\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Basis properties of eigenfunctions \hfil EJDE--2000/28} {EJDE--2000/28\hfil P. E. Zhidkov \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~28, pp.~1--13. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems \thanks{ {\em Mathematics Subject Classifications:} 34L10, 34L30, 34L99. \hfil\break\indent {\em Key words and phrases:} Riesz basis, nonlinear eigenvalue problem, \hfill\break\indent Sturm-Liouville operator, completeness, basis. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted November 17, 1999. Published April 13, 2000.} } \date{} % \author{ P. E. Zhidkov } \maketitle \begin{abstract} We consider three nonlinear eigenvalue problems that consist of $$-y''+f(y^2)y=\lambda y$$ with one of the following boundary conditions: $$\displaylines{ y(0)=y(1)=0 \quad y'(0)=p \,,\cr y'(0)=y(1)=0 \quad y(0)=p\,, \cr y'(0)=y'(1)=0 \quad y(0)=p\,, }$$ where $p$ is a positive constant. Under smoothness and monotonicity conditions on $f$, we show the existence and uniqueness of a sequence of eigenvalues $\{\lambda _n\}$ and corresponding eigenfunctions $\{y_n\}$ such that $y_n(x)$ has precisely $n$ roots in the interval $(0,1)$, where $n=0,1,2,\dots$. For the first boundary condition, we show that $\{y_n\}$ is a basis and that $\{y_n/\|y_n\|\}$ is a Riesz basis in the space $L_2(0,1)$. For the second and third boundary conditions, we show that $\{y_n\}$ is a Riesz basis. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \section{Introduction} We consider three eigenvalue problems which consist of the nonlinear equation $$-y''+f(y^2)y=\lambda y \quad\mbox{on }(0,1) \eqno (1)$$ with one of the following three boundary conditions: $$\displaylines{ \hfill y(0)=y(1)=0 \quad y'(0)=p\,, \hfill\llap{(2a)} \cr \hfill y'(0)=y(1)=0 \quad y(0)=p\,, \hfill\llap{(2b)} \cr \hfill y'(0)=y'(1)=0 \quad y(0)=p\,.\hfill\llap{(2c)} }$$ Hereafter all quantities are real, including $\lambda$ which is a spectral parameter and $p$ which is an arbitrary constant. However, we consider only $p>0$, because when $p<0$ the substitution of $y(x)$ by $-y(x)$ leads to a positive $p$. We assume that $f$ is a continuously differentiable function on ${\mathbb R}$; so that the standard theorems, such as local existence, uniqueness, and continuous dependence on parameters are valid for (1). A pair $(\lambda,y)$ with $\lambda$ real and $y=y(x)$ twice continuously differentiable for all $x\in [0,1]$, that satisfies (1) and one of the boundary conditions (2) is called an {\it eigenvalue} and corresponding {\it eigenfunction}. Since each of the boundary conditions (2) contains Cauchy data and an additional condition, by the standard uniqueness theorem to each value $\lambda$ there corresponds at most one eigenfunction. By the {\it spectrum} we mean the set of all eigenvalues for each problem defined by (1)-(2). For any $g\in L_2=L_2(0,1)$, by the {\it normalized function} $g$ we mean the normalization of this function to $1$ in the above space. In the present paper, we investigate the properties of the eigenvalues and eigenfunctions of each problem defined by (1)-(2). In particular, we prove that for problems (1)-(2b) and (1)-(2c) the eigenvectors form Riesz bases, and that the normalized eigenvectors of problem (1)-(2a) also form a Riesz basis. For problem (1)-(2b), these questions have already been considered in \cite{z1}. The proof that eigenfunctions form a Riesz basis was done using Bary's theorem. In the present paper the author establishes a direct proof (not based on Bary's theorem) of the result in \cite{z1}, and then studies problems (1)-(2a) and (1)-(2c). According to Bary's theorem \cite{b1,b2,g1,m1}, a linearly independent system of functions $\{h_n\}$ which is $L_2(a,b)$ quadratically close to a Riesz basis in this space is a Riesz basis in $L_2(a,b)$. This result in \cite{b1,b2} is presented in a weaker form. However the proof in \cite{b2} applies without modifications to the above statement. Note that an orthonormal basis in $L_2(a,b)$ is also a Riesz basis in $L_2$ (see definitions in Section 2). A Riesz basis that is quadratically close to an orthonormal basis is called Bary basis \cite{g1}. Important properties of Riesz and Bary bases are presented in \cite{b1,b2,g1}. In the present paper, we shall prove linear independence and quadratic closeness of the normalized eigenfunctions of all problems (1)-(2) to orthonormal bases in the space $L_2$. We also obtain estimates of the form $\overline c<\|y_n\|_{L_2}<\overline C$ for the eigenfunctions of problems (1)-(2b) and (1)-(2c) with constants $0<\overline c<\overline C$ independent of $n$. Thus, eigenfunctions of problems (1)-(2b) and (1)-(2c) and normalized eigenfunctions of the problem (1)-(2a) are Riesz bases. Moreover, the latter system is a Bary basis in $L_2$. We present a direct simple and short proof, not based on Bary's theorem, of these systems being Riesz bases in $L_2$. Several papers have presented proofs of completeness, of being a basis, and of other properties for eigenfunctions of nonlinear boundary-value problems; see for example \cite{m1}. In \cite{z2}, the main result consists of proving that the eigenfunctions of a nonlinear Sturm-Liouville-type problem form a basis in $L_2$. These results are also announced (without proofs) in \cite{z3}. However, \cite{z2} contains errors which have been corrected in \cite{z4}, where also an analog of the Fourier transform over eigenfunctions of nonlinear Sturm-Liouville-type problems on a half-line is considered. An independent (and shorter) proof based on the Bary's theorem of the result from \cite{z2} is done in \cite{z5}. Lately, in \cite{z6} it is shown that the eigenfunctions form a basis in $H^s$, where $s$ is a negative constant. This is done for a boundary-value similar to (1), without the spectral parameter, and with zero Dirichlet boundary conditions. Concerning the applications of our results, the author hopes that once developed our results be used in Galerkin and Fourier methods. In addition, our results could be applied in those numerous areas of quantum physics where various modifications of the nonlinear Schr\"odinger equation arise. \section{Main results} First we introduce some notation. Let $L_2=L_2(0,1)$ (or $L_2(a,b)$ where $a0$ has the points $1/(2n) + k/n$ as roots; while $y_0$ is identically equal to $p$, thus it has no roots. In each of these three cases $k=0,\pm 1,\pm 2,\dots$. \item{(f)} For any eigenfunction $y$ of problems (1)-(2) and for all $x\in{\mathbb R}$, $y(c+x)=-y(c-x)$ if $y(c)=0$, and $y(d-x)=y(d+x)$ if $y'(d) =0$. \item{(g)} The eigenvalues of problems (1)-(2) satisfy $\lambda _0<\lambda _1<\lambda _2<\dots$. \end{description} \end{theorem} Parts of Theorem 1 are known (see, for example, \cite{k1}) but the author does not know a reference for all of this theorem. The author does not know of any function $g$ that is an eigenfunction of a problem (1)-(2) and is representable as a superposition of elementary functions. The main result of the paper is stated as follows. \begin{theorem} \label{thm2} Under Assumption {\rm (F)} the eigenfunctions of problems (1)-(2b) and (1)-(2c) form Riesz bases in $L_2$. Also under Assumption {\rm (F)}, the eigenfunctions of problem (1)-(2a) form a basis in $L_2$ and they form a Riesz basis after being normalized. \end{theorem} \paragraph{Remark 1.} It follows from (17) in the proof of Theorem 2 that for problem (1)-(2a), $\|y_n\|\to 0$ as $n\to \infty$. Therefore, this sequence of eigenfunctions is not a Riesz basis in $L_2$. \section{Proofs} First we prove the statement of Theorem 1 for problem (1)-(2a). Consider the Cauchy problem $$\displaylines{ \hfill -y''+f(y^2)y=\lambda y \hfill\llap{(3)} \cr \hfill y(0)=0, \quad y'(0)=p\,. \hfill\llap{(4)} }$$ Since $p>0$, for any $\lambda \in {\mathbb R}$ the solution $y$ satisfies $y'(x)>0$ in a neighborhood of zero. It is well known that equation (3) can be solved by quadratures and our proof of Theorem 1 is based on this fact (see below). Clearly, if a solution $y(x)$ of the problem (3)-(4) can be continued onto a segment $[0,d]$, and if $y'(x)>0$ for $x\in [0,d)$, then the inverse function $x(y)$ with domain $[0,y(d)]$ has the form $$x(y)=\int_0^y \frac {dr}{\sqrt {U(\lambda ,r^2)}}\,, \eqno (5)$$ where $U(\lambda ,r^2)=p^2+F(r^2)-\lambda r^2$, $F(r)=\int_0^r f(t)\, dt$, $y\in [0,y(d)]$, and $x(y(d))=d$. Note that if $y'(d)=0$, then $x'(y(d)-0)=+\infty$. If there exists $\overline \lambda$ such that for $\lambda >\overline \lambda$ the function $U(\lambda ,r^2)$ is positive in a half-open interval $r\in [0,b)$, where $b>0$, then the solution $y$ is defined on the closed interval $[0,x(b)]$. In this case $y(x(b))=b$, $y'(x)>0$ for $x\in [0,x(b))$ and the function $x(y)$ with the domain $[0,b]$ is inverse of $y(x)$ on $[0,x(b)]$. In addition, $y'_x(x(b))=0$ if $x(b)<+\infty$ and $x'(b-0)=+\infty$. We denote by $\Lambda$ the set of values $\lambda$ for which there exists $r>0$ such that $U(\lambda ,r^2)=0$. In view of condition {\rm (F)}, $\inf \Lambda \geq f(0)$. It is clear that if $\lambda \in \Lambda$ and $\tilde {\lambda} > \lambda$ then $\tilde {\lambda}$ also belongs to $\Lambda$. Therefore, either $\Lambda =(\overline \lambda,+\infty)$ (the case A) or $\Lambda =[\overline \lambda,+\infty)$ (the case B) for some $\overline \lambda\geq f(0)$. For each $\lambda \in \Lambda$ we denote by $z(\lambda)$ the greatest lower bound of the set of values $r>0$ for which $U(\lambda ,r^2)=0$. Clearly, $z(\lambda)>0$ and $U(\lambda ,z^2(\lambda ))=0$. Furthermore, $z(\lambda )$ is a monotonically decreasing function. Therefore, if $\lambda >\overline \lambda$, then $$[U(\lambda ,r^2)]'_r\big|_{r= z(\lambda)}=2z(\lambda)f(z^2(\lambda))-2\lambda z(\lambda)< 2z(\lambda )(f(z^2(\lambda '))-\lambda ')\leq 0$$ for any $\lambda '\in (\overline \lambda,\lambda )$ (because $0\geq U'_r(\lambda ',r^2)\big|_{r=z (\lambda ')}=2z(\lambda ')(f(z^2(\lambda '))-\lambda ')$, where $z(\lambda ')>0$); hence by the implicit function theorem $z(\lambda)$ is a continuously differentiable function and $z'(\lambda)<0$ for $\lambda >\overline \lambda$. Also, in the case B $\lim_{\lambda \to \overline {\lambda}+0} z(\lambda )=z(\overline {\lambda} )$. Let us prove now that for any $d>0$ there exists $\lambda >\overline \lambda$ such that $$J(\lambda)=\int_0^{z(\lambda)}\frac {dr}{\sqrt {U(\lambda ,r^2)}}=d\,, \eqno (6)$$ where the integral in the right-hand side is understood as improper in a neighborhood of the point $r=z(\lambda )$. Note that as indicated above, $[U(\lambda ,r^2)]'_r\big|_{r=z(\lambda)}<0$ for $\lambda >\overline \lambda$; thus for $\lambda >\overline \lambda$ the improper integral in the right-hand side of (6) converges, and $J(\lambda)$ is a continuous function. Let us consider the case A: when $\Lambda =(\overline \lambda,+\infty)$. It is clear that $z(\overline \lambda+0)=+\infty$, because otherwise in view of the continuity of the function $U(\lambda ,r^2)$ of the argument $(\lambda,r)$ we should get that $\overline \lambda \in \Lambda$, and that $z(+\infty)=0$. This easily implies that $\overline \lambda= \lim_{r\to \infty}f(r^2)<+\infty$ and the latter fact yields that $$\lim_{\lambda \to \overline \lambda+0} J(\lambda)=+\infty \quad {\rm and} \quad \lim_{\lambda \to +\infty} J(\lambda)=0\,. \eqno (7)$$ Therefore, in case A the existence of $\lambda >\overline \lambda$ such that (6) holds is proved. Let us consider the case B: when $\Lambda =[\overline \lambda,+\infty)$. We have as in the case A $\lim_{\lambda \to + \infty} J(\lambda )=0$. Hence, in view of the continuity of the function $J(\lambda)$, we have only to prove that $\lim_{\lambda \to \overline \lambda+0} J(\lambda)=+\infty$. It follows from the implicit function theorem that $[U(\overline \lambda ,r^2)]'_r\big|_{r=z(\overline \lambda)}=0$ (because otherwise by this theorem there exist values of $\lambda$ smaller than $\overline \lambda$ and belonging to $\Lambda$). This fact implies the required property because $\lim_{\lambda \to \overline \lambda+0}J(\lambda) \geq J(\overline \lambda)$ (indeed, $U(\overline \lambda,r^2)>0$ for $r\in (0,z(\overline \lambda))$ and it is clear that for any $y\in (0,z(\overline \lambda ))$ $$\int_0^y \frac {dr}{\sqrt {U(\lambda ,r^2)}} \to \int_0^y \frac {dr}{\sqrt {U(\overline \lambda ,r^2)}}$$ as $\lambda \to \overline \lambda +0$ and also, $U(\overline \lambda,r^2)= {\rm O}((z(\overline \lambda)-r)^2)$ as $r\to z(\overline \lambda)-0$). In view of the above arguments, it is proved that for any $d>0$ there exists $\lambda \in {\mathbb R}$ such that the corresponding solution $y(x)$ of the problem (3)-(4) can be continued on the segment $[0,d]$, $y'(x)>0$ for $x\in [0,d)$ and $y'(d)=0$ where $y(d)=z(\lambda )$. Let us prove that for any $d>0$ there exists precisely one value $\lambda$ possessing this property. On the contrary, suppose that for some $d>0$ there exist two values $\lambda _1<\lambda _2$ such that the corresponding solutions $y_1(x)$ and $y_2(x)$ of the problem (3)-(4) can be continued on the segment $[0,d]$, $y'_i(x)>0$ for $x\in [0,d)$ and $y'_i(d)=0$ ($i=1,2$). In view of the above arguments $\lambda _1>\overline \lambda$. Multiply equation (3), with $y(x)=y_1(x)$, by $y_2(x)$. Multiply equation (3), with $y(x)=y_2(x)$, by $y_1(x)$. Then subtract these equalities from each other and integrate over the segment $[0,d]$. In view of the initial data (4), applying the integration by parts, we get $$0=\int_0^d y_1(x) y_2(x) [\lambda _1-\lambda _2-f(y_1^2(x))+f(y_2^2 (x))] dx\,. \eqno (8)$$ By (5) the inverse functions $x_1(y)$ and $x_2(y)$ of $y_1(x)$ and $y_2(x)$ respectively, satisfy $x_1(y)y_2(x)$ for all $x\in (0,d]$. Then, in view of the assumption {\rm (F)}, the right-hand side in (8) is negative, which is a contradiction. Thus, it is proved that for any $d>0$ there exists a unique $\lambda$ such that the corresponding solution $y(x)$ of the problem (3)-(4) satisfies $y'(x)>0$ if $x\in [0,d)$ and $y'(d)=0$. Finally, in view of the autonomy of the equation (3) and its invariance with respect to the changes of variables $y(x)\to -y(x)$ and $x\to c-x$, where $c$ is an arbitrary constant, we obtain that if a solution $y(x)$ of the problem (3)-(4) can be continued onto the segment $[0,d]$ and satisfies the conditions $y'(x)>0$ for $x\in [0,d)$ and $y'(d)=0$, then it can be continued onto the whole real line, and $y(d+x)=y(d-x)$ (in particular $y(2d)=0$) and $y(2kd+x)=-y(2kd-x)$ for all $x\in{\mathbb R}$ and for $k=0,\pm 1, \pm 2,\dots$. Therefore, Theorem 1, except the statements (d) and (g), for the problem (1)-(2a) is proved. The statement (d) follows from the fact that the function $z(\lambda)$ for $\lambda >\overline \lambda$, coinciding with $y(d)$, decreases monotonically. Since, as it is proved earlier, for any $d >0$ there exists a unique $\lambda \in{\mathbb R}$ (where $\lambda >\overline \lambda$) such that $y'(x)>0$ for $x\in [0,d)$ and $y'(d)=0$, where $y(x)$ is the solution of the problem (3)-(4) corresponding to this value of the parameter $\lambda$, and since $d>0$ continuously depends on $\lambda$ (because $d=J( \lambda)$), by the properties (7), which, as it is proved earlier, hold in each of the cases A and B, the value $d>0$ is a monotonically decreasing function of the argument $\lambda \in (\overline \lambda,+\infty)$. The statement (g) of Theorem 1 for the problem (1)-(2a) immediately follows from this fact and the above arguments. For problems (1)-(2b) and (1)-(2c) the proof of the statements of Theorem 1 can be made by analogy. Thus, Theorem 1 is proved.\hfill $\diamondsuit$ \smallskip Next, we turn to the proof of Theorem 2. We associate with three problems (1)-(2), respectively, the following three linear eigenvalue problems: $$\displaylines{ \hfill -u''=\mu u, \quad x\in (0,1), \quad u=u(x),\quad u(0)=u(1)=0, \hfill\llap{(9a)}\cr \hfill -u''=\mu u, \quad x\in (0,1), \quad u=u(x),\quad u'(0)=u(1)=0, \hfill\llap{(9b)}\cr \hfill -u''=\mu u, \quad x\in (0,1), \quad u=u(x), \quad u'(0)=u'(1)=0, \hfill\llap{(9c)} }$$ where all quantities are real and $\mu$ is the spectral parameter. For each of the problems (9) by $\{e_n\}$ ($n=0,1,2,\dots$) we denote the orthonormal basis consisting of its normalized eigenfunctions. More precisely, $e_n(x)=\sqrt {2}\sin \pi (n+1)x$ for problem (1)-(2a), $e_n(x)=\sqrt {2}\cos{\pi (2n+1)x \over 2}$ for problem (1)-(2b) (with $n=0,1,2,\dots$), and $e_0(x)\equiv 1$, $e_n(x)=\sqrt {2}\cos \pi nx$ (with $n=1,2,3,\dots$) for the problem (1)-(2c). The corresponding eigenvalues are the numbers $\mu _n=(\pi (n+1))^2$ for problem (9a), the numbers $\mu _n=(\pi (2n+1)/2)^2$ for problem (9b), and the numbers $\mu _n=(\pi n)^2$ for problem (9c) ($n=0,1,2,\dots$). For each of the problems (1)-(2) we set $v_n(x)=y_n(x)/ \|y_n\|$. \begin{lemma} \label{Lemma1} For each of the problems (1)-(2) and an arbitrary integer $n\geq 0$ the following expansion in the Fourier series, understood in the sense of the space $L_2$, takes place $$v_n(x)=\sum_{k=0}^\infty d_{n,k}e_k,$$ where coefficients $d_{n,k}$ are real, $d_{n,0}=\dots =d_{n,n-1}=0$ and $d_{n,n}>0$. \end{lemma} \paragraph{Proof.} For arbitrary integers $n\geq 0$ consider the spaces $L_2(0,I_n)$, where $I_n=1/(n+1)$ for problem (1)-(2a), and $I_n=1/(2n+1)$ for problem (1)-(2b) (i.\ e., $I_n$ is the smallest positive root of $v_n(x)$, see Theorem 1(e)). It is clear that in each of these two cases functions from the system $\{e_k\}$ equal to zero at the point $x=I_n$ ($k=0,1,\dots$) form an orthogonal basis of the space $L_2(0,I_n)$. We also note that for any integer $n\geq 0$ the minimal integer $k\geq 0$, for which $e_k(I_n)=0$, is $k=n$. Hence, $$v_n(x)=\sum_{k=0}^\infty d_{n,k}e_k, \quad n=0,1,2,\dots \eqno (10)$$ in $L_2(0,I_n)$, where $d_{n,k}=0$ if $e_k(I_n)\not =0$, and in view of the above arguments $d_{n,0}=\dots =d_{n,n-1}=0$. In addition, $d_{n,n}>0$ because $v_n(x)>0$ and $e_n(x)>0$ for $x\in (0,I_n)$. We also note that in the case of the problem (1)-(2b) the functions $v_n(x)$ and $e_k(x)$ are even with respect to the point $x=0$, therefore the expansions (10) also hold in the space $L_2(-I_n,I_n)$. Further, one can easily verify that if $e_k(I_n)=0$ for some value of the index $k$, then this function $e_k(x)$, continued for all $x\in{\mathbb R}$, is equal to zero at all points of the real line which are roots of the function $v_n(x)$. Therefore, since clearly any function $e_k(x)$ is odd with respect to an arbitrary its root, by Theorem 1(f) we easily get that expansions (10) of the functions $v_n(x)$ ($n=0,1,2,\dots$) take place in each space $L_2(I)$, too, where $I$ is the interval between two arbitrary nearest roots of the function $v_n(x)$. Hence, the expansions (10) are also valid in the space $L_2$. For problem (1)-(2c) using the same procedure, one can obtain $v_n(x)=\sum_{k=0}^\infty d_{n,k}e_k(x)$, $n=1,2,3,\dots$, in the space $L_2$, where $d_{n,0}=\dots =d_{n,n-1}=0$ and $d_{n,n}>0$, and, in addition, that $v_0(x)=pe_0(x)$. Thus, Lemma 1 is proved. \hfill$\diamondsuit$ \paragraph{Remark 2.} An example of expansions of the form as in Lemma 1 is given in \cite{z6}. In that example the matrix $D=(d_{n,k})$ is upper-triangular with positive elements on the main diagonal, which shows that, generally speaking, the completeness (in particular, the property of being a basis) of the system of functions $\{v_n\}$ does not follow from the indicated properties of the matrix $D$. \begin{lemma} \label{Lemma2} There exists a positive constant $C$ such that $$\|v_n-e_n\|\leq C(n+1)^{-1}, \quad n=0,1,2,\dots$$ for each of the problems (1)-(2). \end{lemma} \paragraph{Proof.} Let $t_n(x)=p e_n(x)/(\sqrt {2} \pi (n+1))$ for problem (1)-(2a), \\ $t_n(x)=p e_n(x)/\sqrt {2}$ for the problem (1)-(2b) (with $n=0,1,2,\dots$), and $t_0(x)=pe_0(x)$, $t_n(x)=p e_n(x)/\sqrt {2}$ (with $n=1,2,3,\dots$) for problem (1)-(2c). We note that by the standard comparison theorem \cite{l1}, Theorem 1(d) immediately implies the existence of $C_1>0$ such that $$|\lambda _n-\mu _n|\leq C_1 \eqno (11)$$ for $n=0,1,2,\dots$ for each of three problems (1)-(2). Therefore, by Theorem 1(d), for each of three problems (1)-(2) we get for $u_n(x)=y_n(x)-t_n(x)$ $$-u''_n=W_n(x)+\mu _n u_n,\quad x\in (0,1)\,. \eqno (12)$$ In addition, for problems (1)-(2a) and (1)-(2c) by Theorem 1(f), $$u_n(0)=u'_n(0)=u_n(1)=u'_n(1)=0\,\eqno (13)$$ where $W_n(x)=(\lambda _n-\mu _n)y_n(x)-f(y_n^2(x))y_n(x)$ is a sequence of continuous functions uniformly bounded with respect to $x\in [0,1]$. For problem (1)-(2a) or (1)-(2c) for each number $n$ we multiply the equality (12) by $2xu'_n(x)$ and integrate over the segment $[0,1]$ with the application of the integration by parts. Then, in view of (13) and the obvious inequality $2ab\leq a^2+b^2$ we get $$\mu _n\|u_n\|^2=2\int_0^1xW_n(x)u'_n(x)dx-\int_0^1[u'_n(x)] ^2 dx \leq \int_0^1W_n^2(x) dx. \eqno (14)$$ Hence, there exists $C_2>0$ such that $$\mu _n\|u_n\|^2\leq C_2 \eqno (15)$$ for all integer $n\geq 0$. Since in problem (1)-(2c) we have $\|t_0\|=p, \quad \|t_n\|={p\over \sqrt {2}}$, where $n=1,2,3,\dots$, it follows from (15) that $\left| \quad \|y_n\|-{p\over \sqrt {2}}\right|\leq C'_2(n+1)^{-1}$ for some $C'_2>0$ independent of $n$. Therefore, taking into account (15), we get the statement of Lemma 2 for problem (1)-(2c). For problem (1)-(2a) we have $$\|t_n\|={p\over \sqrt {2} \pi (n+1)} \quad (n=0,1,2,\dots ); \eqno (16)$$ therefore, (15) implies the existence of $C_3>0$ such that $$\|y_n\|\leq C_3(n+1)^{-1} \eqno (17)$$ for all integer $n\geq 0$. In view of Theorem 1(d), (17) yields the existence of $C_4>0$ such that $\|W_n\|\leq C_4(n+1)^{-1}$ for all $n=0,1,2,\dots$. So, in problem (1)-(2a) we get from the latter estimate and (14) that $$\|u_n\|\leq C_5(n+1)^{-2}$$ with a constant $C_5>0$ independent of the number $n=0,1,2,\dots$. The statement of Lemma 2 for problem (1)-(2a) follows from this estimate together with (16) as for problem (1)-(2c). For problem (1)-(2b) the proof of the statement in Lemma 2 can be done as for problem (1)-(2c). One should only take into account the estimate for the eigenfunctions of the problem (1)-(2b) $$\left| y'_n(1)-t'_n(1) \right| \leq C_6$$ with a constant $C_6>0$ independent of $n=0,1,2,\dots$ (this estimate follows from (11), the properties of the function $t_n(x)$, Theorem 1(e), and the identities $$-[y'_n(x)]^2+F(y^2_n(x))-F(p^2)=\lambda _ny_n^2(x)-\lambda _np^2 \quad (n=0,1,2,\dots )\,.$$ These identities can be obtained by multiplying (1), with $y(x)=y_n(x)$, by $2y'_n(x)$, and then integrating from $0$ to $x$). Thus, Lemma 2 is proved.\hfill$\diamondsuit$ \begin{lemma} \label{Lemma3} For each of the problems (1)-(2) the system of functions $\{v_n\}$ is linearly independent in the space $L_2$. \end{lemma} \paragraph{Proof.} On the contrary suppose that $$\sum_{n=0}^\infty c_nv_n=0 \eqno (18)$$ in the space $L_2$ where $c_n$ are real coefficients, $c_0=\dots =c_{l-1} =0$ and $c_l\not =0$ for a number $l$. Multiply (18) in the space $L_2$ by the function $e_l(x)$. Then, in view of Lemma 1, we get $d_{l,l}c_l=0$. But according to Lemma 1 $d_{l,l}>0$, which is a contradiction that proves the present lemma. \hfill$\diamondsuit$ \paragraph {Remark 3.} In view of Lemmas 2 and 3, it follows from Bary's theorem that for each of the problems (1)-(2) the system of functions $\{v_n\}$ is a Riesz basis in the space $L_2$. Moreover, since $\{e_n\}$ is an orthonormal basis in $L_2$, the functions $\{v_n\}$ form a Bary basis in $L_2$ for each case. As stated in the introduction, we shall establish a direct proof of this fact without using Bary's theorem. In what follows, unless otherwise stated, all three problems (1)-(2) are considered simultaneously. For an arbitrary number $N\geq 0$, denote by $L^N$ the closure in $L_2$ of the set of all finite linear combinations of the functions $\{e_n\}_{n\geq N}$. Denote by $A$ the linear operator defined by the rule $Ae_n={(d_{n,n})}^{-1}v_n$. Then $A$ is defined for the linear combinations of $\{e_n\}$. In view of Lemma 1, it is clear that $A$ maps the subspace $L^N$ into itself, and that $A$ is defined on dense subsets of $L_2$ and of the subspaces $L^N$. Let $A_N$ denote the restriction of the operator $A$ to the subspace $L^N$ ($N=0,1,2,\dots$). Then, in view of Lemma 1 $A_N=\Lambda _N+G_N$ where $\Lambda _N$ is the identity operator in $L^N$ and $G_N$ is an operator defined on all finite linear combinations of $\{e_n\}_{n\geq N}$. In addition, $G_Ne_n={(d_{n,n})}^{-1}\sum_{k>n}d_{n,k}e_k$ so that $A_Ne_n={(d_{n,n})}^{-1}v_n$ for $n=N,N+1,N+2,\dots$. In view of Lemma 2, the inequality $d_{n,n}>0$, and the condition $\|v_n\|=1$ for $n=0,1,2,\dots$, there exists $\overline d\in (0,1)$ such that for $n\geq 0$, $$\overline d\leq d_{n,n}\leq 1\,. \eqno (19)$$ Let $\|G_N\|=\sup\{\|G_Ng\| :g=\sum_{n=N}^M c_ne_n\mbox {and }\|g\|=1\}$, where supremum is taken over all numbers $M=N,N+1,N+2,\dots$ and all real coefficients $c_n$. \begin{lemma} \label{Lemma4} For each of three problems (1)-(2), $\|G_N\|<\infty$ for all numbers $N\geq 0$ and $\lim_{N\to \infty}\|G_N\|=0$. \end{lemma} \paragraph{Proof.} Let $g=\sum_{n=N}^\infty c_ne_n$ where only finitely many coefficients $c_n$ are non-zero, and $\|g\|=1$. It follows from Lemma 2 and (19) that $\sum_{n,k=0 \atop n \not = k}^\infty {(d_{n,n})}^{-2}d_{n,k}^2 <\infty$. Therefore, \begin{eqnarray*} \|G_Ng\|^2&=&\sum_{k=N+1}^\infty \big[\sum_{n=N}^{k-1} c_n{(d_{n,n})}^{-1}d_{n,k}\big]^2 \\ &\leq& \sum_{k=N+1}^\infty \big[\sum_{n=N}^{k-1}c_n^2\big] \times \big[\sum_{n=N}^{k-1}{(d_{n,n})}^{-2}d^2_{n,k}\big]\\ &\leq& \|g\|^2 \sum_{n,k=N \atop n\not =k}^\infty {(d_{n,n})}^{-2} d^2_{n,k} \to 0 \quad \mbox{as } N\to \infty, \end{eqnarray*} and the proof of Lemma 4 is complete. \hfill$\diamondsuit$\smallskip Let $N$ be a positive integer such that $\|G_N\|<1/2$. As shown in \cite{l2}, the linear operator $G_N$ can be uniquely extended to the whole space $L^N$. For this extension we keep the same symbol $G_N$, and we have $\|G_N\| < 1/2$. In view of the latter fact, the linear operator $A_N=\Lambda _N+G_N$, with $G_N\in {\cal L}(L^N;L^N)$) belongs to ${\cal L}(L^N;L^N)$ and possesses a bounded inverse: $$A_N^{-1}=\Lambda _N+\sum_{m=1}^\infty (-1)^m(G_N)^m\,.$$ The proof of this statement can be found in \cite{l2}. The following statement is in fact proved in \cite{b2,g1}. However, since its proof is simple and short, we present it here. \begin{lemma} \label{Lemma5} Let $N$ be the number fixed above. Then for each of the problems (1)-(2), the functions $\{v_n\}_{n\geq N}$ form a Riesz basis in $L^N$. \end{lemma} \paragraph{Proof.} Obviously, the functions $\{e_n\}_{n\geq N}$ form an orthonormal basis in $L^N$. Take an arbitrary function $u\in L^N$ and let $v=A^{-1}_Nu=\sum_{n=N}^\infty c_ne_n$ in the space $L^N$ where $c_n$ are real coefficients. We have $u=A_Nv=\sum_{n=N}^\infty c_nA_Ne_n=\sum _{n= N}^\infty c_n{(d_{n,n})}^{-1}v_n$ where all series converge in $L^N$. Therefore, in view of Lemma 3, the system of functions $\{v_n\}_{n\geq N}$ is a basis in the space $L^N$. Further, with the same notation, $\sum_{n=N}^\infty c_n^2<\infty$ because the series $\sum_{n= N}^\infty c_ne_n$ converges in the space $L^N$. Hence, if $\sum_{n=N}^\infty c_n^2<\infty$, then the series $\sum _{n=N}^\infty c_n{(d_{n,n})}^{-1}v_n$ converges in the space $L^N$. Conversely, let the series $\sum _{n=N}^\infty c_nv_n$ converge in $L^N$. Then, $A_N^{-1} \sum_{n=N}^\infty c_nv_n=\sum_{n=N}^\infty c_nA_N^{-1}v_n= \sum_{n=N}^\infty c_nd_{n,n}e_n$ in the space $L^N$, therefore, $\sum _{n=N}^\infty c_n^2d_{n,n}^2<\infty$. So, in view of the estimates (19), Lemma 5 is proved.\hfill$\diamondsuit$ \begin{lemma} \label{Lemma6} For each of the three problems (1)-(2) the functions $\{v_n\}_{n=0,1,2,\dots }$ form a Riesz basis in $L_2$. \end{lemma} \paragraph{Proof.} Let $N$ be the number fixed above, and $L^N_\bot$ be the finite-dimensional subspace in $L_2$ spanned by $e_0,\dots ,e_{N-1}$, and $P_N$ be the orthogonal projector in the space $L_2$ onto the subspace $L^N_\bot$. We set $w_n=P_Nv_n$, with $0\leq n \leq N-1$. Then $$w_n=v_n-\sum_{k=N}^\infty d_{n,k}e_k\,. \eqno (20)$$ Further, the dimension of the subspace $L^N_\bot$ is obviously equal to $N$ and, in view of Lemma 1, the system of functions $\{w_n\}_{0 \leq n \leq N-1}$ is linearly independent in $L_2$, therefore, it is a basis in $L^N_\bot$. Hence, in view of Lemma 5, the system of functions $\{w_n\}_{0 \leq n \leq N-1} \cup \{v_n\}_{n\geq N}$ is a basis in $L_2$. For an arbitrary $u\in L_2$, $$u=\sum_{n=0}^{N-1}c_nw_n+\sum_{n=N}^\infty c_nv_n \eqno (21)$$ in $L_2$ where $c_n$ are real coefficients. Substituting the expressions for functions $w_n$ from (20) into (21), we get $$u=\sum_{n=0}^{N-1}c_nv_n+\sum_{n=N}^\infty \left(c_n-\sum _{m=0}^{N-1}c_md_{m,n}\right)e_n$$ in $L_2$, hence the functions $\{v_n\} _{n=0,1,2,\dots }$ form a basis for $L_2$. Finally, that these functions form a Riesz basis in $L_2$ follows from Lemma 5, and concludes the present proof. \hfill$\diamondsuit$\smallskip For problem (1)-(2c), the estimate (15) and the arguments following the estimate imply the existence of constants \$0from Lemma 6 and these estimates. \begin{thebibliography}{19} {\frenchspacing \bibitem{b1} N. K. Bary, {\it On bases in Hilbert space}, Doklady Akad. Nauk SSSR, 1946, Vol. {\bf 54}, No 5, pp. 383-386 (in Russian). \bibitem{b2} N. K. Bary, {\it Biorthogonal systems and bases in Hilbert space}, U\v cenye zapiski Moskovskogo Gos. Universiteta, 1951, Vol. {\bf 148}, Matematika, No 4, pp. 69-107 (in Russian). \bibitem{g1} I. C. Gohberg, M. G. Krein, {\it Introduction in the theory of non self-adjoint operators}, Moscow, Nauka, 1965 (in Russian). \bibitem{k1} E. Kamke, {\it Reference book on ordinary differential equations}, Moscow, Nauka, 1976 (in Russian). \bibitem{l1} B. M. Levitan, I. S. 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Modern Phys. {\bf A}, 1997, Vol. {\bf 12}, No 1, pp. 295-298. \bibitem{z4}P. E. Zhidkov, {\it Eigenfunction expansions associated with a nonlinear Schr\"odinger equation on a half-line}, Preprint of the Joint Institute for Nuclear Research, E5-99-144, Dubna, 1999. \bibitem{z5}P. E. Zhidkov, {\it Eigenfunction expansions associated with a nonlinear Schr\"odinger equation}, Communications of the Joint Institute for Nuclear Research, E5-98-61, Dubna, 1998. \bibitem{z6} P. E. Zhidkov, {\it On the property of being a basis for a denumerable set of solutions of a nonlinear Schr\"odinger-type boundary-value problem}, Preprint of the Joint Institute for Nuclear Research, E5-98-109, Dubna, 1998; to appear in Nonlinear Analysis: Theory, Methods \& Applications. }\end{thebibliography} \medskip \noindent{\sc Peter E. Zhidkov} \\ Bogoliubov Laboratory of Theoretical Physics \\ Joint Institute for Nuclear Research \\ 141980 Dubna (Moscow region), Russia \\ email: zhidkov@thsun1.jinr.ru \end{document}