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\markboth{\hfil Riesz bases in $L_2$ \hfil EJDE--2001/74}
{EJDE--2001/74\hfil Peter E. Zhidkov \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No.~74, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Sufficient conditions for functions to form Riesz bases
in $L_2$ and applications to nonlinear boundary-value problems
%
\thanks{ {\em Mathematics Subject Classifications:} 41A58, 42C15, 34L10, 34L30.
\hfil\break\indent
{\em Key words:} Riesz basis, infinite sequence of solutions,
nonlinear boundary-value problem.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted September 24, 2001. Published December 4, 2001.} }
\date{}
%
\author{Peter E. Zhidkov}
\maketitle
\begin{abstract}
We find sufficient conditions for systems of functions to be
Riesz bases in $L_2(0,1)$. Then we improve a theorem
presented in \cite{z7} by showing that a ``standard'' system
of solutions of a nonlinear boundary-value problem, normalized
to 1, is a Riesz basis in $L_2(0,1)$.
The proofs in this article use Bari's theorem.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\section{Introduction}
Early results in the study of basis properties of
eigenfunctions of nonlinear ordinary differential operators
can be found in the monograph by Makhmudov \cite{m1}.
Because of its difficulty and the small number of publications
on this question, basis properties has been established only
for very simple nonlinear ordinary differential equations.
Among the results in this direction, we have the following.
In \cite{z1,z2}, Zhidkov presents an analysis of the equation
\begin{eqnarray*}
& -u''+f(u^2)u=\lambda u, \quad u=u(x), \quad x\in (0,1),&\\
& u(0)=u(1)=0, \quad \int_0^1 u^2(x)\,dx =1\,, &
\end{eqnarray*}
where $\lambda$ is a spectral parameter, $f(s)$ is a smooth
nondecreasing function for $s\geq 0$, and all quantities are real.
In these two publications, it is proved that the
eigenfunctions $\{u_n\}$ ($n=0,1,2,\dots$) of this problem have
precisely $n$ zeros in $(0,1)$. Furthermore, each eigenfunction
is unique up to the coefficient $\pm 1$.
The main result states that the sequence of eigenfunctions $\{u_n\}$
($n=0,1,2,\dots$) is a Bari basis in $L_2=L_2(0,1)$, i.e.,
it is a basis and there exists an orthonormal basis $\{e_n\}$ ($n=0,1,2,\dots$)
in $L_2$ for which $\sum_{n=0}^\infty \|u_n-e_n\|_{L_2}^2<\infty $.
Note that in \cite{z1} there are some errors which have been corrected
in \cite{z3}.
In \cite{z4,z5}, a modified version of the above nonlinear eigenvalue
problem is studied and similar basis properties for their eigenfunctions
are obtained.
In \cite{z6}, an analog to the Fourier transform associated with an
eigenvalue problem for a nonlinear ordinary differential operator on
a half-line is considered.
The aim in the present publication is to improve the result in \cite{z7},
where the following nonlinear problem is considered:
\begin{eqnarray}
& u''=f(u^2)u\,, \quad u=u(x)\,, \quad x\in (0,1)\,, &\label{e1}\\
& u(0)=u(1)=0\,.& \label{e2}
\end{eqnarray}
Here there is no spectral parameter, and all variables are real.
For the rest of this article, we will assume that
\begin{enumerate}
\item[(F)] The function $f(u^2)u$ is a continuously differentiable for
$u\in \mathbb{R}$, $f(0)\geq 0$, and $f(+\infty )=-\infty $.
\end{enumerate}
It is well known now (and partially proved in \cite{z7}) that under
assumption (F): For each integer $n\geq 0$ problem (\ref{e1})--(\ref{e2})
has a solution $u_n$ which
possesses precisely $n$ zeros in $(0,1)$ and that generally speaking this
solution is not unique.
\paragraph{Definition} A sequence $\{u_n\}$ ($n=0,1,2,\dots$) of solutions
to (\ref{e1})--(\ref{e2}) is called standard if the solution $u_n$
has precisely $n$ zeros in $(0,1)$.
The main result in \cite{z7} states that there exists
$s_0<0$ such that for $s0$, $h''_n(x)\leq 0$, and $h'''_n(x)\leq 0$ for all
$x\in \big(0,{1\over 2(n+1)}\big)$
\item[(c)] There exist $00$, $h''(x)\leq 0$ and $h'''(x)\leq 0$ for all
$x\in (0,1/2)$
\end{enumerate}
Then, the sequence of functions $h_n(x)=h((n+1)x)$, where $n=0,1,2,\dots $,
is a Riesz basis in $L_2$.
\end{theorem}
The following statement also follows from Theorem \ref{thm1}, when
applied to problem (\ref{e1})--(\ref{e2}).
\begin{theorem} \label{thm3}
Let assumption (F) be valid and $f(u^2)+2u^2f'(u^2)\leq 0$ for all
sufficiently large $u$. Let $\{u_n\}$ be an arbitrary standard
sequence of solutions of (\ref{e1})--(\ref{e2}).
Then, the sequence
$\left\{\|u_n\|^{-1} u_n\right\}$ is a Riesz basis in $L_2$.
\end{theorem}
To prove this theorem in Section 3, we exploit the following theorem.
\begin{theorem}[Bari's Theorem]
Let $\{e_n\}$ be a Riesz basis in $L_2(a,b)$ and let a system
$\{h_n\}\subset L_2(a,b)$ be $\omega$-linearly independent and
quadratically close to $\{e_n\}$ in $L_2(a,b)$.
Then, the system $\{h_n\}$ is a Riesz basis in $L_2(a,b)$.
\end{theorem}
This theorem, in a weaker form, was proved by N. K. Bari in \cite{b1}.
In its current form it is proved, for example, in \cite{g1} and in \cite{z2}.
We conclude the introduction by pointing out that
the concept of a Riesz basis appeared for the first
time in the middle of last century in the papers of N. K. Bari, as a
result of developments in the general theory of orthogonal series and
bases in infinite-dimensional spaces.
Currently, this concept has important applications in areas
such as wavelet analysis. Readers may consult \cite{c1,n1} for theoretical
aspects of this field and \cite{d1} for applied aspects.
\section{Proof of Theorem \ref{thm1}}
Let $e_n(x)=\sqrt {2} \sin \pi (n+1)x$, $n=0,1,2,\dots $, so that
$\{e_n\}$ is an orthonormal basis in $L_2$.
\begin{lemma} \label{lm1}
Let $g$ satisfy condition (a) of Theorem \ref{thm1} with $n\geq 0$
and let $g$ be positive in $(0,{1\over n+1})$. Then in the expansion
$$
g(\cdot )=\sum_{m=0}^\infty c_m e_m(\cdot ),
$$
understanding in the sense of $L_2$,
one has $c_0=\dots =c_{n-1}=0$ and $c_n>0$.
\end{lemma}
\paragraph{Proof}
We follow the arguments in the proof of a similar statement in \cite{z7}.
We have the above expansion in $L_2(0,{1\over n+1})$ with $c_m=0$
if $m\neq (n+1)(l+1)-1$ for all integers $l\geq 0$ (this occurs because
the functions $\{e_{(n+1)(m+1)-1}\}_m$ form an
orthogonal basis in $L_2( 0,{1\over n+1})$). Therefore,
$c_0=\dots =c_{n-1}=0$. We observe that each
$e_{(n+1)(m+1)-1}$ becomes zero at the points ${1\over n+1},{2\over
n+1},\dots ,1$. Furthermore, due to condition (a) of Theorem \ref{thm1}
the function $g$ is odd with respect to these points and each
function $e_{(n+1)(m+1)-1}(x)$ is odd. Thus
this expansion also holds in each space $L_2( {1\over n+1},{2\over n+1}
),L_2( {2\over n+1},{3\over n+1}),\dots ,L_2({n\over n+1},
1)$. Finally, $c_n>0$ because $e_n(x)$ and $g(x)$ are of the same
sign everywhere. \hfill$\Box$\smallskip
Due to Lemma \ref{lm1}, we have the following sequence of expansions:
\begin{equation}
h_n(\cdot )=\sum_{m=0}^\infty a^n_m e_m(\cdot ) \quad\mbox{in } L_2,
\label{e3}
\end{equation}
with $a^n_0=\dots =a^n_{n-1}=0$ and $a^n_n>0$, for $n=0,1,2,\dots$.
\begin{lemma} \label{lm2}
Under the assumptions of Theorem \ref{thm1}, the coefficients in (\ref{e3})
satisfy
$$(a^n_n)^{-1}|a^n_{(n+1)(m+1)-1}|\leq {\pi \over 2}
(m+1)^{-2}$$
for all $n$ and $m$. In addition, $a^n_{(n+1)(m+1)-1}=0$ if $m=2l+1$ for
$l=0,1,2,\dots $.
\end{lemma}
\paragraph{Proof} The second claim of this lemma is obvious because
$e_{(n+1) (2l+2)-1}(x)$ is odd with respect to the middles of the intervals
$(0,{1\over n+1})$, $({1\over n+1},{2\over n+1}),\dots ,
( {n\over n+1},1 )$
and the function $h_n(x)$ is even so that $a^n_{(n+1)(2l+2)-1}=
(e_{(n+1)(2l+2)-1},h_n)=0$. Let us prove the first claim. Due to
the properties of the functions $h_n$ and $e_{(n+1)(m+1)-1}$, with $m=2l$,
we have
\begin{eqnarray*}
(a^n_n)^{-1}|a^n_{(n+1)(m+1)-1}|
&=&\frac {\left|\int_0^1h_n(x)\sin \pi
(n+1)(m+1)x dx\right|}{\int_0^1h_n(x)\sin \pi (n+1) x dx} \\
&=&\frac {\left| \int_0^{1/2(n+1)}h_n(x)\sin \pi (n+1)(m+1) x dx\right|}
{\int_0^{1/2(n+1)}h_n(x)\sin \pi (n+1)x dx} \\
&=&(m+1)^{-1}\frac {\left| \int_0^{1/2(n+1)} h'_n(x) \cos \pi (n+1)(m+1)
x dx\right| }{\int_0^{1/2(n+1)} h'_n(x)\cos \pi (n+1)x dx} \\
&=&(m+1)^{-1}\frac {\left| \int_0^1 h'_n( {s\over 2(n+1)})
\cos {\pi (m+1)s\over 2} ds\right|}
{\int_0^1 h'_n( {s\over 2(n+1)}) \cos {\pi s\over 2} ds}.
\end{eqnarray*}
Due to the conditions of Theorem \ref{thm1}, $h'_n({s\over 2(n+1)}) $
is a positive non-increasing concave function on $(0,1)$. Therefore,
$$
\int_0^1 h'_n( {s\over 2(n+1)} ) \cos {\pi s\over 2} ds
\geq h'_n(0)\int_0^1 (1-s) \cos {\pi s\over 2} ds={4\over \pi ^2}
h'_n(0).
$$
Using the same properties of $h'_n$, one can easily see on its graph
that
\begin{eqnarray*}
\Big| \int_0^1h'_n( {s\over 2(n+1)} ) \cos {\pi (m+1)s\over 2} ds\Big|
&\leq& h'_n(0)\int_0^{1/(m+1)}\cos {\pi (m+1)s\over 2} ds \\
&=&{2\over \pi (m+1)}h'_n(0)\,.
\end{eqnarray*}
Detailed arguments leading to a similar estimate are considered in
\cite{z7}. We easily obtain now
$$
(a^n_n)^{-1} |a^n_{(n+1)(m+1)-1}|\leq {\pi \over 2}(m+1)^{-2},
$$
which completes the proof. \hfill$\Box$ \smallskip
From the conditions on Theorem \ref{thm1}, we have $00$ be an arbitrary number such that $f(u^2)<0$ and $f(u^2)+
2u^2f'(u^2)\leq 0$ for all $u\geq \overline u$. Let $\{u_n\}$ be an
arbitrary standard system of solutions of problem (\ref{e1})--(\ref{e2}).
We assume that $u_n(x)>0$ for $x\in (0,{1\over n+1})$ for each $n$
which is possible without loss of generality due to the invariance of
(\ref{e1}) when $u(x)$ is replaced by $-u(x)$.
Due to the standard comparison theorem
$\max_{u\in [0,u_n(1/2(n+1))]}|f(u^2)|\to +\infty $ as $n\to
\infty $; hence
$u_n( {1\over 2(n+1)}) \to +\infty $ as $n\to \infty $.
For $n$ sufficiently large, we denote by $x_n\in (0, {1\over 2(n+1)})$
the point for which $u_n(x_n)=\overline u$. Then
$$
u_n({1\over 2(n+1)})-\overline u=\int_{x_n}^{1/2(n+1)} u'_n(x)
dx=u'_n(\tilde x_n)( {1\over 2(n+1)}-x_n)
$$
for some $\tilde x_n\in (x_n,{1\over 2(n+1)})$.
Since $u'_n
(x_n)\geq u'_n(\tilde x_n)$ (because $f(u^2)<0$ for $u>\overline u$ and,
therefore,
$u''_n(x)<0$ for $x\in (x_n,{1\over 2(n+1)})$), we derive
$$
u'_n(x_n)\geq {3\over 2} u_n({1\over 2(n+1)}) (n+1)
\label{e4}
$$
for all sufficiently large $n$. Since in view of (\ref{e1}), $\sup_n
\max_{x\in [0,x_n]} |u''_n(x)|\leq C'$, we have $\min_
{x\in [0,x_n]}|u'_n(x)|\geq u_n( {1\over 2(n+1)} ) (n+1)$
for all sufficiently large $n$. Therefore,
\begin{equation}
00$
and is large enough so that $|\overline u f(\overline u^2)|>|u f(u^2)|$ and
$F(\overline u^2)>F(u^2)$ for all $u\in [0,\overline u)$.
Then from (\ref{e6}), it follows that
\begin{equation}
u'_n(x_n)__x_n u'_n(x_n);
$$
therefore,
\begin{equation}
u'_n(x_n)<{\overline u\over x_n}=l'_n(x)
\label{e8}
\end{equation}
for all sufficiently large $n$.
Take a sufficiently small $\Delta \in (0,{x_n\over 2})$ and define a
continuous function $\omega _1(x)$ equal to $u'''_n(x)$ for $x\in
[x_n, {1\over 2(n+1)}]$ such that $u'''_n(x)\leq \omega _1(x)
\leq 0$ for $x\in [x_n-\Delta ,x_n]$ and $\omega _1(x)=0$ for $x\in [0,x_n
-\Delta )$.
We define $g_1(x)$ to be equal to $u_n(x)$
for $x\in [x_n,{1\over 2(n+1)}]$, and for $x\in [0,x_n)$ to be given
by the rules:
\begin{eqnarray}
g''_1(x)&=& u''_n(x_n)-\int_x^{x_n}\omega _1(t) dt, \nonumber\\
g'_1(x) &=& u'_n(x_n)-\int_x^{x_n}g''_1(t) dt, \label{e9} \\
g_1(x) &=& u_n(x_n)-\int_x^{x_n} g'_1(t) dt. \nonumber
\end{eqnarray}
Then $g_1(x)$ is three times continuously differentiable in
$[0,{1\over 2(n+1)}]$ and satisfies condition (b) of Theorem \ref{thm1}.
It is easy to see that if $\Delta >0$
is sufficiently small, then $g_1(x_n-\Delta )$ and $g'_1(x_n-\Delta )$
are arbitrary close to $u_n(x_n)$ and $u'_n(x_n)$, respectively, and
$g''_1(x)$ is arbitrary close to $u''_n(x_n)$ for all $x\in [0,x_n-\Delta ]$.
Now, due to our choice of $\overline u>0$, for $\Delta >0$ and
sufficiently small, $g_1(0)$ is arbitrary close to
\begin{equation}
u_n(x_n)-x_nu'_n(x_n)+{x_n^2\over 2}u''_n(x_n)\,.
\label{e10}
\end{equation}
This expression is negative because
$$
0=u_n(0)=u_n(x_n)-x_nu'_n(x_n)+\int_0^{x_n} dx \int_x^{x_n}
u''_n(t) dt
$$
where the last term in the right-hand side of this equality is
larger than the last term in (\ref{e10}), due to our choice of $\overline u$
and (\ref{e1}). We have defined
a function $g_1(x)$ satisfying $g_1(0)<0$.
Take now a sufficiently small $\Delta \in (0,{x_n\over 2})$ and
a continuous function $\omega _2(x)\leq 0$ which is equal to $u'''_n(x)$ for
$x\in [x_n,{1\over 2(n+1)}]$ and to $0$ for $x\in [0,x_n-\Delta )$, such that
$$
\int_{x_n-\Delta }^{x_n}\omega _2(x) dx=u''_n(x_n).
$$
Then,
defining the function $g_2(x)$ just as $g_1(x)$ in (\ref{e9}) with the substitution
of $\omega _2$ in place of $\omega _1$ and of $g_2$ in place of $g_1$, we get
that if $\Delta >0$ is sufficiently small, then $g_2(x_n-\Delta )$ and
$g'_2(x_n-\Delta )$ are arbitrary close, respectively, to $u_n(x_n)$ and
$u'_n(x_n)$, and $g''_2(x)=0$ for $0\leq x\leq x_n-\Delta $. Therefore, due
to (\ref{e8}), $g_2(0)>0$ if $\Delta >0$ is sufficiently small, for all
sufficiently large $n$. We have defined a function $g_2(x)$ satisfying $g_2(0)>0$.
Now, consider the family of functions $g_\lambda (x)=\lambda g_1(x)+
(1-\lambda )g_2(x)$ where $\lambda \in [0,1]$. Clearly, there exists a
unique $\lambda _0\in (0,1)$ such that $g_{\lambda _0}(0)=0$.
Extend $g_{\lambda _0}(x)$ continuously on the entire real line by the rules:
$$
g_{\lambda _0}({1\over n+1}+x)=-g_{\lambda _0}(x),\quad
g_{\lambda _0}( {1\over 2(n+1)}+x) =g_{\lambda _0}(
{1\over 2(n+1)}-x)
$$
and denote the obtained function by $h_n(x)$.
This function satisfies conditions (a) and (b) of Theorem \ref{thm1}.
In addition, by Theorem \ref{thm1}(b), $h''_n(x_n)\leq h''_n(x)\leq 0$ for all
$x\in [0,x_n)$.
So far, we have constructed $h_n$ for $n$ sufficient large.
For small values of $n$, we use arbitrary functions $h_n$ satisfying
the conditions of Theorem \ref{thm1}. Therefore the sequence
$\{h_n\}$ ($n=0,1,2,\dots$) satisfies the conditions (a) and (b) of
Theorem \ref{thm1}.
Let $\alpha _n=[ h_n({1\over 2(n+1)}) ]^{-1}$. Then, by
Theorem \ref{thm1}, the system $\{\alpha _nh_n\}$ is a Riesz basis in
$L_2$. Furthermore, by Lemma \ref{lm3}, the system $\{\alpha _n u_n\}$ is
$\omega $-linearly independent in $L_2$. Also, due
to (\ref{e1}) and by construction, there exists $C_1>0$ such that
$$
|u''_n(x)|=\max_{u\in [0,\overline u]} |u f(u^2)|\leq C_1
$$ and
$$
\max_{x\in [0,x_n]} |h''_n(x)|=|h''_n(x_n)|=|u''_n(x_n)|=|\overline
u f(\overline u^2)|\leq C_1
$$
for all $n$ sufficiently large. Hence,
$$
|u'_n(x)-h'_n(x)|\leq C_2x_n
$$
for all $n$ sufficiently large and all $x\in [0,x_n]$. Hence, due to
(\ref{e5}),
$$
\|\alpha _nu_n-\alpha _nh_n\|^2\leq C_3x_n^4\leq C_4(n+1)^{-4}
$$
for all $n$ sufficiently large.
Therefore, the systems $\{\alpha _nu_n\}$ and $\{\alpha _n
h_n\}$ are quadratically close in $L_2$. In view of
Bari's Theorem, the proof of Theorem \ref{thm3} is complete.
\paragraph{Acknowledgment} The author is thankful to
Mrs. G. G. Sandukovskaya for editing the original manuscript.
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\noindent\textsc{Peter E. Zhidkov}\\
Bogoliubov Laboratory of Theoretical Physics, \\
Joint Institute for Nuclear Research, \\
141980 Dubna (Moscow region), Russia \\
e-mail: zhidkov@thsun1.jinr.ru
\end{document}
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