\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 87, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/87\hfil Stability of the basis property]
{Stability of the basis property of eigenvalue
systems of Sturm-Liouville operators with integral boundary condition}

\author[N. S. Imanbaev \hfil EJDE-2016/87\hfilneg]
{Nurlan S. Imanbaev}

\address{Nurlan S. Imanbaev \newline
Institute of Mathematics and Mathematical Modeling,
050010 125 Pushkin str., Almaty, Kazakhstan.  \newline
 Estabushment ``Kazakhstan Engineering and Pedagogical University of Friendship
of  Nations" , 060017, Shymkent, Kazakhstan}
\email{imanbaevnur@mail.ru}

\thanks{Submitted August 27, 2015. Published March 30, 2016.}
\subjclass[2010]{34L20, 34L40}
\keywords{Sturm-Liouville operator basis property; eigenfunction; eigenvalue;
\hfill\break\indent non self-adjoint problem; anti-periodic boundary condition;
perturbation; stability}

\begin{abstract}
 We study a question on stability and instability of the basis
 property of a system of  eigenfunctions of  the  Sturm - Liouville
 operator,  with  an integral perturbation of anti-periodic type
 on the boundary conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}\label{sec1}

Spectral theory of non-self-adjoint  boundary value problems for
ordinary differential equations on a finite interval goes back to
the classical works of  Birkhoff  \cite{b1} and Tamarkin \cite{t1}. They
introduced the concept of  regular boundary conditions and
investigated asymptotic behavior of  eigenvalues and
eigenfunctions of such problems. In their works Malamud and
 Oridoroga \cite{m1,m2} proved completeness of eigenfunctions and
associated functions for a wide class of boundary value problems
which includes regular boundary conditions. In space $ L^2 (0,1) $
we consider an operator $L_0 $, generated by the following
ordinary differential expression:
\begin{equation}\label{e1.1}
L_0 (u) \equiv  - u''(x) + q(x)u(x),\quad q(x) \in C[0,1],\; 0 < x < 1
\end{equation}
 and the boundary value conditions of the form
\begin{equation}\label{e1.2}
U_j (u) = a_{j1} u'(0) + a_{j2} u'(1) + a_{j3} u(0) + a_{j4} u(1) =0,\quad j = 1,2.
\end{equation}

When the boundary conditions \eqref{e1.2} are strongly
regular, the results by  Dunford \cite{d3,d4},
Mikhailov \cite{m3} and  Kesel'man \cite{k2} provide the Riesz basis
property in $ L^2 (0,1) $ of the eigenfunctions and associated
functions ($E$ and $AF$) system  of the problem. In the case when the
boundary conditions are regular but not strongly regular, the
question on basis property of $E$ and $AF$ system is not yet
completely resolved. We introduce the matrix of coefficients of
the boundary conditions \eqref{e1.2}:
$$
A = \begin{pmatrix}
   {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} }  \\
   {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} }
\end{pmatrix}.
$$

By $ A(ij) $ we denote the matrix composed of the i-th and  j-th
columns of the matrix $A $,
$ A_{ij}  = \det A({ij})$. Let the boundary conditions \eqref{e1.2}
be regular but not strongly regular. According to \cite[p. 73]{n1},
if the following conditions hold:
\begin{equation}\label{e1.3}
 A_{12}  = 0,\quad A_{14}  + A_{23}  \ne 0,\quad A_{14}  + A_{23}
=  \mp (A_{13}  + A_{24} ),
\end{equation}
then the boundary conditions \eqref{e1.2} are regular, but not strongly
regular boundary conditions.

 Makin \cite{m4} suggested dividing all regular, but not
strongly regular, boundary conditions into four types:
\begin{itemize}
\item[I] $A_{14}  = A_{23}$, $A_{34}  = 0$;

\item[II] $A_{14}  = A_{23}$, $A_{34}  \ne 0$;

\item[III] $A_{14}  \ne A_{23}$, $A_{34} = 0$;

\item[IV] $A_{14}  \ne A_{23}$, $A_{34} \ne 0$.
\end{itemize}
For example, periodical or antiperiodical boundary
conditions form the type I, and can be determined in the
following form:
$$
A_{14}  = A_{23} ,\quad A_{34}  = 0,
$$
That is, $ a_{11}  =  - a_{12,} a_{13}  = a_{14}  = a_{21}  = a_{22}  =
0 $ and $ a_{23}  =  - a_{24}$.

These conditions will be equivalent to matrix $A$, where the following
two options are possible:
$$
A =\begin{pmatrix}
   1 & { - 1} & 0 & 0  \\
   0 & 0 & 1 & { - 1}  \\
\end{pmatrix}
$$
are periodical or
$$
A = \begin{pmatrix}
   1 & 1 & 0 & 0  \\
   0 & 0 & 1 & 1  \\
\end{pmatrix}
$$
are antiperiodical, and the same boundary conditions with
``the lowest coefficients" form
the  type II. The boundary value conditions defined as $ A_{14}
\ne A_{23} ,\quad A_{34}  = 0$ form the type III. These
conditions are always equivalent to boundary conditions given by
the matrix A:
$$ A = \begin{pmatrix}
   1 & { - 1} & 0 & 0  \\
   0 & 0 & 1 & 0  \\
\end{pmatrix}.
$$
This case will be the aim of our research in this paper. Moreover,
Makin \cite{m4} allocated the one type of non-strongly regular
boundary value conditions, when $E$ and $AF$ systems of the spectral
problem
\begin{equation}
L_0 (u) \equiv  - u''(x) + q(x)u(x)
= \lambda u(x),\quad q(x) \in C[0,1],\; 0 < x <1,
\label{e1.1a}
\end{equation}
with boundary conditions of the form \eqref{e1.2} forms Riesz basis
for any potentials $ q(x) $. When $q(x) \equiv 0$, the problem about
basis property of $E$ and $AF$ system of the problem with general
regular boundary conditions has been completely resolved in \cite{l1}.
In \cite{d1,d2} questions on convergence of eigenfunctions
expansion of the Dirac operator in vector - matrix form and the
Hill operator, forming Riesz basis in $ L^2 (0,1) $, with regular,
but not strongly regular, boundary value conditions have been
considered. For Dirac operators  Mityagin \cite{m6} proved that
periodic (or anti-periodic)  boundary conditions give a rise to a
Riesz system of 2D projections.



\section{Statement of the problem}\label{sec2}

The spectral problem \eqref{e1.1a}-\eqref{e1.2} with boundary conditions of
the type I when $q(x) \equiv 0$ is self-adjoint;
the system of its eigenfunctions is the usual trigonometric
system, forming an orthonormal basis in $ L^2 (0,1)$. For the
case of non-self-adjoint initial operator the question about
preservation of the basis properties with some (weak in a certain
sense) perturbation was shown in the type of several examples in
\cite{i1}.

Riesz basis property of eigenfunctions and associated functions of
periodic and antiperiodic Sturm-Liouville problems was considered
in \cite{v1}. In \cite{i5,s1} questions on stability of basis properties of
the periodic problem for \eqref{e1.1a} were investigated with integral
perturbation of the boundary conditions \eqref{e1.2}, when $j = 2$, of
the type I; that is, $ A_{14}  = A_{23}$, $A_{34}  = 0$.
Moreover, in \cite{m5} similar issues at $q(x) \equiv 0$ have been studied.
 In the present paper we consider a spectral
problem close to research of \cite{m5} when $q(x) \equiv 0$, with integral
perturbation of the boundary conditions \eqref{e1.2}
when $j = 2$, which belong to type I:
\begin{gather}\label{e2.1}
 L_1 (u) \equiv  - u''(x) = \lambda u(x),\quad 0 < x < 1, \\
\label{e2.2}
U_1 (u) \equiv u(0) + u(1) = \int_0^1 {\overline {p(x)} u(x)dx} ,
\quad p(x) \in L^1 ({0,1}), \\
\label{e2.3} U_2 (u) \equiv u'(0) + u'(1) = 0.
\end{gather}
From \cite{s2} it follows that the $E$ and $AF$ system of
the problem \eqref{e2.1}-\eqref{e2.3} is complete and minimal in $ L^2 (0,1)$.
 Moreover, the $E$ and $AF$ system for any $ p(x) $
 forms Riesz basis with brackets. Our aim is to show that the basis property in
$L^2 (0,1) $ of the $E$ and $AF$ system of problem
\eqref{e2.1}-\eqref{e2.3} is unstable with respect to small changes of kernel
$ p(x) $ of integral perturbation.
In \cite{i6} the method of constructing the characteristic determinant of
the spectral problem with integral perturbation of the boundary conditions has
been suggested.

The basis properties in $ L^p ( - 1,1) $ of root functions of the
nonlocal problem for the equations with involution have been
studied in \cite{k3}. Instability of  basis properties of root
functions of the Schrodinger operator with nonlocal perturbation
of the boundary condition has been investigated in \cite{i3}.
In \cite{i2} they studied the question of stability and instability of  basis
property of system of eigenfunctions and associated functions of
the double differentiation operator with an integral perturbation
of Samarskii - Ionkin type boundary conditions. In \cite{m7} they
considered the eigenfunction expansion for Sturm-Liouville
problems with transmission conditions at one interior point.
Boundary value problems with transmission conditions were
investigated extensively in the recent years (see, for example,
\cite{a1,a2,k1,m8}).


\section{Characteristic determinant of a spectral problem}\label{sec3}

In this section we use the method in \cite{i6} to construct
the characteristic determinant of the problem with integral
perturbation of the boundary condition. Applying integration by
parts, for smooth enough complex-valued functions $u(x)$ and $v(x)$ we
obtain the Lagrange formula:
\begin{equation} \label{e3.1}
\begin{aligned}
&\int_0^1 {L_0 (u)\overline {\upsilon (x)} \,dx}
- \int_0^1 {u(x)\overline {L_0^* (\upsilon)} dx}\\
& = [ {u'(0) + u'(1)} ]\overline {\upsilon (0)}
 + u'(1 )[ {\overline {\upsilon (0)}  + \overline
{\upsilon (1)} } ] \\
&\quad - [ {u(0 ) + u(1)} ]\overline {\upsilon '(0)}
- u(1)[ {\overline {\upsilon '(0)} + \overline {\upsilon '(1)} }].
\end{aligned}
\end{equation}
Here $L_0^* (\upsilon )$  is the adjoint differential expression
\begin{equation}\label{e3.2}
L_0^* \upsilon  =  - \upsilon ''(x) + \overline {q(x)} \upsilon (x
), \quad 0 < x < 1.
\end{equation}

Consequently the operator $L_0^*$ corresponding to the operator
$L_0 $ is given by differential expression \eqref{e3.2} and the boundary
conditions
\begin{equation}\label{e3.3}
V_1 (\upsilon ) = \upsilon (0) + \upsilon (1) = 0,\quad
V_2 (\upsilon ) = \upsilon '(0) + \upsilon '(1) = 0.
\end{equation}

Also the operator  $L_1^* $ corresponding to the operator $L_1$ is
given by the loaded differential expression
\begin{equation}\label{e3.4}
L_1^* (\upsilon) =  - \upsilon ''(x) + \overline {q(x)}
\upsilon (x) + p(x)\upsilon '(0),\quad 0 < x < 1,
\end{equation}
and antiperiodic boundary conditions \eqref{e3.3}.
One of the aspects of this problem is
the fact that the adjoint problem to \eqref{e2.1}-\eqref{e2.3} is the spectral
problem for the loaded differential equation
\begin{equation}\label{e3.5}
\begin{gathered}
L_1 ^* (\upsilon) =  - \upsilon ''(x) + p(x)\upsilon '(0)
 = \overline \lambda \upsilon (x),\\
 V_1 (\upsilon ) = \upsilon (0) + \upsilon (1) = 0,\\
 V_2 (\upsilon) = \upsilon '(0) + \upsilon '(1) = 0.
\end{gathered}
\end{equation}

First, we construct the characteristic determinant of the spectral
problem \eqref{e2.1}-\eqref{e2.3}. Presenting the general solution of the
equation \eqref{e2.1} by the formula
 $$
u({x,\lambda }) = C_1 \cos \sqrt \lambda  x + C_2 \sin \sqrt \lambda  x,
$$
and with respect to the boundary conditions \eqref{e2.2}-\eqref{e2.3},
we obtain the following linear system for the coefficients $ C_k$:
\begin{gather*}
C_1 \Big[ {1 + \cos \sqrt \lambda   - \int_0^1 {\overline {p(x)}
\cos \sqrt \lambda  x\,dx} } \Big]
+ C_2 \Big[ {\sin \sqrt \lambda   - \int_0^1 {\overline {p(x)} \sin \sqrt
\lambda  x\,dx} } \Big] = 0, \\
C_1 \Big[ { - \sin \sqrt \lambda  } \Big]
+ C_2 \Big[ {1 + \cos \sqrt \lambda  } \Big] = 0.
 \end{gather*}
Its determinant will be the characteristic determinant of the
spectral problem \eqref{e2.1}-\eqref{e2.3}:
 \begin{equation}\label{e3.6}
\Delta _1 (\lambda)
= \begin{vmatrix}
1 + \cos \sqrt \lambda-\int_0^1\overline{p(x)}\cos\sqrt\lambda x\,dx
& \sin \sqrt \lambda - \int_0^1 \overline {p(x)} \sin \sqrt \lambda  x\,dx  \\
 - \sin \sqrt \lambda
 &  1 + \cos \sqrt \lambda
\end{vmatrix} .
\end{equation}

When $ p(x) = 0 $ we obtain the characteristic
determinant of the unperturbed problem \eqref{e2.1}-\eqref{e2.3}.
It is de denote by $ \Delta _0 (\lambda ) = 2({1 + \cos \sqrt\lambda  })$.
The number $ \lambda _k^0  = ({({2k - 1})\pi })^2 $ is the eigenvalue of
the unperturbed antiperiodic problem, and
$ u_{k0}^0  = \sqrt 2 \cos ( {({2k - 1})\pi x}) $,
$ u_{k1}^0  = \sqrt 2 \sin ({({2k - 1})\pi x })$ are
eigenfunctions. We represent the function $ p(x)$ in
the Fourier series form by the trigonometric system
\begin{equation}\label{e3.7}
p(x) = \sum_{k = 1}^\infty  {[ {a_k \cos ({({2k - 1})\pi x}
) + b_k \sin ({({2k - 1})\pi x})}]} .
\end{equation}

Using \eqref{e3.7}, we find more convenient representation for the
determinant $ \Delta _1 (\lambda )$. To do this, first,
we calculate integrals in \eqref{e3.6}. Simple calculations show that
\begin{gather*}
\int_0^1 {\overline {p(x)} } \cos (\sqrt
\lambda  x) dx = \sum_{k = 1}^\infty  {\frac{{[ {\bar
a_k \sqrt \lambda  \sin \sqrt \lambda   + \bar b_k ({(
{2k - 1})\pi })(\cos \sqrt \lambda   + 1)}
]\,}}{{\lambda  - ({({2k - 1})\pi })^2 }}} ,  \\
\int_0^1 {\overline {p(x
)} } \sin (\sqrt \lambda  x) dx = \sum_{k = 1}^\infty
{\frac{{[ {\bar a_k \sqrt \lambda  (1 + \cos \sqrt \lambda  )
+ \bar b_k ({({2k - 1})\pi })\sin \sqrt
\lambda  } ]\,}}{{\lambda  - ({({2k - 1}
)\pi })^2 }}} .
\end{gather*}

Using  these  results  and standard transformations, the
determinant \eqref{e3.6} is reduced to the form
$$
\Delta _1 ( \lambda ) = \Delta _0 (\lambda ) \cdot A(\lambda ),
$$
where
\begin{equation}\label{e3.8}
 A(\lambda ) = \Big[ {1 + \sum_{k = 1}^\infty {\bar
b_k \frac{{({2k - 1})\pi }}{{\lambda  - (
{({2k - 1})\pi })^2 }}} } \Big].
\end{equation}

Hence, the following theorem is proved.

\begin{theorem}\label{thm3.1}
The characteristic determinant of the spectral
problem \eqref{e2.1}-\eqref{e2.3} with the perturbed boundary value conditions
can be represented in the  form \eqref{e3.8}, where
$ \Delta _0 ( \lambda ) $ is the characteristic determinant of the
unperturbed antiperiodic spectral problem, $ b_k $ are
coefficients of  the expansion \eqref{e3.7} of the function
$ p(x ) $ into trigonometric Fourier series.
\end{theorem}

The function $ A(\lambda )$ in  \eqref{e3.8} has a
first-order pole at the points $ \lambda  = \lambda _k^0$, and
the function $ \Delta _0 (\lambda ) $ has a second
order zero at these points. Therefore, the function $ \Delta _1
(\lambda)$, represented by the formula \eqref{e3.8}, is an
entire analytic function of the variable $ \lambda $. The
characteristic determinant, which is an entire analytical
function, related with the problem on eigenvalues of differential
operator of the third order with nonlocal boundary conditions has
been studied in \cite{i4}.

\section{Particular cases of the characteristic determinant}

The characteristic determinant \eqref{e3.8}  looks simpler when
$$
p(x) = \sum_{k = 1}^N {\big[ {a_k \cos (
{({2k - 1})\pi x}) + b_k \sin ({(
{2k - 1})\pi x})} \big]}.
$$
That is, there exists
such a number $ N $ such that $ a_k  = b_k  = 0 $ for all $ k > N$.
In this case,  formula \eqref{e3.8} takes the form
\begin{equation}\label{e4.1}
\Delta _1 (\lambda ) = \Delta _0 (
\lambda)\Big[ {1 + \sum_{k = 1}^N {\bar b_k
\frac{{({2k - 1})\pi }}{{\lambda  - ({(
{2k - 1})\pi })^2 }}} } \Big].
\end{equation}
From this particular case of  formula \eqref{e3.8}, the
we have the following corollary.

\begin{corollary}  \label{coro4.1}
For any preassigned numbers - a complex $ \hat
\lambda $ and a positive integer $ \hat m $ there always exists a
function $ p(x) $ such that $ \hat \lambda $ will be
an eigenvalue  of the problem \eqref{e2.1}-\eqref{e2.3} of multiplicity
$ \hat m$.
\end{corollary}

From the analysis of  formula \eqref{e4.1} it is easy to see that
$ \Delta_1 ({\lambda _k^0 }) = 0 $ for all $ k > N $. That
is, all eigenvalues $ \lambda _k^0 $, $k > N$, of the unperturbed
periodic problem are the eigenvalues of the perturbed spectral
problem \eqref{e2.1}-\eqref{e2.3}.
It is also not difficult to show that the
multiplicity of the eigenvalues $\lambda _k^0$, $k > N$ is also
preserved. Moreover, from the condition of orthogonality of the
trigonometric system it follows that in this case:
$$
\int_0^1 \overline {p(x)} u_{kj}^0 (x)dx =  0,\quad
j = \overline {0,1} ,\;k > N.\,
$$
Thus, the eigenfunctions $u_{kj}^0 (x)$ of the antiperiodic
problem when $k > N$ satisfy the boundary value conditions
\eqref{e2.2}-\eqref{e2.3} and, therefore, they are eigenfunctions
of the perturbed problem \eqref{e2.1}-\eqref{e2.3}.
 Hence, in this case the system of eigenfunctions of \eqref{e2.1}-\eqref{e2.3} and the
system of eigenfunctions of the periodic problem (an  orthonormal
basis)  differ from each other only in a finite number of  the
first members. Consequently, the system of eigenfunctions of
\eqref{e2.1}-\eqref{e2.3} also forms a Riesz basis in $ L^2({0,1}) $.
The set of functions $ p(x)$, that can be represented as a finite series
\eqref{e3.7}, is dense in $L^1 ({0,1})$. Thus, we have proved the following
result.

\begin{theorem}\label{thm4.1}
Let $ A_{14}  = A_{23}$, $A_{34}  = 0$;
 that is, the boundary conditions \eqref{e2.2}-\eqref{e2.3}
belong to type I with  integral perturbation.
Then the set of  functions $ p(x) \in L^1 ({0,1}) $, such that the
system of eigenfunctions of the perturbed problem \eqref{e2.1}-\eqref{e2.3}
forms Riesz basis in $ L^2 ({0,1})$, is dense in $L^1 ({0,1})$.
\end{theorem}

\section{Instability of the basis property}

Now we show that basis properties of eigenfunctions system of the
perturbed problem \eqref{e2.1}-\eqref{e2.3} is unstable for an arbitrarily
small integral perturbation of the boundary-value condition \eqref{e2.2}.

\begin{theorem}\label{thm5.1}
Suppose that $ A_{14}  = A_{23}$, $A_{34}  =0$; that is, the boundary-value
conditions \eqref{e2.2}-\eqref{e2.3} belong
to  type I. Then the set of functions $ p(x) \in L^1 ({0,1})$, such that
the system of eigenfunctions of the perturbed problem \eqref{e2.1}-\eqref{e2.3}
does not form even a normal basis in $ L^2 ({0,1}) $, is dense in $ L^1 ({0,1})$.
\end{theorem}

\begin{proof}
Let in \eqref{e3.7} the coefficients $ b_k \ne 0 $ for all
sufficiently large $k$. Then from \eqref{e3.8} we note that
 $ \lambda =\lambda _k^0 $ is a
simple eigenvalue of problem \eqref{e2.1}-\eqref{e2.3}.
By direct calculation we get that
$ u_k^1  = b_k \cos ({({2k - 1})\pi})x - a_k \sin ({({2k - 1})\pi })x $
are eigenfunctions of  \eqref{e2.1}-\eqref{e2.3},
corresponding to $ \lambda _k^0  = ({({2k - 1})\pi })^2$.
Moreover, the eigenfunction of the dual
problem \eqref{e3.5}, corresponding to the eigenvalue $ \lambda _k^0 $,
is $ v_k^1 (x) = c_k \cos ({({2k - 1})\pi })x $.

Since the eigenfunctions of the dual problems form biorthogonal
system, then we have the equality of the scalar product
$ ({u_k^1 ,v_k^1 }) = 1 $. Hence, it is easy to obtain
$ b_k \bar c_k = 2$. Therefore,
\begin{equation}\label{e5.1}
\|{u_k^1 } \| \cdot \| {v_k^1 } \|
= \sqrt {1 +| {\frac{{a_k }}{{b_k }}} |^2 }.
\end{equation}

Denote by $ \sigma _N (x) $ partial sum of the Fourier series
\eqref{e3.7}.  It is obvious, that the set of functions, which can be
represented as the infinite series
$$
\tilde p(x) = \sigma _N (x) + \sum_{k = N + 1}^\infty  {[ {\tilde
a_k \cos ({({2k - 1})\pi x}) + \tilde
b_k \sin ({({2k - 1})\pi x})}]},
$$
where $ \tilde a_k  = 2^{ - k}$,
$\tilde b_k  = 2^{ -k}/k$, $k > N$, is dense in $ L^1 ({0,1})$. However,
from \eqref{e5.1}  it  follows that for such kind of functions
$ \tilde p(x) $ for the corresponding eigenfunctions systems
of the direct and conjugate problems there holds:
$ \mathop {\lim }_{k \to \infty } \| {u_k^1 } \|\| {v_k^1 }\| = \infty $.

That is, the condition of uniform minimal property  (see \cite{i1} and
references in it)  of the system does not hold, and therefore, it
does not form even a basis in $ L^2 ({0,1}) $.
\end{proof}

Since  adjoint operators possess the Riesz basis property
of the root functions, we obtain the corollary.

\begin{corollary} \label{coro5.2}
 Suppose that $ A_{14}  = A_{23}$, $A_{34}  = 0$,
that is  boundary
value conditions \eqref{e2.2}-\eqref{e2.3} belongs to type I.
Then the set $ P $ of functions $ p(x) \in L^1 ({0,1})$, such that the
system of eigenfunctions of  \eqref{e3.5} for the loaded
differential equations forms Riesz basis in $L^2 ({0,1})$, is
 everywhere dense in $ L^1 ({0,1})$. The set $ L^1 ({0,1})\backslash P $
is also everywhere dense in $ L^1 ({0,1})$.
\end{corollary}

The results of this paper, in contrast to \cite{s2}, demonstrate
instability of basis properties of the root functions of the
problem with an integral perturbation of the boundary value
conditions of  type  I, which are regular, but not strongly
regular.


\subsection*{Acknowledgements}
This research was financially supported by a grant from the
Ministry of Science and Education of the Republic of Kazakhstan
(Grant No. 0825/GF4).

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