\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations}, 
Vol. 2008(2008), No. 107, pp. 1--30.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/107\hfil Homoclinic orbit solutions]
{Homoclinic orbit solutions of a one Dimensional
 Wilson-Cowan type model}

\author[E. P. Krisner\hfil EJDE-2008/107\hfilneg]
{Edward P. Krisner}

\address{Edward P. Krisner \newline
  B-18 Smith Hall \\
 University of Pittsburgh at Greensburg\\
 Greensburg, PA 15601, USA}
\email{epk15+@pitt.edu}

\thanks{Submitted May 14, 2008. Published August 7, 2008.}
\subjclass[2000]{45K05, 92B99, 34C25} 
\keywords{Shooting; periodic; coupling; integro-differential equation}

\begin{abstract}
 We analyze a time independent integral equation defined on a
 spatially extended domain which arises in the modelling of neuronal
 networks. In this paper, the coupling function is oscillatory and
 the firing rate is a smooth ``heaviside-like'' function.  We will
 derive an associated fourth order ODE and establish that any bounded
 solution of the ODE is also a solution of the integral equation.  We
 will then apply shooting arguments to prove that the ODE has
 $N$-bump homoclinic orbit solutions for any even-valued $N>0$.
 homoclinic orbit.
\end{abstract}

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

\section{Introduction} \label{sec0}

In 1972, Wilson and Cowan \cite{WC1} derived the partial
integro-differential equation
\begin{equation} \label{inteqwt}
  u_t = -u+\int_{-\infty}^{\infty}w(x-y)f(u(y,t)-th)dy
\end{equation}
to describe the behavior of a single layer of neurons \cite{WC1}.
Here, $u(x,t)$ and \\ $f(u(x,t)-th)$ represent the level of
excitation (e.g. voltage) and the firing rate, respectively, of a
neuron at position $x$ and time $t$. The parameter $th \geq 0$
denotes the threshold of excitation. The term $w(x-y)$ determines
the coupling between neurons at positions $x$ and $y$.

 In 1977, Amari \cite{A2} studied pattern
formation in \eqref{inteqwt} for lateral inhibition type couplings.
That is, $w$ is assumed to be continuous, integrable and even, with
$w(0)>0$, and exactly one positive zero. Under the simplifying
assumption that the firing rate $f$ is a Heaviside step function, he
analyzed the existence, multiplicity and stability of stationary
one-bump solutions of the time independent equation
\begin{equation} \label{inteq1}
  u = \int_{-\infty}^{\infty}w(x-y)f(u(y)-th)dy.
\end{equation}
 Equations \eqref{inteqwt} and \eqref{inteq1} have been studied
 with respect to various combinations of firing rate functions and
coupling functions.

Other investigations of \eqref{inteqwt} for lateral inhibition type
couplings include those of Coombes et.~al.~\cite{CLO} and Guo
et.~al.~\cite{GC}.  Coombes et.~al.~\cite{CLO} obtain a closed form
stable $1$-bump homoclinic orbit solution of \eqref{inteq1} with
 \begin{gather*}
  f(u)=H(u), \\
  w(x)=\kappa(1-|x|)e^{-|x|}
 \end{gather*}
  and $H(u)$ is the Heaviside step function.

Guo and Chow \cite{GC} state criteria that result in two 1-bump
solutions of \eqref{inteq1} with
 \begin{gather*}
 f(u)= (\alpha u+\beta)H(u)\quad \textup{and} \\
 w(x) = Ae^{-\alpha|x|}-e^{-|x|}\quad \textup{where }A, a >1.
\end{gather*}


A wide range of techniques have been used to study \eqref{inteqwt}
and \eqref{inteq1}.  Owen et.~al.~employ an Evans function approach
to investigate instabilities of localized solutions of
\eqref{inteqwt} and \eqref{inteq1}.  Kishimoto and Amari \cite{KA}
assume that $f$ has a sigmoidal shape and use the Schauder Fixed
Point Theorem \cite{DHG} to prove the existence of a single bump
stationary solution of \eqref{inteq1}. Ermentrout and McLeod
\cite{EandM} investigate the existence of traveling waves when $w$
is strictly positive and Gaussian shaped, and $f$ is a sigmoidal
function. They use a homotopy argument based on the contraction
mapping theorem to prove the existence of monotonic wave fronts.
Subsequently, Pinto and Ermentrout \cite{PE1} make use of the result
in \cite{EandM} and use singular perturbation methods to study wave
front solutions in a related system of equations.  Ermentrout
\cite{GBE} and Coombes \cite{SCoombes} give an extensive review of
theoretical methods and results.


 In order to analyze more complicated solutions
(e.g. multi-bump solutions), Laing et.~al.~\cite{LTGE}, Coombes et.~
al.~\cite{CLO}, and Guo et.~al.~\cite{GC} derive associated fourth
order ODEs by applying Fourier Transform methods. In \cite{LTGE} and
\cite{CLO} conditions are given which show that when the integral
equation \eqref{inteq1} has a homoclinic orbit satisfying $u(\pm
\infty) = 0$ then that solution also satisfies an associated ODE of
the form
\begin{equation} \label{caneq}
  u''''+q_1u''+h(u)=0,
\end{equation}
where $q_1$ is a real constant and $h$ is a real-valued function.

In this paper we also derive a fourth order ODE associated with
\eqref{inteq1}. This affords us the opportunity to employ the method
of topological shooting to prove the existence of $N$-bump
homoclinic orbit solutions.  The method of topological shooting is a
well known technique, and has been employed to prove the existence
of a wide variety of solutions of two-point boundary value problems.
For example, Shangbing \cite{Shangbing} applies a shooting method to
prove the existence of traveling waves of a bioremediation model.
Peletier and Troy \cite{PT} derive a fourth order ODE by scaling the
Extended Fisher-Kolmogorov equation.  Subsequently, they proved that
the fourth order ODE has two odd 1-bump periodic solutions.

In this paper we extend the results obtained in Krisner
\cite{my2ndpaper}.  In \cite{my2ndpaper}, conditions were given
which guarantee that \eqref{inteq1} has two 1-bump periodic
solutions.  The primary goal in this paper is to develop techniques
which allow us to prove the existence of $N$-bump homoclinic orbit
solutions of \eqref{inteq1} for any even-valued $N>0$. In our survey
the coupling function $w$ is oscillatory shaped and the firing rate
function $f$ is a smooth step-like function.

The outline of the paper is as follows. In Section
\ref{prelimresults}, we define our coupling and firing rate
functions.  These functions were originally introduced in Laing et
al. \cite{LTGE}.  In addition, we will state several previously
established results including
\begin{enumerate}
\item the link between \eqref{inteq1} and a fourth order ODE,
\item the initial conditions of the ODE which yield even solutions,
\item a parameter regime that gives rise to a tractable setting for
our construction of $N$-bump solutions,
\item the existence of infinitely many critical values, and
\item the existence of two $1$-bump periodic solutions.
\end{enumerate}
In Section \ref{contcritvals}, we will show that the critical
numbers are continuous with respect to the initial conditions.  This
analysis will lay the framework for the construction of $N$-bump
periodic solutions which is contained in Section
\ref{nbumpsolutions}.

\section{Preliminary Results} \label{prelimresults}

 In this section we will establish several results that are
necessary for the construction of $N$-bump homoclinic orbit
solutions of the time independent integral equation
\begin{equation} \label{inteqn2}
       u(x)=\int_{-\infty}^{\infty}w(x-y)f(u(y)-th)dy.
 \end{equation}
As in \cite{my2ndpaper} the coupling and firing rate functions are
defined by
 \begin{gather} \label{w}
       w(x) = e^{-b|x|}(b\sin(|x|) + \cos(x)),\quad b>0,\quad \textup{and}\\
 \label{firingrate2}
   f(u-th)=2e^{-r/(u-th)^2}H(u-th),\quad r>0,\; th>0
 \end{gather}
 respectively.
Figure \ref{wandfiringrate} depicts the essential characteristics of
the functions $w$ and $f$.

\begin{figure}[ht]
  \begin{center}
  \includegraphics[width=0.45\textwidth]{fig1a} \quad %{wx}
  \includegraphics[width=0.45\textwidth]{fig1b} %{fumth}   Pulserate
  \end{center}
\caption{Left panel, example of \eqref{w} with $b=1.1$.
  Right panel, example of \eqref{firingrate2} with $r=0.02$, $th=1.75$.}
  \label{wandfiringrate}
\end{figure}

\subsection*{The Associated ODE}
 Here we state an important theorem which establishes a crucial
 connection between the ODE
\begin{equation} \label{THEODE}
  u''''- 2(b^{2}-1)u'' + (b^{2}+1)^{2}u = 4b(b^{2}+1)f(u-th)
\end{equation}
and the integral equation \eqref{inteqn2} with $w$ defined by
\eqref{w} and $f$ defined by \eqref{firingrate2}. Krisner
\cite{mypaper} proves the following result.

\begin{theorem} \label{THETHEOREM}
Suppose that $u$ is a solution of \eqref{THEODE}, and that
$u(t)=o(e^{b|t|})$ as $t \to \pm\infty$. Then $u$ is a solution of
\eqref{inteqn2}.
\end{theorem}

Hence, the goal of this paper will be fulfilled upon proving the
existence of \\ $N$-bump homoclinic orbit solutions of the IVP
\begin{equation} \label{THEODE4}
 \begin{gathered}
u''''-2(b^{2}-1)u''+(b^{2}+1)^{2}u=4b(b^{2}+1)f(u-th) \\
u(0)=\alpha,u'(0)=0,u''(0)=\beta,u'''(0)=0.
\end{gathered}
\end{equation}

\subsection*{Even Solutions of the ODE}
 An immediate consequence of \cite[Lemma 4.1]{my2ndpaper} is the following.

\begin{lemma}\label{usym}
Any solution $u$ of \eqref{THEODE4} satisfies $u(x)=u(-x)$ for all
$x$ in the domain of existence.
\end{lemma}

 Thus, we need only to consider the behavior of solutions
on $[0,\omega)$ for some $\omega>0$ (possibly $\omega=\infty$).

\subsection*{The First Integral Equation}
We now derive an associated 3rd order equation.  By multiplying both
sides of \eqref{THEODE} by $u'$ and integrating over $[0,x]$ we
obtain
\begin{equation} \label{firstintegral}
u'''u' - {(u'')^2 \over 2} -(b^2-1)(u')^2+(b^2+1)^2Q(u)=E
\end{equation}
for some constant $E$ and
\begin{equation} \label{Q_HC}
Q(u)=\int_{0}^{u}\Big(s-\frac{4b}{b^2+1}f(s-th)\Big)ds.
\end{equation}
Since we are interested in solutions that satisfy
$(u,u',u'',u''')\to (0,0,0,0)$ as \\ $x \to \infty$, then $E=0$ in
\eqref{firstintegral}.  Thus, we obtain
\begin{equation} \label{zeroenergyhomo}
u'''u' - {(u'')^2 \over 2} -(b^2-1)(u')^2+(b^2+1)^2Q(u)=0.
\end{equation}

\subsection*{The Initial Conditions}
The conditions $u(0)=\alpha$, $u'(0)=0$, $u''(0)=\beta$, and
$u'''(0)=0$ substituted into \eqref{zeroenergyhomo} yields
\begin{equation} \label{betalphrel}
\beta^2 = 2(b^2+1)^2Q(\alpha)~ \Longleftrightarrow~ \beta = \pm
(b^2+1)\sqrt{2Q(\alpha)}.
\end{equation}
In our construction of even bump solutions we consider $\alpha<0$
and $\beta>0$. This together with \eqref{betalphrel} and the fact
that $Q(u)=\frac{u^2}{2}$ yields
\begin{equation} \label{betalphrelfinal}
\beta=-(b^2+1)\alpha.
\end{equation}
This crucial result reduces our problem to that of one dimensional
shooting.  That is, all solutions of \eqref{THEODE4} that we
consider are uniquely determined by the value of $\alpha$.
Furthermore, we occasionally use the notation $u(\cdot,\alpha)$
whenever its necessary to emphasize the solution's dependence on
$\alpha$. Lastly, we will assume that \eqref{betalphrelfinal} holds
throughout the remainder of this paper.

\subsection*{Range of Parameters}
 In Krisner~\cite{my2ndpaper} we defined a parameter regime
that ensured the existence of 1-bump periodic solutions.  These
periodic solutions will be used to construct the multi-bump
homoclinic orbit solutions.  Hence, the conditions required to
construct the periodic solutions of \cite{my2ndpaper} will also be
required for the construction of the homoclinic orbit solutions.

 We begin by defining the same parameter regime that guaranteed the
existence of periodic solutions.  We will assume throughout the
remainder of this paper that $(r,b,th) \in \Lambda$ where
\begin{equation} \label{lamda2}
 \Lambda = \{(r,b,th) \in X :
 Q(u)=0 \text{ has a unique positive solution}\},
  \end{equation}
  and
  $X = \{(x_1,x_2,x_3) \in \mathbb{R}^3: x_1>0,\; x_2>0,\text{ and }x_3>0\}$,
  (see Figure~\ref{yetanotherQ}).

The following lemma is an immediate consequence of \cite[Theorem
3.5]{my2ndpaper}

\begin{lemma}\label{excontinuum}
 For each fixed $th \in (0,2)$ $\Lambda$ contains a continuum
 provided that $r > 0$ is sufficiently small.
  \end{lemma}
For a proof of the above lemma, see  \cite[Theorem 3.5]{my2ndpaper}.


\subsection*{The Function $Q$}
 As noted above, the function $Q$ defined by \eqref{Q_HC}
plays a pivotal role in determining the parameter regime that we
consider. Figure~\ref{yetanotherQ} depicts a rather typical
qualitative picture of the function $Q$ with $(r,b,th)\in \Lambda$.
We see that $Q'(u)$, where $' = \frac{d}{du}$, has two positive
roots, which we denote by $u_s$ and $u_{ss}$. Specifically, we
define $u_{ss}>0$ so that $Q'(u_{ss})=0<Q(u_{ss})$ and
$u_{s}>u_{ss}$ so that $Q'(u_{s})=Q(u_{s})=0$.

 The function $Q$ with our choice of parameters given in
\eqref{lamda2} also determines the constant solutions of
\eqref{THEODE4}.  Note that $u\equiv 0$ and $u \equiv u_s$ are the
only constant solutions of \eqref{THEODE4}. This is easily seen by
expressing the equation in \eqref{THEODE4} as
\[
u''''-2(b^2-1)u''=-(b^2+1)^2Q'(u).
\]  We can also deduce from
\eqref{zeroenergyhomo} that $u \equiv u_{ss}$ is \textbf{not} a
constant solution.

\begin{figure}[!ht]
  \begin{center}
 \includegraphics[width=0.4\textwidth]{fig2}   % qb1pt44rpt05.eps new boundingbox
  \end{center}
  \caption{The function $Q$ when $(r,b_r,th) \in \Lambda$.
  Here the parameters are $r=.05, b_r=1.44$, and $th=1.5$. Also,
  $u_{ss} \approx 1.76$ and $u_s \approx 3.71.$}
  \label{yetanotherQ}
\end{figure}


\subsection*{Oscillatory Behavior of Solutions} Much of the work in
this paper deals with the behavior of solutions near critical
points.  Thus, it is necessary to guarantee their existence.  The
following theorem, proved in \cite{my2ndpaper}, guarantees the
existence of infinitely many critical points.

\begin{theorem} \label{infmanycritpoints}
Suppose that $(r,b,th) \in \Lambda$ with $r \leq \frac{th^4}{16}$.
Also, let $u$ be a nontrivial solution of \eqref{THEODE4} with
interval of existence $[0,\omega)$. Then $u'$ changes sign on
$(X,\omega)$, for any $X \in (0,\omega)$.
\end{theorem}

For a proof of the above theorem, see \cite[Theorem
5.1]{my2ndpaper}.

\subsection*{Limiting Values as $r \to 0$}

As stated in Lemma~\ref{excontinuum}, ``sufficiently small'' $r$
gave rise to the existence of a continuum in $\Lambda$. In
Theorem~\ref{infmanycritpoints} it was seen that sufficiently small
$r$ ensured the existence of infinitely many critical points. Our
construction of $N$-bump homoclinic orbit solutions will also rely
on sufficiently small $r$ as well as precise limiting values of $b$,
$u_s$ and $u_{ss}$ as $r \to 0$. To emphasize their dependence on
$r$, we write $b_r, u_s(r)$ and $u_{ss}(r)$. In later sections we
will omit the $r$ and simply write $b, u_s$ and $u_{ss}$.

In the following lemma we begin by obtaining the limiting value of
$u_s(r)$ and $b_r$ as $r \to 0^+$.

\begin{lemma}\label{u_sofrlimit}
Suppose that $th \in (0,2)$ is fixed and that $(r,b_r,th)\in
\Lambda$. Then
\begin{gather}
  \label{u_slimit}
u_s(r) \to 2th^+,\quad \textup{and} \\
 \label{b_rlimit}
 b_r \to \frac{2}{th} \pm \frac{\sqrt{4-th^2}}{th}
\end{gather}
as $r \to 0^+$.
\end{lemma}

\begin{proof}
Equation \eqref{u_slimit} is proved in \cite[Lemma 3.4]{my2ndpaper}.
We now prove \eqref{b_rlimit}.  Since $u_s(r)$ satisfies
$Q'(u_s(r))=0$, i.e.,
\[
Q'(u_s(r))=u_s(r)-\frac{8b_r}{b_{r}^{2}+1}e^{-r/(u_s(r)-th)^2}=0,
\]
then it follows from \eqref{u_slimit} that
\begin{equation}\label{bquotient}
\frac{8b_r}{b_{r}^{2}+1}=u_s(r)e^{-r/(u_s(r)-th)^2}\to 2th\quad
\textup{as } r \to 0^+.
\end{equation} Hence, \eqref{b_rlimit} follows
after a little algebra.
\end{proof}

\begin{lemma} \label{lemxchaos}
  Let $0<th<2$ be fixed, and for small $r>0$ let $(r,b_r,th)\in \Lambda$.
  Then there exists
  $r^* > 0$ such that
  \begin{gather}
   \label{ussr}
    0 < u_{ss}(r)-th \leq r^{1/4}\quad \textup{for all } r \in (0,r^*),
    \quad \textup{and} \\
   \label{QussrminusQth}
   Q(u_{ss}(r)) \to Q(th) \quad \textup{as } r \to 0^+.
   \end{gather}
  \end{lemma}

  \textbf{Remarks:} (i)
  The power $r^{1/4}$ in \eqref{ussr} can be
  improved but for our purposes this power will suffice.
  (ii) In our proofs of multi-bump solutions we will
  make use of properties \eqref{ussr} and \eqref{QussrminusQth}.



\begin{proof}[Proof of Lemma \ref {lemxchaos}]
We prove \eqref{ussr} by contradiction.  That is, suppose that there
exists a sequence $\{r_n\}_{n=1}^{\infty}$ such that
\begin{equation} \label{ussboundedaway}
  r_n \to 0^+\quad \textup{and}\quad u_{ss}(r_n) - th >
  r_{n}^{1/4}\quad \textup{for all } n \geq 1.
\end{equation}
Throughout this proof we shall write $b_n$ instead of $b_{r_n}$. To
obtain a contradiction we define
\begin{equation} \label{defnofun}
  u_n = th + r_{n}^{1/4}\quad \textup{for each } n \geq 1.
\end{equation}
By \eqref{ussboundedaway} and \eqref{defnofun} we obtain
\[
th < u_n < u_{ss}(r_n)\quad \textup{for all } n \geq 1.
\]
This means that $0 < Q'(u_n)$ for all $n \geq 1$, or equivalently
\begin{equation} \label{Qpofunesthom}
  0  <  u_n - \frac{8b_n}{b_{n}^{2}+1}e^{-r_n/(u_n-th)^2}
 =  u_n - \frac{8b_n}{b_{n}^{2}+1}e^{-r_{n}^{1/2}}\quad \forall
 n \geq 1.
 \end{equation}
 From \eqref{bquotient} and \eqref{defnofun} it follows that
\[
u_n - \frac{8b_n}{b_{n}^{2}+1}e^{-r_{n}^{1/2}} \to -th < 0\quad
\textup{as } n \to \infty.
\]
  This contradicts \eqref{Qpofunesthom} for large $n > 1$.
  Therefore, \eqref{ussr} must hold as claimed.

  It remains to prove \eqref{QussrminusQth}.
By definition of $Q$ it follows that
\begin{equation} \label{peanutt}
\begin{aligned}
 Q(u_{ss}(r))-Q(th) & =  \frac{u_{ss}^{2}(r)}{2}-\frac{th^2}{2}
 - \frac{8b_r}{b_{r}^{2}+1}\int_{th}^{u_{ss}(r)}e^{-r/(s-th)^2}ds  \\
 & \leq  (u_{ss}(r)-th)\big(\frac{u_{ss}(r)+th}{2}\big).
\end{aligned}
\end{equation}
Since $Q(u_{ss}(r))>Q(th)$, then \eqref{QussrminusQth} is a
consequence of \eqref{ussr} and \eqref{peanutt}. This concludes the
proof.
\end{proof}

\subsection*{Periodic Solutions}

 An important prerequisite of our construction of $N$-bump
homoclinic solutions is the existence of periodic solutions.  Next,
we state Theorem 6.1 of \cite{my2ndpaper}.

\begin{theorem} \label{periodicsoln}
Suppose that $(r,b,th) \in \Lambda$ with $r \leq \frac{th^4}{16}$,
and that $\alpha<0$ and $\beta=-(b^2+1)\alpha$.
  Then, there exists $\alpha^* < \alpha_* < 0$ such that
  $u(\cdot,\alpha^*)$ and $u(\cdot,\alpha_*)$ are 1-bump
  periodic solutions of \eqref{THEODE4}. Moreover, we can choose
  $\alpha^*$ and $\alpha_*$ so that
  \begin{equation} \label{perestimates}
th < ||u(\cdot,\alpha_*)||_{\infty} < u_s <
||u(\cdot,\alpha^*)||_{\infty}
\end{equation}
  where $Q(u_s)=0$.
\end{theorem}

 The existence of these periodic solutions provides a means
of ``controlling'' the bumps of the solution.  For example, if
$u(\cdot,\alpha^*)$ is a periodic solution described in
Theorem~\ref{periodicsoln}, then $N \to \infty$ as $\alpha \to
\alpha^*$.


\section{Continuity of Critical Values} \label{contcritvals}

 In this subsection we will lay the foundation of the shooting
method that we use to prove the existence of $N$-bump homoclinic
orbit solutions for any even $N$. To accomplish this we must first
assume that the conditions of Theorem~\ref{infmanycritpoints} hold.
Recall that this theorem ensures the existence of infinitely many
critical numbers on the domain of existence.  Also, recall that we
will assume that $u(0)=\alpha<0$ and $u''(0)=\beta=-(b^2+1)\alpha$.
Hence, solutions of \eqref{THEODE4} are uniquely determined by
$\alpha$.

 In particular, critical numbers of solutions of
\eqref{THEODE4} are uniquely determined by $\alpha$. The primary
objective of this section is to prove that these critical numbers
{\it continuously} depend on $\alpha$. Hence, it is necessary to
develop a rigorous naming scheme of the critical numbers in a way
that emphasizes their dependence on $\alpha$. As in
Theorem~\ref{infmanycritpoints}, we will assume that $u$ is a
nontrivial solution of \eqref{THEODE4}, i.e., $0 \neq \alpha \neq
u_s$. For notational convenience, we will define the critical
numbers in terms of the sets
\begin{gather*}
 \Gamma_{-}(x_0) = \{x > x_0 ~ | ~ u'(\cdot, \alpha) < 0
~ \textup{on} ~ (x_0,x) \}\\
\Gamma_{+}(x_0) = \{x > x_0 ~ | ~ u'(\cdot, \alpha) > 0 ~
\textup{on} ~  (x_0,x) \}.
  \end{gather*}
Since $u'(0)=0$ and $u''(0)=\beta>0$, then $u'(x,\alpha)>0$ in a
right neighborhood $(0,\delta)$. Thus, for any $u(0)=\alpha<0$ we
define
\begin{equation} \label{critical_point}
  \xi_1(\alpha) =
  \sup\Gamma_+(0),
\end{equation}
\begin{equation} \label{eta_k}
  \eta_k(\alpha) = \begin{cases}
  \sup\Gamma_-(\xi_k(\alpha)) &
  \textup{if } u''(\xi_{k}(\alpha),\alpha)<0
  \textup{ or } u'''(\xi_{k}(\alpha),\alpha)<0 \\
  \xi_k(\alpha) & \textup{if } u''(\xi_{k}(\alpha),\alpha)=0
  \textup{ and } u'''(\xi_{k}(\alpha),\alpha)>0
  \end{cases}
\end{equation}
for $k \geq 1$, and
\begin{equation} \label{xi_k}
  \xi_k(\alpha) =\begin{cases}
  \sup\Gamma_+(\eta_{k-1}(\alpha)) &
  \textup{if } u''(\eta_{k-1}(\alpha),\alpha)>0
  \textup{ or } u'''(\eta_{k-1}(\alpha),\alpha)>0  \\
  \eta_{k-1}(\alpha) & \textup{if } u''(\eta_{k-1}(\alpha),\alpha)=0
  \textup{ and } u'''(\eta_{k-1}(\alpha),\alpha)<0
  \end{cases}
\end{equation}
for $k \geq 2$.

 Theorem~\ref{infmanycritpoints} guarantees that
$\xi_k$ and $\eta_k$ are well defined for all $k \geq 1$ provided
that $u'(x,\alpha)>0$ on an interval of the form $(0,\delta)$. We
will now show that the conditions in the piecewise defined functions
given in \eqref{eta_k} and \eqref{xi_k} encompass all possibilities.
That is, for \eqref{eta_k}, we will show that the negation of the
statement $u''(\xi_{k}(\alpha),\alpha)<0$ or
$u'''(\xi_{k}(\alpha),\alpha)<0$ is $u''(\xi_{k}(\alpha),\alpha) =
0$ and $u'''(\xi_{k}(\alpha),\alpha) > 0$. It follows from
\eqref{critical_point} and \eqref{xi_k} that
$u''(\xi_{k}(\alpha),\alpha) \leq 0$ for all $k \geq 1$. This
reduces the negation of the statement
$u''(\xi_{k}(\alpha),\alpha)<0$ or $u'''(\xi_{k}(\alpha),\alpha)<0$
to $u''(\xi_{k}(\alpha),\alpha) = 0$ and
$u'''(\xi_{k}(\alpha),\alpha) \geq 0$. In the following lemma we
will show that $u'''(\xi_{k}(\alpha),\alpha) > 0$ if
$u''(\xi_{k}(\alpha),\alpha) = 0$.

\begin{lemma} \label{critnumhelper}
Suppose that $u$ is a nontrivial solution of \eqref{THEODE4} with
\\ $u'(x_0)=u''(x_0)=0$ for some $x_0$ in the domain of existence.
Then,
\begin{itemize}
\item[(i)] $u'(x)<0$ in a left
neighborhood of $x_0$ implies that $u'''(x_0)<0$, and
\item[(ii)] $u'(x)>0$ in a left
neighborhood of $x_0$ implies that $u'''(x_0)>0$.
\end{itemize}
\end{lemma}

\begin{proof} of (i): It follows from \eqref{zeroenergyhomo} that
$Q(u(x_0))=0$. That is, $u(x_0)$ is one of the two roots of $Q$,
namely $0$ or $u_s$. Since $u$ is assumed to be nontrivial, then
uniqueness of solutions, (Theorem 7.1 of \cite{CL}), guarantees that
$u'''(x_0)\neq 0$. This together with the assumption that $u'(x)<0$
on $(x_0-\delta,x_0)$ say, leads to $u'''(x_0)<0$.

A similar argument can be used to prove (ii).
\end{proof}

Hence, the negation of $u''(\xi_{k}(\alpha),\alpha)<0$ or
$u'''(\xi_{k}(\alpha),\alpha)<0$ is  $u''(\xi_{k}(\alpha),\alpha) =
0$ and $u'''(\xi_{k}(\alpha),\alpha) > 0$.

Similarly, the negation of $u''(\eta_{k-1}(\alpha),\alpha)>0$ or
$u'''(\eta_{k-1}(\alpha),\alpha)>0$ is \\
  $u''(\eta_{k-1}(\alpha),\alpha)=0$ and
  $u'''(\eta_{k-1}(\alpha),\alpha)<0$. Thus,
  $\xi_k(\alpha)$ and $\eta_k(\alpha)$ are defined in every possible case.

  Another
  question that may arise in the above definitions concerns the need
  for the sign of $u'''(\xi_k(\alpha),\alpha)$ in \eqref{eta_k} and
  $u'''(\eta_{k-1}(\alpha),\alpha)$ in \eqref{xi_k}.  Without
  addressing the sign of the third derivative in these definitions
  it is possible that
  $\xi_k(\alpha)=\xi_{k+1}(\alpha)=\xi_{k+2}(\alpha)\ldots$
  The following lemma ensures that this does not happen.

\begin{lemma} \label{gooddefs}
For any $\alpha < 0$ and any $k \geq 1$ it follows that
\begin{equation}\label{estimatesofcritvals}
 \xi_k(\alpha) \leq \eta_k(\alpha) \leq \xi_{k+1}(\alpha).
 \end{equation}
Furthermore, if $\xi_k(\alpha)=\eta_k(\alpha)$, then
$\eta_k(\alpha)<\xi_{k+1}(\alpha)$. Also, if
$\eta_k(\alpha)=\xi_{k+1}(\alpha)$, then
$\xi_k(\alpha)<\eta_k(\alpha)$.
\end{lemma}

\begin{proof}
First, \eqref{estimatesofcritvals} is a direct consequence of
\eqref{eta_k} and \eqref{xi_k}. Assume that
$\xi_k(\alpha)=\eta_k(\alpha)$. Then \eqref{eta_k} implies that
$u''(\xi_k(\alpha),\alpha)=0$ and $u'''(\xi_k(\alpha),\alpha)>0$.
Thus, $u'''(\eta_k(\alpha),\alpha)>0$ and \eqref{xi_k} result in
$\xi_{k+1}(\alpha)=\sup\Gamma_+(\eta_k(\alpha))>\eta_k(\alpha)$.

Using a similar argument we can show that
$\xi_{k+1}(\alpha)=\eta_k(\alpha)$ implies
$\xi_k(\alpha)<\eta_k(\alpha)$.
\end{proof}

 For a better understanding of \eqref{eta_k} and
\eqref{xi_k} we apply these two definitions to the illustration
given in Figure~\ref{minmaxills}.

\begin{figure}[!ht]
  \begin{center}
 \includegraphics[width=0.6\textwidth]{fig3} % minmaxills.eps
  \end{center}
  \caption{An illustration of a function $u$ such that
  $u''(\xi_k(\alpha),\alpha)=0$ for $k=2, 3, 4$,
  $u'''(\eta_k(\alpha),\alpha)<0$ for $k=1, 2$, and
  $u'''(\xi_4(\alpha),\alpha)>0$.
  The above graph is
  not an actual solution of \eqref{THEODE4}, but merely an interpolating
  polynomial derived using Mathematica.}
  \label{minmaxills}
 \end{figure}


We now prove that $\xi_k$ and $\eta_k$ are continuous functions of
$\alpha$ for all $k \geq 1$. We will do this with the help of the
following lemma.

\begin{lemma} \label{lemxi1cont}
  Suppose that $u(x,\alpha_*)$ is a nonconstant solution of
  \eqref{THEODE4} such that
  $u'(x_*,\alpha_*)=u''(x_*,\alpha_*)=0 \ne u'''(x_*,\alpha_*)$
  for some $x_*>0$ and some $\alpha_* \in \mathbb{R}$.
  Then for any $\epsilon>0$ such that
   \begin{equation} \label{u3nonzerohere}
 u'''(x,\alpha_*) \ne 0\quad \text{on }[x_* - \epsilon,x_* + \epsilon]
   \end{equation}
   it follows that
\begin{itemize}
\item[(i)]
 $u''(x_*-x,\alpha_*)u''(x_*+x,\alpha_*)<0$    on $(0,\epsilon]$.
\end{itemize}
In addition, assume that $\{\alpha_n\}$ is a sequence such that
\begin{equation} \label{alphalimgen}
   \alpha_n \to \alpha_*\quad\text{as }n \to \infty,
 \end{equation}
 and that $u(x,\alpha_n)$ is a nonconstant solution of
 \eqref{THEODE4} for each $n \geq 1$.
 Then there exists $N>0$ such that
  \begin{itemize}
\item[(ii)]
   $u'''(x,\alpha_n)u'''(x,\alpha_*) > 0$
    on $[x_*-\epsilon,x_*+\epsilon]$,
\item[(iii)]
   $u''(x_*-\epsilon,\alpha_n)u''(x_*+\epsilon,\alpha_n)<0$, and
\item[(iv)]    there exists a unique
$\tau_n \in (x_*- \epsilon, x_*+ \epsilon )$ such that
   $u''(\tau_n,\alpha_n)=0$
\end{itemize}
for all $n \geq N$.
   Furthermore, it also follows that
\begin{itemize}
\item[$(v)$] $\tau_n \to x_*$ as $n \to \infty$.
\end{itemize}
\end{lemma}

The above lemma is \cite[Lemma 5.10]{my2ndpaper}.

\begin{theorem} \label{xi1cont}
The function $\xi_1$ as defined in \eqref{critical_point} is a
continuous function of $\alpha$.
\end{theorem}

For a proof of the above theorem, see \cite[Theorem
5.11]{my2ndpaper}.

Theorem~\ref{xi1cont} begins the inductive process that we use to
prove continuity of $\xi_k$ and $\eta_k$. Proving continuity of
$\xi_1$ is sufficient to begin the inductive chain.  That is, we
will assume that $\xi_k$ is continuous to prove that $\eta_k$ is
continuous for arbitrary $k \geq 1$. To complete the induction
process we will show that $\xi_{k+1}$ is continuous under the
assumption that $\eta_k$ is continuous.

We now prove that $\eta_k$ is continuous, given that $\xi_k$ is
continuous, by considering three cases.  Lemma~\ref{gooddefs}
asserts that the three cases are
\begin{itemize}
\item[(1)] $\xi_k(\alpha)<\eta_k(\alpha)<\xi_{k+1}(\alpha)$,
\item[(2)] $\eta_k(\alpha)=\xi_{k+1}(\alpha)$, and
\item[(3)] $\xi_k(\alpha)=\eta_k(\alpha)$.
\end{itemize}
We begin with the simplest case, given by \textbf{(1)}, after the
following lemma.

\begin{lemma}\label{uppisnotzero_HC}
If $u''(\eta_k(\alpha_*),\alpha_*)>0$, then $\eta_k$ is continuous
at $\alpha=\alpha_*$.  Likewise, if
$u''(\xi_k(\alpha_*),\alpha_*)<0$, then $\xi_k$ is continuous at
$\alpha=\alpha_*$.
\end{lemma}

\begin{proof}
Since $\frac{d}{dx}u'(x,\alpha_*)|_{x=\eta_k(\alpha_*)} \ne 0$, then
$\eta_k$ is continuous by way of the Implicit Function Theorem.

A similar argument holds for continuity of $\xi_k$.
\end{proof}

We now state conditions that yield
$u''(\xi_k(\alpha_*),\alpha_*)<0<u''(\eta_k(\alpha_*),\alpha_*)$.

\begin{lemma}\label{killthestrict}
If $\xi_k(\alpha_*)<\eta_k(\alpha_*)<\xi_{k+1}(\alpha_*)$, then
$u''(\eta_k(\alpha_*),\alpha_*)>0$, and $\eta_k$ is continuous at
$\alpha=\alpha_*$. Moreover, if
$\eta_{k-1}(\alpha_*)<\xi_{k}(\alpha_*)<\eta_{k}(\alpha_*)$, then
\\ $u''(\xi_k(\alpha_*),\alpha_*)<0$, and $\xi_k$ is continuous at
$\alpha=\alpha_*$.
\end{lemma}

\begin{proof}
Suppose that $\xi_k(\alpha_*)<\eta_k(\alpha_*)<\xi_{k+1}(\alpha_*)$.
By equation \eqref{eta_k}, it follows that
$u''(\eta_k(\alpha_*),\alpha_*) \geq 0$. For a contradiction, assume
that $u''(\eta_k(\alpha_*),\alpha_*)=0$. Since
$\xi_{k+1}(\alpha_*)>\eta_k(\alpha_*)$ and
$u''(\eta_k(\alpha_*),\alpha_*)=0$, then
$u'''(\eta_k(\alpha_*),\alpha_*)>0$ as a consequence of
\eqref{xi_k}. From this we infer that $u'(x,\alpha_*)>0$ on an
interval of the form $(\eta_k(\alpha_*)-\delta,\eta_k(\alpha_*))$
for some $\delta>0$. But $\xi_k(\alpha_*)<\eta_k(\alpha_*)$ implies
that $u'(x,\alpha_*)<0$ on $(\xi_k(\alpha_*),\eta_k(\alpha_*))$.
Thus, we have obtained a contradiction.  Therefore,
$u''(\eta_k(\alpha_*),\alpha_*)>0$, and continuity of $\eta_k$ at
$\alpha=\alpha_*$ follows from Lemma~\ref{uppisnotzero_HC}.

We can prove that $u''(\xi_k(\alpha_*),\alpha_*)<0$ and continuity
of $\xi_k$ at $\alpha=\alpha_*$ in a similar manner.
\end{proof}

To finish proving continuity of $\eta_k$ at $\alpha=\alpha_*$ we
consider the cases where $\xi_{k+1}(\alpha_*)=\eta_{k}(\alpha_*)$,
and $\xi_{k}(\alpha_*)=\eta_{k}(\alpha_*)$. In the following lemma,
we consider the case where $\xi_{k+1}(\alpha_*)=\eta_{k}(\alpha_*)$.

\begin{lemma} \label{xikrightcont}
Suppose that $\xi_{k}$ is continuous. Then $\eta_k$ is continuous at
any $\alpha=\alpha_*$ such that that
$\xi_{k+1}(\alpha_*)=\eta_{k}(\alpha_*)$, (see
Figure~\ref{xikEetakm1}).
\end{lemma}

\begin{proof}
Let $\epsilon>0$ and suppose that $\{\alpha_n\}$ is a sequence such
that $\alpha_n \to \alpha_*$ as $n \to \infty$. We must show that
there exists $N>0$ such that
\begin{itemize}
\item[(i)] $\eta_{k}(\alpha_n)<\eta_{k}(\alpha_*)+\epsilon$
\item[(ii)] $\eta_{k}(\alpha_n)>\eta_{k}(\alpha_*)-\epsilon$
\end{itemize}
for all $n \geq N$.

(i): For a contradiction, assume that there exists a sequence
$\{\alpha_n\}$ and $\epsilon>0$ such that $\alpha_n \to \alpha_*$ as
$n \to \infty$ and that $\eta_{k}(\alpha_n) \geq
\eta_{k}(\alpha_*)+\epsilon$ for all $n \geq 1$.
Lemma~\ref{gooddefs} and $\xi_{k+1}(\alpha_*)=\eta_{k}(\alpha_*)$
guarantee that $\xi_{k}(\alpha_*)<\eta_{k}(\alpha_*)$, and therefore
$u'(x,\alpha_*)<0$ on $(\xi_{k}(\alpha_*),\eta_{k}(\alpha_*))$.
Since $\xi_{k}(\alpha_*)<\eta_{k}(\alpha_*)$, then \eqref{eta_k}
ensures that
\begin{equation} \label{contrathisstatement}
  u''(\xi_{k}(\alpha_*),\alpha_*)<0\quad \textup{or}\quad
 u'''(\xi_{k}(\alpha_*),\alpha_*)<0.
 \end{equation}

We now show that there exists $N>0$ such that
$\xi_{k}(\alpha_n)<\eta_{k}(\alpha_n)$ for all $n \geq N$.
Otherwise, if there is a subsequence $\{\alpha_{n_j}\}$ such that
$\xi_{k}(\alpha_{n_j})=\eta_{k}(\alpha_{n_j})$ for each $n_j$, then
\eqref{eta_k} asserts that
$u''(\xi_{k}(\alpha_{n_j}),\alpha_{n_j})=0$ and
$u'''(\xi_{k}(\alpha_{n_j}),\alpha_{n_j})>0$ for each $n_j$. By
continuity of $\xi_{k}$ in $\alpha$ and $u''$ and $u'''$ in
$(x,\alpha)$, (Theorem~7.1 of \cite{CL}), it follows that
\begin{gather*}
    0  =  \lim_{n_j\to\infty}
    u''(\xi_{k}(\alpha_{n_j}),\alpha_{n_j})=
    u''(\xi_{k}(\alpha_{*}),\alpha_{*})\quad \textup{and} \\
    0  \leq  \lim_{n_j \to \infty}
    u'''(\xi_{k}(\alpha_{n_j}),\alpha_{n_j})=
    u'''(\xi_{k}(\alpha_{*}),\alpha_{*}).
\end{gather*}
But this contradicts \eqref{contrathisstatement} and proves our
claim that $N>0$ can be chosen such that
$\xi_{k}(\alpha_n)<\eta_{k}(\alpha_n)$ for all $n \geq N$. Without
loss of generality, we can assume that
$\xi_{k}(\alpha_n)<\eta_{k}(\alpha_n)$ for all $n \geq 1$. Thus,
$u'(x,\alpha_n)<0$ on $(\xi_{k}(\alpha_n),\eta_{k}(\alpha_n))$ for
all $n \geq 1$.

Our assumption that $\xi_{k+1}(\alpha_*)=\eta_{k}(\alpha_*)$
together with \eqref{xi_k} imply that
\[
u'''(\eta_{k}(\alpha_*),\alpha_*)<0=u'(\eta_{k}(\alpha_*),\alpha_*)=
u''(\eta_{k}(\alpha_*),\alpha_*).
\]
We can further restrict $\epsilon$ if necessary so that
\[
u'''(x,\alpha_*)<0\quad \textup{on }
[\eta_{k}(\alpha_*)-\epsilon,\eta_{k}(\alpha_*)+\epsilon].
\]
By Lemma~\ref{lemxi1cont}, there exists $N_1>0$ and a sequence
$\{\tau_n\}$ such that
\begin{itemize}
\item[$(a_1)$] $u''(\tau_n,\alpha_n)=0$
\item[$(b_1)$] $\eta_{k}(\alpha_*)-\epsilon<\tau_n<\eta_{k}(\alpha_*)
+\epsilon$
\item[$(c_1)$] $\tau_n \to \eta_{k}(\alpha_*)$ as $n
\to \infty$
\end{itemize}
whenever $n \geq N_1$. Throughout the remainder of this proof we
denote
\[
u_{n}=u(\tau_n,\alpha_n), u_n'=u'(\tau_n,\alpha_n),
~u_n''=u''(\tau_n,\alpha_n),\quad \textup{and}\quad
u_n'''=u'''(\tau_n,\alpha_n).
\]
It follows from continuity of $u'$ and $u'''$ in $(x,\alpha)$
together with $(c_1)$ that \\ $u_n''' \to
u'''(\eta_{k}(\alpha_*),\alpha_*)<0$ and $u_n' \to
u'(\eta_{k}(\alpha_*),\alpha_*)=0$ as $n \to \infty$. Hence, we can
choose $N_2 \geq N_1$ so that
\begin{itemize}
\item[$(a_2)$] $\xi_{k}(\alpha_n)<\xi_{k}(\alpha_*)+\epsilon$,
(by continuity of $\xi_{k}$),
\item[$(b_2)$] $\xi_{k}(\alpha_*)+\epsilon<\tau_n<\eta_{k}(\alpha_*)+\epsilon$, and
\item[$(c_2)$] $u_n'''-(b^2-1)u_n'<0$
\end{itemize}
for all $n \geq N_2$. Now, $(a_2), (b_2)$ and the contradiction
hypothesis $\eta_{k}(\alpha_n) \geq \eta_{k}(\alpha_*)+\epsilon$
ensure that $\xi_{k}(\alpha_n)<\tau_n<\eta_{k}(\alpha_n)$ for each
$n \geq N_2$, and hence $u_n'<0$ for each $n \geq N_2$. To obtain
the desired contradiction, we consider Eq.~\eqref{zeroenergyhomo} at
$x = \tau_n$ which can be written as
\[
u_n'(u_n'''-(b^2-1)u_n')+(b^2+1)^2Q(u_n)=0.
\]
This is a contradiction since $Q(u_n) \geq 0$ for any $u_n$,
$u_n'<0$, and $(c_2)$ for all $n \geq N_2$. This concludes the proof
part (i).

(ii): Since $\xi_k(\alpha_*)<\eta_k(\alpha_*)$ we can restrict
$\epsilon$ so that
$\xi_k(\alpha_*)+\epsilon<\eta_k(\alpha_*)-\epsilon$. By continuity
of $\xi_k$, we can choose $N_1>0$ so that
\begin{equation}\label{xikniscont}
\xi_k(\alpha_*)-\epsilon<\xi_k(\alpha_n)<\xi_k(\alpha_*)+\epsilon
\quad \textup{for all } n \geq N_1.
\end{equation}
To complete the proof of (ii) we must prove that there exists $N_2
\geq N_1$ so that $u'(x,\alpha_n)<0$ on
$(\xi_k(\alpha_n),\eta_k(\alpha_*)-\epsilon]$ whenever $n \geq N_2$.
This will guarantee that
$\eta_k(\alpha_n)>\eta_k(\alpha_*)-\epsilon$ for all $n \geq N_2$.
We will do this in two steps.  We will show that there exists $N_2
\geq N_1$ such that
\begin{itemize}
\item[$(a)$] $u'(x,\alpha_n)<0$ on
$(\xi_k(\alpha_n),\xi_k(\alpha_*)+\epsilon]$ and
\item[$(b)$] $u'(x,\alpha_n)<0$ on
$[\xi_k(\alpha_*)+\epsilon,\eta_k(\alpha_*)-\epsilon]$
\end{itemize}
provided that $n \geq N_2$.

We begin by proving $(a)$. Recall from \eqref{xi_k} that
$u''(\xi_k(\alpha_*),\alpha_*) \leq 0$. First, consider the simpler
of the two cases where $u''(\xi_k(\alpha_*),\alpha_*)<0$. We can
further restrict $\epsilon$ if necessary to ensure that
$u''(x,\alpha_*)<0$ on the compact interval
$[\xi_k(\alpha_*)-\epsilon,\xi_k(\alpha_*)+\epsilon]$. By continuity
of $u''$ in $(x,\alpha)$ we can choose $N_2\geq N_1$ so that
$u''(x,\alpha_n)<0$ on
$[\xi_k(\alpha_*)-\epsilon,\xi_k(\alpha_*)+\epsilon]$ for $n \geq
N_2$. It follows from \eqref{xikniscont} and $N_2 \geq N_1$ that
\[
u'(x,\alpha_n)= \int_{\xi_k(\alpha_n)}^{x}u''(t,\alpha_n)dt<0\quad
\textup{whenever}\quad n \geq N_2,
\]
and $\xi_k(\alpha_n)< x \leq \xi_k(\alpha_*)+\epsilon$. Hence, $(a)$
holds for the case where $u''(\xi_k(\alpha_*),\alpha_*)<0$.

Now assume that $u''(\xi_k(\alpha_*),\alpha_*)=0$. Thus,
\eqref{eta_k} and $\xi_k(\alpha_*)<\eta_k(\alpha_*)$ imply that
$u'''(\xi_k(\alpha_*),\alpha_*)<0$. Again, we restrict $\epsilon$ so
that $u'''(x,\alpha_*)<0$ on the compact interval
$[\xi_k(\alpha_*)-\epsilon,\xi_k(\alpha_*)+\epsilon]$. Continuity of
$u'''$ in $(x,\alpha)$ ensures that we can choose $N_2 \geq N_1$ so
that $u'''(x,\alpha_n)<0$ on
$[\xi_k(\alpha_*)-\epsilon,\xi_k(\alpha_*)+\epsilon]$ whenever $n
\geq N_2$. Now, if $\xi_k(\alpha_n)<x \leq \xi_k(\alpha_n)+\epsilon$
and $n \geq N_2$, then
\begin{equation}\label{Libby}
\begin{aligned}
0 & >   \int_{\xi_k(\alpha_n)}^{x}\int_{\xi_k(\alpha_n)}^{t}
u'''(s,\alpha_n)\,ds\,dt \vspace{.1in} \\
& =
u'(x,\alpha_n)-u''(\xi_k(\alpha_n),\alpha_n)(x-\xi_k(\alpha_n)).
\end{aligned}
\end{equation}
Combining \eqref{Libby} with the fact that
$u''(\xi_k(\alpha_n),\alpha_n) \leq 0$
we obtain \\
$u'(x,\alpha_n)<u''(\xi_k(\alpha_n),\alpha_n)(x-\xi_k(\alpha_n))
\leq 0$ whenever $\xi_k(\alpha_n)<x \leq \xi_k(\alpha_n)+\epsilon$
and $n \geq N_2$. This completes the proof of part $(a)$.

To prove $(b)$ note that $u'(x,\alpha_*)<0$ on
$[\xi_k(\alpha_*)+\epsilon,\eta_k(\alpha_*)-\epsilon]$. Since $u'$
is continuous in $(x,\alpha)$, we can choose $N>0$ so that
$u'(x,\alpha_n)<0$ on the interval
$[\xi_k(\alpha_*)+\epsilon,\eta_k(\alpha_*)-\epsilon]$ for each $n
\geq N$. This completes the proof of part $(b)$ as well as the proof
of the lemma.
\end{proof}

\begin{figure}[!ht]
  \begin{center}
 \includegraphics[width=0.6\textwidth]{fig4}   % xikEetakm1.eps
  \end{center}
 \caption{The case where $\xi_k(\alpha_*)=\eta_{k-1}(\alpha_*)$ for
  $k =2$.}
 \label{xikEetakm1}
 \end{figure}

In the following lemma, we prove continuity of $\eta_k$ at
$\alpha=\alpha_*$ for $\xi_k(\alpha_*)=\eta_k(\alpha_*)$.

\begin{lemma} \label{etakcontxikEetak}
  Assume that $\xi_k$ is continuous.  Then $\eta_k$ is continuous at
  any $\alpha_*$ such that $\xi_k(\alpha_*)=\eta_k(\alpha_*)$.
 \end{lemma}

\begin{proof}
Let $\epsilon>0$ and suppose that $\{\alpha_n\}$ is a sequence such
that $\alpha_n \to \alpha_*$ as $n \to \infty$. We must show that
there exists $N>0$ so that
\begin{itemize}
\item[(i)] $\eta_k(\alpha_n) > \eta_k(\alpha_*)-\epsilon$ and
\item[(ii)] $\eta_k(\alpha_n) < \eta_k(\alpha_*)+\epsilon$
\end{itemize}
for all $n \geq N$. To prove (i) we use continuity of $\xi_k$ and
the fact that \\ $\eta_k(\alpha_n) \geq \xi_k(\alpha_n)$ for any
$\alpha_n$. Hence,
\[
\eta_k(\alpha_n) \geq \xi_k(\alpha_n) \to
\xi_k(\alpha_*)=\eta_k(\alpha_*).
\]
This proves (i).

(ii): We proceed by contradiction and assume that
\begin{equation}\label{conhyp_HC}
  \eta_k(\alpha_n) \geq \eta_k(\alpha_*)+\epsilon.
\end{equation}
By continuity of $\xi_k$, there exists $N>0$ so that
$\xi_k(\alpha_n) < \xi_k(\alpha_*)+\epsilon/2$ whenever $n \geq N$.
Combining this estimate with \eqref{conhyp_HC} yields
\begin{equation}\label{GEST}
 \xi_k(\alpha_n)<
  \xi_k(\alpha_*)+\epsilon/2=\eta_k(\alpha_*)+\epsilon/2
 < \eta_k(\alpha_n)\quad \textup{whenever}\quad n \geq N.
\end{equation}
Therefore, $u'(x,\alpha_n)<0$ on
$(\xi_k(\alpha_n),\eta_k(\alpha_n))$. In particular,
$u'(\xi_k(\alpha_*)+\epsilon/2,\alpha_n)<0$ for all $n \geq N$ as a
consequence of \eqref{GEST}.  Now by Lemma~\ref{gooddefs} and the
assumption that $\xi_k(\alpha_*)=\eta_k(\alpha_*)$ it follows that
$\xi_{k+1}(\alpha_*)>\eta_k(\alpha_*)$. If necessary, we restrict
$\epsilon$ so that $\xi_k(\alpha_*)+\epsilon/2<\xi_{k+1}(\alpha_*)$.
Doing so guarantees that
$u'(\xi_k(\alpha_*)+\epsilon/2,\alpha_*)>0$. This and continuity of
$u'$ in $(x,\alpha)$ contradicts that
$u'(\xi_k(\alpha_*)+\epsilon/2,\alpha_n)<0$ for each $n \geq N$.
That is,
\[
0<   u'(\xi_k(\alpha_*)+\epsilon/2,\alpha_*)= \lim_{n \to \infty}
u'(\xi_k(\alpha_*)+\epsilon/2,\alpha_n) \leq 0.
\]
\end{proof}

We now summarize Lemmas~\ref{killthestrict}, \ref{xikrightcont}, and
\ref{etakcontxikEetak} with the following theorem.
\begin{theorem}\label{etakiscontgeneral}
If $\xi_k$ is continuous, then $\eta_k$ is continuous.
\end{theorem}

We now prove that $\xi_{k+1}$ is continuous given that $\eta_k$ is
continuous.  As we did in proving continuity of $\eta_k$, we will
consider three separate cases.  The case in which
$\eta_k(\alpha)<\xi_{k+1}(\alpha)<\eta_{k+1}(\alpha)$ follows from
Lemma~\ref{killthestrict}.  Thus, all that remains are the cases
where $\eta_k(\alpha)=\xi_{k+1}(\alpha)$ and
$\xi_{k+1}(\alpha)=\eta_{k+1}(\alpha)$. The proofs that follow are
very similar to those in Lemmas~\ref{xikrightcont} and
\ref{etakcontxikEetak}.  For this reason we will merely sketch the
proofs of continuity of $\xi_{k+1}$.

\begin{lemma} \label{xikP1iscont1}
Suppose that $\eta_k$ is continuous.  Then $\xi_{k+1}$ is continuous
at any $\alpha=\alpha_*$ such that
$\eta_k(\alpha_*)=\xi_{k+1}(\alpha_*)$.
\end{lemma}

\begin{proof}
Let $\epsilon>0$ and $\{\alpha_n\}$ be a sequence such that
$\alpha_n \to \alpha_*$ as $n \to \infty$. We must show that there
exists $N>0$ such that
\begin{itemize}
\item[(i)] $\xi_{k+1}(\alpha_n)>\xi_{k+1}(\alpha_*)-\epsilon$
\item[(ii)] $\xi_{k+1}(\alpha_n)<\xi_{k+1}(\alpha_*)+\epsilon$
\end{itemize}
whenever $n \geq N$.

(i):  Note that $\xi_{k+1}(\alpha_n) \geq \eta_k(\alpha_n)$ for any
$\alpha_n$. This together with our assumption that $\eta_k$ is
continuous yield
\[
\xi_{k+1}(\alpha_n) \geq \eta_k(\alpha_n) \to \eta_k(\alpha_*) =
\xi_{k+1}(\alpha_*)\quad \textup{as } n \to \infty.
\]
This completes the proof of (i).

(ii): To obtain a contradiction, assume that $\xi_{k+1}(\alpha_n)
\geq \xi_{k+1}(\alpha_*)+\epsilon$. It follows from
Lemma~\ref{gooddefs} and $\eta_{k}(\alpha_*)=\xi_{k+1}(\alpha_*)$
that $\xi_{k+1}(\alpha_*)<\eta_{k+1}(\alpha_*)$, and hence
$u'(x,\alpha_*)<0$ on $(\xi_{k+1}(\alpha_*),\eta_{k+1}(\alpha_*))$.
We will restrict $\epsilon$ if necessary so that
$\xi_{k+1}(\alpha_*)+\epsilon/2<\eta_{k+1}(\alpha_*)$. This
guarantees that $u'(\xi_{k+1}(\alpha_*)+\epsilon/2,\alpha_*)<0$. Now
choose $N>0$ so that
$\eta_k(\alpha_n)<\eta_{k}(\alpha_*)+\epsilon/2=
  \xi_{k+1}(\alpha_*)+\epsilon/2$ for $n \geq N$. Hence,
  $\eta_k(\alpha_n)<\xi_{k+1}(\alpha_*)+\epsilon/2<\xi_{k+1}(\alpha_n)$
for each $n \geq N$. But this implies that
$u'(\xi_{k+1}(\alpha_*)+\epsilon/2,\alpha_n)>0$ for each $n \geq N$.
This and $u'(\xi_{k+1}(\alpha_*)+\epsilon/2,\alpha_*)<0$ lead to a
contradiction. This completes the proof of (ii) as well as the
lemma.
\end{proof}

All that remains is the case where
$\eta_{k+1}(\alpha_*)=\xi_{k+1}(\alpha_*)$.

\begin{lemma}\label{xikP1cont2}
Suppose that $\eta_k$ is continuous.  Then $\xi_{k+1}$ is continuous
at any $\alpha=\alpha_*$ such that
$\eta_{k+1}(\alpha_*)=\xi_{k+1}(\alpha_*)$.
\end{lemma}

\begin{proof}
Let $\epsilon>0$ and $\{\alpha_n\}$ be a sequence such that
$\alpha_n \to \alpha_*$ as $n \to \infty$. We must show that there
exists $N>0$ such that
\begin{itemize}
\item[(i)] $\xi_{k+1}(\alpha_n)>\xi_{k+1}(\alpha_*)-\epsilon$
\item[(ii)] $\xi_{k+1}(\alpha_n)<\xi_{k+1}(\alpha_*)+\epsilon$
\end{itemize}
whenever $n \geq N$.

(i): We first show that there exists $N>0$ so that
$\eta_k(\alpha_n)<\xi_{k+1}(\alpha_n)$ for all $n \geq N$. Since
$\xi_{k+1}(\alpha_*)=\eta_{k+1}(\alpha_*)$, then
Lemma~\ref{gooddefs} implies that
$\eta_k(\alpha_*)<\xi_{k+1}(\alpha_*)$. Hence, by \eqref{xi_k}, it
follows that either $u''(\eta_k(\alpha_*),\alpha_*)>0$ or
$u'''(\eta_k(\alpha_*),\alpha_*)>0$. It now follows from continuity
of $\eta_k$ in $\alpha$ and $u''$ and $u'''$ in $(x,\alpha)$ that
there exists $N>0$ so that either $u''(\eta_k(\alpha_n),\alpha_n)>0$
or $u'''(\eta_k(\alpha_n),\alpha_n)>0$ for all $n \geq N$. Thus,
$\eta_k(\alpha_n)<\xi_{k+1}(\alpha_n)$ for all $n \geq N$, and
$u'(x,\alpha_n)>0$ on $(\eta_k(\alpha_n),\xi_{k+1}(\alpha_n))$ for
all $n \geq N$. Without loss of generality, assume that
$\eta_k(\alpha_n)<\xi_{k+1}(\alpha_n)$ for all $n \geq 1$.

Our goal now is to show that there exists $N>0$ so that
$u'(x,\alpha_n)>0$ on
$(\eta_k(\alpha_n),\xi_{k+1}(\alpha_*)-\epsilon]$ for all $n \geq
N$. This will be done in two steps.  That is, we will show that
there exists $N>0$ so that
\begin{itemize}
\item[$(a)$] $u'(x,\alpha_n)>0$ on $(\eta_k(\alpha_n),\eta_k(\alpha_*)+\epsilon]$,
          and
\item[$(b)$] $u'(x,\alpha_n)>0$ on
          $[\eta_k(\alpha_*)+\epsilon,\xi_{k+1}(\alpha_*)-\epsilon]$
\end{itemize}
for each $n \geq N$. For technical purposes we restrict $\epsilon$
so that $\eta_k(\alpha_*)+\epsilon<\xi_{k+1}(\alpha_*)-\epsilon$. By
continuity of $\eta_k$ there exists $N_1>0$ so that
$\eta_k(\alpha_*)-\epsilon<\eta_k(\alpha_n)<\eta_k(\alpha_*)+\epsilon$
for all $n \geq N_1$.

By \eqref{eta_k} it follows that $u''(\eta_k(\alpha_*),\alpha_*)
\geq 0$. We begin proving $(a)$ by assuming that
$u''(\eta_k(\alpha_*),\alpha_*)>0$. We further restrict $\epsilon$
so that $u''(x,\alpha_*)>0$ on the interval
$[\eta_k(\alpha_*)-\epsilon,\eta_k(\alpha_*)+\epsilon]$. By
continuity of $u''$ there exists $N_2\geq N_1$ so that
 $u''(x,\alpha_n)>0$ on
$[\eta_k(\alpha_*)-\epsilon,\eta_k(\alpha_*)+\epsilon]$ whenever $n
\geq N_2$. Hence,
\[
u'(x,\alpha_n) = \int_{\eta_k(\alpha_n)}^{x}u''(t,\alpha_n)dt>0
\]
whenever $\eta_k(\alpha_n)< x \leq \eta_k(\alpha_*)+\epsilon$ and $n
\geq N_2$. This proves $(a)$ for the case
$u''(\eta_k(\alpha_*),\alpha_*)>0$.

Now assume that $u''(\eta_k(\alpha_*),\alpha_*)=0$. Then
$u'''(\eta_k(\alpha_*),\alpha_*)>0$ as a consequence of
$\xi_{k+1}(\alpha_*)>\eta_k(\alpha_*)$ and \eqref{xi_k}. Again we
restrict $\epsilon$ to ensure that $u'''(x,\alpha_*)>0$ on
$[\eta_k(\alpha_*)-\epsilon,\eta_k(\alpha_*)+\epsilon]$. If follows
by continuity of $u'''$ in $(x,\alpha)$ that there exists $N_2 \geq
N_1$ so that $u'''(x,\alpha_n)>0$ on
$[\eta_k(\alpha_*)-\epsilon,\eta_k(\alpha_*)+\epsilon]$ whenever $n
\geq N_2$. Thus,
\begin{equation}\label{upppposcon}
\begin{aligned}
 0 <  \int_{\eta_k(\alpha_n)}^{x}\int_{\eta_k(\alpha_n)}^{t}
 u'''(s,\alpha_n)\,ds\,dt  \\
& =
u'(x,\alpha_n)-u''(\eta_k(\alpha_n),\alpha_n)(x-\eta_k(\alpha_n))
 \end{aligned}
\end{equation}
 whenever $\eta_k(\alpha_n)<x \leq \eta_k(\alpha_*)+\epsilon$ and
$n \geq N_2$.
  Since $u''(\eta_k(\alpha_n),\alpha_n) \geq 0$,
  then $u'(x,\alpha_n)>0$ on $(\eta_k(\alpha_n),\eta_k(\alpha_*)+\epsilon]$
  follows from \eqref{upppposcon}.  This completes $(a)$.

 To prove $(b)$, note that $u'(x,\alpha_*)>0$ on
 $[\eta_k(\alpha_*)+\epsilon,\xi_{k+1}(\alpha_*)-\epsilon]$. Then continuity
 of $u'$ in $(x,\alpha)$ ensures that there exists $N>0$ so that
 $u'(x,\alpha_n)>0$ on
 $[\eta_k(\alpha_*)+\epsilon,\xi_{k+1}(\alpha_*)-\epsilon]$ whenever
 $n \geq N$. This completes the proof of part (i).


(ii): For a contradiction, assume that $\xi_{k+1}(\alpha_n) \geq
\xi_{k+1}(\alpha_*)+\epsilon$ for all $n \geq 1$. Our premise that
$\eta_{k+1}(\alpha_*)=\xi_{k+1}(\alpha_*)$ and Lemma~\ref{gooddefs}
imply that $\eta_k(\alpha_*)<\xi_{k+1}(\alpha_*)$. For technical
purposes we restrict $\epsilon$ so that
\begin{equation}\label{helpperMe}
 \eta_k(\alpha_*)+\epsilon<\xi_{k+1}(\alpha_*)-\epsilon.
 \end{equation}
Since $\eta_k$ is continuous, there exists $N_1>0$ so that
\begin{equation}\label{usecontofetak}
\eta_k(\alpha_n)<\eta_k(\alpha_*)+\epsilon\quad \textup{for all }
 n \geq N_1.
 \end{equation} This together with our contradiction premise,
$\xi_{k+1}(\alpha_n) \geq \xi_{k+1}(\alpha_*)+\epsilon$, yield that
$\eta_k(\alpha_n)<\xi_{k+1}(\alpha_n)$ for all $n \geq N_1$, and
therefore $u'(x,\alpha)>0$ on
$(\eta_k(\alpha_n),\xi_{k+1}(\alpha_n))$ for all $n \geq N_1$.

 By \eqref{eta_k},
$\eta_{k+1}(\alpha_*)=\xi_{k+1}(\alpha_*)$ guarantees that
$u'''(\xi_{k+1}(\alpha_*),\alpha_*)>0$. We can further restrict
$\epsilon$ so that $u'''(x,\alpha_*)>0$ on
$[\xi_{k+1}(\alpha_*)-\epsilon,\xi_{k+1}(\alpha_*)+\epsilon]$. It
now follows from Lemma~\ref{lemxi1cont} that there exists $N_2 \geq
N_1$, and a sequence $\{\tau_n\}$ so that
\begin{itemize}
\item[$(a)$] $u''(\tau_n,\alpha_n)=0$,
\item[$(b)$] $\xi_{k+1}(\alpha_*)-\epsilon<
\tau_n<\xi_{k+1}(\alpha_*)+\epsilon$, and
\item[$(c)$] $\tau_n \to \xi_{k+1}(\alpha_*)$ as $n \to \infty$
\end{itemize}
for all $n \geq N_2$.

Combining the contradiction premise with \eqref{helpperMe},
\eqref{usecontofetak}, and $(b)$ yield that
$\eta_k(\alpha_n)<\tau_n<\xi_{k+1}(\alpha_n)$ for each $n \geq N_2$.
Using the notation
\[u_{n}=u(\tau_n,\alpha_n), u_n'=u'(\tau_n,\alpha_n),
~u_n''=u''(\tau_n,\alpha_n),\quad \textup{and}\quad
u_n'''=u'''(\tau_n,\alpha_n),\] we obtain $u_n'>0$ for each $n \geq
N_2$.

Combining $(c)$ with continuity of $u', u'''$ results in $u_n' \to
u'(\xi_{k+1}(\alpha_*),\alpha_*)=0$ and $u_n''' \to
u'''(\xi_{k+1}(\alpha_*),\alpha_*)>0$. Thus, we can choose $N_3\geq
N_2$ so that \\$u_n'''-(b^2-1)u_n'>0$ for all $n \geq N_3$. Now
Eq.~\eqref{zeroenergyhomo} can be written as
\[
u_n'(u_n'''-(b^2-1)u_n')+(b^2+1)^2Q(u_n)=0.
\]
But the left side of the above equation is strictly positive for all
$n \geq N_3$. This is the desired contradiction.  This concludes the
proof of the lemma.
\end{proof}

We now summarize Lemmas~\ref{killthestrict}, \ref{xikP1iscont1}, and
\ref{xikP1cont2} in the following theorem.

\begin{theorem} \label{xikP1general}
If $\eta_k$ is continuous, then $\xi_{k+1}$ is continuous.
\end{theorem}

Theorems~\ref{xi1cont}, \ref{etakiscontgeneral} and
\ref{xikP1general} show that $\xi_k$ and $\eta_k$ are continuous for
all $k \geq 1$.

\section{N-Bump Solutions} \label{nbumpsolutions}

In this section we prove the existence of $N$-bump solutions of
\eqref{THEODE4} for any even valued $N>0$. Recall that a solution is
an $N$-bump solution if there are exactly $N$ disjoint intervals in
which $u>th$, i.e., $u$ exceeds threshold, and $u$ must also satisfy
the limiting property $(u,u',u'',u''')\to (0,0,0,0)$. We will
continue to assume that the initial values of \eqref{THEODE4} are
$\alpha<0$ and $\beta=-(b^2+1)\alpha>0$. In addition, we will assume
that the conditions of Theorem~\ref{infmanycritpoints} hold.  This
guarantees that the functions $\eta_k$ and $\xi_k$ in
\eqref{critical_point}, \eqref{eta_k}, and \eqref{xi_k} are well
defined.

We now proceed to develop a shooting method to prove the existence
of multi-bump homoclinic solutions. This shooting method requires
that we precisely determine the behavior of $u(x,\alpha)$ at (or
near) the parameter value $\alpha = \alpha'$ where
\begin{itemize}
  \item[$(a)$] $u(\xi_k(\alpha'),\alpha')=u_s, (\textup{or}\quad
  u(\xi_k(\alpha'),\alpha')=0)$,
  for some $k \geq 1$,
\item[$(b)$] $u(a_0,\alpha')=th$, where $a_0 = \pm \xi_k(\alpha')$.
  \end{itemize}

\subsection*{Solutions with the critical values $u=0$ or $u=u_s$}

Recall that when  $(r,b,th)\in \Lambda$, then \eqref{THEODE4} has
exactly two constant solutions, namely $u \equiv 0$ and $u \equiv
u_s>0$. Recall also that these constant solutions correspond to the
only two roots of the function $Q$ defined by \eqref{Q_HC}. That is,
$Q(0)=Q(u_s)=0$, (see Figure~\ref{yetanotherQ}).

 In Lemma~\ref{adjpairs} we let $u_*$ denote one of
the two roots of the function $Q$, i.e., either $u_*=0$ or
$u_*=u_s$. It will be shown that if $u(\xi_k(\alpha),\alpha)$
crosses the line $u=u_*$ from below at some $\alpha=\alpha'<0$ and
$k \geq 1$, then $u(\eta_k(\alpha),\alpha)$ also crosses the line
$u=u_*$ from below. In other words, if
$u(\xi_k(\alpha'),\alpha')=u_*$ with
$u'''(\xi_k(\alpha'),\alpha')>0$, (see Figure~\ref{x}), then
$\xi_k(\alpha')=\eta_k(\alpha')$ and $u(\xi_k(\alpha),\alpha)-u_*$
has the same sign as $u(\eta_k(\alpha),\alpha)-u_*$ on an interval
of the form $(\alpha'-\delta,\alpha')$ for some $\delta>0$. Using
similar arguments we can show that $u(\xi_k(\alpha),\alpha)-u_*$ has
the same sign as $u(\eta_k(\alpha),\alpha)-u_*$ on an interval of
the form $(\alpha',\alpha'+\delta)$. There are other cases which
occur.  There is the case where $u(\xi_k(\alpha),\alpha)$ crosses
the line $u = u_*$ at $\alpha=\alpha'$ from above. That is,
$u(\xi_k(\alpha'),\alpha')=u_*$ with
$u'''(\xi_k(\alpha'),\alpha')<0$, and hence
$\xi_k(\alpha')=\eta_{k-1}(\alpha')$. In this case, we can apply
similar arguments to show that $u(\eta_{k-1}(\alpha),\alpha)$ must
also cross $u = u_*$ from above.  The method employed in the
following lemma can be easily modified to prove the other cases.

\begin{figure}[!ht]
  \begin{center}
\includegraphics[width=0.42\textwidth]{fig5a} \hspace{0.5in} %  plateau2.eps
\includegraphics[width=0.42\textwidth]{fig5b}  %  plateau.eps
\end{center}
\begin{center}
\includegraphics[width=0.42\textwidth]{fig5c}  %  plateau3.eps
\end{center}
  \caption{See text}
  \label{x}
 \end{figure}

\begin{lemma} \label{adjpairs}
Suppose that $(r,b,th) \in \Lambda$, (recall Eq.~\eqref{lamda2}),
and that $Q(u_*)=0$, (that is, $u_*=0\quad \textup{or}\quad
u_*=u_s$). Also, assume that $\alpha'<0$ and
$\beta'=-(b^2+1)\alpha'>0$ determine a solution $u(\cdot,\alpha')$
of \eqref{THEODE4} such that
  \[u(\xi_k(\alpha'),\alpha')=u_*\quad \textup{for some}\quad k \geq 1.\]
 If there exists $\delta>0$ such that
  \begin{equation} \label{eqadj}
    u(\xi_k(\alpha),\alpha)>u_*\quad \textup{for}\quad
    \alpha'-\delta<\alpha<\alpha',
  \end{equation}
then there exists $\epsilon>0$ so that
  \begin{itemize}
 \item[(i)]
    $u(\xi_k(\alpha),\alpha)>u(\eta_k(\alpha),\alpha)>u_*\quad
  \textup{for}\quad \alpha' - \epsilon < \alpha < \alpha'$ \\
    given that $u'''(\xi_k(\alpha'),\alpha')>0$, and
 \item[(ii)]
 $u(\xi_k(\alpha),\alpha)>u(\eta_{k-1}(\alpha),\alpha)>u_*,\quad \textup{(provided that}\quad k
 \geq 2\textup{),}\quad
  \textup{for}\quad \alpha'-\epsilon < \alpha < \alpha'$ \\
    given that $u'''(\xi_k(\alpha'),\alpha')<0$.
 \end{itemize}
\end{lemma}

  \begin{proof} (i):
Assume that $u'''(\xi_k(\alpha'),\alpha')>0$. Since
$u(\xi_k(\alpha'),\alpha')=u_*$, \\ $Q(u_*)=0$, and
$u'(\xi_k(\alpha'),\alpha')=0$, then Eq.~\eqref{zeroenergyhomo}
shows that $u''(\xi_k(\alpha'),\alpha')=0$. Hence,
$\xi_k(\alpha')=\eta_k(\alpha')$ as a consequence of \eqref{eta_k}.

The proof of (i) is by contradiction, i.e.,
 assume that there is an increasing sequence $\alpha_n$ such that
 \begin{itemize}
 \item[$(a)$]
 $\alpha_n \to \alpha'$ as $n \to \infty$,
 \item[$(b)$]
  $\alpha'-\delta<\alpha_n< \alpha'$, and
 \item[$(c)$]
 $u(\eta_k(\alpha_n),\alpha_n) \leq u_*$ for all $n \geq 1$.
 \end{itemize}

 Our first task is to show that $u(\eta_k(\alpha_n),\alpha_n) = u_*$
 is impossible whenever $n$ is sufficiently large.  For a contradiction
 assume that there is a subsequence $\alpha_{n_i}$ such that
 $u(\eta_k(\alpha_{n_i}),\alpha_{n_i})=u_*$ for each $i \geq 1$.
 At $x=\eta_k(\alpha_{n_i})$ Eq.~\eqref{zeroenergyhomo} gives
 \begin{equation} \label{u20homo}
    u''(\eta_k(\alpha_{n_i}),\alpha_{n_i})=0\quad \textup{for each }
     i \geq 1.
 \end{equation}
 Since $(b)$ holds, then \eqref{eqadj} ensures that
 $u(\xi_k(\alpha_{n_i}),\alpha_{n_i})> u_*$ for each $i \geq 1$.
 Hence, $u'(x,\alpha_{n_i})<0$ on $(\xi_k(\alpha_{n_i}),\eta_k(\alpha_{n_i}))$.
 By combining this result with Lemma~\ref{critnumhelper} and \eqref{u20homo}
 we see that
 $u'''(\eta_k(\alpha_{n_i}),\alpha_{n_i}) < 0$ for each $i \geq 1$.
 Continuity of $u'''$ in $(x,\alpha)$ and $\eta_k$ in $\alpha$
 yields
  \begin{equation} \label{u3homo}
    \lim_{i \to \infty}u'''(\eta_k(\alpha_{n_i}),\alpha_{n_i})
    = u'''(\eta_k(\alpha'),\alpha') \leq 0.
  \end{equation}
  But this is impossible since
  $u'''(\eta_k(\alpha'),\alpha')=u'''(\xi_k(\alpha'),\alpha')>0$.
  Therefore, we may assume that
  \begin{equation}\label{lessthanu*}
  u(\eta_k(\alpha_n),\alpha_n)<u_*\quad \textup{for all } n \geq 1.
  \end{equation}

  Next we show that
  \begin{equation}\label{Qbiggerthan0}
  Q(u(\eta_k(\alpha_n),\alpha_n))>0\quad \textup{and}\quad
  Q(u(\xi_k(\alpha_n),\alpha_n))>0\quad \textup{for all } n \geq 1.
  \end{equation}
Recall that $u_*$ denotes one of the two roots of $Q$, namely
$u_*=0$ or $u_*=u_s$. First assume that $u_*=u_s$. Then
$Q(u(\xi_k(\alpha_n),\alpha_n))>0$ follows from \eqref{eqadj} and
$(b)$. Continuity of $\eta_k$ in $\alpha$ and $u$ in $(x,\alpha)$,
gives
\[
\lim_{n \to \infty}u(\eta_k(\alpha_n),\alpha_n)=
                           u(\eta_k(\alpha'),\alpha')=
                           u(\xi_k(\alpha'),\alpha')=
                           u_s>0.
 \]
This and \eqref{lessthanu*} show that there exists $N>0$ so that
$u_s>u(\eta_k(\alpha_n),\alpha_n)>0$ for all $n \geq N$. Thus,
$Q(u(\eta_k(\alpha_n),\alpha_n))>0$ for all $n \geq N$. If $u_*=0$,
then a similar argument shows that $N>0$ exists so that
$u_s>u(\xi_k(\alpha_n),\alpha_n)>0$ whenever $n \geq N$. This
guarantees that $Q(u(\xi_k(\alpha_n),\alpha_n))>0$ whenever $n \geq
N$. Without loss of generality, we assume that \eqref{Qbiggerthan0}
holds for all $n \geq 1$.

 We now show that there exists
$x_n \in (\xi_k(\alpha_n),\eta_k(\alpha_n))$
 at which
   \begin{equation} \label{lya}
   -\frac{(u''(x_n,\alpha_n))^2}{2}+(b^2+1)^2Q(u(x_n,\alpha_n))=0
   \end{equation}
   for all $n \geq 1$.
An immediate consequence of \eqref{Qbiggerthan0} and
\eqref{zeroenergyhomo} at $x=\xi_k(\alpha_n)$ is that
$u''(\xi_k(\alpha_n),\alpha_n) < 0$ for each $n \geq 1$. Likewise,
it follows that $u''(\eta_k(\alpha_n),\alpha_n)>0$ for each $n \geq
1$. Thus, there exists a value $y_n \in
(\xi_k(\alpha_n),\eta_k(\alpha_n))$ such that
 $u''(y_n,\alpha_n)=0$.
 Note that $u(\xi_k(\alpha_n),\alpha_n)>u_*>u(\eta_k(\alpha_n),\alpha_n)$
 implies the existence of
 $z_n \in (\xi_k(\alpha_n),\eta_k(\alpha_n))$ such that
 $u(z_n,\alpha) = u_*$, hence $Q(u(z_n,\alpha_n))=0$ for each
 $n \geq 1$. Thus, we have
 \begin{equation} \label{piecewisehomo}
   -\frac{(u''(x,\alpha_n))^2}{2}+(b^2+1)^2Q(u(x,\alpha_n)) \ \
\left\{
\begin{array}{ll}
  \leq 0\quad \textup{at}\quad x=z_n \\
  \geq 0\quad \textup{at}~x=y_n.
 \end{array}
\right.
\end{equation}
 This ensures the existence of a value $x_n$ between $y_n, z_n$ for which
 \eqref{lya} holds.  Also,
 $\xi_k(\alpha_n)<x_n<\eta_k(\alpha_n)$ implies that
 $u'(x_n,\alpha_n) < 0$ for each $n \geq 1$. This fact together
 with \eqref{zeroenergyhomo} yields
 \begin{equation} \label{uconsthomo}
   u'''(x_n,\alpha_n)-(b^2-1)u'(x_n,\alpha_n)=0\quad \textup{for each}
   \quad n \geq 1.
 \end{equation}
 Since
 $\xi_k(\alpha_n) \to \xi_k(\alpha')$
 and $\eta_k(\alpha_n) \to \eta_k(\alpha')=\xi_k(\alpha')$ as
 $n \to \infty$,
 then  $\xi_k(\alpha_n)<x_n<\eta_k(\alpha_n)$ implies that
 $x_n \to \xi_k(\alpha')$ as $n \to \infty$.
Therefore,
 \[
u'(x_n,\alpha_n) \to u'(\xi_k(\alpha'),\alpha')= 0\quad \textup{as }
 n \to \infty
\]
 by continuity of $u'$ in $(x,\alpha)$.
 This and \eqref{uconsthomo} imply that
 \begin{equation} \label{u3hom}
  u'''(x_n,\alpha_n) \to 0\quad \textup{as } n \to \infty.
  \end{equation}
But this contradicts $u'''(x_n,\alpha_n) \to
u'''(\xi_k(\alpha'),\alpha')>0$ as $n \to \infty$.
  This concludes the proof of (i).

  We can prove (ii) in a similar fashion.
\end{proof}

\subsection*{Solutions with a local maximum at $u=th$}

 A solution that plays a crucial role in our shooting argument is
one that satisfies
  \begin{equation} \label{bigamplitudehom}
  u(a_0,\alpha')= th,\quad \textup{where } a_0 = \pm \xi_k(\alpha')
  \end{equation}
  for some $k \geq 1$ (see Figure~\ref{qlemxguideII}).
  The object of Lemmas~\ref{genxlemma}-\ref{genlemx}
  is to show that  $|u(\zeta)| > th$ for every $\zeta > a_0$ such that
  $u'(\zeta)=0$. More precisely, Lemma~\ref{genxlemma}
  is used in the proof of Lemma~\ref{genlemx} to establish the sharper
  estimate
  $u(b_0) \leq -u_{ss}(r) < -th$ where
  \[
  b_0 = \sup\{x>a_0~|~u'(x)<0\quad \textup{on } (a_0,x) \}.
 \]

  Throughout Lemmas~\ref{genxlemma}-\ref{nextlemx}
  it is not necessary to continually refer to the initial condition
  parameter $\alpha$. Hence, we omit the initial condition parameter and
  write $u(x)$ instead of $u(x,\alpha)$.

  The following lemma relies on several preliminary results
  established in Section~\ref{prelimresults}. First,
  Lemma~\ref{u_sofrlimit} states that for any fixed $0<th<2$ there
  exists  $b_r > 1$ such that
  $(r,b_r,th) \in \Lambda$ whenever $r > 0$ is sufficiently small.
  and
  \[
  b_r \to \frac{2}{th} + \frac{\sqrt{4-th^2}}{th}\quad
\textup{as } r \to 0^+.
\]
 The fact that
\[
\frac{2}{th} + \frac{\sqrt{4-th^2}}{th}>1
\]
 for any fixed $0<th<2$ implies that $b_r$ is bounded away from $1$
 as $r \to 0^+$. In particular, there exists a value
 $R>0$ such that
   \begin{equation} \label{brawayfrom1hom}
   b_r \geq 1 + m\quad \textup{for all }  r \in (0,R)
   \end{equation}
   where
\begin{equation} \label{halfdis}
  m =
  \frac{1}{2}\Big(\frac{2}{th} + \frac{\sqrt{4-th^2}}{th} - 1\Big).
\end{equation}
In addition, Lemma~\ref{genxlemma} relies on the fact that
 \begin{equation} \label{ussr2hom}
   u_{ss}(r)-th \to 0^+\quad \textup{as } r \to 0^+,
 \end{equation}
which follows immediately from Lemma~\ref{lemxchaos}.

 \begin{lemma} \label{genxlemma}
 Suppose that $u$ is a nontrivial solution of
 \eqref{THEODE4} where $0 < th < 2$
 is fixed.  Choose $R>0$ so that \eqref{brawayfrom1hom}
 holds, and that
 \begin{equation} \label{rsosmallthat}
    u_{ss}(r)-th \leq \frac{2(b_{r}^{2}-1)th}{b_{r}^{2}+1}
    \quad \textup{for all } r \in (0,R).
  \end{equation}
  Assume that $(-\omega,\omega)$ is the maximal interval in
  $\mathbb{R}$ on which $u$ exists and
  that  $a_0 \in (-\omega,\omega)$ satisfies
 \begin{equation} \label{lemxhyp2}
    u(a_0)=th, u'(a_0)=0, u''(a_0) = -(b_{r}^{2}+1)th\quad \textup{and}
    \quad u'''(a_0) \leq 0.
  \end{equation}
  Then there exists $\bar{x}>a_0$ such that
  \begin{itemize}
  \item[(i)] $u'(x)<0\quad \textup{on } (a_0,\bar{x}]$,
  \item[(ii)] $u(x)>0$ on $(a_0,\bar{x})$, and $u(\bar{x})=0$, and
  \item[$(iii)$] $u''(\bar{x}) \leq -(b_{r}^{2}+1)u_{ss}(r)$.
  \end{itemize}
\end{lemma}

\textbf{Remark:}  Throughout the proof, we shall write $b$ and
$u_{ss}$ instead of $b_r$ and $u_{ss}(r)$.

\begin{proof}[Proof of Lemma \ref{genxlemma}]
By \eqref{THEODE4}, $u''''(a_0) < 0$, thus $u''', u''~,u'~,u$ are
decreasing in a right neighborhood of $x = a_0$. Note that
\eqref{THEODE4} also implies that $u'''' < 0$ as long as $0 < u \leq
th$, and $u'' < 0$. Hence, there exists $\bar{x}>a_0$ such that (i)
and (ii) hold.

We prove $(iii)$ by contradiction.  That is, assume that
\begin{equation} \label{conudub}
  u''(x) > -(b^{2}+1)u_{ss}\quad \textup{on } (a_0,\bar{x}].
\end{equation}
By (\ref{THEODE4}) it follows that
\begin{equation} \label{lin4}
   u'''' = 2(b^2-1)u''-(b^2+1)^2u
   < 2(b^2-1)u''\quad \textup{on } (a_0,\bar{x}).
\end{equation}
Recalling that $u(a_0)=th, u''(a_0)=-(b^2+1)th$ and $u'''(a_0) \leq
0$ we obtain
\begin{equation} \label{2ndinthomo}
  u''(x) \leq (1-3b^2)th + 2(b^2-1)u
\end{equation}
upon two integrations of \eqref{lin4} over $(a_0,x) \subset
(a_0,\bar{x})$. Combining \eqref{conudub} with \eqref{2ndinthomo}
yields
\[
  -(b^2+1)u_{ss} < (1-3b^2)th+2(b^2-1)u\quad \textup{on } (a_0,\bar{x}),
\]
   and
   \begin{equation} \label{becums}
     -(b^2+1)u_{ss} < (1-3b^2)th\quad \textup{at}\quad x = \bar{x}.
 \end{equation}
Denoting $m = u_{ss}-th$ \eqref{becums} is equivalent to
 \[
  -(b^2+1)(th+m)+(3b^2-1)th < 0
 \]
  which results in
 \[
     \frac{2(b^2-1)th}{b^2+1} < m = u_{ss}-th.
 \]
 This contradicts \eqref{rsosmallthat}, hence the proof
 is complete.
 \end{proof}

 In the following lemma, we consider a solution $u$ of
 \eqref{THEODE4} that has a critical point
 $(a_0,th)$, i.e., $u'(a_0)=0$ and $u(a_0)=th$.
 Estimates of $u$ at subsequent critical numbers are given.

\begin{lemma} \label{genlemx}
   Suppose that $u$ is a nontrivial solution of
   \eqref{THEODE4} with $0<th<2$.
   Choose $R>0$ so that for all $r < R$,
    $b \geq 1+m$ where $m$ is defined by \eqref{halfdis}
   and
\[
  u_{ss}-th \leq \frac{2(b^{2}-1)th}{b^{2}+1}.
\]
   Assume that $(-\omega,\omega)$ is the maximal interval in
   $\mathbb{R}$ on which $u$ exists and
   that  $a_0 \in (-\omega,\omega)$ satisfies
   \begin{equation} \label{lemxhyp3}
  u(a_0)=th, u'(a_0)=0, u''(a_0) \leq 0\quad \textup{and}\quad
           u'''(a_0) \leq 0.
    \end{equation}
    Then, $u(b_0) < -u_{ss}$ and $u(a_1)>u_s$ where
 \begin{equation} \label{bnot}
b_0 = \sup\{x > a_0~|~u'(t) \ne 0\quad \textup{on } (a_0,x) \},
\quad \textup{and}
     \end{equation}
 \begin{equation} \label{steaksauce}
a_1 = \sup\{x > b_0~|~u'(t) \ne 0\quad \textup{on } (b_0,x) \}.
 \end{equation}
Furthermore,
 for any $\zeta > a_0$ where $u'(\zeta)=0$ we have
 \begin{equation} \label{lemxresult}
    |u|>th, sgn~ u'' = -sgn ~u,\quad \textup{and}\quad sgn~ u''' = -sgn~ u
    \quad \textup{at}\quad x = \zeta.
    \end{equation}
  \end{lemma}

  In particular, if a local maximum occurs at the critical number
  $x=\zeta$, then $u(\zeta)>u_s$.

\begin{figure}[!ht]
  \begin{center}
\includegraphics[width=0.4\textwidth]{fig6}  %  alphath.eps
  \end{center}
  \caption{A solution with critical value $u=th$.}
  \label{qlemxguideII}
\end{figure}

\begin{proof}
  To prove this lemma we invoke an iterative procedure to which we apply
  Induction.

To start, we determine the exact value of $u''(a_0)$. By
substituting $u(a_0)=th$ and $u'(a_0)=0$ into \eqref{zeroenergyhomo}
we find that $u''(a_0)= \pm (b^2+1)th$. Specifically, the assumption
$u''(a_0) \leq 0$ implies that $u''(a_0)= -(b^2+1)th$. Thus,
$u'(x)<0$ on $(a_0,b_0)$ and $u'(b_0)=0$. The fact that
$u''''(a_0)<0$ implies that $u'''(x)<0$ in a right-neighborhood of
$x=a_0$, and hence
   \begin{equation} \label{why_not}
     y_0=\sup\{x>a_0~|~u'''(t)<0\quad \textup{on } (a_0,x) \}
     \end{equation}
     is well defined and $y_0 \in (a_0,b_0)$.

 We now show that $u(b_0) < -u_{ss}$ and that
 \eqref{lemxresult} holds at $x = b_0$.

 The fact that $u''''(a_0)<0$ implies that
 $u''''$ vanishes on the interval $(a_0,y_0)$. By \eqref{THEODE4}
 and the fact that $u < th$ on $(a_0,b_0)$ it follows that
 $u(\bar{x})=0$ for some $a_0 < \bar{x} < y_0$.
 Since $u'''(y_0)=0$ the first integral equation
 \eqref{zeroenergyhomo} leads to
 \[
   - {(u'')^2 \over 2} -(b^2-1)(u')^2+(b^2+1)^2Q(u) \geq 0
   \quad \textup{at}\quad x = y_0,
\]
 or since $b > 1$,
   \begin{equation} \label{byyytch}
    2(b^2+1)^2Q(u(y_0)) \geq  (u''(y_0))^2.
  \end{equation}
Furthermore, $u'''(x)<0$ on $(a_0,y_0)$ yields
$0>u''(a_0)>u''(\bar{x})>u''(y_0)$, hence
  \[
 (u''(y_0))^2 > (u''(\bar{x}))^2.
 \]
 This fact together with Lemma~\ref{genxlemma} and
 \eqref{byyytch} give
\[
 2(b^2+1)^2Q(u(y_0)) > (b^{2}+1)^2u_{ss}^2.
\]
The fact that
\[
    Q(u) = \frac{u^2}{2}\quad \textup{on } (-\infty,th]
\]
   leads to $u(y_0) < -u_{ss}$. Since $u$ decreases further until it
       reaches the next critical number, $x = b_0$, we must have
  \begin{equation} \label{est3homo}
     u(b_0) < u(y_0) < -u_{ss}.
  \end{equation}

  It remains to show that $u''(b_0) > 0$ and $u'''(b_0) > 0$.
  The fact that $u'(b_0)=0$ and $u'(x)<0$ in a left-neighborhood of
  $x = b_0$ implies that $u''(b_0) \geq 0$. Also, since $u(b_0)<-u_{ss}$
  it follows from \eqref{zeroenergyhomo} that
  \[\frac{(u''(b_0))^2}{2}=(b^2+1)^2Q(u(b_0)) \ne 0,\]
  thus $u''(b_0)>0$.

  To show that $u'''(b_0)>0$ first note that
  $u''(y_0) < 0 < u''(b_0)$. Thus a value $x_0 \in (y_0,b_0)$ exists
  such that $u''(x_0)=0$ and $u'''(x_0) \geq 0$. Also,
  $u'(x) < 0$ on $(a_0,b_0)$ and $\bar{x} < y_0 < x_0$, imply that $u(x)<0$
  on $(x_0,b_0)$. An integration of $(\ref{THEODE4})$ over $(x_0,b_0)$ results in
  \[
  u'''(b_0) = -(b^2+1)\int_{x_0}^{b_0}udx - 2(b^2-1)u'(x_0)+u'''(x_0) > 0
  \]
  as desired.

 In a similar fashion, we shall show that $u(a_1)>u_s$ and
  \eqref{lemxresult} at $x = a_1$ where $a_1$ is defined by
  \eqref{steaksauce}.  First, recall that $u''(b_0)>0$ and $u'''(b_0)>0$
  so that $u'(x)>0$ on $(b_0,a_1)$. Also, note that $u'(x)>0$ holds so long
  as $u'''(x)>0$. Thus,
\begin{equation} \label{y1}
  y_1 = \sup\{x > b_0~|~ u'''(t)>0\quad \textup{on } (b_0,x) \}
\end{equation}
is well defined and $y_1 \in (b_0,a_1)$.
  Since $b > 1$ and $u'''(y_1)=0$, then
  \begin{equation} \label{esta1}
     -\frac{(u'')^2}{2}+(b^2+1)^2Q(u) > 0\quad \textup{at}\quad
 x=y_1
  \end{equation}
  follows from $(\ref{zeroenergyhomo})$.
  Because $u''(b_0)>0$ and $u'''(x)>0$ on $(b_0,y_1)$, it follows
  that
  \begin{equation} \label{udubevenbigger}
    (u''(y_1))^2 > (u''(b_0))^2.
  \end{equation}
Combining $(\ref{esta1})$ with $(\ref{udubevenbigger})$ yields
  \begin{equation} \label{somethin}
      Q(u(y_1)) > \frac{(u''(y_1))^2}{2(b^2+1)^2}>
      \frac{(u''(b_0))^2}{2(b^2+1)^2}=Q(u(b_0)).
  \end{equation}
Now recall from \eqref{est3homo} that
  $u(b_0) < -u_{ss}$, thus
  \begin{equation} \label{udbevenbigger2}
  Q(u(b_0)) > Q(-u_{ss}).
  \end{equation}
  Since
  \[
Q(u)=\frac{u^2}{2}-\frac{8b}{b^2+1}\int_{0}^{u}e^{-r/(s-th)^2}H(s-th)ds,
\]
  then $Q(-u)>Q(u)$ for all $u>th$. Specifically, $Q(-u_{ss})>Q(u_{ss})$ and
  by \eqref{udbevenbigger2} we obtain
 \[
 Q(u(b_0)) > Q(u_{ss}).
 \]
 This and \eqref{somethin} yield
 \begin{equation} \label{wayup}
    Q(u(y_1))> Q(u_{ss}).
    \end{equation}
  The fact that $u'(x)>0$ on $(b_0,a_1)$ together with \eqref{somethin}
  and \eqref{wayup} yield
\begin{equation} \label{desrd}
  u(a_1) > u(y_1) > u_s.
\end{equation}

  It remains to show that $u''(a_1)<0$ and $u'''(a_1)<0$.
  We start by showing that $u''(a_1)<0$. The fact that
  $u'(x)>0$ on $(b_0,a_1)$ with $u'(a_1)=0$ guarantees that
  $u''(a_1) \leq 0$. Eq.~\eqref{zeroenergyhomo} along with
  $u(a_1)>u_s$ leads to
  \[
 \frac{(u'')^2}{2}=(b^2+1)^2Q(u) > 0\quad \textup{at}\quad x=a_1.
  \]
 Therefore, $u''(a_1)<0$ as desired.

  We proceed to show that $u'''(a_1)<0$. First, note that
  $u''(b_0)>0>u''(a_1)$, hence there exists a value
  $x_1 \in (y_1,a_1)$ such that $u''(x_1)=0$ and $u'''(x_1) \leq 0$.
  Here we use the fact that
  \[
Q'(u) = u-\frac{4b}{b^2+1}f(u-th) > 0\quad \textup{for all } u >
u_s.
\]
  By \eqref{somethin} and \eqref{wayup} it follows that $u(y_1)>u_s$.
  Since $u'(x)>0$ on $(b_0,a_1)$ and $x_1>y_1$, then $u(x)>u_s$ on $(x_1,a_1)$.
  Thus, $Q'(u(x))>0$ on $(x_1,a_1)$. An integration of \eqref{THEODE4}
  over $(x_1,a_1)$ gives
  \[
  u'''(a_1)=u'''(x_1) -2(b^2-1)u'(x_1)-(b^2+1)^2\int_{x_1}^{a_1}Q'(u(s))ds < 0
  \]
  as desired.

  In a similar fashion, one can show that $(\ref{lemxresult})$ holds
  at $x = b_1$ where
 \[
    b_1 = \sup\{x > a_1~|~u'(t)<0\quad \textup{on } (a_1,x) \}.
  \]
  In general, this procedure can be continued inductively to show that
  \eqref{lemxresult} holds for all $\zeta > a_0$ for which $u'(\zeta)=0$.
\end{proof}

Lemma~\ref{genlemx} addresses the behavior of a solution $u$ that
satisfies
\[
u(a_0)=th, u'(a_0)=0, u''(a_0) \leq 0,\quad \textup{and}\quad
u'''(a_0) \leq 0.
\]
In the following lemma we show that there is no loss in generality
in assuming that $u'''(a_0) \leq 0$ if $a_0 > 0$.

\begin{lemma} \label{nextlemx}
   Suppose that $u$ is a non-constant solution of
   \eqref{THEODE4} with $0<th<2$.
   Choose $R>0$ so that for all $r < R$,
   $b \geq 1+\delta$ for some $\delta > 0$, and
\[
u_{ss}-th \leq \frac{2(b^{2}-1)th}{b^{2}+1}.
\]
   Assume that $(-\omega,\omega)$ is the maximal interval in
   $\mathbb{R}$ on which $u$ exists and
   that  $a_0 \in (0,\omega)$ satisfies
  \begin{equation} \label{lemxhyp}
  u(a_0)=th, u'(a_0)=0,\quad \textup{and}\quad u''(a_0) \leq 0.
    \end{equation}
    Then  $u'''(a_0) \leq 0$.
\end{lemma}

\begin{proof} Assume on the other hand that $u'''(a_0)>0$. Define
$v(x)=u(-x)$. By symmetry of $u$, (see Lemma~\ref{usym}),
$v(x)=u(x)$, and hence $v$ is a non-constant solution of
 \eqref{THEODE4}.  Furthermore,
\[
  v(-a_0)=th, v'(-a_0)=0, v''(-a_0) \leq 0,\quad \textup{and}\quad
  v'''(-a_0) \leq 0.
\]
  That is, $v$ satisfies the condition of Lemma~\ref{genlemx}.
  Thus, we conclude from Lemma~\ref{genlemx} that for any $\zeta > -a_0$
  where $v'(\zeta)=0$ we have that $|v(\zeta)| > th$. Particularly,
  for $\zeta = a_0$, we have $th = u(a_0) = v(a_0) > th$ a contradiction.
 Therefore, $u'''(a_0) \leq 0$ and the lemma is proved.
 \end{proof}

\subsection*{Construction of $N$-Bump Homoclinic Solutions}

We now proceed in proving the existence of $N$-bump homoclinic orbit
solutions for any even $N$. As usual, we consider $\alpha < 0$ and
$\beta=-(b^2+1)\alpha>0$. This choice of initial conditions ensures
that $\alpha=u(0,\alpha)<0$ and that $u(t,\alpha)$ increases on
$(0,\xi_1(\alpha))$.

The shooting argument we use to construct $N$-bump solutions for
even $N$ is based on the periodic solution proved in
Theorem~\ref{periodicsoln} of Section~\ref{prelimresults}.
  This theorem guarantees the existence of a one bump periodic
 solution $u(\cdot,\alpha^*)$ such that
 \begin{gather*}
 \alpha^* = u(0,\alpha^*)=u(\eta_k(\alpha^*),\alpha^*)<0,
 \quad \textup{and}  \\
 u(\xi_1(\alpha^*),\alpha^*)=u(\xi_k(\alpha^*),\alpha^*)>u_s
 \quad \textup{for all } k \geq 1
 \end{gather*}

Our construction of even bump solutions by shooting methods will
make use of the following technical lemma.

 \begin{lemma} \label{eventechlem}
 If $u(\cdot,\alpha)$ is a solution of \eqref{THEODE4}, then
\[
 u(\xi_1(\alpha),\alpha) \to 0^+\quad \textup{as } \alpha
 \to 0^{-}.
\]
\end{lemma}

  \begin{proof}  Since $\alpha<0$, then the equation in
  \eqref{THEODE4} is linear so long as $u \leq th$.
  Hence, we find that
 \begin{gather} \label{CdFrm}
       u(x,\alpha)=\alpha(\cosh(bx)\cos(x)-b\sinh(bx)\sin(x)),\quad \textup{and}\\
 \label{CdFrm1}
 u'(x,\alpha)=-\alpha(b^2+1)\cosh(bx)\sin(x)
 \end{gather}
so long as $u(x,\alpha) \leq th$. By \eqref{CdFrm} we obtain that
$u(x,\alpha)\leq th$ on $[0,\pi]$ whenever $|\alpha|$ is
sufficiently small. Therefore, $\xi_1(\alpha) = \pi$ is an immediate
consequence of \eqref{CdFrm1}.
\end{proof}

Lemmas \ref{eveninductlem} and \ref{eveninductlem2} below will be
used to generate the induction process.  These lemmas make repeated
use of Lemma~\ref{adjpairs}.  Before stating and proving
Lemmas~\ref{eveninductlem} and \ref{eveninductlem2}, we will prove
the following lemma which yields an equivalent condition of
Lemma~\ref{adjpairs}.

\begin{lemma} \label{adjpairsaltcond}
  Suppose that $Q(u(\eta_k(\alpha),\alpha))=0$ for some $\alpha<0$ and
  $k \geq 1$. That is, either $u(\eta_k(\alpha),\alpha)=0$, or
  $u_s$.
  Then, either $\eta_k(\alpha)=\xi_k(\alpha)$ or
  $\eta_k(\alpha)=\xi_{k+1}(\alpha)$. In particular,
  \begin{itemize}
\item[(i)] $\eta_k(\alpha)=\xi_{k+1}(\alpha)$ if and only if
$u'''(\eta_k(\alpha),\alpha)<0$, and
\item[(ii)] $\eta_k(\alpha)=\xi_k(\alpha)$ if and only if
$u'''(\eta_k(\alpha),\alpha)>0$.
  \end{itemize}
\end{lemma}

\begin{proof}
The condition $Q(u(\eta_k(\alpha),\alpha))=0$ and
\eqref{zeroenergyhomo} imply that $u''(\eta_k(\alpha),\alpha)=0$. By
Lemma~\ref{critnumhelper} it follows that
$u'''(\eta_k(\alpha),\alpha)\ne 0$. Now, \eqref{eta_k} leads to
$\eta_k(\alpha)=\xi_{k+1}(\alpha)$ if and only if
$u'''(\eta_k(\alpha),\alpha)<0$, and \eqref{xi_k} yields that
$\eta_k(\alpha)=\xi_{k}(\alpha)$ if and only if
$u'''(\eta_k(\alpha),\alpha)>0$.
\end{proof}

Lemma~\ref{adjpairsaltcond} implies that if
$u(\eta_{k-1}(\alpha_0),\alpha_0)<0$, for example, and
\\ $u(\xi_{k}(\alpha_0),\alpha_0)=0$, then
$u'''(\xi_k(\alpha_0),\alpha_0)>0$ and
$\xi_k(\alpha_0)=\eta_k(\alpha_0)$.
 We now prove the following two induction lemmas.

\begin{lemma} \label{eveninductlem}
Let $k \geq 2$. Suppose that $[c, d] \subset (\alpha^*, 0)$ is an
interval such that
\[
  th = u(\xi_k(c),c) > u(\xi_k(\alpha),\alpha) > u(\xi_k(d),d) = 0
  \quad \textup{for all } \alpha \in (c,d),
\]
  and
  \[
u(\eta_{k-1}(\alpha),\alpha)<0\quad \textup{for all } \alpha \in
[c,d].
 \]
  Then there exists an interval
  $[\hat{c},\hat{d}] \subset (c,d)$ such that
\[
th=u(\xi_{k+1}(\hat{c}),\hat{c})>u(\xi_{k+1}(\alpha),\alpha)
  >u(\xi_{k+1}(\hat{d}),\hat{d})=0\quad \textup{for all } \alpha \in
  (\hat{c},\hat{d}),
\]
  and
\[
u(\eta_{k}(\alpha),\alpha)<0
  \quad \textup{for all } \alpha \in [\hat{c},\hat{d}].
 \]
\end{lemma}

\textbf{Remark:} See the left panel of Figure~\ref{inductlemguide}
for an illustration of the main ideas of the following proof.

\begin{proof}[Proof of \ref{eveninductlem}]
Since $u(\eta_{k-1}(d),d)<0=u(\xi_k(d),d)$, then we deduce from
Lemma~\ref{adjpairsaltcond} that $\eta_k(d)=\xi_k(d)$ and
$u'''(\xi_k(d),d)>0$. Hence, Lemma~\ref{adjpairs} and our assumption
that $u(\xi_k(\alpha),\alpha)>0$ for each $\alpha \in (c,d)$ imply
that $u(\eta_k(\alpha),\alpha)>0$ whenever $d-\alpha>0$ is
sufficiently small.  Furthermore, the assumption that
$u(\xi_k(c),c)=th$ and Lemma~\ref{genlemx} lead to
$u(\eta_k(c),c)<0$. It now follows by continuity of
$u(\eta_k(\alpha),\alpha)$ in $\alpha$ that
\[
  b = \sup\{\hat{\alpha}>c~|~u(\eta_k(\alpha),\alpha)<0\quad \textup{on }
  (c,\hat{\alpha}) \}
\]
 is well defined, $u(\eta_k(b),b)=0$, and $c<b<d$.
This result together with Lemma~\ref{adjpairsaltcond} and the
assumption that $u(\xi_k(\alpha),\alpha)>0$ for each $\alpha \in
(c,d)$ all lead to  $\xi_{k+1}(b)=\eta_k(b)$ and
$u'''(\xi_{k+1}(b),b)<0$. Another application of
Lemma~\ref{adjpairs} yields that $u(\xi_{k+1}(\alpha),\alpha)<0$
whenever $b-\alpha>0$ is sufficiently small. Once again we invoke
Lemma~\ref{genlemx} to obtain $u(\xi_{k+1}(c),c)>th$. Thus, it
follows from continuity of $u(\xi_{k+1}(\alpha),\alpha)$ in $\alpha$
that
\[
  \hat{d} = \sup\{\hat{\alpha}>c~|~u(\xi_{k+1}(\alpha),\alpha)>0\quad
  \textup{on } (c,\hat{\alpha})\}
  \]
and
\[
\hat{c} =
\inf\{\hat{\alpha}<\hat{d}~|~u(\xi_{k+1}(\alpha),\alpha)<th\quad
  \textup{on } (\hat{\alpha},\hat{d})\}
\]
  are well defined, $th=u(\xi_{k+1}(\hat{c}),\hat{c})>
  u(\xi_{k+1}(\alpha),\alpha)>u(\xi_{k+1}(\hat{d}),\hat{d})=0$, for
  all $\alpha\in (\hat{c},\hat{d})$.
  Also note that $\hat{d}<b$, and therefore, $u(\eta_k(\alpha),\alpha)<0$
  for all $\alpha \in [\hat{c},\hat{d}]$. This concludes the proof
  of the lemma.
  \end{proof}


 \begin{lemma} \label{eveninductlem2}
  Assume that $\alpha^*<0$ denotes the initial value that gave
  rise to the one bump periodic solution of
  Theorem~\ref{periodicsoln}.  That is,
    \begin{gather*}
    u(0,\alpha^*)=u(\eta_k(\alpha^*),\alpha^*)<0\quad \textup{and}\\
    u_s<u(\xi_1(\alpha^*),\alpha^*)=u(\xi_k(\alpha^*),\alpha^*)
    \end{gather*}
    for all $k \geq 1$.
    Suppose that there exists $A_k \in (\alpha^*,0)$
  and $b_{k} \in (\alpha^*,A_k)$ such that
  \begin{itemize}
    \item[(i)] $u(\xi_k(\alpha),\alpha)>u_s$ on $[\alpha^*,A_k)$ and
       $u(\xi_k(A_k),A_k)=u_s$, and
    \item[(ii)] $u(\eta_{k}(\alpha),\alpha)<0$ on $[\alpha^*,b_k)$
    and
    $u(\eta_{k}(b_{k}),b_{k})=0$.
  \end{itemize}
  Then there exists $A_{k+1} \in (\alpha^*,b_k)$ and
  $b_{k+1} \in (\alpha^*,A_{k+1})$ such that
  \begin{itemize}
    \item[(i)] $u(\xi_{k+1}(\alpha),\alpha)>u_s$ on
    $[\alpha^*,A_{k+1})$ and
       $u(\xi_{k+1}(A_{k+1}),A_{k+1})=u_s$, and
    \item[(ii)] $u(\eta_{k+1}(\alpha),\alpha)<0$ on $[\alpha^*,b_{k+1})$
    and
    $u(\eta_{k+1}(b_{k+1}),b_{k+1})=0$.
  \end{itemize}
  \end{lemma}

  \textbf{Remark:} See the right panel of
  Figure~\ref{inductlemguide} for an illustration of the main ideas
  of the following proof.

 \begin{proof}[Proof of Lemma \ref{eveninductlem2}]
 Since $u(\eta_k(b_k),b_k)=0<u_s<u(\xi_k(b_k),b_k)$,
 then $\eta_k(b_k)=\xi_{k+1}(b_k)$ and $u'''(\xi_{k+1}(b_k),b_k)<0$
 as a result of Lemma~\ref{adjpairsaltcond}.  Thus, it follows from
 Lemma~\ref{adjpairs} that $u(\xi_{k+1}(\alpha),\alpha) < 0$ whenever
 $b_{k} - \alpha > 0$ is sufficiently small.  Further, since
 $u(\xi_{k+1}(\alpha^*),\alpha^*) > u_s$, then continuity of
 $u(\xi_{k+1}(\alpha),\alpha)$ in $\alpha$ implies that
  \[
  A_{k+1} = \sup\{\hat{\alpha}>\alpha^*~|~u(\xi_{k+1}(\alpha),\alpha)>u_s
  \quad \textup{on } (\alpha^*,\hat{\alpha}) \}
 \]
  is well defined and $u(\xi_{k+1}(A_{k+1}),A_{k+1})=u_s$.
  Our definition of $A_{k+1}$ together with the fact that
  $u(\xi_{k+1}(\alpha),\alpha) < 0$ whenever $b_{k} - \alpha > 0$
  is sufficiently small ensures that $\alpha^*<A_{k+1}<b_{k}<A_k$.

Since $u(\eta_k(A_{k+1}),A_{k+1})<0$, then
Lemma~\ref{adjpairsaltcond} implies that
$\eta_{k+1}(A_{k+1})=\xi_{k+1}(A_{k+1})$ and
$u'''(\xi_{k+1}(A_{k+1}),A_{k+1})>0$. We once again apply
Lemma~\ref{adjpairs} which guarantees that
$u(\eta_{k+1}(\alpha),\alpha)>u_s$
  for sufficiently small $A_{k+1} - \alpha > 0$.
  Since  $u(\eta_{k+1}(\alpha^*),\alpha^*)<0$, then continuity of
  $u(\eta_{k+1}(\alpha),\alpha)$ in $\alpha$ guarantees that
    \[
  b_{k+1} = \sup\{\hat{\alpha}>\alpha^*~|~u(\eta_{k+1}(\alpha),\alpha)<0
  \quad \textup{on } (\alpha^*,\hat{\alpha}) \}
  \]
  is well defined and $u(\eta_{k+1}(b_{k+1}),b_{k+1})=0$.
  The fact that \\ $u(\eta_{k+1}(\alpha),\alpha)>u_s > 0$
  for sufficiently small $A_{k+1} - \alpha > 0$ implies that
   $b_{k+1}<A_{k+1}$ concluding the proof of the lemma.
  \end{proof}

\begin{figure}[ht]
  \begin{center}
\includegraphics[width=0.46\textwidth]{fig7a} \ \ \  % Speghetti
\includegraphics[width=0.46\textwidth]{fig7b} % Speghetti2
   \end{center}
\caption{An illustration of the behavior of the critical values
$u(\eta_{k-1}(\alpha)), u(\eta_k(\alpha)), u(\xi_k(\alpha)), $ and
$u(\xi_{k+1}(\alpha))$. The left panel corresponds to
Lemma~\ref{eveninductlem}, and the right panel corresponds to
Lemma~\ref{eveninductlem2}.}
  \label{inductlemguide}
\end{figure}


  We proceed by proving the existence of $A_1 \in (\alpha^*,0)$
  and $b_{1} \in (\alpha^*,A_1)$ that satisfy the hypotheses $(i), (ii)$ of
  Lemma~\ref{eveninductlem2}.  First, it follows from Lemma~\ref{eventechlem}
  that $u(\xi_1(\alpha),\alpha)<th<u_s$ for sufficiently small $-\alpha>0$.
  Since $u(\xi_1(\alpha^*),\alpha^*)>u_s$, then continuity of
  $u(\xi_1(\alpha),\alpha)$ in $\alpha$ ensures that
  \begin{equation} \label{steaksauce2}
    A_1 = \sup\{\hat{\alpha}>\alpha^*~|~u(\xi_1(\alpha),\alpha)>u_s\quad
    \textup{on } (\alpha^*,\hat{\alpha}) \}
    \end{equation}
    is well defined and $u(\xi_1(A_1),A_1)=u_s$. Furthermore,
    $A_1<0$ follows from the fact that
    $u(\xi_1(\alpha),\alpha)<th<u_s$ for sufficiently small $-\alpha>0$.

    To construct $b_1$ we note that $\eta_1(A_1)=\xi_1(A_1)$, and
    $u'''(\xi_1(A_1),A_1)>0$. Thus, $u(\eta_1(\alpha),\alpha)>u_s$
    whenever $A_1 - \alpha > 0$ is sufficiently small as a consequence
    of Lemma~\ref{adjpairs}.  Continuity of $u(\eta_1(\alpha),\alpha)$ in
    $\alpha$ and the fact that $u(\eta_1(\alpha^*),\alpha^*)<0$
    implies that
    \begin{equation} \label{barbecuesauce}
    b_1 = \sup\{\hat{\alpha}>\alpha^*~|~u(\eta_1(\alpha),\alpha)<0
    \quad \textup{on } (\alpha^*,\hat{\alpha}) \}
    \end{equation}
    is well defined and $u(\eta_1(b_1),b_1)=0$. The fact that
    $u(\eta_1(\alpha),\alpha)>u_s$ for sufficiently small
     $A_1 - \alpha > 0$ gives $\alpha^* < b_1 < A_1$ as desired.


 We have now shown that the hypothesis of Lemma~\ref{eveninductlem2}
 for $j=1$, i.e., $A_1$ and $b_1$ exist as required.  Proceeding
 inductively, it follows that the entire sequence
 $\{A_j\}_{j=1}^{\infty}$ and $\{b_j\}_{j=1}^{\infty}$ exist and
 satisfy Lemma~\ref{eveninductlem2}.


Throughout the remainder of this paper we define $k \in
\mathbb{Z}^+$ by $2k=N$. To obtain an $N$-bump homoclinic orbit
solution we must ensure that $\alpha_N=u(0,\alpha_N) < 0$ exists
such that $u(\xi_j(\alpha_N),\alpha_N)>th$ for $j = 1,\ldots,k$ and
that $u(\xi_j(\alpha_N),\alpha_N)<th$ for all $j > k$. In
particular, the $N$-bump solution that we construct will have the
properties
\begin{gather} \label{Nbmpprop1even}
u(\xi_j(\alpha_N),\alpha_N)>u_s>th\quad \textup{for}\quad j=1,\ldots,k,\\
 \label{Nbmpprop2even}
0<u(\xi_j(\alpha_N),\alpha_N)<th\quad \textup{for}\quad  j >
k,\quad \textup{and} \\
 \label{Nbmpprop3even}
  u(\eta_j(\alpha_N),\alpha_N)<0\quad \textup{for all }  j \geq 1.
\end{gather}
We will implement a topological shooting method that generates a
nested sequence $\{I_j\}_{j=1}^{\infty}$ of non-empty compact
intervals which possess the following properties:
\begin{itemize}
  \item[(P1)] $I_{j+1} \subset I_{j}^{0}$ for all $j \geq 1$,
  ($I_{j}^{0}$ denotes the interior of $I_j$),
  \item[(P2)] for each $\alpha \in I_1$ and $j = 1,\ldots, k$,
  $u(\xi_j(\alpha),\alpha) > u_s$ and $u(\eta_{j}(\alpha),\alpha) \leq 0$.
  Moreover, $u(\eta_j(\alpha),\alpha)<0$ for all $\alpha \in I_{1}^{0}$
  and $j = 1, \ldots, k$,
  \item[(P3)] for each $j>k$, $0 < u(\xi_j(\alpha),\alpha)<th$
  for all $\alpha \in I_{j}^{0}$ and
  $u(\eta_{j-1}(\alpha),\alpha)<0$ for all $\alpha \in I_{j}$.
\end{itemize}

We proceed by constructing the first interval, $I_1$. First, recall
from $(\ref{steaksauce2})$ that $u(\xi_1(\alpha),\alpha) > u_s$ for
all $\alpha \in  [\alpha^*,A_1)$ and $u(\xi_1(A_1),A_1) = u_s$.
Furthermore,  $u(\eta_1(\alpha),\alpha)<0$ for $\alpha \in
[\alpha^*,b_1)$ and $u(\eta_1(b_1),b_1)=0$ where $b_1 \in
(\alpha^*,A_1)$ is defined by $(\ref{barbecuesauce})$. By
Lemma~\ref{eveninductlem2}, there exists $A_2 \in (\alpha^*,b_1)$
and $b_2 \in (\alpha^*,A_2)$ such that
\begin{itemize}
\item[(i)] $u(\xi_j(\alpha),\alpha) > u_s$ on $[\alpha^*,A_2)$
and $u(\xi_2(A_2),A_2) = u_s$,
\item[(ii)] $u(\eta_j(\alpha),\alpha) \leq 0$
on $[\alpha^*,b_2)$ and $u(\eta_2(b_2),b_2)=0$
\end{itemize}
for $j=1,2$.
  An inductive application of
Lemma~\ref{eveninductlem2} yields the following lemma.

 \begin{lemma} \label{goodlemmaeven}
  For $2k \equiv N$ and $j=1,2,\ldots,k$ it follows
  that
 \begin{gather*}
  u(\xi_j(\alpha),\alpha)>u_s\quad \textup{for}\quad \alpha \in [\alpha^*,A_k),
\quad \textup{and}\quad u(\xi_k(A_k),A_k)=u_s,   \\
u(\eta_{j}(\alpha),\alpha)\leq 0\quad \textup{for}\quad \alpha \in
[\alpha^*,b_k), \quad \textup{and}\quad u(\eta_k(b_k),b_k)=0.
 \end{gather*}
\end{lemma}
An immediate consequence of Lemma~\ref{goodlemmaeven} and the fact
that $[\alpha^*,b_k] \subset [\alpha^*,A_k)$, is that property
$(P2)$ holds for
 \begin{equation} \label{I1Ngt2even}
    I_1 \equiv [\alpha^*,b_k].
 \end{equation}

 The aim of the following lemma is to construct the second interval, $I_2$.
 Subsequently, we will inductively employ Lemma~\ref{eveninductlem} to define
 $I_3,I_4,\ldots$

 \begin{lemma} \label{I_2evenNgt2}
 Let $d_2=A_{k+1}$ where $A_{k+1}$ is defined by Lemma~\ref{eveninductlem2} and
 define
 \[
 c_2 = \inf\{\hat{\alpha}<d_2~|~u(\xi_{k+1}(\alpha),\alpha)<th\quad \textup{on }
  (\hat{\alpha},d_2) \}.
  \]
 Also, define $I_2=[c_2,d_2]$. Then $I_2 \subset I_{1}^{0}$,
 \[
\begin{gathered}
  0=u(\xi_{k+1}(d_2),d_2)<u(\xi_{k+1}(\alpha),\alpha)
   <u(\xi_{k+1}(c_2),c_2)=th\quad \textup{for all } \alpha \in
   I_{2}^{0},\\
  u(\eta_{k}(\alpha),\alpha)<0\quad \textup{for all } \alpha \in I_{2}^{0}.
  \end{gathered}
   \]
   \end{lemma}

 \begin{proof}
 It follows from Lemma~\ref{eveninductlem2} that
 $d_2=A_{k+1}<b_k$. Also, $\alpha^*<c_2$ is a consequence of
 $u(\xi_{k+1}(\alpha^*),\alpha^*)>u_s>th$,
 $u(\xi_{k+1}(A_{k+1}),A_{k+1})=0$, and the fact that
 $u(\xi_{k+1}(\alpha),\alpha)$ is continuous with respect to $\alpha$.
 Thus, $I_2 \subset I_{1}^{0}$.

Now,
\[
0=u(\xi_{k+1}(d_2),d_2)<u(\xi_{k+1}(\alpha),\alpha)
   <u(\xi_{k+1}(c_2),c_2)=th\quad \textup{for all } \alpha \in
   I_{2}^{0}
\]
follows immediately from the definitions of $c_2$  and $d_2$, and
\[
  u(\eta_{k}(\alpha),\alpha)<0\quad \textup{for all } \alpha \in I_{2}
\]
follows by Lemma~\ref{goodlemmaeven} and the fact that
   $I_1 \subset I_{2}^{0}$. This concludes the proof of the
   lemma.
\end{proof}


In Lemma~\ref{I_2evenNgt2} we defined $I_2 = [c_2,d_2]$ and
described the properties of solutions $u(\cdot,\alpha)$ with $\alpha
\in I_2$. Lemma~\ref{eveninductlem} guarantees the existence of $I_3
= [c_3,d_3] \subset I_{2}^{0}$ such that
\begin{gather*}
  0=u(\xi_{k+2}(d_3),d_3)<u(\xi_{k+2}(\alpha),\alpha)<
  u(\xi_{k+2}(c_3),c_3)=th \\
 \textup{for all } \alpha \in I_{3}^{0},\quad \textup{and}\quad
  u(\eta_{k+1}(\alpha),\alpha)<0\quad \textup{for all } \alpha \in [c_3,d_3].
  \end{gather*}
An inductive application of Lemma~\ref{eveninductlem} yields $I_j
=[c_j,d_j]$ for $j \geq 2$ such that
\[
\begin{gathered}
  0=u(\xi_{k+j-1}(d_j),d_j)<u(\xi_{k+j-1}(\alpha),\alpha)
 <u(\xi_{k+j-1}(c_j),c_j)=th \\
  \textup{for all } \alpha \in (c_j,d_j),\quad \textup{and}\\
  u(\eta_{k+j-2}(\alpha),\alpha)<0\quad \textup{for all } \alpha \in
  [c_j,d_j].
  \end{gathered}
  \]

 Hence, to construct our $N$-bump solution, we let
 $\alpha_N \in \bigcap_{j=1}^{\infty} I_j$ and consider
 $u(\cdot,\alpha_N)$. We have just shown that the nested
 sequence of intervals $I_j$ exists which satisfies properties
 $(P1)-(P3)$ described above.
 In particular,
 $u(\xi_j(\alpha_N),\alpha_N)>u_s$ for $j = 1,2,\ldots,
 k$, $u(x,\alpha_N)<th$ on $[\eta_k(\alpha_N),\infty)$,
 and $u(x,\alpha_N)$ satisfies the linear equation with constant
 coefficients
 \[
  u''''-2(b^2-1)u''+(b^2+1)^2u=0\quad \textup{on }
 [\eta_k(\alpha_N),\infty).
  \]
  The general solution of this linear equation is
   \[
  u(x,\alpha_N) =
  c_1e^{-bx}\cos(x)+c_2e^{-bx}\sin(x)+c_3e^{bx}\cos(x)+c_4e^{bx}\sin(x)
  \quad \textup{on } [\eta_k(\alpha),
  \infty)
  \]
  for some constants $c_1-c_4$. The
  only way $u$ can remain below $th$ on $[\eta_k(\alpha_N),\infty)$
  is that $c_3=c_4=0$.
  From this we conclude that
  \[
    u(x,\alpha_N) \to 0\quad \textup{as } x \to \infty.
    \]
This completes the proof of existence of a N-bump solution.

    \section*{Numerical Solutions} %\label{conclusion}
 The solutions in Figure~\ref{twobumpsoln} were obtained by using Mathematica.
 Our choice of parameters are $r=0.02$, $th=1.75$ and $b = 1.095669029014$.
    Furthermore, Mathematica was used to verify that $(r,b,th)\in
    \Lambda$ (see \eqref{lamda2}) for this choice of parameters.
    The solution in the left panel was obtained by manually
    applying the shooting method as described in this paper.
    Special thanks to the referee for computing the solution in the
    right panel.  These numerics suggest that both solutions are highly
    sensitive to the initial conditions as indicated by the large number of
    decimal places that are {\it required} for the value $\alpha$. For this reason, we
    predict that the $N$-bump solutions are unstable.

\begin{figure}[!ht]
  \begin{center}
\includegraphics[width=0.45\textwidth]{fig8a} \ \ \  %  largetwobump.eps
\includegraphics[width=0.45\textwidth]{fig8b}  %  smalltwobump.eps
  \end{center}
  \caption{Two 2-bump solutions with $r=0.02, th=1.75$, and
  $b=1.095669029014$. The
  initial conditions in the left panel are $\alpha=-0.192218655$, and $\beta=0.422975$.
   In the right panel the initial conditions are
   $\alpha=-0.166290142$, and $\beta=0.36592.$}
  \label{twobumpsoln}
 \end{figure}

Lastly, as mentioned in the previous section, the solution is of the
form
\[
  u(x,\alpha_2) =
  c_1e^{-bx}\cos(x)+c_2e^{-bx}\sin(x)+c_3e^{bx}\cos(x)+c_4e^{bx}\sin(x)
  \quad \textup{on } [\eta_1(\alpha),\infty)
\]
  since $u<th$ on $[\eta_1(\alpha),\infty)$. This is only
  possible if $c_3=c_4=0$. Specifically, our computations
  estimate that $c_3 =c_4= 0$, $c_1 \approx 2729.389291$, and
  $c_2 \approx 4900.231464$ for the solution shown in the left
  panel and $c_3=c_4= 0$, $c_1 \approx -41.07182903$, and
  $c_2 =-217.0976993$ for the solution on the right.

 \section*{Conclusion} %\label{conclusion}

In this paper we have analyzed a subclass of stationary solutions of
\eqref{inteqwt}. In previous studies, (see \cite{CLO, LTGE}), the
Fourier transform was applied to both sides of \eqref{inteq1} to
obtain a fourth order ODE.  Then ODE methods were implemented to
obtain a thorough numerical investigation of homoclinic orbit
solutions.  For technical reasons, the Fourier transform does not
give rise to other types of interesting solutions such as periodic,
heteroclinic, or chaotic solutions. The fundamental aim of this
paper was to use the results of Krisner \cite{mypaper,my2ndpaper} to
prove that \eqref{inteqwt} does have $N$-bump homoclinic orbit
solutions. In fact, under the parameter regime derived in Section
\ref{prelimresults}, it was shown in Section \ref{nbumpsolutions}
that \eqref{inteqwt} has $N$-bump homoclinic orbit solutions for any
positive even-valued integer $N$.

A natural extension of this result would be to prove that $N$-bump
solutions exist for a larger parameter regime.  In particular,
Lemma~\ref{genlemx} was proved under the assumption that $b>1$. It
still remains to prove that $N$-bump homoclinic orbit solutions
exist for $0<b<1$ either by modifying the proof of
Lemma~\ref{genlemx} or by adopting an entirely different approach.

Another extension is to incorporate ``noise'' into the Wilson-Cowan
model \eqref{inteqwt} to account for random fluctuations.  It would
be interesting to investigate the existence of special solutions,
such as periodic and/or homoclinic orbit solutions, with the
presence of noise.

\subsection*{Acknowledgements}
 We thank the referee for providing several {\it very} helpful
suggestions which greatly helped the presentation of this paper.


 \begin{thebibliography}{99}

\bibitem{A2} {S. Amari,} \emph{Dynamics of pattern formation in
lateral-inhibition type neural fields,} Biol. Cybern.
 \textbf{27} (1977), pp.~77-87.

\bibitem{Br} {P. C. Bressloff,} \emph{
  Traveling fronts and wave propagation failure in an inhomogeneous neural
  network,} Physica D.
\textbf{155} (2001), pp.~83-100.

\bibitem{SCoombes} {S Coombes,} \emph{2005 Waves, bumps, and patterns in neural field
theories,} Biological Cybernetics, Vol 93, 91-108.

 \bibitem{CLO} {S Coombes, G. J. Lord, and M. R. Owen,}
 \emph{Waves and bumps in neuronal networks with axo-dendritic
 synaptic interactions,} Physica D, \textbf{178(3)} (2003), pp.~219-241.

\bibitem{CL} {E.A. Coddington and N. Levinson,}
\emph{Theory of Ordinary Differential Equations,} McGraw-Hill, New
York, 1955.

\bibitem{DHG} {D.H. Griffel,}
\emph{Applied Functional Analysis,} Halsted Press, New York, 1981

\bibitem{GBE} {G. B. Ermentrout.} \emph{Neural networks as
spatio-temporal pattern forming systems.} Rep. Prog. Phys.
\textbf{61}(4) (1998), pp.~353-430.

\bibitem{EandM} {G. B. Ermentrout, J. B. McLeod,} \emph{Existence and
uniqueness of traveling waves for a neural network,} Rep. Prog.
Phys. \textbf{123A} (1993), pp.~461-478.

\bibitem{GC} {Y.~Guo, C.~Chow,} \emph{Existence and Stability of
Standing Pulses in Neural Networks: {I}. Existence,} SIAM J.~
Appl.~Dyna.~Sys. \textbf{4} (2005), pp.~217-248.

\bibitem{GC2} {Y.~Guo, C.~Chow,} \emph{Existence and Stability of
Standing Pulses in Neural Networks: {I}. Stability,} SIAM J.~
Appl.~Dyna.~Sys. \textbf{4} (2005), pp.~248-281.

\bibitem{KA} {K. Kishimoto \& S. Amari,} \emph{Existence and stability of
local excitations in homogeneous neural fields,} J. Math. Biol.
 \textbf{7} (1979), pp.~303-318.

  \bibitem{mypaper} {E. Krisner} \emph{The link between integral equations
  and higher order ODEs.} J. Math. Anal. and App.,
  \textbf{291(1)} (March 2004), pp.~165-179

  \bibitem{my2ndpaper} {E. Krisner} \emph{Periodic Solutions of a
  One Dimension Wilson-Cowan Type Model.} Elec. J. Diff. Eqns.,
  \textbf{102} (2007), pp.~1-22

\bibitem{LTGE} {C. R. Laing, W. C. Troy, B. Gutkin \& G. B. Ermentrout,}
\emph{Multiple bumps in a neuronal model of working memory,}
  SIAM J. Appl. Math. \textbf{63(1)} (2002), pp.~62-97.

  \bibitem{LT}{C. R. Laing, \& W. C. Troy,}
  \emph{PDE methods for non-local models,}
  Physica D, to appear.

\bibitem{OLC} {M.~R.~Owen, C.~R.~Laing, \& S.~Coombes,}
\emph{Bumps and rings in a two-dimensional neural field: splitting
and rotational instabilities,
  }  New J.~Phys (2007) \textbf{9} 378

\bibitem{PT} {L.A.~Peletier and W.C.~Troy,}
\emph{Spatial Patterns: Higher Order Models in Physics and
Mechanics,} Birkhauser, Boston, 2001.

\bibitem{PE1} {D. J. Pinto
\&   G. B. Ermentrout,} \emph{Spatially structured activity in
synaptically coupled neuronal networks: I. Traveling fronts and
pulses,
  }  SIAM J. Appl. Math. \textbf{62(1)} (2001), pp. 206-225

\bibitem{PE2} {D. J. Pinto
\&   G. B. Ermentrout,} \emph{Spatially structured activity in
synaptically coupled neuronal networks: II. Lateral inhibition and
 standing pulses,
  } SIAM J. Appl. Math. \textbf{62(1)} (2001), pp.226-243.

\bibitem{TL} {C. R. Laing  \& W. C. Troy,}
\emph{Two--bump solutions of Amari's model of working memory,}
Physica D, \textbf{178} (2003), pp.~190-218.

\bibitem{Shangbing} {A. I. Shangbing,} \emph{Traveling waves in a bioremediation model,}
SIAM J. Appl. Math. \textbf{68} (2007), pp.~680-693

\bibitem{Shangbing2} {A.~I.~Shangbing \& S.~P.~Hastings,}
\emph{A shooting approach to layers and chaos in a forced duffing
equation,
  } J.~Diff.~Eqns. \textbf{185(2)} (2002), pp.389-436.

  \bibitem{WC1} {H. R. Wilson \& J. D. Cowan,} \emph{Excitatory
  and inhibitory interactions in localized
  populations of model neurons,} Biophysical J.
  \textbf{12} (1972), pp.~1-24.

\end{thebibliography}


\end{document}