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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 121, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/121\hfil Delay differential equations]
{Existence, uniqueness and constructive results for
 delay differential equations}

\author[P. W. Eloe, Y. N. Raffoul, C. C. Tisdell\hfil EJDE-2005/121\hfilneg]
{Paul W. Eloe, Youssef N. Raffoul, Christopher C. Tisdell} % in alphabetical order

\address{Paul W. Eloe \hfill\break
Department of Mathematics\\
University of Dayton\\
Dayton, OH, USA}
\email{paul.eloe@notes.udayton.edu}

\address{Youssef N. Raffoul \hfill\break
Department of Mathematics\\
University of Dayton\\
Dayton, OH, USA}
\email{youssef.raffoul@notes.udayton.edu}

\address{Christopher C. Tisdell \hfill\break
School of Mathematics\\
The University of New South Wales\\
Sydney NSW 2052, Australia}
\email{cct@maths.unsw.edu.au}

\date{}
\thanks{Submitted July 21, 2005. Published October 27, 2005.}
\subjclass[2000]{34K10, 34K07}
\keywords{Delay differential equation; boundary value problem;
\hfill\break\indent
 existence of solutions; A-solvable; uniqueness of solutions}

\begin{abstract}
 Here, we investigate boundary-value problems (BVPs) for systems
 of second-order, ordinary, delay-differential equations.
 We introduce some  differential inequalities such that all solutions
 (and their derivatives) to a certain family of BVPs satisfy some
 a priori bounds.  The results are then applied, in conjunction
 with topological arguments, to prove the existence of solutions.
 We then apply earlier abstract theory of Petryshyn to formulate
 some constructive results under which solutions to BVPs for systems
 of second-order, ordinary, delay-differential equations
 are A-solvable and may be approximated via a Galerkin method.
 Finally, we provide some differential inequalities such that
 solutions to our equations are unique.
\end{abstract}

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

\section{Introduction}

This paper considers the so-called system of delay-differential
equations
\begin{equation}\label{e1.1}
x''(t)=f(t, x(t), x(h(t)), x'(t)), \quad t \in [0,T],
\end{equation}
subject to the boundary conditions
\begin{gather}
x(t) = \phi(t), \quad t \in [-V,0], \label{e1.2}\\
x(T) = B, \label{e1.3}
\end{gather}
where $T > 0$, $f:[0,T]  \times \mathbb{R}^{3d} \to
\mathbb{R}^{d}$, $V \geq 0$, $h:  [0,T]\to [-V,T]$,
$t-V\le h(t) \le t$, $\phi : [-V,0]\to \mathbb{R}^{d}$
 and $B \in \mathbb{R}^{d}$. We call \eqref{e1.1}--\eqref{e1.3} a
boundary-value problem (BVP) for delay-differential equations. The
given functions $f, h $ and $\phi$ are continuous and we use
(\ref{e1.2}) to solve (\ref{e1.1}) forward in time. A solution
$x= x(t)$ to \eqref{e1.1}--\eqref{e1.3} is a function
$x:[-V, T]\to \mathbb{R}^{d}$ satisfying  (\ref{e1.1}) for all
$t \in [0,T]$, (\ref{e1.2}) for all $t \in [-V,0]$ and (\ref{e1.3})
for $t = T$ with
$$
x \in S := C\big([-V,T]; \mathbb{R}^{d}\big)\cap C^2
\big([0,T]; \mathbb{R}^{d}\big).
$$
If $p$ is a vector then $\|p\|$ denotes the Euclidean norm of $p$ and
$\langle \cdot, \cdot \rangle$  denotes the usual scalar product.

If $\mathcal{I}$ is an interval in $\mathbb{R}$, then we define the
notation $\|x\|_\mathcal{I}$ by
$$
\|x \|_\mathcal{I} := \sup_{t \in \mathcal{I}} \|x(t)\|.
$$
Finally, we define the special norm
$$
\|x\|_1 := \max \{ \|x\|_{[-V,T]}, \|x'\|_{[0,T]} \}.
$$

BVPs for delay-differential equations arise, for example, in
control theory where variational problems are complicated by
the effect of time delays in signal transmission and the lead to a
BVP of the type \eqref{e1.1}--\eqref{e1.3}
(see \cite{S} and references therein).

In the literature, existence, uniqueness and numerical results
concerning solutions to BVPs for delay-differential equations
appear in \cite{LO, TN, GS68, GS70, S, R, C, NS} and
references therein.

The existence results in \cite{LO} and \cite{TN} use topological
transversality involving a priori bounds on solutions.  The a
priori bound theory involves a Lyapunov-type function, maximum
principles and Nagumo-type conditions.  The existence results in
\cite{GS68, S, GS70} employ the Schauder-Tychonoff fixed point
theorem involving a modification method and a priori bounds on
solutions. The a priori bound theory involves upper and lower
solutions and Nagumo-type conditions.

The uniqueness results of \cite{GS70} involve Lipschitz
conditions.
The numerical results in \cite{R, C, NS} employ the finite
difference method and the shooting method.

In comparison with the above works, the contribution that this
paper makes to the field of delay-differential equations is
two-fold:  The methods used herein are new in the
delay-differential equation setting (adapted from \cite{Hale});
and the results contained herein are quite general.

For example, concerning the existence of solutions, our methods
for a priori bounds on solutions do not involve maximum principles,
rather they rely on new inequalities in the delay-differential
 setting of the type
$$
\|f(t,x,x_h,p)\| \le 2 \alpha [ \langle x,f(t,x,x_h,p \rangle + \|p\|^2] +
K.
$$
In addition, concerning uniqueness of solutions, our methods
employ maximum principles, rather than
Lipschitz conditions.

Furthermore, our new constructive methods involve A-proper
mappings and the Galerkin method applied to delay-differential
equations, rather than a finite difference or shooting method.

Additionally, our results apply to systems of equations and
therefore are quite general.  In fact, it appears that even in the
scalar case the results contained herein are new (excluding
Theorem \ref{thm1}).

For more on the theory of delay-differential equations we refer
the reader to \cite{Hale}.


\section{A General Existence Theorem} \label{sec2}

This section contains the general existence theorem that we will
rely on throughout the remainder of the paper. The theorem can be
found in papers such as \cite{LO}, \cite{TN}.

Consider the family of delay-differential equations
\begin{equation}\label{e2.1}
x''(t)= \lambda f(t, x(t), x(h(t)), x'(t)), \quad  t \in [0,T],
\end{equation}
subject to the family of boundary conditions
\begin{gather}
x(t) = \lambda \phi(t), \quad  t \in [-V,0], \label{e2.2}\\
x(T) = \lambda B, \label{e2.3}
\end{gather}
where $\lambda \in [0,1]$.

\begin{theorem} \label{thm1}
Let $f, h$ and $\phi$ be continuous
functions. If all possible solutions $x$ of
\eqref{e2.1}--\eqref{e2.3} satisfy $\|x\|_1 < R$ for some positive
constant $R$ with $R$ independent of $\lambda $, then for each
$\lambda\in [0,1]$, the BVP \eqref{e2.1}--\eqref{e2.3} has a solution
$x$.  In particular, the BVP \eqref{e1.1}-\eqref{e1.3}
has a solution, for $\lambda = 1$.
\end{theorem}

\begin{proof}
We provide a proof for two reasons: First, for the convenience of
the reader; and second, to introduce some concepts that will be
used later in the paper. It can be easily checked that the BVP
\eqref{e2.1}--\eqref{e2.3} is equivalent to the problem of finding
$x\in S$ such that
\begin{equation*} x(t) =
\begin{cases}
\int^T_0\big[-G(t,s) \lambda f(s, x(s), x(h(s)),
x'(s))\big]ds
 + \lambda k(t), & t\in [0,T],\\
\lambda \phi(t), & t\in [-V,0],
\end{cases}
\end{equation*}
where $\lambda \in [0,1]$,
\begin{equation*}
G(t,s) :=\begin{cases}
\frac{(T-t)s}{T}, & 0 \leq s \leq t \leq T, \\
\frac{t(T-s)}{T}, & 0 \leq t \leq s \leq T,
\end{cases}
\end{equation*}
and
$$
k(t):= \frac {T \phi(0) + (B - \phi(0))t}{T}, \quad t\in [0,T].
$$
Define an operator
$J:C([-V,T];\mathbb{R}^d) \cap C^1([0,T];\mathbb{R}^d) \to C\big([-V,T];
 \mathbb{R}^{d}\big)$ by
\begin{equation*}
(Jx)(t) := \begin{cases}
\int^T_0\big[-G(t,s)f(s, x(s), x(h(s)), x'(s))\big]ds
 +  k(t),  & t\in [0,T],\\
  \phi(t), & t\in [-V,0],
\end{cases}
\end{equation*}
and define a set
$$
\Omega:= \Big\{x \in C([-V,T];\mathbb{R}^d) \cap C^1([0,T];\mathbb{R}^d) :
\; \|x\|_1 < R\Big\}.
$$
Since $h, f$ and $\phi$ are continuous, we see that for each
$\lambda\in [0,1]$, $J$ is continuous and completely
continuous (by Arzela-Ascoli Theorem).
Thus, $J:\overline{\Omega}\to C\big([-V,0]; \mathbb{R}^{d})$ is
a compact map since $J$ is restricted to the closure of a
bounded, open set $\Omega$.

Consider the family of problems
$$
(I - \lambda J)(x) = 0,\quad \lambda \in [0,1],
$$
 which is equivalent to \eqref{e2.1}--\eqref{e2.3}. Since $J$ is compact and
$\|x\|_1 < R$ (so $x \notin \partial \Omega $) with $R$ independent of $\lambda$, the
following Leray-Schauder degrees are defined and a homotopy
principle applies
 \begin{align*}
d(I - \lambda J), \Omega, 0)
&= d(I - J, \Omega, 0)\\
&= d(I, \Omega, 0)\\
&= 1\neq  0,
\end{align*}
since $0 \in\Omega$. Therefore,
$$
(I - \lambda J)(x) = 0
$$
has a solution in $x \in C([-V,T];\mathbb{R}^d) \cap C^1([0,T];\mathbb{R}^d)$
for each $\lambda \in[0,1]$. By elementary methods the solution is
 also in $C^2([0,T])$. This
concludes the proof.
\end{proof}


\section{On the equation $x''(t)=f(t, x(t), x(h(t)))$}

Theorem \ref{thm1} shows that bounds on solutions to families of BVPs for
delay-differential equations play an important role in developing
existence results. This section introduces some new inequalities
for delay-differential equations such that all solutions $x$ to
the family of equations
\begin{equation}\label{e3.1}
x''(t)= \lambda f(t, x(t), x(h(t))), \quad  t \in
[0,T],
\end{equation}
subject to the family of boundary conditions
\begin{gather}
x(t) = \lambda \phi(t), \quad t \in [-V,0], \label{e3.2}\\
x(T) = \lambda B, \label{e3.3}
\end{gather}
where $\lambda \in [0,1]$, satisfy $\|x\|_1 < R$ with $R$
independent of $\lambda$. The result is then applied, in
conjunction with Theorem \ref{thm1}, to give some existence results for
the system of equations
\begin{equation}\label{e3.4}
x''(t)= f(t, x(t), x(h(t))), \quad  t \in [0,T],
\end{equation}
subject to the boundary conditions
\begin{gather}
x(t) = \phi(t), \quad  t \in [-V,0], \label{e3.5}\\
x(T) = B. \label{e3.6}
\end{gather}
Since (\ref{e3.1}) is independent of $x'$, once a priori bounds on $x$
are obtained then a priori bounds
on $x'$ naturally follow, with these bounds also independent of $\lambda$.
Define
$$
\beta : = \max\big\{\|\phi(0)\|, \|B\| \big\}.
$$
\begin{theorem} \label{thm2}
Let $f,h$ and $\phi$ be continuous.  Assume
there exist scalar constants $\alpha \geq 0$, $K \geq 0$, such
that
\begin{equation}\label{e3.7} \|f(t, w, y)\| \leq 2 \alpha
\langle w, f(t, w, y) \rangle + K,\quad  \mbox{for all } t \in [0,T],
\; (w,y) \in \mathbb{R}^{2d}.
\end{equation}
Then all solutions $x$ of \eqref{e3.1}--\eqref{e3.3} satisfy
$$
\|x\|_{[-V,T]} \leq \max\big \{\alpha \beta^2 + \beta
+ \frac{KT^2}{8}, \|\phi\|_{[-V,0]}\big \} := M.
$$
\end{theorem}

\begin{proof}
Let $0\le \lambda\le 1$.  See that if (\ref{e3.7}) holds, then multiplying
both sides by $\lambda$, we obtain
\begin{equation}
\| \lambda f(t, w, y)\| \leq  2 \alpha \langle w, \lambda f(t, w, y) \rangle
+ \lambda K 
\leq 2 \alpha \langle w, \lambda f(t, w, y) \rangle + K. \label{e3.75}
\end{equation}
Now consider the family of
BVPs (\ref{e3.1})-(\ref{e3.3}) and its equivalent integral
representation
\begin{equation} \label{e3.8}
 x(t) = \lambda (Jx)(t): =
\begin{cases}
\int^T_0\big[-G(t,s) \lambda f(s, x(s), x(h(s)))\big]ds
+ \lambda k(t), & t\in [0,T],\\
\lambda \phi(t), & t\in [-V,0],
\end{cases}
\end{equation}
where $G$ and $k$ are given in the proof of Theorem \ref{thm1}. Since
$\lambda \in[0,1]$, it is easy to see that
$$
\|x\|_{[-V,0]}\leq \|\phi\|_{[-V,0]}.
$$
Now since $\lambda \in [0,1]$ and $G\geq 0$, taking norms in
(\ref{e3.8}) and using (\ref{e3.75}) we obtain
\begin{align*}
\|x(t)\|
&\leq  \int^T_0 G(t,s) \|\lambda f(s, x(s),x(h(s)))\| ds
+  \|\lambda k(t)\|, \quad t\in [0,T],\\
&\leq  \int^T_0G(t,s)\big[2\alpha \langle x(s),\lambda f(s, x(s), x(h(s)))
\rangle + K\big]ds + \beta, \\
&\leq  \int^T_0G(t,s)\big[2\alpha \langle x(s), \lambda f(s, x(s), x(h(s)))
 \rangle + 2\alpha \|x'(s)\|^2 + K\big]ds + \beta, \\
&= \int^T_0 \big( G(t,s)\big[\alpha [\|x(s)\|^2\big]''
+ K \big) ds + \beta, \\
&\leq \alpha \int^T_0G(t,s)[\|x(s)\|^2]''ds +\frac{KT^2}{8} +\beta,
\end{align*}
where we have used the identity
\begin{equation} \label{e3.9}
[\|x(t)\|^2]'' = 2 \langle x(t), x''(t) \rangle + 2 \|x'(t)\|^2,
\end{equation}
and the inequality
$$
\int^T_0G(t,s)ds \leq \frac{T^2}{8}, \ \ \ t \in [0,T].
$$
The above inequality is readily obtained since the explicit form of $G$
is known. Continuing to employ the explicit form of $G$,
\begin{align*}
\|x(t)\| &\leq  \frac{(T-t)\alpha}{T}\int^t_0 s
[\|x(s)\|^2]''ds,\\
&\quad + \frac{t\alpha}{T}\int^T_t (T-s) [\|x(s)\|^2]''ds
+\frac{KT^2}{8} +\beta, \\
&= I_1 + I_2 + \frac{KT^2}{8} +\beta,
\end{align*}
where
\begin{gather*}
I_1 :=\frac{(T-t)\alpha}{T}\int^t_0 s [\|x(s)\|^2]''ds, \\
I_2 :=\frac{t\alpha}{T}\int^T_t (T-s) [\|x(s)\|^2]''ds.
\end{gather*}
A simple integration by parts on $I_1$ and $I_2$ gives
\begin{gather*}
I_1 = \big( \frac{T-t}{T} \big) \alpha [ t (\|x(t)\|^2)'
- \|x(t)\|^2 + \|x(0)\|^2 ], \\
I_2 = \frac{t \alpha}{T} [ -(T-t)(\|x(t)\|^2)' + \|x(T)\|^2 - \|x(t)\|^2 ].
\end{gather*}
Therefore, adding $I_1$ and $I_2$ and noting some cancellation of terms
and the non-negativity of $\|x(t)\|^2$ we obtain,
\begin{align*}
I_1 + I_2
&\leq \alpha \big[ \big(\frac{T-t}{T}\big) \|x(0)\|^2 + \frac{t}{T}
 \|x(T)\|^2 \big], \\
&\le \alpha \big[ \big( \frac{T-t}{T} \big) \beta^2 + \frac{t}{T} \beta^2
 \big] =  \alpha \beta^2.
\end{align*}
Hence
$$
\|x\|_{[-V,T]} \leq \max \Big\{\alpha \beta ^2 + \beta + \frac{KT^2}{8},
\|\phi\|_{[-V,0]}\Big\}= M.
$$
This concludes the proof.
\end{proof}

\begin{theorem} \label{thm3}
Let the conditions of Theorem \ref{thm2} hold. Then the BVP
\eqref{e3.1}--\eqref{e3.3} has a solution for each
$\lambda \in[0,1]$. In particular, the BVP \eqref{e3.4}--\eqref{e3.6} has
a solution.
\end{theorem}

\begin{proof} From Theorem \ref{thm2} there exists an $M \ge 0$ such that
$\|x\|_{[-V,T]} \le M$ with $M$ independent of $\lambda$.
The a priori bound on $x'$ now naturally follows by differentiating
the first line of (\ref{e3.8}) and taking norms.
This bound $P \ge 0$ on $x'$ will depend on $M$ and is independent
of $\lambda$.
Hence there is a constant $R>0$ such that
$$
\|x\|_1 < \max \{ M, P \} + 1 =: R,
$$
and by  Theorem \ref{thm1} the result follows. This concludes the proof.
\end{proof}

The proof of the next result is immediate from Theorem \ref{thm3} and
will be needed to develop constructive theory in Section \ref{sec5}.

\begin{corollary} \label{coro1}
Consider the BVP \eqref{e3.1}--\eqref{e3.3} with $\phi(0) = 0 = B$. If the
conditions of Theorem \ref{thm3} hold then the BVP
\eqref{e3.1}--\eqref{e3.3} has a solution $x$, for each
$\lambda \in [0,1]$ and the bound on $x$ is given by
$$
\|x\|_{[-V,T]} \leq \max \big\{\frac{KT^2}{8},\ \|\phi\|_{[-V,0]}\big\}.
$$
\end{corollary}

\begin{corollary} \label{coro2}
Let $h$, $f$ and $\phi$ be continuous, scalar-valued ($d=1$) functions. Assume
there exist constants $\alpha \geq 0, K \geq 0$ such that
\begin{equation} \label{Y}
|f(t, w, y)| \leq 2\alpha wf(t, w,y)+K,  \quad \mbox{for all }
 t \in[0,T], (w, y) \in \mathbb{R}^2.
\end{equation}
Then the BVP \eqref{e3.1}--\eqref{e3.3} has a solution $x$ for each
$\lambda \in [0,1]$. In particular, \eqref{e3.4}--\eqref{e3.6} has a
solution.
\end{corollary}

The result in the above corollary is a special case of Theorem \ref{thm3}.

We now present some examples to illustrate the theory of this section.

\begin{example}  \rm
Consider the scalar BVP
\begin{gather*}
x''(t) = t [x(t)]^3 \exp \{-[x(h(t))]^2\}, \quad t \in [0,T],\\
x(t) = 1, \quad t \in [-V,0], \;  x(T) = 0.
\end{gather*}
See that $|w|^3 \le w^4 + 1$ for all $w$, so multiplying both
sides of this inequality by $t \exp \{-[y]^2\}$ we obtain for
$\alpha = 1/2$ and $K = T$,
\begin{align*}
|f(t,w,y)| &\leq t\exp \{-[y]^2\}(w^4 + 1)\\
&=  wf(t,w,y) + t\exp \{-[y]^2\}\\
&\leq wf(t,w,y) + T.
\end{align*}
Hence (\ref{Y}) will hold for the choices $\alpha = 1/2$ and $K = T$.
Therefore, by Corollary \ref{coro2}, the BVP will have a solution.
\end{example}


\begin{example} \rm  Consider the BVP
\begin{gather*}
x''(t) = t^2 \cos(x(t)) \exp\{-x(h(t))\}, \quad  t \in [0,1],\\
x(t) = \phi(t), \quad  t \in [-2,0], \;  x(1) = 0.
\end{gather*}
See that for $\alpha = 0$ and $K = 1$,
\[
|f(t,w,y)| =   |t^2 \cos(w) \exp\{-[y]^2\}|,
\leq 1 = 2 \alpha wf(t,w,y) + K.
\]
Hence (\ref{Y}) will hold for the choices $\alpha = 0$ and $K = 1$.
Therefore, by Corollary \ref{coro2}, the BVP will have a solution.
\end{example}

Note that in both examples, $h$ continuous and $t-T\le h(t)\le t$
is sufficient.  Also note that (\ref{e3.7}) is useful in the case
that $f$ is a polynomial in $w$ and bounded in $y$.

\section{On the equation $x''(t)=f(t, x(t), x(h(t)),x'(t))$}

In this section we consider the class of BVPs for
delay-differential equations \eqref{e1.1}--\eqref{e1.3} and its
corresponding family of problems \eqref{e2.1}--\eqref{e2.3}. Since
$f$ depends on $x'$ the a priori bounds on $x$ of Section 3
may not directly imply bounds on $x'$. Therefore we need to impose additional assumptions
to obtain these bounds on $x'$.

\begin{theorem} \label{thm4}
Let $h, f$ and $\phi$ be continuous.
Assume there exist constants $\alpha \geq 0,  K \geq 0$ such that
\begin{equation}\label{e4.1}
\|f(t, w, y, z)\| \leq 2 \alpha \Big[ \langle w,f(t, w, y, z) \rangle
+ \|z\|^2\Big] + K,
\end{equation}
for all $t\in[0,T]$,  $(w,y,z) \in \mathbb{R}^{3d}$.
Then all solutions $x$ of
\eqref{e2.1}--\eqref{e2.3} satisfy $\|x\|_{[-V,T]} \leq M $ where
$M$ is independent of $\lambda \in [0,1]$ and is defined in the
proof of Theorem \ref{thm2}.
\end{theorem}

The proof of the above theorem is almost identical to that of
Theorem \ref{thm2} and so is omitted.

\begin{theorem} \label{thm5}
Let the conditions of Theorem \ref{thm4} hold.  If, in addition,
$2\alpha M < 1$ then, for all solutions $x$ of
\eqref{e2.1}--\eqref{e2.3},
$$
\|x'\|_{[0,T]} \leq \frac{M[2 \alpha M + KT^2/8]}{T(1 - 2\alpha M)/2}: = N,
$$
with $N$ independent of $\lambda \in [0,1]$.
\end{theorem}

\begin{proof}
 Let $x$ be a solution of \eqref{e2.1}--\eqref{e2.3}. For
$t\in[0, \frac{T}{2}]$ apply Taylor's formula to obtain
 $$
x(t + \frac{T}{2}) - x(t) - \frac{T}{2}x'(t)
 = \int^{t + \frac{T}{2}}_t(t + \frac{T}{2}-s)x''(s)ds.
$$
Thus,
 $$- \frac{T}{2}x'(t)= x(t) -x(t + \frac{T}{2})  +
  \int^{t
 + \frac{T}{2}}_t(t + \frac{T}{2}-s)\lambda f(s, x(s), x(h(s)), x'(s)) ds.
$$
Now taking norms and using:
$\lambda \in [0,1], \|x\|_{[0,T]} \leq M, t + \frac{T}{2}-s \geq 0$,
(\ref{e3.9}) and (\ref{e4.1})
we obtain
\begin{align*}
\frac{T}{2} \|x'(t)\|
&\leq  2 M + \int^{t + \frac{T}{2}}_t(t + \frac{T}{2}-s)\{\alpha
[\|x(s)\|^2]'' + K\} ds, \ \ \ t \in [0,\frac{T}{2}]\\
&=  2 M + \alpha\int^{t
 +\frac{T}{2}}_t(t + \frac{T}{2}-s)[\|x(s)\|^2]''ds
+ K\int^{t + \frac{T}{2}}_t(t + \frac{T}{2}-s)ds.
 \end{align*}
  Using Taylor's formula once more for
  $[\|x(s)\|^2]''$ we obtain
\begin{align*}
\frac{T}{2} \|x'(t) \|
&\leq  2M + \alpha \Big[\|x(t + \frac{T}{2})\|^2 -
\|x(t)\|^2-\frac{T}{2}[\|x(t)\|^2]'\Big]+ \frac{KT^2}{8}, \quad
 t \in [0,\frac{T}{2}],\\
&\leq  2M + \alpha \Big[M^2 - T \langle x(t), x'(t) \rangle \Big]
+ \frac{KT^2}{8}
\\
&\leq  2M + \alpha \Big[M^2 + M T\|x'(t)\|\Big]+
\frac{KT^2}{8}.
\end{align*}
So rearranging we have
$$
\|x^\prime\|_{[0, \frac{T}{2}]} \leq \frac{M(2 + \alpha M) +
KT^2/8}{T(1 - 2 \alpha M)/2}:= N,
$$
 with $N$ independent of $\lambda \in[0,1]$.
 For $t \in [\frac{T}{2},T]$ we use the Taylor formula
  $$
x(t) -x(t- \frac{T}{2}) - \frac{T}{2}x'(t) =
  -\int^{t}_{t -\frac{T}{2}}  (t - \frac{T}{2}-s)x''(s)ds.
$$
By arguing in a similar fashion to the case $ t \in
[0,\frac{T}{2}]$ we obtain $\|x'\|_{[\frac{T}{2},
\;T]}\leq N$ with $N$ defined above and independent of $\lambda
\in [0,1]$. This concludes the proof.
\end{proof}

\begin{theorem} \label{thm6}
 Let the conditions of Theorem \ref{thm4}
hold. If $2 \alpha M < 1$ (with $M$ and $\alpha $ defined in
Theorem \ref{thm2}) then \eqref{e2.1}--\eqref{e2.3} has a solution $x$ for
all $\lambda \in [0,1]$. In particular,
\eqref{e1.1}--\eqref{e1.3} has a solution.
\end{theorem}

\begin{proof} The conditions of Theorem \ref{thm4} guarantee the existence
 of constants $M$, $N$ such that
 $$
\|x\|_1 < \max \{ M,N,\|\phi\|_{[-V,0]} \} + 1 : = R.
$$
 with $R$ independent of $\lambda \in [0,1]$.  Now applying
Theorem \ref{thm1} the result follows and this concludes the proof.
\end{proof}

The proof of the next result is immediate from Theorem \ref{thm5}.

\begin{corollary} \label{coro3}
Consider the BVP \eqref{e2.1}--\eqref{e2.3} with $\phi(0) = 0 = B$. If the
conditions of Theorem \ref{thm3} hold then the BVP
(\ref{e3.1})-(\ref{e3.3}) has a solution for each $\lambda
\in[0,1]$ and the bound on $x$ is given by
$$
\|x\|_{[-V,T]} \leq \max \big\{\frac{KT^2}{8}, \|\phi\|_{[-V,0]}\big\}.
$$
\end{corollary}

\begin{corollary} \label{coro4}
Let $h, f$ and $\phi$ be continuous, scalar-valued (d=1) functions such that
$$|f(t, w, y, z)| \leq 2\alpha\Big[ wf(t, w, y, z)+ |z|^2\Big]+K, \ \mbox{for all} \ t\in[0,T],\ (w,y,z)\in \mathbb{R}^3,$$
and  for some constants $\alpha \geq 0, K \geq 0$.
Then the scalar BVP \eqref{e2.1}--\eqref{e2.3} has a solution $x$ for
each $\lambda \in[0,1]$. In particular,
\eqref{e1.1}--\eqref{e1.3} has a solution.
\end{corollary}

The above corollary is a special case of Theorem \ref{thm6}.

\section{Some Constructive Results} \label{sec5}

In this section we develop some constructive results for the BVP
\eqref{e1.1}--\eqref{e1.3}. The ideas rely on the a priori bounds on
solutions of previous sections and an abstract result due to
Petryshyn \cite{P}.  To apply these results, we consider
homogeneous boundary conditions $\phi (0)=0$, $B=0$.

We introduce the following notation so we can readily apply
Petryshyn's abstract result. Let $X$ and $Y$ denote real Banach
spaces. let $L:D(L) \subset X\to Y$ denote a Fredholm map of index $0$;
in particular, $L$ is linear.  Let Null$(L)$ denote be the null
space of $L$ and let Rank$(L)$ denote the rank of $L$.  Let
$P:D(P) \subset X\to Y$ denote a nonlinear map.

Let $\{X_n\} \subset X, \{Y_n\} \subset Y$, be sequences of
finite-dimensional spaces and for each $n \in \mathbb{Z}^+$ let
$Q_n:Y\to Y_n$ denote a  linear projection.  Define the
scheme $\Gamma = \{X_n, Y_n, Q_n\}$ as \textbf{admissible} for
maps $X\to Y$ provided:
\\
$\dim X_n = \dim Y _n$, for each $n$, and
\\
$\mathop{\rm dist}(x, X_n): = \inf\{\|x - v\|_X: v \in X_n\}\to 0$ as
$ n \to \infty$, for each $x \in X_n$.

For given maps $L$ and $P$, the equation
$$
Lx = Px,\quad  x\in D(L) \cap D(P),
$$
is said to be strongly (feebly) {\bf A-solvable} with respect to
$\Gamma$ if there exists an $N_0 \in \mathbb{Z}^+$ such that the
finite dimensional equation
$$
Q_nLx = Q_n Px, \quad x \in (D(L) \cap D(P)) \cap X_{n},
$$
has a solution $x \in D(L) \cap D(P)) \cap X_{n}$ for each $n \geq N_0$
such that $x_n \to x \in X$, $(x_{n_j} \to x \in X)$ and $Lx = Px$.

If the equation $Lx = Px$ is strongly A-solvable then we follow the lead of Petryshyn and say that the Galerkin method applies.

The mapping $L-P : D(L) \cap D(P) \subset X \to Y$ is said to be {\bf A-proper}  with respect to $\Gamma$ if
\[
Q_n L - Q_n P : (D(L) \cap D(P)) \cap X_n \subset X_n \to Y_n
 \]
is continuous for each $n \in \mathbb{Z}^+$ and if
$\{ x_{n_j} : x_{n_j} \in (D(L) \cap D(P)) \cap X_n\}$ is any bounded sequence in $X$ such that
\[
Q_{n_j} L x_{n_j} - Q_{n_j} Px_{n_j} \to 0 \quad {\rm in } Y,
 \]
then there is a subsequence $\{ x_{n_k}\}$ of
$\{ x_{n_j}\}$ and $x\in D(L) \cap D(P)$ such that
\[
x_{n_k} \to x \quad {\rm in} \ X \quad \mbox{and} \quad Lx = Px.
\]
Since $L$ is a Fredholm map of index zero, there exists closed subspaces
$X_1 \subset X$ and $Y_2 \subset Y$ such that
$X =\mathop{\rm Null}(L) \oplus X_{1}$ and
$Y=Y_{2}\oplus\mathop{\rm Ran} (L)$.
Let $Q$ be the linear
projection of $Y$ onto $Y_2$ and assume there exists a continuous
bilinear form $[\cdot, \cdot]$ on $Y \times X$ mapping $(y,  x)$
into $[y, x]$ such that
$$
y \in \mathop{\rm Ran}(L) \mbox{ iff } [y, x]
=0, \; \mbox{ for all}\; x \in \mathop{\rm Null}(L).
$$
We first need the following result of Petryshyn \cite[Theorem A]{P}, .

\begin{theorem} \label{thmA}
Let $L$ be a Fredholm map of index zero. Assume
$\mathop{\rm Null}(L)=\{0\}$.  Assume there exists a bounded open
ball $G\subset X$ with $0\in G$ such that
\begin{enumerate}
\item[(a)] $P({\bar G})$ is bounded,
\item[(b)]$L - \lambda P: \overline{G} \to Y$ is $A$-proper w.r.t. $\Gamma$ for each $\lambda \in [0,1]$,
\item[(b)]$Lx \ne \lambda Px$ for $x\in \partial G$ and $\lambda \in (0,1]$.
\end{enumerate}
Then $L-P$ is feebly $A$-solvable with respect to $\Gamma$. In
particular the BVP \eqref{e2.1}--\eqref{e2.3} has a solution $x$.
If that solution $x$ is unique in $G$, then $L-P$ is strongly
$A$-solvable with respect to $\Gamma$ and the Galerkin
method is applicable to the BVP \eqref{e2.1}--\eqref{e2.3}.
\end{theorem}

\begin{remark} \rm
Petryshyn's  Theorem A is more general than Theorem 5.1
(see  \cite[Remark 1.2,]{P}).  We assume
$\mathop{\rm Null}(L)=\{ 0\}$ both for simplicity of statement and for the
specific application with $\phi (0)=0, B=0$.
\end{remark}

\begin{theorem}
 Let the conditions of Theorem \ref{thm6} hold. Then the BVP
\eqref{e1.1}--\eqref{e1.3} (with $\phi(0) = 0$) is feebly $A-solvable$ with
respect to $\Gamma$. If $x$ is the unique solution then the BVP
\eqref{e1.1}--\eqref{e1.3} (with $\phi(0) = 0$) is strongly
$A-solvable$ with respect to $\Gamma$. That is, the Galerkin Method is
applicable.
\end{theorem}

\begin{proof} Let
\begin{gather*}
Lx :=\begin{cases}
x''(t), & t \in [0, T]\\
x(t), & t\in [-V,0];
\end{cases}\\
 Px :=\begin{cases}
f(t, x(t), x(h(t)), x'(t)), &  t \in [0,  T]\\
\phi (t),\quad  t\in [-V,0];
\end{cases} \\
X: = \Big\{x\in C([-V, T];\mathbb{R}^d)\cap C^1([0,T];\mathbb{R}^d]):
 x(T) = x(0) = 0,\ x = \phi \mbox{ on }[-V, 0]\Big\};
\\
Y:= C([-V, T];\mathbb{R}^d).
\end{gather*}
Set
$$
G=\{ x \in X:  \|x\|_1<\max\{ M,N,\|\phi\|_{[-V,0]} \} +1\},
$$
where $M$ and $N$ are given in the proof of Theorem \ref{thm6}.

It is easy to see that $P({\bar G})$ is bounded.
Now $L$ is A-proper with respect to $\Gamma$ by \cite{P2}.
Since $P: X \to Y$ is continuous see that $N$ is also completely continuous
 because $X$ is compactly embedded into $C^1([0,T]; \mathbb{R}^d)$ and
therefore $P$ is A-proper with respect to $\Gamma$ (see \cite{P}).
Hence $L - \lambda P: \overline{G} \to Y$ is $A$-proper
with respect to $\Gamma$
for each $\lambda \in [0,1]$.

Finally,  we see that
$$
Lx \ne \lambda Px \quad \mbox{for all } x\in \partial \Omega
\mbox{ and  all } \lambda \in [0,1],
$$
since the a priori bound theory of Theorems \ref{thm4} and \ref{thm5}
is applicable.
Therefore, all of the conditions of Theorem \ref{thmA} are satisfied
and the result follows.  This concludes the proof.
\end{proof}

\section{On Uniqueness of Solutions}\label{sec6}

This brief section provides some results which guarantee the uniqueness
of solutions to the BVPs for delay differential equations.
Our interest here is twofold.  Firstly, the constructive results
of Section \ref{sec5} rely on uniqueness of solutions.
Secondly, BVPs with deviating arguments  can introduce solutions which
do not appear for the ``associated'' non-deviating BVP (see \cite{GS70}).


\begin{theorem} \label{thm7}
If $f$ satisfies
\begin{equation} \label{uni}
\langle u - v, f(t,u,z, u') - f(t,v,w, v') \rangle>0,
\quad \mbox{for all } t \in [0,T],
\end{equation}
and $u,z,u',v,w,v' \in \mathbb{R}^{d}$ with $u \neq v$,   and
$\langle u - v, u' -v' \rangle = 0$,  then \eqref{e1.1} has, at most, one
solution satisfying \eqref{e1.2}-\eqref{e1.3}.
\end{theorem}

\begin{proof}
Assume $u$ and $v$ are solutions to the BVP \eqref{e1.1}--\eqref{e1.3}.
Then $u-v$ satisfies the BVP
\begin{gather*}
u'' (t) - v'' (t) =  f(t,u(t),u(h(t)),u'(t)) - f(t,v(t),v(h(t)),v'(t)),
\quad t \in [0,T],\\
u(t) - v(t) = 0, \quad  t \in [-V,0], \\
   u(T) - v(T) = 0.
\end{gather*}
Consider $r(t) := \|u(t)-v(t)\|^2$, $t\in[-V,T]$.  Now $r$ must have a
positive maximum at some point $c \in [-V,T]$.
From the boundary conditions, $c \in (0,T)$.
Therefore, by a maximum principle we must have
\begin{equation} \label{GB2}
r'(c) = 0, \quad r''(c) \le 0.
\end{equation}
So using the product rule on $r$ we have
\[
r''(c) \ge 2 \big\langle u(c) - v(c), f(c, u(c), u(h(c)),u'(c))- f(c,
v(c), v(h(c)),v'(c))\big\rangle  > 0,
\]
which contradicts \eqref{GB2}.
Therefore $r(t) = \|u(t) - v(t)\|^2 = 0$ for all $t \in [-V,T]$, and
solutions of the BVP \eqref{e1.1}--\eqref{e1.3}  must be unique.
\end{proof}

\subsection*{Acknowledgements}
C. C. Tisdell gratefully acknowledges the financial support of the
Australian Research Council's Discovery Projects (Grant DP0450752).

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\end{document}