We begin with introducing the problem studied. Let ${\mathbb{R}}^{+}\triangleq [0,+\infty )$, $Y\left(t\right)=\left\{y_i\left(t\right)\right|i\in \Lambda \triangleq \{1,2,\cdots ,p\}\}$, $f:{\mathbb{R}}^{+}\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$, $y_i:{\mathbb{R}}^{+}⟶{\mathbb{R}}^n$ and $J_i:{\mathbb{R}}^n⟶{\mathbb{R}}^n$ ($i=1$, 2, ⋯, p) are given maps. Consider the following differential equations with impulses at variable times The main purpose of this study are (i) to provide a sufficient condition to the existence and uniqueness of solution x for the impulsive system (1.1); and (ii) to give the necessary and sufficient condition to the exact times for the solution x meets the curve Y; (iii) to present properties of solution relative to the initial value.

There are some interesting phenomena for the impulsive system (1.1). All of first, it is clear that the system $\dot{x}\left(t\right)=x\left(t\right)+u\left(t\right)$ is controllable, but the following impulsive system is not controllable. Similarly, the system $\dot{x}\left(t\right)=-x\left(t\right)$ is stable, but the impulsive system is not stable when the initial value $x\left(0\right)\ge 1$. Let us look at the third example. Denote by $x(·;0,x_0)$ the solution of the following impulsive differential system with the initial value $(0,x_0)$. Then, we have This implies that the impulsive system (1.1) never has not continuous of solution to respect to the initial value in $L^1$. In addition to, we can also use simple cases to show that the implsive system (1.1) may has not global solution.

The motivation to study this problem is as follows. First of all, many physical phenomena and application models are characterized by (1.1). For example, Integrate-and-Fire models derived from the physical oscillation circuit [1,2] is widely used in neuroscience, neuroscience research concerns current-voltage relations at which the states can be reset once the voltage reaches a threshold level [3,4]. Again, in the application, it is crucial to choose appropriate threshold levels for making decisions to trigger or suspend an impulsive intervention: [5] use glucose threshold level guided injections of insulin; [6] use the time that is economic threshold is reached by the amount of pests as the time of impulsive intervention. Secondly, the theory of impulsive differential equations has been an object of increasing interest because of its wide applicability in biology, in medicine and in more and more fields (see [7] and the its references). Significant interest in the investigation of differential equations with impulse effect is explained by the development of equipment in which significant role is played by complex systems [8,9,10]. Particularly, the qualitative theory of the impulsive system (1.1) has not been systematically established. It is natural to ask for the present the qualitative theory of the impulsive system (1.1). We will discuss the existence and uniqueness of global solution and its properties for the nonlinear ordinary differential equations with impulses at variable times (1.1) under weaker conditions. It is worth pointing out that the solutions of differential systems with impulses may experience the pulse phenomena, namely the solutions may hit a given surface finite or infinite number of times causing rhythmical beating. This situation presents difficulties in the investigation of properties of solutions of such systems. In addition to, it is not suitable to the stronger conditions for control problem. Consequently, it is desirable to find weaker conditions that guarantee the absence or presence of pulse phenomena. More generally, it is significant to find conditions that the solution meets a given surface just has $k\in \mathbb{N}$ times ($\mathbb{N}$ denote the natural numbers set).

Before concluding this section, we review the previous literature to qualitative analysis on the impulsive differential equations. In fact, the qualitative analysis on the impulsive differential equations can at least be traced back to the works by N.M. Kruylov and N.N. Bogolyubov [11] in 1937 in their classical monograph to $\mathbf{NonlinearMechanics}$. Mathematical formulation of the differential equation with impulses at fixed times was firstly presented by A.M. Samoilenko and N.A. Perestyuk [12] in 1974. Since then, the qualitative theory for differential equation with impulses at fixed times in finite (or infinite) dimensional spaces has been extensively studied (see [13,14,15,16] and the references therein). For the differential equations with impulses at variable times, A.M. Samoilenko and N.A. Perestyuk [17] in 1981 given the mathematical model Later relevant works were made by D.D. Bainov and A.B. Dishliev [18] in 1984, S. Hu [19] in 1989, etc. For more details, one can see the monographs of V.Lakshmikantham [20] in 1989, A.M. Samoilenko [21] in 1995, D.D. Bainov [22] in 1995 and M. Benchohra [23] in 2006, and so on. In a word, these works established the qualitative theory of (1.2) under stronger conditions. Specially, it is not suitable to the stronger conditions for control problem and impulsive differential equation in infinite dimensional spaces. At the same time, when $y_i,(\forall i\in \Lambda )$ is a one to one mapping, $x\left(t\right)=y_i\left(t\right)$ be equal to $t=y_i^{-1}\left(x\left(t\right)\right)$. Hence, (1.2) can be treated as a simplified case of (1.1). For the linear case of (1.1), Peng etc [24] obtained the existence and uniqueness of solution and its properties.

The rest of the paper is organized as follows. Section 2 presents the main results. In Section 3, 4, 5, the proofs of the three main theorems are given in turns. Periodicity of autonomous impulsive system is presented in Section 6. As an application, variation of solution relative to control is presented in Section 7, which is foundation to study optimal control problems of systems governed by the differential equations with impulses at variable times. Finally, some new phenomena of impulsive differential system are summarized.

We are going to present our main results in the section. To state the first one, some preliminaries will be introduced. Throughout this paper, we fix $T>0$ and approve that $T=+\infty$. We firstly introduce several definitions. We define the function set: $P{C}_Y\left([0,T),{\mathbb{R}}^n\right)=\{x:[0,T)⟶{\mathbb{R}}^n|x$ is continuous at t when $x\left(t\right)\notin Y\left(t\right)$, x is left continuous at t and the right limit $x(t+)$ exists when $x\left(t\right)\in Y\left(t\right)\}$. For $x\in P{C}_Y\left([0,T),{\mathbb{R}}^n\right)$, $t\in [0,T)$ is called to be an irregular point if $x\left(t\right)\in Y\left(t\right)$. Otherwise, t is called to be a regular point. One can directly verify that the function set $P{C}_Y\left([0,T),{\mathbb{R}}^n\right)$ is not linear. Denoted by $B(z,{\theta }^2)$ the closed ball (in ${\mathbb{R}}^n$) centered at z and of radius ${\theta }^2>0$.

Meanwhile, we introduced the following basic assumptions.

[F] (1) $f:{\mathbb{R}}^{+}\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$ is measurable in t on ${\mathbb{R}}^{+}$ and locally Lipschitz continuous in x i.e. for any $\rho >0$, there exits $L\left(\rho \right)>0$ such that for all $x,y\in {\mathbb{R}}^n$ with $\left|x\right|$, $\left|y\right|\le \rho$, we have

(2) There exists a constant $\tilde{k}>0$ such that

(3) f is continuous partially differentiable in x and $f_x(·,x)\in L_{loc}^1\left({\mathbb{R}}^{+},{\mathbb{R}}^{n\times n}\right)$.

[Y](1) $y_i\in C\left({\mathbb{R}}^{+},{\mathbb{R}}^n\right)$ and $y_i\left(t\right)\ne y_j\left(t\right)$ for all $t\in {\mathbb{R}}^{+}$ and $i\ne j$ ($i,j\in \Lambda$).

(2) $y_i\in C^1([0,T],{\mathbb{R}}^n)$ and $f\left(t,y_i\left(t\right)\right)\ne {\dot{y}}_i\left(t\right)$ ($i\in \Lambda$).

[J](1) $J_i:{\mathbb{R}}^n⟶{\mathbb{R}}^n$ is continuous and

(2) $J_i:{\mathbb{R}}^n⟶{\mathbb{R}}^n$ is continuous partially differentiable.

It is clear that when the assumptions [F](1)(2) hold, for any fix $(s,z_s)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^n$, the differential equation has a unique solution $z(·;s,z_s)\in C\left([s,+\infty ),{\mathbb{R}}^n\right)$ given by We define several functions: and where $\langle ·,·\rangle$ denotes the inner product in ${\mathbb{R}}^n$.

The first main result is presented as follows.

We have to point out that the necessary and sufficient conditions of pulse phenomenon also is given in Theorem 2.2. Not only that, for the existence of solution of the system (1.2), in order to ensure $t_k={\tau }_k\left(x\right)$ monotonous with respect to k in [20], it requires that ${\tau }_k\left(x\right)$ to be smooth and satisfy the corresponding inequality conditions. However, using Theorem 2.2, we can obtain immediately the following result.

Now, we state second and third main results. It follows from Theorem 2.2 that for any fixed sufficient small $\theta >0$, (1.1) has a unique approximate solution $x_{\theta }$ provided that the assumptions [F](1)(2), [Y](1) and [J](1) hold. Let $v\in {\mathbb{R}}^n$, $x_{\theta }(·;\theta ,x_0+\theta v)$ be an approximate $PC-$solution of the equation (1.1) corresponding to $(\theta ,x_0+\theta v)$. We note that (1.1) is not well posed. Thus, we can never expect to have the continuity of solution to respect to the initial value. We have to modify the classical definition of continuous and differentiability, respectively.

Let we state the following main results.

Throughout this section, we define the function $r:(0,+\infty )⟶{\mathbb{R}}^{+}$ given by where ${{\rm Y}}_j$ is defined by (2.1). It is easy from the assumptions [J](1) and [Y](1) to see ${{\rm Y}}_i\in C\left([0,T],{\mathbb{R}}^n\right)$ for all $i\in \Lambda$. Hence, there exists a constant $M\left(T\right)$ such that and

To claim the existence and uniqueness of solution of (1.1), we need the following Lemma.

By Lemma 3.1, one can prove that for any $T>0$, the equation (1.1) has a unique $PC-$solution. Thus, the equation (1.1) admits a unique $PC-$solution $x(·;0,x_0)$ on ${\mathbb{R}}^{+}$.

Next, we discus the number of irregular point for the solution x of (1.1) over ${\mathbb{R}}^{+}$. It is easy that the solution x of (1.1) has no irregular point over ${\mathbb{R}}^{+}$ if and only if for all $i\in \Lambda$, $x(t;u,0,x_0)\ne y_i\left(t\right)$ for any $t\in {\mathbb{R}}^{+}$. Together with (2.4), we have following result.

Now, we prove the necessity on (2) of Theorem 2.2. For convenience, we let $x(·)=x(·;0,x_0)$ and $\left\{t_i|0\le t_1<t_2<\cdots <t_k<+\infty \right\}$ stand for the irregular point set of x over ${\mathbb{R}}^{+}$. Then there firstly exists $l_1\in \Lambda$ such that Together with (2.4), we can affirm For the second irregular point $t_2$ of x, there exists $l_2\in \Lambda$ such that Together with (2.5), it follows Similarly, for the irregular point $t_k$ of x, there is a $l_k\in \Lambda$ such that Moreover, we can see from Lemma 3.2 that x has not irregular point on $(t_k,+\infty )$ if and only if Combined with (2.5), it is easy from the assumptions [J] and [Y] to see $F_{l_{k}j}\left(·;t_k,{{\rm Y}}_{l_k}\left(t_k\right)\right)\in C\left([t_k,+\infty )\right)$ and Therefore, together with (3.7), this means (2.7) and (2.8) hold.

To the sufficient on (2) of Theorem 2.2, suppose $\left\{t_i|0\le t_1<t_2<\cdots <t_k<+\infty \right\}$ satisfies (2.7) and (2.8). Then it is not difficult to claim that $t_j$ is the irregular point x of the (1.1) over ${\mathbb{R}}^{+}$. Simultaneously, one can conclude for (2.8) and Lemma 3.1 that $\left\{t_i|0\le t_1<t_2<\cdots <t_k<+\infty \right\}$ is the irregular point set of x over ${\mathbb{R}}^{+}$. This completes the proof.

Throughout this section, we fix $T>0$. It follows from Theorem 2.2 that the irregular point to the $PC-$solution x of (1.1) at most a finite number of times on the interval $[0,T]$. There are only two possibilities which are Case (1): x has no irregular point on $[0,T]$ and Case (2): x has at least one irregular point on $[0,T]$.

In Case (1), the $PC-$solution x has continuous dependence relative to the initial value in the sense of classical definition, i.e.

In Case (2), if $x_0=y_i\left(0\right)$ for some $i\in \Lambda$, we only study the $PC-$ solution $x\left(·;0+,{{\rm Y}}_i\left(0\right)\right)$. Consequently, we may assume that $x\left(·;0,x_0\right)$ meets the warning wall $Y(·)$, k times in $[0,T]$ and let ${\overline{t}}_j^i$ be the moments of $x\left(·;0,x_0\right)$ hits the warning line $y_i(·)$ ($i\in \Lambda$, $j=1$, 2, ⋯, k), $0<{\overline{t}}_1^i<{\overline{t}}_2^j<\cdots <{\overline{t}}_k^r<T$. By Theorem 2.2, we can prove that there exists $\overline{\delta }>0$ such that when $0\le \theta <\overline{\delta }$, the impulsive differential equation (1.1) has a unique approximate $PC-$solution $x_{\theta }(·;\theta ,x_0+\theta v)$ corresponding to the initial value $(\theta ,x_0+\theta v)$.

Let $t_j^i\left(\theta \right)$ be the moments of $x_{\theta }(·;\theta ,x_0+\theta v)$. For sufficient small $\epsilon >0$, it is obvious from the definition of solution (see (2.1)) that there is an ${\delta }_1>0$ with ${\delta }_1<\overline{\delta }$ such that for any $\theta >0$ with $\theta <{\delta }_1$, This means Together with the continuity of $J_i$, we have It follows from (2.2) that where Consequently, combine with (2.1), there is an ${\delta }_2>0$ with ${\delta }_2<{\delta }_1$ such that for any $\theta >0$ with $\theta <{\delta }_2$, Let In general, repeat the above process, one can show that there is an ${\delta }_{j+1}>0$ with ${\delta }_{j+1}<{\delta }_j$ such that for any $\theta >0$ with $\theta <{\delta }_{j+1}$, and where

In short, for any sufficient small $\epsilon >0$, there exists an $\delta >0$ such that where This completes the proof.

Throughout this section, we fix $T>0$. It follows from Theorem 2.6 that there are only two possibilities which are Case $\left(i\right)$: $x\left(·;0,x_0\right)$ has no irregular point on $[0,T]$ and Case $\left(ii\right)$: $x\left(·;0,x_0\right)$ has at least one irregular point on $[0,T]$.

In Case $\left(i\right)$, one can directly check that $x\left(·;0,x_0\right)$ of (1.1) is G$\widehat{a}$teaux differentiable and its G$\widehat{a}$teaux derivative $\phi$ is a weak solution of the following differential equation To discuss Case $\left(ii\right)$, we define function $h_t$ given by Here, where $\tilde{y}(t,\epsilon )={\tilde{y}}_i(t,\epsilon )$ for some $i\in \Lambda$, ${\tilde{y}}_i(t,\epsilon )\in \partial B\left(y_i\left(t\right),{\epsilon }^2\right)$. By Theorem 2.6, when $x\left(t;0,x_0\right)=y_i\left(t\right)$, there is an $\delta >0$ such that the definition (5.2) is well for all $\epsilon \in [0,\delta ]$, that is, $h_t:[0,\delta ]⟶O\left(t\right)$ is a function and $h_t\left(0\right)=t$, where $O\left(t\right)$ denote some neighborhood of t. For convenience, let $\left\{t_j^i\right|0<t_1^i<t_2^j<\cdots <t_k^r<T\}$ denote the irregular point set of $x\left(·;0,x_0\right)$ on $[0,T]$. If $y_i\in C^1([0,T],{\mathbb{R}}^n)$, it follows from Theorem 2.6 and (5.3) that there is an $\delta >0$ such that and Further, when $f\left(t_j^i,y_i\left(t_j^i\right)\right)\ne {\dot{y}}_i\left(t_j^i\right)$ ($j=1,2,\cdots ,k$, $i\in \Lambda$), we have where $j=1$, 2, ⋯, k. Let $f={\left(f^1,f^2,\cdots ,f^n\right)}^{\top }$, $y_i={\left(y_i^1,y_i^2,\cdots ,y_i^n\right)}^{\top }$ ($i\in \Lambda$). Without loss of generality, we suppose We introduce the following functions then We let We firstly claim the following lemma.