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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)}
 \vspace{\bigskipamount} \\
 %
  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}
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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 $s<s_0$ any standard sequence of solutions
$\{u_n\}$ is a basis in $H^s(0,1)$. In addition, the sequence
$\{ u_n/\|u_n\|_{H^s(0,1)}\}$ is a Riesz basis in $H^s(0,1)$.
Here $H^s(0,1)$ is the usual Sobolev space with negative index $s$.
In the present paper, we improve this result by showing the above
properties of a standard system $\{u_n\}$ in $L_2$ (see
Theorem \ref{thm3} below), by first obtaining a general
result on bases in $L_2$ (see Theorem \ref{thm1} below).
We believe that this result is of a separate interest.

\subsection*{Notation}
By $c,C,C_1,C_2,C',C'',\dots $ we denote positive constants.
By $L_2(a,b)$ we denote the standard Lebesgue space  of
square integrable functions on the interval $(a,b)$.
In this space we introduce the standard inner product, and norm:
$$(g,h)_{L_2(a,b)}=\int_a^b g(x)h(x) dx\,, \quad
\|g\|_{L_2(a,b)}=(g,g)_{L_2(a,b)}^{1/2}\,.
$$
For short notation we will use $(\cdot ,\cdot )$ and
 $\|\cdot \|$ respectively.

Let $l_2$ be the space of square summable sequences
of real numbers. For a Banach space $X$
with a norm $\|\cdot \|_X$, let ${\cal L} (X;X)$ be the linear space of
linear bounded operators acting from $X$ into $X$, equipped with the norm
$$\|A\|_{{\cal L}(X;X)}=\sup_{x\in X: \ \|x\|_X=1}\|Ax\|_X\,.$$
We also set $\|\cdot \|=\|\cdot \|_{{\cal L}(L_2;L_2)}$ for short notation.

Now, for convenience of readers, we define some well-known terms.


\paragraph{Definition}
A system $\{e_n\}\subset L_2(a,b)$
is called a basis in $L_2(a,b)$ if for any $g\in L_2(a,b)$
there exists a unique sequence $\{a_n\}$ of real numbers such that
$g=\sum_{n=0}^\infty a_n e_n$ in $L_2(a,b)$.


There are several definitions of Riesz bases. In accordance with
the classical paper by N. K. Bari \cite{b1}, where this concept was
introduced for the first time, we use the following definition.


\paragraph{Definition}
A basis $\{e_n\}$ in $L_2(a,b)$ is called a Riesz basis in this space
when  the series
$\sum_{n=0}^\infty a_n e_n$, with real coefficients $a_n$, converges
in $L_2(a,b)$ if and only if  $\sum_{n=0}^\infty a_n^2<\infty $.


\paragraph{Remark}
It is proved in \cite{b1} (see also \cite{z2})
that if $\{e_n\}$ is a Riesz basis in $L_2(a,b)$ in the sense of
this definition, then there exist constants $0<c<C$ such that
$$
c\sum_{n=0}^\infty a_n^2\leq \Big\| \sum_{n=0}^\infty
a_ne_n \Big\| ^2_{L_2(a,b)}\leq C\sum_{n=0}^\infty a_n^2
$$
for all $\overline a=(a_0,a_1,a_2,\dots )\in l_2$. These estimates have been often used
to define Riesz bases.


\paragraph{Definition}
 A system of functions  $\{g_n\}$ in $L_2(a,b)$ is
called $\omega $-linearly independent in $L_2(a,b)$ when
$\sum_{n=0}^\infty a_ng_n=0$, with $a_n$ are real numbers,
holds in $L_2(a,b)$  if and only if $0=a_0=a_1=a_2=\dots $.


\paragraph{Definition}
Two systems of functions $\{h_n\}$ and $\{e_n\}$  in  $L_2(a,b)$
are called quadratically close in $L_2(a,b)$, if
$\sum_{n=0}^\infty \|h_n-e_n\|^2_{L_2(a,b)}<\infty $.


\subsection*{Results}

\begin{theorem} \label{thm1}
 Let $\{h_n\}$ be a system of real-valued,
three-times continuously differentiable functions.
Assume that for each integer $n\geq 0$ the following holds:
\begin{enumerate}
\item[(a)]  $h_n\big(x+{1\over n+1}\big)=-h_n(x)$ and
$h_n\big({1\over 2(n+1)}+x\big)=h_n\big({1\over 2(n+1)}-x\big)$
for all $x\in \mathbb{R}$

\item[(b)] $h'_n(x)>0$, $h''_n(x)\leq 0$, and $h'''_n(x)\leq 0$ for all
$x\in \big(0,{1\over 2(n+1)}\big)$

\item[(c)] There exist $0<c<C$ such that $c<h_n\big({1\over 2(n+1)}\big)<C$
for all $n$.
\end{enumerate}
Then, the system $\{h_n\}$ is a Riesz basis in $L_2$.
\end{theorem}


\paragraph{Remark}
Clearly, it follows from Theorem \ref{thm1} that if a system of
functions $\{h_n\}$ satisfies all the conditions of this theorem,
except maybe  (c), then it is a basis in $L_2$.


The next result follows from Theorem \ref{thm1} by taking $h_n(x)=
h((n+1)x)$.

\begin{theorem} \label{thm2}
Let $h(x)$ be a real-valued three-times continuously
differentiable function satisfying:
\begin{enumerate}
\item[(a)] $h(1+x)=-h(x)$ and $h(1/2+x)=h(1/2-x)$ for all $x\in \mathbb{R}$
\item[(b)] $h'(x)>0$, $h''(x)\leq 0$ 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 $0<c\leq |a^n_n|\leq C$.
Then to prove this theorem, it suffices to prove that the system
$\{\overline h_n
\}$ with $\overline h_n=(a^n_n)^{-1}h_n$ is a Riesz basis in $L_2$.


\begin{lemma} \label{lm3}
Let $\{g_n\}$ be a sequence of functions
such that  $g_n$ satisfies condition (a) on Theorem \ref{thm1}
and is positive in $( 0,{1\over n+1} )$. Then the system
$\{g_n\}$ is $\omega$-linearly independent in $L_2$.
\end{lemma}

The proof of this lemma is rather simple. We refer the reader to the
proof of similar  statements in \cite{z2,z5,z7}. \hfill $\Box$ \smallskip

Let $b^n_m=(a^n_n)^{-1}a^n_m$, and let $\mathop{\rm Id}$ be the unit operator
in $L_2$. For positive integers $m$, let $B_m$
be the operator mapping $e_n$ into $b^n_{(n+1)(m+1)-1}e_{(n+1)(m+1)-1}$,
$B_m\in {\cal L}(L_2;L_2)$.
Also let $B=\sum_{m=1}^\infty B_m$. Then for each $m$,
$$
\|B_m\|\leq \sup_n |b^n_{(n+1)(m+1)-
1}|=b_m.
$$
Furthermore, by Lemma 2,
$$
\sum_{m=1}^\infty b_m\leq {\pi \over 2}
\sum_{l=1}^\infty (2l+1)^{-2}\leq {\pi \over 2}
\int_{1/2}^\infty (2x+1)^{-2} dx =\pi /8\,;
$$
hence, $B\in {\cal L}(L_2;L_2)$ and $\|B\|
\leq \pi /8<1$. Therefore, the operator $A=\mathop{\rm Id}+B$ has a bounded inverse
$A^{-1}=\mathop{\rm Id}+\sum_{n=1}^\infty (-1)^nB^n$. Note also that
$Ae_n=\overline h_n$. Hence, as proved in \cite{g1}, $\{\overline h_n\}$
is a Riesz basis in $L_2(0,1)$.
For the convenience of the reader, we present a short proof of this statement.

Take an arbitrary $v\in L_2$ and let $u=A^{-1}v=\sum_{n=0}
^\infty c_ne_n\in L_2$ where $c_n$ are real coefficients. Then,
$\sum_{n=0}^\infty c_n^2<\infty $ because $\{e_n\}$ is an
orthonormal basis in $L_2$.
Since the series $\sum_{n=0}^\infty c_ne_n$ converges in $L_2$, we have
$v=Au=\sum_{n=0}^\infty c_nAe_n=\sum_{n=0}^\infty c_n\overline h_n$
where all infinite sums also converge in $L_2$.
Therefore, in view of Lemma \ref{lm3}, the system $\{\overline h_n\}$
is a basis in $L_2$ and, if $\sum_{n=0}^\infty c_n^2<\infty $, then the
series $\sum_{n=0}^\infty c_nh_n$ converges in $L_2$.
Conversely, let a series $u=\sum_{n=0}^\infty c_n\overline h_n$ converge in
$L_2$.
Then  $A^{-1}u=\sum_{n=0}^\infty c_ne_n$ in $L_2$; hence
$\sum_{n=0}^\infty c_n^2<\infty $. Thus, $\{\overline h_n\}$
is a Riesz basis in $L_2$, and the proof of Theorem \ref{thm1} is
complete.

\section{Proof of Theorem \ref{thm3}}


As was proved in \cite{z7}, any solution $u_n$
of problem (\ref{e1})--(\ref{e2}), that possesses precisely $n$ zeros
in $(0,1)$, satisfies condition (a) of Theorem \ref{thm1}.
In addition, $u_n$ is strictly monotone ($u'_n(x)\not =0$) in the interval
$(0,{1\over 2(n+1)})$.
Let $\overline u>0$ 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}
0<x_n\leq (n+1)^{-1}\big[u_n({1\over 2(n+1)}) \big]^{-1}
\label{e5}
\end{equation}
for all sufficiently large $n$.

Using $u_n$ and $n$ large, we now want to construct a function $h_n$
that satisfies the conditions of Theorem \ref{thm1}.
Introduce the linear function $l_n(x)={\overline u\over x_n}x$ which is
equal to $0$ at $x=0$ and to $\overline u=u_n(x_n)$ at $x=x_n$.
Multiply (\ref{e1}), with $u=u_n$, by $2u'_n(x)$ and integrate the result
from $0$ to $x$. Then
\begin{equation}
\{[u'_n(x)]^2+F(u^2_n(x))\}'=0, \quad  x\in \mathbb{R},
\label{e6}
\end{equation}
where $F(s)=-\int_0^s f(t) dt$. Due to condition (F),
$F(u^2)\to +\infty $ as $u\to \infty $,
therefore, without loss of generality, we can assume that $\overline u>0$
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)<u'_n(x), \quad x\in [0,x_n),
\label{e7}
\end{equation}
for all sufficiently large $n$. By (\ref{e7}), we have
$$
\overline u=\int_0^{x_n}u'_n(x) dx >x_n u'_n(x_n);
$$
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}