\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \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} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \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: $$h_n(\cdot )=\sum_{m=0}^\infty a^n_m e_m(\cdot ) \quad\mbox{in } L_2, \label{e3}$$ 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, 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 u'_n(x_n)x_n u'_n(x_n); $$therefore, $$u'_n(x_n)<{\overline u\over x_n}=l'_n(x) \label{e8}$$ for all sufficiently large n. Take a sufficiently small \Delta \in (0,{x_n\over 2}) and define a continuous function \omega _1(x) equal to u'''_n(x) for x\in [x_n, {1\over 2(n+1)}] such that u'''_n(x)\leq \omega _1(x) \leq 0 for x\in [x_n-\Delta ,x_n] and \omega _1(x)=0 for x\in [0,x_n -\Delta ). We define g_1(x) to be equal to u_n(x) for x\in [x_n,{1\over 2(n+1)}], and for x\in [0,x_n) to be given by the rules: \begin{eqnarray} g''_1(x)&=& u''_n(x_n)-\int_x^{x_n}\omega _1(t) dt, \nonumber\\ g'_1(x) &=& u'_n(x_n)-\int_x^{x_n}g''_1(t) dt, \label{e9} \\ g_1(x) &=& u_n(x_n)-\int_x^{x_n} g'_1(t) dt. \nonumber \end{eqnarray} Then g_1(x) is three times continuously differentiable in [0,{1\over 2(n+1)}] and satisfies condition (b) of Theorem \ref{thm1}. It is easy to see that if \Delta >0 is sufficiently small, then g_1(x_n-\Delta ) and g'_1(x_n-\Delta ) are arbitrary close to u_n(x_n) and u'_n(x_n), respectively, and g''_1(x) is arbitrary close to u''_n(x_n) for all x\in [0,x_n-\Delta ]. Now, due to our choice of \overline u>0, for \Delta >0 and sufficiently small, g_1(0) is arbitrary close to $$u_n(x_n)-x_nu'_n(x_n)+{x_n^2\over 2}u''_n(x_n)\,. \label{e10}$$ This expression is negative because$$ 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. \begin{thebibliography}{20} \frenchspacing \bibitem{b1} Bari, N.K., Biorthogonal systems and bases in Hilbert space. {\it Moskov. Gos. Univ. 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Eqns.}, 2000, {\bf 2000}, No 28, 1-13. \bibitem{z6} Zhidkov, P.E. An analog of the Fourier transform associated with a nonlinear one-dimensional Schr\"odinger equation. {\it Preprint of the JINR}, E5-2001-63, Dubna, 2001; to appear in {\it Nonlinear Anal.: Theory, Methods and Applications}''. \bibitem{z7} Zhidkov, P.E. On the property of being a basis for a denumerable set of solutions of a nonlinear Schr\"odinger-type boundary-value problem. {\it Nonlinear Anal.: Theory, Methods and Applications}, 2001, {\bf 43}, No 4, 471-483. \end{thebibliography} \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}