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\markboth{ A Second look at the first result of Landesman-Lazer type }
{ Alan C. Lazer }
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 113--119\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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A Second look at the first result of Landesman-Lazer type
%
\thanks{ {\em Mathematics Subject Classifications:} 34C25, 34C27.
\hfil\break\indent
{\em Key words:} Globally asymptotically stable, Brouwer fix-point theorem,
polar coordinates.
\hfil\break\indent
\copyright 2000 Southwest Texas State University.
\hfil\break\indent Published October 25, 2000. } }
\date{}
\author{ Alan C. Lazer \\[12pt]
{\em Dedicated to Paul Frederickson }}
\maketitle
\begin{abstract}
We discuss some results concerning periodic and almot periodic solutions
of ordinary differential equations which are precursors of a result on
weak solutions of a semilinear elliptic boundary due to E. M. Landesman
and the author. It is observed that in the earliest of these, if one
looks for periodic solutions instead of almost peridic solutions,
then the conditions can be relaxed.
\end{abstract}
Let $g:{\mathbb R} \to {\mathbb R} $ be a continuous function and assume that the limits
\[
g(\pm\infty )= \lim _{s\to \pm \infty } g(s)
\]
exist and are finite, and that $\forall \xi \in {\mathbb R}$,
$g(-\infty )0\}$, and
$D^-=\{x\in D : \varphi _k(x)<0\}$,
then the condition
\begin{eqnarray*}
g(-\infty )\int _{D^+} \varphi _kdx+g(\infty )
\int _{D^-}\varphi _kdx &<& \int _D h\varphi _kdx \\
&<&g(\infty ) \int_{D^+}\varphi _kdx+g(-\infty ) \int _{D^-}\varphi _kdx
\end{eqnarray*}
is both necessary and sufficient for the existence of a weak solution of (1.1).
Only a short time before D.E. Leach and the author [5] considered the
ordinary differential equation
$$
u''+n^2u+g(u)=e(t)=e(t+2\pi )\,, \eqno{(1.2)}
$$
where $n>0$ is an integer and $g$ satisfies the same conditions as
before, it was shown that if
\[ A=\int ^{2\pi}_0 e(t)\cos (nt)\, dt \quad\mbox{and}\quad
B=\int ^{2\pi }_0 e(t)\sin(nt)\, dt\,,
\]
then the inequality $2(g(\infty )-g(-\infty ))>\sqrt{A^2+B^2}$ is both
necessary and sufficient for the existence of a $2\pi $-periodic solution of
(1.2).
It is natural to ask what happens if $g$ satisfies the same conditions and
$n=0$ in (1.2). If we consider more generally
$$ u''+cu' +g(u)=e(t)=e(t+2\pi )\,, \eqno{(1.3)}
$$
where $c\in {\mathbb R} $, and we assume that there is a $2\pi $-periodic
solution $\hat{u} (t)$, then integration from 0 to $2\pi $ shows that
\[
\int ^{2\pi }_0 (e(t)-g(\hat{u} (t)))\,dt=0\,.
\]
Therefore, if
\[
e_0\equiv \frac{1}{2\pi } \int ^{2\pi }_0 e(t)\,dt\,,
\]
then a necessary condition for the existence of a $2\pi $-periodic solution
of (1.3) is
$$ g(-\infty ) 0$ there exists a
unique $T$-periodic solution of (1.7) which is globally asymptotically
stable.
In [7] Reissig considered (1.6) under the assumptions that $F$ is a
continuous, strictly increasing function, $E(t)$ is a continuous periodic
function with least period $T>0$, and that for arbitrary $x_0,y_0$, and
$t_0$, there is a unique solution of (1.6) with $x(t_0)=x_0$,
$x'(t_0)=y_0$. Reissig showed that the condition
\[ F(\infty )-F(-\infty)>\max E(t)-\min E(t)
\]
implies the existence and uniqueness of a
$T$-periodic solution of (1.6) and that this solution is globally
asymptotically stable.
In [2] the assumption that $e(t)$ is almost periodic implies the existence
of
\[ M[e(t)\exp (-it)]=\lim _{T\to \infty }
\frac{1}{T} \int ^{t_0+T}_{t_0} e(t)\exp (-it)\,dt
\]
uniformly with respect to $t_0\in {\mathbb R} $. It was shown in [2] that if
$f$ is continuous and strictly positive except at isolated points, so that
its antiderivative $F$ is strictly increasing, then
$$
F(\infty )-F(-\infty )>\pi \big|M[e(t)\exp (-it)]\big| \eqno{(1.8)}
$$
is both necessary and sufficient for the existence of an almost
periodic solution of (1.7). Moreover, if this condition holds, there is a
unique almost periodic solution which is globally asymptotically stable.
In [2] the assumption that $F$ is strictly increasing was used in an
essential way. The proof was based on a result of Amerio [1]. To use this
result it was necessary to show that if whenever $\{h_m\}^\infty _1$ is a
sequence of real numbers such that
\[ e^*(t)= \lim _{m\to \infty } e(t+h_m)
\]
exists uniformly with respect to $t\in {\mathbb R} $, then
the differential equation
\[ x''+f(x)x' +x=e^*(t)
\]
has a unique solution bounded on ${\mathbb R} $. Since $e^*(t)$ is almost
periodic and
\[ \big|M[e^*(t)\exp (-it)]\big|=\big|M[e(t)\exp (-it)]\big|,
\]
verification of Amerio's condition for (1.7) amounts to showing that
condition (1.8) and the other assumptions on $f$ imply that (1.7) itself has a
unique solution bounded on ${\mathbb R} $. Examination of the arguments given in
[2] shows that it is enough to assume the existence of the limits
$F(\infty)$ and $F(-\infty )$ and the condition (1.8) in order to ensure the
existence of at least one solution of (1.7) which is bounded on
${\mathbb R}$. It is the proof of uniqueness
which depends on the strict monotonicity of $F$. This was accomplished by
noting that if $x_1(t)$ and $x_2(t)$ are two solutions of (1.7) and we set
$$\displaylines{
x' _1(t)=y_1(t)-F(x_1(t)), \cr
x' _2(t)=y_2(t)-F(x_2(t)), \cr
d(t) \equiv \sqrt{(x_1(t)-x_2(t))^2+(y_1(t)-y_2(t))^2},
}$$
then
$$
d(t)d' (t)\equiv -(x_1(t)-x_2(t))(F(x_1(t))-F(x_2(t)))\leq 0\,.
$$
Actually, for reasons of exposition, we have simplified what was done in
[2]. A more complicated system which contains (1.6) and (1.7) as special
cases was considered but the same type of reasoning described above was
used.
What we would like to point out is in the case that $e(t)$ is a
$2\pi$-periodic function, the assumptions that $F$ is strictly increasing can
be replaced by the assumption that the limits $F(\infty )$ and $F(-\infty )$
exist and that $\forall \xi \in {\mathbb R}$,
\[ F(-\infty )\sqrt{A^2+B^2}
\]
is necessary and sufficient for the existence of a $2\pi $-periodic solution
of (1.7). \medskip
We sketch the proof of the sufficiency using the same reasoning as in [2]
and the Brouwer fixed-point theorem.
If $x(t)$ and $y(t)$ satisfy
$$ x' =y-F(x), \quad y' =-x+e(t) \eqno{(1.9)}
$$
then $x(t)$ will be a solution of (1.7) so we look for a
$2\pi$-periodic solution of this system.
If $r(t)=\sqrt{x(t)^2+y(t)^2}$ and $r(t)\neq 0$, then
\[
r'(t)=-x(t)F(x(t))/r(t)+y(t)e(t)/r(t),
\] so, by the boundedness of $F$, there exists a constant $M>0$ such that
$r(t)\neq 0$ implies $|r'(t)|\leq M$. From this we infer the existence of
$r_0>0$ such that $x(0)^2+y(0)^2\geq r^2_0$, implies $x(t)^2+y(t)^2>0$
for $0\leq t\leq 2\pi$.
If $x(0)^2+y(0)^2\geq r^2_0$ and $t\in [0,2\pi ]$ we can set
$x(t)=r(t)\sin \theta (t),$ \linebreak $y(t)=r(t)\cos \theta (t)$, where
$$\displaylines{
\hfill r' (t)=-F(r(t)\sin \theta (t))\sin \theta (t)+e(t)\cos \theta(t)
\hfill\llap{(1.10)} \cr
\hfill \theta ' (t) = 1-\frac{F(r(t)\sin \theta (\theta )\cos\theta
(t)}{r(t)}- \frac{e(t)\sin \theta (t)}{r(t)} \,.\hfill \llap{(1.11)}
}$$
If for $c\geq r_0$ and $\varphi \in {\mathbb R} $, $r(t,c,\varphi )$ and
$\theta (t,c,\varphi )$ denote the components of the solution of the
system (1.10)-(1.11) such that $r(0,c, \varphi )=c$, $\theta
(0,c,\varphi )=\varphi $, then
\[ r(t,c,\varphi )=c+O(1) \quad \mbox{as } c\to \infty
\]
uniformly with respect to $t\in [0,2\pi ]$ and
$\varphi \in {\mathbb R}$. Therefore, integration of (1.11) yields
$\theta(t,c,\varphi )=t+\varphi +0(1/c)$ as $c\to \infty $ uniformly with
respect to $t\in [0,2\pi ]$ and $\varphi \in {\mathbb R} $.
Since
\begin{eqnarray*}
\lefteqn{ r(2\pi ,c,\varphi )-c} \\
&=& \int ^{2\pi}_0 -F(r(t,c,\varphi)\sin
\theta (t,c,\varphi ))\sin \theta (t,c,\varphi )+e(t)\cos \theta
(t,c,\varphi )\,dt
\end{eqnarray*}
the asymptotic estimates for $r(t,c,\varphi )$ and
$\theta (t,c,\varphi )$ together with the assumptions on $F$ imply that
\[
r(2\pi ,c,\varphi )-c\to \int ^{2\pi }_0 e(t)\cos (t+\varphi
)dt-2[F(\infty )-F(-\infty )]
\] as $c\to \infty $ uniformly with respect to $\varphi \in {\mathbb R} $.
Since
\[ \int ^{2\pi}_0 e(t)\cos(t+\varphi )dt=A\cos \varphi -B\sin \varphi
\leq \sqrt{A^2+B^2},
\]
our basic assumption implies the existence of $c_*\geq r_0$ such that if
$c\geq c_*$, then $r(2\pi ,c,\varphi )0$ and $v' <0$ respectively, then
\begin{eqnarray*}
\sqrt{A^2+B^2}& =& \int^{2\pi }_0 e(t)v(t)\,dt\\
&=&-\int ^{2\pi }_0 F(u(t))v' (t)\,dt \\
&<&-F(\infty ) \int _N v' (t)\,dt -F(-\infty )\int_P v'(t)\,dt \\
&=&2(F(\infty )-F(-\infty )). \
\end{eqnarray*}
If $E(t)$ is $2\pi $-periodic and continuous and $F$ is a locally
Lipschitzian function of the type considered above, then a necessary and
sufficient condition for (1.6) to have a $2\pi $-periodic solution is
\[
2(F(\infty )-F(-\infty ))>\sqrt{C^2+D^2}
\]
where
\[ C={ \int ^{2\pi}_0}E(t)\cos t\,dt, \;\; D={ \int ^{2\pi }_0}E(t)\sin t\,dt
\]
The proof follows from considering the system
\[ x' =y, \quad y' =-F(y)-x+e(t)\,.
\]
After introducing polar coordinates, one
can obtain asymptotic estimates and show that the period map maps a closed
disk into itself.
That Frederickson was the major contributor to [2] was acknowledged in
[5]. After a third of a century his contribution to the development to
what are called Landesman-Lazer type results needs to be acknowledged
again.
\begin{thebibliography}{0} {\frenchspacing
\bibitem{a1} Amerio, L., {\em Soluzioni quasi-periodiche, o limitate, di sistemi
differenziali non lineari quasi-periodici, o limitati}, Ann Mat. Pura
Appl. 39 (1955), 97--119.
\bibitem{f1} Frederickson, P.O. and Lazer, A.C., {\em Necessary and sufficient damping
in a second order oscillator}, J. Differential Eqs. 5 (1969),
262--270.
\bibitem{l1} Landesman, E.M. and Lazer, A.C., {\em Nonlinear perturbations of linear
elliptic boundary value problems at resonance}, J. Math. Mech. 19
(1970), 609--623.
\bibitem{l2} Lazer, A.C., {\em On Schauder's fixed point theorem and forced
second-order nonlinear oscillations}, f J. Math. Anal. Appl. 21 (1968),
421--425.
\bibitem{l3} Lazer, A.C. and Leach, D.E., {\em Bounded perturbations of forced
harmonic oscillators at resonance}, Ann. Mat. Pura Appl. 82 (1969),
49--68.
\bibitem{l4} Levenson, N., {\em On a nonlinear differential equation of the second
order}, J. Math. Phys. 22 (1943), 181--187.
\bibitem{r1} Reissig, R., {\em Uber eine nichtlinear Differentialgliechung
2. Ordnung}, Math. Nach. 14 (1955), 65--71.
}\end{thebibliography} \medskip
\noindent{\sc Alan C. Lazer} \\
Department of Mathematics \\
University of Miami \\
Coral Gables, FL 33124, USA. \\
e-mail: A.Lazer@math.miami.edu
\end{document}