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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 278, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/278\hfil Basicity in $L_p$ of root functions]
{Basicity in $L_p$ of root functions for differential equations
 with involution}

\author[L. V. Kritskov, A. M. Sarsenbi \hfil EJDE-2015/278\hfilneg]
{Leonid V. Kritskov, Abdizhahan M. Sarsenbi}

\address{Leonid V. Kritskov \newline
Lomonosov Moscow State University,
Faculty of Computational Mathematics and Cybernetics,
119899 Moscow, Russia}
\email{kritskov@cs.msu.ru}

\address{Abdizhahan M. Sarsenbi \newline
Auezov South-Kazakhstan State University,
Department of Mathematical Methods and Modeling, 
160012 Shymkent Kazakhstan.\newline
Institute of Mathematics and Mathematical Modeling, 
050010 Almaty, Kazakhstan}
\email{abzhahan@mail.ru}

\thanks{Submitted October 17, 2015. Published November 4, 2015.}
\subjclass[2010]{34K08, 34L10, 46B15}
\keywords{ODE with involution; nonlocal boundary-value problem;
\hfill\break\indent basicity of root functions}

\begin{abstract}
 We consider the differential equation
 \[
 \alpha u''(-x)-u''(x)=\lambda u(x), \quad -1<x<1,
 \]
 with the nonlocal boundary conditions $u(-1)=0$, $u'(-1)=u'(1)$ where
 $\alpha\in (-1,1)$. We prove that if $r=\sqrt{(1-\alpha)/(1+\alpha)}$
 is irrational then the system of its eigenfunctions is complete and minimal
 in $L_p(-1,1)$ for any $p>1$, but does not constitute a basis.
 In the case of a rational value of $r$ we specify the way of choosing
 the associated functions which provides the system of all root functions
 of the problem forms a basis in $L_p(-1,1)$.
\end{abstract}

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

\section{Introduction and statement of results}

This article continues the research started in \cite{k6} where a full spectral
analysis in $L_2(-1,1)$  is given to the problem
\begin{equation}
\begin{gathered}
\alpha u''(-x)-u''(x) =\lambda u(x),\quad -1<x<1,\\
u(-1)=0,\quad u'(-1)=u'(1).
\end{gathered} \label{e1}
\end{equation}
The differential expression in \eqref{e1} contains the involution transform
of the argument $x$ while the parameter $\alpha$ belongs to $(-1,1)$.

When $\alpha$ equals zero the problem \eqref{e1} becomes the
 known Samarskii-Ionkin problem \cite{i3} which gives the classical example
of a boundary-value problem with regular, but not strongly regular
boundary conditions. It has an infinite number of associated functions,
and these functions could be tuned to produce (together with eigenfunctions)
an unconditional basis in $L_2(-1,1)$.

Such problems have a typical instability. Both the basicity of root
functions and the equiconvergence of the related spectral decomposition with
the Fourier trigonometric series could disappear at either of the following
situations:
(a) after a small change of associated functions in their root
subspaces \cite{i2};
(b) after a perturbation of the differential expression by
adding subordinate terms $a_1(x)u'(x)+a_2(x)u(x)$ with sufficiently small
coefficients \cite{i1,m1};
(c) after a small shift of the boundary conditions;
 e.g., of the form $u'(0)=u'(1)+\varepsilon u(1)$, $\varepsilon\in (0,1)$
\cite{i2}.
Vladimir A. Il'in called such instability the essential
 nonself-adjointness of the problem.

The considered boundary-value problem \eqref{e1} for the differential
equation with involution encapsulates the same instability but with respect
to its parameter $\alpha$.


\begin{proposition}[\cite{k6}] \label{prop1}
Denote
\begin{equation}
r=\sqrt{(1-\alpha)/(1+\alpha)}. \label{e2}
\end{equation}
Then
\begin{itemize}

\item[(1)] for any positive $r$, the system of root functions of \eqref{e1}
is complete and minimal in $L_2(-1,1)$;

\item[(2)] if $r$ is irrational then there are no associated functions while
the eigenfunctions of \eqref{e1} do not constitute a basis in $L_2(-1,1)$;

\item[(3)] if $r$ is rational then there is an infinite number of associated
 functions which could be chosen to make the whole system of root functions
of \eqref{e1} an unconditional basis in $L_2(-1,1)$.
\end{itemize}
\end{proposition}

In this paper we obtain an analogous result in any Lebesgue space
 $L_p(-1,1)$, $1<p<\infty$. We prove the following results.


\begin{theorem} \label{thm1}
Let $r$ in \eqref{e2} be a positive irrational number.
Then the system of eigenfunctions of \eqref{e1} is complete and minimal
in $L_p(-1,1)$, $1<p<\infty$, but is not uniformly minimal, and therefore
does not constitute a basis in $L_p(-1,1)$.
\end{theorem}

\begin{theorem} \label{thm2}
Let $r$ be rational. Then the system of root functions of \eqref{e1}
is complete and minimal in $L_p(-1,1)$, $1<p<\infty$, and the associated
functions could be chosen in such a way that the whole system forms a basis
in $L_p(-1,1)$.
\end{theorem}

The functional-differential equations with involutions evoked interest
of mathematicians in early 1940s. Since 1970s the qualitative theory of
first-order differential equations with involution is cultivated rather
extensively (see, e.g., books by Przeworska-Rolewicz \cite{p1},
Wiener \cite{w2} and the recent research by Watkins \cite{w1}).
Boundary-value problems for second and higher order equations have been
studied in \cite{g2,o1,p1,w2}.
 Cabada and Tojo added a new element in the previous
studies: the construction of the Green function \cite{c1,c2}. Spectral
topics (the basicity of root functions, equiconvergence of spectral expansions)
for first- and second-order operators which contain involution in their main
terms are discussed in \cite{k3,k4,k5,s1,s2}.

Since the pioneering paper by Ionkin \cite{i3} for the heat flow equation
and the introduction of a new approach to these problems by Il'in,
there have been many research papers on non-local boundary value problems
(see the overview in \cite{i2}). Among the recent ones -- the papers
of Aleroev, Kirane and Malik \cite{a1}, Ashyralyev and Sarsenbi \cite{a3},
Furati, Iyiola and Kirane \cite{f1}, Kerimov \cite{k2},
Makin \cite{m2}, Mokin \cite{m4}, Sarsenbi \cite{s3}, Sarsenbi
and Tengaeva \cite{s4}.

For non-Hilbert spaces, the spectral properties of conventional differential
operators were considered in \cite{a2,b2,g1,k7,s5}.


\section{The case of irrational number $r$}

As in \cite{k6}, one can easily calculate the spectrum of \eqref{e1}:
\begin{equation}
\sigma = \{ 0; \pi^2 (1\pm\alpha)n^2 ,\;  n\in\mathbb{N} \} \label{e3}
\end{equation}
and the corresponding eigenfunctions:
\begin{equation}
\begin{gathered}
\lambda_0=0: u_0(x)=x+1,\quad
\lambda'_l=\pi^2(1+\alpha) l^2: u^{(1)}_l(x) =\sin(\pi lx),\\
\lambda''_k=\pi^2(1-\alpha) k^2:
 u^{(2)}_k(x) = \cos(\pi kx)+ \frac{\cos \pi k}{\sin (\pi rk)} \sin(\pi rkx),
\quad l,k\in\mathbb{N}.
\end{gathered}\label{e4}
\end{equation}

The dual system is formed by eigenfunctions of  the adjoint problem
\begin{equation}
\begin{gathered}
\alpha v''(-x)-v''(x) =\lambda v(x),\quad -1<x<1,\\
v(-1)=v(1),\quad \alpha v'(-1)=-v'(1),
\end{gathered} \label{e5}
\end{equation}
namely,
\begin{equation}
\begin{gathered}
\lambda_0=0: v_0(x)=1/2,\quad
\lambda''_k=\pi^2(1-\alpha)k^2: v^{(2)}_k(x) =\cos(\pi kx),\\
\lambda'_l=\pi^2(1+\alpha)l^2:
v^{(1)}_l(x) = \sin(\pi lx)+ \frac{\cos \pi l}{r\sin (\pi l/r)}
 \cos\Bigl(\frac{\pi lx}{r}\Bigr),\quad l,k\in\mathbb{N}.
\end{gathered} \label{e6}
\end{equation}

Recall that the system $\{ e_n\}$ in a Banach space $\mathcal{B}$ is called
\textit{complete} in $\mathcal{B}$ if it spans
$\mathcal{B}$ and is \textit{minimal} if neither element in this system belongs
to the span of others.

It is known  \cite[pp. 6--8]{k1} that
\begin{itemize}
\item the system $\{ e_n\}$ is minimal in $\mathcal{B}$ if and only
if it has the dual system $\{ e^*_n\}$ in $\mathcal{B}^*$;

\item if $\mathcal{B}$ is reflexive then the system $\{ e_n\}$ is complete
in $\mathcal{B}$ if and only if it is total, i.e. the relations
$e^*(e_n)=0$ for all $n$ with a given $e^*\in\mathcal{B}^*$ yield  $e^*=0$.
\end{itemize}


\begin{lemma} \label{lem1}
Both systems \eqref{e4} and \eqref{e6} are complete and minimal in
$L_p(-1,1)$ for any $p>1$.
\end{lemma}


\begin{proof}
The minimality of the systems \eqref{e4} and \eqref{e6} is provided by
their mutual biorthogonality. Their completeness follows from totality.
For instance, consider a function $f\in L_q(-1,1)$, $q^{-1}+p^{-1}=1$,
 which is orthogonal to each function in \eqref{e4}.
Then, as $f(x)$ is orthogonal to the functions $u_k^{(1)}(x)$, and due to
the fact that the trigonometric system forms a basis in $L_q$
\cite[p.128]{k1}, the function $f(x)$ a.e. coincides with an even function.
Thus, we have
$$
0=\int_{-1}^1 f(x)u_k^{(2)}(x)  dx
= \sin(\pi rk) \int_{-1}^1 f(x)\cos(\pi kx)  dx
$$
and, since $r\not\in\mathbb{Q}$, $f(x)$ is orthogonal to $\cos(\pi kx)$, $k\in\mathbb{N}$,
and therefore, it is a.e. a constant function on $[-1,1]$.
The relation $\int_{-1}^1 f(x)u_0(x)  dx=0$ provides $f(x)$ vanishes
a.e. on $[-1,1]$.
The proof is complete.
\end{proof}

The system $\{ e_n\}\subset \mathcal{B}$ is called \textit{uniformly minimal}
in $\mathcal{B}$ \cite{m3} if its dual system $\{ e^*_n\}\subset \mathcal{B}^*$
satisfies the relation
\begin{equation}
\sup_n \Bigl(\| e_n\|\cdot\| e^*_n\|\Bigr) <\infty. \label{e7}
\end{equation}


\begin{lemma} \label{lem2}
Neither system \eqref{e4} nor  \eqref{e6} is uniformly minimal
in $L_p(-1,1)$, $p>1$.
\end{lemma}

\begin{proof}
Let us consider the system \eqref{e4} in the space $L_p(-1,1)$.
Taking into account that the $L_q(-1,1)$-norms of functions $v_k^{(2)}(x)$
in \eqref{e6} ($q^{-1}+p^{-1}=1$) satisfy the estimates
\begin{equation}
2^{1/q}\ge \| v_k^{(2)}\|_q\ge 2^{-1/p} \| v_k^{(2)}\|_1 \ge 2^{-1/p}, \label{e8}
\end{equation}
we show that there exists such a sequence $k_n$ of positive integers such that
the norm $\| u_{k_n}^{(2)}\|_p$ tends to infinity. Evaluating the $L_1$-norm
of the function $u_{k}^{(2)}(x)$:
\begin{equation}
\begin{aligned}
\int_{-1}^1 |u_{k}^{(2)}(x)|  dx
&\ge \frac{1}{|\sin(\pi rk)|} \int_{-1}^1 |\sin(\pi rkx)|  dx -2\\
&\ge \frac{1}{|\sin(\pi rk)|} \Bigl( 1-\frac{\sin(2\pi kr)}{2\pi kr}  \Bigr) -2
\end{aligned} \label{e9}
\end{equation}
one  notes (see \cite[p.25]{s6}) that the inequality
$| \frac{1}{r}-\frac{k}{s}| < 1/s^2$ has infinitely many solutions
$k=k_n$, $s=s_n\in\mathbb{N}$. Hence $ |\pi rk_n -\pi s_n| < \pi r/s_n$ and
$|\sin (\pi rk_n)|<|\sin(\pi r/s_n)|$.
Therefore, the right-hand side of inequality \eqref{e9} blows
up as $k=k_n\to\infty$ which means that the norm
\[
\| u_{k_n}^{(2)}\|_p\ge 2^{(1-p)/p} \| u_{k_n}^{(2)}\|_1
\]
 also tends to infinity.

Together with estimate \eqref{e8}, this shows that the condition of
uniform minimality \eqref{e7} is not valid for the functions $u_{k_n}^{(2)}(x)$
and $v_{k_n}^{(2)}(x)$.
The proof is complete.
\end{proof}

A system $\{ e_n\}\subset \mathcal{B}$ is called a \textit{ basis}
in $\mathcal{B}$ if, for any $f\in\mathcal{B}$, there exists a unique
convergent to $f$ series: $\sum_{n=1}^\infty \alpha_n e_n = f$.
In this case the series is called the biorthogonal series for $f$ and
$\alpha_n=e^*_n(f)$ for any $n$. Any basis in $\mathcal{B}$ is a uniformly
minimal system \cite{m3}.

It follows from Lemma \ref{lem2} that the systems \eqref{e4} and \eqref{e6}
do not form bases in $L_p(-1,1)$ whatever $1<p<\infty$.
Then the proof of Theorem \ref{thm1} is complete.


\section{The case of rational number $r$}

Now let $r$ be equal to the irreducible fraction $\frac{m_1}{m_2}$
($m_1,m_2\in\mathbb{N}$). Then the spectrum \eqref{e3} of
problem \eqref{e1} contains two subsequences that glue to each other:
\begin{equation}
\lambda_n^* \equiv \lambda'_{m_1n} = \lambda''_{m_2n}\quad \forall n\in\mathbb{N}.
\label{e10}
\end{equation}
These eigenvalues have multiplicity $2$ and there are one eigenfunction
and one associated function corresponding to them in each problem
\eqref{e1} and \eqref{e5}. The straightforward calculation shows that
the biorthogonal pairs are formed by the functions
(we use notation from \eqref{e4} and \eqref{e6})
\begin{equation}
\begin{gathered}
u_0(x),\quad u^{(1)}_l(x),\quad  l\not\equiv 0 \pmod{m_1},\\
u^{(2)}_k(x),\quad  k\not\equiv 0 \pmod{m_2},\\
u^*_n(x) =\sin (\pi m_1nx),\\
u^*_{n,1}(x) =\frac{ x\cos(\pi m_1nx)+(-1)^{(m_1+m_2)n}
\cos(\pi m_2nx)}{2(1+\alpha)\pi m_1n} + a_n u^*_n(x),
\quad n\in\mathbb{N},
\end{gathered} \label{e11}
\end{equation}
for  problem \eqref{e1}, and
\begin{equation}
\begin{gathered}
v_0(x),\quad v^{(1)}_l(x),\quad  l\not\equiv 0\ \pmod{m_1},\\
v^{(2)}_k(x),\quad  k\not\equiv 0 \pmod{m_2},\\
v^*_n(x) =2(1+\alpha)\pi m_1n(-1)^{(m_1+m_2)n} \cos(\pi m_2nx),\\
v^*_{n,1}(x) =-r^{-1}(-1)^{(m_1+m_2)n} x\sin(\pi m_2nx)+\sin(\pi m_1nx)
-a_n v^*_n(x),\quad n\in\mathbb{N},
\end{gathered}\label{e12}
\end{equation}
for  problem \eqref{e5}.
The functions $u^*_n(x),u^*_{n,1}(x)$ in \eqref{e11} and
$v^*_n(x),v^*_{n,1}(x)$ in \eqref{e12} for each $n\in\mathbb{N}$ are the eigen- and  associated
functions which correspond to the sequence $\{\lambda_n^*\}$ in \eqref{e10}.
The constants $a_n\in\mathbb{R}$ could be taken arbitrarily.

\begin{lemma} \label{lem3}
Systems \eqref{e11} and \eqref{e12} are complete and minimal
in $L_p(-1,1)$, $p>1$.
\end{lemma}

The proof of Lemma \ref{lem3} mimics the proof of Lemma \ref{lem1}, 
with minor changes. We omit it.

\begin{lemma} \label{lem4}
If $a_n=O(1/n)$, $n\to\infty$, then the systems \eqref{e11} and
\eqref{e12} are uniformly minimal in $L_p(-1,1)$, $p>1$.
If $\lim_{n\to\infty} n{a_n}=\infty$ then these systems are not uniformly
 minimal and, therefore, do not form bases.
\end{lemma}

\begin{proof}
 We start with eigenfunctions of the biorthogonal pair $u_l^{(1)}(x)$ and
$v_l^{(1)}(x)$, $l\not\equiv 0 \pmod{m_1}$. Their norms satisfy the estimates:
$$
\| u_l^{(1)}\|_p\le 2^{1/p},\quad
\| v_l^{(1)}\|_q\le 2^{1/q}\Bigl( 1+\Bigl( r|\sin(\pi l/r)|\Bigr)^{-1}\Bigr).
$$
The right-hand part of the second estimate is bounded because for
$l\not\equiv 0 \pmod{m_1}$ the number $l/r=lm_2/m_1$ is not integer
and hence $|\sin(\pi l/r)|\ge \sin(\pi/m_1)$.

Similarly one can prove the boundedness of 
 $\| u_k^{(2)}\|_p\cdot \| v_k^{(2)}\|_q$ for
$k\not\equiv 0  \pmod{m_2}$.

In the case  $\lambda=\lambda_n^*$ the biorthogonal pairs are formed
 by the functions $u_n^*(x), v_{n,1}^*(x)$ and $u_{n,1}^*(x), v_n^*(x)$.
For all $n\in\mathbb{N}$ the relations
\begin{equation}
c_1\le \| u_n^*\|_p\le c_2,\quad c_1n\le \| v_n^*\|_q\le nc_2 \label{e13}
\end{equation}
are valid with some positive constants $c_1,c_2$.

If $a_n=O(1/n)$ then
$$
\| u_{n,1}^*\|_p\le \frac{c_3}{n}, \quad \| v_{n,1}^*\|_q\le c_3,
$$
and, by virtue of \eqref{e13}, the uniform minimality condition \eqref{e7}
is satisfied.

If $\lim_{n\to\infty} n{a_n}=\infty$ then we come to the estimates
$$
\| u_{n,1}^*\|_p\ge c_4 |a_n|>0, \quad \| v_{n,1}^*\|_q\ge c_4|a_n|n>0,
$$
which mean that $\| u_n^*\|_p\cdot \| v_{n,1}^*\|_q$ and
$\| u_{n,1}^*\|_p\cdot \| v_n^*\|_q$ disagree with \eqref{e7}.
The proof is complete.
\end{proof}

Further we consider the uniformly minimal systems \eqref{e11} and
\eqref{e12} and for simplicity suppose that $a_n\equiv 0$  for any $n$.
Since the natural normalization of the functions $u_{n,1}^*(x)$ and $v_{n}^*(x)$
 makes these systems uniformly bounded on $[-1,1]$, the known result of
Gaposhkin \cite{m3} provides they could form only conditional bases in
$L_p(-1,1)$ for $p\ne 2$. Therefore, in order to study their basis properties
we should specify the order of root functions in \eqref{e11} and \eqref{e12}.
In $L_2(-1,1)$ the order of root functions is irrelevant since they
form an unconditional basis \cite{k6}.

The proposed order will correspond to the order of functions in the
classical trigonometric system.
The biorthogonal system which consists of root functions of the problem
\eqref{e1} and the related root functions of the adjoint problem \eqref{e5}
starts with the pair
$$
\begin{bmatrix} u_0(x) \\ v_0(x) \end{bmatrix}
 = \begin{bmatrix} x+1 \\ 1/2 \end{bmatrix},
$$
which is followed by the juxtaposed  blocks ($k=1,2,\ldots$) of coupled pairs
\begin{align*}
&\begin{bmatrix} u_k^{(1)}(x) & u_k^{(2)}(x) \\ v_k^{(1)}(x) & v_k^{(2)}(x)
\end{bmatrix} \\
&=\begin{bmatrix} \sin(\pi kx) & \cos(\pi kx)+ \frac{\cos \pi k}{\sin \pi rk}
 \sin(\pi rkx) \\
\sin(\pi kx)+ \frac{\cos \pi k}{r\sin \frac{\pi k}{r}} \cos(\frac{\pi kx}{r})
& \cos(\pi kx) \end{bmatrix}.
\end{align*}
However if $k\equiv 0  \pmod{m_1}$ then the first column of the block should
be replaced by the column
$$
\begin{bmatrix} \sin(\pi kx) \\
\sin(\pi kx)-r^{-1}(-1)^{(1+r)k/r} x\sin(\frac{\pi kx}{r}) \end{bmatrix};
$$
if $k\equiv 0  \pmod{m_2}$ then the second column is also replaced by the column
$$
\begin{bmatrix}
(2(1+\alpha)\pi kr)^{-1} \Bigl[ (-1)^{(1+r)k}\cos (\pi kx) + x\cos(\pi krx)  \Bigr] \\
2(1+\alpha)\pi kr (-1)^{(1+r)k}\cos (\pi kx) \end{bmatrix}.
$$

Hence the partial sums of the biorthogonal series with respect to the root
functions of the problem \eqref{e1} take the form (we use the notation
$K_1=m_1\mathbb{N}$ and $K_2=m_2\mathbb{N}$)
\begin{equation}
\begin{aligned}
S_N(x,f)
& = (f,v_0) u_0(x) +
\sum_{\substack{1\le k\le N \\ k\not\in K_1}} (f,v_k^{(1)})  u_k^{(1)}(x)
+ \sum_{\substack{1\le k\le N \\ k\not\in K_2}} (f,v_k^{(2)})  u_k^{(2)}(x)\\
&\quad +\sum_{\substack{1\le k\le N \\ k\in K_1}}
\Bigl( f(t), \sin(\pi kt) -r^{-1}(-1)^{(1+r)k/r} t\sin\bigl(\frac{\pi kt}{r}\bigr)
 \Bigr)   \sin(\pi kx) \\
&\quad +\sum_{\substack{1\le k\le N \\ k\in K_2}}
\Bigl( f(t), \cos(\pi kt)\Bigr)  \Bigl[ \cos (\pi kx)
+ (-1)^{(1+r)k} x\cos(\pi krx)\Bigr] .
\end{aligned} \label{e14}
\end{equation}
This sum evidently contains the partial sum of the Fourier trigonometric series:
\begin{equation}
\begin{aligned}
&S_N^{(0)}(x,f) \\
&= (f,1/2) + \sum_{k=1}^N \Bigl\{ (f(t),\cos(\pi kt)) \cos(\pi kx)
+ (f(t),\sin(\pi kt)) \sin(\pi kx) \Bigr\},
\end{aligned}\label{e15}
\end{equation}
the remaining items group into the following sums:
\begin{gather}
S_N^{(1)}(x,f) = \sum_{\substack{1\le k\le N \\ k\not\in K_1}}
\frac{\cos \pi k}{r\sin\frac{\pi k}{r}} \Bigl( f(t),
\cos\big(\frac{\pi kt}{r}\big) \Bigr)  \sin(\pi kx), 
 \nonumber \\
S_N^{(2)}(x,f) = \sum_{\substack{1\le k\le N \\ k\not\in K_2}}
\frac{\cos \pi k}{\sin(\pi kr)} \Bigl(f(t), \cos(\pi kt) \Bigr)  
\sin(\pi krx), \nonumber \\
S_N^{(3)}(x,f) = -\sum_{\substack{1\le k\le N \\ k\in K_1}}
r^{-1}(-1)^{(1+r)k/r} \Bigl( f(t),t\sin\big(\frac{\pi kt}{r}\big) \Bigl)  
\sin(\pi kx), \nonumber \\
S_N^{(4)}(x,f) = \sum_{\substack{1\le k\le N \\ k\in K_2}}
(-1)^{(1+r)k} \Bigl( f(t),\cos(\pi kt)\Bigr)  x\cos(\pi krx). \label{e16}
\end{gather}

To analyze these four sums, we decompose $f(x)$ into the sum of its even
and odd components
$$
f(x) =f_+(x)+f_-(x) \equiv \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}
$$
and note that for the odd component $f_-(x)$ all the sums in \eqref{e16} vanish.

In $S_N^{(3)}(x,f_+)$ we make the substitution $k=m_1n$ and for simplicity
suppose that $m_1+m_2$ is even. Then this sum takes the form
$$
S_N^{(3)}(x,f_+) = -r^{-1} \sum_{\substack{1\le k\le N \\ k=m_1n}}
\int_0^1 f_+(t)t \sin(\pi m_2nt)  dt \cdot \sin(\pi m_1nx)
$$
and further substitutions $\tau=m_2t, y=m_1x$ transform it into the sum
$$
S_N^{(3)}(x,f_+) = -(rm_2^2)^{-1} \sum_{\substack{1\le k\le N \\ k=m_1n}}
\int_0^{m_2} f_+\Bigl(\frac{\tau}{m_2} \Bigr)\tau \sin(\pi n\tau)  d\tau
\cdot \sin(\pi ny).
$$
It could be easily interpreted as a sum of $m_2$ partial sums of Fourier
trigonometric series for functions which $L_p$-norms are $O(1)\| f\|_p$.
A similar conclusion could be made about $S_N^{(4)}(x,f_+)$.

The sum $S_N^{(2)}(x,f_+)$ naturally splits into $m_2-1$ items in accordance
with the remainder $k_1=k\pmod{m_2}$, $k_1=\overline{1,m_2-1}$.
We suppose, for simplicity, that $k_1$ and $m_1+m_2$ are even.
Then the corresponding parts of the sum equal
\begin{align*}
S_N^{(2,k_1)}(x,f_+)
&= \frac{1}{\sin(\pi k_1r)} \sum_{\substack{1\le k\le N \\ k=k_1+m_2n}}
\Bigl\{
\int_0^1 f(t)\cos(\pi k_1t) \cos(\pi m_2nt)  dt  \\
&\quad\times \Bigl[ \cos(\pi m_1nx)\sin(\pi k_1rx)
 + \sin(\pi m_1nx)\cos(\pi k_1rx) \Bigr] \\
&\quad -\int_0^1 f(t)\sin(\pi k_1t) \sin(\pi m_2nt)  dt  \\
&\quad \times \Bigl[ \cos(\pi m_1nx)\sin(\pi k_1rx)
+ \sin(\pi m_1nx)\cos(\pi k_1rx) \Bigr]
\Bigg\} .
\end{align*}

Similar to $S_N^{(3)}(x,f_+)$ this expression consists of four items
which are combinations of the partial sums of Fourier trigonometric
series for functions which $L_p$-norms are $O(1)\| f\|_p$,
and of the partial sums of conjugate trigonometric series which
converge in $L_p(0,1)$ to functions which $L_p$-norms are also
$O(1)\| f\|_p$ by  Riesz theorem \cite[p. 566]{b1}.
The remaining sum $S_N^{(1)}(x,f_+)$ is considered similarly.

It is known \cite[pp.593--594]{b1} that if $F(x)\in L_p$ then the partial
sums $\sigma_N(x,F)$ of its Fourier trigonometric series and the
 partial sums $\sigma_N^*(x,F)$ of its conjugate series satisfy the estimate
$$
\| \sigma_N(x,F)\|_p \le c\| F\|_p,\quad \| \sigma_N^*(x,F)\|_p \le c\| F\|_p
$$
uniformly with respect to $N$.

It follows from \eqref{e14}--\eqref{e16} that
\begin{equation}
\| S_N(x,f)\|_p \le \| (f,1/2)  x\|_p +\| S_N^{(0)}(x,f)\|_p
+ \sum_{j=1}^4 \| S_N^{(j)}(x,f_+)\|_p = O(1) \| f\|_p \label{e17}
\end{equation}
uniformly with respect to $N$.

The system of root functions of the problem \eqref{e1} is complete
and minimal in $L_p(-1,1)$ (Lemma \ref{lem3}), therefore
(see, e.g.,  \cite[p. 11]{k1})
the estimate \eqref{e17} is sufficient for its basicity in $L_p(-1,1)$
for $p>1$. Theorem \ref{thm2} is proved.

\subsection*{Acknowledgments}
This research was supported by the Committee of Science of the Ministry
for Education and Science of the Kazakhstan Republic (Grant N 0971/GF).

\begin{thebibliography}{99}

\bibitem{a1} T. S. Aleroev, M. Kirane, S. A. Malik;
 \emph{Determination of a source term for a time fractional diffusion equation
with an integral type over-determining condition},
\newblock{Electron. J. Differ. Equ.}, \textbf{2013:270} (2013), 1--16.

\bibitem{a2} Z. S. Aliyev, N. B. Kerimov;
 \emph{Basis properties of a spectral problem with spectral parameter in
the boundary condition},
\newblock{Sbornik: Mathematics}, \textbf{197}, No.10 (2006), 1467--1487.

\bibitem{a3} A. Ashyralyev, A. M. Sarsenbi;
\emph{Well--posedness of a parabolic equation with nonlocal boundary condition},
\newblock{Boundary Value Problems}, \textbf{2015}, article ID 38, 1--11.

\bibitem{b1} N. K. Bari;
 \emph{A Treatise on trigonometric series}, New York, 1964.

\bibitem{b2} H. E. Benzinger;
 \emph{The $L^p$ behavior of eigenfunction expansions},
\newblock{Trans. Amer. Math. Soc.}, \textbf{174} (1972), 333--344.

\bibitem{c1} A. Cabada, F. A. F. Tojo;
 \emph{Existence results for a linear equation with reflection, non-constant
coefficient and periodic boundary conditions},
\newblock{J. Math. Anal. Appl.}, \textbf{412}, No.1 (2014), 529--546.

\bibitem{c2} A. Cabada, F. A. F. Tojo;
\emph{Solutions and Green's function of the first order linear equation with
reflection and initial conditions},  \newblock{Boundary Value Problems},
\textbf{2014:99} (2014).

\bibitem{g1} A. M. Gomilko, G. V. Radzievskii;
 \emph{Equivalence in $L_p[0,1]$ of the system $e^{i2\pi kx}$ ($k=0,\pm 1,\ldots$)
and the system of the eigenfunctions of an ordinary functional-differential operator},
\newblock{Math. Notes}, \textbf{49}, No.1 (1991), 34--40.

\bibitem{g2} C. P. Gupta;
 \emph{Two-point boundary value problems involving reflection of the argument},
\newblock{Int. J. Math. and Math. Sci.}, \textbf{10}, No.2 (1987), 361--371.

\bibitem{f1} K. M. Furati, O. S. Iyiola, M. Kirane;
 \emph{An inverse problem for a generalized fractional diffusion},
\newblock{Appl. Math. Comp.}, \textbf{249} (2014), 24--31.

\bibitem{i1} V. A. Il'in;
 \emph{Basis property and equiconvergence with the trigonometric series
of root function expansions and the form of boundary conditions for a
nonself-adjoint differential operator},
\newblock{Differ. Equ.}, \textbf{30}, No.9 (1994), 1402--1413.

\bibitem{i2} V. A. Il'in and L. V. Kritskov;
\emph{Properties of spectral expansions corresponding to nonself-adjoint
differential operators},
\newblock{J. Math. Sci. (NY)}, \textbf{116}, No.5 (2003), 3489--3550.

\bibitem{i3} N. I. Ionkin;
\emph{Solution of a boundary-value problem in heat conduction with a nonclassical
boundary condition},
\newblock{Differ. Equ.}, \textbf{13}, No.2 (1977), 204--211.

\bibitem{k1} B. S. Kashin, A. A. Saakyan;
 \emph{Orthogonal series}, Transl. of Math. Monographs, AMS, vol.75, 1989.

\bibitem{k2} N. B. Kerimov;
 \emph{On a boundary value problem of N. I. Ionkin type},
\newblock{Differ. Equ.}, \textbf{49}, No.10 (2013), 1233--1245.

\bibitem{k3} A. P. Khromov, V. P. Kurdyumov;
 \emph{Riesz bases formed by root functions of a functional-differential
equation with a reflection operator},
\newblock{Differ. Equ.}, \textbf{44}, No.2 (2008), 203--212.

\bibitem{k4} A. A. Kopzhassarova, A. L. Lukashov, A. M. Sarsenbi;
\emph{Spectral properties of non-self-adjoint perturbations for
a spectral problem with involution},
\newblock{Abstr. Appl. Anal.}, \textbf{2012}, article ID 590781, 1--5.

\bibitem{k5} A. A. Kopzhassarova, A. M. Sarsenbi;
 \emph{Basis properties of eigenfunctions of second-order differential
operators with involution},
\newblock{Abstr. Appl. Anal.}, \textbf{2012}, article ID 576843, 1--6.

\bibitem{k6} L. V. Kritskov, A. M. Sarsenbi;
 \emph{Spectral properties of a nonlocal problem for the differential equation
with involution}, \newblock{Differ. Equ.}, \textbf{51}, No.8 (2015), 984--990.

\bibitem{k7} V. M. Kurbanov;
 \emph{On an analog of the Riesz theorem and the basis property of
the system of root functions of a differential operator in $L_p$},
\newblock{Differ. Equ.}, I: \textbf{49}, No.1 (2013), 7--19;
II: \textbf{49}, No.4 (2013), 437--449.

\bibitem{m1} A. S. Makin;
 \emph{A class of boundary value problems for the Sturm-Liouville operator},
\newblock{Differ. Equ.}, \textbf{35}, No.8 (1999), 1067--1076.

\bibitem{m2} A. S. Makin;
 \emph{A class of essentially nonself-adjoint boundary value problems},
\newblock{Dokl. Math.}, \textbf{89}, No.3 (2014), 351--353.

\bibitem{m3} V. D. Mil'man;
 \emph{Geometric theory of Banach spaces. Part I},
\newblock{Russian Math. Surveys}, \textbf{25}, No.3 (1970), 111--170.

\bibitem{m4} A. Yu. Mokin;
 \emph{On a family of initial-boundary value problems for the heat equation},
\newblock{Differ. Equ.}, \textbf{45}, No.1 (2009), 126--141.

\bibitem{o1} D. O'Regan;
 \emph{Existence results for differential equations with reflection of the argument},
\newblock{J. Aust. Math. Soc.}, A \textbf{57}, No.2 (1994), 237--260.

\bibitem{p1} D. Przeworska-Rolewicz;
 \emph{Equations with transformed argument. Algebraic approach.}
\newblock Elsevier Sci. Publ.: Amsterdam, PWN: Warsawa, 1973.

\bibitem{s1} M. F. Sadybekov, A. M. Sarsenbi;
 \emph{Criterion for the basis property of the eigenfunction system of a multiple
differentiation operator with an involution},
\newblock{Differ. Equ.}, \textbf{48}, No.8 (2012), 1112--1118.

\bibitem{s2} M. F. Sadybekov, A. M. Sarsenbi;
\emph{Mixed problem for a differential equation with involution under
boundary conditions of a general form}, \newblock{in: {\em AIP Conf. Proc.}},
{\bf 1470} (2012), 225--227.

\bibitem{s3} A. M. Sarsenbi;
 \emph{Unconditional bases related to a nonclassical second-order differential
operator},
\newblock{Differ. Equ.}, \textbf{46}, No.4 (2010), 509--514.

\bibitem{s4} A. M. Sarsenbi, A. A. Tengaeva;
 \emph{On the basis properties of root functions of two generalized eigenvalue
problems}, \newblock{Differ. Equ.}, \textbf{48}, No.2 (2012), 306--308.

\bibitem{s5} V. Serov;
 \emph{Green's function and convergence of Fourier series for elliptic
differential operators with potential from Kato space},
\newblock{Abstr. Appl. Anal.}, \textbf{2010}, article ID 902638, 1--10.

\bibitem{s6} A. B. Shidlovskii;
 \emph{Transcendental numbers}. Berlin, New York, 1989.

\bibitem{w1} W. Watkins;
 \emph{Asymptotic properties of differential equations with involutions},
\newblock{Int. J. Pure Appl. Math.}, \textbf{44}, No.4 (2008), 485--492.

\bibitem{w2} J. Wiener;
 \emph{Generalized solutions of functional differential equations.}
\newblock World Sci.: Singapore, New Jersey, London, Hong Kong, 1993.

\end{thebibliography}

\end{document}