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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 219, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/219\hfil Third-order operator-differential equations]
{Third-order operator-differential equations with discontinuous coefficients
and operators in the boundary conditions}

\author[A. R. Aliev, N. L. Muradova \hfil EJDE-2013/219\hfilneg]
{Araz R. Aliev, Nazila L. Muradova}  % in alphabetical order

\address{Araz R. Aliev \newline
Baku State University,
Institute of Mathematics and Mechanics of NAS of Azerbaijan,
Baku, Azerbaijan}
\email{alievaraz@yahoo.com}

\address{Nazila L. Muradova \newline
Nakhchivan State University,
Nakhchivan, Azerbaijan}
\email{nazilamuradova@gmail.com}

\thanks{Submitted June 7, 2013. Published October 4, 2013.}
\subjclass[2000]{47E05, 34B40, 34G10}
\keywords{Operator-differential equation; discontinuous coefficient;
\hfill\break\indent operator-valued boundary condition; self-adjoint operator;
 regular solvability;  
\hfill\break\indent Sobolev-type space; intermediate derivative operators}

\begin{abstract}
 We study a third-order operator-differential equation
 on the semi-axis with a discontinuous coefficient and
 boundary conditions which include an abstract
 linear operator. Sufficient conditions for the
 well-posed and unique solvability are found by means of
 properties of the operator coefficients in a Sobolev-type space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

It is known that many problems of partial differential equations can be
reduced to problems for differential equations whose coefficients are
unbounded operators in a Hilbert space.
Many articles are dedicated to the study of problems with operators
in the boundary conditions for operator-differential equations of
second order (see, for example, \cite{a1,a7,f1,g3,m2,m3,m4,y2}
and the references therein);
however, these studies are far from the full completion.
Note that only a few papers are dedicated to the study of such boundary-value
problems  for operator-differential equations of third order
(see, for example, \cite{a5}).

This article is dedicated to the study of boundary-value problem for
a class of third-order operator-differential equations with a
discontinuous coefficient; one of the boundary conditions
includes an abstract linear operator.
Such equations cover some non-classical problems of mathematical
physics (see \cite{b1}), investigated in inhomogeneous environments.

Let $H$ be a separable Hilbert space with the scalar product
$(x,y)$, $x,y\in H$ and let $A$ be a self-adjoint positive-definite
operator in $H$ ($A=A^{*} \ge cE$, $c>0$, $E$ is the identity operator).
By $H_{\gamma } $ ($\gamma \ge 0$) we denote the scale of Hilbert spaces
generated by the operator $A$; i.e., $H_{\gamma } =D(A^{\gamma } )$,
$(x,y)_{\gamma } =(A^{\gamma } x,A^{\gamma } y)$,
 $x, y\in D(A^{\gamma } )$, for $\gamma =0$ we consider that $H_0 =H$,
 $(x,y)_0 =(x,y)$, $x,y\in H$.

We denote by $L_2 ([a,b];H)$, $-\infty \le a<b\le +\infty $,
the Hilbert space of all vector functions defined on $[a,b]$
with values in $H$ and endowed with the norm
\[
\| f \|_{L_2 ( {[ {a,b} ];H} )}
= \Big( {\int_a^b {\| {f(t)} \|_H^2 dt} } \Big)^{1/2}.
\]
Following the book \cite{l1}, we introduce the Hilbert space
\[
W_2^3 ([a,b];H)=\{u(t): u'''(t) \in L_2 ([a,b];H) , A^3 u(t)\in L_2 ([a,b];H)\}
\]
endowed with the norm
\[
\| u\| _{W_2^3 ([a,b];H)} =
\big(\| u'''\| _{L_2 ([a,b];H)}^2 +\| A^3 u\| _{L_2 ([a,b];H)}^2 \big)^{1/2} .
\]
Hereafter, derivatives are understood in the sense of distributions
in a Hilbert space \cite{l1}. The spaces $L_2 ((-\infty ,+\infty );H)$,
$W_2^3 ((-\infty ,+\infty );H)$, $L_2 ([0,+\infty );H)$ and
$W_2^3 ([0,+\infty );H)$ will be denoted by $L_2 (R;H)$, $W_2^3 (R;H)$,
$L_2 (R_{+};H)$ and $W_2^3 (R_{+} ;H)$, respectively.

Further, we denote by $L(X,Y)$ the space of all linear bounded operators
 acting from a Hilbert space $X$ to another Hilbert space $Y$,
and we denote by $\sigma (\cdot )$ the spectrum of the operator $(\cdot )$.

Consider the boundary value problem in the Hilbert space $H$
\begin{gather} \label{e1}
-u'''(t)+\rho (t)A^3 u(t)+\sum _{j=1}^3A_{j} \frac{d^{3-j} u(t)}{dt^{3-j} }
=f(t), t\in R_{+} , \\
 \label{e2}
u'(0)=0, \quad u''(0)= Ku(0),
\end{gather}
where $A=A^{*} \ge cE$, $c>0$, $K\in L(H_{5/2} ,H_{1/2} )$, $\rho (t)=\alpha $,
if $0\le t\le 1$, $\rho (t)=\beta $, if $1<t<+\infty $, here $\alpha $,
$\beta $ are positive numbers, $f(t)\in L_2 (R_{+} ;H)$,
$u(t)\in W_2^3 (R_{+} ;H)$.

\begin{definition} \label{def1} \rm
If a vector function $u(t)\in W_2^3 (R_{+} ;H)$ satisfies \eqref{e1}
 almost everywhere in $R_{+} $, then it is called a regular solution
of equation \eqref{e1}.
\end{definition}

\begin{definition} \label{def2} \rm
If for any $f(t)\in L_2 (R_{+} ;H)$ there exists a regular solution
of  \eqref{e1}, which satisfies the boundary conditions \eqref{e2}
in the sense that
\[
\lim_{t\to 0} \|  u'(t) \| _{H_{3/2}} =0, \quad
\lim_{t\to 0} \|  u''(t)-Ku(t) \| _{H_{1/2}} =0
\]
and the following inequality holds
\[
\| u\| _{W_2^3  (R_{+} ; H)} \le \text{const } \| f\| _{L_2 (R_{+} ; H)} ,
\]
then we say that the problem \eqref{e1}, \eqref{e2} is regularly solvable.
\end{definition}

Similar kind of problems on a semi-axis for elliptic operator-differential
equations of the second order is considered in papers \cite{g3,m2,m3}.
We should especially note the work \cite{s2} which considers the non-local
boundary value problems for second order elliptic operator-differential
equations on the interval with the coefficients belonging to a broader
class of discontinuous functions, while the coefficients in the
boundary conditions are complex numbers.
In \cite{a2,b1,g1,g2,m1,n1,s1,s3,s4}
along with other problems investigated the solvability of boundary-value
problems for elliptic operator-differential equations of the general
form when the coefficients in the boundary conditions are complex numbers
and the equations do not contain discontinuous coefficients.
Such case also is considered in \cite{a3,a6} for the third and fourth orders
equations with multiple characteristics. Note that
 problem \eqref{e1}, \eqref{e2} is investigated in the case
$A_3 =0$, $K=0$ in \cite{a4} and when $\rho (t)\equiv 1$, $t\in R_{+} $,
$K=0$ in \cite{m1}.

In this article, we obtain the conditions of regular solvability of
the boundary-value problem \eqref{e1}, \eqref{e2} by means of properties
of operator coefficients.


\section{Main results}

Before proceeding to the consideration of the question posed,
let us introduce additional notation.
Let 
\[
W_{2,K}^3 (R_{+} ;H)=\{ u(t):u(t)\in W_2^3 (R_{+} ;H),  u'(0)=0,
 u''(0)=Ku(0)\}
\]
and  denote by $P_0 $, $P_1 $ and $P$ the operators acting from
the space $W_{2,K}^3 (R_{+} ;H)$ into the space $L_2 (R_{+} ;H)$
by the following rules, respectively:
\begin{gather*}
P_0 u(t)=-u'''(t)+\rho (t)A^3 u(t),\\
P_1 u(t)=A_1 u''(t)+A_2 u'(t)+A_3 u(t),\\
Pu(t)=P_0 u(t)+P_1 u(t), \quad u(t)\in W_{2,K}^3 (R_{+} ;H).
\end{gather*}
Put $B=A^{1/2}KA^{-5/2}$,
$\kappa (c_1 ,c_2 ,c_3 )=c_1 \sqrt[3]{\beta ^2 }
+c_2 \sqrt[3]{\alpha \beta } +c_3 \sqrt[3]{\alpha ^2 } $
and
\begin{align*}
K_{\alpha ,\beta }
&=(E+\frac{1}{\sqrt[3]{\alpha ^2 } } \omega _2 A^{-2} K)
 (\kappa (1,1,1)\omega _2 E-\kappa (1,\omega _2 ,\omega _1 )
e^{\sqrt[3]{\alpha } (\omega _2 -1)A} )\\
&\quad + \Big(E+\frac{1}{\sqrt[3]{\alpha ^2 } } \omega _1 A^{-2} K\Big)
\Big(\kappa (1,\omega _1 ,\omega _2 )e^{\sqrt[3]{\alpha }
(\omega _1 -1)A} -\kappa (1,1,1)\omega _1 E\Big),
\end{align*}
where $\omega _1 =-\frac{1}{2} + \frac{\sqrt{3} }{2} i$,
$\omega _2 =-\frac{1}{2} -\frac{\sqrt{3} }{2} i$.

\begin{lemma} \label{lem1} 
Let $A=A^{*} \ge cE$, $c>0$, $K\in L(H_{5/2},H_{1/2})$,
$-\sqrt[3]{\alpha ^2 } \omega _2 \notin \sigma (B)$ and the
operator $K_{\alpha ,\beta } $ have a bounded inverse operator in $H_{5/2}$.
Then the equation $P_0 u=0$ has only the trivial solution in the space
$W_{2,K}^3 (R_{+} ;H)$.
\end{lemma}

\begin{proof} The general solution of the equation $P_0 u(t)=0$ in the
space $W_2^3 (R_{+} ;H)$ has the following form \cite{y1}:
\[
u_0 (t)=\begin{cases}
u_{0,1} (t)=e^{\sqrt[3]{\alpha } \omega _1 tA} \varphi _0
+e^{\sqrt[3]{\alpha } \omega _2 tA} \varphi _1
+e^{-\sqrt[3]{\alpha } (1-t)A} \varphi _2 , & 0\le t<1, \\
u_{0,2} (t)=e^{\sqrt[3]{\beta } \omega _1 (t-1)A} \varphi _3
 +e^{\sqrt[3]{\beta } \omega _2 (t-1)A} \varphi _{4} ,&1<t<+\infty ,
\end{cases}
\]
where the vectors $\varphi _{k} \in
H_{5/2}$, $k=0,1,2,3,4$, are determined from the boundary conditions
\eqref{e2} and the condition $u_0 (t)\in W_2^3 (R_{+} ;H)$.
Therefore, to determine the vectors $\varphi _{k} $, $k=0,1,2,3,4$,
we have the following relations:
\begin{gather*}
u'_{0,1} (0)=0,\quad  u''_{0,1} (0)=Ku_{0,1} (0), \quad
u_{0,1} (1)=u_{0,2} (1),\\
u'_{0,1} (1)=u'_{0,2} (1),\quad  u''_{0,1} (1)=u''_{0,2} (1).
\end{gather*}
From these relations we obtain the following system of equations with
respect to $\varphi _{k} $, $k=0,1,2,3,4$:
\begin{equation} \label{e3}
\begin{gathered}
\omega _1 \varphi _0 +\omega _2 \varphi _1 +e^{-\sqrt[3]{\alpha } A}
\varphi _2 =0\, ,
\\
\omega _1^2 \varphi _0 +\omega _2^2 \varphi _1
 +e^{-\sqrt[3]{\alpha } A} \varphi _2
=\frac{1}{\sqrt[3]{\alpha ^2 } } A^{-2} K(\varphi _0 +\varphi _1
+e^{-\sqrt[3]{\alpha } A} \varphi _2 ),
\\
e^{\sqrt[3]{\alpha } \omega _1 A} \varphi _0 +e^{\sqrt[3]{\alpha }
\omega _2 A} \varphi _1 +\varphi _2
=\varphi _3 +\varphi _{4} \, , \\
\sqrt[3]{\alpha } \omega _1 e^{\sqrt[3]{\alpha } \omega _1 A} \varphi _0
+\sqrt[3]{\alpha } \omega _2 e^{\sqrt[3]{\alpha } \omega _2 A} \varphi _1
 +\sqrt[3]{\alpha } \varphi _2 =\sqrt[3]{\beta } \omega _1 \varphi _3
 +\sqrt[3]{\beta } \omega _2 \varphi _{4} \, ,
\\
\sqrt[3]{\alpha ^2 } \omega _1^2 e^{\sqrt[3]{\alpha } \omega _1 A}
\varphi _0 +\sqrt[3]{\alpha ^2 } \omega _2^2 e^{\sqrt[3]{\alpha }
\omega _2 A} \varphi _1 +\sqrt[3]{\alpha ^2 } \varphi _2
=\sqrt[3]{\beta ^2 } \omega _1^2 \varphi _3 +\sqrt[3]{\beta ^2 }
\omega _2^2 \varphi _{4} \, .
 \end{gathered}
\end{equation}
Taking into account $\omega _1 \omega _2 =1$, $\omega _1 +\omega _2 =-1$,
$\omega _1^2 =\omega _2 $, $\omega _2^2 =\omega _1 $, from the system
 \eqref{e3} after simple transformations with respect to $\varphi _0 $ we have
\begin{align*}
&(E+\frac{1}{\sqrt[3]{\alpha ^2 } } \omega _2 A^{-2} K)
(\kappa (1,1,1)\omega _2 E-\kappa (1,\omega _2 ,\omega _1 )
e^{\sqrt[3]{\alpha } (\omega _2 -1)A} )\varphi _0 \\
&+(E+\frac{1}{\sqrt[3]{\alpha ^2 } }
\omega _1 A^{-2} K)(\kappa (1,\omega _1 ,\omega _2 )
e^{\sqrt[3]{\alpha } (\omega _1 -1)A} -\kappa (1,1,1)\omega _1 E)
\varphi _0 =0.
\end{align*}
Consequently,
\begin{equation} \label{e4}
\begin{aligned}
K_{\alpha ,\beta } \varphi _0
&\equiv \Big[\big(E+\frac{1}{\sqrt[3]{\alpha ^2 } } \omega _2 A^{-2} K\big)
\big(\kappa (1,1,1)\omega _2 E-\kappa (1,\omega _2 ,\omega _1 )
e^{\sqrt[3]{\alpha } (\omega _2 -1)A} \big) \\
&\quad +\big(E+\frac{1}{\sqrt[3]{\alpha ^2 } } \omega _1 A^{-2} K\big)
\big(\kappa (1,\omega _1 ,\omega _2 )e^{\sqrt[3]{\alpha } (\omega _1 -1)A}
-\kappa (1,1,1)\omega _1 E\big)]\varphi _0 =0.
\end{aligned}
\end{equation}
By the assumption of this lemma, $K_{\alpha ,\beta } $ has a
bounded inverse operator in the space $H_{5/2}$, then from equation
\eqref{e4} follows that $\varphi _0 =0$. Considering $\varphi _0 =0$
in the first and second equations of \eqref{e3}, we obtain
\begin{gather} \label{e5}
\varphi _1 =-\omega _1 e^{-\sqrt[3]{\alpha } A} \varphi _2 ,\\
 \label{e6}
(E+\frac{1}{\sqrt[3]{\alpha ^2 } } \omega _1 A^{-2} K)e^{-\sqrt[3]{\alpha }
A} \varphi _2 =0.
\end{gather}
In turn, by  assumption $-\sqrt[3]{\alpha ^2 } \omega _2 \notin \sigma (B)$
from \eqref{e6} it follows that $\varphi _2 =0$, and therefore,
from \eqref{e5} $\varphi _1 =0$. Now, considering
 $\varphi _0 =\varphi _1 =\varphi _2 =0$ in the fourth and fifth equations
of \eqref{e3}, we obtain:
\begin{gather*}
\omega _1 \varphi _3 +\omega _2 \varphi _{4} =0,\\
\omega _2 \varphi _3 +\omega _1 \varphi _{4} =0.
\end{gather*}
And from these equations we have that $\varphi _3 =\varphi _{4} =0$.
Thus, $u_0 (t)=0$.
The proof is complete.
\end{proof}

Let us consider the question of regular solvability of problem
 \eqref{e1}, \eqref{e2} when $A_1 =A_2 =A_3 =0$.

\begin{lemma} \label{lem2}
In the conditions of Lemma \ref{lem1}, the problem
\begin{gather} \label{e7}
-u'''(t)+\rho (t)A^3 u(t)=f(t), \quad t\in R_{+}, \\
\label{e8}
u'(0)=0, \quad  u''(0)= Ku(0)
\end{gather}
is regularly solvable.
\end{lemma}

\begin{proof}
 We show that the equation $P_0 u(t)=f(t)$ has a solution
$u(t)\in W_{2,K}^3 (R_{+} ;H)$ for any $f(t)\in L_2 (R_{+} ;H)$.
First, we continue the vector-function $f(t)$ in such a way that
$f(t)=0$ for $t<0$. We denote the new function by $g(t)$.
Let $\hat{g}(\xi )$ be the Fourier transform of the vector-function $g(t)$;
 i.e.,
\[
\hat{g}(\xi )=\frac{1}{\sqrt{2\pi } } \int _{-\infty }^{+\infty }
g(t)e^{-i\xi t} dt ,
\]
where the integral on the right side is understood in the sense of
convergence on the average in $H$. Then, using the direct and inverse
Fourier transforms, it is clear that the vector-functions
\begin{gather*}
\upsilon _1 (t)=\frac{1}{2\pi } \int _{-\infty }^{+\infty }
(i\xi ^3 E+\alpha A^3 )^{-1}
\Big(\int _0^{+\infty }f(s)e^{-i\xi s} ds \Big)e^{it\xi } d\xi ,\quad
t\in R,
\\
\upsilon _2 (t)=\frac{1}{2\pi }
\int _{-\infty }^{+\infty }(i\xi ^3 E+\beta A^3 )^{-1}
\Big(\int _0^{+\infty }f(s)e^{-i\xi s} ds \Big)e^{it\xi } d\xi ,\quad t\in R,
\end{gather*}
satisfy the equations
\begin{gather*}
-\frac{d^3 \upsilon (t)}{dt^3 } +\alpha A^3 \upsilon (t)=g(t),\\
-\frac{d^3 \upsilon (t)}{dt^3 } +\beta A^3 \upsilon (t)=g(t),
\end{gather*}
respectively, almost everywhere in $R$. We prove that $\upsilon _1 (t)$
and $\upsilon _2 (t)$ belong  to $W_2^3 (R;H)$. By Plancherel's theorem
\begin{align*}
\| \upsilon _1 (t)\| _{W_2^3 (R;H)}^2
& =\| \upsilon '''_1 (t)\| _{L_2 (R;H)}^2
 +\| A^3 \upsilon _1 (t)\| _{L_2 (R;H)}^2 \\
&=\| -i\xi ^3 \hat{\upsilon }_1 (\xi )\| _{L_2 (R;H)}^2
 +\| A^3 \hat{\upsilon }_1 (\xi )\| _{L_2 (R;H)}^2 ,
\end{align*}
where $\hat{\upsilon }_1 (\xi )$ is the Fourier transform of the function
$\upsilon _1 (t)$.
Since
\[
\hat{\upsilon }_1 (\xi )=(i\xi ^3 E+\alpha A^3 )^{-1} \hat{g}(\xi ),
\]
we have
\begin{equation} \label{e9}
\begin{aligned}
\| -i\xi ^3 \hat{\upsilon }_1 (\xi )\| _{L_2 (R;H)}
&=\| -i\xi ^3 (i\xi ^3 E+\alpha A^3 )^{-1} \hat{g}(\xi )\| _{L_2 (R;H)} \\
&\le \sup_{\xi \in R} \| -i\xi ^3 (i\xi ^3 E+\alpha A^3 )^{-1} \| _{H\to H}
  \| \hat{g}(\xi )\| _{L_2 (R;H)} \\
&=\sup_{\xi \in R} \| -i\xi ^3 (i\xi ^3 E+\alpha A^3 )^{-1}
 \| _{H\to H} \| g(t)\| _{L_2 (R;H)} .
\end{aligned}
\end{equation}
It follows from the spectral theory of self-adjoint operators that
\begin{equation}
\begin{aligned}
\| -i\xi ^3 (i\xi ^3 E+\alpha A^3 )^{-1} \|
&=\sup_{\sigma \in \sigma (A)}
\big|-i\xi ^3 (i\xi ^3 +\alpha \sigma ^3 )^{-1} \big|\\
&= \sup_{\sigma  \in \sigma (A)} \frac{{| \xi  |^3 }}{{( {\xi ^6
+ \alpha ^2 \sigma ^6 } )^{1/2} }} \le 1.
\end{aligned}
\end{equation}
Therefore, from \eqref{e9} it follows that
 $-i\xi ^3 \hat{\upsilon }_1 (\xi )\in L_2 (R;\, H)$. Since
\begin{equation} \label{e10}
\begin{aligned}
\| A^3 \hat{\upsilon }_1 (\xi )\| _{L_2 (R;H)}
& =\| A^3 (i\xi ^3 E+\alpha A^3 )^{-1} \hat{g}(\xi )\| _{L_2 (R;H)} \\
&\le \sup_{\xi \in R} \| A^3 (i\xi ^3 E+\alpha A^3 )^{-1} \| _{H\to H}
\| \hat{g}(\xi )\| _{L_2 (R;H)} \\
&=\sup_{\xi \in R} \| A^3 (i\xi ^3 E+\alpha A^3 )^{-1} \| _{H\to H}
\| g(t)\| _{L_2 (R;H)} ,
\end{aligned}
\end{equation}
again from the spectral theory of self-adjoint operators, we have
\[
\| A^3 (i\xi ^3 E+\alpha A^3 )^{-1} \| =\sup_{\sigma \in \sigma (A)}
|\sigma ^3 (i\xi ^3 +\alpha \sigma ^3 )^{-1} |
=\sup_{\sigma \in \sigma (A)} \frac{\sigma ^3 }{(\xi ^6
+\alpha ^2 \sigma ^6 )^{1/2} }
 \le \frac{1}{\alpha } .
\]
Thus, from \eqref{e10} it follows that
$A^3 \hat{\upsilon }_1 (\xi )\in L_2 (R;\, H)$. Hence,
$\upsilon _1 (t)\in W_2^3 (R;H)$. Thus $\upsilon _2 (t)\in W_2^3 (R;H)$.

Let us denote the restriction of the vector-function $\upsilon _1 (t)$ on
$[0,1)$ by $u_{\alpha } (t)$ and the restriction of the vector-function
$\upsilon _2 (t)$ on $(1,+\infty )$ by $u_{\beta } (t)$.
It is obvious, that $u_{\alpha } (t)\in W_2^3 ([0,1);H)$,
$u_{\beta } (t)\in W_2^3 ((1,+\infty );H)$. Then, from the theorem on
traces \cite[ch. 1]{l1} it follows that $\frac{d^{s} u_{\alpha } (0)}{dt^{s} } $,
$\frac{d^{s} u_{\alpha } (1)}{dt^{s} } $, $\frac{d^{s} u_{\beta } (0)}{dt^{s} } $,
 $\frac{d^{s} u_{\beta } (1)}{dt^{s} } \in H_{{5/2}-s} $, $s=0,1,2$.

Now, we denote
\[
u(t)=\begin{cases}
u_1 (t)=u_{\alpha } (t)+e^{\sqrt[3]{\alpha } \omega _1 tA} \psi _0
+e^{\sqrt[3]{\alpha } \omega _2 tA} \psi _1 +e^{-\sqrt[3]{\alpha } (1-t)A}
\psi _2 ,& 0\le t<1, \\
u_2 (t)=u_{\beta } (t)+e^{\sqrt[3]{\beta } \omega _1 (t-1)A} \psi _3
+e^{\sqrt[3]{\beta } \omega _2 (t-1)A} \psi _{4} ,& 1<t<+\infty ,
\end{cases}
\]
where $\psi _{k} \in H_{5/2}$, $k=0,1,2,3,4$. The function $u(t)$ belongs
to  $W_{2,K}^3 (R_{+} ;H)$, so the vectors $\psi _{k}$, $k=0,1,2,3,4$,
can be determined from the following relations:
\[u'_1 (0)=0, \quad u''_1 (0)=Ku_1 (0),  \quad
u_1 (1)=u_2 (1),\quad  u'_1 (1)=u'_2 (1),\quad  u''_1 (1)=u''_2 (1).
\]
From here with respect to $\psi _{k} $, $k=0,1,2,3,4$, we have the
system of equations
\begin{gather*}
u'_{\alpha } (0)+\sqrt[3]{\alpha } \omega _1 A\psi _0 +\sqrt[3]{\alpha }
\omega _2 A\psi _1 +\sqrt[3]{\alpha }
Ae^{-\sqrt[3]{\alpha } A} \psi _2 =0\, ,
 \\
u''_{\alpha } (0)+\sqrt[3]{\alpha ^2 } A^2
 \big(\omega _1^2 \psi _0 +\omega _2^2 \psi _1
 +e^{-\sqrt[3]{\alpha } A} \psi _2 \big)
=Ku_{\alpha } (0)+K\big(\psi _0 +\psi _1 +e^{-\sqrt[3]{\alpha } A} \psi _2 \big),
\\
u_{\alpha } (1)+e^{\sqrt[3]{\alpha } \omega _1 A} \psi _0
+e^{\sqrt[3]{\alpha } \omega _2 A} \psi _1 +\psi _2
=u_{\beta } (1)+\psi _3 +\psi _{4} \, ,
 \\
\begin{aligned}
& u'_{\alpha } (1)+\sqrt[3]{\alpha } \omega _1 Ae^{\sqrt[3]{\alpha }
 \omega _1 A} \psi _0 +\sqrt[3]{\alpha } \omega _2 Ae^{\sqrt[3]{\alpha }
 \omega _2 A} \psi _1 +\sqrt[3]{\alpha } A\psi _2\\
& =u'_{\beta } (1)+\sqrt[3]{\beta } \omega _1 A\psi _3
 +\sqrt[3]{\beta } \omega _2 A\psi _{4} \, ,
\end{aligned} \\
\begin{aligned}
&u''_{\alpha } (1)+\sqrt[3]{\alpha ^2 } \omega _1^2 A^2
 e^{\sqrt[3]{\alpha } \omega _1 A} \psi _0 +\sqrt[3]{\alpha ^2 }
 \omega _2^2 A^2 e^{\sqrt[3]{\alpha } \omega _2 A} \psi _1
 +\sqrt[3]{\alpha ^2 } A^2 \psi _2\\
&=u''_{\beta } (1)+\sqrt[3]{\beta ^2 } \omega _1^2 A^2 \psi _3
 +\sqrt[3]{\beta ^2 } \omega _2^2 A^2 \psi _{4} \,.
\end{aligned}
 \end{gather*}
From this system we obtain:
\begin{equation} \label{e11}
\begin{gathered}
\sqrt[3]{\alpha } \omega _1 \psi _0 +\sqrt[3]{\alpha }
\omega _2 \psi _1 +\sqrt[3]{\alpha } e^{-\sqrt[3]{\alpha } A} \psi _2
=-A^{-1} u'_{\alpha } (0),
 \\
\begin{aligned}
&\sqrt[3]{\alpha ^2 } (\omega _1^2 \psi _0 +\omega _2^2 \psi _1
+e^{-\sqrt[3]{\alpha } A} \psi _2 )-A^{-2} K(\psi _0
+\psi _1 +e^{-\sqrt[3]{\alpha } A} \psi _2 )\\
&=A^{-2} (Ku_{\alpha } (0)-u''_{\alpha } (0))\, ,
\end{aligned} \\
e^{\sqrt[3]{\alpha } \omega _1 A} \psi _0 +e^{\sqrt[3]{\alpha }
\omega _2 A} \psi _1 +\psi _2 -\psi _3 -\psi _{4}
=u_{\beta } (1)-u_{\alpha } (1),
\\
\begin{aligned}
&\sqrt[3]{\alpha } \omega _1 e^{\sqrt[3]{\alpha } \omega _1 A} \psi _0
+\sqrt[3]{\alpha } \omega _2 e^{\sqrt[3]{\alpha } \omega _2 A} \psi _1
 +\sqrt[3]{\alpha } \psi _2 -\sqrt[3]{\beta } \omega _1 \psi _3
-\sqrt[3]{\beta } \omega _2 \psi _{4} \\
&=A^{-1} (u'_{\beta } (1)-u'_{\alpha } (1))\, ,
\end{aligned}\\
\begin{aligned}
&\sqrt[3]{\alpha ^2 } \omega _1^2 e^{\sqrt[3]{\alpha }
\omega _1 A} \psi _0 +\sqrt[3]{\alpha ^2 } \omega _2^2
e^{\sqrt[3]{\alpha } \omega _2 A} \psi _1
+\sqrt[3]{\alpha ^2 } \psi _2 -\sqrt[3]{\beta ^2 }
\omega _1^2 \psi _3 -\sqrt[3]{\beta ^2 } \omega _2^2 \psi _{4}\\
& =A^{-2} (u''_{\beta } (1)-u''_{\alpha } (1))\, .
\end{aligned}
\end{gathered}
\end{equation}
Since $u_{\alpha } (t)\in W_2^3 ([0,1);H)$ and
$u_{\beta } (t)\in W_2^3 ((1,+\infty );H)$,  by the
theorem on traces \cite[ch. 1]{l1}
$A^{-1} u'_{\alpha } (0)$, $A^{-2} (Ku_{\alpha } (0)-u''_{\alpha } (0))$,
 $u_{\beta}(1)-u_{\alpha}(1)$, $A^{-1} (u'_{\beta } (1)-u'_{\alpha } (1))$
and $A^{-2} (u''_{\beta } (1)-u''_{\alpha } (1))$ belong to $H_{5/2}$.
Then by these values acting also as in the system \eqref{e3}, in this case,
taking into account that the operator $K_{\alpha ,\beta } $ has a
bounded inverse operator in the space $H_{5/2}$ and
$-\sqrt[3]{\alpha ^2 } \omega _2 \notin \sigma (B)$, obviously,
from \eqref{e11} it is possible to find the vectors $\psi _{k} $, $k=0,1,2,3,4$,
where all $\psi _{k} \in H_{5/2}$, $k=0,1,2,3,4$.
Therefore, $u(t)\in W_2^3 (R_{+};H)$ satisfies equation \eqref{e7}
almost everywhere in $R_{+} $ and conditions \eqref{e8}.

By lemma \ref{lem1}, the problem
\begin{gather*}
-u'''(t)+\rho (t)A^3 u(t)=0,\\
u'(0)=0, \quad u''(0)= Ku(0)
\end{gather*}
has only the trivial solution in the space $W_{2,K}^3 (R_{+} ;H)$.

Now we  show that the operator $P_0 :W_{2,K}^3 (R_{+} ;H)\to L_2 (R_{+} ;H)$
is bounded. Indeed, for $u(t)\in W_{2,K}^3 (R_{+} ;H)$ we have
\begin{align*}
&\| P_0 u\| _{L_2 (R_{+} ;H)}^2\\
&=\| u'''\| _{L_2 (R_{+} ;H)}^2 +\| \rho (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
-2\operatorname{Re}(u''',\rho (t)A^3 u)_{L_2 (R_{+} ;H)} \\
&\le \| u'''\| _{L_2 (R_{+} ;H)}^2
+\| \rho (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
 +2\| u'''\| _{L_2 (R_{+} ;H)} \| \rho (t)A^3 u\| _{L_2 (R_{+} ;H)} \\
&\le 2\big(\| u'''\| _{L_2 (R_{+} ;H)}^2
+\| \rho (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 \big)\\
&\le 2\max (1;\alpha ^2 ;\beta ^2 )\| u\| _{W_{_2 }^3 (R_{+} ;H)}^2 .
\end{align*}
Thus, according to the Banach theorem on the inverse operator,
there exists $P_0^{-1} :L_2 (R_{+} ;H)\to W_{2,K}^3 (R_{+} ;H)$
and it is bounded. Hence, it follows that
\[
\| u\| _{W_2^3  (R_{+} ; H)} \le \text{const} \| f\| _{L_2^{}
 (R_{+}\, H)} .
\]
The proof is complete.
\end{proof}

On the basis of Lemmas \ref{lem1} and \ref{lem2} we obtain the following conclusion.

\begin{theorem} \label{thm1}
Let the conditions of Lemma \ref{lem1} be satisfied. Then the operator $P_0 $
is an isomorphism between the spaces $W_{2,K}^3 (R_{+} ;H)$ and
$L_2 (R_{+} ;H)$.
\end{theorem}

Let us prove the following coercive inequality which will be used further.

\begin{lemma} \label{lem3}  Let $\operatorname{Re}(B)\ge 0$.
Then for any $u(t)\in W_{2,K}^3 (R_{+} ;H)$, the following inequality holds
\begin{equation} \label{e12}
\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
+\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
\le  \frac{1}{\min (\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)}^2 .
\end{equation}
\end{lemma}

\begin{proof}
 Consider the following equalities:
\begin{equation} \label{e13}
\begin{aligned}
(P_0 u,A^3 u)_{L_2 (R_{+} ;H)}
& =(-u'''+\rho (t)A^3 u,A^3 u)_{L_2 (R_{+} ;H)} \\
&=(-u''',A^3 u)_{L_2 (R_{+} ;H)} +(\rho (t)A^3 u,A^3 u)_{L_2 (R_{+} ;H)}\\
&=(-u''',A^3 u)_{L_2 (R_{+} ;H)} +\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 ,
\end{aligned}
\end{equation}
\begin{equation} \label{e14}
\begin{aligned}
(P_0 u,-\rho ^{-1} (t)u''')_{L_2 (R_{+} ;H)}
&=(-u'''+\rho (t)A^3 u,-\rho ^{-1} (t)u''')_{L_2 (R_{+} ;H)} \\
&=(-u''',-\rho ^{-1} (t)u''')_{L_2 (R_{+} ;H)} -(A^3 u,u''')_{L_2 (R_{+} ;H)} \\
&=\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2 -(A^3 u,u''')_{L_2 (R_{+} ;H)} .
\end{aligned}
\end{equation}
Note that by integrating by parts for $u(t)\in W_{2,K}^3 (R_{+} ;H)$, we have
\begin{equation} \label{e15}
-\operatorname{Re}(u''',A^3 u)_{L_2 (R_{+} ;H)}
=\operatorname{Re}(BA^{5/2} u(0),A^{5/2} u(0)).
\end{equation}
By  \eqref{e13} and \eqref{e14} and taking into account \eqref{e15}, we obtain
\begin{equation} \label{e16}
\begin{aligned}
&(P_0 u,A^3 u-\rho ^{-1} (t)u''')_{L_2 (R_{+} ;H)}\\
&=\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
+\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
+2\operatorname{Re}(BA^{5/2} u(0),A^{5/2} u(0)).
\end{aligned}
\end{equation}
Applying the Cauchy-Schwarz inequality to the left side  and then the
 Young's inequality and taking into account \eqref{e15}, we obtain
\begin{equation} \label{e17}
\begin{aligned}
&(P_0 u,A^3 u-\rho ^{-1} (t)u''')_{L_2 (R_{+} ;H)} \\
&\le \|\rho ^{-1/2} (t) P_0 u\| _{L_2 (R_{+} ;H)} \| \rho ^{1/2} (t)A^3 u-\rho ^{-1/2} (t)
u'''\| _{L_2 (R_{+} ;H)} \\
&\le \frac{1}{2\min (\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)}^2
 +\frac{1}{2} \| \rho ^{1/2} (t)A^3 u-\rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2\\
&=\frac{1}{2\min (\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)}^2
+\frac{1}{2} \| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 \\
&\quad +\frac{1}{2} \| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
+\operatorname{Re}\big(BA^{5/2} u(0),A^{5/2} u(0)\big).
\end{aligned}
\end{equation}
Taking into account \eqref{e17} into \eqref{e16}, we have
\begin{equation} \label{e18}
\begin{aligned}
&\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
 +\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
+2\operatorname{Re}(BA^{5/2} u(0),A^{5/2} u(0))\\
&\le \frac{1}{\min (\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)}^2 .
\end{aligned}
\end{equation}
Since $\operatorname{Re}B\ge 0$, then from inequality \eqref{e18},
 we obtain the validity of inequality \eqref{e12}.
The proof is complete.
\end{proof}

Theorem \ref{thm1} implies that the norm $\| P_0 u\| _{L_2 (R_{+} ;H)} $ 
is equivalent
to the norm $\| u\| _{W_2^3 (R_{+} ;H)} $ in the space $W_{2,K}^3 (R_{+} ;H)$.
Therefore, the norms of the intermediate derivative operators
$A^{j} \frac{d^{3-j} }{dt^{3-j} } :W_{2,K}^3 (R_{+} ;H)\to L_2 (R_{+} ;H)$,
$j=1,2,3$, can be estimated with respect to $\| P_0 u\| _{L_2 (R_{+} ;H)} $
(by the continuity of these operators \cite{l1}). Methods for solution of
equations with scalar boundary conditions are often inapplicable to the
problems with boundary conditions which include abstract operators.
 For example, when $K=0$, operator pencil factorization method for the
estimation of the norms of intermediate derivative operators has been
developed in \cite{a4} (this method was first mentioned in \cite{m1}
 when considering
 operator-differential equations with constant coefficients).
 The estimates for the norms of intermediate derivative operators are
playing an important role in obtaining solvability conditions.
But, the method of \cite{a4} is not applicable to the boundary value problems
for odd order operator-differential equations with the boundary conditions
 which include abstract operators. In this work, to estimate the norms
of intermediate derivative operators we use the classical inequalities
 of mathematical analysis and the coercive inequality \eqref{e12}.

\begin{theorem} \label{thm2}
Let $\operatorname{Re}B\ge 0$. Then for any $u(t)\in W_{2,K}^3 (R_{+} ;H)$
the following inequalities hold:
\begin{equation} \label{e19}
\| A^{j} \frac{d^{3-j} u}{dt^{3-j} } \| _{L_2 (R_{+} ;H)}
 \le a_{j} \| P_0 u\| _{L_2 (R_{+} ;H)} ,\quad j=1,2,3,
\end{equation}
where
\[
a_1 =\frac{2^{1/3} \max ^{1/3} (\alpha ;\beta )}{3^{1/2}
 \min ^{2/3} (\alpha ;\beta )} ,\quad
a_2 =\frac{2^{1/3} \max ^{1/6} (\alpha ;\beta )}{3^{1/2}
 \min ^{5/6} (\alpha ;\beta )} ,\quad
a_3 =\frac{1}{\min (\alpha ;\beta )} .
\]
\end{theorem}

\begin{proof} Let $u(t)\in W_{2,K}^3 (R_{+} ;H)$. Integrating by parts
and applying the Cauchy-Schwarz inequality, and then the Young's inequality,
 we obtain
\begin{equation} \label{e20}
\begin{aligned}
\| Au''\| _{L_2 (R_{+} ;H)}^2 
&=\int _0^{+\infty }(Au'',Au'')_{H} dt \\
&=(Au',Au'')_{H}|_0^{+\infty } -\int _0^{+\infty }(Au',Au''')_{H} dt \\
&=-\int _0^{+\infty }(A^2 u',u''')_{H} dt \le \| A^2 u'\| _{L_2 (R_{+} ;H)}
\| u'''\| _{L_2 (R_{+} ;H)} \\
&\le \max_{t} \rho ^{1/2} (t)\| A^2 u'\| _{L_2 (R_{+} ;H)}
\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)} \\
&\leq \frac{\varepsilon }{2} \max (\alpha ;\beta )\| A^2
u'\| _{L_2 (R_{+} ;H)}^2 +\frac{1}{2\varepsilon } \|
 \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2 ,
\end{aligned}
\end{equation}
with $\varepsilon >0$.
Proceeding in a similar manner, we have
\begin{equation} \label{e21}
\begin{aligned}
\| A^2 u'\| _{L_2 (R_{+} ;H)}^2
&=\int _0^{+\infty }(A^2 u',A^2 u')_{H} dt \\
&= (A^2 u,A^2 u')_{H}|_0^{+\infty }
  -\int _0^{+\infty }(A^2 u,A^2 u'')_{H} dt\\
& =-\int _0^{+\infty }(A^3 u,Au'')_{H} dt \\
&\le \| A^3 u\| _{L_2 (R_{+} ;H)}  \| Au''\| _{L_2 (R_{+} ;H)} \\
&\le \max_{t} \rho ^{-1/2} (t)\| Au''\| _{L_2 (R_{+} ;H)} \|
 \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)} \\
&\le \frac{\eta }{2} \frac{1}{\min (\alpha ;\beta )}
\| Au''\| _{L_2 (R_{+} ;H)}^2 +\frac{1}{2\eta } \|
\rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 ,
\end{aligned}
\end{equation}
with $\eta >0$.
Taking into account inequality \eqref{e21} in \eqref{e20}:
\begin{equation} \label{e22}
\begin{aligned}
\| Au''\| _{L_2 (R_{+} ;H)}^2 
&\le \frac{\varepsilon }{2} \max (\alpha ;\beta )(\frac{\eta }{2}
\frac{1}{\min (\alpha ;\beta )} \| Au''\| _{L_2 (R_{+} ;H)}^2\\
&\quad +\frac{1}{2\eta } \| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 )
+\frac{1}{2\varepsilon } \| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2 .
\end{aligned}
\end{equation}
From this inequality we obtain
\begin{equation} \label{e23}
\begin{aligned}
\big(1-\frac{\varepsilon \eta \max (\alpha ;\beta )}{4\min (\alpha ;\beta )}
\big)\| Au''\| _{L_2 (R_{+} ;H)}^2 
&\le \frac{\varepsilon \max (\alpha ;\beta )}{4\eta }
\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2\\
&\quad +\frac{1}{2\varepsilon } \| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2 .
\end{aligned}
\end{equation}
Choosing $\eta =\frac{\varepsilon ^2 \max (\alpha ;\beta )}{2} $,
from inequality \eqref{e23} we have
\begin{align*}
&\| Au''\| _{L_2 (R_{+} ;H)}^2 \\
&\le \frac{4\min (\alpha ;\beta )}{8\varepsilon
\min (\alpha ;\beta )-\varepsilon ^{4} \max ^2 (\alpha ;\beta )}
\big[\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
+\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2 \big].
\end{align*}
Then, by minimizing $\varepsilon $, we find
$\varepsilon =\sqrt[3]{2\min (\alpha ;\beta )/\max ^2 (\alpha ;\beta )}$. 
Therefore,
\begin{equation} \label{e24}
\begin{aligned}
&\| Au''\| _{L_2 (R_{+} ;H)}^2 \\
&\le  \frac{2^{2/3} \max ^{2/3} (\alpha ;\beta )}{3\min ^{1/3}
(\alpha ;\beta )} \big[\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
+\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2 \big].
\end{aligned}
\end{equation}
Now, taking into account inequality \eqref{e12}, from inequality \eqref{e24}
 we obtain
\[
\| Au''\| _{L_2 (R_{+} ;H)}^2 \le \frac{2^{2/3}
\max ^{2/3} (\alpha ;\beta )}{3\min ^{4/3} (\alpha ;\beta )}
 \| P_0 u\| _{L_2 (R_{+} ;H)}^2 .
\]
As a result,
\[
\| Au''\| _{L_2 (R_{+} ;H)} \le \frac{2^{1/3}
\max ^{1/3} (\alpha ;\beta )}{3^{1/2} \min ^{2/3}
(\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)} .
\]
To estimate the norm $\| A^2 u'\| _{L_2 (R_{+} ;H)} $, we take into
account  \eqref{e20} in \eqref{e21}:
\begin{equation} \label{e25}
\begin{aligned}
&\big(1-\frac{\varepsilon \eta \max (\alpha ;\beta )}{4\min (\alpha ;\beta )}
\big)\| A^2 u'\| _{L_2 (R_{+} ;H)}^2\\
&\le \frac{\eta }{4\varepsilon \min (\alpha ;\beta )}
 \| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
+\frac{1}{2\eta } \| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 .
\end{aligned}
\end{equation}
Choosing $\varepsilon =\eta ^2/(2\min (\alpha ;\beta )) $,
from inequality \eqref{e25} we have
\begin{align*}
&\| A^2 u'\| _{L_2 (R_{+} ;H)}^2\\
& \le  \frac{4\min ^2 (\alpha ;\beta )}{8\eta \min ^2 (\alpha ;\beta )
-\eta ^{4} \max (\alpha ;\beta )} 
\big[\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
+\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 \big].
\end{align*}
In this case, minimizing $\eta $, we find
$\eta =\sqrt[3]{2\min ^2 (\alpha ;\beta )/\max (\alpha ;\beta )} $.
Therefore,
\begin{equation} \label{e26_}
\begin{aligned}
&\| A^2 u'\| _{L_2 (R_{+} ;H)}^2\\
& \le \frac{2^{2/3} \max ^{1/3} (\alpha ;\beta )}{3\min ^{2/3}
(\alpha ;\beta )} [\| \rho ^{-1/2} (t)u'''\| _{L_2 (R_{+} ;H)}^2
+\| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2 ].
\end{aligned}
\end{equation}
From this inequality, taking into account inequality \eqref{e12}, we obtain
\[
\| A^2 u'\| _{L_2 (R_{+} ;H)}^2
\le \frac{2^{2/3} \max ^{1/3} (\alpha ;\beta )}{3\min ^{5/3}
(\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)}^2 .
\]
Thus,
\[
\| A^2 u'\| _{L_2 (R_{+} ;H)}
\le \frac{2^{1/3} \max ^{1/6} (\alpha ;\beta )}{3^{1/2} \min ^{5/6}
(\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)} .
\]
Now we estimate the norm $\| A^3 u\| _{L_2 (R_{+} ;H)} $.
From inequality \eqref{e12} we have
\[
\frac{1}{\min (\alpha ;\beta )} \| P_0 u\| _{L_2 (R_{+} ;H)}^2
\ge \| \rho ^{1/2} (t)A^3 u\| _{L_2 (R_{+} ;H)}^2
\ge \min (\alpha ;\beta )\| A^3 u\| _{L_2 (R_{+} ;H)}^2 .
\]
Hence, we obtain
\[
\| A^3 u\| _{L_2 (R_{+} ;H)}^2 \le \frac{1}{\min ^2 (\alpha ;\beta )}
\| P_0 u\| _{L_2 (R_{+} ;H)}^2
\]
or
\[
\| A^3 u\| _{L_2 (R_{+} ;H)} \le \frac{1}{\min (\alpha ;\beta )}
 \| P_0 u\| _{L_2 (R_{+} ;H)} .
\]
The proof is complete.
\end{proof}

Now, we prove the boundedness of the operator
$P_1 :W_{2,K}^3 (R_{+} ;H)\to L_2 (R_{+} ;H)$.

\begin{lemma} \label{lem4}
Let $A_{j} A^{-j} \in L(H,H)$, $j=1,2,3$.
Then $P_1 $ is a bounded operator from the space $W_{2,K}^3 (R_{+} ;H)$
into the space $L_2 (R_{+} ;H)$.
\end{lemma}

\begin{proof} For any $u(t)\in W_{2,K}^3 (R_{+} ;H)$ we have
\begin{align*}
\| P_1 u\| _{L_2 (R_{+} ;H)}
&=\| A_1 u''+A_2 u'+A_3 u\| _{L_2 (R_{+} ;H)} \\
&\le \| A_1 A^{-1} \| _{H\to H} \| Au''\| _{L_2 (R_{+} ;H)} 
 +\| A_2 A^{-2} \| _{H\to H} \| A^2 u'\| _{L_2 (R_{+} ;H)}\\
&\quad +\| A_3 A^{-3} \| _{H\to H} \| A^3 u\| _{L_2 (R_{+} ;H)} .
\end{align*}
Applying the theorem on intermediate derivatives \cite[ch. 1]{l1}, we obtain
from the last inequality that
\[
\| P_1 u\| _{L_2 (R_{+} ;H)} \le const\| u\| _{W_2^3 (R_{+} ;H)} .
\]
The proof is complete.
\end{proof}

Let us consider the question of regular solvability of problem
\eqref{e1}, \eqref{e2}.

\begin{theorem} \label{thm3}
Let $A=A^{*} \ge cE$, $c>0$, $K\in L(H_{5/2} ,H_{1/2} )$,
$-\sqrt[3]{\alpha ^2 } \omega _2 \notin \sigma (B)$,
the operator $K_{\alpha ,\beta }$ has a bounded inverse in the space
$H_{5/2}$, $\operatorname{Re}B\ge 0$ and $A_{j} A^{-j} \in L(H,H)$,
$j=1,2,3$, moreover, the following inequality holds
\[
a_1 \| A_1 A^{-1} \| _{H\to H} +a_2 \| A_2 A^{-2} \| _{H\to H}
+a_3 \| A_3 A^{-3} \| _{H\to H} <1,
\]
where the numbers $a_{j} $, $j=1,2,3$, are defined in Theorem \ref{thm2}.
Then the boundary value problem \eqref{e1}, \eqref{e2}
is regularly solvable.
\end{theorem}

\begin{proof}
 Boundary value problem \eqref{e1}, \eqref{e2} can be represented in
the operator form
\[
P_0 u(t)+P_1 u(t)=f(t),
\]
where $f(t)\in L_2 (R_{+} ;H)$, $u(t)\in W_{2,K}^3 (R_{+} ;H)$.

Under conditions $A=A^{*} \ge cE$, $c>0$, $K\in L(H_{5/2} ,H_{1/2} )$,
$-\sqrt[3]{\alpha ^2 } \omega _2 \notin \sigma (B)$, the operator
$K_{\alpha ,\beta } $ has a bounded inverse in the space $H_{5/2}$,
by Theorem \ref{thm1} the operator $P_0$ has a bounded inverse $P_0^{-1} $
acting from the space  $L_2 (R_{+} ;H)$ into the space $W_{2,K}^3 (R_{+} ;H)$.
If we put $v(t)=P_0u(t)$ we obtain the following equation in $L_2 (R_{+} ;H)$:
\[
(E+P_1 P_0^{-1} )v(t)=f(t).
\]
We show that under the conditions of the theorem, the norm of the operator
$P_1 P_0^{-1} $ is less than unity. Taking into account inequalities \eqref{e19},
we have
\begin{align*}
&\| P_1 P_0^{-1} v\| _{L_2 (R_{+} ;H)}\\
& =\| P_1 u\| _{L_2 (R_{+} ;H)} \\
&\le \| A_1 u''\| _{L_2 (R_{+} ;H)} +\| A_2 u'\| _{L_2 (R_{+} ;H)}
+\| A_3 u\| _{L_2 (R_{+} ;H)} \\
&\le \| A_1 A^{-1} \| _{H\to H} \| Au''\| _{L_2 (R_{+} ;H)} 
 +\| A_2 A^{-2} \| _{H\to H} \| A^2 u'\| _{L_2 (R_{+} ;H)}\\
&\quad +\| A_3 A^{-3} \| _{H\to H} \| A^3 u\| _{L_2 (R_{+} ;H)} \\
&\le  a_1 \| A_1 A^{-1} \| _{H\to H} \| P_0 u\| _{L_2 (R_{+} ;H)}
 +a_2 \| A_2 A^{-2} \| _{H\to H} \| P_0 u\| _{L_2 (R_{+} ;H)} \\
&\quad +a_3 \| A_3 A^{-3} \| _{H\to H} \| P_0 u\| _{L_2 (R_{+} ;H)} \\
&=\sum _{j=1}^3a_{j} \| A_{j} A^{-j} \| _{H\to H} \| v\| _{L_2 (R_{+} ;H)}  .
\end{align*}
Thus,
\[
\| P_1 P_0^{-1} \| _{L_2 (R_{+} ;H)\to L_2 (R_{+} ;H)}
\le \sum _{j=1}^3a_{j} \| A_{j} A^{-j} \| _{H\to H}  <1.
\]
Therefore, the operator $E+P_1 P_0^{-1} $ is invertible in the space
$L_2 (R_{+} ;H)$ and $u(t)$ is defined by the formula
\[
u(t)=P_0^{-1} (E+P_1 P_0^{-1} )^{-1} f(t),
\]
moreover
\begin{align*}
&\| u\| _{W_2^3 (R_{+} ;H)} \\
&\le \| P_0^{-1} \| _{L_2 (R_{+} ;H)\to W_2^3 (R_{+} ;H)}
\| (E+P_1 P_0^{-1} )^{-1} \| _{L_2 (R_{+} ;H)\to L_2 (R_{+} ;H)}
\| f\| _{L_2 (R_{+} ;H)} \\
&\le \text{const} \| f\| _{L_2 (R_{+} ;H)} .
\end{align*}
The proof is complete.
\end{proof}

\begin{corollary} \label{coro1}
In the conditions of Theorem \ref{thm3}, the operator $P$ is an isomorphism
between the spaces $W_{2,K}^3 (R_{+} ;H)$ and $L_2 (R_{+} ;H)$.
\end{corollary}

In conclusion, we remark that our solvability results imply the results
of \cite{a4} when $K=0$ and $A_3 =0$, and the results of \cite{m1}
 when $K=0$ and $\alpha =\beta =1$.

\subsection*{Acknowledgements}
We are very grateful to the referees for their careful reading of the
original manuscript, for their helpful comments
which led to the improvement of this article.


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\end{document}