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\AtBeginDocument{{\noindent\small
Sixth Mississippi State Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conference 15 (2007),  pp. 399--415.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{399}
\title[\hfilneg EJDE-2006/Conf/15\hfil Existence, multiplicity, and bifurcation]
{Existence, multiplicity, and bifurcation in systems of ordinary
differential equations}
\author[J. R. Ward Jr.\hfil EJDE/Conf/15 \hfilneg]
{James R. Ward Jr.}

\address{James R. Ward Jr. \newline
Department of Mathematics\\
University of Alabama at Birmingham\\
Birmingham, AL 35294, USA}
\email{jrw87@math.uab.edu}

\dedicatory{Dedicated to my friend Klaus Schmitt}

\thanks{Published February 28, 2007.}
\thanks{Supported by grant INT 0204032 from the NSF }
\subjclass[2000]{34B15, 47J10, 47J15} 
\keywords{Global bifurcation; rotation number;  Leray-Schauder degree;
\hfill\break\indent nonlinear boundary value problems}

\begin{abstract}
 We prove new non-resonance conditions for boundary value problems for two
 dimensional systems of ordinary differential equations. We apply these results
 to the existence of solutions to nonlinear problems. We then study global
 bifurcation for such systems of ordinary differential equations Rotation
 numbers are associated with solutions and are shown to be invariant along
 bifurcating continua. This invariance is then used to analyze the global
 structure of the bifurcating continua, and to demonstrate the existence of
 multiple solutions to some boundary value problems.
\end{abstract}

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

\section{Introduction}

The purpose of this paper is to prove some existence, bifurcation, and
multiplicity results for boundary value problems for two dimensional systems
of ordinary differential equations. In this respect the paper is a
continuation of \cite{Wrot}. Consider the parameter dependent family of
boundary-value problems
\begin{gather}
\frac{dw}{dt}=F(\lambda,w,t),\quad t\in[ 0,\omega] \label{eq1} \\
Bw=0 \label{bc1}
\end{gather}
where $F=(F_{1},F_{2})\in C(\mathbb{R}\mathbb{R}^{2}\times[ 0,\omega],
\mathbb{R}^{2})$, $t\in[ 0,\omega]$, $w=(u,v)\in
\mathbb{R}^{2}$, and $\lambda\in \mathbb{R}$ is a parameter.
 We concentrate here on $Bw:=(u(0),u(\omega))$, which we
will call the Dirichlet problem. Our methods work just as well with many
other boundary conditions, including the periodic problem and
$Bw=(u(0),v(\omega))$. The most general form we allow for the function $F$
will usually be
\begin{equation}
F(\lambda,w,t)=B(\lambda,t)w+g(\lambda,w,t) \label{F}
\end{equation}
with $B(\lambda,t)=\lambda J+A(t)$ or $B(\lambda,t)=\lambda A(t)$, where
\[
J=\begin{pmatrix}
0 & -1\\
1 & 0 \end{pmatrix}, \quad
A(t)=\begin{pmatrix}
0 & -p(t)\\
q(t) & 0 \end{pmatrix}
\]
with $p,q\in L^{\infty}(0,\omega)$ and
$g(\lambda,w,t)=o(|
w| )$ as $| w| \to 0$ (or $\infty)$,
uniformly with respect to $\lambda$ and $t$ in compact sets.

If $w=w(t)=(u(t),v(t))$ is an $\omega$-periodic solution of (\ref{eq1}) with
$w(t)\neq0$ for all $t$, then the mapping $t\mapsto\frac{w(t)}{|
w(t)| }$ defines a mapping from the circle $S^{1}$ into itself. If
$\varphi$ denotes this mapping then the Brouwer degree $\deg(\varphi)$ is
defined. It is the same as the rotation number of $w(t)$ (with respect to the
origin). If $\theta=\tan^{-1}(\frac{v}{u})$ then
\[
\frac{d\theta}{dt}=\frac{v'u-vu'}{u^{2}+v^{2}}
\]
and the rotation number of such an $\omega$-periodic solution is
\[
\mathop{\rm rot}(w)=\frac{1}{2\pi}\int_{0}^{\omega}\frac{uv'-vu'
}{u^{2}+v^{2}}\,dt=\frac{1}{2\pi}\int_{0}^{\omega}\frac{F_{2}u-vF_{1}}
{u^{2}+v^{2}}\,dt.
\]
For most of our results we will use the rotation number to distinguish
solutions and branches of solutions. This idea was used in \cite{Wrot} to
study global bifurcation from zero and solution multiplicity. In \cite{Wrot}
only problems with $B(\lambda,t)=\lambda J$ in (\ref{F}) were considered, and
bifurcation from infinity was not studied, as it is here. Rotation numbers can
be assigned to solutions of non-periodic boundary value problems, such as the
two already mentioned, by appropriately extending the solutions to a larger
interval on which the extension is periodic and the rotation number is an
integer. In the non-systems case of second order scalar Sturm-Liouville
boundary value problems on an interval $[0,\omega]$, bifurcating branches can
be distinguished by the number of solution nodal points in $[0,\omega[ $
\cite{Rab1}. Our methods are based upon Leray-Schauder degree and change of
degree as the parameter $\lambda\in \mathbb{R}$ varies.
 Krasnosel'skii \cite{Kra} first used Leray-Schauder degree to prove
the existence of bifurcation at eigenvalues of odd multiplicity and Rabinowitz
\cite{Rab1} later showed global bifurcation from these eigenvalues and proved
fundamental results on global structure of bifurcating continua. These ideas
and results have been applied and extended by subsequent researchers in deep
and ingenious ways to understand bifurcations and solution structure for
nonlinear boundary value problems. The reader is referred to the fundamental
paper \cite{Rab1} or the expositions in \cite{Rab2} or \cite{Brown} for the
fundamental ideas. The rotation numbers of solutions have been used before to
analyze global solution structure for boundary value problems, see
\cite{CHMZ}, \cite{CMZ1}. The paper \cite{Ber} improves some results of
\cite{Wrot} for superlinear systems, and makes interesting use of rotation
number in connection with the Capietto-Mawhin-Zanolin continuation theorem
(see \cite{CMZ1}).

In \S 2 we study linear systems. In \S 3 we apply the ideas of \S 2 to study
existence under nonresonance conditions. In \S 4 we study bifurcation from a
line of trivial solutions, making use of rotation number to characterize
branches. In \S 5 we prove results on bifurcation from infinity. In \S 6 we
obtain conditions for bifurcating branches to bend to the left or to the
right. In \S 7 we apply the earlier results to prove a theorem on multiplicity
of solutions.

In the sequel, for $x=(x_{1},x_{2},\dots,x_{n})^{T}\in\mathbb{R}^{n}$
we let $| x| :=(\sum_{i=1}^{n}x_{i}^{2})^{1/2}$; with
$T$ indicating the transpose. We will sometimes omit the $T$, so by $w=(u,v)$
we usually mean the column vector. For $w\in C([A,B],
\mathbb{R}^{n})$, $\| w\| :=\max_{[A,B]}| w(t)| $,
and for $w\in L^{p}((A,B), \mathbb{R}^{n})$,
$1\leq p\leq\infty$, let $\| w\| _{p}:=(\int_{A}^{B}| w(t)| ^{p}\,dt)^{1/p}$.

\section{Linear systems}

We begin by considering linear systems of the form
\begin{equation} \label{LS}
\begin{aligned}
\frac{du}{dt}   =-p(t)v\\
\frac{dv}{dt}   =q(t)u
\end{aligned}
\end{equation}
for $t\in[ 0,\omega]$ where $p,q\in L^{\infty}(0,\omega)$, together with
boundary conditions
\begin{equation}
B(u,v)=(0,0). \label{LBC}
\end{equation}
The boundary operator in (\ref{LBC}) is linear and could represent
$T$-periodic boundary conditions, $B(u,v)=(u(\omega),v(\omega))-(u(0),v(0))$,
the boundary operator $B(u,v)=(u(0),u(\omega))$, or $Bw=(u(0),v(\omega))$, or
possibly others. The admissible boundary conditions are those that allow a
well defined rotation number to be associated with nontrivial solutions to
\eqref{LS}, (5), (\ref{LBC}). If $w=(u,v)^{T}$ is a nontrivial solution with
$w(\omega)-w(0)=0$ then there is an integer rotation number defined by
\[
\mathop{\rm rot}(w)=\frac{1}{2\pi}\int_{0}^{\omega}\frac{q(t)u^{2}+p(t)v^{2}
}{u^{2}+v^{2}}\,dt.
\]


In the case of non periodic boundary conditions such as $Bw=(u(0),u(\omega
))=(0,0)$, more care must be taken to obtain an integer rotation number. In
the latter case, extend $p(t)$ and $q(t)$ respectively to functions
$\widetilde{p}(t),\widetilde{q}(t)$, on $[-\omega,\omega]$, so that both are
even and extend $u(t)$ to $\widetilde{u}(t)$, odd on $[-\omega,\omega]$ and
$v(t)$ to $\widetilde{v}(t)$, even on $[-\omega,\omega]$. Then $\widetilde
{w}=(\widetilde{u},\widetilde{v})^{T}$ satisfies
\begin{gather*}
\frac{d\widetilde{u}}{dt}   =-\widetilde{p}(t)\widetilde{v}\\
\frac{d\widetilde{v}}{dt}   =\widetilde{q}(t)\widetilde{u}
\end{gather*}
on $[-\omega,\omega]$. Moreover $\widetilde{w}$ satisfies the periodic
conditions $\widetilde{w}(\omega)-\widetilde{w}(-\omega)=0$. We will
henceforth refer to $\widetilde{w}$ as the \textit{odd/even} extension of $w$.
We define the rotation number of $w$ to be the rotation number of
$\widetilde{w}$:
\begin{equation}
\mathop{\rm rot}(w):=\mathop{\rm rot}(\widetilde{w})=\frac{1}{2\pi}
\int_{-\omega}^{\omega}\frac{\widetilde{q}(t)\widetilde{u}^{2}
+\widetilde{p}(t)\widetilde
{v}^{2}}{\widetilde{u}^{2}+\widetilde{v}^{2}}\,dt \label{rot2}
\end{equation}
is a well defined integer. Notice that the rotation number has the properties
of Brouwer degree. Indeed, in the periodic case it is the same as the degree
of the map from $S^{1}(=[0,\omega]/\{0,\omega])\to  S^{1}$ defined by
$t\longmapsto w(t)/| w(t)| $, with a similar identification
in the second boundary condition considered above. One may also associate a
rotation number with nontrivial solutions satisfying the boundary condition
$u(0)=0$, $v(\omega)=0$, and others.

We wish to compare the rotation numbers associated with solutions of two
different systems. Let $p_{j},q_{j}\in L^{\infty}([0,T],\mathbb{R})$
for $j=1,2$ and let $w_{j}=(u_{j},v_{j})$ ($j=1,2$) be non-trivial
solutions of the $j$th problem, so
\begin{equation}
\frac{du_{j}}{dt}=-p_{j}(t)v_{j},\quad \frac{dv_{j}}{dt}=q_{j}
(t)u_{j},\quad (u_{j}(T),v_{j}(T))=(u_{j}(0),v_{j}(0)). \label{LSJ}
\end{equation}


\begin{lemma} \label{lem1}
Let $p_{j},q_{j}\in L^{\infty}([0,T],\mathbb{R}),
w_{j}=(u_{j},v_{j})$ ($j=1,2$) be a non trivial solution of \eqref{LSJ}
 for $j=1,2$ respectively. Suppose that we have
\begin{equation}
a(t):=\max(p_{1}(t),q_{1}(t))\leq b(t):=\min(p_{2}(t),q_{2}(t))\quad
\text{a.e.}
\label{ineq1}
\end{equation}
Then $\mathop{\rm rot}(w_{1})\leq\mathop{\rm rot}(w_{2})$. If there is a set
$E\subset[ 0,T]$ of positive Lebesgue measure such that strict inequality
holds in either inequality \eqref{ineq1} for $t\in E$, then
$\mathop{\rm rot}(w_{1})<\mathop{\rm rot}(w_{2})$.
\end{lemma}

\begin{proof}
We have
\begin{align*}
\mathop{\rm rot}(w_{1})
&  =\frac{1}{2\pi}\int_{0}^{2\pi}\frac{q_{1}(t)u_{1}
^{2}+p_{1}(t)v_{1}^{2}}{u_{1}^{2}+v_{1}^{2}}\,dt\\
&\leq\frac{1}{2\pi}\int_{0}^{2\pi}a(t)\,dt\\
&  \leq\frac{1}{2\pi}\int_{0}^{2\pi}b(t)\,dt\\
&\leq\frac{1}{2\pi}\int_{0}^{2\pi
}\frac{q_{2}(t)u_{2}^{2}+p_{2}(t)v_{2}^{2}}{u_{2}^{2}+v_{2}^{2}}
dt=\mathop{\rm rot}(w_{2})
\end{align*}
which proves the first part of the claim. If there were a set of positive
measure $E$ on which $a(t)<b(t)$ for $t\in E$, then the second integral
inequality would be strict also, and this would imply that $\mathop{\rm rot}
(w_{1})<\mathop{\rm rot}(w_{2})$.
\end{proof}

The same conclusion holds if we impose the boundary conditions $u(0)=0,u(\pi
)=0$ or $u(0)=0,v(\pi)=0$. In those cases we consider the appropriate periodic
extensions of the nontrivial solutions $w$ to define $\mathop{\rm rot}(w)$. For
instance, in the $u(0)=0,u(\pi)=0$ case, we extend $u$ to be odd on $[-\pi
,\pi]$, and $v$ to be even on $[-\pi,\pi]$, and then both to be $2\pi
-$periodic; we extend $p$ and $q$ to be even on $[-\pi,\pi]$ and then also
$2\pi$-periodic. Then the rotation number of $w$ is well-defined. Using this
and similar arguments one may establish the conclusion of the lemma if these
other boundary conditions are imposed on \eqref{LS}.

In this paper we will, for the sake of concreteness, in the main consider the
boundary conditions $B(u,v):=(u(\pi),u(0))=(0,0)$.

We now consider nonresonance conditions. Let $\mu\in\mathbb{R}$ and
consider the problem
\begin{equation} \label{LSC}
\begin{gathered}
\frac{du}{dt}    =-\mu v\\
\frac{dv}{dt}    =\mu u
\end{gathered}
\end{equation}
with boundary conditions
\begin{equation}
(u(\pi),u(0))=(0,0). \label{DBC}
\end{equation}
It is easy to check that the eigenvalues of the above system consists of the
set of integers $\mathbb{Z}$, and for $n=0$ an eigenfunction is
$w_{0}=(0,1)$, while for each $n\in\mathbb{Z}\backslash\{0\}$,
and eigenfunction is $w_{n}(t)=(\sin(nt),-\cos(nt))$.
 The associated rotation numbers (as defined above) are
$\mathop{\rm rot}(w_{n})=n$.

We introduce a notation useful here. Suppose $F$ and $G$ are real valued
Lebesgue measurable functions on an interval $I$. The notation $F(t)\lesssim
G(t)$ on $I$ (or just $F\lesssim G$ on $I$) will mean that $F(t)\leq G(t)$
a.e. on $I$ and there is a set of positive measure in $I$ on which the
inequality is strict. We can now state and prove the lemma

\begin{lemma} \label{lem2}
Consider the problem \eqref{LS} with $\omega=\pi$ and $p,q\in L^{\infty}
(0,\pi)$ and boundary conditions $(u(\pi),u(0))=(0,0)$. Suppose that there is
$n\in\mathbb{Z}$ such that $n\lesssim p(t)\lesssim(n+1)$ and
$n\lesssim q(t)\lesssim(n+1)$ on $[0,\pi]$. Then the problem has
no non-trivial solutions.
\end{lemma}

\begin{proof}
Suppose there is a nontrivial solution $w=(u,v)^{T}$. Let $\widetilde{w}$ be
the odd/even extension of $w$. Then $\mathop{\rm rot}(w):=\mathop{\rm rot}
(\widetilde{w})$ is well defined and is an integer. However the rotation
numbers of (odd/even extended) solutions to the system (\ref{LSC}),
(\ref{DBC}) with $\mu=n$ and $\mu=n+1$ are $n$ and $n+1$, respectively, and
this implies $n<\mathop{\rm rot}(w)<n+1$, which is impossible.
This proves the lemma.
\end{proof}

It should be clear that analogous results are true with other boundary
conditions, or on other intervals, with similar proofs.

We have an immediate corollary regarding the problem
\begin{gather}
\begin{gathered}
\frac{du}{dt}+p(t)v    =f(t)\\
\frac{dv}{dt}-q(t)u    =g(t)
\end{gathered} \label{LBVP}\\
u(0)=0,\quad u(\pi)=0. \label{BC1}
\end{gather}


\begin{corollary} \label{coro3}
Let $p,q\in L^{\infty}(0,\pi)$ satisfy the conditions of the lemma, and let
$f,g\in L^{1}(0,\pi)$. Then there is a unique solution to (\ref{LBVP}),
\eqref{BC1}.
\end{corollary}

\begin{remark} \label{rmk4} \rm
By a solution to (\ref{LBVP}), \eqref{BC1} we mean a pair of functions
$(u,v)$, each absolutely continuous on $[0,\pi]$, satisfying \eqref{BC1} and
also satisfying (\ref{LBVP}) $a.e$.
\end{remark}

\begin{remark} \label{rmk5} \rm
The lemmas and corollary hold with appropriate modifications for many other
boundary value problems, such as the periodic one.
\end{remark}

Now suppose $p,q\in L^{\infty}(0,\pi)$ satisfy
\begin{equation}
-M\lesssim p(t),q(t)\lesssim M\text{ a.e.} \label{M}
\end{equation}
for some $M>0$.  Let
\[
J=\begin{pmatrix}
0 & 1\\
-1 & 0 \end{pmatrix}, \quad
A(t)=\begin{pmatrix}
0 & p(t)\\
-q(t) & 0 .\end{pmatrix}
\]
We will now study parameter dependent linear systems of the forms
\begin{equation}
\frac{dw}{dt}+\lambda Jw+A(t)w=0 \label{ls1}
\end{equation}
 and
\begin{equation}
\frac{dw}{dt}+\lambda A(t)w=0. \label{ls2}
\end{equation}
with the boundary conditions \eqref{BC1}, i.e.,
\begin{equation}
u(0)=0,\quad u(\pi)=0,\text{ where }w(t)=(u(t),v(t)). \label{BC1a}
\end{equation}
We first analyze the problem \eqref{ls1} with boundary conditions
\eqref{BC1a}. Extend $p$ and $q$ to $P$ and $Q$, respectively, even on
$[-\pi,\pi]$, and then to be $2\pi$-periodic. This is equivalent to letting
$P(t+\pi):=p(\pi-t)$ and $Q(t+\pi):=q(\pi-t)$for $0\leq t\leq\pi$ and then
extending $P$ and $Q$ to be $2\pi$ periodic on $\mathbb{R}$.
If $w=(u,v)$ satisfies the boundary conditions, we extend $u$ to be even
with respect to $0$ and $2\pi$ periodic on $\mathbb{R}$,
and extend $v$ to be odd with respect to zero and also $2\pi$ periodic. As
before, we call this the \textit{odd/even extension of }$w$. For each
$\mu\in \mathbb{R}$ let $W_{\mu}=(U_{\mu},V_{\mu})$ be the solution
 to the initial-value problem
\begin{equation} \label{IV}
\begin{gathered}
\frac{dU}{dt}+\mu V+P(t)V    =0\\
\frac{dV}{dt}-\mu U-Q(t)U    =0\\
U(0)   =0,\quad V(0)=1.
\end{gathered}
\end{equation}
We can still define a real valued function $\Psi$ by the equation
\begin{equation} \label{eq3}
\begin{aligned}
\Psi(\mu)  &  :=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(\mu+P(t))V_{\mu}^{2}
+(\mu+Q(t))U_{\mu}^{2}}{U_{\mu}^{2}+V_{\mu}^{2}}\,dt\\
&  =\mu+\frac{1}{2\pi}\int_{0}^{2\pi}\frac{P(t)V_{\mu}^{2}+Q(t)U_{\mu}^{2}
}{U_{\mu}^{2}+V_{\mu}^{2}}\,dt.
\end{aligned}
\end{equation}
If  also $U_{\mu}(\pi)=0$ then $W_{\mu}$ will be $2\pi$-periodic and have an
integral rotation number (this is true because if $(U,V)$ is any solution on
$[0,\pi]$ satisfying the boundary conditions, then it has a $2\pi$ periodic
extension satisfying the differential equations, as was shown in Section 2.
But there is only one solution satisfying the initial conditions). Conversely,
if $\Psi(\mu)$ is an integer, then the change in angle of $W_{\mu}(t)$ with
respect to the origin over the interval $0\leq t\leq2\pi$ is an integral
multiple of $2\pi$, and hence $W_{\mu}(t)$ must be $2\pi$-periodic. From this
and that $P(t)$ and $Q(t)$ are even and $2\pi$- periodic one can deduce that
$U(t)$ is odd and $V(t)$ even, and hence that $U(\pi)=0$. Thus we have the
following result.

\begin{lemma} \label{lem6}
$W_{\mu}$ satisfies the boundary conditions \eqref{BC1a} if
and only if $\Psi(\mu)\in\mathbb{Z}$.
\end{lemma}

The solution $W_{\mu}$ to the initial value problem (\ref{IV}) varies
continuously with respect to the parameter $\mu\in\mathbb{R}$ and
therefore $\Psi$ is a continuous function. It follows from (\ref{M})
that
\[
-M<\int_{0}^{2\pi}\frac{P(t)V_{\mu}^{2}+Q(t)U_{\mu}^{2}}{U_{\mu}^{2}+V_{\mu
}^{2}}<M
\]
and hence
\[
\mu-M<\Psi(\mu)<\mu+M
\]
for all $\mu\in \mathbb{R}$. Thus $\Psi(\mu)\to \pm\infty$
as $\mu\to \pm\infty$, and
there is a doubly infinite sequence $\{\mu_{n}:n\in \mathbb{Z}\}$
such that $\Psi(\mu_{n})=n$ and a nontrivial solution $W_{n}$ to
\eqref{ls1}, \eqref{BC1a} with $\lambda=\mu_{n}$ and
$\mathop{\rm rot}(W_{n})=n$.
 Notice also that since $\Psi(\mu_{n})=n$ we have $\mu_{n}-M<n<\mu_{n}+M$. We
do not know if there can be more than one solution $\mu$ to the equation
$\Psi(\mu)=n$. We will refer to the set of all such solutions $\mu$ for
$n\in\mathbb{Z}$ as the set of eigenvalues for
\eqref{ls1}, \eqref{BC1a}. If $P=Q$ then
$\Psi(\mu)=\mu+\overline{P}$ where $\overline{P}$ denotes the mean value of
$P$. In this special case $\mu_{n}=\overline{P}-n$ is unique. In this case we
also note that $U^{2}(t)+V^{2}(t)$ is constant. It follows from the general
theory of compact linear operators that the set of eigenvalues has no finite
limit point. From this and the structure of $\Psi$ it follows that there can
be at most finitely many eigenvalues associated with any given rotation number.

Each eigenvalue has a one dimensional eigenspace since if it were two
dimensional the eigenspace would be a basis for all solutions to
 \eqref{ls1}, and then all solutions would have to satisfy $u(0)=0$.

We have proven the following result.

\begin{theorem} \label{thm7}
Let $p,q\in L^{\infty}(0,\pi)$ satisfy \eqref{eq2}. Then the problem
\eqref{ls1}, \eqref{BC1a} has a doubly infinite sequence of eigenvalues
$\{\mu_{n}:n\in \mathbb{Z}\}$.
Moreover the eigenspace associated with each eigenvalue is one
dimensional and if $w\neq0$ is a function in the eigenspace for some
eigenvalue $\mu$ then there is an $n\mathbb{Z}$ such that
$\Psi(\mu)=n$ and $\mathop{\rm rot}(w)=n$, where the latter denotes
the rotation number associated with $w$ as defined earlier in \eqref{rot2}.
There are at most finitely many eigenvalues associated with
the same rotation number.
\end{theorem}

We now consider parameter dependent linear systems of the form \eqref{ls2}
with boundary conditions \eqref{BC1a}. We again assume $p,q\in L^{\infty
}(0,\pi)$ and make the additional assumption that there is a $\beta\in
L^{\infty}(0,\pi)$ and $M>0$ such that
\begin{equation}
0\lesssim\beta(t)\leq\min(p(t),q(t))\leq M\text{ for a.a. }t\in[
0,\pi]. \label{beta}
\end{equation}


We again extend $p$ and $q$ to $P$ and $Q$, respectively, even on
$[-\pi,\pi ]$, and then $2\pi$-periodic on the real line.
If $w=(u,v)$ satisfies
\eqref{BC1a}, we make the odd/even $2\pi$-periodic extension of $w$.  Let
$W_{\mu}$ be the solution to the initial-value problem
\begin{gather*}
\frac{dU}{dt}+\mu P(t)V   =0\\
\frac{dV}{dt}-\mu Q(t)U   =0\\
U(0)    =0,\quad V(0)=1.
\end{gather*}


We can as before define a real valued function $\Psi$ by the equation
\begin{align*}
\Psi(\mu)  & :=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\mu P(t)V_{\mu}^{2}+\mu
Q(t)U_{\mu}^{2}}{U_{\mu}^{2}+V_{\mu}^{2}}\,dt\\
&  =\frac{\mu}{2\pi}\int_{0}^{2\pi}\frac{P(t)V_{\mu}^{2}+Q(t)U_{\mu}^{2}
}{U_{\mu}^{2}+V_{\mu}^{2}}\,dt.
\end{align*}
The Lemma is valid here, so $\Psi(\mu)\in \mathbb{Z}$ if and only
if $W_{\mu}$ satisfies the boundary conditions. Clearly $\Psi$
is a continuous real valued function. Let $\overline{\beta}$ be the mean value
of $\beta$ over $[0,2\pi\}$. For $\mu>0$ we have $\Psi(\mu)\geq\mu
\overline{\beta}$ and for $\mu<0$ we have $\Psi(\mu)<\mu\overline{\beta}$.
Thus the range of $\Psi$ is the set of real numbers and for each
$n\in\mathbb{Z}$ there is at least one $\mu\in\mathbb{R}$ with
$\Psi(\mu)=n$. We will refer to the set of all such solutions $\mu$ for
$n\in\mathbb{Z}$ as the set of eigenvalues for
\eqref{ls2}, \eqref{BC1a}. If $P=Q$ then
$\Psi(\mu)=\mu\overline{P}$ where $\overline{P}$ denotes the mean value of
$P$. In this special case $\mu_{n}=n/\overline{P}$ is unique. Note that in
this case $U^{2}(t)+V^{2}(t)$ is constant. As in the previous theorem, there
can be at most finitely many eigenvalues associated with any given rotation
number.

Each eigenvalue has a one dimensional eigenspace since if it were two
dimensional the eigenspace would be a basis for all solutions to \eqref{ls2},
and then all solutions would have to satisfy $u(0)=0$.

We have proven the following result.

\begin{theorem} \label{thm8}
Let $p,q\in L^{\infty}(0,\pi)$ satisfy \eqref{beta}. Then the problem
\eqref{ls2}, \eqref{BC1a} has a doubly infinite sequence of eigenvalues
$\{\mu_{n}:n\in\mathbb{Z}\}$. Moreover the eigenspace associated with
each eigenvalue is one
dimensional and if $w\neq0$ is a function in the eigenspace for some
eigenvalue $\mu$ then there is an $n\in\mathbb{Z}$ such that
 $\Psi(\mu)=n$ and $\mathop{\rm rot}(w)=n$, where the latter denotes
the rotation number associated with $w$ as defined earlier in \eqref{rot2}.
There are at most finitely many eigenvalues associated with the same
rotation number.
\end{theorem}

\section{Nonresonance and existence}

We now consider nonlinear problems of the form
\begin{equation}  \label{QL}
\begin{gathered}
\frac{du}{dt}+p(t,u,v)v    =f(t,u,v)\\
\frac{dv}{dt}-q(t,u,v)u    =g(t,u,v)
\end{gathered}
\end{equation}
with the boundary conditions \eqref{BC1}; that is, the conditions are:
\[
u(0)=0,\quad u(\pi)=0.
\]
We will assume in this section that $p,q,f,g$ satisfy Carath\'{e}odory
conditions. That is, we assume that for almost all $t\in[ 0,\pi]$ the
maps $p(t,.,.)$, $q(t,.,.)$, $f(t,.,.)$, $g(t,.,.)$ are continuous on
$\mathbb{R}^{2}$, and for each $(u,v)\in\mathbb{R}^{2}$, the maps
$p(.,u,v)$, $q(.,u,v)$, $f(.,u,v)$, $g(.,u,v)$ are Lebesgue measurable
on $[0,\pi]$.
We also assume there is a function $S_{1}\in L^{\infty}(0,\pi)$
such that $| p(t,u,v)| +| q(t,u,v)| \leq S_{1}(t)$ for all
$(u,v)\in\mathbb{R}^{2}$ and $a.a$. $t\in[ 0,\pi]$, and for
each $R\geq0$ there is a
function $M_{R}\in L^{1}(0,\pi)$ such that
$| f(t,u,v)|+| g(t,u,v)| \leq M_{R}(t)$ for all
$|(u,v)| \leq R$ and $a.a$. $t\in[ 0,\pi]$.

We now can state an existence theorem.

\begin{theorem} \label{thm9}
Let $p,q,f,g$ be as described above. In addition assume:
\begin{enumerate}
\item There are functions $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in
L^{\infty}(0,\pi)$ and $N\in\mathbb{Z}$ such that for all
$(u,v)\in\mathbb{R}^{2}$,
\begin{gather*}
N   \lesssim\alpha_{1}(t)\leq p(t,u,v)\leq\beta_{1}(t)\lesssim N+1, \\
N  \lesssim\alpha_{2}(t)\leq q(t,u,v)\leq\beta_{2}(t)\lesssim
N+1\quad
\end{gather*}
hold on $[0,\pi]$.

\item There is a function $m\in L^{1}(0,\pi)$ such that for each
$\varepsilon>0$ there is an $R(\varepsilon)\geq0$ for which the following hold
$a.e.$:
\begin{gather*}
| f(t,u,v)|   \leq\varepsilon m(t)|(u,v)| ,\\
| g(t,u,v)|   \leq\varepsilon m(t)|(u,v)| .
\end{gather*}
Then there is at least one solution to \eqref{QL}, \eqref{BC1}.
\end{enumerate}
\end{theorem}

\begin{proof}
The proof uses degree theory. We sketch the argument. Define functions
$A_{i}$, $i=1,2$ by $A_{i}:=\frac{1}{2}(\alpha_{i}+\beta_{i})$.
Then $n\lesssim A_{i}\lesssim n+1$ so that for each pair
$k_{1},k_{2}\in L^{1}=L^{1}(0,\pi)$
there is a unique solution $w=(u,v)\in C=C([0,\pi],\mathbb{R}^{2})$
to the boundary problem
\begin{gather*}
\frac{du}{dt}+A_{1}(t)v   =k_{1}(t)\\
\frac{dv}{dt}-A_{2}(t)u   =k_{2}(t)
\end{gather*}
with boundary conditions \eqref{BC1}. Let $\Gamma$ denote the linear mapping
$(k_{1},k_{2})\mapsto w=(u,v)$ from $L^{1}\times L^{1}$ into $C\times C$. The
mapping $\Gamma$ is compact.

Let $C_{0}$ denote the Banach space of all pairs of continuous functions
$w=(u,v)$ on $[0,\pi]$ satisfying $u(0)=0=u(\pi)$, with norm $\|
w\| :=\max_{[0,\pi]}| w(t)| $. Define a mapping
$N:C_{0}\to  L^{1}\times L^{1}$ by
\[
N(w)(t):=\begin{pmatrix}
-A_{1}(t)v(t)+p(t,u(t),v(t))v(t)-f(t,u(t),v(t))\\
A_{2}(t)u(t)-q(t,u(t),v(t))u(t)-g(t,u(t),v(t))
\end{pmatrix}
\]
for $t\in[ 0,\pi]$. The mapping $N$ is continuous and maps bounded sets
into bounded sets. The boundary value problem \eqref{QL}, \eqref{BC1} is
equivalent to the equation
\begin{equation}
w+\Gamma N(w)=0 \label{E}
\end{equation}
in $C_{0}$. We now apply a homotopy to (\ref{E}) and use
 Leray-Schauder degree.

The mapping $\Gamma N:C_{0}\to  C_{0}$ is completely continuous: it is
continuous, and maps bounded sets into relatively compact ones. We consider
the parameterized family of equations:
\begin{equation}
w+\lambda\Gamma N(w)=0. \label{EL}
\end{equation}
We shall show that there is a number $R^{\ast}>0$ such that if $w$ is a
solution to (\ref{EL}) for any $\lambda\in[ 0,1]$ then $\|
w\| <R^{\ast}$. It will then follow that the degree $\deg
_{LS}(I+\lambda\Gamma N,B(0,R^{\ast}),0)$ is independent of $\lambda\in
[ 0,1]$, and from this we will be able to deduce that a solution to
(\ref{E}) exists. We proceed. .

Suppose that there is no such number $R^{\ast}$. It then follows that there
is a sequence $\{(\lambda_{n},w_{n})\}$ in $[0,1]\times C_{0}$ such that
$\| w_{n}\| \to \infty$ and for each $n\in\mathbb{N}$ we have
$w_{n}+\lambda_{n}\Gamma Nw_{n}=0$. Then $w_{n}=(u_{n}v_{n})$
satisfies
\begin{equation} \label{SL}
\begin{gathered}
\frac{du_{n}}{dt}+(1-\lambda_{n})A_{1}(t)v_{n}+\lambda_{n}p_{n}(t)v_{n}
=\lambda_{n}f(t,u_{n},v_{n})\\
\frac{dv_{n}}{dt}-(1-\lambda_{n})A_{2}(t)u_{n}-\lambda_{n}q_{n}(t)u_{n}
=\lambda_{n}g(t,u_{n},v_{n})
\end{gathered}
\end{equation}
and $u_{n}(\pi)=0=u_{n}(0)$, where $p_{n}(t)=p(t,u_{n},v_{n})$,
and $q_{n}(t)=q(t,u_{n},v_{n}).$Let $\widetilde{w}_{n}=w_{n}/\|
w_{n}\| $. It follows from (\ref{SL}) and the properties of the terms
in the equation that there is a constant $c_{1}>0$ such that for all
$n\in\mathbb{N}$, $\| \widetilde{w}_{n}\| _{L^{1}}<c_{1}$.
 From the latter
inequality it follows that there is a subsequence of
$\{(\lambda_{n},\widetilde{w}_{n})\}$ convergent in $[0,1]\times C_{0}$ to some
$(\widetilde{\lambda},\widetilde{W})=(\widetilde{\lambda},(\widetilde
{u},\widetilde{v}))$. We relabel that convergent subsequence as
$\{(\lambda_{n},\widetilde{w}_{n})\}$. Now for all $n\in\mathbb{N}$,
\begin{gather*}
N    \lesssim\alpha_{1}(t)\leq p_{n}(t)\leq\beta_{1}(t)\lesssim N+1, \\
N    \lesssim\alpha_{2}(t)\leq q_{n}(t)\leq\beta_{2}(t)\lesssim
N+1,\quad \text{a.e. }
\end{gather*}
The sequence $\{(p_{n},q_{n})\}$ is bounded in $L^{\infty}\times L^{\infty}$
and hence in $L^{2}\times L^{2}$, so there is a subsequence weakly convergent
in $L^{2}\times L^{2}$ to some $(\widetilde{p},\widetilde{q})$ which must also
satisfy the inequalities
\begin{gather*}
N    \lesssim\alpha_{1}(t)\leq\widetilde{p}(t)\leq\beta_{1}(t)\lesssim
N+1, \\
N   \lesssim\alpha_{2}(t)\leq\widetilde{q}(t)\leq\beta
_{2}(t)\lesssim N+1,\quad \text{a.e. }
\end{gather*}
Integrating the differential equations from $0$ to $t$ and taking limits we
can conclude that
\begin{gather*}
\frac{d\widetilde{u}}{dt}+(1-\widetilde{\lambda})A_{1}(t)\widetilde
{v}+\widetilde{\lambda}\widetilde{p}(t)\widetilde{v}    =0\\
\frac{d\widetilde{v}}{dt}-(1-\widetilde{\lambda})A_{2}(t)\widetilde
{u}-\widetilde{\lambda}\widetilde{q}(t)\widetilde{u}    =0
\end{gather*}
and $(\widetilde{u}(\pi),\widetilde{u}(0))=(0,0)$. Moreover we must have
\begin{equation} \label{contra}
\begin{gathered}
N   \lesssim\alpha_{1}(t)\leq(1-\widetilde{\lambda})A_{1}(t)+\widetilde
{\lambda}\widetilde{p}(t)\leq\beta_{1}(t)\lesssim N+1\\
N    \lesssim\alpha_{2}(t)\leq(1-\widetilde{\lambda}
)A_{2}(t)+\widetilde{\lambda}\widetilde{q}(t)\leq\beta_{2}(t)\lesssim
N+1,\quad \text{a. e.}
\end{gathered}
\end{equation}
which implies, by Lemma \ref{lem2}, that $\widetilde{w}=(\widetilde{u},\widetilde
{v})=(0,0)$, a contrary to $\| \widetilde{w}\| =1$, and we
have reached a contradiction. It follows that there indeed must be a number
$R^{\ast}>0$ such that if $(\lambda,w)$ is a solution of (\ref{EL}) then
$\| w\| <R^{\ast}$. Therefore by the homotopy invariance of
Leray-Schauder degree $\deg_{LS}(I+\lambda\Gamma N,B(0,R^{\ast}),))$ is
independent of $\lambda\in[ 0,1]$ and
\[
d_{LS}(I+\Gamma N,B(0,R^{\ast}),0)=\deg_{LS}(I,B(0,R^{\ast}),0)=1
\]
and hence there is a $w^{\ast}\in B(0,R^{\ast})\subset C_{0}$ satisfying
$w^{\ast}+\Gamma Nw^{\ast}=0$. This proves the theorem.
\end{proof}

\begin{remark} \label{rmk10} \rm
Theorem \ref{thm9} recalls other theorems, some going back as far
 as \cite{Dolph}; the
paper \cite{LazerLeach} inspired many others to look seriosly at nonresonance
conditions for nonlinear differential equations. Conditions of the form
$\lambda_{N}\lesssim p(t)\lesssim\lambda_{N+1}$ have been used mainly in
boundary value problems for second order ordinary and partial differential
equations. This kind of condition does not seem to have been earlier used in
the situation of Theorem \ref{thm9}, perhaps because the proof of
Lemma \ref{lem2} differs from
proofs given for analogous lemmas in the second order case.
\end{remark}

\section{Bifurcation from zero}

Suppose $p,q\in L^{\infty}(0,\pi)$ satisfy
\begin{equation}
-M\lesssim p(t),q(t)\lesssim M\text{ a.e.} \label{eq2}
\end{equation}
for some $M>0$.  Let
\begin{equation}
J=\begin{pmatrix}
0 & 1\\
-1 & 0\end{pmatrix},\quad
A(t)=\begin{pmatrix}
0 & p(t)\\
-q(t) & 0
\end{pmatrix}. \label{A}
\end{equation}
Let $g\in C(\mathbb{R}\times[ 0,\pi]\times
\mathbb{R}^{2},\mathbb{R}^{2})$ (more generally, $g$ may be a
Carath\'{e}odory function) with
$g(\lambda,t,0)=0$, and $g(\lambda,t,w)=o(w)$\ We will apply the results on
\eqref{ls1} and \eqref{ls2} to study bifurcation and multiplicity questions
for the systems
\begin{equation}
\frac{dw}{dt}+\lambda Jw+A(t)w=g(\lambda,t,w) \label{BIF1}
\end{equation}
and
\begin{equation}
\frac{dw}{dt}+\lambda A(t)w=g(\lambda,t,w)  \label{BIF2}
\end{equation}
with the boundary conditions \eqref{BC1}, i.e.,
\begin{equation}
u(0)=0,\quad u(\pi)=0,\quad \text{where }w(t)=(u(t),v(t)). \label{BC1b}
\end{equation}
We now consider bifurcation from zero. Let $p,q\in L^{\infty}(0,\pi)$ satisfy
\eqref{eq2} and let $A(t)$ be as defined in \eqref{A}.
Let $g:\mathbb{R}\times[ 0,\pi]\times\mathbb{R}^{2}\to\mathbb{R}^{2}$
be a Carath\'{e}odory function. That is, for each
$(\lambda,w)\in\mathbb{R}\times\mathbb{R}^{2}$ the map
$t\mapsto g(\lambda,t,w)$ is Lebesgue measurable, and for almost
all $t\in[ 0,\pi]$ the map $(\lambda,w)\mapsto g(\lambda,t,w)$ is
continuous. Moreover, for each $r\geq0$ there is $\alpha_{r}\in L^{1}(0,\pi)$
such that $| g(\lambda,t,w)| \leq\alpha_{r}(t)$ a.e. for
$| \lambda| +| w| \leq r$. We also
assume $g(\lambda,0,t)=0$ and $| g(\lambda,w,t)|=o(| w| )$ as
$| w| \to 0$,
uniformly with respect to $\lambda$ and $t$ in compact sets. Let $w=(u,v)^{T}$
and consider the boundary value problem \eqref{BIF1}, \eqref{BC1b}.
\begin{gather*}
\frac{dw}{dt}+\lambda Jw+A(t)w    =g(\lambda,t,w)\\
u(0)   =0,\quad u(\pi)=0,
\end{gather*}
where $w=(u,v)^{T}$. First we write an abstract version of
\eqref{BIF1}, \eqref{BC1b}. Let $Y=L^{1}(0,\pi)$ and $X$
the Banach space of $\mathbb{R}^{2}$ valued functions $w=(u,v)$
continuous on $[0,\pi]$ with $u(0)=0=u(\pi)$
with norm $\| w\| =\max_{t\in[ 0,\pi]}|
w(t)| $. Let $D=\{w\in X:w'\in Y\}$. That is, functions in
$D$ satisfy the boundary conditions and are absolutely continuous. Now let
$0\leq c<1$ be a number such that the problem
\begin{equation} \label{linbif}
\begin{gathered}
\frac{dw}{dt}+cJw+A(t)w  =0\\
u(0)=0,\quad u(\pi)=0
\end{gathered}
\end{equation}
has no non-trivial solution.  In this case we define $L:D\to  Y$ be
defined by
\[
Lw:=\frac{dw}{dt}+cJw+A(t)w
\]
for $w\in D$. The linear operator $L$ has a compact inverse $L^{-1}$.
Let $G:\mathbb{R}\times X\to  Y$ be defined by
$G(\lambda,w):=g(\lambda,\cdot,w(\cdot))$
for $(\lambda,w)\in\mathbb{R}\times X$. The map $G$ is continuous and
take bounded sets to bounded sets.
Now our problem is equivalent to the equation in $X$ given by.
\[
w+(\lambda-c)L^{-1}Jw=L^{-1}G(\lambda,w)
\]
or
\begin{equation}
w+\mu L^{-1}Jw=L^{-1}\widetilde{G}(\mu,w) \label{AbsBVP}
\end{equation}
where $\mu=\lambda-c$ and $\widetilde{G}(\mu,w)=G(\mu+c,w)$. Now
$\lambda^{\ast}$ is an eigenvalue of \eqref{ls1}, \eqref{BC1b} (equivalently,
$\Psi(\lambda^{\ast})\notin\mathbb{Z}$) if and only if
$\mu^{\ast}=\lambda^{\ast}-c$ is a characteristic value of
$L^{-1}J$. If $\mu$ is not a characteristic value of $L^{-1}J$ then the
Leray-Schauder degree $\deg_{LS}(I+\mu L^{-1}J,B(r),0)$ is defined for $r>0$
(where $B(r)=\{w\in X:\| w\| <r\}$). Now if $\mu^{\ast}$ is a
characteristic value of $L^{-1}J$ then the null space of $I+\mu^{\ast}L^{-1}J$
is one dimensional, and thus the Leray-Schauder degree $\deg_{LS}(I+\mu
L^{-1}J,B(r),0)$ changes sign as $\mu$ crosses $\mu^{\ast}$.

We now consider the global bifurcation question for the problem \eqref{BIF1},
\eqref{BC1b}, which is equivalent to \eqref{AbsBVP}. Let $\sigma$ denote the
characteristic values of $L^{-1}J$, so that $\mu\in\sigma$ if and only if
$\lambda=\mu+c$ is an eigenvalue of the problem \eqref{ls1}, \eqref{BC1}. Let
$\mathcal{S}_{0}$ denote the set of all non-trivial solutions $(\mu,w)$ of
\eqref{AbsBVP} and let $\mathcal{S}$ denote the closure of $\mathcal{S}_{0}$
in $\mathbb{R}\times X$. A point $(\mu^{\ast},0)$ is a bifurcation point from the line of
trivial solutions if every neighborhood of $(\mu^{\ast},0)$ contains a member
of $\mathcal{S}_{0}$.

\begin{theorem} \label{thm12}
Assume $p,q\in L^{\infty}(0,\pi)$ satisfy \eqref{eq2} with $A$ as in
\eqref{A}. Let
$g:\mathbb{R}\times[ 0,\pi]\times\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a
Carath\'{e}odory function as described above with $g(\lambda ,t,0)=0$ and
$g(\lambda,t,w)=o(w)$ as $| w| \to 0$
uniformly with respect to $\lambda,t$ in bounded sets. Then:
\begin{itemize}
\item[(c1)] Each $\mu^{\ast}\in\sigma$ is a bifurcation point of
\eqref{AbsBVP}, and
thus $\lambda^{\ast}=\mu^{\ast}+c$ is a bifurcation point of \eqref{BIF1},
\eqref{BC1b}.

\item[(c2)] For $\mu^{\ast}\in\sigma$ let $\mathcal{C}(\mu^{\ast})$ denote the
component of $\mathcal{S}$ which contains $(\mu^{\ast},0)$. Then
$\mathcal{C}(\mu^{\ast})$ is either unbounded in
$\mathbb{R}\times X$ or $\mathcal{C}(\mu^{\ast})$ meets another
point $(\widehat{\mu},0)$
with $\widehat{\mu}\in\sigma\backslash\{\mu^{\ast}\}$. Moreover
$\mathop{\rm rot}(w)$ is defined and constant for all
$(\mu,w)\in\mathcal{C}(\mu^{\ast})$ for $w\neq0$, and is the same as
the rotation number associated with the
eigenfunctions of \eqref{ls1}, \eqref{BC1} at $\lambda^{\ast}=\mu^{\ast}+c$.
Thus $\mathcal{C}(\mu^{\ast})$ can only meet another bifurcation point
$(\widehat{\mu},0)$ if $\widehat{\lambda}=\widehat{\mu}+c$ is associated with
same same rotation number as is $\lambda^{\ast}$.
\end{itemize}
\end{theorem}

\begin{proof}
The proof of this theorem is based upon the Rabinowitz global bifurcation
theorem and the properties of Leray-Schauder degree. The details are similar
to the proof in \cite[Theorem 3]{Wrot} and will not be repeated here.
\end{proof}

We now consider the global bifurcation question for the problem \eqref{BIF2},
\eqref{BC1a}. We assume \eqref{beta} holds. From this and Theorem \ref{thm8}
it follows that there is a number $c>0$ such that \eqref{ls2} has
no eigenvalues in the
half-open interval $(0,c]$. Let $X=C([0,\pi],\mathbb{R}^{2})$, $Y=L^{1}(0,\pi)$,
 and
\[
D=\{w\in X:w=(u,v)\text{ is absolutely continuous and }u(0)=u(\pi)=0\}.
\]
Define $L:D\to  Y$ by $Lw:=w'+cAw$. The operator $L$ has a
compact inverse. Assume $g$ satisfies the conditions of the preceding theorem
and let the nonlinear operator $G$ also be as defined earlier. The problem
\eqref{BIF2}, \eqref{BC1a} is equivalent to
\begin{equation}
w+\mu L^{-1}Aw=L^{-1}\widetilde{G}(\mu,w) \label{absbvp2}
\end{equation}
where $\mu=\lambda-c$ and $\widetilde{G}(\mu,w)=G(\mu+c,w)$. Let $\sigma$
denote the characteristic values of $L^{-1}A$, so that $\mu\in\sigma$ if and
only if $\lambda=\mu+c$ is an eigenvalue of the problem \eqref{ls2},
\eqref{BC1}. Let $\mathcal{S}_{0}$ denote the set of all non-trivial solutions
$(\mu,w)$ of \eqref{absbvp2} and let $\mathcal{S}$ denote the closure of
$\mathcal{S}_{0}$ in $\mathbb{R}\times X$. A point $(\mu^{\ast},0)$
is a bifurcation point from the line of trivial solutions if every
neighborhood of $(\mu^{\ast},0)$ contains a member of $\mathcal{S}_{0}$.

\begin{theorem} \label{thm11}
Assume $p,q\in L^{\infty}(0,\pi)$ satisfy \eqref{beta} with $A$ as in
\eqref{A}. Let
$g:\mathbb{R}\times[ 0,\pi]\times\mathbb{R}^{2}\to\mathbb{R}^{2}$
be a Carath\'{e}odory function as described above with
$g(\lambda,t,0)=0$ and $g(\lambda,t,w)=o(w)$ as $| w| \to 0$
uniformly with respect to $\lambda,t$ in bounded sets. Then:
\begin{itemize}
\item[(c1)] Each $\mu^{\ast}\in\sigma$ is a bifurcation point of
\eqref{absbvp2}, and
hence $\lambda^{\ast}=\mu^{\ast}+c$ is a bifurcation point for \eqref{BIF2},
\eqref{BC1a}.

\item[(c2)] For $\mu^{\ast}\in\sigma$ let $\mathcal{C}(\mu^{\ast})$ denote the
component of $\mathcal{S}$ which contains $(\mu^{\ast},0)$. Then
$\mathcal{C}(\mu^{\ast})$ is either unbounded in
$\mathbb{R}\times X$ or $\mathcal{C}(\mu^{\ast})$ meets another
 point $(\widehat{\mu},0)$
with $\widehat{\mu}\in\sigma\backslash\{\mu^{\ast}\}$. Moreover
$\mathop{\rm rot}(w)$ is defined and constant for all
$(\mu,w)\in\mathcal{C}(\mu^{\ast })$ for $w\neq0$, and is the same
as the rotation number associated with the
eigenfunctions of \eqref{ls2}, \eqref{BC1} at $\lambda^{\ast}=\mu^{\ast}+c$.
Thus $\mathcal{C}(\mu^{\ast})$ can only meet another bifurcation point
$(\widehat{\mu},0)$ if $\widehat{\lambda}=\widehat{\mu}+c$ is associated with
same same rotation number as is $\lambda^{\ast}$.
\end{itemize}
\end{theorem}

\begin{proof}
The proof is based upon the Rabinowitz global bifurcation theory, making use
of the properties established for \eqref{ls2} and properties of Leray-Schauder
degree. See  \cite[Theorem 3]{Wrot} for a related result and proof.
\end{proof}

\section{Bifurcation from infinity}

 We shall study bifurcation from infinity in systems of the form
\begin{equation}
\frac{dw}{dt}+\lambda A(t)w=g(\lambda,w,t),\quad t\in[ 0,\pi],
\label{eqinf}
\end{equation}
where $w=(u,v)^{T}$ satisfies the boundary conditions \eqref{BC1}:
\[
u(0)=0=u(\pi),
\]
We assume that $A(t)$ has the form \eqref{A} and satisfies \eqref{beta}, and
that $g:\mathbb{R}\times[ 0,\pi]\times\mathbb{R}^{2}\to\mathbb{R}^{2}$
is a Carath\'{e}odory function. That is, for each
$(\lambda,w)\in\mathbb{R}\times\mathbb{R}^{2}$ the map
$t\mapsto g(\lambda,t,w)$ is Lebesgue measurable, and for almost
all $t\in[ 0,\pi]$ the map $(\lambda,w)\mapsto g(\lambda,t,w)$ is
continuous. Moreover, for each $r\geq0$ there is $\alpha_{r}\in L^{1}(0,\pi)$
such that $| g(\lambda,t,w)| \leq\alpha_{r}(t)$ a.e. for
$| \lambda| +| w| \leq r$. In addition
we assume
\begin{equation}
\lim_{| w| \to \infty}\frac{|g(\lambda,t,w)| }{| w| }=0 \label{ginf}
\end{equation}
uniformly with respect to $\lambda$ and $t$ in bounded sets.

We will say that $(\lambda^{\ast},\infty)$ (or $\lambda^{\ast}$) is a
bifurcation point at infinity if there is a sequence $\{(\lambda_{n},w_{n})$
of solutions to \eqref{eqinf}, \eqref{BC1} with $\lambda_{n}\to
\lambda^{\ast}$ and $\| w_{n}\| \to \infty$ as
$n\to \infty$. We apply Leray-Schauder degree to prove the existence of
continua bifurcating from infinity.

Let $X=C([0,\pi],\mathbb{R}^{2})$, $Y=L^{1}([0,\pi],\mathbb{R}^{2})$ and
let $D$ be the set of all absolutely continuous $w=(u,v)^{T}\in X$
satisfying $u(0)=u(\pi)=0$. Let $0<c\leq1$ be such that \eqref{ls2} has no
eigenvalues in the interval $,c]$. $L:D\subset X\to  Y$ by
\[
Lw:=\frac{dw}{dt}+cA(t)w
\]
for $w\in D$. It is not difficult to verify that $L$ is a bijection from
$D(L)$ onto $Y$, and its inverse $L^{-1}:Y\to  X$ is compact. Let
$\widetilde{G}(\mu,w)(t):=g(\mu+c,t,w(t))$ for $\mu\in\mathbb{R},t\in[ 0,\pi]$,
and $w\in X$. Then $\widetilde{G}$ is a continuous
mapping from $X$ into $Y$, and takes bounded sets to bounded sets.
 The problem
\eqref{eqinf}, \eqref{BC1} is equivalent to the abstract equation
\begin{equation}
w+\mu L^{-1}Aw=L^{-1}\widetilde{G}(\mu,w) \label{infb}
\end{equation}
where $\mu=\lambda-c$.

Let $\mathcal{S}$ denote the set of solutions $(\mu,w)$ to \eqref{infb} and
adjoin to $\mathcal{S}$ the set of $(\mu^{\ast},\infty)$ such that $(\mu
^{\ast},\infty)$ is a bifurcation point at infinity. Denote this set by
$\mathcal{S^{\ast}}$. By the component $\mathcal{C}(\mu^{\ast})$ of
$\mathcal{S^{\ast}}$ containing $(\mu^{\ast},\infty)$, we mean the union of
all components of $\mathcal{S}$ which contain sequences $\{(\mu_{n},w_{n})\}$
with $\mu_{n}\to \mu^{\ast}$ and $||w_{n}||\to \infty$. We now
show that each $(\lambda^{\ast}+c,\infty)$, $\lambda^{\ast}\in\sigma$ (the
eigenvalues of \eqref{ls2}) is a bifurcation point at infinity for
\eqref{infb}. Thus each $(\lambda^{\ast},\infty)$ is a bifurcation point at
infinity for \eqref{eqinf}, \eqref{BC1}, with associated component
$\mathcal{C}_{\lambda^{\ast}}=\{(\lambda^{\ast},w):(\lambda^{\ast}
+c,w)\in\mathcal{C}(\lambda^{\ast}+c)\}$.

\begin{theorem} \label{thm13}
Let $g$ satisfy \eqref{ginf}. Then each $\mu^{\ast}\in\sigma+c$
 is a point of
bifurcation at infinity for \eqref{infb}, and thus each $\lambda^{\ast}
\in\sigma$ is a point of bifurcation at infinity for \eqref{eqinf},
\eqref{BC1}. Let $\mathcal{C(\mu}^{\ast})$ denote the component of
$\mathcal{S^{\ast}}$ containing $(k,\infty)$; then $\mathcal{C}(\mu^{\ast
})-\{(\mu^{\ast},\infty)\}$ is unbounded. Moreover at least one of the
following holds:
\begin{itemize}
\item[(C1)] The projection of $\mathcal{C}(\mu^{\ast})$ on $\mathbb{R}$ is unbounded, or

\item[(C2)] $\mathcal{C}(\mu^{\ast})$ meets another bifurcation point at infinity
$(\widehat{\mu},\infty)$, $\widehat{\mu}\neq\mu^{\ast}$.
\end{itemize}
\end{theorem}

\noindent\textit{Remark:} It may happen that there is a line of trivial solutions, say
$\mathcal{L}=\{(\lambda,0):\lambda\in\mathbb{R}\}$. If so, and $\mathcal{C}
(\mu^{\ast})$ meets $\mathcal{L}$ at some point $(\lambda^{\ast},0)$, then
$\mathcal{L}\subset\mathcal{C}(\mu^{\ast})$ and the projection of
$\mathcal{C}(\mu^{\ast})$ on $\mathbb{R}$ is unbounded.

\begin{proof}[Proof of Theorem \ref{thm13}]
Bifurcation from infinity in this case may be proven by converting the problem
to one of bifurcation from zero by means of the inversion
$y=w/\|w\| ^{2}$ as in \cite{Rabinfty}. For $y\in X$ let
\[
H(\mu,y):=\| y\| ^{2}\widetilde{G}(\mu,y/\|
y\| ^{2})\quad \text{for }y\neq0,\; H(\mu,0):=0.
\]
One can show that $H$ is continuous and takes bounded sets to bounded sets.
Additionally,
\[
\lim_{\| y\| \to 0}\frac{H(\mu,y)}{\|
y\| }=0
\]
with the limit uniform w.r.t $\mu$ in bounded sets. Letting $y=w/\|
w\| ^{2}$ in \eqref{infb} converts that equation to
\begin{equation}
y+\mu L^{-1}Ay=L^{-1}H(\mu,y). \label{invert}
\end{equation}
The Rabinowitz global theory for bifurcation from the line of trivial
solutions $\{(\mu,0):\mu\in\mathbb{R}\}$ applies to (\ref{invert}).
Meeting each point $(\mu^{\ast},0)$ with
$\mu^{\ast}$ a characteristic value of $L^{-1}A$ there is a continuum
$\widetilde{\mathcal{C}}(\mu^{\ast})$ of nontrivial solutions, and
$\widetilde{\mathcal{C}}(\mu^{\ast})$ is either unbounded in
$\mathbb{R}\times X$ or meets another bifurcation point
$(\widehat{\mu},0)$,
$\widehat{\mu}\neq\mu^{\ast}$. The inversion mapping $(\mu,y)\mapsto
(\mu,y/\| y\| ^{2})$ maps each $\widetilde{\mathcal{C}}
(\mu^{\ast})$ to a continuum $\mathcal{C}(\mu^{\ast})$ of solutions to
\eqref{infb} meeting $(\mu^{\ast},\infty)$. The points $(\mu,w)$ in
$\mathcal{C}(\mu^{\ast})$ produce solutions $(\mu-c,w)=(\lambda,w)$ to
\eqref{eqinf}, \eqref{BC1}; these solutions form a continuum $\mathcal{C}
_{1}(\lambda^{\ast})$, $\lambda^{\ast}=\mu^{\ast}-c$. By examination of the
solutions close to the bifurcation point $(\lambda^{\ast},\infty)$ one can
show that the rotation of such solutions is that associated with the
eigenfunctions of \eqref{ls2}, \eqref{BC1} at $\lambda=\lambda^{\ast}$. This
may not be continued on the entire continuum, since solutions may pass through
the origin. The properties of the continua and the global bifurcation theorem
implies the conclusions of the theorem.
\end{proof}

\begin{remark} \label{rmk14} \rm
One could prove a similar bifurcation from infinity result for equation
\eqref{BIF1} but that will be omitted. Besides the paper \cite{Rabinfty} of
Rabinowitz on bifurcation from infinity, the reader is referred to the paper
\cite{SchmittWang} of Schmitt and Wang.
\end{remark}

\section{Global behavior of continua}

In this section we give conditions which imply stronger conclusions regarding
the global behavior of the continua $\mathcal{C}$ bifurcating from
$(\lambda^{\ast},0)$ or $(\lambda^{\ast},\infty)$ found in the preceding
sections. We show that under some conditions on the signs of the
nonlinearities the $\mathcal{C}$ bend to the left or to the right of
$\lambda^{\ast}$. Consider again systems of the form
\begin{equation}
\frac{dw}{dt}+\lambda A(t)w=g(\lambda,t,w) \label{globbeh}
\end{equation}
with boundary conditions \eqref{BC1} as before, and $w=(u,v)^{T}$. We will
assume that $p=q$, i.e.,
\[
A(t)=\begin{pmatrix}
0 & p(t)\\
-p(t) & 0 \end{pmatrix}.
\]
Assume $g$ satisfies the Carath\'{e}odory conditions. We place some additional
structural conditions on $g$:
\[
g(\lambda,t,u,v)=\begin{pmatrix}
-vg_{1}(\lambda,t,u,v)\\
ug_{2}(\lambda,t,u,v) \end{pmatrix}.
\]
Thus our system takes the form
\begin{equation} \label{NS}
\begin{gathered}
\frac{du}{dt}  =-\lambda p(t)v-vg_{1}(\lambda,u,v,t)\\
\frac{dv}{dt}  =\lambda p(t)u+ug_{2}(\lambda,u,v,t).
\end{gathered}
\end{equation}

\begin{theorem} \label{thm15}
Assume that $p$ and $g=(-vg_{1},ug_{2})$ satisfy the conditions of
Theorem \ref{thm12}.
Also suppose that $p\geq0$, and $g_{1}(\mu,u,v,t),g_{2}(\mu,u,v,t)\leq0$
($\geq0$) for all
$(\mu,u,v,t)\in\mathbb{R}\times\mathbb{R}^{2}\times[ 0,\pi]$.
Let $\mathcal{C}\subset\mathbb{R}\times X$ denote the continuum
bifurcating from a point $(\lambda^{\ast},0)$.
 Then $\mathcal{C}\subset[ \lambda^{\ast},\infty)\times X$
($(-\infty,\lambda^{\ast}]\times X$).
\end{theorem}

\begin{proof}
Suppose $g_{1}(\mu,u,v,t),g_{2}(\mu,u,v,t)\leq0$ for all
$(\mu,u,v,t)\in\mathbb{R}\times\mathbb{R}^{2}\times[ 0,\pi]$.
Let $(\lambda,w)=(\lambda,(u,v))\in\mathcal{C}$,
with $(u,v)\neq(0,0)$, and let $P(t)=\lambda p(t)+g_{1}(\mu,u(t),v(t),t)$ and
$Q(t)=\lambda p(t)+g_{2}(\mu,u(t),v(t),t)$. Then $u,v$ satisfy
\begin{gather*}
\frac{du}{dt}   =-P(t)v\\
\frac{dv}{dt}   =Q(t)u.
\end{gather*}
Suppose $\lambda<\lambda^{\ast}$. Then $P(t)\leq\lambda p(t)\lesssim
\lambda^{\ast}p(t)$ and$,Q(t)\leq\lambda p(t)\lesssim\lambda^{\ast}p(t)$. Thus
if $\mathop{\rm rot}(w)$ is the is the rotation number of $w$,
by Lemma \ref{lem1} we must
have $\mathop{\rm rot}(w)<d_{1}$, where $d_{1}$ is the rotation
associated with solutions to
\begin{gather*}
\frac{du}{dt}    =-\lambda^{\ast}p(t)v\\
\frac{dv}{dt}    =\lambda^{\ast}p(t)u.
\end{gather*}
satisfying \eqref{BC1}. But by Theorem \ref{thm12} $d_{1}$ must be the the rotation
number of all solutions $w$ with $(\lambda,w)\in\mathcal{C}$. This is a
contradiction and implies that $\mu\geq\lambda^{\ast}$, and thus
$\mathcal{C}\subset[ \lambda^{\ast},\infty)\times X$.

If $g_{1}(\mu,u,v,t),g_{2}(\mu,u,v,t)\geq0$ for all
$(\mu,u,v,t)\in \mathbb{R}\times\mathbb{R}^{2}\times[ 0,\pi]$
 then very similar arguments show that
$\mathcal{C}\subset(-\infty,\lambda^{\ast}]\times X$.
\end{proof}

\begin{remark} \label{rmk16} \rm
If we assume that $p=q$, and $g=(g_{1},g_{2})^{T}$ satisfy the conditions for
bifurcation from infinity given in Theorem \ref{thm13} and $p$, $g_{1},g_{2}$ satisfy
the sign conditions of the preceding theorem, then we can show that the
continua bifurcating from infinity have the same containment properties.
\ Similar results hold for the systems \eqref{BIF1}.
\end{remark}

\section{Multiple solutions}

In \cite{Wrot} the author applied results on bifurcation from the line of
trivial solutions together with bending \ directions of the bifurcating
continua to prove multiplicity of solutions under appropriate conditions. In
this section we apply our results on bifurcation from infinity to obtain
related results. We prove the existence multiple structurally distinct
solutions to boundary value problems of the form
\begin{gather} \label{inf sol}
\begin{gathered}
\frac{du}{dt}   =-f(\lambda,t,u,v)v\\
\frac{dv}{dt}  =g(\lambda,t,u,v)u
\end{gathered} \\
u(0)=0=u(\pi) \label{BC}
\end{gather}
where $w=(u,v)$. To obtain the existence of infinitely many solutions we apply
our earlier bifurcation from infinity results to the family of problems
\begin{equation} \label{inf bif}
\begin{aligned}
\frac{du}{dt}   =-\lambda p(t)-g_{1}(\lambda,t,u,v)v\\
\frac{dv}{dt}   =\lambda p(t)+g_{2}(\lambda,t,u,v)u
\end{aligned}
\end{equation}
for $\lambda\in\mathbb{R}$ with boundary conditions
\eqref{BC}. Let $p\in L^{\infty}(0,\pi)$,$p\gtrsim0$ on $[0,\pi]$ Let
\[
A(t):=\begin{pmatrix}
0 & p(t)\\
-p(t) & 0 \end{pmatrix}.
\]
Suppose $g:=(g_{1},g_{2})^{T}$ satisfy the Carath\'{e}odory conditions and
\begin{equation}
\lim_{| w| \to \infty}\frac{|
g(\lambda,t,w)| }{| w| }=0)\quad  \label{lim}
\end{equation}
uniformly with respect to $\lambda$, $t$ in compact sets. Recall that we call
$\widetilde{w}$ the odd/even extension of $w$ provided
$w=(u,v)\in C([0,\pi],\mathbb{R}^{2})$, $u(0)=0=v(0)$ and
 $\widetilde{w}=(\widetilde{u},\widetilde{v})$ where
$\widetilde{u}$, $\widetilde{v}$ are, respectively, the odd extension of $u$
and the even extension of $v$ to $[-\pi,\pi]$. \

\begin{theorem} \label{thm17}
In addition to the above, suppose solutions to initial value problems are
unique, $g(\lambda,t,0)=0$, $g_{i}(\lambda,t,w)\leq0$, for $\lambda\geq0$ and
$g_{i}(\lambda,t,w)\geq0$, for $\lambda\leq0$ $(i=1,2)$, and
\[
\lim_{| w| \to 0}\frac{| \lambda
A(t)w-g(\lambda,t,w)| }{| w| }=0
\]
uniformly with respect to $(\lambda,t)$ in bounded sets.  Let
$\lambda_{n},n\in\mathbb{Z}$ denote the $n$th eigenvalue of \eqref{ls2},
\eqref{BC}, $N$ be a positive
integer. Then \eqref{inf sol}, \eqref{BC} with $\lambda=\lambda^{\ast}$ has at
least $N$ topologically distinct solutions if $\lambda^{\ast}>\lambda_{N}$ or
$\lambda^{\ast}<\lambda_{-N}$ . Indeed, in this case there is for each
$k\in\mathbb{N}$ there is a solution $w_{k}=(u_{k},v_{k})$ such that
the odd/even $2\pi$
periodic extension $\widetilde{w}_{k}$ of $w_{k}$ has rotation number $k$.
\end{theorem}

\begin{proof}
The eigenvalues of the problem linearized at infinity are the
$\lambda_{n}$, $n\in\mathbb{Z}$. By Theorem \ref{thm13} problem
(\ref{inf bif}), \eqref{BC} has at each
$\lambda_{n}$ a continuum $\mathcal{C}_{n}$ of solutions bifurcating
 from $(\lambda_{n},\infty)$, and $\mathcal{C}_{n}$ is either unbounded
in $\mathbb{R}\times X$, or else meets another, distinct, bifurcation
point $(k,\infty)$ ($k\neq n$). Now solutions to initial value problems
are unique and $g(\lambda,t,0)=0$. Therefore the rotation number of solutions near the
bifurcation point must be continued to all nontrivial solutions in
$\mathcal{C}_{n}$ Note that in this case, since $p=q$, there is exactly one
eigenvalue associated with rotation number $n\in\mathbb{Z}$, and it is
$\lambda_{n}$. Thus $\mathcal{C}_{n}$ cannot meet any other
bifurcation point at infinity. The sign conditions on the $g_{i}$ imply that
for $n\geq1$, $\mathcal{C}_{n}\subset[ \lambda_{n},\infty)$ and for
$n\leq-1$, $\mathcal{C}_{n}\subset(-\infty,\lambda_{n}]$ (this follows from
noticing the sign conditions of Theorem \ref{thm15} can be used discriminately, based
in this case on the sign of $\lambda$). Now the only bifurcation point from
the line of trivial solutions is $(\lambda,0)=(0,0)$. It follows that no
$\mathcal{C}_{n}$ with $n\neq0$ can meet the line of trivial solutions. On the
other hand the projection of these $\mathcal{C}_{n}$ (with $n\neq0$) on
$\mathbb{R}$ must be unbounded. Thus for $n\geq1$, $\mathcal{C}_{n}$
contains a nontrivial solution $(\mu,w)$ for each $\mu>\lambda_{n}$,
and for $n\leq-1$,
$\mathcal{C}_{n}$ contains a nontrivial solution $(\mu,w)$ for each
$\mu<\lambda_{n}$. These nontrivial solutions have rotation number $n$. It
follows now that problem (\ref{inf bif}), \eqref{BC} with $\lambda
=\lambda^{\ast}$ has at least $N$ solutions if $\lambda^{\ast}>\lambda_{N}$ or
$\lambda^{\ast}<\lambda_{-N}$.
\end{proof}

\begin{example} \label{exa18} \rm
The following system, with $p=q=1$,
\begin{gather*}
\frac{du}{dt}    =-\lambda v+\frac{\lambda v}{1+u^{2}+v^{2}}+\frac
{\lambda\sin^{2}(t)u^{2}v}{1+u^{4}+v^{4}}\\
\frac{dv}{dt}    =\lambda u-\frac{\lambda u}{1+u^{2}+v^{2}}-\frac{\lambda
\cos^{2}(t)v^{4}u}{2+u^{6}+v^{6}}
\end{gather*}
satisfies the conditions of the preceding theorem.
\end{example}

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\end{document}