\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 87, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2004/87\hfil Nonlinear Sturm-Liouville problems]
{Convergence of eigenfunction expansions
 corresponding to nonlinear Sturm-Liouville operators} 

\author[A. S. Makin, H. B. Thompson\hfil EJDE-2004/87\hfilneg]
{Alexander S. Makin, H. Bevan Thompson}  % in alphabetical order

\address{Alexander S. Makin\hfill\break
Moscow State Academy of Instrument-Making and Informatics\\
Stromynka 20, 107846, Moscow, Russia}
\email{alexmakin@yandex.ru}

\address{H. Bevan Thompson\hfill\break
Department of Mathematics \\
The University of Queensland \\
Queensland 4072, Australia}
\email{hbt@maths.uq.edu.au}

\date{}
\thanks{Submitted October 10,2003. Published June 29, 2004.}
\subjclass[2000]{34L10, 34B15}
\keywords{Sturm-Liouville operator; basis property; eigenfunction}


\begin{abstract}
 It is well known that the classical linear Sturm-Liouville
 eigenvalue problem is self-adjoint and possesses a family 
 of eigenfunctions which form an orthonormal basis for the 
 space $L_2$. A natural question is to ask if a similar result
 holds for nonlinear problems. In the present paper, 
 we examine the basis property for eigenfunctions of nonlinear 
 Sturm-Liouville equations subject to general linear, separated 
 boundary conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction} 

We consider the nonlinear eigenvalue problem
\begin{gather}
u''-q(x, u, \lambda)u+\lambda u=0 ,\quad x\in[0, 1], \label{e1}\\
\alpha_1u(0)+\beta_1u'(0)=0 , \quad\alpha_2 u(1)+\beta_2u'(1)=0, \label{e2}
\end{gather}
where $|\alpha_i|+|\beta_i|>0$, for $i=1, 2$.
Here $x$ and $\lambda$ are real variables and $q$ is a real-valued
function defined on $\Omega=[0, 1]\times\mathbb{R}^2$. By an eigenfunction
 of \eqref{e1}-\eqref{e2} corresponding to an eigenvalue $\lambda$ we mean a
twice continuously differentiable, real-valued function $u(x)$,
$(u(x)\not \equiv 0)$, satisfying \eqref{e1} on
$[0, 1]$ and \eqref{e2}.

The main result is as follows.


\begin{theorem} \label{thm1}
Assume that
\begin{itemize}
\item[(1)] $q(x, u, \lambda)$ is continuous on the set $\Omega$
\item[(2)] There exist constants $M_0$ and $M$ such that
$|q(x, u, \lambda)|\le M$ on the set  $\Omega_0=\{ 0\le x\le
1, |u|\le M_0, -\infty < \lambda <\infty\}$
\item[(3)] For any $\lambda$, $-\infty <\lambda <\infty$, and any $x$,
$0\le x\le 1$,
$\frac{\partial}{\partial\lambda} q(x, 0, \lambda)=0$.
 \end{itemize}
Then there exists a system $\{u_n(x)\}$ $(n=0, 1, \dots)$ of
eigenfunctions of problem \eqref{e1}-\eqref{e2} which forms a Riesz
basis for the space $L_2(0,1)$.
\end{theorem}

Note that when the function $q(x, u, \lambda)$ does not depend
on $\lambda$, conditions (2) and (3) of Theorem \ref{thm1} can be omitted.


Before proving Theorem \ref{thm1} we would like to compare it with existing
results in the literature. Since $q(x,0,\lambda)=q_0(x)$ we see
that problem \eqref{e1}-\eqref{e2} can be written in the operator
equation form
$$
Lu+N(u,\lambda)+\lambda u=0
$$
where $L$ is the linear operator generated by the differential
expression $Lu=u''-q_0(x)u$ and boundary conditions \eqref{e2} and
$N(u,\lambda)=(q_0(x)-q(x,u,\lambda))u$. Obviously $L$
is a self-adjoint operator with discrete spectrum and an
orthonormal system of eigenfunctions which is complete in
$L_2(0,1)$. Moreover all the eigenvalues are simple. Thus in order
to construct an eigenfunction system for the above spectral
problem which forms a basis for $L_2(0,1)$ one could invoke the
Crandall-Rabinowitz approach, \cite{c2}, and the method by Brown \cite{b2}.
However to apply this approach requires strong assumptions on the
nonlinear operator $N$, namely that $q(x,u,\lambda)\in C^1$. Here we
weaken this smoothness assumption on the function
$q(x,u,\lambda)$. In addition any eigenfunction system
$\{u_n(x)\}$ constructed by that method lies in the neighborhood
of zero, that is, $\lim_{n\to\infty}\|u_n(x)\|_{L_2(0,1)}=0$. Thus
the system $\{u_n(x)\}$ may be an unconditional basis but it
cannot be a Riesz basis since in this context a Riesz basis is an
almost normalized system of functions.

Nonlinear eigenvalue problems have a long history; we refer the reader
to \cite{m1} and its reference list for more information.

\section{Proof of Theorem \ref{thm1}}

The proof divides naturally into the following parts.

\noindent (a) We use a standard method to show that, without loss of
generality, we may make certain assumptions about the function
$q(x,u,\lambda)$.

\noindent (b) We prove a simple technical lemma giving estimates for
solutions of the initial value problem for equation \eqref{e1}.

\noindent (c) We employ a polar coordinates technique to establish the
existence of eigenvalues of problem \eqref{e1}-\eqref{e2}. Alternative
methods such as fixed point theorems could have been used here.
However we prefer the classical Pr\" ufer transformation since it
gives the eigenfunctions in more explicit form. In particular, we
obtain the eigenfunctions with prescribed initial data and this is
important later in the proof. Note that since the function
$q(x,u,\lambda)$ is merely continuous it follows that solutions of
initial value problems for equation \eqref{e1} may not be unique. To
overcome this obstacle we apply the generalized Kneser's theorem
in the form given by Pugh in \cite{p2}. This application requires the
right hand side of our system of equations to have compact
support. We use cut-off functions to produce a new system of
equations with compact support and show there is a solution of our
new system satisfying the boundary conditions \eqref{e2} which is a
solution of problem \eqref{e1}-\eqref{e2}. Then we use the Sturm comparison
theorem to obtain a two-side estimate for the eigenvalues of
problem \eqref{e1}-\eqref{e2}.

\noindent (d) We prove that the constructed eigenfunction system of problem
\eqref{e1}-\eqref{e2} divided by a suitable number $p$ is quadratically
close to a complete orthonormal eigenfunction system of the
self-adjoint eigenvalue problem for the linearized equation
subject to the same boundary conditions. In this part of the proof
we use well known asymptotic formulae for eigenvalues of the
linear Sturm-Liouville operator, an integral representation for
solutions of the initial-value problem, and $\lambda$-independent
relations between the $L_{\infty}$ and $L_2$-norms of
eigenfunctions and their derivatives \cite{t1}.

\noindent (e) We apply well known theorems of functional analysis to
establish the basis property for the constructed eigenfunction
system.

One can see that the arguments in steps a), b) and e) are short
and simple, while those in steps c) and d) are more complicated. \smallskip

\noindent{\bf (a)}  Let us consider the following eigenvalue problem for the
linearized equation
\begin{equation}
u''-q_0(x)u+\lambda u=0 \label{e1'}
\end{equation}
with boundary conditions \eqref{e2}. From \cite{k2}, all eigenvalues of problem
\eqref{e1'}-\eqref{e2} are real and, moreover, there exists a smallest
eigenvalue, $\lambda_0^{(0)}$. We set
$\lambda^*=2M+1-\lambda_0^{(0)}$ and
$\tilde{q}(x, u, \lambda)=q(x, u, \lambda-\lambda^*)+\lambda^*$.
Clearly the function $\tilde{q}(x, u, \lambda)$ satisfies all the
conditions of the theorem with the inequality in condition (2)
replaced by the inequality
$M-\lambda_0^{(0)}+1\le\tilde{q}(x, u, \lambda)\le 3M-\lambda_0^{(0)}+1$.
We denote $M_1=3M+|\lambda_0^{(0)}|+1$. Since
$$
\tilde{u}''_n-\tilde{q}(x, \tilde{u}_n,  \tilde{\lambda}_n)\tilde{u}_n+\tilde{\lambda}_n\tilde{u}_n
=\tilde{u}''_n-q(x, \tilde{u}_n, \tilde{\lambda}_n-\lambda^*)\tilde{u}_n+(\tilde{\lambda}_n-\lambda^*)
\tilde{u}_n
$$
it follows that an eigenfunction $\tilde{u}_n(x)$ of the problem
\begin{gather*}
\tilde{u}''-\tilde{q}(x, \tilde{u}, \tilde{\lambda})\tilde{u}+\tilde{\lambda}\tilde{u}=0,\\
\alpha_1\tilde{u}(0)+\beta_1\tilde{u}'(0)=0 ,\quad \alpha_2\tilde{u}(1)+\beta_2\tilde{u}'(1)=0
\end{gather*}
corresponding to an eigenvalue $\tilde{\lambda}_n$ is an eigenfunction of
problem \eqref{e1}-\eqref{e2} corresponding to the eigenvalue
$\lambda_n=\tilde{\lambda}_n-\lambda^*$. Thus without loss of generality we may
assume that the function $q(x, u, \lambda)$ satisfies the inequalities
\begin{equation} \label{e3}
|q(x, u, \lambda)-q_0(x)|\le 2M, \qquad |q(x, u, \lambda)|\le M_1
\end{equation}
on the set $\Omega_0$ 
 and that the smallest eigenvalue of problem
\eqref{e1'}, \eqref{e2}, $\lambda_0^{(0)}$, is $2M+1$. Also without loss of
generality we may assume  that
$|\alpha_1|+|\beta_1|\le 1$. We will assume
later that $\lambda >0$ and set $\mu=\sqrt{\lambda}$. \smallskip

\noindent {\bf (b)} Let $\eta(t)\in C^\infty(R)$ be a cut off function
$$\eta(t)=\begin{cases}
1&\mbox{if }|t|\le\frac{M_0}2\\
0&\mbox{if }|t|\ge M_0,\end{cases}
$$
and $0\le\eta(t)\le 1$. Consider the initial-value problem
\begin{equation} \label{e4}
\begin{gathered}
u''-q(x, u, \lambda)\eta(u)u+\lambda u=0 ,\quad x\in[0, 1],\\
u(0)=b\beta_1,\qquad u'(0)=-b\alpha_1,
\end{gathered}\end{equation}
where $b$ is an arbitrary number; if $\beta_1=0$ then we set
$u(0)=0$ and $u'(0)=b\mu$.

\begin{lemma} \label{lm1}
If $\lambda\ge 1$ and $b$ be given, then any solution $u(x)$ of
problem \eqref{e4} on  $[0, 1]$ satisfies
$$|u(x)|\le |b|e^{M_1}.$$
\end{lemma}

\begin{proof} It follows from \cite{b1} that a solution $u(x)$
of problem \eqref{e4} exists on  $[0, 1]$. By
\cite[Ch. 3, Sec. 6, Th. 6.4]{c1}, any solution of \eqref{e4} satisfies the integral
equation
\begin{equation} \label{e5}
u(x)=u(0)\cos\mu x+u'(0)\frac{\sin\mu x}{\mu}
+\frac1{\mu}\int_0^x\sin\mu(x-t)q(t, u(t), \lambda
)\eta (u(t))u(t)dt.
\end{equation}
From inequality \eqref{e3} it follows that
$$
|q(t, u(t), \lambda)|\eta(u(t)) \le M_1\quad \forall (t,u,\lambda)\in \Omega.
$$
From this it follows that
$$
|u(x)|\le|R(x)|+\frac{M_1}{\mu}\int_0^x |u(t)|dt
$$
where $R(x)$ is the sum of two first terms
on the right-hand side of \eqref{e5}.
From the last inequality and Gronwall's lemma, \cite{b1}, we see that
\begin{equation} \label{e6}
|u(x)|\le|R(x)|+\frac{M_1}{\mu}\int_0^x
e^{\frac{M_1}{\mu}(x-t)}|R(t)|dt.
\end{equation}
Since $|R(x)|\le b$ it follows from \eqref{e6} that
$$
|u(x)|\le|b|e^{M_1}.
$$
Let
\begin{equation} \label{e7}
0\le|b|\le b_0
\end{equation}
where $b_0=\min\big(1,\frac{M_0}{2e^{M_1}}\big)$. It
follows from Lemma \ref{lm1} that $|u(x)|\le\frac{M_0}2$.

From this and the definition of the cut off function $\eta(t)$ it
follows that if condition \eqref{e7} is satisfied, then any solution of
problem \eqref{e4} is a solution of initial-value problem
\begin{equation} \label{e4'}
\begin{gathered}
u''-q(x, u, \lambda)u+\lambda u=0 ,\quad x\in[0, 1],\\
u(0)=b\beta_1 ,  u'(0)=-b\alpha_1;
\end{gathered}
\end{equation}
if $\beta_1=0$ then $u(0)=0$ and $u'(0)=b\mu$. 
\end{proof}

\noindent{\bf (c)} Now we consider two linear eigenvalue problems
\begin{equation} \label{e8}
\begin{gathered} u''+Q_1(x, \lambda,d)u=0,\\
\alpha_1u(0)+\beta_1u'(0)=0,\\ \alpha_2 u(1)+\beta_2u'(1)=0
\end{gathered}
\end{equation}
and
\begin{equation} \label{e9}
\begin{gathered} u''+Q_2(x, \lambda,d)u=0,\\
\alpha_1u(0)+\beta_1u'(0)=0,\\ \alpha_2 u(1)+\beta_2u'(1)=0
\end{gathered}
\end{equation}
where $Q_1(x, \lambda,d)=-q(x, 0, \lambda)-d+\lambda$,
$Q_2(x, \lambda,d)=-q(x, 0, \lambda)+d+\lambda$, and $d$ is an arbitrary
positive number. Let $\theta_1(x, \lambda, d)$ be a
solution of the equation
$\theta_i'=\cos^2\theta_i+Q_i(x, \lambda, d)\sin^2\theta_i$,
moreover,
$\theta_i(0, \lambda, d)=\arctan (-\frac{\alpha_1}{\beta_1})$
if $\beta_1\ne 0$ and $\theta_i(0, \lambda, d)=0$ $(i=1, 2)$ if
$\beta_1=0$. By the oscillation theorem, \cite{h1}, problem \eqref{e8} has
eigenvalues $\lambda_0^{(1)}(d)<\lambda_1^{(1)}(d)<\dots$, and problem \eqref{e9}
has eigenvalues $\lambda_0^{(2)}(d)<\lambda_1^{(2)}(d)<\dots$. Clearly, it
follows  that $\lambda_n^{(1)}(d)=\lambda_n^{(2)}(d)+2d$,
$\lambda_0^{(2)}(2M)=1$. From \cite{h1} it also follows that
\begin{equation} \label{e10}
\theta_i(\lambda_n^{(i)}(d), 1, d)=\arctan (-\frac{\alpha_2}{\beta_2}
)+\pi n,
\end{equation}
where $n=0, 1, \dots$. Here and throughout the remaining part of
the paper we assume that $\arctan 
(-\frac{\alpha_2}{\beta_2})=\pi$ if $\beta_2=0$.

\begin{lemma} \label{lm2}
 For each $b\ne 0$ satisfying condition \eqref{e7} and  $n\ (n=0,1,\dots)$
there exists an eigenvalue $\lambda_n(b)$ of problem \eqref{e1}-\eqref{e2}
satisfying 
\begin{equation} \label{e11}
\lambda_n^{(2)}(2M)\le\lambda_n(b)\le\lambda_n^{(1)}(2M).
\end{equation}
Moreover, the corresponding eigenfunction $u_n(x,\lambda_n(b))$
is a solution of initial-value problem \eqref{e4'}.
\end{lemma}

\begin{proof}
Let $u(x, \lambda)$ be a solution of problem \eqref{e4}. Using the classical
Pr\" ufer transformation \cite{h1},
$$
u(x)=r(x)\sin\varphi(x) ,\quad   u'(x)=r(x)\cos\varphi(x)
$$
it follows that for any $\lambda$ problem \eqref{e4} is equivalent the system
of equations
\begin{equation} \label{e12}
\begin{gathered}
\varphi'=\cos^2\varphi+\left[\Lambda-q(x, r\sin\varphi, \Lambda)\eta(r\sin\varphi)\right]
\sin^2\varphi\\
r'=\frac12 r\left[1-\Lambda+q(x, r\sin\varphi, \Lambda)\eta(r\sin\varphi)\right]\sin 2\varphi\\
\Lambda'=0
\end{gathered}
\end{equation}
with initial conditions
\begin{equation} \label{e13}
\varphi(0)=\varphi_0 ,\quad r(0)=r_0 ,\quad \Lambda(0)=\lambda
\end{equation}
where $\varphi_0=\arctan  (-\frac{\alpha_1}{\beta_1})$ and
$r_0= |b|\sqrt{\alpha_1^2+\beta_1^2}$ if $\beta_1\ne0$
while $\varphi_0=0$ and $r_0=|b|\mu$ if $\beta_1=0$.

Let us consider also the system of equations
\begin{equation} \label{e12'}
\begin{gathered}
\varphi'=\left[\cos^2\varphi+(\Lambda-q(x, r\sin\varphi, \Lambda)\eta(r\sin\varphi)\sin^2
\varphi\right]\tilde{\eta}(\varphi, r, \Lambda)\\
r'=\frac12\left[r(1-\Lambda+q(x, r\sin\varphi, \Lambda)\eta(r\sin\varphi)\sin
2\varphi\right]\tilde{\eta}(\varphi, r, \Lambda)\\
\Lambda'=0
\end{gathered}
\end{equation}
subject to the same initial conditions \eqref{e13}, where
$\tilde{\eta}(\varphi, r, \Lambda)=\eta_0(\varphi)\eta_0(r)\eta_0(\Lambda)$
and where the cut off function $\eta_0(t)=1$ if $|t|\le H$, 
$\eta_0(t)=0$ if $|t|\ge H+1$, $0\le\eta_0(t)\le1$,
$\eta_0(t)\in C^\infty(R)$, $H$ is an arbitrary number such that
$H\ge\mu e^{1+\lambda+M_1}+\pi$.

It is easy to see that right hand side of system \eqref{e12'} has
compact support. From inequality \eqref{e3} if follows that for any $x$,
\begin{equation} \label{e14}
|q(x, r\sin\varphi, \Lambda)|\eta(r\sin\varphi)\le M_1.
\end{equation}
From \eqref{e14} and the first of equations \eqref{e12'} it follows that
$|\varphi'|\le 1+\lambda+M_1$. Since $|\varphi(0)|<\pi$
we have $|\varphi(x)|\le 1+\pi+\lambda+M_1$ for $0\le x\le 1$.

 From the second of equations \eqref{e12'} it follows that
$$
r(x)=r(0)\exp\big(\frac12\int_0^xh(t)d t\big),
$$
where
$$
h(t)=\left[1-\Lambda+q(t, r(t)\sin\varphi(t),
\Lambda)\eta(r(t)\sin\varphi(t))\sin 2\varphi(t)\right]\tilde{\eta}
(\varphi(t), r(t), \Lambda)dt.
$$
Evaluating right-hand part of the last equation we obtain
$$
r(x)\le r(0)e^{\frac12(1+\lambda+M_1)}\le|b|\mu
e^{\frac12(1+\lambda+M_1)}.
$$
By the definitions of $H$ and $\eta(\varphi, r, \Lambda)$, any solution 
of the Cauchy problem \eqref{e12'}-\eqref{e13} is a solution of the
Cauchy problem \eqref{e12}-\eqref{e13}.

Let $(\widetilde{\varphi}(x), \tilde{r}(x), \wedge)$ be an arbitrary solution of
Cauchy problem \eqref{e12'}-\eqref{e13} for an arbitrary $\lambda$,
$1\le\lambda\le\lambda_n^{(1)}(2M)$, where 
$H=\sqrt{\lambda_n^{(1)}(2M)} e^{1+\lambda_n^{(1)}(2M)+M_1}+\pi$, and 
$b$ satisfies condition \eqref{e7}.
Since any solution of Cauchy problem \eqref{e12'}-\eqref{e13} is a
solution of Cauchy problem \eqref{e12}-\eqref{e13} we have
$$
\widetilde{\varphi}'=\cos^2\widetilde{\varphi}+(\lambda-q(x, \tilde{r}\sin\widetilde{\varphi}, \lambda)\eta(\tilde{r}\sin\widetilde{\varphi}))
\sin^2\widetilde{\varphi}\,.
$$ 
Moreover, the function
$\tilde{u}(x)=\tilde{r}(x)\sin\widetilde{\varphi}(x)$ is a solution of problem \eqref{e4}. Then by
Lemma \ref{lm1}, $|\tilde{u}(x)|\le\frac{M_0}2$; therefore,
$\eta(\tilde{u}(x))=1$. By virtue of \eqref{e3} we have inequality
$$
Q_1(x, \lambda, 2M)\le-q(x, \tilde{u}(x), \lambda)\eta(\tilde{u}(x))+\lambda\le Q_2(x, \lambda, 2M).
$$ 
From the last inequality and the
comparison theorem, \cite{h1} it follows that
$$
\theta_1(x, \lambda, 2M)\le\widetilde{\varphi}(x, \lambda)\le\theta_2(x, \lambda, 2M).
$$
From this and \eqref{e10} it follows that
\begin{equation} \label{e15}
\widetilde{\varphi}(1, \lambda_n^{(2)}(2M))\le\arctan (-\frac{\alpha_2}{\beta_2}
)+\pi n\le\widetilde{\varphi}(1, \lambda_n^{(1)}(2M)).
\end{equation}
Further we consider Cauchy problem \eqref{e12'}-\eqref{e13}. It
is clear that the set of initial conditions
$$
P=\{0, \varphi_0, r_0, \lambda\}
$$
where $\lambda_n^{(2)}(2M)\le\lambda\le\lambda_n^{(1)}(2M)$ is a connected
compact set in $R^4$. Hence by the generalised Kneser's theorem,
\cite{p2}, the set $K_1(P)=\bigcup_{p\in P}K_1(p)$ where $K_1(p)$
is the funnel section at the point $x=1$ of the set of solutions
of system \eqref{e12'} subject to the initial conditions
$p=(0, \varphi_0, r_0, \lambda)$ is a nonempty connected compact set in
$R^3$. From \eqref{e15} it follows that the set $K_1(P)$ contains points
above and below the plane
$\varphi=\arctan (-\frac{\alpha_2}{\beta_2})+\pi n$, therefore the set
$K_1(P)$ has a point of intersection with this plane.
Therefore, for some $\lambda=\lambda_n(b)$ satisfying \eqref{e11},
there exists a solution
$(\varphi_n(x, \lambda), r_n(x, \lambda),\lambda_n(b))$  to  the Cauchy problem
\eqref{e12'}-\eqref{e13} such that
\begin{equation} \label{e16}
\varphi_n(1, \lambda_n(b))=\arctan (-\frac{\alpha_2}{\beta_2})+\pi n.
\end{equation}
Now solutions of problem \eqref{e12'}-\eqref{e13} are solutions of
problem \eqref{e12}-\eqref{e13}, problem \eqref{e12}-\eqref{e13} is equivalent 
problem \eqref{e4}, and any solution of problem \eqref{e4} is a solution of 
problem \eqref{e4'}. Hence for any $b$ satisfying condition \eqref{e7} there is a
function $\lambda_n(b)$ satisfying inequality \eqref{e11} and a corresponding
solution
$$
u_n(x, \lambda_n(b))=r_n(x, \lambda_n(b))\sin\varphi_n (x, \lambda_n(b))
$$ 
of problem \eqref{e4'}. From \eqref{e16} it
follows that the function $u_n(x, \lambda_n(b))$ satisfies boundary
conditions \eqref{e2}; that is, the function $u_n(x, \lambda_n(b))$ is an
eigenfunction of problem \eqref{e1}-\eqref{e2} corresponding to the
eigenvalue $\lambda_n(b)$. 
\end{proof}

\noindent {\bf (d)} Let $\{\overset{\circ}{u}_n(x)\}$ be a complete orthonormal
system of eigenfunctions in $L_2(0, 1)$ of the linear
self-adjoint problem \eqref{e1'}-\eqref{e2}, $\lambda_n^{(0)}$ be the
corresponding eigenvalues, and $\mu_n^{(0)}=\sqrt{\lambda_n^{(0)}}$,
for $n=0, 1, \dots$. Above it was shown that $\lambda_n^{(0)}\ge 2M+1$. 
From \cite{t1} it follows that $\max_{0\le x\le1}|\overset{\circ}{u}_n(x)|\le C_0$, 
$\max_{0\le x\le1}|\overset{\circ}{u}_n'(x)|\le \mu_n^{(0)} C_0$; therefore,
$\overset{\circ}{u}_n(0)=\beta_1\gamma_n$ and $\overset{\circ}{u}_n'(0)=\zeta_n\mu_n^{(0)}$, 
where $0\le|\beta_1\gamma_n|$, $|\zeta_n|\le C_0$ and 
$|\beta_1\gamma_n|+|\zeta_n|>0$. If
$\beta_1\ne0$ it follows from boundary condition \eqref{e2} that
$\zeta_n=\frac{-\alpha_1\gamma_n}{\overset{\circ}{\mu}_n}$  and therefore
$\gamma_n\ne0$. If $\beta_1=0$ then $\zeta_n\ne0$. By \cite{h1} for any
$d>0$
\begin{equation} \label{e17}
\lambda_n^{(2)}(d)\le\lambda_n^{(0)}\le\lambda_n^{(1)}(d).
\end{equation}

\begin{lemma}
There exist $p\ne 0$ and an eigensystem
$\{u_n(x, \lambda_n, p)\}$, $n=0, 1, \dots$ for problem \eqref{e1}-\eqref{e2}
satisfying initial conditions
$u_n(0, \lambda_n, p)=p\gamma_n\beta_1$, 
$u'_n(0, \lambda_n, p)=-p\gamma_n\alpha_1$ 
if $\beta_1\ne 0$ and initial conditions
$u_n(0, \lambda_n, p)=0$, $u'_n(0, \lambda_n, p)=p\zeta_n\sqrt{\lambda_n}$
if $\beta_1=0$  such that
$$
\sum_{n=0}^\infty\left\|\frac{u_n(x, \lambda_n, p)}p
-\overset{\circ}{u}_n(x)\right\|^2_{L_2(0,1)}<1.
$$
\end{lemma}

\begin{proof}  Let $p$ be an arbitrary number satisfying 
\begin{equation} \label{e*}
0<|p|<\frac{b_0|\beta_1|}{C_0}\mbox{ if }\beta_1\ne0
\quad \mbox{and}\quad 0<|p|<\frac{b_0}{C_0} 
\mbox{ if }\beta_1=0. 
\end{equation} 
Denote $p_n=p\gamma_n$ if
$\beta_1\ne0$ and $p_n=p\zeta_n$ if $\beta_1=0$. It is easy to see
that in both cases $0<|p_n|<b_0$. Notice that
$|p_n|\le \frac{C_0}{|\beta_1|}|p|$ if
$\beta_1\ne0$ and $|p_n|\le C_0|p|$ if
$\beta_1=0$.

For every $n=0, 1, \dots$, let $\lambda_n(p_n)$ be an eigenvalue of
problem \eqref{e1}-\eqref{e2} satisfying inequality \eqref{e15} and let
$u_n(x, \lambda_n(p_n))$ be a corresponding eigenfunction satisfying
the initial conditions $u_n(0, \lambda_n(p_n))=p_n\beta_1$,
$u'_n(0, \lambda_n(p_n))=-p_n\alpha_1$ if $\beta_1\ne 0$ and
satisfying the initial conditions $u_n(0, \lambda_n(p_n))=0$,
$u_n'(0, \lambda_n(p_n))=p_n\sqrt{\lambda_n(p_n)}$ if $\beta_1=0$.
 From inequalities \eqref{e11}-\eqref{e17} it follows that
$|\lambda_n(p_n)-\lambda_n^{(0)}|\le 4M$. From the last
inequality and the asymptotic formulae $\mu_n^{(0)}=\pi
n+O(n^{-1})$ if $\beta_1\ne0, \beta_2\ne0$ or $\beta_1=\beta_2=0$
and $\mu_n^{(0)}=\pi(n+1/2)+O(n^{-1})$ if $\beta_1=0, \beta_2\ne0$
given in \cite{k2}  it follows that there exists $N_0>0$ such that for
all $n>N_0$
\begin{equation} \label{e18}
|\mu_n(p_n)-\mu_n^{(0)}|\le\frac{4M}{\mu_n(p_n)+\mu_n^{(0)}}
\le\frac{5M}{n},
\end{equation}
where $\mu_n(p_n)=\sqrt{\lambda_n(p_n)}$ and
$\mu_n^0=\sqrt{\lambda_n^0}$.
 Thus there exists $N_1 > N_0$ such that for all $n > N_1$
\begin{equation} \label{e19}
\mu_n^{(0)}>\frac{n}2 ,\quad \mu_n(p_n)>\frac{n}2.
\end{equation}
Let $\phi(x, \lambda),\psi(x, \lambda)$ be the fundamental system of
solutions of equation \eqref{e1'} satisfying $\phi(0, \lambda)=0$,
$\phi_x'(0, \lambda)=\mu$, $\psi(0, \lambda)=1$, $\psi_x'(0, \lambda)=0$.
Now
\begin{equation} \label{e20}
\phi(x, \lambda)=\sin\mu
x+O(\frac1{\mu}) ,\quad \psi(x, \lambda)=\cos\mu x+O(\frac1{\mu})
\end{equation}
uniformly in $x$, $0\le x\le 1$; see \cite{m1}. Therefore,
\begin{equation} \label{e21}
\begin{gathered}
|\phi(x, \lambda)|\le C_1 ,\quad  |\psi(x, \lambda)| \le C_1\\
|\phi(x, \lambda_2)-\phi(x, \lambda_1)|\le|\mu_2-\mu_1|
+O(\frac1{\mu_1})+O(\frac1{\mu_2}),\\
|\psi(x, \lambda_2)-\psi(x, \lambda_1)|\le |\mu_2-\mu_1|
+O(\frac1{\mu_1})+O(\frac1{\mu_2}).
\end{gathered}
\end{equation}
uniformly in $x$, $0\le x\le 1$.  From \cite{p1} it follows that the
functions $\phi(x, \lambda)$ and $\psi(x, \lambda)$ are continuous with
partial derivatives in $(x, \lambda)$ and hence
\begin{equation} \label{e22}
\begin{gathered}
|\phi(x, \lambda_2)-\phi(x, \lambda_1)|\le C_2(\widehat{\lambda})|\lambda_2-\lambda_1|,\\
|\psi(x, \lambda_2)-\psi(x, \lambda_1)|\le C_2(\widehat{\lambda})|\lambda_2-\lambda_1|.
\end{gathered}
\end{equation}
for $0\le x\le 1$, $1\le \lambda_1, \lambda_2\le\widehat{\lambda}$, for any $\widehat{\lambda}\ge 1$
where $C_2(\widehat{\lambda})$ depends on $\widehat{\lambda}$.
Let $W(x, \lambda)$ be the Wronskian for functions $\phi(x, \lambda)$ and
$\psi(x, \lambda)$. We denote
$$
K(x, t, \lambda)=\frac{\psi(x, \lambda)\phi(t, \lambda)-\phi(x, \lambda)\psi(t, \lambda)}{W(t, \lambda)}.
$$
An easy calculation gives $W(0, \lambda)=-\mu$. From this and
Liouville's formula it follows that for any $x$, $W(x, \lambda)=-\mu$. 
From the last equality and estimates \eqref{e21} it follows that
\begin{equation} \label{e23}
|K(x, t, \lambda)|\le\frac{2C_1^2}\mu.
\end{equation}
It is easy to see that
\begin{equation} \label{e24}
\overset{\circ}{u}_n(x)=\beta_1\gamma_n\psi(x, \overset{\circ}{\lambda}_n)+\zeta_n\phi(x, \overset{\circ}{\lambda}_n).
\end{equation}
By \cite[Ch. 3, Sec. 6, Th. 6.4]{c1} solutions $u_n(x, \lambda_n(p_n))$ of
the initial value problem \eqref{e4'} satisfy
\begin{equation} 
\begin{aligned}  \label{e25}
&u_n(x, \lambda_n(p_n))\\
&=p\left[\beta_1\gamma_n\psi(x, \lambda_n(p_n))+\zeta_n\phi(x, \lambda_n(p_n))\right]\\
&\quad +\int_0^x K(x, t, \lambda_n(p_n))(q(t, u_n
(t, \lambda_n(p_n)), \lambda_n(p_n))-q_0(t)) u_n(t, \lambda_n(p_n))dt.
\end{aligned}
\end{equation}
Now, we establish an upper bound for $u_n(x, \lambda_n(p_n))$. Clearly
\begin{equation} \label{e26}
|p(\beta_1\gamma_n\psi(x, \lambda_n(p_n))+\zeta_n\phi(x, \lambda_n(p_n)))|
\le C_3|p|.
\end{equation}
 From Lemma \ref{lm1} and estimates for the $p_n$ it follows that for any
$p$ satisfying condition \ref{e*},
$|u_n(x, \lambda_n(p_n))|\le\frac{M_0}2$. Thus from \eqref{e3} it
follows that
\begin{equation} \label{e27}
|q(t, u_n(x, \lambda_n(p_n)), \lambda_n(p_n))-q_0(t) |\le 2M.
\end{equation}
 Applying  Gronwall's lemma to \eqref{e25} using \eqref{e23}, \eqref{e26}, 
and \eqref{e27}, we see that
\begin{equation} \label{e28}
|u_n(x, \lambda_n(p_n))|\le C_4|p|.
\end{equation} 
 From \eqref{e18}, \eqref{e19}, \eqref{e20}, \eqref{e21}, \eqref{e27}, and 
\eqref{e28} it follows that for $n>N_1$
\begin{align*}
&\big|\frac{u_n(x, \lambda_n(p_n))}p-\overset{\circ}{u}_n(x)\big| \\
&\le|\beta_1\gamma_n| \big|\psi(x, \lambda_n(p_n))-\psi(x, \overset{\circ}{\lambda}_n)\big|
+|\zeta_n|\big| \phi(x, \lambda_n(p_n))-\phi(x, \overset{\circ}{\lambda}_n)\big| \\
&\quad +\frac1p\int_0^x|K(x, t, \lambda_n(p_n))\|q
(t, u_n(t, \lambda_n(p_n)), \lambda_n(p_n))-q_0(t)||u_n(t,\lambda_n(p_n))|dt\\
&\le \frac{\widetilde{C}}n.
\end{align*}
Clearly there exists a number $N_2>N_1$ such that
$$
\widetilde{C}^2\sum_{n=N_2}^\infty \frac1{n^2}<\frac12.
$$
From the last two inequalities and any $p$ satisfying condition
\ref{e*} it follows that
\begin{equation} \label{e29}
\sum_{n=N_2}^\infty\Big|\frac{u_n(x, \lambda_n(p_n))}p -\overset{\circ}{u}(x)\Big|^2<\frac12.
\end{equation}
Consider the case $0\le n\le N_2-1$. From \eqref{e11} it follows that
\begin{equation} \label{e30}
\lambda_n(p_n)\le\lambda_n^{(1)}(2M)\le\lambda_{N_2}^{(1)}(2M).
\end{equation}
Let $\varepsilon$ be an arbitrary positive number. From condition 1) of
Theorem \ref{thm1} it follows that there exists $\delta$, $0<\delta<1$,
such that for $|u|<\delta$,
$1\le\lambda\le\lambda_{N_2}^{(1)}(2M)$, $0\le t\le 1$,
\begin{equation} \label{e31}
|q(t, u, \lambda)-q_0(t)|<\varepsilon.
\end{equation}
Let $p$ satisfy the supplementary condition
$$
0<|p|<\frac{\delta}{C_4+1}.
$$
Then from \eqref{e28} we obtain $|u_n(x, \lambda_n(p_n))|<\delta$. From
the last inequality, \eqref{e30}, and \eqref{e31} it follows that
$$
Q_1(x, \lambda_n(p_n), \varepsilon)\le-q(x, u_n(x, \lambda_n(p_n)), \lambda_n(p_n))+
\lambda_n(p_n)\le Q_2(x, \lambda_n(p_n), \varepsilon).
$$
From this and the comparison theorem, \cite{h1}, we obtain
$$
\theta_1(1, \lambda_n(p_n), \varepsilon)\le\varphi_n(1, \lambda_n(p_n))\le\theta_2
(1, \lambda_n(p_n), \varepsilon).
$$ 
Moreover, we have the equality
$$
\theta_1(1, \lambda_n^{(1)}(\varepsilon), \varepsilon)=\varphi_n(1, \lambda_n(p_n))
=\theta_2(1, \lambda_n^{(2)}(\varepsilon), \varepsilon)=\arctan 
(-\frac{\alpha_2}{\beta_2})+\pi n.
$$
 From the last two relations it follows that
\begin{gather*}
\theta_1(1, \lambda_n(p_n), \varepsilon)\le\theta_1(1, \lambda_n^{(1)}(\varepsilon),\varepsilon),\\
\theta_2(1, \lambda^{(2)}_n(\varepsilon), \varepsilon) \le\theta_2(1, \lambda_n(p_n), \varepsilon).
\end{gather*}
 From this and the monotonicity of the functions
$\theta_i(1, \lambda, \varepsilon)$, \cite{h1}, it follows that
$$
\lambda_n^{(2)}(\varepsilon)\le\lambda_n(p_n)\le\lambda_n^{(1)}(\varepsilon).
$$
From the last inequality and \eqref{e17} we obtain
\begin{equation} \label{e32}
|\lambda_n(p_n)-\lambda_n^{(0)}|\le 2\varepsilon.
\end{equation}
 From \eqref{e22}, \eqref{e23}, \eqref{e28}, \eqref{e31}, and \eqref{e32} 
it follows that
\begin{align*}
&\big|\frac{u_n(x, \lambda_n(p_n))}p -\overset{\circ}{u}_n(x)\big|  \\
&\le |\beta_1\gamma_n|\big|\psi(x, \lambda_n(p_n))-\psi(x, \overset{\circ}{\lambda}_n)\big|
+|\zeta_n|\big|\phi(x, \lambda_n(p_n))-\phi(x, \overset{\circ}{\lambda}_n)\big|\\
&\quad +\frac1p\int_0^x|K(x, t, \lambda_n(p_n))\|u_n(t,
\lambda_n(p_n))||q(t, u_n(t, \lambda_n(p_n)), \lambda_n(p_n))-q_0(t)|dt\\
&\le C_5\varepsilon\,,
\end{align*}
where the constant $C_5$ does not depend on $\varepsilon$.

Choosing $\varepsilon=1/(C_5\sqrt{2N_2})$ we obtain
$$
\sum_{n=0}^{N_2-1}\big|\frac{u_n(x, \lambda_n(p_n))}p
-\overset{\circ}{u}_n(x)\big|^2<\frac12.
$$
 From the last inequality and \eqref{e29} it follows that
$$
\sum_{n=0}^\infty\left\|\frac{u_n(x, \lambda_n(p_n))}p-\overset{\circ}{u}_n(x)
\right\|^2_{L_2(0, 1)}<1.
$$ 
\end{proof}

\noindent{\bf (e)} Obviously, the system
$\{\overset{\circ}{u}_n(x)\}$ forms an orthonormal basis for the space
$L_2(0,1)$. From the last inequality it follows that the
eigenfunction systems $\{u_n(x,\lambda_n(p_n))/p\}$ and
$\{u_n(x,\lambda_n(p_n))\}$ are a Bari basis and a Riesz basis,
respectively \cite{g1,k1}.

\subsection*{Acknowledgements}
This work was carried out while the first author was visiting the University
of Queensland. The first author was supported by a Raybould Fellowship.

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\end{document}