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     {\flushleft\it Vol. 20, No. 2, pp. 144--145 (2023) \hspace{\fill}{\rm\small УДК \udcs \hspace{-4mm}}}
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\title[Eigenfunction expansions of impulsive $q$-Sturm--Liouville problems]{\large Eigenfunction expansions of impulsive $q$-Sturm--Liouville problems}


\author[Bilender P. Allahverdiev]{{\bf Bilender P. Allahverdiev}\protect\orcidicon{0000-0002-9315-4652}}

\address{Bilender P. Allahverdiev  % Контактные данные всех авторов и место работы указываются только на английском языке
	\newline\hphantom{iii} Department of Mathematics, Khazar University,
	\newline\hphantom{iii} AZ1096 Baku, Azerbaijan, and 
	\newline\hphantom{iii} Research Center of Econophysics,
	\newline\hphantom{iii} UNEC-Azerbaijan State University of Economics,
	\newline\hphantom{iii} Baku, Azerbaijan}%
\email{\textcolor{blue}{bilenderpasaoglu@gmail.com}}%

\author[H\"{u}seyin Tuna]{{\bf H\"{u}seyin Tuna}\protect\orcidicon{0000-0001-7240-8687}}

\address{H\"{u}seyin Tuna  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Department of Mathematics, Mehmet Akif Ersoy University,
\newline\hphantom{iii} 15030 Burdur, Turkey and,
\newline\hphantom{iii} Research Center of Econophysics,
\newline\hphantom{iii} UNEC-Azerbaijan State University of Economics,
\newline\hphantom{iii} Baku, Azerbaijan}%
\email{\textcolor{blue}{hustuna@gmail.com}}%

\author[Hamlet A. Isayev]{{\bf Hamlet A. Isayev}\protect\orcidicon{0000-0002-6383-0883}}

\address{Hamlet A. Isayev % Контактные данные всех авторов и место работы указываются только на английском языке
	\newline\hphantom{iii} Department of Mathematics, Khazar University,
	\newline\hphantom{iii}  AZ1096 Baku, Azerbaijan}%
	\email{\textcolor{blue}{hamlet@khazar.org}}%
	

\thanks{}
\thanks{\copyright \ }
\thanks{\rm )}
\thanks{\it Received January, 1, 2023, Published December, 31, 2023}%


\semrtop \vspace{1cm}
\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc P.P. Petrov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} In this work, an impulsive $q$-Sturm--Liouville problem is studied. The
existence of a countably infinite set of eigenvalues and eigenfunctions is
proved and a uniformly convergent expansion formula in the eigenfunctions is established.\medskip

\noindent{\bf Keywords:} Difference equations, impulsive conditions,
$q$-Sturm--Liouville problems, eigenfunction expansions.
 \end{quote}
}

\bigskip

\section{Introduction}
The Fourier method is one of the important
methods used in the solution of partial differential equations. While solving
with this method, the partial differential equation considered is reduced to
Sturm--Liouville problems. The eigenvalues and eigenfunctions of these
problems are investigated. Meanwhile, the problem of uniform convergence of
eigenfunction expansions arises. For this reason, eigenvalues, eigenfunctions,
and eigenfunction expansions of Sturm--Liouville problems have been
extensively studied in the theory of differential equations (see
\cite{ankitap,32,02,ber,g2,levinson,lev,nai,o12,stone,tit}).

Quantum calculus, which has a long history, has recently started to attract
the attention of many researchers ( see \cite{er}). In 2005 Annaby and Mansour
studied the $q$-analogue of the classical Sturm--Liouville equations (see
\cite{ankitap}). Classical results are obtained by considering the problem in
the regular case. Thereupon, the researchers began to study whether the
results obtained in the classical Sturm--Liouville theory were valid for the
$q$-Sturm--Liouville problems. In this context, the existence of countably
infinite sets of eigenvalues and eigenvectors of impulsive $q$%
-Sturm--Liouville problems and uniform convergence of eigenfunction expansions
will be investigated. The Uniform convergence problem will be investigated
using Steklov's method (see \cite{32,g2,stek}).

\section{Preliminaries}

In this section, the basic concepts of $q$-calculus that will be used in the
article will be given. For more detailed information, the following sources
can be examined,\  \cite{kac,ankitap,er}.

Let $q\in \left(  0,1\right)  \ $and let $A\subset \mathbb{R\ }$be a
$q$-geometric set, i.e., if $q\zeta \in A$ for all $\zeta \in A.$ We begin by
defining the \textit{operator} $\mathcal{D}_{q}$ by%
\[
\mathcal{D}_{q}f\left(  \zeta \right)  =\left \{
\begin{array}
	[c]{cc}%
	\frac{f\left(  q\zeta \right)  -f\left(  \zeta \right)  }{\left(  q-1\right)
		\zeta}, & \zeta \neq0\\
	\lim_{n\rightarrow \infty}\frac{f\left(  q^{n}\xi \right)  -f\left(  0\right)
	}{q^{n}\xi}, & \zeta=0,
\end{array}
\right.  \text{ }%
\]
where $\zeta,\xi \in A.\ $We define the \textit{Jackson} $q$%
-\textit{integration} by%
\[
\int_{0}^{\zeta}f\left(  \gamma \right)  d_{q}\gamma=\zeta \left(  1-q\right)
\sum_{n=0}^{\infty}q^{n}f\left(  q^{n}\zeta \right)  ,
\]
where $\zeta \in A.$ From \cite{han1}, we have%
\[
\int_{0}^{\infty}f\left(  \gamma \right)  d_{q}\gamma=\sum_{n=-\infty}^{\infty
}q^{n}f\left(  q^{n}\right)  .
\]
Through the remainder of the paper, we deal only with functions $q$-regular at
zero, i.e., functions satisfying%
\[
\lim_{n\rightarrow \infty}f\left(  \zeta q^{n}\right)  =f\left(  0\right)  ,
\]
for every $\zeta \in A.$

\section{Main Results}

Consider the following problem%
\begin{equation}
	(\tau y)\left(  \zeta \right)  :=-\frac{1}{q}D_{q^{-1}}D_{q}y\left(
	\zeta \right)  +v\left(  \zeta \right)  y\left(  \zeta \right)  =\lambda y\left(
	\zeta \right)  ,\  \zeta \in(0,d)\cup(d,a) \label{1.1}%
\end{equation}
with the boundary and impulsive conditions
\begin{equation}%
	\begin{tabular}
		[c]{l}%
		$y\left(  0\right)  -h_{1}D_{q^{-1}}y\left(  0\right)  =0,$\\
		$y\left(  a\right)  +h_{2}D_{q^{-1}}y\left(  a\right)  =0,$%
	\end{tabular}
	\  \  \label{1.3}%
\end{equation}%
\begin{align}
	y\left(  d-\right)   &  =\eta y\left(  d+\right)  ,\text{ }\label{i1}\\
	\mathcal{D}_{q^{-1}}y\left(  d-\right)   &  =\frac{1}{\eta}\mathcal{D}%
	_{q^{-1}}y\left(  d+\right)  , \label{i2}%
\end{align}
where $h_{1},h_{2},\eta>0,\  \lambda$ is a complex eigenvalue parameter, $v$ is
a real-valued continuous functions on $[0,d)\cup(d,q^{-1}a]$ and has finite
limits $v(d\pm),$ $v\left(  \zeta \right)  \geq0,\  \zeta \in \lbrack
0,d)\cup(d,q^{-1}a],$ $a>0.$

$H=L_{q}^{2}(0,d)\overset{\cdot}{+}L_{q}^{2}\left(  d,a\right)  $ is a Hilbert
space endowed with the following inner product
\[
\mathcal{\langle}y,z\rangle:=\int_{0}^{d}y^{\left(  1\right)  }\overline
{z^{\left(  1\right)  }}d_{q}\zeta+\int_{d}^{a}y^{\left(  2\right)  }%
\overline{z^{\left(  2\right)  }}d_{q}\zeta,
\]
where
\[
y(\zeta)=\left \{
\begin{array}
	[c]{cc}%
	y^{\left(  1\right)  }(\zeta), & \zeta \in(0,d)\\
	y^{\left(  2\right)  }(\zeta), & \zeta \in(d,a)
\end{array}
\right.
\]
and%
\[
z(\zeta)=\left \{
\begin{array}
	[c]{cc}%
	z^{\left(  1\right)  }(\zeta), & \zeta \in(0,d)\\
	z^{\left(  2\right)  }(\zeta), & \zeta \in(d,a).
\end{array}
\right.
\]


Let%
\[
\mathcal{T}:\mathcal{D}\subset H\rightarrow H,\text{ }\mathcal{T}y=\tau
y,\ y\in \mathcal{D},
\]
where
\[
\mathcal{D}=\left \{  y\in H:%
\begin{array}
	[c]{c}%
	y\  \text{and\ }D_{q}y\  \text{are }q\text{-regular at zero,}\\
	y\left(  d\pm \right)  \text{\ and\ }\mathcal{D}_{q^{-1}}y\left(  d\pm \right)
	\  \text{exist, }\\
	y\left(  0\right)  -h_{1}D_{q^{-1}}y\left(  0\right)  =0,\\
	y\left(  a\right)  +h_{2}D_{q^{-1}}y\left(  a\right)  =0,\\
	y\left(  d-\right)  =\eta y\left(  d+\right)  ,\\
	\mathcal{D}_{q^{-1}}y\left(  d-\right)  =\frac{1}{\eta}\mathcal{D}_{q^{-1}%
	}y\left(  d+\right)  ,\text{ and}\\
	\tau y\in H
\end{array}
\right \}  .
\]


Let $y,z\in \mathcal{D}.\ $Then we have%
\[
\int_{0}^{a}\left[  (\tau y)(x)\overline{z(x)}-y(x)\overline{(\tau
	z)(x)}\right]  d_{q}x
\]%
\begin{equation}
	=[y,z]\left(  a\right)  -[y,z]\left(  d+\right)  +[y,z]\left(  d-\right)
	-[y,z]\left(  0\right)  , \label{lag}%
\end{equation}
where
\[
\lbrack y,z]:=y(\overline{D_{q^{-1}}z})-(D_{q^{-1}}y)\overline{z}.\
\]


\begin{theorem}
	$\mathcal{T}$\textit{\ is a\ positive self-adjoint operator in }$H$\textit{.}
\end{theorem}

\begin{proof}
	Let $y,z\in \mathcal{D}.\ $It follows from conditions (\ref{1.3})-(\ref{i2})
	and (\ref{lag}) that%
	\begin{equation}
		\langle \mathcal{T}y,z\rangle=\langle y,\mathcal{T}z\rangle. \label{s}%
	\end{equation}
	Since $\mathcal{D}$ is a dense subset in $H,$ we see that $\mathcal{T}%
	$\textit{\ is a\ self-adjoint operator.}
	
	Let
	\[
	y(\zeta)=\left \{
	\begin{array}
		[c]{cc}%
		y^{\left(  1\right)  }(\zeta), & \zeta \in(0,d)\\
		y^{\left(  2\right)  }(\zeta), & \zeta \in(d,a),
	\end{array}
	\right.  y\in \mathcal{D}.
	\]
	Then we get%
	\[
	\langle \mathcal{T}y,y\rangle=\int_{0}^{d}\left[  -\frac{1}{q}D_{q^{-1}}%
	D_{q}y^{\left(  1\right)  }\left(  \zeta \right)  +v\left(  \zeta \right)
	y^{\left(  1\right)  }\left(  \zeta \right)  \right]  \overline{y^{\left(
			1\right)  }\left(  \zeta \right)  }d_{q}\zeta
	\]%
	\[
	+\int_{d}^{a}\left[  -\frac{1}{q}D_{q^{-1}}D_{q}y^{\left(  2\right)  }\left(
	\zeta \right)  +v\left(  \zeta \right)  y^{\left(  2\right)  }\left(
	\zeta \right)  \right]  \overline{y^{\left(  2\right)  }\left(  \zeta \right)
	}d_{q}\zeta
	\]%
	\[
	=\int_{0}^{d}-\frac{1}{q}\left(  D_{q^{-1}}D_{q}y^{\left(  1\right)  }\left(
	\zeta \right)  \right)  \overline{y^{\left(  1\right)  }\left(  \zeta \right)
	}d_{q}\zeta+\int_{0}^{d}v\left(  \zeta \right)  \left \vert y^{\left(  1\right)
	}\left(  \zeta \right)  \right \vert ^{2}d_{q}\zeta
	\]%
	\[
	+\int_{d}^{a}-\frac{1}{q}\left(  D_{q^{-1}}D_{q}y^{\left(  2\right)  }\left(
	\zeta \right)  \right)  \overline{y^{\left(  2\right)  }\left(  \zeta \right)
	}d_{q}\zeta+\int_{d}^{a}v\left(  \zeta \right)  \left \vert y^{\left(  2\right)
	}\left(  \zeta \right)  \right \vert ^{2}d_{q}\zeta
	\]%
	\[
	=-\frac{1}{q}\int_{0}^{d}D_{q}\left[  D_{q}y^{\left(  1\right)  }\left(
	q^{-1}\zeta \right)  \right]  \overline{y\left(  \zeta \right)  }d_{q}\zeta
	+\int_{0}^{d}v\left(  \zeta \right)  \left \vert y^{\left(  1\right)  }\left(
	\zeta \right)  \right \vert ^{2}d_{q}\zeta
	\]%
	\[
	-\int_{d}^{a}D_{q}\left[  D_{q}y^{\left(  2\right)  }\left(  q^{-1}%
	\zeta \right)  \right]  \overline{y^{\left(  2\right)  }\left(  \zeta \right)
	}d_{q}\zeta+\int_{d}^{a}v\left(  \zeta \right)  \left \vert y^{\left(  2\right)
	}\left(  \zeta \right)  \right \vert ^{2}d_{q}\zeta
	\]%
	\begin{align*}
		&  =-\frac{1}{q}\left[  D_{q}y^{\left(  1\right)  }\left(  q^{-1}d-\right)
		\overline{y^{\left(  1\right)  }\left(  d-\right)  }\right]  +\frac{1}%
		{q}\left[  D_{q}y^{\left(  1\right)  }\left(  0\right)  \overline{y^{\left(
				1\right)  }\left(  0\right)  }\right] \\
		& \\
		&  -\frac{1}{q}\left[  D_{q}y^{\left(  2\right)  }\left(  q^{-1}a\right)
		\overline{y^{\left(  2\right)  }\left(  a\right)  }\right]  +\frac{1}%
		{q}\left[  D_{q}y^{\left(  2\right)  }\left(  d+\right)  \overline{y^{\left(
				2\right)  }\left(  d+\right)  }\right] \\
		& \\
		&  +\frac{1}{q}\int_{0}^{d}\left \vert D_{q}y^{\left(  1\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta+\int_{0}^{d}v\left(  \zeta \right)
		\left \vert y^{\left(  1\right)  }\left(  \zeta \right)  \right \vert ^{2}%
		d_{q}\zeta \\
		& \\
		&  +\frac{1}{q}\int_{d}^{a}\left \vert D_{q}y^{\left(  2\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta+\int_{d}^{a}v\left(  \zeta \right)
		\left \vert y^{\left(  2\right)  }\left(  \zeta \right)  \right \vert ^{2}%
		d_{q}\zeta
	\end{align*}%
	\begin{align*}
		&  =\frac{1}{q}h_{1}\left \vert D_{q}y^{\left(  1\right)  }\left(  0\right)
		\right \vert ^{2}+\frac{1}{q}h_{2}\left \vert D_{q}y^{\left(  2\right)  }\left(
		q^{-1}a\right)  \right \vert ^{2}\\
		& \\
		&  +\frac{1}{q}\int_{0}^{d}\left \vert D_{q}y^{\left(  1\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta+\int_{0}^{d}v\left(  \zeta \right)
		\left \vert y^{\left(  1\right)  }\left(  \zeta \right)  \right \vert ^{2}%
		d_{q}\zeta \\
		& \\
		&  +\frac{1}{q}\int_{d}^{a}\left \vert D_{q}y^{\left(  2\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta+\int_{d}^{a}v\left(  \zeta \right)
		\left \vert y^{\left(  2\right)  }\left(  \zeta \right)  \right \vert ^{2}%
		d_{q}\zeta>0,
	\end{align*}
	due to $h_{1},h_{2}>0$ and $v\left(  \zeta \right)  \geq0$ for $\zeta \in
	\lbrack0,q^{-1}a].$
\end{proof}

Let
\[
u\left(  \zeta \right)  =\left \{
\begin{array}
	[c]{cc}%
	u^{\left(  1\right)  }(\zeta), & \zeta \in(0,d)\\
	u^{\left(  2\right)  }(\zeta), & \zeta \in(d,a)
\end{array}
\right.
\]
$\ $and$\ $%
\[
\chi \left(  \zeta \right)  =\left \{
\begin{array}
	[c]{cc}%
	\chi^{\left(  1\right)  }(\zeta), & \zeta \in(0,d)\\
	\chi^{\left(  2\right)  }(\zeta), & \zeta \in(d,a)
\end{array}
\right.
\]
$\ $be solutions of the problem%
\[
-\frac{1}{q}D_{q^{-1}}D_{q}y\left(  \zeta \right)  +v\left(  \zeta \right)
y\left(  \zeta \right)  =0,
\]
\[
y\left(  d-\right)  =\eta y\left(  d+\right)  ,
\]%
\[
\mathcal{D}_{q^{-1}}y\left(  d-\right)  =\frac{1}{\eta}\mathcal{D}_{q^{-1}%
}y\left(  d+\right)  ,
\]
satisfying
\begin{align*}
	u^{\left(  1\right)  }\left(  0\right)   &  =h_{1},\ D_{q^{-1}}u^{\left(
		1\right)  }\left(  0\right)  =1,\\
	\chi^{\left(  2\right)  }\left(  a\right)   &  =-h_{2},\ D_{q^{-1}}%
	\chi^{\left(  2\right)  }\left(  a\right)  =1.
\end{align*}


\begin{lemma}
	Zero is not an eigenvalue of the operator $\mathcal{T}.$
\end{lemma}

\begin{proof}
	Let $y\in H$ and $\mathcal{T}y=0.$ Then we have%
	\[
	-\frac{1}{q}D_{q^{-1}}D_{q}y\left(  \zeta \right)  +v\left(  \zeta \right)
	y\left(  \zeta \right)  =0,
	\]
	and%
	\[
	y\left(  \zeta \right)  =\left \{
	\begin{array}
		[c]{cc}%
		c_{1}u^{\left(  1\right)  }\left(  \zeta \right)  +c_{2}\chi^{\left(  1\right)
		}\left(  \zeta \right)  , & \zeta \in(0,d)\\
		c_{3}u^{\left(  2\right)  }\left(  \zeta \right)  +c_{4}\chi^{\left(  2\right)
		}\left(  \zeta \right)  , & \zeta \in(d,a),
	\end{array}
	\right.
	\]
	where $c_{1},c_{2},c_{3}$ and $c_{4}$ are constants$.\ $From conditions
	(\ref{1.3})-(\ref{i2}), we infer that $y=0.$
\end{proof}

\begin{definition}
	The $q$-Wronskian of $y~$and$\ z$ is defined as%
	\[
	W_{q}\left(  y,z\right)  :=yD_{q}z-zD_{q}y.
	\]
	
\end{definition}

\begin{theorem}
	Let%
	\begin{equation}
		G\left(  \zeta,t\right)  =-\frac{1}{W_{q}\left(  u,\chi \right)  }\left \{
		\begin{array}
			[c]{cc}%
			u\left(  \zeta \right)  \chi \left(  t\right)  , & 0\leq \zeta \leq t\leq a,\text{
			}\zeta \neq d,\ t\neq d,\\
			u\left(  t\right)  \chi \left(  \zeta \right)  , & 0\leq t\leq \zeta \leq a,\text{
			}\zeta \neq d,\ t\neq d.
		\end{array}
		\right.  \label{29}%
	\end{equation}
	Then $G\left(  \zeta,t\right)  $ is a $q$-Hilbert--Schmidt kernel, i.e.,
	\[
	\int_{0}^{d}\int_{0}^{d}\left \vert G\left(  \zeta,t\right)  \right \vert
	^{2}d_{q}\zeta d_{q}t<\infty,\  \int_{d}^{a}\int_{d}^{a}|G(\zeta,t)|^{2}%
	d_{q}\zeta d_{q}t<\infty.
	\]
	
\end{theorem}

\begin{proof}
	By (\ref{29}), we see that
	\[
	\int_{0}^{d}d_{q}\zeta \int_{0}^{d}\left \vert G\left(  \zeta,t\right)
	\right \vert ^{2}d_{q}t<\infty,
	\]%
	\[
	\int_{d}^{a}d_{q}\zeta \int_{d}^{a}\left \vert G\left(  \zeta,t\right)
	\right \vert ^{2}d_{q}t<\infty
	\]
	due to $u\left(  .\right)  \chi \left(  .\right)  \in H\times H.$ Then, we
	obtain%
	\begin{equation}
		\int_{0}^{d}\int_{0}^{d}\left \vert G\left(  \zeta,t\right)  \right \vert
		^{2}d_{q}\zeta d_{q}t<\infty,\  \int_{d}^{a}\int_{d}^{a}|G(\zeta,t)|^{2}%
		d_{q}\zeta d_{q}t<\infty. \label{31}%
	\end{equation}
	
\end{proof}

\begin{theorem}
	[\cite{nai}]Let $A$ be an operator defined as
	\[
	A\{ \zeta_{i}\}=\{y_{i}\},
	\]
	where $i\in \mathbb{N}:=\{1,2,3,...\}$ and%
	\begin{equation}
		y_{i}=\sum_{k=1}^{\infty}a_{ik}\zeta_{k}. \label{30}%
	\end{equation}
	If
	\begin{equation}
		\sum_{i,k=1}^{\infty}\left \vert a_{ik}\right \vert ^{2}<\infty, \label{28}%
	\end{equation}
	then $A$ is compact in $l^{2}.$
\end{theorem}

\begin{theorem}
	Let $K:H\rightarrow H$ be an operator defined as
	\begin{equation}
		(Kf)\left(  \zeta \right)  =\left \{
		\begin{array}
			[c]{cc}%
			\int_{0}^{d}G\left(  \zeta,\gamma \right)  f^{\left(  1\right)  }(\gamma
			)d_{q}\gamma, & \zeta \in \lbrack0,d)\\
			\int_{d}^{a}G\left(  \zeta,\gamma \right)  f^{\left(  2\right)  }(\gamma
			)d_{q}\gamma, & \zeta \in(d,a],
		\end{array}
		\right.  \label{300}%
	\end{equation}
	where%
	\[
	f(\zeta)=\left \{
	\begin{array}
		[c]{cc}%
		f^{\left(  1\right)  }(\zeta), & \zeta \in \lbrack0,d)\\
		f^{\left(  2\right)  }(\zeta), & \zeta \in(d,a],
	\end{array}
	\right.  f\in H.
	\]
	Then $K$ is a compact operator.
\end{theorem}

\begin{proof}
	Let
	\[
	\phi_{i}=\phi_{i}\left(  \zeta \right)  =\left \{
	\begin{array}
		[c]{cc}%
		\phi_{i}^{\left(  1\right)  }(\zeta), & \zeta \in \lbrack0,d)\\
		\phi_{i}^{\left(  2\right)  }(\zeta), & \zeta \in(d,a]
	\end{array}
	\right.  \ (i\in \mathbb{N)}%
	\]
	be a complete, orthonormal basis of $H.\ $Let$\ i,k\in \mathbb{N}.$ If we set%
	\[
	\zeta_{i}=\langle f,\phi_{i}\rangle=\int_{0}^{d}f^{\left(  1\right)  }\left(
	\zeta \right)  \overline{\phi_{i}^{\left(  1\right)  }\left(  \zeta \right)
	}d_{q}\zeta
	\]%
	\[
	+\int_{d}^{a}f^{\left(  2\right)  }\left(  \zeta \right)  \overline{\phi
		_{i}^{\left(  2\right)  }\left(  \zeta \right)  }d_{q}\zeta,
	\]%
	\[
	y_{i}=\langle g,\phi_{i}\rangle=\int_{0}^{d}g^{\left(  1\right)  }\left(
	\zeta \right)  \overline{\phi_{i}^{\left(  1\right)  }\left(  \zeta \right)
	}d_{q}\zeta
	\]%
	\[
	+\int_{d}^{a}g^{\left(  2\right)  }\left(  \zeta \right)  \overline{\phi
		_{i}^{\left(  2\right)  }\left(  \zeta \right)  }d_{q}\zeta,
	\]%
	\[
	a_{ik}=\int_{0}^{d}\int_{0}^{d}G\left(  \zeta,t\right)  \overline{\phi
		_{i}^{\left(  1\right)  }\left(  \zeta \right)  }\overline{\phi_{k}^{\left(
			1\right)  }\left(  t\right)  }d_{q}\zeta d_{q}t\text{ }%
	\]%
	\[
	+\int_{d}^{a}\int_{d}^{a}G\left(  \zeta,t\right)  \overline{\phi_{i}^{\left(
			2\right)  }\left(  \zeta \right)  }\overline{\phi_{k}^{\left(  2\right)
		}\left(  t\right)  }d_{q}\zeta d_{q}t.
	\]
	Then, $H\ $ is mapped isometrically $l^{2}.$ Then, $K$ transforms into $A$
	defined as (\ref{30}) in $l^{2}\ $and (\ref{31}) is translated into
	(\ref{28}). From Theorem 3, $A$ is compact. Thus, $K$ is compact.
\end{proof}

Since$\ K=\mathcal{T}^{-1},\ $the completeness of the system of all
eigenvectors of $\mathcal{T}$ is equivalent to the completeness of the system
of all eigenvectors of $K$. By the Hilbert--Schmidt theorem, we obtain the
following theorem.

\begin{theorem}
	For the boundary-value problem (\ref{1.1})-(\ref{i2}), there exists an
	orthonormal basis $\left \{  \psi_{k}\right \}  _{k\in \mathbb{N}}\ $in $H.$ For
	$f\in H,$ we have%
	\begin{equation}
		f\left(  \zeta \right)  =\sum_{k=1}^{\infty}c_{k}\psi_{k}\left(  \zeta \right)
		, \label{ser}%
	\end{equation}
	where
	\[
	c_{k}=\langle f,\psi_{k}\rangle,\text{ }k\in \mathbb{N}.
	\]
	Thus, we get
	\begin{equation}
		\lim_{N\rightarrow \infty}\left \{
		\begin{array}
			[c]{c}%
			\int_{0}^{d}\left \vert f^{\left(  1\right)  }\left(  \zeta \right)  -\sum
			_{k=1}^{N}c_{k}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  \right \vert
			^{2}d_{q}\zeta \\
			+\int_{d}^{a}\left \vert f^{\left(  2\right)  }\left(  \zeta \right)
			-\sum_{k=1}^{N}c_{k}\psi_{k}^{\left(  2\right)  }\left(  \zeta \right)
			\right \vert ^{2}d_{q}\zeta
		\end{array}
		\right \}  =0, \label{38}%
	\end{equation}
	Moreover, it follows from (\ref{38}) that
	\begin{equation}
		\int_{0}^{d}\left \vert f^{\left(  1\right)  }\left(  \zeta \right)  \right \vert
		^{2}d_{q}\zeta+\int_{d}^{a}\left \vert f^{\left(  2\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta=\sum_{k=1}^{\infty}\left \vert
		c_{k}\right \vert ^{2}. \label{39}%
	\end{equation}
	
\end{theorem}

Now let's prove the main result of the article

\begin{theorem}
	Let $f,D_{q}f:[0,q^{-1}a]\rightarrow \mathbb{R}$ be continuous functions on
	$[0,d)\cup(d,q^{-1}a],$ has finite limits $f(d\pm),$ $D_{q^{-1}}f(d\pm)$ and
	satisfying conditions (\ref{1.3})-(\ref{i2}). Then the series
	\begin{equation}
		f\left(  \zeta \right)  =\sum_{k=1}^{\infty}c_{k}\psi_{k}\left(  \zeta \right)
		, \label{40}%
	\end{equation}
	where
	\[
	c_{k}=\langle f,\psi_{k}\rangle,\text{ }k\in \mathbb{N},
	\]
	converges uniformly to $f\ $on the set $[0,d)\cup(d,a].$
\end{theorem}

\begin{proof}
	Let
	\begin{align}
		S\left(  y\right)   &  :=\frac{1}{q}h_{1}\left \vert D_{q}y^{\left(  1\right)
		}\left(  0\right)  \right \vert ^{2}+\frac{1}{q}h_{2}\left \vert D_{q}y^{\left(
			2\right)  }\left(  q^{-1}a\right)  \right \vert ^{2}\nonumber \\
		& \nonumber \\
		&  +\frac{1}{q}\int_{0}^{d}\left \vert D_{q}y^{\left(  1\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta+\int_{0}^{d}v\left(  \zeta \right)
		\left \vert y^{\left(  1\right)  }\left(  \zeta \right)  \right \vert ^{2}%
		d_{q}\zeta \nonumber \\
		& \nonumber \\
		&  +\frac{1}{q}\int_{d}^{a}\left \vert D_{q}y^{\left(  2\right)  }\left(
		\zeta \right)  \right \vert ^{2}d_{q}\zeta+\int_{d}^{a}v\left(  \zeta \right)
		\left \vert y^{\left(  2\right)  }\left(  \zeta \right)  \right \vert ^{2}%
		d_{q}\zeta, \label{func}%
	\end{align}
	and $S(y)\geq0$. If we take%
	\[
	y=f\left(  \zeta \right)  -\sum_{k=1}^{N}c_{k}\psi_{k}\left(  \zeta \right)
	\]
	in (\ref{func}), we conclude that%
	\[
	S\left(  f\left(  \zeta \right)  -\sum_{k=1}^{N}c_{k}\psi_{k}\left(
	\zeta \right)  \right)
	\]%
	\[
	=\frac{1}{q}h_{1}\left[  D_{q}f^{\left(  1\right)  }\left(  0\right)
	-\sum_{k=1}^{N}c_{k}D_{q}\psi_{k}^{\left(  1\right)  }\left(  0\right)
	\right]  ^{2}%
	\]%
	\[
	+\frac{1}{q}h_{2}\left[  D_{q}f^{\left(  2\right)  }\left(  q^{-1}a\right)
	-\sum_{k=1}^{N}c_{k}\left(  D_{q}\psi_{k}^{\left(  2\right)  }\left(
	a\right)  \right)  \right]  ^{2}%
	\]%
	\[
	+\frac{1}{q}\int_{0}^{d}\left(  D_{q}f^{\left(  1\right)  }\left(
	\zeta \right)  -\sum_{k=1}^{N}c_{k}D_{q}\psi_{k}^{\left(  1\right)  }\left(
	\zeta \right)  \right)  ^{2}d_{q}\zeta
	\]%
	\[
	+\frac{1}{q}\int_{d}^{a}\left(  D_{q}f^{\left(  2\right)  }\left(
	\zeta \right)  -\sum_{k=1}^{N}c_{k}D_{q}\psi_{k}^{\left(  2\right)  }\left(
	\zeta \right)  \right)  ^{2}d_{q}\zeta
	\]%
	\begin{align*}
		&  +\int_{0}^{a}v\left(  \zeta \right)  \left(  f^{\left(  1\right)  }\left(
		\zeta \right)  -\sum_{k=1}^{N}c_{k}\psi_{k}^{\left(  1\right)  }\left(
		\zeta \right)  \right)  ^{2}d_{q}\zeta \\
		& \\
		&  +\int_{d}^{a}v\left(  \zeta \right)  \left(  f^{\left(  2\right)  }\left(
		\zeta \right)  -\sum_{k=1}^{N}c_{k}\psi_{k}^{\left(  2\right)  }\left(
		\zeta \right)  \right)  ^{2}d_{q}\zeta
	\end{align*}%
	\[
	=\frac{1}{q}h_{1}\left[  D_{q}f^{\left(  1\right)  }\left(  0\right)  \right]
	^{2}+\frac{1}{q}h_{2}\left[  D_{q}f^{\left(  2\right)  }\left(  q^{-1}%
	a\right)  \right]  ^{2}%
	\]%
	\[
	-2\frac{1}{q}\sum_{k=1}^{N}c_{k}\left[
	\begin{array}
		[c]{c}%
		-h_{1}D_{q}f^{\left(  1\right)  }\left(  0\right)  D_{q}\psi_{k}^{\left(
			1\right)  }\left(  0\right) \\
		\\
		-h_{2}D_{q}f^{\left(  2\right)  }\left(  q^{-1}a\right)  D_{q}\psi
		_{k}^{\left(  2\right)  }\left(  a\right)
	\end{array}
	\right]
	\]%
	\[
	-\frac{1}{q}\sum_{k,m=1}^{N}c_{k}c_{m}\left[
	\begin{array}
		[c]{c}%
		-h_{1}D_{q}\psi_{k}^{\left(  1\right)  }\left(  0\right)  D_{q}\psi
		_{m}^{\left(  1\right)  }\left(  0\right) \\
		\\
		-h_{2}D_{q}\psi_{k}^{\left(  2\right)  }\left(  a\right)  D_{q}\psi
		_{m}^{\left(  2\right)  }\left(  a\right)
	\end{array}
	\right]
	\]%
	\begin{align*}
		&  +\frac{1}{q}\int_{0}^{d}(D_{q}f^{\left(  1\right)  }\left(  \zeta \right)
		)^{2}d_{q}\zeta+\int_{0}^{d}v\left(  \zeta \right)  f^{\left(  1\right)
			2}\left(  \zeta \right)  d_{q}\zeta \\
		& \\
		&  +\frac{1}{q}\int_{d}^{a}(D_{q}f^{\left(  2\right)  }\left(  \zeta \right)
		)^{2}d_{q}\zeta+\int_{d}^{a}v\left(  \zeta \right)  f^{\left(  2\right)
			2}\left(  \zeta \right)  d_{q}\zeta
	\end{align*}%
	\[
	-2\frac{1}{q}\sum_{k=1}^{N}c_{k}\left[
	\begin{array}
		[c]{c}%
		\int_{0}^{d}D_{q}f^{\left(  1\right)  }\left(  \zeta \right)  D_{q}\psi
		_{k}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta \\
		+\int_{d}^{a}D_{q}f^{\left(  2\right)  }\left(  \zeta \right)  D_{q}\psi
		_{k}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\end{array}
	\right]
	\]%
	\[
	-2\sum_{k=1}^{N}c_{k}\left[
	\begin{array}
		[c]{c}%
		\int_{0}^{d}v\left(  \zeta \right)  f^{\left(  1\right)  }\left(  \zeta \right)
		\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta \\
		+\int_{d}^{a}v\left(  \zeta \right)  f^{\left(  2\right)  }\left(
		\zeta \right)  \psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\end{array}
	\right]
	\]%
	\[
	+\frac{1}{q}\sum_{k,m=1}^{N}c_{k}c_{m}\left[
	\begin{array}
		[c]{c}%
		\int_{0}^{d}D_{q}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  D_{q}%
		\psi_{m}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta \\
		+\int_{d}^{a}D_{q}\psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  D_{q}%
		\psi_{m}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\end{array}
	\right]
	\]%
	\[
	+\sum_{k,m=1}^{N}c_{k}c_{m}\left[
	\begin{array}
		[c]{c}%
		\int_{0}^{d}v\left(  \zeta \right)  \psi_{k}^{\left(  1\right)  }\left(
		\zeta \right)  \psi_{m}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta \\
		+\int_{d}^{a}v\left(  \zeta \right)  \psi_{k}^{\left(  2\right)  }\left(
		\zeta \right)  \psi_{m}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\end{array}
	\right]  .
	\]
	Applications of (\ref{1.3})-(\ref{i2}) and $q$-integration by parts give%
	\[
	\frac{1}{q}\int_{0}^{d}D_{q}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)
	D_{q}f^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta+\int_{0}%
	^{d}v\left(  \zeta \right)  f^{\left(  1\right)  }\left(  \zeta \right)
	\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta
	\]%
	\[
	+\frac{1}{q}\int_{d}^{a}D_{q}\psi_{k}^{\left(  2\right)  }\left(
	\zeta \right)  D_{q}f^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	+\int_{d}^{a}v\left(  \zeta \right)  f^{\left(  2\right)  }\left(
	\zeta \right)  \psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\]%
	\[
	=\frac{1}{q}D_{q}\psi_{k}^{\left(  1\right)  }\left(  q^{-1}d-\right)
	f^{\left(  1\right)  }\left(  d-\right)  -\frac{1}{q}D_{q}\psi_{k}^{\left(
		1\right)  }\left(  0\right)  f^{\left(  1\right)  }\left(  0\right)
	\]%
	\[
	+\frac{1}{q}D_{q}\psi_{k}^{\left(  2\right)  }\left(  q^{-1}a\right)
	f^{\left(  2\right)  }\left(  a\right)  -\frac{1}{q}D_{q}\psi_{k}^{\left(
		2\right)  }\left(  d+\right)  f^{\left(  2\right)  }\left(  d+\right)
	\]%
	\[
	-\frac{1}{q}\int_{0}^{d}f^{\left(  1\right)  }\left(  \zeta \right)  D_{q^{-1}%
	}\left(  D_{q}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  \right)
	d_{q}\zeta
	\]%
	\[
	-\frac{1}{q}\int_{d}^{a}f^{\left(  2\right)  }\left(  \zeta \right)  D_{q^{-1}%
	}\left(  D_{q}\psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  \right)
	d_{q}\zeta
	\]%
	\[
	+\int_{0}^{d}v\left(  \zeta \right)  f^{\left(  1\right)  }\left(
	\zeta \right)  \psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}%
	\zeta+\int_{d}^{a}v\left(  \zeta \right)  f^{\left(  2\right)  }\left(
	\zeta \right)  \psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\]%
	\[
	=-\frac{1}{q}h_{2}D_{q^{-1}}f^{\left(  2\right)  }\left(  a\right)  D_{q^{-1}%
	}\psi_{k}^{\left(  2\right)  }\left(  a\right)
	\]%
	\[
	-\frac{1}{q}h_{1}D_{q^{-1}}f^{\left(  1\right)  }\left(  0\right)  D_{q}%
	\psi_{k}^{\left(  1\right)  }\left(  0\right)
	\]%
	\[
	+\int_{0}^{d}f^{\left(  1\right)  }\left(  \zeta \right)  \left[  -\frac{1}%
	{q}D_{q^{-1}}\left(  D_{q}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)
	\right)  +v\left(  \zeta \right)  \psi_{k}^{\left(  1\right)  }\left(
	\zeta \right)  \right]  d_{q}\zeta
	\]%
	\[
	+\int_{d}^{a}f^{\left(  2\right)  }\left(  \zeta \right)  \left[  -\frac{1}%
	{q}D_{q^{-1}}\left(  D_{q}\psi_{k}^{\left(  2\right)  }\left(  \zeta \right)
	\right)  +v\left(  \zeta \right)  \psi_{k}^{\left(  2\right)  }\left(
	\zeta \right)  \right]  d_{q}\zeta
	\]%
	\[
	=-h_{2}\frac{1}{q}D_{q^{-1}}f^{\left(  2\right)  }\left(  a\right)  D_{q^{-1}%
	}\psi_{k}^{\left(  2\right)  }\left(  a\right)
	\]%
	\[
	-h_{1}\frac{1}{q}D_{q^{-1}}f^{\left(  1\right)  }\left(  0\right)  D_{q}%
	\psi_{k}^{\left(  1\right)  }\left(  0\right)  +\lambda_{k}c_{k},
	\]
	and%
	\[
	\frac{1}{q}\int_{0}^{d}D_{q}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)
	D_{q}\psi_{m}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}\zeta
	\]%
	\[
	+\frac{1}{q}\int_{d}^{a}D_{q}\psi_{k}^{\left(  2\right)  }\left(
	\zeta \right)  D_{q}\psi_{m}^{\left(  2\right)  }\left(  \zeta \right)
	d_{q}\zeta
	\]%
	\[
	+\int_{0}^{d}v\left(  \zeta \right)  \psi_{k}^{\left(  1\right)  }\left(
	\zeta \right)  \psi_{m}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}%
	\zeta+\int_{d}^{a}v\left(  \zeta \right)  \psi_{k}^{\left(  2\right)  }\left(
	\zeta \right)  \psi_{m}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta
	\]%
	\[
	=\frac{1}{q}D_{q}\psi_{m}^{\left(  1\right)  }\left(  q^{-1}d-\right)
	\psi_{k}^{\left(  1\right)  }\left(  d-\right)  +\frac{1}{q}D_{q}\psi
	_{m}^{\left(  2\right)  }\left(  q^{-1}a\right)  \psi_{k}^{\left(  2\right)
	}\left(  a\right)
	\]%
	\[
	-\frac{1}{q}D_{q}\psi_{m}^{\left(  1\right)  }\left(  0\right)  \psi
	_{k}^{\left(  1\right)  }\left(  0\right)  -\frac{1}{q}D_{q}\psi_{m}^{\left(
		2\right)  }\left(  d+\right)  \psi_{k}^{\left(  2\right)  }\left(  d+\right)
	\]%
	\[
	+\int_{0}^{d}\psi_{k}^{\left(  1\right)  }\left(  \zeta \right)  \left[
	-\frac{1}{q}D_{q^{-1}}\left(  D_{q}\psi_{m}^{\left(  1\right)  }\left(
	\zeta \right)  \right)  +v\left(  \zeta \right)  \psi_{m}^{\left(  1\right)
	}\left(  \zeta \right)  \right]  d_{q}\zeta
	\]%
	\[
	+\int_{d}^{a}\psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  \left[
	-\frac{1}{q}D_{q^{-1}}\left(  D_{q}\psi_{m}^{\left(  2\right)  }\left(
	\zeta \right)  \right)  +v\left(  \zeta \right)  \psi_{m}^{\left(  2\right)
	}\left(  \zeta \right)  \right]  d_{q}\zeta
	\]%
	\[
	=\frac{1}{q}\psi_{k}^{\left(  2\right)  }\left(  a\right)  D_{q^{-1}}\psi
	_{m}^{\left(  2\right)  }\left(  a\right)  -\frac{1}{q}\psi_{k}^{\left(
		1\right)  }\left(  0\right)  D_{q^{-1}}\psi_{k}^{\left(  1\right)  }\left(
	0\right)
	\]%
	\[
	+\lambda_{k}\left[  \int_{0}^{d}\psi_{k}^{\left(  1\right)  }\left(
	\zeta \right)  \psi_{m}^{\left(  1\right)  }\left(  \zeta \right)  d_{q}%
	\zeta+\int_{d}^{a}\psi_{k}^{\left(  2\right)  }\left(  \zeta \right)  \psi
	_{m}^{\left(  2\right)  }\left(  \zeta \right)  d_{q}\zeta \right]
	\]%
	\[
	=-\frac{1}{q}h_{1}D_{q^{-1}}\psi_{k}^{\left(  2\right)  }\left(  a\right)
	D_{q^{-1}}\psi_{m}^{\left(  2\right)  }\left(  a\right)
	\]%
	\[
	-\frac{1}{q}h_{2}D_{q^{-1}}\psi_{k}^{\left(  1\right)  }\left(  0\right)
	D_{q^{-1}}\psi_{m}^{\left(  1\right)  }\left(  0\right)  +\lambda_{k}%
	\delta_{km},
	\]
	where
	\[
	\delta_{km}:=\left \{
	\begin{array}
		[c]{cc}%
		1, & \text{if\ }k=m\\
		0, & \text{if }k\neq m.
	\end{array}
	\right.
	\]
	Hence%
	\begin{align*}
		&  S\left(  f\left(  \zeta \right)  -\sum_{k=1}^{N}c_{k}\psi_{k}\left(
		\zeta \right)  \right)  =\frac{1}{q}h_{1}\left[  D_{q}f^{\left(  1\right)
		}\left(  0\right)  \right]  ^{2}\\
		& \\
		&  +\frac{1}{q}h_{2}\left[  D_{q}f^{\left(  2\right)  }\left(  q^{-1}a\right)
		\right]  ^{2}+\frac{1}{q}\int_{0}^{d}(D_{q}f^{\left(  1\right)  }\left(
		\zeta \right)  )^{2}d_{q}\zeta \\
		& \\
		&  +\int_{0}^{d}v\left(  \zeta \right)  f^{\left(  1\right)  2}\left(
		\zeta \right)  d_{q}\zeta+\frac{1}{q}\int_{d}^{a}(D_{q}f^{\left(  2\right)
		}\left(  \zeta \right)  )^{2}d_{q}\zeta \\
		& \\
		&  +\int_{d}^{a}v\left(  \zeta \right)  f^{\left(  2\right)  2}\left(
		\zeta \right)  d_{q}\zeta-\frac{1}{q}\sum_{k=1}^{N}\lambda_{k}c_{k}^{2}.
	\end{align*}
	Then we get%
	\[
	\sum_{k=1}^{\infty}\lambda_{k}c_{k}^{2}\leq h_{1}\left[  D_{q}f^{\left(
		1\right)  }\left(  0\right)  \right]  ^{2}+h_{2}\left[  D_{q}f^{\left(
		2\right)  }\left(  q^{-1}a\right)  \right]  ^{2}%
	\]%
	\begin{align}
		&  +\int_{0}^{d}(D_{q}f^{\left(  1\right)  }\left(  \zeta \right)  )^{2}%
		d_{q}\zeta+q\int_{0}^{d}v\left(  \zeta \right)  f^{\left(  1\right)  2}\left(
		\zeta \right)  d_{q}\zeta \nonumber \\
		& \nonumber \\
		&  +\int_{d}^{a}(D_{q}f^{\left(  2\right)  }\left(  \zeta \right)  )^{2}%
		d_{q}\zeta+q\int_{d}^{a}v\left(  \zeta \right)  f^{\left(  2\right)  2}\left(
		\zeta \right)  d_{q}\zeta. \label{bes}%
	\end{align}
	due to $S$ is nonnegative for all $N$.\ Thus, the convergence of the series
	\[
	\sum_{k=1}^{\infty}\lambda_{k}c_{k}^{2}%
	\]
	follows.
	
	Now,\ we shall prove that the series
	\begin{equation}
		\sum_{k=1}^{\infty}\left \vert c_{k}\psi_{k}\left(  \zeta \right)  \right \vert
		\label{63}%
	\end{equation}
	is uniformly convergent on $[0,d)\cup(d,a].$ Since $\mathcal{T}\psi
	_{k}=\lambda_{k}\psi_{k},$ $k\in \mathbb{N},$ we have
	\[
	\psi_{k}\left(  \zeta \right)  =\lambda_{k}(\mathcal{T}^{-1}\psi_{k})\left(
	\zeta \right)  =\lambda_{k}\langle G\left(  \zeta,t\right)  ,\psi_{k}%
	\rangle,\text{ }k\in \mathbb{N}.
	\]
	If we rewrite the series (\ref{63}), we conclude that%
	\begin{equation}
		\sum_{k=1}^{\infty}\left \vert c_{k}\psi_{k}\left(  \zeta \right)  \right \vert
		=\sum_{k=1}^{\infty}\lambda_{k}\left \vert c_{k}\Upsilon_{k}\left(
		\zeta \right)  \right \vert , \label{65}%
	\end{equation}
	where%
	\[
	\Upsilon_{k}\left(  \zeta \right)  =\langle G\left(  \zeta,t\right)  ,\psi
	_{k}\rangle,\text{ }k\in \mathbb{N}.
	\]
	This can be regarded as the Fourier coefficients of $G\left(  \zeta,t\right)
	$ as a function of $t.$ From (\ref{bes}), we find
	\[
	\sum_{k=1}^{\infty}\lambda_{k}\Upsilon_{k}^{2}\left(  \zeta \right)  \leq
	h_{1}\left[  D_{q}G^{\left(  1\right)  }\left(  \zeta,0\right)  \right]
	^{2}+h_{2}\left[  D_{q}G^{\left(  2\right)  }\left(  \zeta,q^{-1}a\right)
	\right]  ^{2}%
	\]%
	\[
	+\int_{0}^{d}(D_{q}G^{\left(  1\right)  }\left(  \zeta,t\right)  )^{2}%
	d_{q}t+q\int_{0}^{d}v\left(  t\right)  G^{\left(  1\right)  2}\left(
	\zeta,t\right)  d_{q}t
	\]%
	\[
	+\int_{d}^{a}(D_{q}G^{\left(  2\right)  }\left(  \zeta,t\right)  )^{2}%
	d_{q}t+q\int_{0}^{a}v\left(  t\right)  G^{\left(  2\right)  2}\left(
	\zeta,t\right)  d_{q}t.
	\]
	Since all the functions appearing under the integral sign are bounded, we
	deduce that%
	\[
	\sum_{k=1}^{\infty}\lambda_{k}\Upsilon_{k}^{2}\left(  \zeta \right)  \leq C,
	\]
	where $\ C$ is a constant. Applying the Cauchy--Schwartz inequality to the
	series (\ref{65}), we see that%
	\begin{align}
		\sum_{k=n}^{n+m}\lambda_{k}\left \vert c_{k}\Upsilon_{k}\left(  \zeta \right)
		\right \vert  &  \leq \sqrt{\sum_{k=n}^{n+m}\lambda_{k}c_{k}^{2}}\sqrt
		{\sum_{k=n}^{n+m}\lambda_{k}\Upsilon_{k}^{2}\left(  \zeta \right)  }\nonumber \\
		& \nonumber \\
		&  \leq \sqrt{C}\sqrt{\sum_{k=n}^{n+m}\lambda_{k}c_{k}^{2}}. \label{67}%
	\end{align}
	From (\ref{bes}) and (\ref{67}), the series (\ref{63})\ is uniformly
	convergent on $[0,d)\cup(d,a].$ Since
	\[
	\left \vert \sum_{k=1}^{\infty}c_{k}\psi_{k}\left(  \zeta \right)  \right \vert
	\leq \sum_{k=1}^{\infty}\left \vert c_{k}\psi_{k}\left(  \zeta \right)
	\right \vert ,
	\]
	the series (\ref{40}) is also uniformly convergent on$\ [0,d)\cup(d,a].$
	
	Let
	\begin{equation}
		f_{1}\left(  \zeta \right)  =\sum_{k=1}^{\infty}c_{k}\psi_{k}\left(
		\zeta \right)  . \label{68}%
	\end{equation}
	Since the series (\ref{68}) is uniformly convergent on $[0,d)\cup(d,a],$ we
	get%
	\[
	\int_{0}^{d}f_{1}^{\left(  1\right)  }\left(  \zeta \right)  \psi_{k}^{\left(
		1\right)  }\left(  \zeta \right)  d_{q}\zeta+\int_{d}^{a}f_{1}^{\left(
		2\right)  }\left(  \zeta \right)  \psi_{k}^{\left(  2\right)  }\left(
	\zeta \right)  d_{q}\zeta=c_{k}\text{ }(k\in \mathbb{N}).
	\]
	Consequently, the Fourier coefficients of $f$ and $f_{1}$ are the same.
	Applying the Parseval equality (\ref{39}) to the function $f-f_{1},$\ we find
	$f-f_{1}=0,$ due to the Fourier coefficients of the function $f-f_{1}$ are
	zero. This finishes the proof.
\end{proof}

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