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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 67, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/67\hfil Boundedly solvable delay differential operators]
{Boundedly solvable extensions of delay differential operators}
\author[B. \"O. G\"uler, B. Y{\i}lmaz, Z. I. Ismailov \hfil EJDE-2017/67\hfilneg]
{Bahad{\i}r \"O. G\"uler, B{\"u}lent Y{\i}lmaz, Zameddin I. Ismailov}
\address{Bahad{\i}r \"O. G\"uler \newline
Department of Mathematics,
Karadeniz Technical University, Turkey}
\email{boguler@ktu.edu.tr}
\address{B\"ulent Y{\i}lmaz \newline
Department of Mathematics,
Marmara University, Turkey}
\email{bulentyilmaz@marmara.edu.tr}
\address{Zameddin Ismailov \newline
Department of Mathematics,
Karadeniz Technical University, Turkey}
\email{zameddin.ismailov@gmail.com}
\dedicatory{Communicated by Ludmila S. Pulkina}
\thanks{Submitted January 10, 2017. Published March 6, 2017.}
\subjclass[2010]{47A20, 47B38}
\keywords{Delay differential expression; boundedly solvable operator}
\begin{abstract}
We describe all boundedly solvable extensions of minimal operators
generated by first-order linear delay differential operators
in Hilbert spaces of vector-functions on finite intervals.
Also, we study the structure of spectrum of these extensions.
To do this we use methods from operator theory.
\end{abstract}
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\section{Introduction}
It is known that many solvability problems arising in life sciences can be
expressed as boundary value problems for linear functional
(time delay, time proportional, neutral, advanced etc.) equations in
corresponding functional spaces. The general theory of linear functional equations
can be found in \cite{b1,e1,h1}.
The solvability of the considered problems may be seen as boundedly solvability
of linear differential operators in corresponding functional Banach spaces.
Note that the theory of boundedly solvable extensions of a linear densely
defined closed operator in Hilbert spaces was presented in the important
works of Vishik in \cite{v1,v2}.
Let us recall that an operator $S:D(S)\subset H\to H$ on any Hilbert space $H$
is called boundedly solvable, if $S$ is one-to-one and onto, and $S^{-1}\in L(H)$.
The main aim of this work is to describe of all boundedly solvable extensions
of the minimal operator generated by first-order linear delay differential-operator
expression in the Hilbert space of vector-functions
at finite interval in terms of boundary conditions.
Lastly, the structure of spectrum of these extensions will be investigated.
\section{Description of solvable extensions}
In the Hilbert space $L^2(H,(a,b)), a,b\in\mathbb{R}$ of H-valued vector-functions
consider the linear delay differential-operator expression of first order
in form
\begin{equation} \label{e2.1}
l(u)=(\alpha(t)u(t))'+A(t)u(t-\tau)
\end{equation}
where:
\begin{itemize}
\item[(1)] $H$ is a separable Hilbert space;
\item[(2)] the function $\alpha:[a,b]\to \mathbb{R}_{+}$ is Lebesgue measurable;
\item[(3)] there are positive reel numbers $c$ and $C$ such that for $x\in[a,b]$,
\[
c\leq\alpha(x)\leq C;
\]
\item[(4)] the operator-function $A(\cdot):[a,b]\to L(H)$ is continuous on the
uniform operator topology;
\item[(5)] $\frac{\|A(t)\|}{\alpha(t)}\in L^1(H,(a,b))$;
\item[(6)] $0\leq\tau