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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 181, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/181\hfil Bounded solvability]
{Bounded solvability of mixed-type functional differential operators
for first order}

\author[B. Y{\i}lmaz, Z. I. Ismailov \hfil EJDE-2016/181\hfilneg]
{B\"ulent Y{\i}lmaz, Zameddin I. Ismailov}

\address{B\"ulent Y{\i}lmaz \newline
Marmara University, Department of Mathematics,
Kad\i k\"{o}y, 34722,\.{I}stanbul, Turkey}
\email{bulentyilmaz@marmara.edu.tr}

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


\thanks{Submitted May 18, 2016. Published July 10, 2016.}
\subjclass[2010]{47A10, 34K06}
\keywords{Mixed-type functional differential expression;
\hfill\break\indent bounded solvable operator; spectrum}

\begin{abstract}
 In this article, we study all boundedly solvable extensions generated 
 by linear  mixed-type (forward-backward) functional differential-operator 
 expressions  of first order in the Hilbert space of vector-functions 
 on a finite interval.  Our main tools are  methods of operator theory. 
 Also we study the structure of the spectrum of these extensions.
\end{abstract}

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

\section{Introduction}

Mixed-type (forward-backward) differential equations are a large class of
functional differential equations in which time derivative may depend both on
past and future values of the argument. The fundamental interest and analysis
to such equations are motivated by applications in different fields, for
example in control theory, population genetics, population growth,
epidemiology, nerve conduction theory, economy, physics, electrodynamics,
observer theory, spatial lattice and etc.(see \cite{i1} and references therein). The
qualitative analysis of mixed-type functional differential equations is quite
complicated. On the other hand the analysis of considered boundary or initial
value problem for mixed-type functional differential equations for even order
is really complicated.

Mallet-Paret \cite{m1} established an existence theory for such
equations using a Fredholm theory and the implicit function theory. Some
spectral investigations of such equations can be found in \cite{m1,m2,r1}, and
references therein. The numerical approach to these problems can be seen in
 Ford,  Lumb, Teodoro, Lima et al \cite{f1}.

Since analytical expression of solutions, eigenvalues and corresponding
eigenfunctions is very difficult (they are ill-posed), then the methods of
numerical analysis play significant role in this theory. It is known that an
operator
\[
A:D(A)\subset H\to H
\]
in a Hilbert space $H$ is called boundedly solvable, if $A$ is one-to-one,
\[
AD(A)=H\quad \text{and}\quad A^{-1}\in L(H).
\]
Firstly using methods of operator theory, we describe
all boundedly solvable extensions of minimal operator generated
by some mixed-type differential operator expression for first order
in the Hilbert space of vector-functions on a finite interval.
This is done in terms of the boundary values.
Lastly, the structure of spectrum of these extensions is investigated.

\section{Description of boundedly solvable extensions}

Consider the simplest scalar mixed type functional differential equation in
nonhomogeneous form
\begin{equation} \label{e2.0}
\begin{gathered}
\dot{x}(t) =ax(t)+bx(t-\alpha)+cx(t+\tau)+h(t)\\
\alpha,\tau   \geq0,\quad a,b,c\in  \mathbb{C}, \;t\in[ t_1,t_2] \\
x(t)=\varphi(t),\quad t\in[ t_1-\alpha,t_1], \quad
x(t)=\psi (t),\quad t\in[ t_2,t_2+\tau]
\end{gathered}
\end{equation}
 Note that without loss of generality it can be assumed $a=0$. For
this it is sufficient to use the substitution
$v(t)=e^{at}u(t),t\in[a,b]$ .

In addition to the above boundary problem functions $\varphi$ and $\psi$ can
be chosen as
\begin{gather*}
\varphi(t)=0,t\in[ t_1-\alpha,t_1], \\
\psi(t)=0,t\in[ t_2,t_2+\tau]
\end{gather*}
For this it is sufficient to use the substitution
\[
v(t)=\begin{cases}
\varphi(t), & t_1-\alpha\leq t<t_1\\
u(t), & t_1\leq t\leq t_2\\
\psi(t), & t_2<t\leq t_2+\tau
\end{cases}
  -\begin{cases}
\varphi(t), & t_1-\alpha\leq t<t_1\\
0, & t_1\leq t\leq t_2\\
\psi(t), & t_2<t\leq t_2+\tau
\end{cases}
\]
Hence in this situation it is sufficient to consider the following
nonhomogeneous mixed type functional differential equation
\[
u'(t)=bu(t-\alpha)+cu(t+\tau)+h(t),\quad t\in[ t_1,t_2]
\]
with boundary conditions
\[
u(t)=0,\quad t\in[ t_1-\alpha,t_1)\cup(t_2,t_2+\tau]
\]


Now consider the  mixed-type differential-operator expression of the form
\[
l(u)=u'(t)+A(t)u(t-\alpha)+B(t)u(t+\tau)
\]
in the Hilbert space of vector-functions on a finite interval $L^2(H,(a,b))$, where
\begin{itemize}

\item[(1)] $\alpha\geq0,\tau\geq0;$

\item[(2)] $H$ is a separable Hilbert space with inner product $(.,.)_{H}$ and norm
$\|\cdot\|_{H}$;

\item[(3)] the operator- functions
$A:[a,b]\to L(H)$ and $B:[a,b]\to L(H)$
are continuous on the uniform operator topology.
\end{itemize}

We remark that when
\begin{itemize}
\item[(1)] $\alpha>0$ and $\tau=0$,

\item[(2)] $\alpha=0$ and $\tau>0,$

\item[(3)] $\alpha>0$ and $\tau>0$
\end{itemize}
the differential expression $l(.)$ is expressed as a retarded, advanced
and mixed-type delay differential expression in $L^2(H,(a,b))$.

Now let us introduce the  special operators:
$S_{\alpha}^{-}:L^2(H,(a,b))\to L^2(H,(a,b))$,
\[
S_{\alpha}^{-}u(t)=\begin{cases}
0 &\text{if }  a<t<a+\alpha,\\
u(t-\alpha), &\text{if }  a+\alpha<t<b
\end{cases}
\]
and $S_{\tau}^{+}:L^2(H,(a,b))\to L^2(H,(a,b))$,
\[
S_{\tau}^{+}u(t)=\begin{cases}
u(t+\tau), & \text{if }  a<t<b-\tau,\\
0, &\text{if }  b-\tau<t<b.
\end{cases}
\]
According to differential expression $l(\cdot)$ we will consider the
differential-operator-expression in a direct sum
\[
\mathcal{H}=L^2(H,(a-\alpha,a))\oplus L^2(H,(a,b))\oplus L^2
(H,(b,b+\tau))
\]
\begin{equation}
k(u)=(u_1,l(u_2),u_{3}), \quad u=(u_1,u_2,u_{3}) \label{e2.1}
\end{equation}
where
\[
l(u_2)=u_2'(t)+A(t)S_{\alpha}^{-}u_2(t)+B(t)S_{\tau}^{+}u_2(t)
\]
On the other hand here we will consider also the simply differential expression
\begin{equation}
m(v)=v'(t) \label{e2.2}
\end{equation}
in $L^2(H,(a,b))$. By the standard way minimal
$M_0$ and maximal $M$ operators  corresponding to
differential expression \eqref{e2.2} can be defined in $L^2(H,(a,b))$
(see \cite{h1}).

For the operators $S_{\alpha}^{-}$ and $S_{\tau}^{+}$ we have
\begin{align*}
\|S_{\alpha}^{-}u\|_{L^2(H,(a,b))}^2
&=\int_{a}^{b}(S_{\alpha} u(t),S_{\alpha}u(t))_{H}dt \\
&=\int_{a+\alpha}^{b}(u(t-\alpha),u(t-\alpha))_{H}dt \\
&=\int_{a}^{b-\alpha}(u(t),u(t))_{H}dt \\
&\leq\int_{a}^{b}\|u(t)\|_{H}^2dt=\|u\|_{L^2(H,(a,b))}^2
\end{align*}
and
\begin{align*}
\|S_{\tau}^{+}u\|_{L^2(H,(a,b))}^2
&=\int_{a}^{b}(S_{\tau}u(t),S_{\tau}u(t))_{H}dt \\
&=\int_{a}^{b-\tau}(u(t+\tau),u(t+\tau))_{H}dt \\
&=\int_{a+\tau}^{b}(u(t),u(t))_{H}dt \\
&\leq\int_{a}^{b}\|u(t)\|_{H}^2dt=\|u\|_{L^2(H,(a,b))}^2
\end{align*}
for all $u\in L^2(H,(a,b))$.
Then $\|S_{\alpha}^{-}\|\leq1,\|S_{\tau}^{+}\|\leq1$. That is,
\[
S_{\alpha}^{-},S_{\tau}^{+}\in L(L^2(H,(a,b))),\quad \alpha,\tau\geq0\,.
\]
In this work we define
\begin{gather*}
C_{\alpha\tau}(t)=A(t)S_{\alpha}^{-}+B(t)S_{\tau}^{+},\quad a<t<b; \\
L_0:=M_0+C_{\alpha\tau}(t), \\
L_0:{\mathaccent"7017 W}_2^1 (H,(a,b))\subset L^2(H,(a,b))\to L^2(H,(a,b));\\
L:=M+C_{\alpha\tau}(t), \\
D(L):W_2^1(H,(a,b))\subset L^2(H,(a,b))\to L^2(H,(a,b)).
\end{gather*}
Then the operators
\begin{gather*}
K_0:=E_1\oplus L_0\oplus E_2, \\
K:=E_1\oplus L\oplus E_2
\end{gather*}
are called the minimal and maximal operators corresponding to differential
expressions \eqref{e2.1} respectively. Here $E_1$ and $E_2$ are identity
operators in $L^2(H,(a-\alpha,a))$ and $L^2(H,(b,b+\tau))$ respectively.

It is important to note that the solvability of boundary value problem
\eqref{e2.0} is equivalent to a solvability of operator equation
\[
k(u)=H
\]
where $u=(u_1,u_2,u_{3})$, $H=(\varphi,h,\psi)$ in the direct sum of
Hilbert spaces
\[
L^2(t_1-\alpha,t_1)\oplus L^2(t_1,t_2)\oplus L^2(t_2,t_2+\tau)
\]
In this paper the solvability of problem \eqref{e2.0} will be investigated
from this point of view in more general case of equation and space.

Now let $U(t,s)$, $t,s\in[ a,b]$, be the family of evolution operators
corresponding to the homogeneous differential equation
\begin{gather*}
U_t'(t,s)f+C_{\alpha\tau}(t)U(t,s)f=0,\quad t,s\in(a,b)\\
U(s,s)f=f,\quad f\in H
\end{gather*}

The operator $U(t,s),t,s\in[ a,b]$ is a linear continuous boundedly
invertible in $H$ and
\[
U^{-1}(t,s)=U(s,t),s,t\in[ a,b]
\]
(for more detail analysis of this concept see \cite{k1}).

Let us introduce the operator
\[
Uz(t):=U(t,a)z(t),\text{ }U:L^2(H,(a,b))\to L^2(H,(a,b)).
\]
In this case, it is easy to see that the following  relation
for the differentiable vector-function
$z\in L^2(H,(a,b))$, $z:[a,b]\to H$ is valid 
\[
l(Uz)=Uz'(t)+(U_t^{'}+C_{\alpha\tau}(t)U)z(t)=Um(z)
\]
From this $U^{-1}lU(z)=m(z)$. Hence it is clear that if $\widetilde{L}$ is an
extension of the minimal operator $L_0$; that is, $L_0\subset
\widetilde{L}\subset L$, then
\[
U^{-1}L_0U=M_0,\quad M_0\subset U^{-1}LU=\widetilde{M}\subset M, \quad U^{-1}LU=M.
\]
For example, one can easily prove the validity of last relation. It is known
that
\[
D(M)=W_2^1(H(a,b),\quad D(M_0)={\mathaccent"7017 W}_2^1(H(a,b)).
\]
If $u\in D(M)$, then
\[
l(Uz)=Um(z)\in L^2(H,(a,b));
\]
that is, $Uu\in D(L)$. From last relation $M\subset U^{-1}LU$. Contrary, if a
vector-function $u\in D(L)$, then
\[
m(U^{-1}v)=U^{-1}l(v)\in L^2(H,(a,b));
\]
that is, $U^{-1}v\in D(M)$. From last relation $U^{-1}L\subset MU$; that is
$U^{-1}LU\subset M$. Hence $U^{-1}LU=M$.


\begin{theorem} \label{thm2.1}
$\ker L_0=\{0\}$ and $\overline{R}(L_0)\neq L^2(H,(0,1))$.
\end{theorem}

\begin{theorem} \label{thm2.2}
Each boundedly solvable extension $\widetilde{L}$ of
the minimal operator $L_0$ in $L^2(H,(a,b))$ is generated by the
differential-operator expression $l(.)$ and boundary condition
\begin{equation}
(B+E)u(a)=BU(a,b)u(b), \label{e2.3}
\end{equation}
where $B\in L(H)$ and $E$ is a identity operator in $H$.
The operator $B$ is determined uniquely by the extension $\widetilde{L}$, i.e $\widetilde{L}=L_B$.

On the contrary, the restriction of the maximal operator $L_0$ to the
manifold of vector-functions satisfy the condition \eqref{e2.3} for some bounded
operator $B\in L(H)$ is a boundedly solvable extension of the minimal operator
$L_0$ in the $L^2(H,(a,b))$.
\end{theorem}

\begin{proof}
Firstly, it is described all boundedly solvable extensions
$\widetilde{M}$ of the minimal operator $M_0$ in $L^2(H,(a,b))$ in terms
of boundary values.

Consider the following so-called Cauchy extension $M_c$,
\begin{gather*}
M_cu=u'(t), \\
M_c:D(M_c)=\{u\in W_2^1H(a,b):u(a)=0\}\subset L^2
(H,(a,b))\to L^2(H,(a,b))
\end{gather*}
of the minimal operator $M_0$. It is clear that $M_c$ is a boundedly
solvable extension of $M_0$ and
\[
M_c^{-1}:=L^2(H,(a,b))\to L^2(H,(a,b)),\text{ }M_c
^{-1}f(t)=\int_{a}^{t}f(x)dx,\quad f\in L^2(H,(a,b)).
\]
Now we assume that $\widetilde{M}$ is a solvable extension of the minimal
operator $M_0$ in $L^2(H,(a,b))$. In this case it is known that domain of
$\widetilde{M}$ can be written in direct sum in form
\[
D(\widetilde{M})=D(M_0)\oplus(M_c^{-1}+B)V,
\]
where $V=\ker M=H$, $B\in L(H)$ (see \cite{v1}). Therefore for each
$u(t)\in D(\widetilde{M})$ it holds
\[
u(t)=u_0(t)+M_c^{-1}f+Bf,\quad u_0\in D(M_0),\quad f\in H.
\]
That is,
\[
u(t)=u_0(t)+tf+Bf,\quad u_0\in D(M_0),\quad f\in H.
\]
Hence
\[
u(a)=Bf,\quad u(b)=f+Bf=(B+E)f
\]
and from these relations it follows that
\begin{equation}
(B+E)u(a)=Bu(b). \label{e2.4}
\end{equation}

On the other hand uniqueness of operator $B\in L(H)$ is clear from the work
\cite{v1}. Therefore $\widetilde{M}=M_B$. This completes the necessary part of
this assertion.

On the contrary, if $M_B$ is a operator generated by differential expression
\eqref{e2.2} and boundary condition \eqref{e2.4}, then $M_B$ is
boundedly invertible and
\begin{gather*}
M_B^{-1}:=L^2(H,(a,b))\to L^2(H,(a,b)), \\
M_B^{-1}f(t)=\int_{a}^{t}f(x)dx+B\int_{a}^{b}f(x)dx,\quad
f\in L^2(H,(a,b))
\end{gather*}

Consequently, all boundedly solvable extensions of the minimal operator
$M_0$ in $L^2(H,(0,1))$ are generated by differential expression \eqref{e2.2} and
boundary condition \eqref{e2.4} with any linear bounded operator $B$.

Now consider the general case. For the this in the $L^2(H,(a,b))$ introduce
an operator
$U:L^2(H,(a,b))\to L^2(H,(a,b))$, by
\[
(Uz)(t):=U(t,a)z(t),\quad z\in L^2(H,(a,b))
\]
From the properties of family of evolution operators $U(t,s),t,s\in[a,b]$ imply that a operator $U$ is a linear continuous boundedly invertible
and
\[
(U^{-1}z)(t)=U(a,t)z(t).
\]
On the other hand from the relations
\[
U^{-1}L_0U=M_0,\quad U^{-1}\widetilde{L}U=\widetilde{M},\quad U^{-1}LU=M
\]
it implies that an operator $U$ is one-to-one between of sets of boundedly
solvable extensions of minimal operators $L_0$ and $M_0$ in
$L^2(H,(a,b))$.

Extension $\widetilde{L}$ of the minimal operator $L_0$ is boundedly
solvable in $L^2(H,(a,b))$ if and only if the operator
$\widetilde{M}=U^{-1}\widetilde{L}U$ is an extension of the minimal $M_0$ in
$L^2(H,(a,b))$. Then $u\in D(\widetilde{L})$ if and only if
\[
(B+E)U(a,a)u(a)=BU(a,b)u(b);
\]
that is,
\[
(B+E)u(a)=BU(a,b)u(b).
\]
\end{proof}
This proves the validity of the claims in theorem.

From the above theorem, we have following assertion.

\begin{theorem} \label{thm2.3}
 Every boundedly solvable extension $\widetilde{K}$ of the
minimal operator $K_0$ in $\mathcal{H}$ is generated by
differential-operator expression \eqref{e2.1} and boundary condition
\[
(B+E)u_2(a)=BU(a,b)u_2(b)
\]
where $B\in L(H)$ and $E$ are a identity operators in $H$. The operator $B$ is
determined uniquely by the extension $\widetilde{K}$, i.e.
$\widetilde{K}=K_B$ And vice versa.
\end{theorem}

\begin{corollary} \label{coro1}
The resolvent operator $R_{\lambda}(K_B),\lambda \in\rho(K_B)$ of any
boundedly solvable operator $K_B$ of the minimal operator $K_0$,
generated by differential expression \eqref{e2.1} with boundary
condition
\[
(B+E)u_2(a)=BU(a,b)u_2(b),\quad B\in L(H)
\]
is of the form
\[
R_{\lambda}(K_B)=(E_1,R_{\lambda}(L_B),E_2),
\]
where $R_{\lambda}(K_B):\mathcal{H}$ $\to\mathcal{H}$ ,
\begin{align*}
R_{\lambda}(L_B)f(t)
&=U(t,a)(E+B(1-e^{\lambda}))^{-1}B\int_{a}
^{b}e^{\lambda(b-s)}U(a,s)f(s)ds\\
&\quad + \int_{a}^{t}e^{\lambda(b-s)}U(a,s)f(s)ds,\quad
f\in L^2 (H,(a,b)).
\end{align*}
\end{corollary}

\begin{example} \label{examp1} \rm
 Consider the  forward-backward differential
equation
\begin{gather*}
u=(u_1,u_2,u_{3})\in L^2(t_1-1,t_1)\oplus L^2(t_1,t_2)\oplus L^2(t_2,t_2+1) \\
u_2'(t)  =au_2(t)+bu_2(t-1)+cu_2(t+1), \quad t\in[ t_1,t_2],\; a,b,c\in \mathbb{C}
\end{gather*}
with boundary conditions
\begin{gather*}
u_2(t)=\varphi(t),\quad t_1-1\leq t\leq t_1, \\
u_2(t)=\psi(t),\quad t_2\leq t\leq t_2+1,
\end{gather*}
where $\varphi$ $\in C[t_1-1,t_1]$ and $\psi\in C[t_2,t_2+1]$
(see \cite{f1}).
\end{example}

It is clear that from Theorem \ref{thm2.2}, that all $L^2$-boundedly solvable
boundary value problem in this case can be written in the form
\begin{gather*}
u_2'(t)=au_2(t)+bu_2(t-1)+cu_2(t+1)+f_2(t)\\
(\gamma+1)U_2(t_1)=\gamma U(t_2,t_1)u_2(t_2)
\end{gather*}
and in this case solutions have the form
\[
u=(\varphi,u_2,\psi),u_2(t)=U(t,t_1)\gamma\int_{t_1}^{t_2}U(t_1,s)f_2
(s)ds+\int_{t_1}^{t}U(t_1,s)f_2(s)ds,
\]
where: $\gamma\in\mathbb{C}$,
$U(t,s)=\exp(aE+bS_1^{-}+cS_1^{+})(t-s)$, $t_1\leq t$, $s\leq t_2$, and
$f_2\in L^2(t_1,t_2)$.


\section{Spectrum of boundedly solvable extensions}

 In this section we study the spectrum structure of boundedly solvable extension
$K_B=E_1\oplus L_B\oplus E_2$ of the minimal operator $K_0
=E_1\oplus L_0\oplus E_2$ in Hilbert space
\[
\mathcal{H=}L^2(H,(a-\alpha,a))\oplus L^2(H,(a,b))\oplus L^2
(H,(b,b+\tau))\,.
\]

Firstly, note that as in \cite{i2} for the spectrum 
$\sigma (L_B)$ of any boundedly solvable extension $L_B$ of $L_0$ the following
assertion can be proved.

\begin{theorem} \label{thm3.1}
If $L_B$ is a boundedly solvable extension of
the minimal operator $L_0$ in the Hilbert space $L^2(H,(a,b))$, then
spectrum set of $L_B$ has the form
\begin{align*}
\sigma(L_B)=\big\{ &\lambda\in\mathbb{C}:\lambda=\ln| \frac{\mu+1}{\mu}|
 +i\arg(\frac{\mu+1}{\mu })+2n\pi i,\\
 & \mu\in\sigma(B)\backslash\{0,-1\},\; n\in \mathbb{Z}\big\}
\end{align*}
\end{theorem}

The following assertion follows from a result in  \cite{o1}.

\begin{theorem} \label{thm3.2}
 If $\alpha,\tau\geq0$, $\alpha+\tau>0$ and 
$K_B =E_1\oplus L_B\oplus E_2$ is any boundedly solvable extension on of the
minimal operator $K_0=E_1\oplus L_0\oplus E_2$ in $\mathcal{H}$, then
\begin{gather*}
\sigma_{p}(K_B)=\sigma_{p}(L_B)\cup\{1\}, \\
\sigma_c(K_B)=\big[  \big( (  \sigma_{p}(L_B))  ^{c}
\cap\sigma_c(L_B)\cap\sigma_{r}(L_B)\big)  ^{c}\big]  \backslash \{1\},
\\
\sigma_{r}(K_B)=[  (  \sigma_{p}(L_B))  ^{c}\cap\sigma
_{r}(L_B)]  \backslash\{1\},
\\
\rho(K_B)=\rho(L_B)\backslash\{1\},
\end{gather*}
where: $\sigma_{p}(.),\sigma_c(.),\sigma_{r}(.)$, and $\rho(.)$  denote
point, continuous, residual and resolvent sets of an operator respectively.
\end{theorem}


We remark that when 
 $\alpha\tau=0$, i.e.
\begin{itemize}
\item[(1)] $\alpha  =0$  and $\tau>0$,  advanced type,
\item[(2)] $\alpha  >0$ and  $\tau=0$,  retarded type, 
\item[(3)] $\alpha  =0$ and  $\tau=0$,  ordinary type
\end{itemize}
the differential expression spectrum of boundedly solvable extensions is
easy to be investigated as in the above theorem.


\begin{thebibliography}{10}

\bibitem{f1} N. J. Ford, P. M. Lumb;
{Mixed-Type Functional Differential Equations: a numerical approach},
 \emph{Research in Mathematics and Applications},
Chester Research Online, 2012 or J. Comp. Appl. Math, 229(2), pp. 471-479, (2009).

\bibitem{h1} L. H\"{o}rmander;
{On the theory of general partial differential operators},
\emph{Acta Mathematica}. 94, pp 161-248, (1955).

\bibitem{i1}  V. Iakovleva, C. J. Vanegas;
{Spectral Analysis of the Semigroup Associated
to a Mixed Functional Differential Equation},
 \emph{Int. J.Pure and App. Math.},
V. 72 (4), p. 491-499, ( 2011).

\bibitem{i2} Z. I. Ismailov, P. Ipek;
{Spectrums of Solvable Pantograph Differential- Operators
for First Order},
\emph{Abstract and Applied Analysis}, v. 2014, p. 1-8, (2014).

\bibitem{k1} S. G. Krein;
{Linear differential operators in Banach space},
\emph{Translations of Mathematical Monographs}, Vol. 29. American
Mathematical Society, Providence, RI, (1971).

\bibitem{m1} J. Mallet-Paret;
{The Fredholm Alternative for Functional Differential
Equations of Mixed Type},
\emph{Journal of Dynamics and Differential Equations}, 11, 1-47, (1999).

\bibitem{m2} J. Mallet-Paret, S. M. Verduyn Lunel;
{Mixed-Type Functional Differential Equations, Holomorphic Factorization
and Applications},
\emph{Proceedings Equadiff 2003}, Hasselt 2003, World Scientific,
Singapore, 73-89, (2005).

\bibitem{o1} E. Otkun \c{C}evik, Z. I. Ismailov;
{Spectrum of the direct sum of operators},
\emph{Electronic Journal of Differential Equations}, Vol. 2012, No. 210,pp. 1--8,
(2012).

\bibitem{r1}  A. Rustichini;
 {Functional Differential Equations of Mixed Type: The Li-near
 Autonomous Case},
\emph{Journal of Dynamics and Differential Equations}, V. 1 (2), pp 121-143, (1989).

\bibitem{v1} M. I. Vishik;
{On general boundary problems for elliptic differential equations},
\emph{Amer. Math. Soc. Transl. II}, Vol 24 , 107-172, (1963).



\end{thebibliography}

\end{document}