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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 149, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/149\hfil Two classical periodic problems]
{Two classical periodic problems on time scales}

\author[P. Amster, C. C. Tisdell\hfil EJDE-2007/149\hfilneg]
{Pablo Amster, Christopher C. Tisdell}  % in alphabetical order

\address{Pablo Amster \newline
Departamento de Matem\'atica \\
Facultad de Ciencias Exactas y Naturales\\
Universidad de Buenos Aires\\
Ciudad Universitaria, Pabell\'on I,
(1428) Buenos Aires, Argentina. \newline
Consejo Nacional de Investigaciones
Cient\'\i ficas y T\'ecnicas (CONICET), Argentina}
\email{pamster@dm.uba.ar}

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

\thanks{Submitted April 3, 2007. Published November 9, 2007.}
\thanks{Supported by grants: PIP 5477 from CONICET, Argentina,
and DP0450752 from the \hfill\break\indent
Australian Research Council's Discovery Projects}

\subjclass[2000]{39A12, 39A99}
\keywords{Time scale; boundary value problem; forced pendulum equation;
\hfill\break\indent  Landesman-Lazer conditions; existence of solutions}

\begin{abstract}
 We consider the generalization of two classical
 periodic problems to the context of time scales.
 On the one hand, we generalize a celebrated result by Castro
 for the forced pendulum equation. On the other hand, we
 extend a well-known result by Nirenberg to a resonant
 system of equations on time scales.  Furthermore, the results are
 new even for  classical difference equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{proposition}[theorem]{Proposition}
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{definition}[theorem]{Definition}
 \newtheorem{remark}[theorem]{Remark}
 \newtheorem{example}[theorem]{Example}

\section{Introduction}

In recent years there has been an increasing interest in dynamic
equations on time scales. The concept of time scales (also known
as {\sl measure chains}) was introduced in 1990 by Hilger
\cite{H90} with  the motivation of providing a unified approach to
continuous and discrete calculus. Thus, the notion of a
generalized derivative $y^{\Delta}(t)$ was introduced, where the
domain of the function $y(t)$ is an arbitrary closed non-empty
subset of $\mathbb{T} \subset \mathbb{R}$. If $\mathbb{T} =\mathbb{R}$ then the usual derivative
is retrieved, that is $y^{\Delta}(t) = y'(t)$. On the other hand,
if the time scale is taken to be $\mathbb{Z}$ then the generalized
derivative reduces to the usual forward difference, that is
$y^{\Delta}(t) = \Delta y(t)$.

The field of dynamic equations on time scales allows us to  model
hybrid processes where time may flow continuously in one part of
the process (with the model leading to a differential equation)
and then time may flow discretely in another part of the process
(leading to a difference equation).  Moreover, these types of
stop-start hybrid processes occur naturally and for more on the
current and future applications of dynamic equations on time
scales the reader is referred to the cover story of New Scientist
\cite{ns} or the monographs by Bohner and Peterson \cite{BP01} and
Bohner et al \cite{BP03}.

The field of dynamic equations on time scales is not only  about
unification. It is important to emphasize that by researching
dynamic equations on time scales, new advances can be made into
each of the theories of differential and difference equations in
their own right.  For example, once a result is proved in the
general time scale setting, special cases of the new results may
give new theorems for each of the theories of differential and
difference equations.

In this work, we consider a generalization of two classical
resonant periodic problems to the context of time scales. On the
one hand, we study the forced pendulum equation
\begin{equation}
\label{pend}
y^{\Delta\Delta} +  a\sin (y^\sigma) = p(t),\quad t \in [0,T]_{\mathbb T}
\end{equation}
where $a$ is a positive constant.

For the continuous case $\mathbb{T} = \mathbb{R}$, Castro proved in
\cite[Theorem A]{C}  that if $a\le (\frac{2\pi}T)^2$ and
$p_0 = p - c$ with $c= \overline {p}:=\frac 1T\int_0^T p(t)dt$,
then there exist two real numbers $ d(p_0)$ and $D(p_0)$ with
$$
-a\le d(p_0) \le 0\le D(p_0)\le a
$$
such that equation (\ref{pend}) admits $T$-periodic solutions if and only if
$$
d(p_0) \le c\le D(p_0).
$$
A more general result has been obtained by Mawhin and Willem in \cite{mw}, and
by Fournier and Mawhin in \cite{FM}, using topological methods.

Also, we investigate
the existence of periodic solutions
$y:[0,\sigma^2(T)]_{\mathbb T}\to \mathbb{R}^N$
to the following nonlinear system of second
order differential equations on time scales
\begin{equation} \label{sys}
y^{\Delta\Delta} = f(t,y^\sigma), \quad  t \in [0,T]_{\mathbb T};
\end{equation}
under Landesman-Lazer type conditions.
We shall assume that the nonlinearity
$f:[0,T]_{\mathbb{T}}\times \mathbb{R}^{N} \to
\mathbb{R}^N$ is bounded and continuous although, unlike the pendulum
equation, $f(t,z)$ will be typically a non-periodic function of $z$.

By investigating the general equation (\ref{sys}), special cases
of our results give novel results for (classical) difference
equations and also for non-classical difference equations, such as
$q$-difference equations (used in physics).  Thus this article not
only makes a new  contribution to time scales, it also provides
new results for difference equations.


There exists a vast literature on Landesman-Lazer type conditions for
resonant problems, starting at the pioneering work \cite{ll} for a
second order elliptic (scalar) differential
equation under Dirichlet conditions. For a
survey on Landesman-Lazer conditions see e.g. \cite{ma2}. In
\cite{n}, Nirenberg extended the Landesman-Lazer conditions
to a system of elliptic
equations. Nirenberg's result can be adapted for a system of
periodic ODE's in the following way:

\begin{theorem}\label{nir}
Let $p\in C([0,T],\mathbb{R}^N)$ and let $g:\mathbb{R}^N\to \mathbb{R}^N$ be continuous
and bounded. Further, assume that the radial limits $g_v:=
\lim_{r\to +\infty} g(rv)$ exist uniformly respect to $v\in
S^{N-1}$, the unit sphere of $\mathbb{R}^N$. Then the problem
$$y'' + g(y) = p(t)$$
 has at least one $T$-periodic solution if
the following conditions hold:

\begin{itemize}
\item
$g_v\neq \overline p:=\frac 1T \int_0^T p(t)dt$ for any $v\in
S^{N-1}$.

\item
The degree of the mapping $\theta:S^{N-1}\to S^{N-1}$ given by
$$\theta (v)= \frac{g_v-\overline p}{|g_v-\overline p|}$$
is non-zero.
\end{itemize}
\end{theorem}

For completeness, let us introduce the essential terminology of time scales.

\begin{definition}\label{def:1.1} \rm
A time scale $\mathbb{T}$ is a non-empty, closed subset of $\mathbb{R}$,
equipped with the topology induced from the standard topology on $\mathbb{R}$.
\end{definition}

\begin{definition}\label{def:1.2} \rm
The forward (backward) jump operator
$\sigma(t)$ at $t$ for $t< \sup \mathbb{T}$
(respectively $\rho(t)$ at $t$ for $t> \inf \mathbb{T}$) is given by
\[
\sigma (t) = \inf \{ \tau > t: \tau \in \mathbb{T}\}, \quad
(\rho(t) = \sup \{ \tau < t: \tau \in \mathbb{T}\}, ) \ \mbox{for all $t \in \mathbb{T}.$}
\]
Additionally $\sigma (\sup \mathbb{T}) = \sup \mathbb{T}$, if $\sup \mathbb{T} < \infty$, and
$\rho(\inf \mathbb{T}) = \inf \mathbb{T}$, if $\inf \mathbb{T} > - \infty$.
Furthermore, denote $\sigma^2(t) = \sigma (\sigma (t))$ and
$y^\sigma (t) = y(\sigma (t))$.

\end{definition}


\begin{definition} \rm
If $\sigma(t)>t$ then the point $t$ is called right-scattered; while
if $\rho (t) < t $ then $ t$ is termed left-scattered.  If $t< \sup \mathbb{T}$ and $\sigma (t) = t$
then the point $t$ is called right-dense; while if
$t> \inf \mathbb{T}$ and $\rho(t) =t$ then we say $t$ is left-dense.
\end{definition}

If $\mathbb{T}$ has a left-scattered maximum at $m$ then we define
$\mathbb{T}^k = \mathbb{T} - \{m\}$.  Otherwise $\mathbb{T}^k = \mathbb{T}$.

\begin{definition}\label{def:1.3} \rm
Fix $t\in \mathbb{T}^k $ and let $y: \mathbb{T} \to \mathbb{R}^n$.  Then
$y^\Delta (t)$ is the vector (if it exists) with the property that given
$\epsilon > 0$ there is a neighborhood $U $ of $t$ such that,
for all $s\in U$ and each $i =1,\dots,n$
\[ |[y_i (\sigma(t)) -y_i(s)] -y_i^\Delta (t)[\sigma(t)-s]| \le \epsilon | \sigma (t) -s|. \]
Here $y^\Delta (t)$ is termed the (delta) derivative of $y(t)$ at $t$.
\end{definition}

\begin{theorem}[\cite{H90}] \label{thm:1.1}
Assume that $y: \mathbb{T} \to \mathbb{R}^n$ and let $t\in \mathbb{T}^k$.
\begin{itemize}
\item[(i)] If
$y$ is differentiable at $t$ then $y$ is continuous at $t$.
\item[(ii)]If $y$ is continuous at $t$ and $t$ is right-scattered then $y$ is
differentiable at $t$ and
\[
y^\Delta (t) = \frac{y(\sigma(t))-y(t)}{\sigma(t)-t}.
 \]
\item[(iii)] If $y$ is differentiable and $t$ is right-dense then
\[
y^\Delta (t) = \lim\limits_{s\to t} \frac{y(t)-y(s)}{t-s}.
 \]
\item[(iv)] If $y$ is differentiable at $t$ then
$y(\sigma(t))=y(t)+\mu(t)y^\Delta (t)$.
\end{itemize}
\end{theorem}

\begin{definition}\label{def:1.5} \rm
The function $y$ is said to be right-dense continuous, that
is $y\in C_{rd}(\mathbb{T};\mathbb{R}^n)$ if:
\begin{itemize}
\item[(a)] $y$ is continuous at every right-dense point $t\in \mathbb{T}$, and
\item[(b)] $\lim_{s\to t^-} y(s)$
exists and is finite at every left-dense point $t\in \mathbb{T}$.
\end{itemize}
\end{definition}

\begin{proposition}\label{1.4}
For any right-dense continuous function $y$ there exists an
antiderivative; i.e., a differentiable function $Y$ such that
$Y^\Delta (t) =y(t)$. Moreover, $Y$ is
unique up to a constant term, and the time
scale integral of $y$ is thus defined by
\[
\int_a^t y(s)\Delta s = Y(t) - Y(a).
\]
\end{proposition}

We shall use the standard notation for
the different intervals in $\mathbb{T}$.
For example, if $a,b\in \mathbb{R}$ with $a< b$, then
the closed interval of numbers between
$a$ and $b$ will be denoted by
$[a,b]_{\mathbb{T}} := \{ t\in \mathbb{T} : a\le t\le b\}$.

In this context, the
periodic boundary conditions for problems (\ref{pend}) and
(\ref{sys}) read:
\begin{equation} \label{per}
y(0) = y(\sigma^2(T)), \quad y^\Delta(0) = y^\Delta(\sigma(T)).
\end{equation}


The paper is organized as follows.
In Section \ref{prelim} we introduce some preliminary
results concerning the Lebesgue integral on time scales,
and the associated linear problem for (\ref{pend}) and (\ref{sys}).

In the third section, we study the periodic problem for
equation (\ref{pend}).
Following the ideas in \cite{FM}, we generalize Castro's result
for an equation on time scales.

Finally, in Section \ref{llaz} we study an extension of the standard
Landesman-Lazer conditions for system (\ref{sys}). We shall
obtain a general result that extends Theorem \ref{nir}
for a system of differential equations in
time scales.


\section{Preliminary results} \label{prelim}

Let us define a measure in the following way.
For $a<b\in \mathbb{T}$, consider
$\mathcal{A}\subset
\mathcal{P}([a,b)_{\mathbb{T}})$ 
the completion of the 
Borel $\sigma$-algebra
generated by the family
$$
\{ [x,y)_{\mathbb{T}}: a\le x<y\le b, x,y\in \mathbb{T}\}.
$$
Hence, there is a unique $\sigma$-additive
measure $\mu:\mathcal{A}\to \mathbb{R}^+$
defined over this basis as:
$\mu([x,y)_{\mathbb{T}}) = y-x$.

In the following lemma we establish the equivalence between
the Lebesgue integral with respect to $\mu$ and the Cauchy
integral on time scales when the integrand is right-dense
continuous. A more
precise result is given in \cite{G}; in particular, it is proved that
any right-dense continuous function is Lebesgue integrable.

\begin{lemma}\label{leb}
If $\varphi\in C_{rd}([a,b]_\mathbb{T})$, then
$$
\int_a^b \varphi(t)\Delta t = \int_{[a,b)}\varphi d\mu :=
\int_a^b \varphi\,d\mu.
$$
\end{lemma}

\begin{proof}
For $t\in [a,b)_\mathbb{T}$, define
$\phi(t) = \int_{[a,t)}\varphi \,d\mu$.
If $t$ is right-dense, take $s\in [a,b)_\mathbb{T}-\{t\}$.
Assume for example that $s>t$, then:
$$
\frac{\phi(s)-\phi(t)}{s-t} - \varphi(t) =
\frac 1{s-t} \int_{[t,s)} \varphi-\varphi(t)\,d\mu.
$$
A similar equality holds for $s<t$,
and by continuity of $\varphi$ it is immediate
to prove that
$\frac{\phi(s)-\phi(t)}{s-t} - \varphi(t) \to 0$ as $s\to t$.
Hence $\phi$ is $\Delta$-differentiable at $t$ and
$\phi^\Delta(t) = \varphi(t)$.

If $t$ is right-scattered,
it is clear that $\phi$ is continuous at $t$, and
$$
\phi(\sigma(t)) - \phi(t)
= \int_{\{t\}}\varphi \,d\mu = \varphi(t)(\sigma(t)-t).
$$
It follows that $\phi^\Delta(t)=\varphi(t)$.
We conclude that $\phi$ is an antiderivative of $\varphi$, and
the result holds.
\end{proof}

\begin{remark} \label{rmk22} \rm
It follows from the previous lemma that all the
theorems for the Lebesgue integral theory
such as dominated convergence or Fatou Lemma
hold.
\end{remark}

\begin{lemma} \label{Green}
Let $\varphi\in C_{rd}([0,T]_\mathbb{T})$ and $s\in \mathbb{R}$.
Then there exists a unique solution of the problem
\begin{equation}
\begin{gathered}
y^{\Delta\Delta}(t) = \varphi(t) \quad\mbox{in } [0,T]_\mathbb{T}\\
y(0) = y(\sigma^2(T)) = s.
\end{gathered}
\end{equation}
Furthermore,
$$
y(t) = s + \int_0^{\sigma(T)} G(t,s)\varphi(s)\Delta s,
$$
where the Green function $G$ is given by
\begin{equation}
\label{green}
G(t,s) =
\begin{cases}
\frac {-t}{\sigma^2(T)}(\sigma^2(T)-\sigma(s)) &
\hbox{if }  t\le s \\
\frac {-\sigma(s)}{\sigma^2(T)}(\sigma^2(T)- t)
& \hbox{if }  t\ge \sigma(s).
\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
By integration, it follows that
$$
y(t) = s+\int_0^t \phi(r)\Delta r -
\frac t{\sigma^2(T)}\int_0^{\sigma^2(T)}\phi(r)\Delta r,
$$
where
$\phi(r) = \int_0^r \varphi(s)\Delta s$.
From the previous lemma, we have that
$$
\int_0^r \varphi(s)\Delta s =
\int_{[0,\sigma(T))} \varphi(s).\chi_{[0,r)}(s)d\mu =
\int_{[0,\sigma(T))} \varphi(s).
\chi_{(s,\sigma^2(T))}(r)d\mu.
$$
If $s$ is right-scattered then $\chi_{(s,\sigma^2(T))}=
\chi_{[\sigma(s),\sigma^2(T))}$. On the other hand, if
$s$ is right-dense, then $\mu(\{ s\})=0$ and we conclude that
$$
\phi(r) = \int_{[0,\sigma(T))} \varphi(s).
\chi_{[\sigma(s),\sigma^2(T))}(r)d\mu.
$$
Hence,
$$
y(t) = s+ \int_0^{\sigma(T)} \varphi(s)
\Big(
\int_0^{\sigma^2(T)}
\chi_{[\sigma(s),\sigma^2(T))}(r)
\big[\chi_{[0,t)}(r)- \frac{t}{\sigma^2(T)}\big]
\Delta r
\Big) \Delta s,
$$
and the result holds.
\end{proof}



\section{The forced pendulum equation} \label{forcedpendulum}

In this section we extend Castro's result to the context of time
scales. More precisely:

\begin{theorem}\label{interv}
Assume that $p_0$ is $rd$-continuous, and that
$\overline {p_0} = 0$, where
$$
\overline {p_0} := \frac 1{\sigma(T)}\int_0^{\sigma(T)} p_0(t)\Delta t.
$$
Then there exist two real numbers $ d(p_0)$ and $D(p_0)$ with
$$
-a\le d(p_0) \le D(p_0)\le a
$$
such that problem (\ref{pend}-\ref{per}) for $p=p_0+c$
admits at least one solution if and only if
$d(p_0) \le c\le D(p_0)$.
\end{theorem}

\begin{remark} \label{rm3.2}\rm
It may be noticed that no condition on $a$ is assumed.
Thus, Theorem \ref{interv} is indeed
a generalization of a Mawhin-Willem result (see \cite{mw}).
\end{remark}

\begin{remark} \rm
In the continuous case $\mathbb{T} = \mathbb{R}$ a standard variational argument
shows that $d(p_0) \le 0\le D(p_0)$. The matter of extending this
result to a general time scale was considered in \cite{AmTi}. 
\end{remark}



\begin{proof}[Proof of Theorem \ref{interv}]
Let us introduce the
function $P_0 (t) = \int_0^{\sigma(T)} G(t,s)p_0(s)\Delta s$,
where $G$ is given by (\ref{green}), and consider
the following equivalent problem for $u= y-P_0$:
\begin{equation}\label{pend2}
u^{\Delta\Delta} +  a\sin (u^\sigma + P_0^\sigma) = c,
\end{equation}
under periodic conditions.
Define $c:C_{rd}([0,T]_\mathbb{T})\to\mathbb{R}$ by
$$
c(u) = \frac a{\sigma(T)} \int_0^{\sigma(T)}
\sin(u^\sigma+P_0^\sigma)\Delta t
$$
and consider the following integro-differential equation on time scales:
\begin{equation}\label{intdif}
u^{\Delta\Delta} +  a\sin (u^\sigma+
P_0^\sigma) = c(u)
\end{equation}

\noindent{\bf Claim}: For each $r\in\mathbb{R}$ problem (\ref{intdif}-\ref{per})
admits at least one solution $u$ such that $u(0) = r$.

Indeed, for $v\in C_{rd}([0,T]_\mathbb{T})$ let us define $u:=T^rv$ as the
unique solution of the linear problem
\begin{gather*}
u^{\Delta\Delta} + a \sin (v^\sigma+P_0^\sigma)) = c(v)\\
u(0) = u(\sigma^2(T)) = r.
\end{gather*}
 From Lemma \ref{Green}, we have that
$$
T^r(v)(t) = r + \int_0^{\sigma(T)} G(t,s)
(c(v)-a\sin (v^\sigma+P_0^\sigma))\Delta s,
$$
and it follows from Arzel\'a-Ascoli Theorem that
$T^r:C_{rd}([0,T]_\mathbb{T})\to C_{rd}([0,T]_\mathbb{T})$
is compact. Furthermore,
$\|T^r(v)\|_{C_{rd}([0,T]_\mathbb{T})} \le C$ for some constant $C$, and
by Schauder Theorem $T^r$ has a fixed point $u$.
Integrating the equation, it follows that
$u^\Delta(0) = u^\Delta(\sigma(T))$,
and then $u$ is a solution of (\ref{intdif}-\ref{per}).

Next, define the set
$$
E = \{ u: u \hbox{ solves (\ref{pend2}-\ref{per}) for some $c$}
\}.
$$
It is clear that $u \in E$ if and only if $u$ is a
solution of (\ref{intdif}-\ref{per}), with $c=c(u)$.
Thus, $E$ is nonempty, and
it suffices to prove that $I(p_0):= c(E)$ is a compact interval.

 From the periodicity of the equation it is immediate that
$c(E) = c(E_{2\pi})$, where
$$
E_{2\pi} = \{ u\in E: u(0)\in [0,2\pi]\}.
$$

Let $\{u_n\}$ be a sequence in $E_{2\pi}$.
Using the above Green representation,
it follows that
$\| u_n - u_n(0)\|_{C_{rd}([0,T]_\mathbb{T})}  \le C$
for some constant $C$ independent of $n$.
Moreover, as $u_n-u_n(0) = T^0(u_n)$ and $u_n(0)\in [0,2\pi]$,
there exists a
subsequence that converges to a function
$u$ for the $C_{rd}$-norm. By Lemma \ref{leb} and dominated
convergence we obtain that
$u = u(0) + T^0(u)$, and hence
$u\in E_{2\pi}$. By continuity of the function $c$,
compactness of $I(p_0)$ follows.

In order to see that $I(p_0)$ is connected, assume
that $c_1,c_2\in I(p_0)$,
$c_1< c_2$, and let $c\in (c_1,c_2)$. Choose $u_i\in E$ such that
$$
u_i^{\Delta\Delta} + a\sin(u_i^\sigma+P_0^\sigma) = c_i.
$$
As $u_1$ and $u_2$ are bounded, adding a multiple of $2\pi$ if necessary, we may
assume that
$u_1\ge u_2$. Hence
$$
u_1^{\Delta\Delta} + a\sin(u_1^\sigma+P_0^\sigma) \le c \le
u_2^{\Delta\Delta} + a\sin(u_2^\sigma+P_0^\sigma).
$$
It follows that $(u_2,u_1)$ is an ordered couple of a lower and an
upper solution of the problem $u^{\Delta\Delta} + a\sin(u^\sigma+P_0^\sigma) =
c$, and the proof follows from Theorem 5 in \cite{stehlik}.
\end{proof}


The following proposition gives some bounds for the numbers
$d(p_0)$ and $D(p_0)$.

\begin{proposition} \label{prop1}
Let $K=\sup_{t\in [0,\sigma(T)]_\mathbb{T}}
\int_0^{\sigma(T)}|G(t,s)|\;\Delta s$, and
$$R(p_0) = \Big[
\Big(\int_0^{\sigma(T)} \cos (P_0^\sigma)\;\Delta t\Big)^2 +
\Big(\int_0^{\sigma(T)} \sin (P_0^\sigma)\;\Delta t\Big)^2
\Big]^{1/2}.
$$
Then
$$
d(p_0) \le -a\big( \frac{R(p_0)}{\sigma(T)} - 2aK\big)
$$
and
$$
D(p_0) \ge a\big(\frac{R(p_0)}{\sigma(T)} - 2aK \big).
$$
In particular, if $a< \frac{R(p_0)}{2K\sigma(T)}$,
then $d(p_0)< 0 < D(p_0)$.
\end{proposition}

\begin{proof}
Let $v$ be a solution of
(\ref{intdif}-\ref{per}).
A simple computation shows that
$$
|c(v)- c(v(0))| \le \frac a{\sigma(T)}
\int_0^{\sigma(T)}|v^\sigma - v(0)|\;\Delta t
\le a \|v-v(0)\|_{C_{rd}([0,\sigma(T)])},
$$
and
$$
|v-v(0)| \le
\int_0^{\sigma(T)} |G(t,s)|.|c(v)-a\sin (v^\sigma+P_0^\sigma)|\;\Delta t
\le 2aK.
$$
Thus, $|c(v)-c(v(0))|\le 2a^2K$, and it follows that
$$
d(p_0)=\inf_{v \in E} c(v)\le \inf_{v \in E} c(v(0)) +
2a^2K =
\inf_{x \in [0,2\pi]} c(x) + 2a^2K.
$$
In the same way,
$$
D(p_0)\ge \sup_{x \in [0,2\pi]} c(x) - 2a^2K.
$$
For $x\in\mathbb{R}$,
\begin{align*}
c(x) &= \frac a{\sigma(T)} \int_0^{\sigma(T)}\sin(x+P_0^\sigma)\;\Delta t \\
&= \frac a{\sigma(T)} \Big(
\sin x \int_0^{\sigma(T)}\cos(P_0^\sigma)\;\Delta t
+\cos x \int_0^{\sigma(T)}\sin(P_0^\sigma)\;\Delta t
\Big).
\end{align*}
Thus,
$$\sup_{x \in [0,2\pi]} c(x) = -
\inf_{x \in [0,2\pi]} c(x) = \frac a{\sigma(T)}R(p_0),$$
and the proof is complete.
\end{proof}

\begin{remark} \label{rmk3.4} \rm
The smallness condition on $a$ in the previous proposition
may be improved by observing that:
$$
|v - v(0)| \le K_2 \Big( \int_0^{\sigma(T)} [c(v)-a\sin(v^\sigma
+ P_0^\sigma)]^2\;\Delta t \Big)^{1/2},
$$
where
$$
K_2 = \sup_{t\in [0,\sigma(T)]_\mathbb{T}}
\Big(\int_0^{\sigma(T)}G(t,s)^2\;\Delta s\Big)^{1/2}.
$$
As
\begin{align*}
&\int_0^{\sigma(T)} [c(v)-a\sin(v^\sigma
 + P_0^\sigma)]^2\;\Delta t\\
& = -a \int_0^{\sigma(T)} \sin(v^\sigma+P_0^\sigma) [c(v)-a\sin(v^\sigma
 + P_0^\sigma)]\;\Delta t \\
& \le a\sigma(T)^{1/2}
\Big( \int_0^{\sigma(T)} [c(v)-a\sin(v^\sigma
+ P_0^\sigma)]^2\;\Delta t \Big)^{1/2},
\end{align*}
it follows that $|v - v(0)| \le K_2 a\sigma(T)^{1/2}$.
Hence,
\begin{gather*}
d(p_0)\le -a \Big( \frac{R(p_0)}{\sigma(T)} - aK_2\sigma(T)^{1/2}\Big),\\
D(p_0)\ge a \Big( \frac{R(p_0)}{\sigma(T)} - aK_2\sigma(T)^{1/2}\Big),
\end{gather*}
and the smallness condition for $a$ reads:
$$
a < \frac {R(p_0)}{K_2\sigma(T)^{3/2}}.
$$
For example, if
$\mathbb{T} = \mathbb{R}$ then $K = T^2/8$, and $K_2 = \frac{T^{3/2}}{4\sqrt{3}}$, and
thus $2K\sigma(T)^{1/2} > K_2\sigma(T)^{3/2}$.
\end{remark}



\begin{remark} \label{rmk3.5} \rm
It follows from the proof of Theorem \ref{interv} that $E$
is infinite. However, the interval $I({p_0}) = [d(p_0),D(p_0)]$
might reduce to a single point $c_0$;
in this case the equation is called singular,
and problem (\ref{pend}-\ref{per}) with $p= p_0+c_0$ admits
infinitely many solutions.

The problem of finding $p_0$
for which (\ref{pend}-\ref{per}) is singular,
or proving that such a $p_0$ does not exist, is still
open.
For the standard case $\mathbb{T} = \mathbb{R}$,
Ortega and Tarallo have proved in
\cite{OT} that the following statements are equivalent:
\begin{itemize}

\item[(i)] $I({p_0}) = \{0\}$.


\item[(ii)] For any $r\in\mathbb{R}$ there exists a unique
$T$-periodic solution $u_r$
of (\ref{pend}-\ref{per}) for $p=p_0$
such that $u_r(0)= r$.


\item[(iii)] There exists a continuous path $r\to u_r$ which
satisfies
$$\lim_{r\to\pm\infty} u_r(t) = \pm \infty$$
uniformly in $t$.
\end{itemize}
\end{remark}

When $a$ is small, the following proposition
gives a necessary condition for singularity.


\begin{proposition} \label{prop2}
Let $a< \frac 1K$, where $K$ is defined as before,
and assume that $I(p_0) = \{ c_0\}$. Then every solution of the
problem
\begin{gather*}
u^{\Delta\Delta}+a\sin(u^\sigma) = p_0 + c_0 \\
u(0) = u(\sigma^2(T))
\end{gather*}
also satisfies: $u^\Delta(0) = u^\Delta(\sigma(T))$.
\end{proposition}

\begin{proof}
For $s,c\in\mathbb{R}$ define $u_{s,c}$ as the unique solution of the
problem
\begin{gather*}
u^{\Delta\Delta}+a\sin(u^\sigma) = p_0+ c_0 + c \\
u(0) = u(\sigma^2(T))= s.
\end{gather*}
We claim that the operator given by
$(s,c)\to u_{s,c}$ is well defined and
continuous. Indeed,
if $u$ and $v$ are solutions of the previous problem, it follows
that
$$
(u-v)(t) =
-a\int_0^{\sigma(T)}G(t,s)[\sin(u^\sigma(s))-\sin(v^\sigma(s))]\Delta s,
$$
and hence
$$
\| u-v\|_{C_{rd}([0,\sigma(T)])} \le a K
\|u-v\|_{C_{rd}([0,\sigma(T)])}.
$$
As $aK <1$, it follows that $u=v$.
Moreover, if $c\to \hat c$ and $s\to \hat s$,
then
$$
(u_{s,c}-u_{\hat s,\hat c})(t) = s-\hat s +
a\int_0^{\sigma(T)}G(t,\xi)[c-\hat c -
\sin(u_{s,c}^\sigma(\xi))+\sin(u_{\hat s,\hat c}^\sigma(\xi))]\Delta \xi.
$$
Thus,
$$
(1-aK)\|u_{s,c}-u_{\hat s,\hat c}\|_{C_{rd}([0,\sigma(T)])}
\le |s-\hat s|
+ aK|c-\hat c|
$$
and continuity follows. Next, define
$\theta(s,c) = u^\Delta_{s,c}(\sigma(T)) - u^\Delta_{s,c}(0)$.
By definition of $u_{s,c}$ it is clear that
$$
\theta(s,c) = \int_0^{\sigma(T)}[p_0+ c_0+
c-a\sin(u_{s,c}^\sigma)]\Delta t =
c\sigma(T) - a \int_0^{\sigma(T)}\sin(u_{s,c}^\sigma) \Delta t.
$$
It follows that $\theta$ is continuous, and
$$
\theta(s,a) \ge 0\ge \theta(s,-a).
$$
We conclude that for each $s$ there exists a number $c(s)$ such that
$\theta(s,c(s))=0$. As the problem is singular, we deduce that
$c(s)=0$, and it follows that $u_{s,0}$ also satisfies:
$u_{s,0}^\Delta(\sigma(T))- u_{s,0}^\Delta(0)=0$.
\end{proof}


\section{Landesman-Lazer conditions for a resonant system}
\label{llaz}

In this section we shall give an existence result for problem
(\ref{sys}-\ref{per}), which may be regarded as an extension of
Theorem \ref{nir}.


\begin{remark} \label{rmk4.1} \rm
A different existence result for (\ref{sys}-\ref{per})
is given in
\cite{ART} Theorem 3.3,
assuming that $f$ satisfies the Hartman-type condition (see
\cite{H}):
$$
\langle f(t,z),z\rangle > 0
\quad \hbox{for $z\in \mathbb{R}^N$ with } |z| = R.
$$
\end{remark}

Our Landesman-Lazer type condition reads as follows.


\begin{description}
\item[Condition (F1)]
 There exists a family $\{ (U_j,w_j)\}_{j=1,\dots, K}$
where $U_j$ is an open subset of $S^{N-1}$ and
$w_j \in S^{N-1}$, such that $\{U_j\}$ covers
$S^{N-1}$ and the limit
\begin{equation} \label{h1}
\limsup_{s\to +\infty}
\left\langle f(t,su), w_j \right\rangle :=
\overline f_{u,j}(t)
\end{equation}
exists uniformly for $u\in U_j$.
\end{description}

\begin{remark} \label{rmk4.2}\rm
If condition (F1) holds,  then a straightforward computation
shows that the mapping
$u \mapsto \overline f_{u,j}(t)$ is continuous in $U_j$
for each fixed $t$.
\end{remark}

\begin{theorem} \label{LL}
Assume that $f$ is bounded, and that condition (F1)
holds. Then the periodic boundary value problem
(\ref{sys}-\ref{per})
admits at least one solution, provided that
\begin{enumerate}
\item \label{ll1}
For each $u\in S^{N-1}$ there exists $j$ such that $u\in U_j$ and
$$
\int_0^{\sigma(T)} \overline f_{u,j}(t) d\mu <0,
$$
where $\mu$
is the measure introduced in section \ref{prelim}.

\item \label{deg}
There exists a constant $R_0$ such that
$d_B(F, B_R,0)\neq 0$ for any $R\ge R_0$,
where $d_B$ is the Brouwer degree,
$B_R\subset \mathbb{R}^N$ denotes
the open ball of radius $R$ centered at $0$,
and $F:\mathbb{R}^N\to \mathbb{R}^N$ is defined by
$$
F(y) = \int_0^{\sigma(T)} f(t,y)\Delta t.
$$
\end{enumerate}
\end{theorem}

\begin{remark}\label{rmk4.4} \rm
It follows from the proof below that
$F(y)\ne 0$ for $y\in \mathbb{R}^N$ with $|y|$ large.
Thus, the Brouwer
degree in condition \ref{deg} is well defined.
\end{remark}

\begin{proof}[Proof of Theorem \ref{LL}]
For $\lambda\in [0,1]$, let us define the compact operator
$T_\lambda:C_{rd}([0,T]_\mathbb{T})\to C_{rd}([0,T]_\mathbb{T})$ given by
$$
T_\lambda y(t) = y(0) + \overline{f(\cdot,y^\sigma)}
+ \lambda\int_0^{\sigma(T)} G(t,s)f(s,y^\sigma(s))\Delta s.
$$
For $\lambda\neq 0$, if $y = T_\lambda y$ then
evaluating at $t=0$ it follows that
$\overline{f(\cdot,y^\sigma)}=0$.
Moreover,
$y(\sigma^2(T)) = y(0)$, and
$y^{\Delta\Delta}(t) = \lambda f(t,y^\sigma)$.
Integrating this last equation,
we deduce that also $ y^\Delta(0) = y^\Delta(\sigma(T))$.

We claim that the solutions of the equation
$y = T_\lambda y$ are a priori bounded. Indeed,
if $y_n = T_{\lambda_n} y_n$
with $\lambda_n\in (0,1]$ and $\|y\|_{C_{rd}([0,\sigma(T)])}\to \infty$,
then
$$
\|y_n - y_n(0)\|_{C_{rd}([0,\sigma(T)])}\le K\|f\|_C,
$$
and $y_n(0)\to \infty$.
Let $z_n(t)= \frac{y_n(t)}{|y_n(t)|}$, then taking a subsequence if necessary
we may assume that $z_n(t)\to u\in S^{N-1}$ as $n\to \infty$, uniformly in $t$.
Thus, for some $j$ we have by Fatou's Lemma that
$$
0 = \int_0^{\sigma(T)}
\langle f(t,y_n^\sigma),w_j\rangle d\mu < 0
$$
for $n$ large, a contradiction.

On the other hand, if $y= T_0 y$
then $y$ is constant and
$F(y) = 0$. As before, if we suppose that
$F(y_n) = 0$ with $|y_n|\to \infty$, a contradiction yields.
We conclude that if
$\Omega = B_R(0) \subset C_{rd}([0,T]_\mathbb{T})$ with $R$ large enough, then
the Leray-Schauder degree $d_{LS}(I-T_\lambda,\Omega, 0)$
is well defined and
$d_{LS}(I-T_1,\Omega, 0) = d_{LS}(I-T_0,\Omega, 0)$. Moreover,
as $T_0 y = y(0)+ \overline{f(\cdot,y^\sigma)}\in \mathbb{R}^N$
for any $y$, it follows that
$$
d_{LS}(I-T_0,\Omega, 0) = d_{B}((I-T_0)|_{\mathbb{R}^N},\Omega\cap \mathbb{R}^N, 0).
$$
As $(I-T_0)|_{\mathbb{R}^N} = -\sigma(T) F$,
this last degree is non-zero.
We conclude that the equation
$y= T_1y$ admits a solution in $\Omega$, which corresponds
to a solution of (\ref{sys}-\ref{per}).
\end{proof}

Some examples are now provided to illustrate the main ideas of the paper.

\begin{example} \label{exa4.5} \rm
If $f(t,y) = p(t) - g(y)$
and $g_v:= \lim_{r\to +\infty} g(rv)$ exist uniformly
respect to $v\in S^{N-1}$, then for any
$w\in S^{N-1}$ we have that
$\langle f(t,sv), w\rangle \to
\langle p - g_v, w\rangle$ uniformly in $S^{N-1}$.
If $\overline p\neq g_v$,
then for any $v_0\in S^{N-1}$ there exists
$w\in S^{N-1}$ such that
$\langle \overline p - g_v, w\rangle < 0$ in a neighborhood of $v_0$.
By compactness, (F1) and
the first condition of Theorem \ref{LL} are fulfilled.
Furthermore, if the degree of the mapping
$\theta (v)= \frac{g_v-\overline p}{|g_v-\overline p|}$ is non-zero,
it is immediate to see that
$F(y) = \int_0^{\sigma(T)} p(t) - g(y)\Delta t =
\sigma(T) (\overline p - g(y))$ satisfies: $d_B(F,B_R,0)\neq 0$
when $R$ is large. Thus, Theorem \ref{nir} can be regarded as a particular
case of Theorem \ref{LL} for $\mathbb{T}=\mathbb{R}$.
\end{example}


\begin{example} \label{exa4.6} \rm
Let $f=(f_1,\dots,f_N)$ with
$f_i(t,y) = \frac{\psi_i(t,y)}{|y|^2 + 1} + \xi_i(t)
\hbox{\rm arctan}(y_i)$,
where $\psi_i$ is continuous such that
$|\psi_i(t,y)|\le A|y|^r + B$ for some $r<2$, and $\xi$ is rd-continuous.
Then
(\ref{sys}-\ref{per})
admits at least one solution, provided that
$\int_0^{\sigma(T)} \xi_i \Delta t \neq 0$ for $i=1,\cdots ,N$.
Indeed, for $y \in \mathbb{R}^N$ with $y_i\neq 0$, set
$k = \mathop{\rm sgn}\big(\mathop{\rm sgn}(y_i)\int_0^{\sigma(T)}
 \xi_i \Delta t\big)$ and
$w_i = ke_i$.
Then
$$
\lim_{s\to +\infty} \langle f(t,sy),w_i\rangle
= k\mathop{\rm sgn}(y_i)\frac {\pi}2
\xi_i(t):= \overline f_{y,w_i}(t),
$$
and
$$
\int_0^{\sigma(T)} \overline f_{y,w_i}(t) d\mu =
k\mathop{\rm sgn}(y_i) \int_0^{\sigma(T)} \xi_i \;\Delta t <0.
$$
Moreover, it is easy to see that if $|y_i| \gg 0$ then
$$
F_i(y).F_i(-y) <0.
$$
Thus, the second condition in Theorem \ref{LL} is fulfilled.
\end{example}

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\end{document}


Variational methods for two resonant problems on time scales.