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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 205, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/205\hfil Uniformly finally bounded solutions]
{Uniformly finally bounded solutions to systems of differential
equations with variable structure and impulses}

\author[K. G. Dishlieva, A. B. Dishliev\hfil EJDE-2014/205\hfilneg]
{Katya G. Dishlieva, Angel B. Dishliev }  % in alphabetical order

\address{Katya G. Dishlieva \newline
Faculty of Applied Mathematics and Informatics\\
Technical University of Sofia, Bulgaria}
\email{kgd@tu-sofia.bg}

\address{Angel B. Dishliev \newline
Department of Mathematics\\
University of Chemical Technology and Metallurgy - Sofia, Bulgaria}
\email{dishliev@uctm.edu}

\thanks{Submitted July 17, 2013. Published October 2, 2014.}
\subjclass[2000]{34A37, 65L05}
\keywords{ Variable structure; impulses; switching functions;
\hfill\break\indent   uniformly finally bounded solutions}

\begin{abstract}
 We study nonlinear non-autonomous systems of ordinary differential equations
 with variable structure and impulses. The consecutive changes on right-hand
 sides of this system and the impulsive effects on the solution of the
 corresponding initial problem take place simultaneously at the moment
 when the solution cancels the switching functions.
 We find sufficient conditions for the uniform final boundedness of solutions.
 These results are obtained using a suitable variation of the Lyapunov second method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Gurgula and Perestyuk are the first who apply the Lyapunov second method 
to study the solutions of impulsive equations. In \cite{g2}, they use the ``classical''
continuous Lyapunov functions to study the stability of the ``zero solution''� 
of such equations. Discontinuous Lyapunov functions were introduced by 
 Bainov and  Simeonov \cite{b1}. The solutions qualities of one special class 
differential equations with variable structure and impulses are studied 
for the first time in \cite{m1}. The equations, studied in this paper were
introduced in \cite{c2,d1}. There are numerous applications of equations with 
impulsive effects. We will point the articles 
\cite{a1,a3,b2,d1,d2,n1,n2,m2,p2,s1,s2,w1}.
The applications of differential equations with variable structure are 
mainly in the control theory and engineering practice: 
\cite{a2,d1,f1,g1,h1,p1}.
The main object of this article is to study the following initial problem for 
nonlinear non-autonomous systems of ordinary differential equations 
with variable structure and impulses at non-fixed moments:
\begin{gather}
\frac{dx}{dt}=f_i(t,x),\quad\text{if }\varphi_i\big(x(t)\big)\neq0,\; 
 t_{i-1}<t< t_i; \label{e1}\\
\varphi_i\big(x(t_i)\big)=0,\quad i=1,2,\dots ;\label{e2}\\
x(t_i+0)=x(t_i)+I_i\big(x(t_i)\big);\label{e3}\\
x(t_0)=x_0, \label{e4}
\end{gather}
where: $f_i:\mathbb{R}^+\times \mathbb{R}^n\to\mathbb{R}^n$,
$\varphi_i:\mathbb{R}^n\to\mathbb{R}$,
$I_i:\mathbb{R}^n\to\mathbb{R}^n$,
the initial point $(t_0,x_0)\in \mathbb{R}^+\times \mathbb{R}^n$.

Denote the solution of problem \eqref{e1}--\eqref{e4}
 by  $x(t;t_0,x_0)$ which is left-continuous at each point in the domain. 
In general, this solution has a finite jump discontinuity on the right 
at the moments $t_1,t_2,\dots $. They are named moments of switching. 
The functions $I_1,I_2,\dots $ and $\varphi_1,\varphi_2,\dots $ are named 
impulsive and switching functions, respectively.

We use the following notation:
\begin{itemize}
\item $\Phi_i\big\{x\in \mathbb{R}^n: \varphi_i(x)=0\big\}$, $i=1,2,\dots,$ 
are called switching hypersurfaces;

\item $Id$ is an identity in $\mathbb{R}^n$;

\item $\gamma(t_0,x_0)=\big\{x(t:t_0,x_0),\; t_0\leq t\leq T\big\}$ 
is a trajectory of problem \eqref{e1}--\eqref{e4} for $t_0\leq t\leq T$;

\item $B_{\delta}(x_0)=\big\{x\in \mathbb{R}^n: \|x-x_0\|<\delta\big\}$,
 where $x_0\in \mathbb{R}^n$ and $\delta={\rm const}>0$;

\item $B_{\delta}^c(x_0)=\big\{x\in \mathbb{R}^n: 
\|x-x_0\|\geq\delta\big\}=\mathbb{R}^n\backslash B_{\delta}(x_0)$.
\end{itemize}

\begin{definition} \label{def1} \rm
We say that the solution of system of differential equations with variable 
structure and impulses  \eqref{e1}--\eqref{e3} is:
\begin{itemize}
\item  bounded, if $(\forall t_0\in \mathbb{R}^+)(\forall\alpha=const>0)
\big(\exists\beta=\beta(t_0,\alpha)>0\big)$ such that 
$$
\big(\forall x_0\in B_{\alpha}(x_0)\big) \Rightarrow
\big\|x(t;t_0,x_0)\big\|<\beta,\ t\geq t_0;
$$

\item  uniformly bounded, if
$(\forall t_0\in \mathbb{R}^+)(\forall\alpha={\rm const}>0)
\big(\exists\beta=\beta(\alpha)>0\big)$ such that
$$\big(\forall x_0\in B_{\alpha}(x_0)\big)
\Rightarrow\big\|x(t;t_0,x_0)\big\|<\beta,\ t\geq t_0;
$$

\item  quasi-uniformly finally bounded, if 
$(\exists\beta=const>0): (\forall\alpha=const>0)\big
(\exists T=T(\alpha)>0\big)$ such that
$$
 \big(\forall x_0\in B_{\alpha}(x_0)\big)
\Rightarrow\big\|x(t;t_0,x_0)\big\|<\beta,\ t\geq t_0+T;
$$

\item  uniformly finally bounded, if the solutions are uniformly 
bounded and quasi-uniformly finally bounded.
\end{itemize}
\end{definition}

\begin{definition} \label{def2} \rm
We say that a sequence of scalar piecewise continuous Lyapunov  functions
$$
\big\{V_i,\ V_i:\mathbb{R}^+\times \mathbb{R}^n\to\mathbb{R}^+,\;
 i=1,2,\dots \big\},
$$
corresponds to the system of differential equations with variable structure 
and impulses \eqref{e1}--\eqref{e3} if:
\begin{itemize}
\item[(1)] $V_i\in C[\mathbb{R}^+\times \mathbb{R}^n\backslash\Phi_i,\mathbb{R}^+]$,
$i=1,2,\dots$;
\item[(2)] $V_i(t,0)=0,\ t\in \mathbb{R}^+,\ i=1,2,\dots$;
\item[(3)] For every point $(t,x_{\Phi_i})\in \mathbb{R}^+\times\Phi_i$  and 
for each  $i=1,2,\dots $, there exist the limits:
\begin{gather*}
\lim_{x\to x_{\Phi_i},\Phi_i(x)<0,}V_i(t,x)=V_i(t,x_{\Phi_i}-0)=V_i(t,x_{\Phi_i}),\\
\lim_{x\to x_{\Phi_i},\Phi_i(x)>0,}V_i(t,x)=V_i(t,x_{\Phi_i}+0).
\end{gather*}
\end{itemize}
\end{definition}

We note that in general, 
$$
V_i(t,x_{\Phi_i})=V_i(t,x_{\Phi_i}-0)\neq V_i(t,x_{\Phi_i}+0).
$$

\begin{definition} \label{def3}\rm
Let $\{V_i:\; i=1,2,\dots \}$ be a sequence of scalar piecewise continuous 
Lyapunov  functions.
Then for every point $(t,x)\in \mathbb{R}^+\times \mathbb{R}^n\backslash \Phi_i$ 
and for each  $i=1,2,\dots $, we define the derivative of   
$V_i$ at point  $(t,x)$ with respect to system \eqref{e1}--\eqref{e3}, as  follows:
$$
\dot{V}_i(t,x)=\dot{V}_{i,\eqref{e1}-\eqref{e3}}(t,x)
=\lim_{h\to+0}\frac{1}{h}\big(V_i(t+h,x+hf_i(t,x))-V_i(t,x)\big).
$$
\end{definition}

\begin{remark} \label{rmk1}\rm
It can be shown that for $i=1,2,\dots $  and for every point 
$(t,x)=\big(t,x(t;t_0,x_0)\big)\in[t_0,\infty)\times(\mathbb{R}^n\backslash \Phi_i)$, 
it holds
\begin{align*}
\dot{V}(t,x)
&= D_{\eqref{e1}-\eqref{e3}}^+V_i\big(t,x(t;t_0,x_0)\big) \\
&= \limsup_{h\to+0}\frac{1}{h}\Big(V_i\big(t+h,x(t+h;t_0,x_0)\big)
 -V_i\big(t,x(t;t_0,x_0)\big)\Big).
\end{align*}
In other words, the derivative of each Lyapunov function  $V_i,\ i=1,2,\dots $, 
at every point  $(t,x)=\big(t,x(t;t_0,x_0)\big)$ with respect to the system 
of differential equations \eqref{e1}--\eqref{e3} coincides with the upper 
right Dini derivative at the same point with respect to the solution of 
system under consideration.
\end{remark}

We shall use the following class of scalar functions
$$
K=\big\{a\in C[\mathbb{R}^+,\mathbb{R}^+],\ a\uparrow\uparrow,\ a(0)=0\big\},$$
i.e. $a$ is a strictly monotonically increasing function and $a(0)=0$.
We use the following conditions:
\begin{itemize}
\item[(H1)] The functions $f_i$ belong to 
 $C[\mathbb{R}^+\times \mathbb{R}^n,\mathbb{R}^n]$, $i=1,2,\dots$;

\item[(H2)] There exist the constants $C_{f_i}>0$  such that
$$
\big(\forall(t,x)\in \mathbb{R}^+\times \mathbb{R}^n\big)
\Rightarrow\|f_i(t,x)\|\leq C_{f_i},\ i=1,2,\dots ;
$$

\item[(H3)] The functions $\varphi_i$ belong to $C^1[\mathbb{R}^n,\mathbb{R}]$,
$i=1,2,\dots$;

\item[(H4)] There exist  constants $C_{\operatorname{grad}\varphi_i}>0$  such that
$$
(\forall x\in \mathbb{R}^n)\Rightarrow\big\|\operatorname{grad}\varphi_i(x)\big\|
\leq C_{\operatorname{grad}\varphi_i},\quad i=1,2,\dots ;
$$

\item[(H5)] The functions $I_i$ belong to 
$C[\mathbb{R}^n,\mathbb{R}^n]$, $i=1,2,\dots$;

\item[(H6)] There exist the constants $C_{\varphi_{i+1}(Id+I_i)}>0$  such that
$$
(\forall x\in \Phi_i)\Rightarrow\big|\varphi_{i+1}\big((Id+I_i)(x)\big)\big|
=\big|\varphi_{i+1}\big(x+I_i(x)\big)\big|
\geq C_{\varphi_{i+1}(Id+I_i)},\ i=1,2,\dots;
$$

\item[(H7)] The next inequalities are valid
$$
\varphi_i\big((Id+I_{i-1})(x)\big)\cdot\big\langle \operatorname{grad}\varphi_i(x),f_i(t,x)
\big\rangle<0,\quad (t,x)\in \mathbb{R}^+\times \mathbb{R}^n,\; i=1,2,\dots;
$$

\item[(H8)] The series 
$$
\sum_{i=1}^{\infty}\frac{C_{\varphi_i(Id+I_{i-1})}}{C_{\operatorname{grad}\varphi_i}\cdot C_{f_i}}
$$  
diverges;

\item[(H9)] There exist the constants $C_{\langle \operatorname{grad}\varphi_i,f_i\rangle}>0$  
such that
$$
\big(\forall(t,x)\in \mathbb{R}^+\times \mathbb{R}^n\big)\Rightarrow\big|
\big\langle \operatorname{grad}\varphi_i(x),f_i(t,x)\big\rangle\big|
\geq C_{\langle \operatorname{grad}\varphi_i,f_i\rangle},\ i=1,2,\dots ;
$$
\item[(H10)] For every point $(t_0,x_0)\in \mathbb{R}^+\times \mathbb{R}^n$  
and for each $i=1,2,\dots $, the solution of initial problem
$$
\frac{dx}{dt}=f_i(t,x),\quad x(t_0)=x_0
$$
exists and it is unique for  $t\geq t_0$;

\item[(H11)] There exists constant $C_I>0$  such that
$$
(\forall x\in \mathbb{R}^n)\Rightarrow\big\|I_i(x)\big\|\leq C_I,\quad
 i=1,2,\dots .
$$
\end{itemize}

\section{Preliminary results}

\begin{theorem}[\cite{d3}] \label{thm1} 
Assume  conditions {\rm (H1)--(H7)}. Then
\begin{itemize}
\item[(1)] If the trajectory $\gamma(t_0,x_0)$  of problem \eqref{e1}--\eqref{e4}
 meets consecutively the switching hypersurfaces $\Phi_i$  and $\Phi_{i+1}$,
 then the following estimate is valid for the switching moments $t_i$  and  
 $t_{i+1}$:
$$
t_{i+1}-t_i\geq\frac{C_{\varphi_{i+1}(Id+I_i)}}{C_{\operatorname{grad}\varphi_{i+1}}
 C_{f_{i+1}}},\quad i=1,2,\dots ;
$$
\item[(2)] If the trajectory $\gamma(t_0,x_0)$  meets all the switching 
 hypersurfaces $\Phi_i$, $i=1,2,\dots $, and condition (H8) is satisfied, 
 then the switching moments increase indefinitely; i.e.  
$\lim_{i\to\infty}t_i=\infty$ is satisfied.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk2} \rm
If the following inequalities are satisfied
\begin{gather*}
0<C_{f_1}=C_{f_2}=\dots ;\\
0<C_{\operatorname{grad}\varphi_1}=C_{\operatorname{grad}\varphi_2}=\dots ;\\
0<C_{\varphi_2(Id+I_1)}=C_{\varphi_3(Id+I_2)}=\dots ,
\end{gather*}
then it is easy to establish that
$$
\sum_{j=1}^{\infty}\frac{C_{\varphi_j(Id+I_{j-1})}}{C_{\operatorname{grad}\varphi_j}
 C_{f_j}}=\infty.
$$
Therefore, condition (H8) follows by the equalities above. 
If these equalities and the conditions (H1)--(H7) are satisfied, 
and using the previous theorem, we deduce that the switching moments 
increase indefinitely.
\end{remark}

\begin{theorem}[\cite{d3}] \label{thm2} 
 Assume  conditions {\rm (H1), (H3), (H5), (H7), (H9), (H10)}.
Then the trajectory of problem \eqref{e1}--\eqref{e4} meets every one
of the hypersurfaces $\Phi_i$, $i=1,2,\dots $.
\end{theorem}

Using Theorem \ref{thm1} and condition (H10), we obtain the next theorem.

\begin{theorem} \label{thm3}
Assume conditions {\rm (H1)--(H8), (H10)}.
Then the solution of problem \eqref{e1}--\eqref{e4} exists and it is unique
for  $t\geq t_0$.
\end{theorem}

We introduce a piecewise differentiable function 
$V:[t_0,\infty)\to\mathbb{R}^+$ with points of discontinuity, which coincide 
with the moments of switching $t_1,t_2,\dots $, by 
\begin{equation} \label{e5}
\begin{aligned}
V(t)&=V\big(t,x(t;t_0,x_0)\big)\\
&=\begin{cases}
V_1\big(t,x(t;t_0,x_0)\big),\quad \text{if } t_0\leq t\leq t_1; \\
V_{i+1}\big(t,x(t;t_0,x_0)\big)\\
=V_{i+1}\Big(t,x(t_i+0;t_0,x_0)+\int_{t_i}^{t}f_{i+1}\big(\tau,x(\tau;t_0,x_0)\big)
d\tau\Big)\\
=V_{i+1}\Big(t,x(t_i;t_0,x_0)+I_i\big(x(t_i;t_0,x_0)\big)+\int_{t_i}^{t}f_{i+1}
\big(\tau,x(\tau;t_0,x_0)\big)d\tau\Big),\\
\quad\text{if } t_i<t\leq t_{i+1},\; i=1,2,\dots .
\end{cases}
\end{aligned}
\end{equation}
It is obvious that the function is continuous from the left in its domain.

The next theorem contains sufficient conditions under which function, 
introduced above is monotonically decreasing.

\begin{theorem} \label{thm4} 
Assume that:
\begin{itemize}
\item[(1)] Conditions (H1)--(H10) hold.
\item[(2)] There exists a sequence of scalar piecewise continuous Lyapunov  functions
$$
\big\{V_i;V_i:\mathbb{R}^+\times D\to\mathbb{R}^+,\; i=1,2,\dots \big\},
$$
corresponding to the impulsive system of differential equations 
\eqref{e1}--\eqref{e3}, and constant $\rho>0$  such that
\begin{itemize}
\item[(2.1)] It is satisfied that
$$
V_{i+1}\big(t+0,x+I_i(x)\big)=V_{i+1}\big(t,x+I_i(x)\big)
\leq V_i(t,x),\; (t,x)\in \mathbb{R}^+\times\big(B_{\rho}^c(0)\cap\Phi_i\big),
$$
for $i=1,2,\dots$;

\item[(2.2)] It is satisfied that
$$
\dot{V_i}(t,x)\leq0,\ (t,x)\in \mathbb{R}^+\times\big(B_{\rho}^c(0)
\cap\Phi_i\big),\quad i=1,2,\dots .
$$
\end{itemize}
\end{itemize}
Then function  $V(t)=V\big(t,x(t;t_0,x_0)\big)$, defined in
\eqref{e5}, is monotonically decreasing in every interval  $(t_*,t^*)$,
 for which:  $t_0<t_*$ and
$$
\|x(t;t_0,x_0)\|\geq \rho,\quad t_*<t<t^*.
$$
\end{theorem}

\begin{proof}
 According to Theorem \ref{thm1},  $\lim_{i\to\infty}t_i=\infty$. 
This implies that there are numbers  $k$ and  $p$ such that
$$
t_k\leq t_*<t_{k+1}<\dots <t_{k+p}\leq t^*<t_{k+p+1}.
$$
Within each of the open intervals
\begin{equation} \label{e6}
(t_*,t_{k+1});\; (t_{k+1},t_{k+2});\dots ; (t_{k+p-1},t_{k+p});\; (t_{k+p},t^*)
\end{equation}
in accordance with condition (2.2) of the theorem, it is satisfied that
$$
\frac{d}{dt}V(t)=D_{\eqref{e1}-\eqref{e3}}^+V_i\big(t,x(t;t_0,x_0)\big)
=\dot{V}_{\eqref{e1}-\eqref{e3}}(t,x)=\dot{V}(t,x)\leq 0.
$$
So, we conclude that the function $V=V(t)$  is monotonically decreasing 
in each one of these intervals.

By condition (2.1) for $i=k+1,k+2,\dots ,k+p$, we have
\begin{align*}
V(t_i+0)-V(t_i)
&= V_{i+1}\big(t_i+0,x(t_i+0;t_0,x_0)\big)-V\big(t_i,x(t_i;t_0,x_0)\big)\\
&= V_{i+1}\big(t_i,x(t_i;t_0,x_0)+I_i\big(x(t_i;t_0,x_0)\big)\big)
 -V_i\big(t_i,x(t_i;t_0,x_0)\big)\\
&\leq 0.
\end{align*}
In this way, we obtain that function $V$  is monotonically decreasing in 
the union of intervals \eqref{e6}, i.e. interval  $(t_*,t^*)$.
The proof is complete.
\end{proof}

The next theorem contains sufficient conditions for the function
 $V$, introduced above, to be bounded.

\begin{theorem} \label{thm5}
Assume that:
\begin{itemize}
\item[(1)] Conditions (H1)--(H11) hold; 
\item[(2)] There exists a sequence of scalar piecewise continuous Lyapunov  
functions 
$$
\big\{V_i;V_i:\mathbb{R}^+\times D\to\mathbb{R}^+,\; i=1,2,\dots \big\},
$$
corresponding to the impulsive system of differential equations 
\eqref{e1}--\eqref{e3}, and constant  $\rho>0$ such that 
\begin{itemize}
\item[(2.1)] The functions $a,b\in K$  and constant  $\rho>0$ correspond on 
the upper sequence of Lyapunov  functions satisfy
\begin{itemize}
\item[(2.1.1)] $a(\|x\|)\leq V_i(t,x)\leq b(\|x\|)$,
$(t,x)\in \mathbb{R}^+\times B_{\rho}^c(0)$, $i=1,2,\dots$,
\item[(2.1.2)] $\lim_{u\to\infty}a(u)=\infty$;
\end{itemize}
\item[(2.2)] It is satisfied that
\begin{align*}
V_{i+1}\big(t+0,x+I_i(x)\big)
&= V_{i+1}\big(t,x+I_i(x)\big)\\
&\leq V_{i+1}(t,x),\quad(t,x)\in \mathbb{R}^+\times\big(B_{\rho}^c(0)
\cap\Phi_i\big),\quad i=1,2,\dots ;
\end{align*}

\item[(2.3)] It is satisfied that
$$
\dot{V}_i(t,x)\leq0,\ (t,x)\in \mathbb{R}^+\times\big(B_{\rho}^c(0)
\backslash \Phi_i\big),\quad i=1,2,\dots .
$$
\end{itemize}
\end{itemize}
Then
$$
(\forall\alpha=const\geq\rho+C_i>0)(\forall x_0\in B_{\alpha}(0))
\Rightarrow V(t)=V\big(t,x(t;t_0,x_0)\big)<\alpha(\beta),\ t\geq t_0,
$$
where $\beta=\beta(\alpha)>a^{-1}\big(b(\alpha+C_I)\big)$.
\end{theorem}

\begin{proof}
 Let $\alpha$  be an arbitrary positive constant, satisfying the inequality  
$\alpha\geq\rho+C_I$. From the definition of constant  $\beta$, it follows:
$$
\alpha(\beta)>b(a+C_I)>a(\alpha+C_I)\ and\ \beta>\alpha+C_I.
$$
Let $(t_0,x_0)\in \mathbb{R}^+\times B_{\alpha}(0)$.
Assume that
$$
(\exists t^*>t_0):\ V(t^*)=V\big(t^*,x(t^*;t_0,x_0)\big)\geq\alpha(\beta).
$$
We will note that, it is possible for point $t^*$  to be a switching moment, 
i.e. to exist a number $k\in N$ such that  $t^*=t_k$. From the assumptions 
made and the conditions of theorem, we obtain
$$
b\big(\big\|x(t^*;t_0,x_0)\big\|\big)\geq V(t^*)\geq\alpha(\beta)>b(\alpha),
$$
i.e.
$\big\|x(t^*;t_0,x_0)\big\|>\alpha$.
From the inequality above and having in mind that the solution of problem 
is continuous on the left (including the switching points), we conclude 
that there is a point $t_*$  such that
\begin{gather}
t_0<t_*<t^*; \label{e7}\\
\rho<\alpha-C_I\leq\big\|x(t_*;t_0,x_0)\big\|\leq\alpha; \label{e8}\\
\alpha\leq\big\|x(t;t_0,x_0)\big\|,\ t_*<t\leq t^*. \label{e9}
\end{gather}
Using successively condition (2.2) of this Theorem, inequalities 
\eqref{e7} and \eqref{e8}, Theorem \ref{thm4}, and finally the assumptions made, 
we conclude that
$$
b\big(\big\|x(t_*;t_0,x_0)\big\|\big)\geq V\big(t_*,x(t_*;t_0,x_0)\big)
=V(t_*)\geq V(t^*)>b(\alpha),
$$
from there we have $\big\|x(t_*;t_0,x_0)\big\|>\alpha,$ which contradicts
\eqref{e8}.
The proof is complete.
\end{proof}

\section{Main results}

The main objective of  this section is finding  sufficient 
conditions for the uniform final boundedness of the solutions of system 
\eqref{e1}--\eqref{e3}. The conditions are obtained by using the sequences 
of peacewise continuous scalar Lyapunov functions.

\begin{theorem} \label{thm6}
Assume that {\rm (H1)--(H11)} and conditions (2.1) and (2.2) of 
Theorem \ref{thm5} are 
satisfied. 
Also assume that
$$
\dot{V}_i(t,x)\leq -c(\|x\|),\quad
 (t,x)\in \mathbb{R}^+\times\big(B_{\rho}^c(0)\backslash\Phi_i\big),\;
 i=1,2,\dots ,
$$
where the function  $c\in K$.
Then the solutions of system of differential equations with variable 
structure and impulses \eqref{e1}--\eqref{e3} are uniformly finally bounded.
\end{theorem}

\begin{proof} 
According to Theorem \ref{thm5}, the solution of system  \eqref{e1}--\eqref{e4} 
is uniformly bounded.
More precisely, let:
\begin{gather}
\alpha\geq\rho; \label{e10}\\
\beta=\beta(\alpha)>\max\big\{\alpha+C_I,a^{-1}\big(b(\alpha)\big)\big\}; \label{e11}\\
(t_0,x_0)\in \mathbb{R}^+\times B_{\alpha}(0). \label{e12}
\end{gather}
Then
\begin{equation}
\big\|x(t;t_0,x_0)\big\|<\beta,\quad t\geq t_0. \label{e13}
\end{equation}
The statement, formulated above will be valid if we replace the
inequalities \eqref{e10} and \eqref{e11} with
\begin{gather}
\alpha\geq\rho+C_I; \label{e14}\\
\beta=\beta(\alpha)>a^{-1}\big(b(\alpha+C_I)\big). \label{e15}
\end{gather}
Indeed, inequality \eqref{e10} obviously follows from
\eqref{e14}. Using \eqref{e15}, we obtain
\begin{gather}
\beta>a^{-1}\big(b(\alpha+C_I)\big)>a^{-1}\big(b(\alpha)\big); \label{e16}\\
a(\beta)>b(\alpha+C_I)>a(\alpha+C_I)\Leftrightarrow\beta>\alpha+C_I. \label{e17}
\end{gather}
From these two inequalities,  we obtain \eqref{e11}.
Finally, by \eqref{e12}, \eqref{e14} and \eqref{e15} follows \eqref{e13}.

Let $B=\rho+C_I$. We shall show that
\begin{align*}
&(\forall\alpha=const>\rho+C_I)\big(\exists T=T(\alpha)>0\big):
 \big(\forall x_0\in B_{\alpha}(0)\big) \\
&\Rightarrow\|x(t;t_0,x_0)\|<\beta,\ t\geq t_0+T,
\end{align*}
whence, it follows that the solutions of the system are quasi-uniformly 
finally bounded.

There is $\beta>B$  and therefore $a(\beta)>a(B)$. Let $\nu$  be a natural 
number such that
$$
\nu-1\leq\frac{a(\beta)-a(B)}{a(B)}<\nu.
$$
Denote
$$
\theta_k=t_0+k\frac{a(B)}{c(\rho)},\quad k=0,1,\dots ,\nu.
$$
We will show that regardless of the the choice of initial point 
$x_0\in B_{\alpha}(0)$, the next estimates are valid:
\begin{equation} \label{e18}
V(t)=V\big(t,x(t;t_0,x_0)\big)<(\nu+1-k)a(B),\quad
 t\geq\theta_k,\; k=0,1,\dots ,\nu.
\end{equation}
We shall prove the statement by induction. For  $k=0$, i.e. for
$t\geq \theta_0=t_0$, using Theorem \ref{thm5} we have
$$
V(t)=V\big(t,x(t;t_0,x_0)\big)<a(\beta)<(\nu+1-0)a(B),\quad t\geq t_0=\theta_0.
$$
Assume that
\begin{equation} \label{e19}
V(t)<(\nu+1-k)a(B),\quad t\geq\theta_k.
\end{equation}
We shall show that
\[
V(t)<(\nu-k)a(B),\ t\geq\theta_{k+1}.
\]
If the opposite is true, i.e.
$$
(\exists t^*\geq\theta_{k+1}): V(t^*)\geq(\nu-k)a(B),
$$
then by \eqref{e19} for  $t=\theta_k$, the inequality above and monotony
of function  $V$ (see Theorem \ref{thm4}), we obtain
\begin{equation} \label{e20}
V(\theta_k)-V(\theta_{k+1})\leq V(\theta_k)-V(t^*)\leq a(B).
\end{equation}
On the other hand, using the fact that function $V$  is a piecewise
differentiable, and also the inequality of condition (2) of this theorem,
we arrive at the estimate
\begin{align*}
V(\theta_{k+1})
&\leq V(\theta_k)+\int_{\theta_k}^{\theta_{k+1}}\frac{d}{dt}V(\tau)d\tau\\
&< V(\theta_k)-c(\rho)(\theta_{k+1}-\theta_k)\\
&= V(\theta_k)-a(B).
\end{align*}
which contradicts \eqref{e20}.
We substitute
$$
T=T(\alpha)=\frac{a(B)}{c(\rho)}\nu.
$$
Then
$$
t\geq t_0+T=t_0+\frac{a(B)}{c(\rho)}\nu=\theta_{\nu}.
$$
Then by \eqref{e18} and condition (2.1.1) of Theorem \ref{thm5},
finally we obtain
\[
a\big(\big\|x(t;t_0,x_0)\big\|\big)
\leq V(t)
=V\big(t,x(t;t_0,x_0)\big)<a(B),
\]
i.e.
$\|x(t;t_0,x_0)\|<\beta$.
The proof is complete.
\end{proof}

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\end{document}