] \\
&\geq  (|p^3|) + [|p| + 1] \\
&\geq  |f(t,p)|  \quad \text{for all }   (t,p) \in [0,1] \times \mathbb{R}
\end{align*}
and thus \eqref{WO} holds. It is easy to see that \eqref{ER}, \eqref{SAN}
and \eqref{SHI} hold for $\beta = 1/2$, $L=0$, $N=1$, $A=1$, $B=-2$. Thus,
all of the conditions of Theorem \ref{npthm3.2} hold and the solvability
follows.  The
theorems of \cite{LSN, Skora, Skora2}, for example, do not apply to the
above because of the wider class of boundary conditions.
\end{proof}

We now consider an example involving a system of differential equations.


\begin{example} \label{eg2.5.3} \rm
Consider \eqref{i1.1}--\eqref{i1.3} with $n=2$ and $f$ given by
\begin{equation}
\begin{aligned}
f(t,p) &=  (h(t,y,z), j(t,y,z)), \quad t \in [0,1],   \\
&:= ((t+1)y^3 + ye^{-z^2} + 1, (t+1)z^3 + ze^{-y^2}) .
\end{aligned} \label{sys}
\end{equation}
and
$$ (y(0),z(0)) - 2 (y(1),z(1)) = (0,0)
$$
with
$$
(y(t_1^+),z(t_1^+)) = (y(t_1^-),z(t_1^-))  + (y(t_1)/2,z(t_1)/2).
$$
The above impulsive BVP has at least
one solution.
\end{example}

\begin{proof}
We show that above $f$ satisfies the conditions of Theorem \ref{npthm3.2}.
Note that for all $(t,p) \in [0,1]\times \mathbb{R}^2$ we have
\begin{align*}
\|f(t,p)\|
&\leq  |h(t,y,z)| +  |j(t,y,z)| \\
&\leq  2|y|^3 + |y|e^{-z^2} + 2|z|^3 + |z|e^{-y^2} +1.
\end{align*}
Below, we will need the following simple inequalities:
\begin{gather*}
w^4 \ge |w|^3 - 1, \quad w^4 + w \ge |w|^3 -10, \quad \text{for all }
 w \in \mathbb{R}, \\
d^2e^{-c^2} \ge |d|e^{-c^2} -1,  \quad \text{for all }  (c,d) \in \mathbb{R}^2.
\end{gather*}
For $a \ge 0$ and $b \ge 0$ to be chosen below, consider for
$(t,p) \in [0,1] \times \mathbb{R}^2$,
\begin{align*}
2a \langle p,f(t,p) \rangle + b
&\geq  2 a \left[ y^4 + y + y^2e^{-z^2} + z^4 + z^2e^{-y^2} \right] + b \\
&\geq  2 a \left[ |y|^3 -10 + |y|e^{-z^2} - 1 + |z|^3 -1 + |z|e^{-y^2} - 1 \right] + b \\
&\geq  2|y|^3 + |y|e^{-z^2} + 2|z|^3 + |z|e^{-y^2} + 1, \quad \text{for} \ a = 1, \ b = 27  \\
&\geq  \|f(t,p)\|
\end{align*}
Thus $f$ satisfies the conditions of Theorem \ref{npthm3.2} for the
choices $a = 1$ and $b=27$.

It is not difficult to verify that the remaining conditions of
Theorem \ref{npthm3.2} hold with $\beta = 1/2$, $N=1$ and $L=0$.
Thus we conclude that our problem has at least one solution.
\end{proof}

The theorems in
\cite{CTY, FN00, FNp2,LNS, LSN, N02NA, N02JMAA, N97, QL, Skora, Skora2}
do not directly apply to the previous example as: the
growth of $\|f(t,p)\|$ in $\|p\|$ is super--linear; the boundary conditions
are a wider range or a different class;  and the problem involves a system
of  impulsive BVPs.

\subsection*{Acknowledgement}
C.C. Tisdell was supported by grant DP0450752 from The Australian
Research Council's Discovery Projects.
 J.J. Nieto was supported by grants MTM2004-06652-C03-01
from  the Ministerio de Educaci\'{o}n y Ciencia and FEDER,
and  PGIDIT05PXIC20702PN from  Xunta de Galicia
and FEDER.

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