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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 36, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/36\hfil Solvability of multipoint differential operators]
{Solvability of multipoint differential operators of first order}

\author[Z. I. Ismailov, P. Ipek \hfil EJDE-2015/36\hfilneg]
{Zameddin I. Ismailov, Pembe Ipek}

\dedicatory{In memory of acad. M. G. Gasymov (1939-2008)}

\address{Zameddin I. Ismailov \newline
Karadeniz Technical University, Department of Mathematics,
61080, Trabzon, Turkey}
\email{zameddin.ismailov@gmail.com}

\address{Pembe Ipek \newline
Karadeniz Technical University, Institute of Natural Sciences,
61080 Trabzon, Turkey}
\email{pembeipek@hotmail.com}

\thanks{Submitted January 2, 2015. Published February 10, 2015.}
\subjclass[2000]{47A10}
\keywords{Differential operators; solvable extension; spectrum}

\begin{abstract}
 Based on methods of operator theory, we describe all boundedly solvable
 extensions of the minimal operator generated by a linear multipoint
 functional differential-operator expression of first order, in a direct
 sum of Hilbert space of vector-functions. We also study the
 structure of spectrum of these extensions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks



\section{Introduction}

The general theory of extension of densely defined linear operator in Hilbert
spaces was started by von Neumann with his important work \cite{n2} in 1929.
Later in 1949 and 1952 Vishik in \cite{v1,v2} studied
the boundedly (compact, regular and normal) invertible extensions
of any unbounded linear densely defined operator in Hilbert spaces
Generalization of these results to the nonlinear and complete additive
Hausdorff topological spaces in abstract terms have been done by
Kokebaev, Otelbaev and Synybekov in \cite{k1,o2}.
By Dezin \cite{d1} another approach to the description of regular extensions
for some classes of linear differential operators in Hilbert spaces
of vector-functions at finite interval has been offered.

 On other hand  the role of the two point and multipoint theory
of functional differential equations in our lives is indisputable.
The general theory of delay differential equations is presented in many books
(for example \cite{e2,s1}). Applications of this theory can be found in economy,
biology, control theory, electrodinamics, chemistry, ecology, epidemiology,
tumor growth, neural networks and etc. (see \cite{e1,o1}).

 In addition note that oscillation and boundness properties of solutions of
different classes of dynamic equations have been investigated in the
book by  Agarwal, Bohner and Li \cite{a1}.

Let us remember that an operator $S:D(S)\subset H \to  H$ on Hilbert spaces
is called boundedly invertible, if $ S $ is one-to-one, $ SD(S)=H $ and
$ S^{-1}\in L(H)$.

 The main goal of this work is to describe all boundedly solvable extensions
of the minimal operator generated by linear multipoint functional
differential-operator expression for first order in the direct sum of
 Hilbert spaces of vector-functions at finite intervals and investigate
the structure of spectrum of these extensions.
In the second section all boundedly solvable extensions of the above mentioned
minimal operator are described. Structure of spectrum of these extensions
is studied in third section
Finally, the obtained results are illustrated by applications.

\section{Description of Boundedly Solvable Extensions}

Throughout this work $H_n$ is a separable Hilbert space,
$ \mathcal{H}_n=L^2(H_n,\Delta_n) $,
$ \Delta_n=(a_n,b_n)\subset \mathbb{R}  $ for each $n \geq 1  $ with
property $ -\infty < \inf_{n\geq 1} a_n <\sup_{n\geq 1}b_n<\infty $,
 $\inf_{n\geq 1}| \Delta_n|>0  $ and
$ \mathcal{H}=\oplus_{n=1}^\infty \mathcal{H}_n $.

 We consider the  linear multipoint functional differential-operator
expression of first order in $ \mathcal{H} $ of the form
\begin{equation*}
l(u)=(l_n(u_n)),~ u=(u_n),
\end{equation*}
where:
\begin{itemize}
\item[(1)] $ l_n(u_n)=u_n'(t)+A_n(t)u(\alpha_n(t))$, $n \geq 1$;

\item[(2)] operator-function $A_n(\cdot):[a_n,b_n]\to L(H_n)$,
$n\geq 1$ is continuous on the uniformly operator topology and
$\sup_{n\geq 1}\sup_{t\in \Delta_n} \| A_n(t) \| <\infty $;

\item[(3)] For any $ n \geq 1$, $\alpha_n:[a_n,b_n] \to [a_n,b_n]$
is invertible and
$\alpha_n, (\alpha_n^{-1})'\in C[a_n,b_n]$,
also $\sup_{n\geq 1}(\| (\alpha_n^{-1}(t))'
\|_{\infty} )^{1/2}<\infty$
\end{itemize}

By standard way the minimal $ L_{n0} $ and maximal $ L_n $ operators
corresponding to the differential expression $ l_n(\cdot ) $ in
$ \mathcal{H}_n $ can be defined for any $ n \geq 1 $.
It is clear that for every $ n\geq 1 $ domains of minimal $ L_{n0} $
 and maximal $ L_n $ operators in $ \mathcal{H}_n $ are in the forms
 $$
 D(L_{n0})=\stackrel{_o}{W}_{2}^{1}(H_n,\Delta_n), \quad\text{and}\quad
 D(L_n)={W}_{2}^{1}(H_n,\Delta_n),
 $$
respectively.
For any scalar function $\varphi:[a_n,b_n]\to[a_n,b_n]$ and
$ n\geq 1$ now define an operator $P_\varphi$ in $\mathcal{H}_n$ in the form
\[
P_\varphi u_n(t)=u_n(\varphi(t)), \quad u_n\in \mathcal{H}_n\,.
\]
If a function  $ \varphi\in C^{1}[a_n,b_n] $ and
$\varphi'(t)>0 \ (<0)$ for $t\in[a_n,b_n]$, then for any
$u_n\in \mathcal{H}_n$,
\begin{align*}
\|P_\varphi u_n\|^2_{\mathcal{H}_n}
&= \int_{a_n}^{b_n}\|u_n(\varphi(t))\|^2_{H_n} \,dt  \\
&= \int_{\varphi(a_n)}^{\varphi(b_n)}\|u_n(\varphi(x))\|^2_{H_n}
 (\varphi^{-1})'(x)dx   \\
&\leq  \big| \int_{\varphi(a_n)}^{\varphi(b_n)}\|u_n(x)\|^2_{H_n}
 |(\varphi^{-1})'(x)|dx\big|  \\
&\leq  \|(\varphi^{-1})'\|_\infty\int_{a_n}^{b_n}\|u_n(x)\|^2_{H_n} dx  \\
&= \|(\varphi^{-1})'\|_\infty\|u_n\|^2_{\mathcal{H}_n}
\end{align*}
Consequently, for any strictly monotone function $ \varphi \in C^{1}[a_n,b_n] $,
the operator $P_\varphi$ belongs to $L(\mathcal{H}_n)$
and $\|P_\varphi\|\leq\sqrt{\|(\varphi^{-1})'\|_\infty}$.
In actually, the expression $ l(\cdot ) $ in $ \mathcal{H} $ can
be written in the form
\begin{equation} \label{e1}
l(u)=u'(t)+A_{\alpha}(t)u(t),
\end{equation}
where $ u=(u_n) $,
\[
A(t)= \begin{pmatrix}
   A_1(t)\\
    & A_2(t) & & 0\\
    & & \ddots\\
    & 0 & & A_n(t)\\
    & & & & \ddots
 \end{pmatrix}, \quad
P_{\alpha}= \begin{pmatrix}
   P_{\alpha_{1}}\\
    &  P_{\alpha_{2}} & & 0\\
    & & \ddots\\
    & 0 & &  P_{\alpha_n}\\
    & & & & \ddots
 \end{pmatrix}
\]
and $ A_{\alpha} (t)=A(t)P_{\alpha}, \ A_{\alpha} \in L(\mathcal{H})$.

The operators $ L_0(M_0) $ and $ L(M) $ be a minimal and maximal
operators corresponding to \eqref{e1} (with $ m(\cdot )=\frac{d}{\,dt} $) in
$ \mathcal{H} $, respectively. Let us define
\begin{gather*}
L_0=M_0+A_{\alpha}(t), \quad L=M+A_{\alpha}(t), \\
L_0: D(L_0)\subset \mathcal{H}\to\mathcal{H}, \quad
L:D(L)\subset\mathcal{H}\to \mathcal{H}, \\
D(L_0)=\{ (u_n)\in \mathcal{H}:  u_n\in \stackrel{_o}{W}_{2}^{1}(H_n,\Delta_n),
\; n\geq1, \; \sum_{n=1}^\infty \| L_{n0}u_n \|^2_{\mathcal{H}_n} < \infty \},
\\
D(L)=\{ (u_n)\in \mathcal{H}: u_n\in {W}_{2}^{1}(H_n,\Delta_n), \; n\geq1, \;
 \sum_{n=1}^\infty \| L_nu_n \|^2_{\mathcal{H}_n}<\infty \},
\end{gather*}

The main goal in this section is to describe all boundedly solvable extensions
of the minimal operator $ L_0 $ in $ \mathcal{H} $ in terms in the boundary values.
Before that, we prove the validity the following assertion.

\begin{lemma} \label{lem2.1}
The kernel and image sets of $ L_0 $ in $ \mathcal{H} $ satisfy
$ \ker L_0=\{0\} $ and $ \overline{\operatorname{Im}(L_0)}\neq \mathcal{H}$.
\end{lemma}

\begin{proof}
 Firstly, we prove that for any $ n\geq 1 $ $ \ker L_{n0}=\{0\}$.
Consider the  boundary values problems
\begin{gather*}
 u_n'(t)+A_n(t)P_{\alpha_n}u_n(t)=0,\\
u_n(a_n)=u_n(b_n)=0, \quad n\geq 1
\end{gather*}
The general solution of these differential equations  are
\[
u_n(t)=\exp \Big( -\int_{a_n}^{t}A_n(s)P_{\alpha_n}\,ds \Big) f_n, \quad n\geq 1
\]
Then from boundary value conditions we have
$f_n=0$ for $n\geq 1$.
Hence $ \ker L_{n0}=\{0\}$ for $n\geq 1$.  Then $ \ker L_0=\{0\}$.

To show that $ \overline{\operatorname{Im}(L_0)}\neq \mathcal{H} $ it is sufficient
to show that $ \overline{\operatorname{Im}(L_{n0})}\neq \mathcal{H}_n $
for some $n\geq 1$.
So we consider the  boundary value problem
\[
L_{n0}^{*}u_n=-u_n'(t)+(A_n(t)P_{\alpha_n})^{*}u_n(t)=0, \quad u_n\in \mathcal{H}_n
\]
The solutions of this equation are
\[
u_n(t)=\exp \Big( \int_{a_n}^{t}(A_n(s)P_{\alpha_n})^{*}\,ds \Big)g_n, \quad
 g_n\in H_n
\]
Consequently,
$H_n\subset \ker L_{n0}^{*}$,  $n\geq 1$
This means that for any $n\geq 1$
$\overline{\operatorname{Im}(L_{n0})}\neq \mathcal{H}_n$.
Then $\overline{\operatorname{Im}(L_0)}\neq \mathcal{H}$.
\end{proof}

\begin{theorem} \label{thm2.2}
If $ \widetilde{L} $ is any extension of $ L_0 $ in $ \mathcal{H}$, then
\[
\widetilde{L}=\oplus_{n=1}^\infty \widetilde{L}_n,
\]
where $ \widetilde{L}_n $ is a extension of $ L_{n0}$, $n\geq 1 $, in
$ \mathcal{H}_n$.
\end{theorem}

\begin{proof}
Indeed, in this case for any $ n\geq 1 $ the linear manifold
\[
\widetilde{M}_n:=\{ u_n\in D(L_n): (u_n)\in D(\widetilde{L}) \}
\]
contains $ D(L_{n0})$. That is,
\[
D(L_{n0})\subset \widetilde{M}_n \subset D(L_n), \quad n\geq 1
\]
Then the  operator
\[
\widetilde{L}_nu_n=l_n(u_n), \quad u_n\in \widetilde{M}_n, \; n\geq 1
\]
is an extensions of $ L_{n0}$. Consequently, for any $ n\geq 1 $,
\[
\widetilde{L}_n: D(\widetilde{L}_n)=\widetilde{M}_n\subset \mathcal{H}_n
\to \mathcal{H}_n
\]
From this, we obtain
\[
\widetilde{L}=\oplus_{n=1}^\infty \widetilde{L}_n\,.
\]
For the boundedly solvable extensions of $ L_0 $ and $ L_{n0}$, $ n\geq 1$,
the following statement holds.
\end{proof}

\begin{theorem} \label{thm2.3}
For the boundedly solvability of any extension
 $ \widetilde{L}=\oplus_{n=1}^\infty \widetilde{L}_n $ of the minimal
 operator $ L_0 $, the necessary and sufficient conditions are the boundedly
solvability of the coordinate operators $ \widetilde{L}_n $ of the minimal
operator  $ L_{n0}$, $n\geq 1 $ and
$ \sup_{n\geq 1} \| \widetilde{L}_n^{-1} \| <\infty$.
\end{theorem}

\begin{proof}
 It is clear that the extension $ \widetilde{L} $ is one-to-one operator
in $ \mathcal{H} $ if and only if the all coordinate
 extensions $ \widetilde{L}_n$, $n\geq 1 $ of $ \widetilde{L} $ are
one-to-one operators in $ \mathcal{H}_n, \ n\geq 1$.
On the other hand for the boundedness  of
$ \widetilde{L}^{-1}=\oplus_{n=1}^\infty \widetilde{L}^{-1}_n $ of
$ \mathcal{H} $ the necessary and sufficient condition is
 $ \sup_{n\geq 1} \| \widetilde{L}_n^{-1} \| <\infty  $
(see \cite{i1,n1}).

Now let $U_n(t,s)$, $t,s\in \Delta_n$, $n\geq 1 $, be the family of evolution
operators corresponding to the homogeneous differential equation
\[
\frac{\partial U_n}{\partial t}(t,s)f+A_n(t)P_{\alpha_n}U_n(t,s)f=0, \quad
 t,s\in \Delta_n
\]
with the boundary condition
\[
U_n(s,s)f=f, \ f\in H_n\,.
\]
The operator $ U_n(t,s)$, $t,s \in \Delta_n$, $n\geq 1 $ is linear
continuous boundedly invertible in $ H_n $ with the property (see \cite{k2})
\[
U_n^{-1}(t,s)=U_n(s,t),~s,t\in \Delta_n \, .
\]
Lets us introduce the operator $U_n:\mathcal{H}_n\to \mathcal{H}_n$ as
\[
U_nz_n(t)=: U_n(t,0)z_n(t), \quad t\in \Delta_n\,.
\]
It is clear that if $ \widetilde{L}_n$ is any extension of the minimal
operator $L_{n0}$, that is, $L_{n0}\subset \widetilde {L}_n \subset L_n$, then
\[
U^{-1}_nL_{n0}U_n=M_{n0},\quad
M_{n0}\subset U^{-1}_n\widetilde{L}_nU_n
=\widetilde{M}_n\subset M_n, \quad
U^{-1}_nL_nU_n=M_n
\]
In addition, for any $ n\geq 1 $,
\begin{align*}
 \| U_n \|
&=  \big\| \exp\Big( - \int_{a_n}^{t} A_n(s) P_{\alpha_n}\,ds\Big) \big\|   \\
&\leq  \exp \Big(  \int_{a_n}^{b_n} \| A_n(s) \| \| P_{\alpha_n} \| \,ds  \Big) \\
&\leq  \exp \Big( | \Delta_n|\| P_{\alpha_n} \|\sup_{t\in \Delta_n}\| A_n(t)\|
\Big)< \infty
\end{align*}
and
\begin{align*}
 \| U^{-1}_n \|
&=  \| \exp\Big(  \int_{a_n}^{t} A_n(s) P_{\alpha_n}\,ds\Big) \|   \\
&\leq  \exp \Big( | \Delta_n|\| P_{\alpha_n} \|\sup_{t\in \Delta_n}\| A_n(t)\|
  \Big)< \infty\,.
\end{align*}
\end{proof}

\begin{theorem} \label{thm2.4}
Let  $ n\geq 1$. Each boundedly solvable extension $ \widetilde{L}_n$ of
the minimal operator $ L_{n0} $ in $ \mathcal{H}_n $ is generated by
the differential-operator expression $ l_n(\cdot ) $ and the boundary condition
\begin{equation} \label{e2}
(K_n+E_n)u_n(a_n)=K_nU_n(a_n,b_n)u_n(b_n),
\end{equation}
where $K_n\in L(H_n)$ and $E_n:H_n\to H_n$ is identity operator.
The operator $K_n$ is determined by the extension $\widetilde{L}_n$ uniquely,
i.e. $\widetilde{L}_n=L_{K_n}$.

 On the contrary, the restriction of the maximal operator $L_n$ in
$ \mathcal{H}_n $ to the linear manifold of vector-functions satisfy the
condition \eqref{e2} for some bounded operator $K_n\in L(H_n), $  is a
 boundedly solvable extension of the minimal operator $L_{n0}$.

On other hand, for $ n\geq 1 $
\[
\| \widetilde{L}^{-1}_n \| \leq | \Delta_n|^{1/2}
\exp\Big( 2| \Delta_n|\| P_{\alpha_n}\| \sup_{t\in \Delta_n}\| A_n(t) \| \Big)
\]
\end{theorem}

\begin{proof}
For any $ n\geq 1 $, the description the all boundedly solvable extensions of
 $ \widetilde{L}_n $ of the minimal operator $L_{n0}$ in $ \mathcal{H}_n$,
$n\geq 1 $ have been given in work \cite{i2}.
From the relation $ U^{-1}_n L_{K_n}U_n=M_{K_n} $ we obtain
\[
L_{K_n}^{-1}=U_nM_{K_n}^{-1} U^{-1}_n,
\]
where $M_{K_n}u_n(t)=u'_n(t)  $ with the boundary condition
\[
 (K_n+E_n)u_n(a_n)=K_nu_n(b_n), \quad n\geq 1\,.
\]
Hence for $ f_n\in L^2(H_n,\Delta_n) $,
\begin{align*}
\| M_{K_n}^{-1}f_n \|^2_{\mathcal{H}_n}
&= \int_{a_n}^{b_n}\| K_n\int_{a_n}^{b_n}f_n(s)\,ds
+\int_{a_n}^{t}f_n(s)\,ds\|^2\,dt   \\ 
&\leq  2 \int_{a_n}^{b_n}\| K_n\int_{a_n}^{b_n}f_n(s)\,ds\|^2\,dt
 +2\int_{a_n}^{b_n}\| \int_{a_n}^{t}f_n(s)\,ds\|^2\,dt  \\
&\leq  2 \| K_n\| ^2  \int_{a_n}^{b_n}
 \int_{a_n}^{b_n}\| f_n(s)\| ^2\,ds| \Delta_n| \,dt\\
&\quad +2\int_{a_n}^{b_n} \Big(\int_{a_n}^{b_n} \| f_n(s)\| ^2\,ds \Big) \,dt | \Delta_n|  \\  \\
&=  \big(2  \| K_n\|^2| \Delta_n| ^2+2 | \Delta_n| ^2\big)
 \| f_n \| ^2_{\mathcal{H}_n}  \\
&= 2 | \Delta_n| ^2 ( 1+ \| K_n \| ^2) \| f_n \| ^2_{\mathcal{H}_n}\,.
 \end{align*}
 Hence,
\[
\| M_{K_n}^{-1}\|\leq \sqrt{2}| \Delta_n|
\big(  1+ \| K_n \| ^2 \big) ^{1/2}, \quad n\geq 1
\]
From this and the properties of  evolution operators,
 the validity of inequality is clear.
This completes proof of theorem.
\end{proof}

\begin{theorem} \label{thm2.5}
Let $ L_{K_n} $ be a boundedly solvable extension of $ L_{n0} $ in
$ \mathcal{H}_n $ and
\[
M_{K_n}=U_n^{-1}L_{K_n}U_n, \quad\text{for }  n\geq1 \,.
\]
To have
\[
\sup_{n\geq1}\| L^{-1}_{K_n}\|<\infty,
\]
the necessary and sufficient condition is
\[
\sup_{n\geq1}\| M^{-1}_{K_n}\|<\infty\,.
\]
\end{theorem}

\begin{theorem} \label{thm2.6}
Assumed that
\begin{gather*}
M_{K_n}:W^{1}_{2}(H_n,\Delta_n)\subset L^2(H_n,\Delta_n)\to L^2(H_n,\Delta_n), \\
M_{K_n}u_n(t)=u'_n(t), \\
(K_n+E_n)u_n(0)=K_nu_n(1)
\end{gather*}
To have
\[
\sup_{n\geq1}\| M^{-1}_{K_n}\|<\infty,
\]
the necessary and sufficient condition is
$\sup_{n\geq1}\| K_n\|<\infty$.
\end{theorem}

\begin{proof}
Indeed, from the proof of Theorem \ref{thm2.4},
\[
\| M_{K_n}^{-1}\|\leq \sqrt{2}| \Delta_n|
\big(  1+ \| K_n \| ^2 \big) ^{1/2}, \quad n\geq 1
\]
From this, if $ \sup_{n\geq1}\| K_n\|<\infty$, then
 $ \sup_{n\geq1}\| M^{-1}_{K_n}\|<\infty$.
On the contrary, assumed that $ \sup_{n\geq1}\| M^{-1}_{K_n}\|<\infty$.
Then in the relation
\[
K_n \int_{a_n}^{b_n}f_n(t)\,dt=M^{-1}_{K_n}f_n(t)
-\int_{a_n}^{t}f_n(t)\,dt, \ f_n\in \mathcal{H}_n, \quad n\geq 1
\]
choosing the functions $ f_n(t)=f_n^{*}$, $t\in \Delta_n$,
$f_n^{*}\in H_n$, $n\geq 1$, we have
\[
\| K_nf_n^{*}\|_{H_n}| \Delta_n|\leq \| M^{-1}_{K_n}f_n^{*}\|_{H_n}
+| \Delta_n| \| f_n^{*}\|_{H_n}, \quad n\geq 1\,.
\]
Then
\[
\| K_n \| \leq \frac{1}{| \Delta_n|} \| M^{-1}_{K_n}\|+1\,.
\]
Consequently, from the above  relation and the condition on
 $ \Delta_n$, $n\geq 1 $, we  obtain
\[
\sup_{n\geq1}\| K_n\|\leq \Big( \inf_{n\geq1}| \Delta_n|\Big) ^{-1}
\sup_{n\geq1}\| M^{-1}_{K_n}\| +1<\infty\,.
\]
\end{proof}

Now using Theorems \ref{thm2.4}--\ref{thm2.6}, we formulate  an assertion on the
description of all  boundedly solvable extensions of to in $ \mathcal{H}$.

\begin{theorem} \label{thm2.7}
Each boundedly solvable extension $ \widetilde{L} $ of the minimal operator
 $  L_0$ in  $ \mathcal{H}  $ is generated by  differential-operator
 expression \eqref{e1} and  boundary conditions
\[
(K_n+E_n)u_n(a_n)=K_nU_n(a_n,b_n)u_n(b_n), \quad n\geq 1,
\]
where $K_n\in L(H_n)$,
$ K=\oplus_{n=1}^{\infty}K_n\in L\big( \oplus_{n=1}^{\infty}H_n\big)  $ and
$E_n:H_n\to H_n$ is identity operator. The operator $K$ is determined by
the extension $\widetilde{L}$ uniquely, i.e. $\widetilde{L}=L_K$ and vice versa.
\end{theorem}

\begin{remark} \label{rmk2.8} \rm
 If in the \eqref{e1}, $ \alpha_n(t)=t$  ($\alpha_n(t)<t$),  $t\in [a_n,b_n] $
for any $ n\geq 1$, then this problem  corresponds to the problem of
theory of multipoint ordinary (delay) differential operators in Hilbert spaces
 of  vector-functions.
 \end{remark}


\section{Structure of spectrum of boundedly solvable extensions}

 In this section the structure of spectrum of boundedly solvable extensions
of minimal operator $ L_0 $ in $ \mathcal{H} $ is investigated.
First we consider  the spectrum for the boundedly solvable extension 
$ L_{K}$, $K=(K_n) $ of the minimal operator $ L_0 $ in $ \mathcal{H} $; that is,
\[
L_{K}u=\lambda u+f, \quad \lambda \in \mathbb{C}, \quad u=(u_n), \quad
 f=(f_n)\in \mathcal{H}
\]
From it follows that
\[
\oplus_{n=1}^\infty (L_{K_n}-\lambda E_n)(u_n)=(f_n)
\]
The last relation is equivalent to the equations
\[
(L_{K_n}-\lambda E_n)u_n=f_n, \ n\geq 1, \quad \lambda \in \mathbb{C}, \quad
 f_n\in \mathcal{H}_n
\]
That is, for any $ n\geq 1 $, we have
\[
U_n(M_{K_n}-\lambda E_n)U^{-1}_nu_n=f_n
\]
Therefore,
\begin{equation} \label{e3}
\sigma_{p}(L_{K_n})=\sigma_{p}(M_n),\quad
\sigma_{c}(L_{K_n})=\sigma_{c}(M_n),\quad
\sigma_{r}(L_{K_n})=\sigma_{r}(M_n)
\end{equation}
Consequently, we consider the the spectrum parts 
of $ M_{K_n}$; that is,
\[
M_{K_n}u_n=\lambda u_n+f_n, \quad \lambda \in \mathbb{C}, \quad
 f_n\in  \mathcal{H}_n, \quad n\geq 1
\]
Then from this we obtain
\begin{gather*}
u'_n=\lambda u_n+f_n,\\
(K_n+E_n)u_n(a_n)=K_nu_n(b_n), \quad n\geq 1
\end{gather*}
Since the general solution of the above differential equation in 
$ \mathcal{H}_n $ has the form
\[
u_n(t,\lambda)=\exp(\lambda (t-a_n))f^{0}_n+\int_{a_n}^t 
\exp(\lambda (t-s))f_n(s)\,ds, \quad t \in \Delta_n,
\]
$f^{0}_n\in H_n$, $ n\geq 1$,
from the boundary condition it is obtained that
\[
\Big( E_n+K_n\big( 1-\exp(\lambda | \Delta_n|)\big) \Big) 
f^{0}_n=K_n\int_{a_n}^{b_n}\exp(\lambda (b_n-s))f_n(s)\,ds, \quad n\geq 1
\]
It is easy to show that 
$ \lambda_{n,m}=\frac{2m\pi i}{| \Delta_n|}\in\rho(M_{K_n})$, 
$ m\in\mathbb{Z}$, $n\geq 1$. 
Then for $ \lambda_{n,m}\neq 2m\pi i/| \Delta_n|$, 
$m\in\mathbb{Z}$, $n\geq 1 $, we have
\begin{align*}
&\Big( K_n-\frac{1}{exp(\lambda | \Delta_n|)-1}E_n \Big)f^{0}_n \\
&= \big(1-\exp(\lambda | \Delta_n|)\big)^{-1}K_n
\int_{a_n}^{b_n} \exp(\lambda(b_n-s))f_n(s)\,ds,
\end{align*}
$f^{0}_n\in H_n$, $f_n\in \mathcal{H}_n$, $n\geq 1$.
From this and relations \eqref{e3} it follows the validity of 
following statement.

\begin{theorem} \label{thm3.1}
In order for $ \lambda$ to belong to 
$\sigma_{p}(L_{K_n})$ $(\sigma_{c}(L_{K_n}), \sigma_{r}(L_{K_n})) $,
it is necessary and sufficient that 
\[
 \mu =\frac{1}{\exp(\lambda | \Delta_n|) -1}
\in\sigma_{p}(K_n)\ (\sigma_{c}(K_n), \quad \sigma_{r}(K_n)).
\]
\end{theorem}

Then a structure of spectrum of boundedly solvable extension $ L_{K_n} $ 
can be formulated in the form

\begin{theorem} \label{thm3.2}
The point spectrum of the boundedly solvable extension $ L_{K_n} $ has the form
\begin{align*}
\sigma_{p}(L_{K_n})=\Big\{ & \lambda \in \mathbb{C}:
 \lambda=\frac{1}{| \Delta_n|}\big\{ \ln|{\frac{\mu+1}{\mu}}|+ i 
\arg({\frac{\mu+1}{\mu}})+2m\pi i\big\} ,\\ 
&\mu\in \sigma_{p}(K_n)\backslash\{0,-1\},\; m\in\mathbb{Z}\Big\}
\end{align*}
Similarly propositions on the continuous $ \sigma_{c}(L_{K_n}) $ and 
residual $ \sigma_{r}(L_{K_n}) $ spectrums are true.
\end{theorem}

Lastly, using the results on the spectrum parts of direct sum of  operators 
in the  direct sum of  Hilbert spaces \cite{o3} in we can proved the following theorem.

\begin{theorem} \label{thm3.3}
For the parts of spectrum of the boundedly solvable extension 
$ L_{K}=\oplus_{n=1}^\infty L_{K_n}$, $K=(K_n) $ in Hilbert spaces
 $ \mathcal{H}=\oplus_{n=1}^\infty \mathcal{H}_n $ the following 
statements are true
\begin{gather*}
\sigma_{p}(L_{K})=\cup_{n=1}^\infty \sigma_{p}(L_{K_n}), \\
\begin{aligned}
\sigma_{c}(L_{K}) 
&= \Big\{ \Big( \cup_{n=1}^\infty \sigma_{p}(L_{K_n})\Big)^{c} 
  \cap \Big( \cup_{n=1}^\infty \sigma_{r}(L_{K_n})\Big)^{c} 
  \cap \Big( \cup_{n=1}^\infty \sigma_{c}(L_{K_n})\Big) \Big\}  \\
&\quad  \cup \Big\{ \lambda \in \cap_{n=1}^\infty \rho (L_{K_n}): 
 \sup_{n\geq1}\| R_{\lambda} (L_{K_n})\|= \infty \Big\},
\end{aligned}\\
\sigma_{r}(L_{K})=\Big( \cup_{n=1}^\infty \sigma_{p}(L_{K_n})\Big)^{c} 
     \cap \Big( \cup_{n=1}^\infty \sigma_{r}(L_{K_n})\Big)
\end{gather*}
\end{theorem}

\section{Applications}

In this section, we  present an application of above results.

\begin{example} \label{examp1} \rm
For any $ n\geq 1 $ let us $ H_n=(\mathbb{C},|\cdot |)$,  $a_n=0$, $b_n=1$, 
$ A_n(t)=c_n$,  $c_n\in\mathbb{C}$, $\sup_{n\geq 1}|c_n|<\infty$, 
$\alpha_n(t)=\alpha_nt$, $0<\alpha_n<1 $ with property 
$ \sup_{n\geq1}( \frac{1}{\alpha_n} )<\infty  $.
 Consider the  pantograph type delay differential expression
\[
l(u)=u'_n(t)+c_nu_n(\alpha_nt)
\]
in  $ \mathcal{H}=\oplus_{n=1}^\infty \mathcal{H}_n$,
 where $\mathcal{H}_n=L^2(0,1)$.

In this case by Theorem \ref{thm2.7}, all boundedly solvable extensions $ L_{k} $ 
of the minimal operator $ L_0 $ generated by $ l(\cdot ) $ in 
$ \mathcal{H} $ are described by the differential expression  
$ l(\cdot ) $ and the boundary conditions
\[
(k_n+1)u_n(0)=k_nU_n(0,1)u_n(1),
\]
where $ k_n\in\mathbb{C}$, $n\geq 1 $, $ \sup_{n\geq 1}|k_n|<\infty $ 
and vice versa.

On other hand in the case  $ k_n\notin \{0,-1\}$, for any $ n\geq 1 $ 
the point, the continuous and residual spectrums of $ L_{k_n} $ in 
$ \mathcal{H}_n=L^2(0,1) $ is of the form
\begin{gather*}
\sigma_{p}(L_{k_n})=\Big\{  \lambda \in \mathbb{C}:\lambda
= ln|{\frac{k_n+1}{k_n}}|+ i \arg({\frac{k_n+1}{k_n}})+2m\pi i ,\; 
  m\in\mathbb{Z}\Big\},
\\
\sigma_{c}(L_{k_n})=\sigma_{r}(L_{k_n})=\emptyset\,.
\end{gather*}
Hence by Theorem \ref{thm3.3} spectrum parts of $ L_{k}=\oplus_{n=1}^\infty L_{k_n} $ 
has the form
\begin{gather*}
\sigma_{p}(L_{k})= \cup_{n=1}^\infty  \cup_{m\in\mathbb{Z} }
\Big\{   ln|{\frac{k_n+1}{k_n}}|+ i \arg({\frac{k_n+1}{k_n}})+2m\pi i \Big\} ,
\\
\sigma_{c}(L_{k})=\Big\{ \lambda \in \cap_{n=1}^\infty \rho (L_{k_n}):
\sup_{n\geq1}\| R_{\lambda} (L_{k_n})\|= \infty \Big\},
\\
\sigma_{r}(L_{k})=\emptyset
\end{gather*}
\end{example}

\begin{example} \label{examp2} \rm
Let $ H_n=(\mathbb{C},|\cdot |)$, $ (a_n)$, a sequence of real numbers, 
$ \sup_{n\geq 1}|a_n|<\infty$, $(b_n)$, $b_n=a_n+1$, $\Delta_n=(a_n,b_n)$, 
$\mathcal{H}_n=L^2(H_n, \Delta_n)$,
\[
l_n(u_n)=u'_n+u_n(\alpha_n(t)), \quad
 \alpha_n(t)=(t-a_n)^2+a_n-\frac{1}{2}, \quad t\in  \Delta_n, \quad n\geq 1,
\]
$ \mathcal{H}=\oplus_{n=1}^\infty \mathcal{H}_n$, 
$l(\cdot )= \oplus_{n=1}^\infty l_n(\cdot )$. 
In this case for any $  n\geq 1 $ the function $ \alpha_n(\cdot ) $ 
is increase, invertible and 
$  \alpha^{-1}_n(t)=a_n+\sqrt{t-a_n+\frac{1}{2}} $ 
 and $(\alpha^{-1}_n(t))'=\frac{1}{2\sqrt{t-a_n+\frac{1}{2}}}$.
Hence $ \sup_{n\geq1} \| (\alpha^{-1}_n)'\|_{\infty} \leq \frac{1}{2}$.
In this case all boundedly solvable extensions of minimal operator 
$ L_0 $ in $ \mathcal{H} $ are described by $ l(\cdot ) $ and boundary conditions
\[
(k_n+1)u_n(a_n)=k_nU_n(a_n,b_n)u_n(b_n), \quad n\geq 1,
\]
where $\sup_{n\geq1}| k_n|<\infty$.
On the other hand for $ k_n\notin \{0,-1\} $ spectrum of each boundedly 
solvable extension $ L_{k_n} $ has the form
\begin{gather*}
\sigma_{p}(L_{k_n})=\Big\{  \lambda \in \mathbb{C}:\lambda
= \ln|{\frac{k_n+1}{k_n}}|+ i \arg({\frac{k_n+1}{k_n}})+2m\pi i , \;
  m\in\mathbb{Z}\Big\} ,
\\
\sigma_{c}(L_{k_n})=\sigma_{r}(L_{k_n})=\emptyset, \quad n\geq 1
\end{gather*}
Hence by Theorem \ref{thm3.3}, the spectrum parts of $ L_{k}=\oplus_{n=1}^\infty L_{k_n} $
 have the form
\begin{gather*}
\sigma_{p}(L_{k})= \cup_{n=1}^\infty  \cup_{m\in\mathbb{Z} }
\big\{   ln|{\frac{k_n+1}{k_n}}|+ i \arg({\frac{k_n+1}{k_n}})+2m\pi i \big\} ,
\\
\sigma_{c}(L_{k})=\big\{ \lambda \in \cap_{n=1}^\infty \rho (L_{k_n}):
\sup_{n\geq1}\| R_{\lambda} (L_{k_n})\|= \infty \big\},
\\
\sigma_{r}(L_{k})=\emptyset
\end{gather*}
\end{example}

\begin{remark} \label{rmk4.3} \rm
Similar to the problems in Example 4.1 and 4.2, we can 
investigate the case 
 $ \alpha_n(t)=b_n-t$, $a_n\leq t \leq b_n$, $n\geq 1$. 
\end{remark}

\subsection*{Acknowledgements}
The authors are grateful to G. Ismailov (Undergraduate Student of Marmara
 University, Istanbul) for his helps in preparing the English version 
of this article and for the technical discussions.

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\end{document}