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\begin{document}
\title[On the basisness in $L_{2}(0,1)$ of the root functions]{On the
basisness in $L_{2}(0,1)$ of the root functions in not strongly regular
boundary value problems}
\author{Khanlar R. Mamedov}
\address[Khanlar R. MAMEDOV]{Mersin University, Science and Arts Faculty,
Mathematics Department, 33343 Ciftlikkoy Campus, Mersin-Turkey}
\email{hanlar@mersin.edu.tr}
\author{Hamza Menken}
\address[Hamza MENKEN]{Mersin University, Science and Arts Faculty,
Mathematics Department, 33343 Ciftlikkoy Campus, Mersin-Turkey}
\email{hmenken@mersin.edu.tr}
\subjclass[2000]{34L10, 34B24, 47E05}
\keywords{Riesz basis, periodic and anti-periodic boundary conditions,
regular boundary conditions, not strongly regular boundary conditions,
non-self adjoint Sturm-Liouville operator, Bari's theorem.}

\begin{abstract}
In the present article we consider the non-self adjoint Sturm-Liouville
operators with periodic and anti-periodic boundary conditions which are not
strongly regular. We obtain the asymptotic formulas for eigenvalues and
eigenfunctions of these boundary value problems, when the potential $q(x)\in
C^{(4)}[0,1]$ is a complex-valued function$.$ Then using these asymptotic
formulas, the Riesz basisness in $L_{2}(0,1)$ of the root functions are
proved.
\end{abstract}

\maketitle

\section{Introduction}

\qquad {\large It is well known that the basisness of the root functions of
a differential operator depends on regularity of boundary conditions
generating the given differential operator. The basisness in the space }$%
L_{2}(0,1)$ {\large of the root functions of a linear differential operator
of order }$n${\large \ with regular (strongly regular, see. \cite{Naim},
p.71) boundary conditions is shown in \cite{Kesel, Mikh}. In \cite{Kesel,
Dun-Sch, Walk} it is shown that the root functions of a boundary problem
which is generated by not strongly regular boundary conditions may not be
form a basis in }$L_{2}(0,1)${\large . In \cite{Ionk}, one non-classical
heat conduction problem in homogeneous rod has been studied. This problem is
reduced to the following boundary value problem }%
\begin{equation*}
-y^{\prime \prime }(x)=\lambda y(x),\text{ \ \ \ \ }0<x<1,
\end{equation*}%
\begin{equation*}
y(0)=0\text{, \ }y^{\prime }(0)=y^{\prime }(1)
\end{equation*}%
{\large whose boundary conditions are regular, but not strongly regular. All
the eigenvalues of this problem starting with the second one are double, the
total number of associated functions is infinite. Nevertheless, in the paper
it was established that the chosen specially system of the root functions
forms an unconditional basis in }$L_{2}(0,1).$

{\large After this work, in \cite{Ker-Mam}, the boundary-value problem
generated by the differential equation }%
\begin{equation}
y^{\prime \prime }+q(x)y=\lambda y  \label{1}
\end{equation}%
{\large and not strongly regular boundary conditions }%
\begin{equation}
y(0)-y(1)=0,\text{ \ \ }y^{\prime }(0)-y^{\prime }(1)=0  \label{2}
\end{equation}%
{\large or }%
\begin{equation}
y(0)+y(1)=0,\text{ \ \ }y^{\prime }(0)+y^{\prime }(1)=0  \label{3}
\end{equation}%
{\large was considered. Here, }$q(x)\in C^{(4)}[0,1]${\large \ \ was a
complex valued function satisfying the condition }$q(0)\neq q(1).${\large \
In this paper, it was shown that the root functions of the boundary problems
(\ref{1}), (\ref{2}) and (\ref{1}), (\ref{3}) formed Riesz basis in }$%
L_{2}(0,1).$

{\large Let us present briefly the main definitions and fact which will be
used in what follows.}

\begin{definition}
{\large \label{Def.1} A system }$\left\{ \varphi _{n}\right\} _{n=1}^{\infty
}${\large \ forms a basis in the space }$X${\large \ if, for any element }$%
f\in X${\large \ there exists a unique expansion of it in the elements of
the system, i.e. the series }$\underset{j=1}{\overset{\infty }{\tsum }}%
c_{j}\varphi _{j}${\large \ convergent to }$f${\large \ in the norm of the
space }$X.$
\end{definition}

\begin{definition}
{\large \cite{Bari, Goh-Kre}\label{Def.2} A system }$\left\{ \varphi
_{n}\right\} _{n=1}^{\infty }${\large \ is called a Riesz basis of the
Hilbert space }$H${\large \ if \ there exists a bounded linear invertible
operator }$A${\large \ such that the system }$\left\{ A\varphi _{n}\right\}
_{n=1}^{\infty }${\large \ forms an orthonormal basis in }$H${\large .}
\end{definition}

\begin{theorem}
{\large \cite{Bari, Goh-Kre}\label{Theo.1}\ If the sequence }$\left\{
\varphi _{j}\right\} _{j=1}^{\infty }${\large \ is complete in the Hilbert
space }$H${\large , there corresponds to it a complete biorthogonal sequence 
}$\left\{ \psi _{j}\right\} _{j=1}^{\infty }${\large , and for any }$f\in H$%
{\large \ one has }$\underset{j=1}{\overset{\infty }{\tsum }}\left\vert
(f,\varphi _{j})\right\vert <\infty ,${\large \ }$\underset{j=1}{\overset{%
\infty }{\tsum }}\left\vert (f,\psi _{j})\right\vert ^{2}<\infty ,${\large \
then the sequence }$\left\{ \psi _{j}\right\} _{j=1}^{\infty }${\large \
forms a Riesz basis in }$H.$
\end{theorem}

{\large We consider the boundary-value problems (\ref{1}), (\ref{2}) and (%
\ref{1}), (\ref{3}), where }$q(x)\in C^{(4)}[0,1]${\large \ is a
complex-valued function. Without loss of generality, we can assume that }$%
\overset{1}{\underset{0}{\tint }}q(x)dx=0.$

{\large In the present paper, in Section 2 we obtain the asymptotic formulas
of eigenvalues and eigenfunctions of the boundary problems (\ref{1}), (\ref%
{2}). In Section 3, using these asymptotic formulas and Theorem \ref{Theo.1}%
, we prove the basisness in }$L_{2}(0,1)${\large \ of the root functions of
the boundary problem (\ref{1}), (\ref{2}). In Section 4, similar results are
obtained for the boundary problem (\ref{1}), (\ref{3}).}

\section{\protect\large The asymptotic formulas for eigenvalues and
eigenfunctions of the periodic problem}

{\large First we shall prove the following lemma.}

\begin{lemma}
{\large \label{Lemma1} All eigenvalues of the boundary-value problem (\ref{1}%
), (\ref{2}), starting from some number, are simple and form two infinite
sequences }$\lambda _{k,1},${\large \ }$\lambda _{k,2},${\large \ }$%
k=N,N+1,\cdot \cdot \cdot ,${\large \ where }$N${\large \ is a positive
integer and }%
\begin{equation}
\lambda _{k,1}=-(2k\pi )^{2}-\frac{q^{\prime }(1)-q^{\prime }(0)+\underset{0}%
{\overset{1}{\tint }}q^{2}(t)dt}{(4k\pi )^{2}}+O(\frac{1}{k^{3}}),  \label{6}
\end{equation}%
\begin{equation}
\lambda _{k,2}=-(2k\pi )^{2}+\frac{q^{\prime }(1)-q^{\prime }(0)-\underset{0}%
{\overset{1}{\tint }}q^{2}(t)dt}{(4k\pi )^{2}}+O(\frac{1}{k^{3}}),  \label{7}
\end{equation}%
{\large and the corresponding eigenfunctions are of the form }%
\begin{equation}
y_{k,1}(x)=\sin 2k\pi x+O(\frac{1}{k}),  \label{8}
\end{equation}%
\begin{equation}
y_{k,2}(x)=\cos 2k\pi x+O(\frac{1}{k}).  \label{9}
\end{equation}
\end{lemma}

\begin{proof}
{\large We assume that }$q(0)=q(1).$ {\large The case }$q(0)\neq q(1)$%
{\large \ was investigated in \cite{Ker-Mam}. Consider the equation (\ref{3}%
) or }%
\begin{equation}
y^{\prime \prime }+q(x)y+\mu ^{2}y=0,  \label{10}
\end{equation}%
{\large where }$\mu =\sqrt{-\lambda }${\large \ \ and }$\sqrt{r}e^{i\varphi
/2}${\large \ for }$-\pi <\varphi \leq \pi .${\large \ From \cite{Naim, Marc}%
, it is well known that the eigenvalues of the boundary problem (\ref{1}), (%
\ref{2}) are asymptotically located in pairs, i.e. }%
\begin{equation*}
\lambda _{k,1}=\lambda _{k,2}+O(k^{1/2})=-(2k\pi )^{2}\left\{ 1+\frac{\xi
_{0}}{k}+O\left( \frac{1}{k^{3/2}}\right) \right\} ,\text{ \ }(k=N,N+1,\cdot
\cdot \cdot )\text{.}
\end{equation*}%
{\large It follows from the last relation that }%
\begin{eqnarray*}
\mu _{k,1} &=&\sqrt{-\lambda _{k,1}}=2k\pi \left\{ 1+\frac{\xi _{0}}{2k}%
+O\left( \frac{1}{k^{3/2}}\right) \right\} ,\text{ \ }(k=N,N+1,\cdot \cdot
\cdot ) \\
\mu _{k,2} &=&\sqrt{-\lambda _{k,2}}=2k\pi \left\{ 1+\frac{\xi _{0}}{2k}%
+O\left( \frac{1}{k^{3/2}}\right) \right\} ,\text{ \ }(k=N,N+1,\cdot \cdot
\cdot ).
\end{eqnarray*}%
{\large Hence, there exists a positive number }$c_{o}${\large \ such that }$%
\left\vert \func{Im}\mu _{k,1}\right\vert \leq c_{o}${\large \ and }$%
\left\vert \func{Im}\mu _{k,2}\right\vert \leq c_{o}${\large . Thus, the
relation }%
\begin{equation*}
\mu _{k,1},\mu _{k,2}\in Q=\left\{ \mu :\func{Re}\mu \geq 0,\left\vert \func{%
Im}\mu \right\vert \leq c_{o}\right\}
\end{equation*}%
{\large holds for all }$k=N,N+1,\cdot \cdot \cdot ${\large . It is easy to
verify that }$Q\subset S_{0}-ic_{o}\equiv T${\large , where }$S_{0}=\left\{
\mu :0\leq \arg \mu \leq \frac{\pi }{2}\right\} ${\large .}

{\large From \cite{Naim, Marc}, it is well known that in a region }$T$%
{\large \ of the complex plane }$\mu $ {\large the equation(\ref{10}) has
two linear independent solutions }$\varphi _{1}(x,\mu ),${\large \ }$\varphi
_{2}(x,\mu )${\large \ satisfying the relations }%
\begin{equation*}
\varphi _{j}(x,\mu )=e^{\mu \omega _{j}x}\left\{ \underset{m=0}{\overset{6}{%
\sum }}\frac{u_{m}(x)}{(2\omega _{j}\mu )^{m}}+O(\frac{1}{\mu ^{7}})\right\}
,\text{ \ }(j=1,2),
\end{equation*}%
\begin{equation*}
\varphi _{j}^{\prime }(x,\mu )=\mu \omega _{j}e^{\mu \omega _{j}x}\left\{
u_{0}(x)+\underset{m=1}{\overset{6}{\sum }}\frac{u_{m}(x)+2u_{m-1}^{\prime
}(x)}{(2\omega _{j}\mu )^{m}}+O(\frac{1}{\mu ^{7}})\right\} ,\text{ }(j=1,2),
\end{equation*}%
{\large where }%
\begin{equation*}
\omega _{1}=-\omega _{2}=i,\text{ }u_{0}(x)\equiv 1,\text{ \ }u_{m}(x)=-%
\overset{x}{\underset{0}{\int }}l\left( u_{m-1}(t)\right) dt,\text{ }%
m=1,2,3,4,5,6.
\end{equation*}%
{\large It follows that}%
\begin{equation*}
\varphi _{j}(0,\mu )=1+O(\frac{1}{\mu ^{7}}),
\end{equation*}%
\begin{eqnarray*}
\varphi _{j}(1,\mu ) &=&e^{\mu \omega _{j}}\left\{ 1-\frac{1}{(2\omega
_{j}\mu )^{3}}[q^{\prime }(1)-q^{\prime }(0)+\underset{0}{\overset{1}{\tint }%
}q^{2}(t)dt]+\frac{1}{(2\omega _{j}\mu )^{4}}[q^{\prime \prime
}(1)-q^{\prime \prime }(0)\right. \\
&&+\frac{5}{2}q^{2}(1)-\frac{3}{2}q^{2}(0)-q(0)q(1)]-\frac{1}{(2\omega
_{j}\mu )^{5}}[q^{\prime \prime \prime }(1)-q^{\prime \prime \prime
}(0)+7q(1)q^{\prime }(1) \\
&&-5q(0)q^{\prime }(0)-q(0)q^{\prime }(1)-q(1)q^{\prime }(0)+(q(1)-q(0))%
\underset{0}{\overset{1}{\tint }}q^{2}(t)dt \\
&&+2\underset{0}{\overset{1}{\tint }}q^{3}(t)dt-\underset{0}{\overset{1}{%
\tint }}q^{\prime ^{2}}(t)dt]+\frac{1}{(2\omega _{j}\mu )^{6}}%
[q^{(4)}(1)-q^{(4)}(0)+9q(1)q^{\prime \prime }(1) \\
&&-7q(0)q^{\prime \prime }(0)-q(0)q^{\prime \prime }(1)-q(1)q^{\prime \prime
}(0)+\frac{11}{2}q^{\prime 2}(1)-\frac{9}{2}q^{\prime 2}(0)-q^{\prime
}(0)q^{\prime }(1) \\
&&+\frac{15}{2}q^{3}(1)-\frac{7}{2}q^{3}(0)-\frac{5}{2}q(0)q^{2}(1)-\frac{3}{%
2}q(1)q^{2}(0) \\
&&\left. +(q^{\prime }(1)-q^{\prime }(0))\underset{0}{\overset{1}{\tint }}%
q^{2}(t)dt+\frac{1}{2}(\underset{0}{\overset{1}{\tint }}q^{2}(t)dt)^{2}]+O(%
\frac{1}{\mu ^{7}})\right\} ,
\end{eqnarray*}%
\begin{eqnarray*}
\varphi _{j}^{\prime }(0,\mu ) &=&\mu \omega _{j}\left\{ 1-\frac{2q(0)}{%
(2\omega _{j}\mu )^{2}}+\frac{2q^{\prime }(0)}{(2\omega _{j}\mu )^{3}}-\frac{%
1}{(2\omega _{j}\mu )^{4}}[2q^{\prime \prime }(0)+2q^{2}(0)]\right. \\
&&+\frac{1}{(2\omega _{j}\mu )^{5}}[2q^{^{\prime \prime \prime
}}(0)+8q(0)q^{\prime }(0)]-\frac{1}{(2\omega _{j}\mu )^{6}}[2q^{(4)}(0) \\
&&\left. +10q^{\prime 2}(0)+12q(0)q^{\prime \prime }(0)+4q^{3}(0)]+O(\frac{1%
}{\mu ^{7}})\right\} ,
\end{eqnarray*}%
\begin{eqnarray*}
\varphi _{j}^{\prime }(1,\mu ) &=&\mu \omega _{j}e^{\mu \omega _{j}}\left\{
1-\frac{q(0)+q(1)}{(2\omega _{j}\mu )^{2}}+\frac{1}{(2\omega _{j}\mu )^{3}}%
[q^{\prime }(1)+q^{\prime }(0)-\underset{0}{\overset{1}{\tint }}%
q^{2}(t)dt]\right. \\
&&-\frac{1}{(2\omega _{j}\mu )^{4}}[q^{\prime \prime }(1)+q^{\prime \prime
}(0)+\frac{3}{2}q^{2}(1)+\frac{3}{2}q^{2}(0)-q(0)q(1)] \\
&&+\frac{1}{(2\omega _{j}\mu )^{5}}[q^{\prime \prime \prime }(1)+q^{\prime
\prime \prime }(0)+5q(1)q^{\prime }(1)+5q(0)q^{\prime }(0)-q(0)q^{\prime }(1)
\\
&&-q(1)q^{\prime }(0)+(q(1)+q(0))\underset{0}{\overset{1}{\tint }}%
q^{2}(t)dt-2\underset{0}{\overset{1}{\tint }}q^{3}(t)dt+\underset{0}{\overset%
{1}{\tint }}q^{\prime 2}(t)dt]
\end{eqnarray*}%
\begin{eqnarray*}
&&-\frac{1}{(2\omega _{j}\mu )^{6}}[q^{(4)}(1)+q^{(4)}(0)+\frac{13}{2}%
q^{\prime 2}(1)+\frac{9}{2}q^{\prime 2}(0)-q^{\prime }(0)q^{\prime }(1) \\
&&+7q(1)q^{\prime \prime }(1)+7q(0)q^{\prime \prime }(0)-q(0)q^{\prime
\prime }(1)-q(1)q^{\prime \prime }(0)+\frac{7}{2}q^{3}(1) \\
&&+\frac{7}{2}q^{3}(0)-\frac{3}{2}q(0)q^{2}(1)-\frac{3}{2}%
q(1)q^{2}(0)+(q^{\prime }(1)+q^{\prime }(0))\underset{0}{\overset{1}{\tint }}%
q^{2}(t)dt \\
&&\left. -\frac{1}{2}(\underset{0}{\overset{1}{\tint }}q^{2}(t)dt)^{2}]+O(%
\frac{1}{\mu ^{7}})\right\} .
\end{eqnarray*}

{\large Let us substitute all these expressions into the characteristic
determinant }%
\begin{equation*}
\Delta (\mu )=\left\vert 
\begin{array}{cc}
U_{1}(\varphi _{1}) & U_{1}(\varphi _{2}) \\ 
U_{2}(\varphi _{1}) & U_{2}(\varphi _{2})%
\end{array}%
\right\vert ,
\end{equation*}%
{\large where }$U_{1}(y)=y(1)-y(0),${\large \ }$U_{2}(y)=y^{\prime
}(1)-y^{\prime }(0)${\large .}\newline
{\large By elementary transformations, we obtain the relation}%
\begin{eqnarray}
(i\mu )^{-1}\Delta (\mu ) &=&e^{2i\mu }\left\{ 1-\frac{2q(0)}{(2i\mu )^{2}}-%
\frac{1}{(2i\mu )^{3}}\underset{0}{\overset{1}{\tint }}q^{2}(t)dt-\frac{1}{%
(2i\mu )^{4}}[2q^{\prime \prime }(0)-\frac{1}{2}q^{2}(1)\right.  \notag \\
&&+\frac{3}{2}q^{2}(0)+q(0)q(1)]-\frac{1}{(2i\mu )^{5}}[q(1)q^{\prime
}(1)-q(0)q^{\prime }(1)-q(0)q^{\prime }(0)  \notag \\
&&+q(1)q^{\prime }(0)-2q(0)\underset{0}{\overset{1}{\tint }}q^{2}(t)dt+2%
\underset{0}{\overset{1}{\tint }}q^{3}(t)dt-\underset{0}{\overset{1}{\tint }}%
q^{\prime 2}(t)dt]  \notag \\
&&-\frac{1}{(2i\mu )^{6}}[2q^{(4)}(0)+\frac{1}{2}q^{\prime 2}(1)+\frac{21}{2}%
q^{\prime 2}(0)-q^{\prime }(0)q^{\prime }(1)-q(1)q^{\prime \prime }(1) 
\notag \\
&&+11q(0)q^{\prime \prime }(0)+q(0)q^{\prime \prime }(1)+q(1)q^{\prime
\prime }(0)-2q^{3}(1)+3q^{3}(0)  \notag \\
&&\left. +3q(0)q^{2}(1)-(\underset{0}{\overset{1}{\tint }}q^{2}(t)dt)^{2}]+O(%
\frac{1}{\mu ^{7}})\right\}  \notag \\
&&-2e^{i\mu }\left\{ 1-\frac{2q(0)}{(2i\mu )^{2}}-\frac{1}{(2i\mu )^{4}}%
[2q^{\prime \prime }(0)+2q^{2}(0)]-\frac{1}{(2i\mu )^{6}}[2q^{(4)}(0)\right.
\notag \\
&&\left. +12q(0)q^{\prime \prime }(0)+10q^{\prime 2}(0)+4q^{3}(0)]+O(\frac{1%
}{\mu ^{7}})\right\}  \notag \\
&&+\left\{ 1-\frac{2q(0)}{(2i\mu )^{2}}+\frac{1}{(2i\mu )^{3}}\underset{0}{%
\overset{1}{\tint }}q^{2}(t)dt-\frac{1}{(2i\mu )^{4}}[2q^{\prime \prime }(0)-%
\frac{1}{2}q^{2}(1)\right.  \notag \\
&&+\frac{3}{2}q^{2}(0)+q(0)q(1)]+\frac{1}{(2i\mu )^{5}}[q(1)q^{\prime
}(1)-q(0)q^{\prime }(0)-q(0)q^{\prime }(1)  \notag \\
&&+q(1)q^{\prime }(0)-2q(0)\underset{0}{\overset{1}{\tint }}q^{2}(t)d+2%
\underset{0}{\overset{1}{\tint }}q^{3}(t)dt-\underset{0}{\overset{1}{\tint }}%
q^{\prime 2}(t)dt]  \notag \\
&&-\frac{1}{(2i\mu )^{6}}[2q^{(4)}(0)+\frac{1}{2}q^{\prime 2}(1)+\frac{21}{2}%
q^{\prime 2}(0)-q^{\prime }(0)q^{\prime }(1)-q(1)q^{\prime \prime }(1) 
\notag \\
&&+11q(0)q^{\prime \prime }(0)+q(0)q^{\prime \prime }(1)+q(1)q^{\prime
\prime }(0)-2q^{3}(1)+3q^{3}(0)  \notag \\
&&\left. +3q(0)q^{2}(1)-(\underset{0}{\overset{1}{\tint }}q^{2}(t)dt)^{2}]+O(%
\frac{1}{\mu ^{7}})\right\} ,  \label{11}
\end{eqnarray}%
{\large for }$\mu \in T${\large \ sufficiently large in absolute value.}

{\large Let }$b(\mu )${\large \ be the coefficient of }$e^{2i\mu }${\large \
in(\ref{11}). Using the expansion}%
\begin{equation*}
\frac{1}{1-x}=1+x+x^{2}+x^{3}+O(x^{4}),\text{ \ }x\rightarrow 0,
\end{equation*}%
{\large it can be easily seen that the relation}

\begin{eqnarray}
b^{-1}(\mu ) &=&1+\frac{2q(0)}{(2i\mu )^{2}}+\frac{1}{(2i\mu )^{3}}\underset{%
0}{\overset{1}{\tint }}q^{2}(t)dt+\frac{1}{(2\omega _{j}\mu )^{4}}%
[2q^{\prime \prime }(0)-\frac{1}{2}q^{2}(1)+\frac{11}{2}q^{2}(0)  \notag \\
&&+q(0)q(1)]+\frac{1}{(2i\mu )^{5}}[q(1)q^{\prime }(1)-q(0)q^{\prime
}(0)-q(0)q^{\prime }(1)+q(1)q^{\prime }(0)  \notag \\
&&+2q(0)\underset{0}{\overset{1}{\tint }}q^{2}(t)dt+2\underset{0}{\overset{1}%
{\tint }}q^{3}(t)dt-\underset{0}{\overset{1}{\tint }}q^{\prime 2}(t)dt]+%
\frac{1}{(2i\mu )^{6}}[2q^{(4)}(0)  \notag \\
&&+\frac{1}{2}q^{\prime 2}(1)+\frac{21}{2}q^{\prime 2}(0)-q^{\prime
}(0)q^{\prime }(1)-q(1)q^{\prime \prime }(1)+19q(0)q^{\prime \prime }(0) 
\notag \\
&&+q(0)q^{\prime \prime }(1)+q(1)q^{\prime \prime
}(0)-2q^{3}(1)+17q^{3}(0)+q(0)q^{2}(1)  \notag \\
&&\left. +4q(1)q^{2}(0)+\frac{1}{2}(\underset{0}{\overset{1}{\tint }}%
q^{2}(t)dt)^{2}]+O(\frac{1}{\mu ^{7}})\right\}  \label{12}
\end{eqnarray}%
{\large holds for }$\mu \in T${\large \ sufficiently large in absolute value.%
}

{\large Thus, for }$\mu \in T${\large \ sufficiently large in absolute
value, the equation }$\Delta (\mu )=0${\large \ is equivalent to the
equation }%
\begin{equation}
(i\mu )^{-1}b^{-1}(\mu )\Delta (\mu )e^{i\mu }=0.  \label{13}
\end{equation}%
{\large Using (\ref{11}), (\ref{12}) and the relations }$q(1)=q(0)${\large \
and }$q^{\prime }(1)\neq q^{\prime }(0)${\large , from the equation(\ref{13}%
) we obtain two equations}%
\begin{equation}
\mu _{k,1}=2k\pi +\frac{q^{\prime }(1)-q^{\prime }(0)+\underset{0}{\overset{1%
}{\tint }}q^{2}(t)dt}{(4k\pi )^{3}}+O(\frac{1}{k^{4}}),  \label{14}
\end{equation}%
\begin{equation}
\mu _{k,2}=2k\pi -\frac{q^{\prime }(1)-q^{\prime }(0)-\underset{0}{\overset{1%
}{\tint }}q^{2}(t)dt}{(4k\pi )^{3}}+O(\frac{1}{k^{4}}).  \label{15}
\end{equation}

{\large By Rouche's theorem, we have asymptotic expressions for the roots }$%
\mu _{k,1}${\large \ and }$\mu _{k,2}${\large \ , }$k=N,N+1,\cdots ,${\large %
\ of the equations (\ref{14}) and (\ref{15}), respectively, where }$N$%
{\large \ is a positive integer }%
\begin{equation}
\mu _{k,1}=2k\pi +\frac{q^{\prime }(1)-q^{\prime }(0)}{(4k\pi )^{3}}+O(\frac{%
1}{k^{4}}),  \label{16}
\end{equation}%
\begin{equation}
\mu _{k,2}=2k\pi -\frac{q^{\prime }(1)-q^{\prime }(0)}{(4k\pi )^{3}}+O(\frac{%
1}{k^{4}}).  \label{17}
\end{equation}%
{\large Note that }$\mu _{k,1}${\large \ and }$\mu _{k,2}${\large \ are
simple roots of the equations (\ref{14}) and (\ref{15}), respectively. From
the relations (\ref{16}), (\ref{17}) and the relations }$\lambda _{k,1}=-$%
{\large \ }$\mu _{k,1}^{2},${\large \ }$\lambda _{k,2}=-${\large \ }$\mu
_{k,2}^{2},${\large \ we obtain the formula (\ref{6}) and observe that these
eigenvalues are simple.}

{\large Let us calculate }$U_{2}(\varphi _{1}(x,\mu _{k,1}))${\large \ and }$%
U_{2}(\varphi _{2}(x,\mu _{k,1})).${\large \ Since}%
\begin{equation*}
e^{i\mu _{k,1}}-1=\frac{q^{\prime }(1)-q^{\prime }(0)+\underset{0}{\overset{1%
}{\tint }}q^{2}(t)dt}{(2i\mu _{k,1})^{3}}+O(\frac{1}{\mu _{k,1}^{4}}),
\end{equation*}%
{\large we have}%
\begin{eqnarray}
U_{2}(\varphi _{1}(x,\mu _{k,1})) &=&\varphi _{1}^{\prime }(1,\mu
_{k,1})-\varphi _{1}^{\prime }(0,\mu _{k,1})  \notag \\
&=&i\mu _{k,1}e^{i\mu _{k,1}}[1-\frac{q(1)+q(0)}{(2i\mu _{k,1})^{2}}+\frac{%
q^{\prime }(1)+q^{\prime }(0)-\underset{0}{\overset{1}{\tint }}q^{2}(t)dt}{%
(2i\mu _{k,1})^{3}}+O(\frac{1}{\mu _{k,1}^{4}})]  \notag \\
&&-i\mu _{k,1}[1-\frac{2q(0)}{(2i\mu _{k,1})^{2}}+\frac{2q^{\prime }(0)}{%
(2i\mu _{k,1})^{3}}+O(\frac{1}{\mu _{k,1}^{4}})]  \notag \\
&=&\frac{q^{\prime }(1)-q^{\prime }(0)}{(2i\mu _{k,1})^{2}}+O(\frac{1}{\mu
_{k,1}^{3}}).  \label{18}
\end{eqnarray}

{\large In a similar way, we obtain}%
\begin{equation*}
U_{2}(\varphi _{2}(x,\mu _{k,1}))=\frac{q^{\prime }(1)-q^{\prime }(0)}{%
(2i\mu _{k,1})^{2}}+O(\frac{1}{\mu _{k,1}^{3}}).
\end{equation*}

{\large Without loss of generality, we can assume that }$q^{\prime
}(1)-q^{\prime }(0)\neq 0.$ {\large Since }$U_{2}(\varphi _{j}(x,\mu
_{k,1}))\neq 0,${\large \ }$j=1,2,${\large \ and }$q^{\prime }(1)-q^{\prime
}(0)\neq 0,${\large \ we seek the eigenfunction }$y_{k,1}(x)${\large \
corresponding to the eigenvalue }$\lambda _{k,1}${\large \ in the form }%
\begin{equation}
y_{k,1}(x)=\frac{(2i\mu _{k,1})^{2}}{2i\left[ q^{\prime }(1)-q^{\prime }(0)%
\right] }\left\vert 
\begin{array}{cc}
\varphi _{1}(x,\mu _{k,1}) & \varphi _{2}(x,\mu _{k,1}) \\ 
U_{2}(\varphi _{1}(x,\mu _{k,1})) & U_{2}(\varphi _{2}(x,\mu _{k,1}))%
\end{array}%
\right\vert .  \label{19}
\end{equation}%
{\large From the equalities}%
\begin{equation*}
\varphi _{j}(x,\mu _{k,1})=e^{\mu _{k,1}\omega _{j}x}\left[ 1+\frac{u_{1}(x)%
}{(2w_{j}\mu _{k,1})}+\frac{u_{2}(x)}{(2w_{j}\mu _{k,1})^{2}}+O(\frac{1}{\mu
_{k,1}^{3}})\right] ,\text{ \ }j=1,2
\end{equation*}%
{\large and the formulas (\ref{18}), (\ref{19}) we obtain }%
\begin{equation*}
y_{k,1}(x)=\sin \mu _{k,1}x+O(\frac{1}{\mu _{k,1}}).
\end{equation*}%
{\large Therefore, the eigenfunction }$y_{k,1}(x)${\large \ satisfies the
asymptotic formula (\ref{10}).}

{\large In a similar way, since }$U_{1}(\varphi _{j}(x,\mu _{k,1}))\neq 0,$%
{\large \ }$j=1,2,${\large \ and }$q^{\prime }(1)-q^{\prime }(0)\neq 0,$%
{\large \ we can seek the eigenfunctions }$y_{k,2}(x)${\large \
corresponding to the eigenvalues }$\lambda _{k,2}${\large \ in the form }%
\begin{equation*}
y_{k,2}(x)=-\frac{(2i\mu _{k,2})^{3}}{4\left[ q^{\prime }(1)-q^{\prime }(0)%
\right] }\left\vert 
\begin{array}{cc}
\varphi _{1}(x,\mu _{k,2}) & \varphi _{2}(x,\mu _{k,2}) \\ 
U_{1}(\varphi _{1}(x,\mu _{k,2})) & U_{1}(\varphi _{2}(x,\mu _{k,2}))%
\end{array}%
\right\vert .
\end{equation*}%
{\large Thus, we obtain }%
\begin{equation*}
y_{k,2}(x)=\cos \mu _{k,2}x+O(\frac{1}{\mu _{k,2}}).
\end{equation*}%
{\large This completes the proof of the lemma.}
\end{proof}

\section{{\protect\large The Riesz basisness in }$L_{2}(0,1)${\protect\large %
\ of the root functions for the periodic problem}}

\begin{theorem}
{\large \label{Theo.2}The root functions of the boundary problem (\ref{1}), (%
\ref{2}) form a Riesz basis in }$L_{2}(0,1).$
\end{theorem}

\begin{proof}
{\large \ The system of the root functions of the boundary problem (\ref{1}%
), (\ref{2}) is complete and minimal in }$L_{2}(0,1)${\large . The
minimality of this system follows from the fact that this system has a
biorthogonal system consisting of the root functions of the adjoint operator 
}%
\begin{eqnarray*}
l^{\ast }(v) &=&v^{\prime \prime }+\overline{q(x)}v, \\
v(1) &=&v(0),\text{ \ }v^{\prime }(1)=v^{\prime }(0).
\end{eqnarray*}%
{\large For any }$f\in L_{2}(0,1),${\large \ with a direct computation we
have that }$\overset{\infty }{\underset{n=N}{\tsum }}${\large \ }$\left\vert
(f,y_{k,1})\right\vert ^{2}<\infty ,${\large \ }$\overset{\infty }{\underset{%
n=N}{\tsum }}${\large \ }$\left\vert (f,y_{k,2})\right\vert ^{2}<\infty $%
{\large . On the other hand, the eigenfunctions of the adjoint operator have
of the form }%
\begin{equation}
\upsilon _{k,1}(x)=2\sin 2k\pi x+O(\frac{1}{k}),  \label{a}
\end{equation}%
\begin{equation}
\upsilon _{k,2}(x)=2\cos 2k\pi x+O(\frac{1}{k}),  \label{b}
\end{equation}%
{\large and the inequalities }$\overset{\infty }{\underset{n=N}{\tsum }}$%
{\large \ }$\left\vert (f,\upsilon _{k,1})\right\vert ^{2}<\infty $ {\large %
and\ }$\overset{\infty }{\underset{n=N}{\tsum }}${\large \ }$\left\vert
(f,\upsilon _{k,2})\right\vert ^{2}<\infty $ \ {\large hold}. {\large %
According to Theorem \ref{Theo.1}, the root functions of the boundary
problem (\ref{1}), (\ref{2}) form a Riesz basis in }$L_{2}(0,1).${\large \
This completes the proof.}
\end{proof}

{\large We note that applying the main result in \cite{Ilin} and using the
asymptotic formulas (\ref{6})-(\ref{9}), (\ref{a}), (\ref{b}) Theorem \ref%
{Theo.2} can be proved.}

\section{{\protect\large The basisness in }$L_{2}(0,1)$ {\protect\large of
the root functions for the anti-periodic boundary-value problem}}

{\large Similarly, the following results are obtained for the boundary
problem (\ref{1}), (\ref{3}).}

\begin{lemma}
{\large \label{Lemma2}\ All eigenvalues of the boundary value problem (\ref%
{1}), (\ref{3}), starting from some number, are simple and form two infinite
sequence }$\lambda _{k,1},${\large \ }$\lambda _{k,2},${\large \ }$%
k=N,N+1,\cdots ,${\large \ where }$N${\large \ is a positive integer and}%
\begin{equation}
\lambda _{k,1}=-\left[ (2k+1)\pi \right] ^{2}+\frac{q^{\prime }(1)-q^{\prime
}(0)-\overset{1}{\underset{0}{\int }}q^{2}(x)dx}{[2(2k+1)\pi ]^{2}}+O(\frac{1%
}{k^{3}}),  \label{20}
\end{equation}%
\begin{equation}
\lambda _{k,2}=-\left[ (2k+1)\pi \right] ^{2}-\frac{q^{\prime }(1)-q^{\prime
}(0)+\overset{1}{\underset{0}{\int }}q^{2}(x)dx}{[2(2k+1)\pi ]^{2}}+O(\frac{1%
}{k^{3}}),  \label{21}
\end{equation}%
{\large and the corresponding eigenfunctions are of the form}%
\begin{equation}
y_{k,1}(x)=\sin (2k+1)\pi x+O(\frac{1}{k}),  \label{22}
\end{equation}%
\begin{equation}
y_{k,2}(x)=\cos (2k+1)\pi x+O(\frac{1}{k}).  \label{23}
\end{equation}
\end{lemma}

\begin{proof}
{\large In the anti-preiodic case, in a similar way to the proof Lemma \ref%
{Lemma1}, we have the relations}

\begin{equation*}
e^{i\mu }+1=\frac{q^{\prime }(1)-q^{\prime }(0)-\overset{1}{\underset{0}{%
\int }}q^{2}(x)dx}{(2i\mu )^{3}}+O(\frac{1}{\mu ^{4}}),
\end{equation*}%
\begin{equation*}
e^{i\mu }+1=-\frac{q^{\prime }(1)-q^{\prime }(0)-\overset{1}{\underset{0}{%
\int }}q^{2}(x)dx}{(2i\mu )^{3}}+O(\frac{1}{\mu ^{4}}).
\end{equation*}%
{\large From these relations we can obtain (\ref{20}) and (\ref{21}). Again
in a similar way to the proof Lemma \ref{Lemma1}, we obtain }%
\begin{eqnarray*}
U_{1}(\varphi _{1}(x,\mu _{k,1})) &=&\varphi _{1}(1,\mu _{k,1})+\varphi
_{1}(0,\mu _{k,1}) \\
&=&\frac{2[q^{\prime }(1)-q^{\prime }(0)]}{(2i\mu _{k,1})^{3}}+O(\frac{1}{%
\mu _{k,1}^{4}}),
\end{eqnarray*}%
\begin{eqnarray*}
U_{1}(\varphi _{2}(x,\mu _{k,1})) &=&\varphi _{2}(1,\mu _{k,1})+\varphi
_{2}(0,\mu _{k,1}) \\
&=&-\frac{2[q^{\prime }(1)-q^{\prime }(0)]}{(2i\mu _{k,1})^{3}}+O(\frac{1}{%
\mu _{k,1}^{4}}).
\end{eqnarray*}%
{\large Since }$U_{1}(\varphi _{j}(x,\mu _{k,1}))\neq 0,${\large \ }$j=1,2$%
{\large , we can seek the eigenfunction }$y_{k,1}(x)${\large \ corresponding
to the eigenvalue }$\lambda _{k,1}${\large \ in the form }%
\begin{equation*}
y_{k,1}(x)=-\frac{(2i\mu _{k,1})^{3}}{\left[ q^{\prime }(1)-q^{\prime }(0)%
\right] }\left\vert 
\begin{array}{cc}
\varphi _{1}(x,\mu _{k,1}) & \varphi _{2}(x,\mu _{k,1}) \\ 
U_{1}(\varphi _{1}(x,\mu _{k,1})) & U_{1}(\varphi _{2}(x,\mu _{k,1}))%
\end{array}%
\right\vert .
\end{equation*}%
{\large Hence, we have }%
\begin{equation*}
y_{k,1}(x)=\sin (2k+1)\pi x+O(\frac{1}{k}),
\end{equation*}%
{\large i.e., the formula (\ref{22}) satisfies. In similar way we can obtain
the formula (\ref{23}).}
\end{proof}

\begin{theorem}
{\large \label{Theo.3}The root functions of the boundary problem (\ref{1}), (%
\ref{3}) form a Riesz basis in }$L_{2}(0,1).$
\end{theorem}

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\end{document}