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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 72, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/72\hfil Periodic problem for an impulsive system]
{Periodic problem for an impulsive system of the loaded hyperbolic equations}

\author[A. T. Assanova, Z. M. Kadirbayeva \hfil EJDE-2018/72\hfilneg]
{Anar T. Assanova, Zhazira M. Kadirbayeva}

\address{Anar T. Assanova \newline
Department of Differential Equations,
Institute of Mathematics and Mathematical Modelling of MES RK,
Pushkin str., 125, 050010  Almaty, Kazakhstan}
\email{assanova@math.kz, anarasanova@list.ru}

\address{Zhazira M. Kadirbayeva \newline
Department of Differential Equations, 
Institute of Mathematics and Mathematical Modelling of MES RK, 
Pushkin str., 125, 050010  Almaty, Kazakhstan. \newline 
Institute of Mathematics,
Physics and Computer Sciences, 
Kazakh  State Women's \newline Teacher Training University, 
Almaty, Kazakhstan, Aiteke bi str., 49, 050012
Almaty,  Kazakhstan}
\email{apelman86pm@mail.ru}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted February 14, 2018. Published March 15, 2018.}
\subjclass[2010]{34A37, 34B37, 35B10, 35L20, 35R12}
\keywords{System of loaded hyperbolic equations; periodic problem;
\hfill\break\indent impulse effects; solvability}

\begin{abstract}
 We consider a periodic problem for a system of the loaded hyperbolic equations 
 with impulse effects. By introducing additional functions, this problem is 
 reduced to an equivalent problem consisting of a family of periodic 
 boundary-value problems of loaded ordinary differential equations with 
 impulse effects and integral relations.  We obtain sufficient conditions 
 for the existence of unique solution to the family of periodic boundary-value 
 problems. Conditions of unique solvability of periodic problem are established 
 in terms of initial data.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction and statement of the problem}

In this article  we consider a periodic problem for the system of second-order 
loaded  hyperbolic equations  with impulse effects
\begin{gather}
\begin{aligned}
\frac{\partial ^2 u}{\partial t \partial x} 
&= A(t,x) \frac{\partial u}{\partial x} + B(t,x)
\frac{\partial u}{\partial t} + C(t,x)u + f(t,x) \\
&\quad + \sum^k_{i=1}M_i(t,x)\frac{\partial u(t_i+0,x)}{\partial x} 
 + \sum^k_{i=1}L_i(t,x)\frac{\partial u(t_i+0,x)}{\partial t} \\
&\quad +  \sum ^k_{i=1}K_i(t,x)u(t_i+0,x), \quad t \neq t_i, 
\end{aligned} \label{e1} \\
u(t,0) = \psi (t),  \quad t \in [0,T], \label{e2} \\
u(0,x)=  u(T,x) + \varphi_0 (x), \quad x\in [0,\omega], \label{e3} \\ 
\frac{\partial u(t_i +0,x)}{\partial x} - \frac{\partial u(t_i-0,x)}{\partial x} 
= \varphi_i(x), \quad i = \overline{1,k}, \label{e4}
\end{gather}
where the domain is ${\Omega} = [0, T] \times [0, \omega] $,
$ u=\mathrm{col}(u_1, u_2, \dots , u_n)$, the $ (n\times n)$ matrices
 $ A(t,x)$, $B(t,x)$, $C(t,x)$, 
 and the $n$-vector function $f(t,x)$ are continuous on $\Omega$, 
the  $(n\times n)$ matrices $ M_i(t,x)$, $L_i(t,x)$,
$K_i(t,x)$, $i=\overline{1,k}$ are continuous on $\Omega$, 
the  $n$-vector function  $\psi (t)$ is continuously differentiable on $[0,T]$, 
the $ n$-vector function $\varphi _0(x)$ is continuously  differentiable
 on $[0,\omega]$, the $ n$-vector functions $\varphi _j(x)$,
$j=\overline{1,k}$ are continuous on $[0,\omega]$,
 $0 < t_1 < t_2 < \dots  < t_k < T$, 
$\|u(t,x)\| =\max _{i=\overline{1,n}}|u_i(t,x)|$.

The data of this problem satisfy the compatibility condition
$\psi(0)=  \psi(T) + \varphi_0 (0)$. 
We denote $t_0 =0$, $t_{k+1}=T$, $\Omega_r= [t_{r-1}, t_r)\times[0,\omega]$, 
$ r=\overline{1,k+1}$,
i.e. $\Omega = \cup ^{k+1}_{r=1}\Omega_r$.

Let  $PC(\Omega, \{t_i\}^k_{i=1},R^n)$ be the space of vector-functions 
$u(t,x)$ piecewise continuous  on $\Omega$  with possible discontinuities 
on the lines $t=t_i$, $i=\overline{1,k}$, and let the norm be
\[
 \|u\|_1 =\max _{i=\overline{1,k+1}}\sup _{(t,x)\in \Omega_i}
\|u(t,x)\|.
\]
A function $u(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n)$ with partial derivatives 
\begin{gather*}
\frac{\partial u(t,x)}{\partial x}\in PC(\Omega, \{t_i\}^k_{i=1},R^n),  \quad
\frac{\partial u(t,x)}{\partial t}\in PC(\Omega, \{t_i\}^k_{i=1},R^n), \\
\frac{\partial ^2 u(t,x)}{\partial t \partial x}\in PC(\Omega, \{t_i\}^k_{i=1},R^n)
\end{gather*}
is called a solution of problem \eqref{e1}--\eqref{e4}  
if it satisfies system \eqref{e1} for all $(t,x)\in \Omega$ (except the
lines $t=t_i$, $ i = \overline{1,k}$), the boundary conditions 
\eqref{e2} and \eqref{e3}, and the conditions of
impulsive effect at fixed times \eqref{e4}.

Periodic boundary-value problems for a system of differential equations
 hyperbolic type have been researched by many authors, see for example 
\cite{5, 18, 23, 29, 31}.
Some classes of boundary-value problems for ordinary and partial differential 
equations with impulse effects were studied in \cite{1, 2, 11, 12, 15, 17, 25, 30}. 
For the prevoius decades the theory of loaded equations has been developed 
intensively in works of many mathematicians, see \cite{20, 21, 22, 24, 26, 27,28}.
A review of results on boundary-value problems for the loaded differential equations 
of various classes can be found in \cite{22, 28}. 
This periodic problem for the system of loaded hyperbolic equations second-order 
with impulse effects is  investigated first time here.

For  investigating and solving the problem, we use the method of introducing 
functional parameters \cite{3}--\cite{16}. This method is a generalization of 
parametrization method \cite{19} for partial differential equations. 
By introducing new unknown functions the periodic problem \eqref{e1}--\eqref{e4} 
is reduced to an equivalent problem  consisting a family of periodic 
boundary-value problems  for system  of loaded ordinary differential
equations with impulse effects and integral relations. 
Then,  we establish the relationship between
the unique solvability of periodic problem and the unique solvability of the 
family of periodic boundary-value problems. Sufficient conditions for
the existence of a unique solution to the family of periodic boundary value 
problems are obtained by the method of introduction of functional parameters. 
Also we present an algorithms for finding the approximate solutions of
these problems. The  results obtained are applied to a periodic problem 
for the system of loaded hyperbolic equations with impulse effects.

\section{Reduction of problem \eqref{e1}--\eqref{e4} to an equivalent problem, and 
algorithm}

In this section we introduce the unknown functions 
\[
 v(t,x)  = \frac{\partial u(t,x)}{\partial x}, \quad
 w(t,x)  = \frac{\partial u(t,x)}{\partial t}
\]
to  reduce problem \eqref{e1}--\eqref{e4} to the equivalent problem 
\begin{gather}
\begin{aligned}
\frac{\partial v}{\partial t}
& = A(t,x)v + \sum ^k_{i=1}M_i(t,x)v(t_i+0,x) \\
&\quad + F(t,x, u(t,x), w(t,x)), \quad  t \neq t_i, \quad x\in[0,\omega], 
\end{aligned} \label{e5} \\
v(0,x)=  v(T,x) + \varphi'_0 (x), \quad x\in [0,\omega], \label{e6} \\
v(t_i +0,x) - v(t_i-0,x)= \varphi_i(x), \quad i = \overline{1,k}, \label{e7}\\
u(t,x) = \psi (t) + \int ^x_0 v(t,\xi)d\xi, \quad
w(t,x) = \dot{\psi} (t) + \int ^x_0 \frac{\partial v(t,\xi)}{\partial t}d\xi,
  \label{e8}
\end{gather}
where  
\begin{align*}
F(t,x, u(t,x), w(t,x)) 
&= f(t,x) + B(t,x) w(t,x) 
 + \sum ^k_{i=1}L_i(t,x)w(t_i+0,x) \\
&\quad + C(t,x)u(t,x) + \sum ^k_{i=1}K_i(t,x)u(t_i+0,x).
\end{align*}

In problem \eqref{e5}--\eqref{e8}, the condition $ u(t,0) = \psi (t)$ 
is taken into account in relations \eqref{e8}.
Condition \eqref{e6} is equivalent to condition \eqref{e3} 
and the compatibility condition.

A triple of functions $\{v(t,x), u(t,x), w(t,x)\}$ piecewise continuous on 
$\Omega$ is called a solution of problem \eqref{e5}--\eqref{e8} if the 
function has a piecewise continuous derivative with respect to
$t$ on $\Omega$ and satisfies a one-parameter family of periodic problems 
for loaded ordinary differential equations with impulse effects 
\eqref{e5}--\eqref{e7}, where the functions $u(t,x)$ and $w(t,x)$ are
connected with $v(t,x)$ and $\frac{\partial v(t,x)}{\partial t}$ 
by the integral relations \eqref{e8}.

Problem \eqref{e5}--\eqref{e7} at fixed  $u (t, x)$ and $w (t, x)$ 
represents a family of periodic  problems for
a system of loaded ordinary differential equations with impulse effects.   
The relations  \eqref{e8}  allows to determine unknown functions  
$u (t, x) $ and $w (t, x) $  by $ v (t, x) $ and its derivative
 $ { \frac {\partial v (t, x)} {\partial t}}$. The solution of problem 
\eqref{e5}--\eqref{e8} is unknown functions $ v (t, x)$, $u (t, x)$ and $w (t, x) $
which will be find by iterative processes  based on the following algorithm.
\smallskip

\noindent\textbf{Step 0.}
 Solving the family of periodic problems with impulse effects 
\eqref{e5}--\eqref{e7} under  the
assumptions that $ u(t,x) = \psi(t)$, $w (t,x)= \dot{\psi}(t)$ 
on the right-hand side of \eqref{e5}, we
find the function   $ v^{(0)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n)$.  
From integral conditions \eqref{e8} at $ v(t,x) = v^{(0)}(t,x)$, 
$\frac{\partial v(t,x)}{\partial t} = \frac{\partial
v^{(0)}(t,x)}{\partial t}$ we determine 
\[
u^{(0)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n), \quad 
w^{(0)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n).
\]
\smallskip

\noindent\textbf{Step 1.}
 Solving the family of periodic problems with impulse effects 
\eqref{e5}--\eqref{e7} under  the
assumptions that $ u(t,x) = u^{(0)}(t,x)$,  $w (t,x)= w^{(0)}(t,x)$ 
on the right-hand side of \eqref{e5}, we find the function 
$ v^{(1)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n)$. From integral
conditions \eqref{e8} at $ v(t,x) = v^{(1)}(t,x)$, 
$\frac{\partial v(t,x)}{\partial t} = \frac{\partial
v^{(1)}(t,x)}{\partial t}$ we determine 
\[ 
u^{(1)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n), \quad 
w^{(1)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n); 
\]
and so on.
\smallskip

\noindent\textbf{Step $m$.}
 Solving the family of periodic problems with impulse effects 
\eqref{e5}--\eqref{e7} under  the
assumptions that $ u(t,x) = u^{(m-1)}(t,x)$, 
$w (t,x)= w^{(m-1)}(t,x)$ on the right-hand side of
\eqref{e5}, we find the function 
$ v^{(m)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n)$. From integral
conditions \eqref{e8} at   $ v(t,x) = v^{(m)}(t,x)$,
 $\frac{\partial v(t,x)}{\partial t} = \frac{\partial
v^{(m)}(t,x)}{\partial t}$  we determine  
\[
u^{(m)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n), \quad 
w^{(m)}(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n), \quad m = 1,2, \dots .
\]

The highlight of the  proposed algorithm is the solvability 
of the family of periodic problems for the system of loaded ordinary 
differential equations with impulse effects
\eqref{e5}--\eqref{e7} for fixed $ u(t,x) $, $ w(t,x)$.
This question will be investigated in the next section. 
Conditions for the convergence of the
algorithm  will be given in Section 4.

\section{Family of periodic problems for loaded ordinary differential equations
 with impulse effects}

For fixed $u(t,x)$ and $w(t,x)$, problem \eqref{e5}--\eqref{e7} requires 
special investigations. Therefore, we
consider a family of periodic problems for the following system of 
loaded differential equations
with impulse effects 
\begin{gather}
\frac{\partial v}{\partial t} = A(t,x)v 
+ \sum ^k_{i=1}M_i(t,x)v(t_i+0,x)+ F(t,x), \quad t \neq t_i, \;
 x\in[0,\omega],\label{e9} \\
v(0,x)=  v(T,x) + \varphi'_0 (x), \quad x\in [0,\omega], \label{e10} \\
v(t_i +0,x) - v(t_i-0,x)= \varphi_i(x), \quad i = \overline{1,k}, \label{e11}
\end{gather} 
where $v=\mathrm{col}(v_1, v_2, \dots ,v_n)$, the $n$-vector function 
$F(t,x)$ is continuous on $\Omega$,  $t_0 < t_1 < t_2 < \dots  < t_k
< t_{k+1}$.

A function $v(t,x) \in PC(\Omega, \{t_i\}^k_{i=1},R^n)$ with the derivative 
\[
\frac{\partial v(t,x)}{\partial t} \in PC(\Omega, \{t_i\}^k_{i=1},R^n)
\] 
is called a solution to the family of
periodic problems with impulse effects \eqref{e9}--\eqref{e11}
if it satisfies system of loaded ordinary
differential equations \eqref{e9} except the lines $t=t_i$, $ i = \overline{1,k}$,
 the periodic condition
\eqref{e10}, and the conditions of impulse effects \eqref{e11} for all 
$x \in [0,\omega]$.

For fixed $x\in [0,\omega]$, problem \eqref{e9}--\eqref{e11} is a periodic 
problem for a system of loaded ordinary differential equations with impulse effects. 
Changing the variable $x$ over $[0,\omega]$,
we get a family of periodic problems for loaded ordinary differential equations 
with impulse effects. The boundary value problems for ordinary differential 
equations with impulse effects were studied by numerous authors. Conditions for 
the existence of solutions of the analyzed problems
were obtained and expressed in different ways by using various  methods and 
approaches \cite{1, 2, 17, 25, 30}. The different problems for loaded differential 
equations were studied by many authors. For
a survey, bibliography, and detailed analysis, see \cite{22, 28}.

The parametrization method is used for the investigation and solving of the 
family of periodic problems with impulse effects. The idea of this method is 
to introduce additional parameters as values of the required function on the 
lines of partition of the domain with respect to the
variable $t$. By the change of unknown functions, the original problem is 
reduced to an equivalent problem with functional parameters. 
The properties of these parameters inherit the properties of
the solutions.

In this section, the partition of the domain $\Omega$ is non-uniform and 
the additional parameters are introduced as values of the required function 
on the lines $t=t_i$, $ i = \overline{0,k}$.

By using the straight lines $t=t_i$, $ i = \overline{1,k}$, we split the domain
 $\Omega$ into the subdomains $\Omega_i$,  $ i = \overline{1,k+1}$. 
Let $v_r(t,x)$ be the restriction of the function
$v(t,x)$ to $\Omega_r$, $ r = \overline{1,k+1}$. 
We introduce the parameters $\lambda_r(x) =v_r(t_{r-1},x)$, 
$ r = \overline{1,k+1}$. By the change of the unknown function
\[
v_r(t,x) = \widetilde{v}_r(t,x) + \lambda_r(x), \quad 
(t,x) \in \Omega_r, \quad r =\overline{1,k+1},
\]
we reduce problem \eqref{e9}--\eqref{e11} to the following equivalent
 periodic problem with parameters: 
\begin{gather}
\begin{aligned}
\frac{\partial \widetilde{v}_r}{\partial t} 
&= A(t,x)\widetilde{v}_r + A(t,x) \lambda_r(x) 
+ \sum ^k_{i=1}M_i(t,x)\lambda_{i+1}(x) \\
&\quad + F(t,x), \quad (t,x)
\in \Omega_r, \; r= \overline{1, k+1}, 
\end{aligned} \label{e12} \\
\widetilde{v}_r(t_{r-1},x) = 0, \quad 
r=\overline{1, k+1}, \quad x\in [0,\omega], \label{e13}\\
\lambda_1(x) = \lim  _{t\to T-0} \widetilde{v}_{k+1}(t,x) + \lambda_{k+1}(x) 
+ \varphi'_0 (x), \quad x\in [0,\omega], \label{e14} \\
\lambda_{i+1}(x) - \lim  _{t\to t_i-0} \widetilde{v}_{i}(t,x) -  \lambda_{i}(x) 
= \varphi_i(x), \quad i = \overline{1,k}. \label{e15}
\end{gather}
The solution of problem \eqref{e12}--\eqref{e15} is
obtained as a system of pairs $(\lambda(x), \widetilde{v}([t],x))$ with 
elements $\lambda(x) = (\lambda_1(x), \lambda_2(x),\dots ,\lambda_{k+1}(x))'$ 
and 
\[
 \widetilde{v}([t],x) = (\widetilde{v}_1(t,x), \widetilde{v}_2(t,x), \dots , 
\widetilde{v}_{k+1}(t,x))',
\]
 where the functions
 $\widetilde{v}_r(t,x)$ are continuous on $\Omega_r$ together with their partial 
derivatives $\frac{\partial \widetilde{v}_r(t,x)}{\partial t}$ on $\Omega_r$,
 $r= \overline{1, k+1}$, have a finite left limit
 $\lim  _{t\to t_r-0} \widetilde{v}_{r}(t,x)$, 
$r= \overline{1, k+1}$, and
satisfy, for $\lambda_r(x) = \lambda^{\ast}_r(x)$, the system of differential 
equations \eqref{e12} and
conditions \eqref{e13}--\eqref{e15}.

Unlike problem \eqref{e9}--\eqref{e11}, here, we get the initial conditions 
\eqref{e13} as values of the unknown
function on the lines $ t=t_{r-1}$, $r= \overline{1, k+1}$. 
For fixed $\lambda_r(x)$, $r=\overline{1, k+1}$, there exist the solutions 
of the Cauchy problem on $\Omega_r$ for system \eqref{e12}
with condition \eqref{e13}.

The Cauchy problem \eqref{e12}, \eqref{e13} is equivalent to the Volterra 
integral equation of second kind
\begin{equation}
\begin{aligned}
\widetilde{v}_{r}(t,x)
&= \int ^t_{t_{r-1}} A(\tau,x)\widetilde{v}_{r}(\tau,x)d\tau + \int
^t_{t_{r-1}} A(\tau,x)d\tau \lambda_r(x) \\
&\quad + \sum ^k_{i=1}  \int ^t_{t_{r-1}} M_i(\tau,x)d\tau \lambda_{i+1}(x)
+ \int ^t_{t_{r-1}}  F(\tau,x)d\tau.
\end{aligned} \label{e16}
\end{equation}
Passing on the right-hand side of \eqref{e16} to the limit as $t \to t_r -0$, 
we obtain $\lim _{t\to t_r-0} \widetilde{v}_{r}(t,x)$, 
$r= \overline{1, k+1}$, $x \in [0,\omega]$. Substituting
these limits in \eqref{e14} and \eqref{e15}, we obtain the following system
 of $(k+1)$ functional equations for
the unknown  vector functions $\lambda_r(x)$, $r= \overline{1, k+1}$: 
\begin{gather}
\begin{aligned}
&\lambda_1(x) - \int^T_{t_k} A(\tau,x)d\tau \lambda_{k+1}(x)
  - \sum ^k_{i=1} \int ^T_{t_k} M_i(\tau,x)d\tau \lambda_{i+1}(x)
  -  \lambda_{k+1}(x) \\
& = \varphi'_0  (x) + \int ^T_{t_k}  F(\tau,x)d\tau 
 + \int^T_{t_k} A(\tau,x)\widetilde{v}_{k+1}(\tau,x)d\tau,
\end{aligned}  \label{e17}\\
\begin{aligned}
&\lambda_{i+1}(x) -  \Big [I + \int ^{t_i}_{t_{i-1}} A(\tau,x)d\tau \Big] \lambda_i(x)
  - \sum ^k_{j=1}  \int ^{t_i}_{t_{i-1}} M_j(\tau,x)d\tau \lambda_{j+1}(x) \\
& =  \varphi_i(x)  + \int ^{t_i}_{t_{i-1}} F(\tau,x)d\tau
  +  \int ^{t_i}_{t_{i-1}} A(\tau,x)\widetilde{v}_{i}(\tau,x)d\tau,
\quad i = \overline{1,k}. 
\end{aligned}\label{e18}
\end{gather}
Now we construct the algorithm for finding the approximate
solution to problem \eqref{e9}--\eqref{e11}. 
The parametrization method splits the process of determination of
unknown functions into two stages: 
(i) determination of the introduced 
functional parameters $\lambda_r(x)$ from the system of equations 
\eqref{e17}, \eqref{e18};
 (ii) determination of the unknown functions
$\widetilde{v}_{r}(t,x)$ from the integral equations \eqref{e16}.

We denote by $Q(x)$  the $(n(k+1)\times n(k+1))$ matrix corresponding 
to the left-hand side of system equations \eqref{e17}, \eqref{e18} 
consists of the coefficients at the parameters
$\lambda_r(x)$, $r=\overline{1,k+1}$.

The following assertion gives the conditions of the unique solvability of
 problem \eqref{e9}--\eqref{e11} and
convergence this algorithm.

\begin{theorem} \label{thm1} 
Suppose the $n(k+1)\times n(k+1) $ matrix $Q(x)$ is invertible for all 
$ x\in [0,\omega ]$ and that the following inequalities are true: 
\begin{itemize}
\item[(a)] $\|[Q(x)]^{-1}\|\leq \gamma (x)$,
where $\gamma (x)$ is a positive  function continuous with respect to 
$x\in [0,\omega]$; 
\item[(b)]
\[
 q(x)=  \gamma (x) \Bigl\{ 1 + \sum _{i=1}^k \max  _{t\in [0,T]}\|M_i(t,x)\|h  
\Bigr\} \Big[e^{\alpha (x)h} - 1 - \alpha (x)h \Big] \leq \chi < 1,
\]
where $\alpha (x) = \max  _{t\in [0,T]}\|A(t,x)\|$, 
$ h = \max _{r=\overline{1,k+1}} (t_r-t_{r-1})$, $\chi$ is a constant.
\end{itemize}
Then the family of periodic problems with impulse effects 
\eqref{e9}--\eqref{e11} has a unique solution
$v^{\ast}(t,x)$.
\end{theorem}

The proof of Theorem \ref{thm1} is analogous to scheme of the proof of
 \cite[Theorem 2]{12}; we omit it.

\section{Main results}

The following statement gives the conditions of the feasibility and the 
convergence of the proposed algorithm, simultaneously guaranteeing 
the existence of unique solution of the equivalent
problem \eqref{e5}--\eqref{e8}.

\begin{theorem} \label{thm2} 
Suppose the $n(k+1)\times n(k+1) $ matrix $Q(x)$ is invertible for all $ x\in
[0,\omega ]$ and that inequalities (a) and (b) of Theorem \ref{thm1} hold.
Then   problem \eqref{e5}--\eqref{e8} is uniquely solvable.
\end{theorem}

The proof of Theorem \ref{thm2} is analogous to the proof of \cite[Theorems 1]{12}, 
taking into account the features of the considered problem. 
From the equivalence  of the problems \eqref{e5}--\eqref{e8} and
\eqref{e1}--\eqref{e4}, the assertion follows.


\begin{theorem} \label{thm3} 
Suppose the $n(k+1)\times n(k+1) $ matrix $Q(x)$ is invertible for all $ x\in
[0,\omega ]$ and that inequalities (a) and (b) of Theorem \ref{thm1} hold.
Then the periodic problem for system of loaded hyperbolic equations 
with impulse effects \eqref{e1}--\eqref{e4}
is uniquely solvable. 
\end{theorem}

\subsection*{Acknowledgements}
 This research was financially supported by  grants from the Ministry of 
Science and Education of Kazakhstan (Grant No. 0822/$\Gamma \Phi $4
and No. AP05131220).


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\end{thebibliography}

\end{document}