\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 128, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/128\hfil On the $\phi_0$-stability] {On the $\phi_0$-stability of impulsive dynamic system on time scales} \author[P. Wang, M. Wu\hfil EJDE-2005/128\hfilneg] {Peiguang Wang, Meng Wu} % in alphabetical order \address{Peiguang Wang \hfill\break College of Electronic and Information Engineering, Hebei University, Baoding, 071002, China} \email{pgwang@mail.hbu.edu.cn} \address{Meng Wu \hfill\break College of Mathematics and Computer Science, Hebei University, Baoding, 071002, China} \email{wumeng919@126.com} \date{} \thanks{Submitted June 20, 2005. Published November 23, 2005.} \thanks{Supported by the Natural Science Foundation of Hebei Province, China (A2004000089)} \subjclass[2000]{34D20, 34A37} \keywords{Time scales; Impulsive dynamic system; $\phi_0$-stability; \hfill\break \indent cone-valued Lyapunov functions} \begin{abstract} This paper concerns impulsive dynamic system on time scales. We obtain sufficient conditions of the $\phi_0$-stability for such systems by employing cone-valued Lyapunov functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Dynamic systems with impulses have been subject of numerous investigations, and are used as mathematical models of various real processes and phenomena in physics, biology, control theory and so on, which during their evolutionary processes experience an abrupt change of state at certain moments of time. Furthermore, the theory of such systems is much richer than the corresponding theory of differential systems without impulses. The past ten years have seen a significant development in the theory of impulsive differential equations. For the basic theory and recent developments, see the monograph \cite{l1} and references cited therein. Recently, the stability theory of dynamic equations on time scales is emerging as an important area of investigation since it demonstrates the interplay of the two different theories, namely, the theories of continuous and discrete dynamic systems \cite{l2}. However, the corresponding theory of such equations is still at an initial stage of its development, especially the impulsive dynamic system on time scales. For the various of stability problem, it is well known that the method of cone-valued Lyapunov functions is beneficial in applications and circumvents the limitations of the useful and well-known method of both the scalar and vector Lyapunov functions. Lakshmikantham and Leela \cite{l3} introduced the concept of cone-valued Lyapunov functions. Since then, Akpan \cite{a3}, Akinyele \cite{a1}, Akinyele and Adeyeye \cite{a2}, Soliman \cite{s1} investigated the stability and $\phi_0$-stability of the differential systems, impulsive control systems, hybrid systems and perturbed impulsive systems by employing the method of cone-valued Lyapunov functions. However, there are few results for the stability of various of impulsive dynamic system on time scales \cite{l4}. In this paper, utilizing the framework of the theory of dynamic systems on time scale, we establish new comparison results in terms of cone-valued Lyapunov functions and investigate impulsive dynamic system on time scales relative to $\phi_0$-stability. \section{Preliminaries} Let $\mathbb{T}$ be a time scale (any subset of $\mathbb{R}$ with order and topological structure defined in a canonical way) with $t_0\geq 0$ as minimal element and no maximal element. The basic concepts on time scales can be seen in \cite{l2}. \begin{definition} \label{def2.1} \rm The mapping $g: \mathbb{T}\to X$, where $X$ is a Banach space, is called rd-continuous if at each right-dense $t\in \mathbb{T}$, it is continuous and at each left-dense $t$, the left-sided limit $g(t^-)$ exists. \end{definition} \begin{definition} \label{def2.2} \rm For each $t\in \mathbb{T}$, let $N$ be a neighborhood of $t$, Then, we define the generalized derivative (or Dini derivative), $D^+ u^\Delta(t)$, to mean that, given $\varepsilon>0$, there exists a right neighborhood $N_\varepsilon\subset N$ of $t$ such that $$ \frac{u(\sigma(t))-u(t)}{\mu(t,s)} t, \quad \mbox{where } \mu(t,s)\equiv \sigma(t)-s. $$ \end{definition} When $t$ is the right-scattered and $u$ is continuous at $t$, we have, as in the case of the derivative $$ D^+ u^\Delta(t)=\frac{u(\sigma(t))-u(t)}{\mu^*(t)}, $$ where $\mu^*(t)=\sigma(t)-t$. \begin{definition} \label{def2.3} \rm A proper subset $K$ of $\mathbb{R}^n$ is called a cone if \begin{itemize} \item[(i)] $\lambda K\subseteq K,\; \lambda\geq 0$; \item[(ii)] $K+K\subseteq K$ \item[(iii)] $K=\overline{K}$ \item[(iv)] $K^0\neq\emptyset$ \item[(v)] $K\cap (-K)=\{0\}$, \end{itemize} where $\overline{K}$ and $K^0$ denote the closure and interior of $K$, respectively, and $\partial K$ denotes the boundary of $K$. The order relation on $\mathbb{R}^n$ induced by the cone $K$ is defined as follows: $x\leq_K y$ iff $y-x\in K$ and $x<_{K^0} y$ if and only if $y-x\in K^0$. \end{definition} \begin{definition} \label{def2.4} \rm The set $K^*=\{\phi\in \mathbb{R}^n:(\phi,x)\geq0,\mbox{ for all }x\in K \}$ is said to be adjoint cone if it satisfies the properties (i)-(v). $$ x\in K^0 \quad \mbox{if} \quad (\phi,x)>0, $$ and $$ x\in \partial k\quad \mbox{if}\quad (\phi,x)=0, \quad \mbox{for some } \phi\in K_0^*,\; K_0=K-\{0\}. $$ \end{definition} \begin{definition} \label{def2.5} \rm A function $g:D\to \mathbb{R}^n$, $D\in \mathbb{R}^n$ is said to be quasimonotone relative to $K$ if $x,y\in D$ and $y-x\in \partial K$ implies that there exists $\phi_0\in K_0^*$ such that $$ (\phi_0,y-x)=0 \quad \mbox{and} \quad (\phi_0,g(y)-g(x))\geq 0. $$ \end{definition} \section{Comparison results} Consider the impulsive dynamic system \begin{equation} \begin{gathered} x^\Delta=f(t,x),\quad t\in \mathbb{T},\; t\neq t_k,\\ x(t^+)=x(t)+I_k(x),\quad t=t_k,\; k\in \mathbb{N} \\ x(t_0^+)=x_0,\quad t_0\geq 0, \end{gathered}\label{e3.1} \end{equation} under the following assumptions: \begin{itemize} \item[(i)] $00$, define $S(\rho)=\{x\in \mathbb{R}^n : \| x \|<\rho\}$. Let $V\in C_{rd}[\mathbb{T}\times S(\rho),K]$. Then $V$ is said to belong to the class $V_0$ if $V$ is locally Lipschitzian in $x$ and rd-continuous in $(t_{k-1},t_k]\times S(\rho)$, and for each $x\in S(\rho)$, $k\in \mathbb{N}$, $\lim_{(t,y)\to(t_k^+,x)}V(t,y)=V(t_k^+,x)$ exists. Following Definition \ref{def2.2}, we say that $V\in C_{rd}[\mathbb{T}\times S(\rho),K]$, $D^+V^\Delta(t,x(t))$ if for each $\varepsilon>0$, there exists a right neighborhood $N_\varepsilon\subset N$ of $t$ such that $$ \frac{1}{\mu(t,s)} [V(\sigma(t),x(\sigma(t)))-V(s,x(\sigma(t)))-\mu(t,s)f(t,x(t))] t$. As before, if $t$ is right-scattered and $V(t,x(t))$ is continuous at $t$, this reduces to $$ D^+V^\Delta(t,x(t))=\frac{V(\sigma(t),x(\sigma(t)))-V(t,x(t))}{\mu^*}. $$ \begin{lemma}[\cite{l1}] \label{lem3.1} Let $m\in C_{rd}[\mathbb{T},\mathbb{R}^n]$ be a mapping that is differentiable for each $t\in\mathbb{T}$ and that satisfies $$ m^{\Delta}(t,x)\leq g(t,m(t)),\;t\in\mathbb{T}, $$ where $g\in C_{rd}[\mathbb{T},\mathbb{R}^n]$ and $g(t,u)\mu^*(t)$ be nondecreasing in $u$ for each $t\in\mathbb{T}$. Then $m(t_0)\leq u_0$ implies $$ m(t)\leq r(t),\quad t\in \mathbb{T}, $$ where $r(t)$ is the maximal solution of $u^{\Delta}=g(t,u)$, $u(t_0)=u_0\geq0$ existing on $\mathbb{T}$. \end{lemma} Consider the comparison system \begin{equation} \begin{gathered} u^\Delta=g(t,u),\;\quad t\in \mathbb{T},\; t\neq t_k,\\ u(t_k^+)=J_k(u(t_k)),\quad k\in \mathbb{N}\\ u(t_0^+)=u_0\geq 0. \end{gathered}\label{e3.2} \end{equation} Assume that \begin{itemize} \item[(i)] $g\in C_{rd}[\mathbb{T}\times K,\mathbb{R}^n]$, $g$ is rd-continuous in $(t_{k-1},t_k]\times K$, $K$ is a cone in $\mathbb{R}^n$, and for each $u\in K$, $\lim_{(t,v)\to(t_k^+,u)}g(t,v)=g(t_k^+,u)$, $g(t,u)\mu^*(t)+u$ is quasimonotone nondecreasing in $u$ relative to $K$ for each $(t,u)$ \item[(ii)] $V\in C_{rd}[\mathbb{T}\times S(\rho),K]$, $V$ is locally Lipschitzian in $x$ relative to $K$ and $$ D^+V^\Delta(t,x)\leq_K g(t,V(t,x)),\quad t\neq t_k $$ \item[(iii)] $J_k\in C_{rd}[K,K]$, and $J_k$ is quasimonotone nondecreasing relative to $K$ such that $$ V(t,x+I_k(x))\leq_K J_k(V(t,x)),\quad t=t_k. $$ \end{itemize} \begin{lemma} \label{lem3.2} Assume (i)--(iii) above, and let $r(t,t_0,u_0)$ be the maximal solution of \eqref{e3.2} existing on $\mathbb{T}$. Then for any solution $x(t)=x(t,t_0,x_0)$ of \eqref{e3.1} which exists on $\mathbb{T}$, we have $$ V(t,x(t))\leq_K r(t,t_0,u_0)\quad \mbox{provided that} \quad V(t_0^+,x_0)\leq_K u_0. $$ \end{lemma} \begin{proof} Let $x(t)=x(t,t_0,x_0)$ be the solution of \eqref{e3.1} existing for $t\geq t_0$, $t\in \mathbb{T}$, and set $m(t)=V(t,x(t))$. Then by assumption (ii), it is easy to derive $$ D^+m^\Delta(t)\leq_K g(t,m(t)).\label{e3.3} $$ For $t\in[t_0,t_1]$, $m(t_0)=V(t_0,x_0)=u_0$. Then we get by Lemma \ref{lem3.1} $$ m(t)\leq_K r_0(t,t_0,u_0),\label{e3.4} $$ where $r_0(t,t_0,u_0)$ is the maximal solution of \eqref{e3.2} with $r_0(t_0^+,t_0,u_0)=u_0$. Since $J_1(u)$ is quasimonotone nondecreasing relative to $K$, and by (iii), $$ m(t_1)\leq_K J_1(m(t_1))\leq_K r_0(t_1,t_0,u_0)=u_1^+,\label{e3.5} $$ where $u_1^+\leq J_1(r_0(t_1,t_0,u_0))$. By \eqref{e3.3}, \eqref{e3.5} and Lemma \ref{lem3.1}, $$ m(t)\leq_K r_1(t,t_1,u_1^+),\quad t\in(t_1,t_2], $$ where $r_1(t,t_1,u_1^+)$ is the maximal solution of \eqref{e3.2} with $r_1(t_1^+,t_1,u_1^+)=u_1^+$. This procedure can be repeated successively to arrive at $$ m(t)\leq_K r_k(t,t_k,u_k^+),\quad t\in(t_k,t_{k+1}], $$ where $r_k(t,t_k,u_k^+)=u_k^+$ for each $k=0,1,2,\dots$. %We shall define %$$ %u(t)=\begin{cases} %u_0,& t=t_0;\\ %r_0(t,t_0,u_0),& t\in (t_0,t_1];\\ %r_1(t,t_1,u_1^+), & t\in (t_1,t_2];\\ %\dots \\ %r_k(t,t_k,u_k^+), & t\in (t_k,t_{k+1}]. % \end{cases} %$$ It is clear that %$u(t)$ is a solution of \eqref{e3.2}, and we obtain $m(t)\leq_K u(t)$, $t\geq t_0$ and $m(t)\leq_K r(t,t_0,u_0)$, $t\geq t_0$. Therefore, $$ V(t,x(t))\leq_K r(t,t_0,u_0),\quad t\geq t_0,\; t\in\mathbb{T}. $$ The proof is complete. \end{proof} \section{Main results} \begin{definition} \label{def4.1} \rm The trivial solution $u=0$ of \eqref{e3.2} is $\phi_0$-equistable if for each $\varepsilon>0$ and $t_0\in\mathbb{T}$, there exists a positive function $\delta=\delta(t_0,\varepsilon)$ that is rd-continuous in $t_0$ for each $\varepsilon$ such that for $\phi_0\in K^*_0$ $$ (\phi_0,u_0)<\delta \quad \mbox{implies} \quad (\phi_0,r(t))<\varepsilon,\quad t\geq t_0,\; t\in\mathbb{T}. $$ \end{definition} In the above definition, and for the rest of this paper, $r(t)$ denotes the maximal solution of \eqref{e3.2} relative to the cone $K\subseteq \mathbb{R}^n$. Other $\phi_0$-stability concepts can be similarly defined. \begin{definition} \label{def4.2} \rm The trivial solution $u=0$ of \eqref{e3.2} is uniformly $\phi_0$-stable if $\delta$ in Definition \ref{def4.1} is independent of $t_0$. \end{definition} \begin{definition} \label{def4.3} \rm The trivial solution $u=0$ of \eqref{e3.2} is $\phi_0$-equi-asymptotically stable if it is $\phi_0$-equistable, and for each $\varepsilon>0,\; t_0\in\mathbb{T}$, there exist $\delta_0=\delta_0(t_0)>0$ and $T=T(t_0,\varepsilon)$ such that $$ (\phi_0,u_0)<\delta_0 \quad \mbox{implies} \quad (\phi_0,r(t))<\varepsilon,\; t\geq t_0+T(t_0,\varepsilon),\; t\in\mathbb{T}. $$ \end{definition} \begin{definition} \label{def4.4}\rm The trivial solution $u=0$ of \eqref{e3.2} is uniformly $\phi_0$-asymptotically stable if $\delta$ and $T$ in Definition \ref{def4.3} is independent of $t_0$. \end{definition} \begin{definition} \label{def4.5} \rm A function $a(r)$ is said to belong to the class $\kappa$ if $a\in C_{rd}[\mathbb{R}_+,\mathbb{R}_+]$, $a(0)=0$, and $a(r)$ is strictly monotone increasing function in $r$. \end{definition} Next we discuss stability criteria for the trivial solution of \eqref{e3.1} under the assumptions \begin{itemize} \item[(iv)] $f(t,0)=0$, $g(t,0)=0$, and for some $\phi_0\in K_o^*$, $(t,x)\in \mathbb{T}\times S(\rho)$, $$ b(\|x\|)\leq (\phi_0,V(t,x))\leq a(\|x\|),\quad a,b\in \kappa. $$ \item[(v)] There exists a $\rho_o$ such that $x\in S(\rho_0)$ implies that $x+I_k(x)\in S(\rho)$ for all $K$. \end{itemize} \begin{theorem} \label{thm4.1} In addition to the hypothesis of Lemma \ref{lem3.2}, we assume that (iv), (v) are satisfied. Then the $\phi_0$-stability properties of the trivial solution of the comparison system \eqref{e3.2} imply the corresponding stability properties of the trivial solution of \eqref{e3.1}. \end{theorem} \begin{proof} Let $\varepsilon\in (0,\rho_0)$ and $t_0\in\mathbb{T}$ be given. Suppose that the trivial solution of \eqref{e3.2} is $\phi_0$-stable. Then given $b(\varepsilon)>0$, and $t_0\in \mathbb{T}$, there exists a $\delta^*=\delta^*(t_0,\varepsilon)>0$ such that $$ (\phi_0,u_0)<\delta^*\quad \mbox{implies}\quad (\phi_0,u(t,t_0,u_0))0$ such that $$ a(\delta)<\delta^*.\label{e4.2} $$ We claim that if $\|x_0\|<\delta$, then $\|x(t)\|<\varepsilon$, $t\in\mathbb{T}$, where $x(t)=x(t,t_0,x_0)$ is any solution of \eqref{e3.1}. If this is not true, there would exist a $t^*\in\mathbb{T}$, $t^*>t_0$ such that $t_k0$ and $T>0$ such that $$ (\phi_0,u_0)<\delta^*\quad \mbox{implies}\quad (\phi_0,u(t,t_0,u_0))0$ and designate by $\delta_0=\delta_0(\rho)$ so that we have $$ x_0<\delta \quad \mbox{implies} \quad \|x(t)\|<\rho,\quad t\geq t_0,\; t\in\mathbb{T}. $$ Setting $$ L(t,x(t))=V(t,x(t))+\int_{t_0}^t c[V(s,x(s))]\Delta s, $$ we see that $$ D^+L^{\Delta}(t,x(t))=D^+V^{\Delta}(t,x)+c[V(t,x)]\leq g(t,V(t,x))+c[V(t,x)]=0. $$ Then \begin{equation} L(t,x(t))\leq L(t_0,x_0)=V(t_0,x_0),\quad t\geq t_0.\label{e4.5} \end{equation} Choose $T=T(\varepsilon)>0$, for any given $\varepsilon>0$, to satisfy $T>a(\delta_0)/c[b(\delta)]$, where $\delta=\delta(\varepsilon)$. Because of the trivial solution of \eqref{e3.1} is uniformly $\phi_0$-stable, it is sufficient to show that there exists a $t^*\in [t_0,t_0+T]$ such that $\|x(t^*)\|<\delta$. If this is not true, we should have $\|x(t)\|\geq\delta$, $t\in[t_0,t_0+T]$ and hence $(\phi_0,V(t,x(t)))\geq b(\delta)$, $t\in[t_0,t_0+T]$. It then follows from \eqref{e4.5} \begin{align*} a(\delta_0)&