Viable periodic solutions in state-dependent impulsive problems

Abstract

Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are examined. The so-called impulsive index is introduced, as a topological tool, and its properties are studied. Some sufficient conditions for its homotopy property are discussed in detail. In a construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier where jumps occur. Several illustrative examples are added.

1 Introduction

Impulsive ordinary, or partial, differential equations became a strongly growing field of research in a last half of a century. Some results in this field are simple modifications and generalizations of analogous results for non-impulsive problems but, beyond them, there are essentially new, nontrivial and interesting questions requiring new methods and proof arguments. We meet this especially in state-dependent impulsive problems where jumps occur in variable times. A study in this direction was initiated by Mil’man and Myshkis in [18]. Problems with impulses in variable times are met in e.g. physical and biological models (cf. [3]), where concerned quantities rapidly change when they attain some barriers or some critical levels in a phase space or in an extended phase space. In an analysis of solutions of such problems, in contrast to those with fixed impulse times, one also has to take into account a geometry or topology of barriers and their relationships with dynamics of the problem. This area is still poorly investigated, especially in the context of applications of known or construction of new appropriate topological tools.

In the paper we focus our attention on detecting a periodic behavior of some of trajectories for state-dependent impulsive ODEs under an additional requirement that these trajectories are viable in a prescribed closed subset $K\subset {\mathbb R}^n$ of a state space. In other words, we are going to find a solution to the following periodic problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \dot{x}(t)=f(x(t)) &{} \hbox {for}\quad \hbox {a.e. } t\ge 0, x\in {\mathbb R}^n,\\ x(t)\in K,\\ x(t^+):=\lim _{s\rightarrow t^+} x(s)=I(x(t)) &{} \hbox {for}\quad x(t)\in M\subset \partial K,\\ x(0)=x(T_x), &{} \hbox {for}\quad \hbox {some} T_x>0, \end{array} \right. \end{aligned}$$

(1.1)

where $f:{\mathbb R}^n\supset \Omega \rightarrow {\mathbb R}^n$ is sufficiently smooth to generate a semiflow, and $I:M\rightarrow {\mathbb R}^n$ is a continuous impulse function. More specific assumptions on $f, M$ and $I$ will be given later. The notion of viability used above is motivated by biological models where some species survives only if it does not leave a region of safety $K$ given as a set of constraints.

There are only a few papers concerning problem (1.1) (see, e.g. [8, 9, 14, 17]). In most of them additional Nagumo-type tangency conditions are assumed to ensure that from every point of $K$ there starts at least one trajectory, even with impulses taken into account, remaining in $K$ for some nontrivial time interval. Then the fixed point theory can be applied for a suitably defined Poincaré–Krasnoselskii operator.

If tangency conditions do not hold on the whole boundary, we allow trajectories for a non-impulsive problem, even all of them, to leave $K$ through a so-called exit set $K^-$. Of course, we want to prevent this escape, so we place a barrier $M$ in $K^-$ trying to return at least some of them to the set $K$ via the impulse function $I$ (see [14]). In some biological models it means an external impact (e.g. restocking the lake). In [14] the impulse function was assumed to satisfy $I(M)\subset K$. Analysis of models in mathematical biology shows that it is worth considering a weaker assumption $I(M)\subset {\mathbb R}^n$, in practice at least $I(M)\cap K\ne \emptyset$. This situation is more complicated and needs deeper local topological tools to study problem (1.1). In the paper we construct a so-called impulsive index, which is a fixed point index of suitably defined continuous map in an exit set $K^-$. Its nontriviality will imply the existence of viable periodic trajectories in $K$.

The paper is organized as follows. In Sect. 2 at first we give basic assumptions, and define the impulsive index. Then properties of the index are presented, discussed and proved. Several examples are given to illustrate the results. In Sect. 3 we present geometric sufficient conditions for a homotopy property of the index with some results, concerning exit sets, interesting in themselves. The Poincaré map technique is adapted to compute the impulsive index in a neighborhood of a periodic point in Sect. 4 (see also [14] where this technique was proposed). The last section (Sect. 5) contains several concluding remarks concerning an alternative construction of the impulsive index. Advantages and disadvantages of this alternative technique are discussed.

2 Impulsive index

Let $K\subset {\mathbb R}^n$ be an arbitrary closed subset, $\Omega \supset K$ an open neighborhood in ${\mathbb R}^n$, and $f:\Omega \rightarrow {\mathbb R}^n$ be a sufficiently smooth vector field such that the equation $\dot{x}=f(x)$ generates a semiflow $\pi$ on $\Omega$. We are interested in viable in $K$ solutions so, without any loss of generality, we can assume in the sequel that $\Omega ={\mathbb R}^n$. Let us recall that by a solution of problem (1.1) we mean a left-continuous function $x:[0,\infty )\rightarrow {\mathbb R}^n$, with a discrete set of discontinuity points, absolutely continuous between these points, and satisfying $\dot{x}(t)=f(x(t))$ in a.e. $t\ge 0$, and $x(t^+)=I(x(t))$ in discontinuity points. The set of functions from $[0,\infty )$ to ${\mathbb R}^n$ which are continuous between discontinuity points taken from a discrete set, and left-continuous in these points will be denoted below by $PC([0,\infty ),{\mathbb R}^n)$.

In each point $x\in K$ we can consider the Bouligand tangent cone

$$\begin{aligned} \displaystyle { T_K(x):=\left\{ v \in {\mathbb R}^n\ | \ \liminf _{h \rightarrow 0^+}dist(x+hv, K) /h =0\right\} }. \end{aligned}$$

Obviously, if the following tangency condition

$$\begin{aligned} f(x)\in T_K(x) \quad \hbox {for every } x\in K \end{aligned}$$

is satisfied, then all solutions starting in $K$ remain in this set forever (see the first results of this type in [19] and [6]). If it is not satisfied, the following exit set for $K$

$$\begin{aligned} K^-=K^-(f):=\{x_0\in K \ | \ \forall \varepsilon >0\, :\, \pi (\{x_0\}\times (0,\varepsilon ))\not \subset K\} \end{aligned}$$

appears. If topological properties of the exit set sufficiently differs from properties of $K$, then still there exists at least one viable trajectory in $K$. This was firstly proved in the celebrated Ważewski paper [20]. More precisely, if $K^-$ is closed and it is not a strong deformation retract of $K$, then a viable solution exists (comp. [11]). In the paper we are interested in even worse case:

$$\begin{aligned} \hbox {All absolutely continuous solutions to} \, \dot{x}=f(x) \, \hbox {leave the set}\, K. \end{aligned}$$

(2.1)

It means, in consequence, that $K$ is strongly deformed via homotopy onto $K^-$. Namely, the exit function $\tau _K:K\rightarrow [0,\infty )$, $\tau _K(x_0):=\sup \{t\ge 0 \ | \ \pi (\{x_0\}\times [0,t])\subset K$ is continuous, and we can define a homotopy $h:K\times [0,1]\rightarrow K$, $h(x_0,\lambda ):=\pi (x_0,\lambda \tau _K(x_0))$. Putting $r(x_0):=h(x_0,1)$ we obtain a (continuous) retraction $r:K\rightarrow K^-$.

Hence, in the whole paper we assume that

$$\begin{aligned} h(x,\alpha ):=\pi (x,\alpha \tau _K(x)), x\in K, \hbox { is a deformation of}\, K\, \hbox {onto}\, K^-. \end{aligned}$$

(2.2)

Of course, (2.2) implies that $K^-$ is a closed subset of $K$. When we consider a family $\pi _\lambda$ of semiflows, we will assume, analogously, that deformations $h_\lambda$ are given, see below.

Let $M\subset K^-$ be an impulse set, i.e., there is an impulse function $I:M\rightarrow {\mathbb R}^n$ moving instantly each trajectory reaching the set $M$ in a point $p\in M$ to a point $I(p)\in {\mathbb R}^n$. In practice, we will be interested in impulse functions satisfying $I(M)\cap K\ne \emptyset$ which means that some trajectories are moving back to $K$. In our considerations we assume that

• (A1) $K^-$ is a closed ANR,

• (A2) $M=cl_{K^-}(int_{K^-}M)$,

• (A3) $I$ is a compact map, i.e. $I(M)$ is relatively compact,

• (A4) $dist(I(M),M)>0$,

where $int_{K^-}A, cl_{K^-}A$ and, in the sequel, $\partial _{K^-}A$ denote the relative interior, closure and boundary of a set $A\subset K^-$ in $K^-$, respectively. Note that, if $M$ is compact, which is often the case, (A4) is implied by $I(M)\cap M=\emptyset$. This condition or (A4) is assumed in many papers concerning impulsive semidynamical systems (see, e.g., [1, 5, 10]). It implies, in particular, that for every $x_0\in K{\setminus }M$ the sequence $\sigma _0(x_0)=0, \sigma _1(x_0)=\phi (x_0), \sigma _{m+1}(x_0)=\sigma _m(x_0) +\phi ((I\circ r)^m(x_0))$ satisfies the condition $\lim _{m\rightarrow \infty } \sigma _m(x_0)=\infty$, where $\phi (x)=\inf \{s>0 \ | \ \pi (\{x\}\times [0,s))\cap M=\emptyset \hbox { and } \pi (x,s)\in M\}$.

We define $M^I:=I^{-1}(K)$ and

$$\begin{aligned} g:M^I\rightarrow K^-, \ g(x):=r(I(x)) \hbox { for }\quad x\in M^I. \end{aligned}$$

(2.3)

Denote

$$\begin{aligned} \mathcal P:=\{x\in PC([0,\infty ),{\mathbb R}^n) \ | \ x \hbox { is a viable periodic solution to}\, (1.1)\}. \end{aligned}$$

Notice that, since every absolutely continuous solution to $\dot{x}=f(x)$ leaves the set $K$, there is a correspondence between elements of $\mathcal P$ and fixed points of iterations of the map $g$ defined in (2.3). Indeed, let $x\in \mathcal P$ start from $x_0\in K$. Then $z=\pi (x_0,\tau _K(x_0))\in M$, and there is a minimal $m\in \mathbb N$ such that $x_0=\pi (I(g^m(z)),s)$ for some $s\in [0,\tau _K(I(g^m(z)))]$. It implies that $z=\pi (x_0,\tau _K(x_0))=g^{m+1}(z)$, so $z$ is a fixed point of $g^{m+1}$. On the other hand, if $z=g^m(z)$, then the map $x(0):=z$, $x(t):=\pi (I(g^k(z)),t-\sigma _k(I(z)))$ for $t\in (\sigma _k(I(z)),\sigma _{k+1}(I(z))], k\ge 0$, is a viable periodic solution to problem (2.2) starting from $z$.

In particular, each fixed point of $g$ generates a periodic viable trajectory in $K$. This motivates us to construct a fixed point index type topological tool detecting fixed points of $g$.

Let $V$ be an arbitrary open subset of $K^-$. Let us define

$$\begin{aligned} U_V:=int_{K^-}(I^{-1}(K)\cap V)\subset V, \end{aligned}$$

(2.4)

and assume that

$$\begin{aligned} Fix\, g \cap U_V \hbox { is a compact set}. \end{aligned}$$

We define an impulsive index of $\pi$ on $V$ as

$$\begin{aligned} \hbox {ind}_K(\pi ,I,V):=\hbox {ind}(g,U_V), \end{aligned}$$

(2.5)

where $\hbox {ind}(g,U_V)$ is a fixed point index for compact maps on ANRs (see [16]).

Below we show the following main properties of the impulsive index.

Theorem 2.1

(Existence) If $\hbox {ind}_K(\pi ,I,V)\ne 0$, then there exists a viable periodic solution to (1.1) reaching the set $V$.

Theorem 2.2

(Additivity) Let $K^-\supset V=V_1\cup V_2$, $V_i$ open and $V_1\cap V_2=\emptyset$. If $Fix\, g \cap (U_V)$ is a compact set, then

$$\begin{aligned} \hbox {ind}_K(\pi ,I,V)=\hbox {ind}_K(\pi ,I,V_1)+\hbox {ind}_K(\pi ,I,V_2). \end{aligned}$$

Theorem 2.3

(Homotopy I) Let $\pi _\lambda$, $\lambda \in [0,1]$, be a continuous family of semiflows, i.e. the map ${\mathbb R}^n\times [0,\infty )\times [0,1]\ni (x,t,\lambda )\mapsto \pi _\lambda (x,t)$ is continuous. Assume that, for every $\lambda \in [0,1]$, the exit set $K_\lambda ^-$ for $\pi _\lambda$ is a closed ANR, there are retractions $r_\lambda :K\rightarrow K_\lambda ^-$ along trajectories, i.e., $r_\lambda (x)=\pi _\lambda (x,\tau _K(x))$, and $M\subset K_\lambda ^-$ satisfies $M=cl_{K_\lambda ^-}(int_{K_\lambda ^-}M)$. Moreover, assume that $X=\bigcup _{\lambda \in [0,1]} K_\lambda ^- \in ANR$, and the set $\bigcup _{\lambda \in [0,1]} r_\lambda (I(M)\cap K)$ is relatively compact in $X$.

Let $V\subset \bigcap _{\lambda \in [0,1]} K_\lambda ^-$ be open in every $K_\lambda ^-$, and such that $U_{V_{\lambda _0}}$ is an open subset of $X$ for some $\lambda _0\in [0,1]$, and $Fix\, g_\lambda \cap U_{V_{\lambda }}$ is compact, where $g_\lambda :=r_\lambda \circ I:M^I\rightarrow K_\lambda ^-$.

Then the numbers $\hbox {ind}_K(\pi _\lambda ,I,V)$ are well defined, and

$$\begin{aligned} \hbox {ind}_K(\pi _0,I,V)=\hbox {ind}_K(\pi _1,I,V). \end{aligned}$$

Theorem 2.4

(Homotopy II) Let $\pi _\lambda$, $\lambda \in [0,1]$, be a continuous family of semiflows, and assume that, for every $\lambda \in [0,1]$, the exit set $K_\lambda ^-$ for $\pi _\lambda$ is a closed ANR, there are retractions $r_\lambda :K\rightarrow K_\lambda ^-$ along trajectories, impulse sets $M_\lambda \subset K_\lambda ^-$ satisfying $M_\lambda =cl_{K_\lambda ^-}(int_{K_\lambda ^-}M_\lambda )$, and impulse functions $I_\lambda \rightarrow {\mathbb R}^n$. Moreover, assume that, for every pair $0\le a\le b\le 1$, the set $X_a^b=\bigcup _{\lambda \in [a,b]} K_\lambda ^-$ is an ANR, and the set $\bigcup _{\lambda \in [a,b]} r_\lambda (I_\lambda (M_\lambda )\cap K)$ is relatively compact in $X_a^b$.

Let $V_\lambda \subset K_\lambda ^-$ be such that $\bigcup _{\lambda \in [0,1]} U_{V_\lambda }\times \{\lambda \}$ is an open subset of $X_0^1$, and such that $F:=\bigcup _{\lambda \in [0,1]} \left( (Fix\, g_\lambda \cap U_{V_{\lambda }})\times \{\lambda \}\right)$ is compact, where $g_\lambda :=r_\lambda \circ I_\lambda :M_\lambda ^I\rightarrow K_\lambda ^-$ and $M_\lambda ^I:=I_\lambda ^{-1}(I_\lambda (M_\lambda )\cap K)$. Then

$$\begin{aligned} \hbox {ind}_K(\pi _0,I_0,V_0)=\hbox {ind}_K(\pi _1,I_1,V_1). \end{aligned}$$

Before proofs, let us comment some of assumptions in the above theorems.

Remark 2.5

If $Fix\, g\cap \partial _{K^-}(U_V)=\emptyset$ in (2.5) and Theorem 2.1, then $Fix\, g \cap U_V$ is a compact set. Indeed, it follows from the continuity of $g$ and compactness of the impulse map in the following way. We have $I(M^I)\subset K$, $K$ is closed in ${\mathbb R}^n$, and $cl(I(M))$ is compact. Hence, $cl(I(M^I))\subset K$ is also compact. Now, $cl (r(I(M^I)))\subset cl(r(cl(I(M^I))))=r(cl(I(M^I)))\subset K^-$, and $r(cl(I(M^I)))$ is compact because $r$ is continuous.

If $I(V\cap M)\subset K$ in Theorem 2.1, then $U_V=int_{K^-}(I^{-1}(K\cap V)= int_{K^-}(V\cap M)$, and

$$\begin{aligned}\hbox {ind}_K(\pi ,I,V)=\hbox {ind}(g,int_{K^-}(V\cap M)).\end{aligned}$$

In particular, if $V\subset M$ is an open subset of $K^-$, then

$$\begin{aligned} \hbox {ind}_K(\pi ,I,V)=\hbox {ind}(g,V), \end{aligned}$$

as in paper [14].

If $V\cap M=\emptyset$ (see Fig. 1) or $I(M)\cap K=\emptyset$ (see Fig. 2), then $U_V=\emptyset$, and we trivially obtain $\hbox {ind}_K(\pi ,I,V)=0$.

Fig. 1

A system with $V\cap M=\emptyset$

Fig. 2

A system with $I(M)\cap K=\emptyset$

The assumption $M\subset \bigcap _{\lambda \in [0,1]} K_\lambda ^-$ in Theorem 2.3 is sensible. One knows that

$$\begin{aligned} K_\Rightarrow ^\lambda :=\{x\in \partial K ; f_\lambda (x)\not \in T_K(x)\}\subset K_\lambda ^-\subset cl K_\Rightarrow ^\lambda , \end{aligned}$$

if semiflows are generated by equations $\dot{x}=f_\lambda (x)$ ([7], Lemma 5.1). Assume that $M\subset \{x\in \partial K ; f_{\lambda _0}(x)\not \in T_K(x)\}$ and $(f_\lambda )$ is a continuous family of maps. Then, for $\lambda$ sufficiently near $\lambda _0$, $M\subset \{x\in \partial K ; f_{\lambda }(x)\not \in T_K(x)\}\subset K_\lambda ^-$.

The existence of a family of retractions $r_\lambda$ in Theorem 2.3 is also natural. Indeed, if $K$ is bounded and $\pi _{\lambda _0}$ does not have a viable trajectory in $K$ for some $\lambda _0\in [0,1]$, then $\pi _{\lambda }$ does not have a viable trajectory in $K$ for $\lambda$ sufficiently close to $\lambda _0$. To check this, let us assume, by contrary, that there is a sequence $\lambda _n\rightarrow \lambda _0$ and a viable trajectory $\pi _{\lambda _n}(\cdot ,x_n)$ in $K$ for every $n\ge 1$. By the compactness of $K$ we can assume, without any loss of generality, that $x_n\rightarrow x\in K$. From the continuity of a family of semiflows it follows that $\pi _{\lambda _0}(\cdot ,x)$ is viable in $K$; a contradiction.

Observe that in Theorem 2.4 we have a family of maps $g_\lambda$ with possibly different domains and codomains, and, in fact, we consider an otopy introduced in [4] in the context of vector fields on manifolds.

Proof of Theorem 2.1

It is sufficient to notice that $0\ne \hbox {ind}_K(\pi ,I,V)=\hbox {ind}(g,U_V)$ implies $Fix\, g\cap U_V\ne \emptyset$ so, there is a point $p\in U_V\subset V$ with $r(I(p))=p$. This means that a solution of (1.1) starting from $I(p)$ is viable in $K$ and periodic, and touches the barrier in $p\in V$.$\square$

Proof of Theorem 2.2

Notice that

$$\begin{aligned} U_V= & {} int_{K^-}(I^{-1}(K)\cap V)\\= & {} int_{K^-}(I^{-1}(K)\cap (V_1\cup V_2))\\= & {} int_{K^-}([I^{-1}(K)\cap V_1]\cup [I^{-1}(K)\cap V_2] \end{aligned}$$

Denote $A:=I^{-1}(K)\cap V_1$ and $B=I^{-1}(K)\cap V_2$. We are going to check that $U_V=U_{V_1}\cup U_{V_2}$, i.e., $int_{K^-}(A\cup B)=int_{K^-}A\cup int_{K^-}B$. Of course, ‘$\supset$’ is always true. It is sufficient to prove that each open set contained in $A\cup B$ can be represented as a sum of open subsets in $A$ and $B$, respectively.

Let $W$ be an open subset of $K^-$ contained in $A\cup B=I^{-1}(K)\cap V$. Define $W_1:=W\cap A$ and $W_2:=W\cap B$. Since $W_1\subset V_1$ and $V_1$ is open in $K^-$, for an arbitrary $x\in W_1$ there exists $r_0>0$ such that $B(x,r_0)\subset V_1$, where $B(x,r_0)$ is an open ball in $K^-$. The set $W$ is open so there is $0<r\le r_0$ such that $B(x,r)\subset W\subset I^{-1}(K)\cap V$. Therefore, $B(x,r)\subset V_1\cap I^{-1}(K)=A$. Similarly we check that $W_2$ is an open set contained in $B$.

Now, from the assumption it follows that $Fix\, g\cap (U_{V_1}\cup U_{V_2})$ is a compact set. Moreover, $U_{V_1}\cup U_{V_2}=\emptyset$ so $Fix\, g\cap U_{V_i}$ is also compact for $i=1,2$. By the additivity property of the fixed point index on ANRs we obtain

$$\begin{aligned} \hbox {ind}_K(\pi ,I,V)= & {} \hbox {ind}(g,U_V)=\hbox {ind}(g,U_{V_1})+\hbox {ind}(g,U_{V_2})\\= & {} \hbox {ind}_K(\pi ,I,V_1)+\hbox {ind}_K(\pi ,I,V_2). \end{aligned}$$

$\square$

Proof of Theorem 2.3

At first we prove that $U_{V_\lambda }=U_{V_{\lambda _0}}$ for every $\lambda \in [0,1]$. To do this we take any open set $W\subset K_\lambda ^-$ contained in $I^{-1}(K)\cap V$. Let $\lambda '\in [0,1]$ be arbitrary. Since $V$ is open in $K_\lambda ^-$ and in $K_{\lambda '}^-$, there is $r_0>0$ such that for every $0<r\le r_0$ we have $B_{K_\lambda ^-}(x,r)\cup B_{K_{\lambda '}^-}(x,r)\subset V$, and consequently, $B_V(x,r)=B_{K_\lambda ^-}(x,r)= B_{K_{\lambda '}^-}(x,r)$, where these balls are in spaces $V, K_\lambda ^-, K_{\lambda '}^-$, respectively. But $W$ is open in $K_\lambda ^-$ so there is $0<\delta \le r_0$ such that $B_{K_\lambda ^-}(x,\delta )\subset W$ which implies that $B_{K_{\lambda '}^-}(x,\delta )=B_{K_\lambda ^-}(x,\delta )\subset W$. Hence, $W$ is open in $K_{\lambda '}^-$.

Consider the maps $g_\lambda$ as $g_\lambda :M^I\rightarrow X$, i.e., with a codomain $X$. By the assumption, the set $U_{V_{\lambda _0}}$ is open in $X$. Therefore, we have a homotopy $g_\lambda :U_{V_{\lambda _0}}\rightarrow X$. Now, we use the contraction and homotopy properties of the fixed point index on ANRs to obtain

$$\begin{aligned} \hbox {ind}_K(\pi _0,I,V)= & {} \hbox {ind}(g_0,U_{V_{\lambda _0}})=\hbox {ind}_X(g_0,U_{V_{\lambda _0}})\\= & {} \hbox {ind}_X(g_1,U_{V_{\lambda _0}})=\hbox {ind}_K(\pi _1,I,V), \end{aligned}$$

where, to distinguish, $\hbox {ind}_X$ stands for the fixed point index on the ANR $X$.$\square$

Remark 2.6

It is not obvious that $U_{V_{\lambda _0}}$, which is open in every $K_\lambda ^-$, is open in $X$, simultaneously. Some sufficient conditions for this property are presented in a separate section (see Sect. 3).

Proof of Theorem 2.4

We show that the index $\hbox {ind}_K(\pi _\lambda ,I_\lambda ,V_\lambda )$ is locally constant, i.e., for every $\lambda _0\in [0,1]$ there exists $\delta >0$ such that $\hbox {ind}_K(\pi _\lambda ,I_\lambda ,V_\lambda ) = \hbox {ind}_K(\pi _{\lambda _0},I_{\lambda _0},V_{\lambda _0})$ for each $\lambda \in (\lambda _0-\delta ,\lambda _0+\delta ) \cap [0,1]$.

Fix any $\lambda _0\in [0,1]$. From the assumption on the set $F$ it follows that $F_{\lambda _0}:=Fix\, g_{\lambda _0} \cap U_{V_{\lambda _0}}$ is a compact set as an image of $F$ by the projection $p:\partial K\times [0,1]\rightarrow \partial K, p(x,\lambda )=x$. For every point $x\in F_{\lambda _0}$ there exists $r(x)>0$ such that $B_X(x,r(x))\times (\lambda _0-r(x),\lambda _0+r(x))\subset \bigcup _{\lambda \in [0,1]} U_{V_\lambda }\times \{\lambda \}$. Hence, we can take a finite covering $\{B_X(x_k,r(x_k))\times (\lambda _0-r(x_k),\lambda _0+r(x_k))\}_{k=1}^m$ of $F_{\lambda _0}\times \{\lambda _0\}$, and its Lebesgue number $l$. Then, denoting $I_{\lambda _0}:=(\lambda _0-l,\lambda _0+l)\cap [0,1]$, we obtain

$$\begin{aligned} F_{\lambda _0}\times \{\lambda _0\}\subset \bigcup _{x\in F_{\lambda _0}}(B_X(x,l)\times I_{\lambda _0})= \left( \bigcup _{x\in F_{\lambda _0}} B_X(x,l)\right) \times I_{\lambda _0}\subset \bigcup _{\lambda \in I_{\lambda _0}}(U_{V_\lambda }\times \{\lambda \}). \end{aligned}$$

Denote $U:=\bigcup _{x\in F_{\lambda _0}} B_X(x,l)$. Then $U\subset U_{V_\lambda }$ for every $\lambda \in I_{\lambda _0}$, and all assumptions of Theorem 2.3 are satisfied with $V:=U$ and $X:=X_a^b$, $a:=\max \{\lambda _0-l,0\}$, $b:=\min \{\lambda _0+l,1\}$. Hence, for every $\lambda \in I_{\lambda _0}$,

$$\begin{aligned} \hbox {ind}_K(\pi _\lambda ,I_\lambda ,V_\lambda )=\hbox {ind}(g_\lambda ,U_{V_\lambda })= \hbox {ind}(g_\lambda ,U)=const. \end{aligned}$$

By the compactness of $[0,1]$ we get $\hbox {ind}_K(\pi _0,I_0,V_0)=\hbox {ind}_K(\pi _1,I_1,V_1)$.$\square$

3 Sufficient conditions for the homotopy property

As noted in Remark 2.6, conditions which are sufficient for the homotopy property (see Theorem 2.3) need a deeper study. We start with a rather obvious observation:

Remark 3.1

If $U$ is an open subset of each $A_i, i\in \{1,\ldots ,m\}$, then $U$ is open in $X:=\bigcup _{i=1}^m A_i$.

Indeed, in a simple proof for every $x\in U$ and $i\in \{1,\ldots ,m\}$ we find $r_i>0$ such that $U\supset B_{A_i}(x,r_i)=B_X(x,r_i)\cap A_i$. Taking $r:=\min \{r_1,\ldots ,r_m\}$ we immediately obtain

$$\begin{aligned} B_X(x,r)=\bigcup _{i=1}^m (B_X(x,r)\cap A_i) =\bigcup B_{A_i}(x,r)\subset U, \end{aligned}$$

which means that $U$ is open in $X$.

It is easily seen that the above arguments fail in the case of infinitely many sets $A_i$.

Below we present several results which give, aside from the main aim of the section, interesting properties of exit sets.

Proposition 3.2

If $(f_\lambda )$, $\lambda \in [0,1]$, is a continuous family of vector fields in ${\mathbb R}^n$ generating, via equations $\dot{x}=f_\lambda (x)$, a continuous family of semiflows $\pi _\lambda$, and $K$ is a closed subset of ${\mathbb R}^n$ with compact exit sets $K_\lambda ^-$, then the map $\phi ^-:[0,1]\multimap {\mathbb R}^n$, $\phi ^-(\lambda )=K_\lambda ^-$ is lsc (lower semicontinuous).

Proof

Suppose that $\phi ^-$ is not lsc. Then there are $\lambda _0\in [0,1]$ and $\varepsilon >0$ such that for every $\delta >0$ we can find $\lambda \in [0,1]$ with $|\lambda -\lambda _0|<\delta$ and $K_{\lambda _0}^-\not \subset N_\varepsilon (K_\lambda ^-)$, where $N_\varepsilon (\cdot )$ denotes an $\varepsilon$-neighborhood of a set. Taking $\delta =\frac{1}{m}$, $m\ge 1$, we obtain, for every $m\ge 1$, a number $\lambda _m\in (\lambda _0-\frac{1}{m},\lambda _0+\frac{1}{m})\cap [0,1]$ and a point $x_m\in K_{\lambda _0}^-$ such that $dist(x_m,K_{\lambda _m}^-)\ge \varepsilon$.

By the compactness of $K_{\lambda _0}$ we obtain, up to subsequence, that $x_m\rightarrow x_0$ for some $x_0\in K_{\lambda _0}^-$. Consider a family of Cauchy problems

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \dot{x}=f_{\lambda _m}(x),\\ x(0)=x_0 \end{array} \right. \end{aligned}$$

(3.1)

with solutions $y_m$ defined on a common interval $[0,T]$, $T>0$. From the convergence $\lambda _m\rightarrow \lambda _0$ it follows that $y_m\rightarrow y$, where $y$ is a solution to

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \dot{x}=f_{\lambda _0}(x),\\ x(0)=x_0. \end{array} \right. \end{aligned}$$

(3.2)

We know that, for every $\eta >0$, there is a positive time $s\in (0,\rho )$ with $y(s)\not \in K$. Take $s$ such that $|x_0-y(t)|<\frac{\varepsilon }{3}$ for every $t\in [0,s]$. By the convergence $y_m\rightarrow y$ and closedness of the set $K$ we get that $y_m(s)\not \in K$ for sufficiently big $m\ge 1$. It implies that $y_m(s_m)\in K_{\lambda _m}^-$ for some $s_m\in [0,s)$.

We can assume that $m$ is so big that $|y_m(t)-y(t)|<\frac{\varepsilon }{3}$ for every $t\in [0,s]$. Then

$$\begin{aligned} |y_m(s_m)-x_m|\le |y_m(s_m)-y(s_m)|+|y(s_m)-x_0|+|x_0-x_m|<\varepsilon , \end{aligned}$$

which contradicts the inequality $dist(x_m,K_{\lambda _m}^-)\ge \varepsilon$. $\square$

Under some regularity assumptions on the set $K$ one obtains the following

Proposition 3.3

Assume that $K\subset {\mathbb R}^n$ is a closed $C^1$-manifold with a boundary $\partial K$, $(f_\lambda )$ is as in Proposition 3.2, and $int_{\partial K} K_\lambda ^-= K_\Rightarrow ^\lambda$ for every $\lambda \in [0,1]$ (see Remark 2.5.4). Then the map $\phi ^+:[0,1]\multimap {\mathbb R}^n$, $\phi ^+(\lambda )=cl(\partial K{\setminus }K_\lambda ^-)$ has a closed graph.

Proof

Assume that $[0,1]\ni \lambda _m\rightarrow \lambda$ and $y_m\rightarrow y$, where $y_m\in cl(\partial K{\setminus }K_{\lambda _m}^-)$. Take a sequence $(x_m)$ such that $x_m\in \partial K{\setminus }K_{\lambda _m}^-$ and $|x_m-y_m|<\frac{1}{n}$, for every $n\ge 1$. Then, obviously, $x_m\rightarrow y$. We shall show that $y\in cl(\partial K{\setminus }K_{\lambda _0}^-)$.

Suppose it is not true. Then $y\in int_{\partial K}K_{\lambda _0}^-$ which implies, by the assumption, that $\langle n(y),f_{\lambda _0}(y)\rangle =:d>0$, where $n(y)$ stands for the outer normal vector to $K$ in $y$. Using the continuity of the family $(f_{\lambda })$, the map $n(\cdot )$ and an inner product, we get, for $m$ big enough,

$$\begin{aligned} \langle n(x_m),f_{\lambda _m}(x_m)\rangle= & {} \langle n(x_m)-n(y),f_{\lambda _m}(x_m)\rangle + \langle u(y),f_{\lambda _0}(y)\rangle \\&+ \langle u(y), f_{\lambda _m}(x_m)-f_{\lambda _0}(y)\rangle \\\ge & {} -|n(x_m)-n(y)|(|f_{\lambda _0}(y)|+1)+d -|n(y)||f_{\lambda _m}(x_m)-f_{\lambda _0}(y)|. \end{aligned}$$

It implies that $\langle n(x_m),f_{\lambda _m}(x_m)\rangle >0$ for sufficiently big $m$. Hence, $x_m\in K_{\lambda _m}^-$; a contradiction. $\square$

We use Propositions 3.2 and 3.3 to prove:

Proposition 3.4

Let assumptions of Proposition 3.3 be satisfied with compact exit sets $K_\lambda ^-$. Assume that $U\subset int_{\partial K}K_{\lambda _0}^-$ is open in $K_{\lambda _0}^-$. Then

$$\begin{aligned} \forall x\in U \ \exists r>0 \ \exists \delta >0 \ \forall \lambda \in (\lambda _0-\delta ,\lambda _0+\delta )\cap [0,1] \ : \ dist(x,\partial _{\partial K}K_\lambda ^-)>r. \end{aligned}$$

Proof

Suppose that for some $x\in U$ we have

$$\begin{aligned} \forall r>0 \ \forall \delta >0 \ \exists \lambda \in (\lambda _0-\delta ,\lambda _0+\delta )\cap [0,1] \ : \ dist(x,\partial _{\partial K}K_\lambda ^-)\le r. \end{aligned}$$

Take $r=\frac{1}{m}$ and $\delta =\frac{1}{m}$. Then, for every $m\ge 1$, we can find $\lambda _m\in (\lambda _0-\frac{1}{m},\lambda _0+\frac{1}{m})\cap [0,1]$ such that $dist(x,,\partial _{\partial K}K_{\lambda _m}^-)\le \frac{1}{m}$ which implies that there exists $x_m\in \partial _{\partial K}K_{\lambda _m}^-$ with $|x-x_m|\le \frac{1}{m}$.

From Proposition 3.2 it follows that for every $m\ge 1$ there exists $\eta >0$ such that $K_{\lambda _0}^-\subset N_{\frac{1}{m}}(K_\lambda ^-)$ for each $\lambda \in [0,1]$ with $|\lambda -\lambda _0|<\eta$. Therefore, we can assume that $K_{\lambda _0}^-\subset N_{\frac{1}{m}}(K_{\lambda _m}^-)$. By Proposition 3.3 the map $\phi ^+:[0,1]\multimap {\mathbb R}^n$, $\phi ^+(\lambda )=cl(\partial K{\setminus }K_\lambda ^-)$ has a closed graph. Hence, since $\lambda _m\rightarrow \lambda _0$, $x_m\rightarrow x$ and $x_m\in \partial _{\partial K}K_{\lambda _m}^-\subset cl(\partial K{\setminus }K_{\lambda _m}^-)$, we obtain that $x\in cl(\partial K{\setminus }K_{\lambda _0}^-)$. But we know that $x\in K_{\lambda _0}^-$, so $x\in \partial _{\partial K}K_{\lambda _0}^-$ which contradicts the assumption that $x\in U\subset int_{\partial K}K_{\lambda _0}^-$. $\square$

Assumptions of Proposition 3.4 are sufficient for one of the main proof arguments in the homotopy property of the impulsive index (see Remark 2.6) as we can see in the following:

Proposition 3.5

Under assumptions of Proposition 3.4 every set $U\subset int_{\partial K} K_{\lambda _0}^-$, which is open in $K_{\lambda _0}^-$ and contained in each $K_\lambda ^-$, is also open in $X=\bigcup _{\lambda \in [0,1]} K_\lambda ^-$.

Proof

It is easy to see that $U$ is open in $\partial K$. Moreover, $U\subset X\subset \partial K$. This completes the proof. $\square$

Open problem. Is the following property true?

If $(f_\lambda )$, $\lambda \in [0,1]$, is a continuous family of vector fields in ${\mathbb R}^n$ generating a continuous family of semiflows $\pi _\lambda$, and $K$ is a closed subset of ${\mathbb R}^n$ with compact exit sets $K_\lambda ^-$, then the map $\phi ^+:[0,1]\multimap {\mathbb R}^n$, $\phi ^+(\lambda )=cl(\partial K{\setminus }K_\lambda ^-)$ has a closed graph.

In other words, is Proposition 3.3 true without $C^1$-regularity of the set and an additional assumption $int_{\partial K} K_\lambda ^-= K_\Rightarrow ^\lambda$ on the exit set?

4 Computation of the index via Poincaré-type maps

In the section we briefly remind the technique of computation of the fixed point index $\hbox {ind}(g_\lambda ,U)$ and, consequently, the impulsive index $\hbox {ind}_K(\pi ,I,V)$, proposed firstly in [14]. The idea is partially based on the Poincaré map and Poincaré section technique used in the study of periodic trajectories of ordinary smooth flows.

Let $K$ be a closed subset of ${\mathbb R}^n$, and we are given a problem (1.1) with the differential equation generating a closed exit set $K^-$ which is a strong deformation retract of $K$ via the homotopy along trajectories (see Sect. 2). Let $M\subset K^-$ be a barrier, and $V\subset K^-$ be any subset open in $K^-$.

In what follows we assume that the vector field $f:{\mathbb R}^n\rightarrow {\mathbb R}^n$ and the impulse function $I:M\rightarrow {\mathbb R}^n$ are of class $C^1$, at least. Let $p\in M\cap V$ be a fixed point of $g=rI:M^I\rightarrow K^-$. Denote $q:=I(p)\in K$ and the time $T:=\tau _K(q)$. Obviously, $p=\pi (q,T)$, where $\pi$ is a flow generated by $f$. We shall also use the assumption that $M$ is a level set $\{x\in {\mathbb R}^n\ | \ \gamma (x)=0\}$ for some smooth function $\gamma :{\mathbb R}^n\rightarrow \mathbb R$.

Finally assume that $p\in \int _{\partial K}(M\cap V)$, the vector $f(p)$ is transversal to $\partial K$, and $f(q)$ is transversal to $I(M)$. Then, for some $N\subset M\cap V$ with $p\in int_{K^-}N$ there exists a $\delta >0$ such that for every $y\in \{\pi (x,t) \ | \ x\in N, |t|<\delta \}$ one can find a unique $\delta (y)\in (-\delta ,\delta )$ with $\pi (y,\delta (y))\in N$. Such a neighborhood $N$ is a Poincaré section for $\pi$.

Under the above assumptions we can check that the exit time function $\tau _K$ is smooth in a neighborhood of $q$. Without any loss of generality we can assume that $I(N)$ is this neighborhood. Since $g(x)=\pi (I(x),\tau _K(I(x)))$, we get

$$\begin{aligned} g_x(x)=\pi _x(I(x),\tau _K(I(x)))I_x(x)+ \pi _t(I(x),\tau _K(I(x)))(\tau _K)_x(I(x))I_x(x), \end{aligned}$$

(4.1)

where, e.g., $g_x(x)$ stands for the state variable derivative of $g$. Remind that (one easily checks it) $t\mapsto f(\pi (q,t))$ is a solution on $[0,T]$ to the linear differential equation $\dot{y}=f_x(\pi (q,t))y$ with an initial condition $y(0)=f(q)$. It is seen that $y(T)=f(p)$. We know also that $t\mapsto \pi _x(q,t)$ is a fundamental matrix solution to the equation $\dot{Y}=f_x(\pi (q,t))Y$ with $Y(0)=\pi _x(q,0)$. Hence, we get

$$\begin{aligned} Y(T)f(q)=\pi _x(q,T)f(q)=f(\pi (q,T))=f(p). \end{aligned}$$

Now we choose in ${\mathbb R}^n$ two bases $\mathcal B_1:=\{f(p),a_2,\ldots ,a_n\}$ and $\mathcal B_2:=\{f(q),b_2,\ldots ,b_n\}$ such that $a_i$ are parallel to $M$ at $p$ (they form a basis in a tangent space to $M$ at $p$) and $b_i$ are parallel to $I(M)$ at $q$. We consider a linear map $\pi _x(q,T)$ in bases $\mathcal B_2$ and $\mathcal B_1$ obtaining the matrix

$$\begin{aligned}\pi _x(q,T)= \left[ \begin{array}{c@{\quad }c} 1&{}A\\ 0&{}B \end{array}\right] , \ \hbox { where}\quad A\in M_{1\times (n-1)} \quad \hbox {and}\quad B\in M_{(n-1)\times (n-1)}. \end{aligned}$$

If $w\in {\mathbb R}^n$ is parallel to $M$ at $p$, then it has the form $w=\left[ \begin{array}{c} 0\\ w_M\end{array}\right]$. Moreover, $I_x(p)\left[ \begin{array}{c} 0\\ w_M\end{array}\right] = \left[ \begin{array}{c} 0\\ w_{IM}\end{array}\right]$, where $\left[ \begin{array}{c} 0\\ w_{IM}\end{array}\right]$ is a vector parallel to $I(M)$ at $q$.

Now, using (4.1) we can compute a derivative of a Poincaré map. Let $w$ be a vector parallel to $M$ at $p$. Then

$$\begin{aligned} Dg(p)w= & {} \pi _x(q,T)I_x(p)\left[ \begin{array}{c} 0\\ w_{M}\end{array}\right] + \pi _t(q,T)(\tau _K)_x(q)I_x(p) \left[ \begin{array}{c} 0\\ w_{M}\end{array}\right] \\= & {} \pi _x(q,T)\left[ \begin{array}{c} 0\\ w_{IM}\end{array}\right] + f(p)(\tau _K)_x(q) \left[ \begin{array}{c} 0\\ w_{IM}\end{array}\right] \\= & {} \left[ \begin{array}{c} Aw_{IM}\\ Bw_{IM}\end{array}\right] + \left[ \begin{array}{c} c\\ 0\end{array}\right] , \end{aligned}$$

for some $c\in \mathbb R$. Since $g$ maps $M$ into $K^-(f)$, we obtain in the basis $\mathcal B_1$

$$\begin{aligned} Dg(p)w=\left[ \begin{array}{c} 0\\ Bw_{IM}\end{array}\right] = \left[ \begin{array}{c@{\quad }c} 0&{}0\\ 0&{}B\cdot T_pI\end{array}\right] w, \end{aligned}$$

where $T_pI:T_pM\rightarrow T_qI(M)$ is the derivative of $I$ treated as a map between manifolds. This is a very good information because in a computation of the degree one uses eigenvalues of the derivative. Here, eigenvalues of $Dg(p)$ correspond to those of the matrix $B\cdot T_pI\in M_{(n-1)\times (n-1)}$.

Taking a sufficiently small neighborhood $U$ of $p$ we obtain

$$\begin{aligned} \hbox {ind}(g,p):=\hbox {ind}(g,U)=\hbox {deg}(i-g,U)=(-1)^\beta , \end{aligned}$$

where $\beta$ is the sum of the multiplicities of the negative real eigenvalues of $\hbox {id}_{T_pM} -B\cdot T_pI$.

To illustrate how the above technique works in practice we present an example which is a modification of Example 4.1 in [14].

Example 4.1

We have a system of equations

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}=1\\ \dot{y}=y \end{array}\right. \end{aligned}$$

generating a global flow $\pi ((x,y),t)=(x+t,ye^t)$ on $\mathbb R^2$. For the set of constraints $K:=[-1,4]\times [-3/2,2]$ we obtain the exit set $K^-=K_1\cup K_2\cup K_3$, where $K_i$ are the faces $x=4$, $y=2$ and $y=-3/2$ of $K$, respectively (see Fig. 3). Let $M:=\{(4,y) \ | \ y\in [-3/2,2]\} = cl_{K^-}\{(4,y) \ | \ y\in (-3/2,2)\}$ and $I:M\rightarrow {\mathbb R}^n$ be an impulse function given by $I(x,y):=(y,y(y-1))$. For simplicity, take $V=\{(4,y) \ | \ y\in (-3/2,2)\}$ which is open in $K^-$.

Fig. 3

Illustration to Example 4.1

It is easy to check that $M^I=I^{-1}(I(M)\cap K) = \{(4,y) \ | \ y\in [-1,2]\}$ and $U_V=int_{K^-}(I^{-1}(I(V\cap M)\cap K)\cap V)=\{(4,y) \ | \ y\in (-1,2)\}$.

Since $g(4,-1)=I(4,-1)=(-1,2)\in K^-$ and $g(4,2)=I(4,2)=(2,2)\in K^-$, we have $\hbox {ind}(g,U_V)=0$. On the other hand, it is evident that $p:=(4,0)$ is a fixed point of $g$ with $q:=I(p)=(0,0)$. Now, as in [14], we take two bases $\mathcal B_1:=\{(1,0), (0,1)\}$ and $\mathcal B_2:=\{(1,0), (1,-1)\}$ of $\mathbb R^2$, and notice that $\pi _x(q,T)=\left[ \begin{array}{c@{\quad }c} 1&{}\quad \! 1\\ 0&{}\quad \! -e^T \end{array}\right]$ in bases $\mathcal B_2$ and $\mathcal B_1$, i.e., $\pi _x(q,T):(\mathbb R^2,\mathcal B_2)\rightarrow (\mathbb R^2,\mathcal B_1)$, where $T=\tau _K(q)=4$.

Since $DI(p)=\left[ \begin{array}{c@{\quad }c} 0&{}0\\ 0&{}1 \end{array}\right]$ in bases $\mathcal B_1$ and $\mathcal B_2$, for any vector $w=\left[ \begin{array}{c} 0\\ v \end{array}\right]$ parallel to $M$ we get

$$\begin{aligned} Dg(p)(w)=Dg(p)\left[ \begin{array}{l} 0\\ v \end{array}\right] =\left[ \begin{array}{l} 0\\ -e^{T}v \end{array}\right] =\left[ \begin{array}{l@{\quad }l} 0&{}0\\ 0&{}-e^T \end{array}\right] \left[ \begin{array}{l} 0\\ v \end{array}\right] . \end{aligned}$$

This implies that $\hbox {ind}(g,p):=\hbox {ind}(g,W)=\hbox {deg}(i-g,W) = \hbox {sgn}(1+e^T)=1$ for some small neighborhood $W$ of $p$ in $U_V$. From the additivity property of the fixed point index we obtain $\hbox {ind}(g,U_V{\setminus }cl_{K^-(f)}W)=\hbox {ind}(g,U_V)-\hbox {ind}(g,W)=-1\ne 0$ which implies the existence of another periodic trajectory.

5 Concluding remarks

We start the section with a discussion on a possible alternative definition of the impulsive index. Instead of conditions (A1)–(A4) from Sect. 2 we assume

• (Z1) $\partial K$ is a closed ANR, and the exit set $K^-$ is closed,

• (Z2) $M=cl_{\partial K}(int_{\partial K}M)$,

• (Z3) $I$ is a compact map, i.e. $I(M)$ is relatively compact,

• (Z4) $dist(I(M),M)>0$.

As before, we assume (2.1) with the retraction $r:K\rightarrow K^-$, $r(x)=\pi (x,\tau _K(x))$. Define $M^I:=I^{-1}(K)$ and $g:M^I\rightarrow \partial K, \ g(x):=r(I(x))$ for $x\in M^I$. Notice that a codomain is different from the one in (2.3). Of course, $g(M^I)\subset K^-$.

Let $V$ be an arbitrary open subset of $\partial K$. We define

$$\begin{aligned} U_V:=int_{\partial K}(I^{-1}(K)\cap V)\subset V, \end{aligned}$$

(5.1)

and assume that

$$\begin{aligned} Fix\, g \cap U_V \hbox { is a compact set}. \end{aligned}$$

Now we put

$$\begin{aligned} \hbox {ind}_K'(\pi ,I,V):=\hbox {ind}(g,U_V), \end{aligned}$$

(5.2)

where $\hbox {ind}(g,U_V)$ is a fixed point index in an ANR $\partial K$, not in $K^-$.

This index has standard properties (see Sect. 2) with the homotopy property under easier assumptions. We give, for instance, an analogue to Theorem 2.3 as the following

Theorem 5.1

(Homotopy I) Let $\pi _\lambda$, $\lambda \in [0,1]$, be a continuous family of semiflows. Assume that, for every $\lambda \in [0,1]$, the exit set $K_\lambda ^-$ for $\pi _\lambda$ is closed, there are retractions $r_\lambda :K\rightarrow K_\lambda ^-$, $r_\lambda (x)=\pi _\lambda (x,\tau _K(x))$, and $M=cl_{\partial K}(int_{\partial K}M)$. Moreover, assume that the set $\bigcup _{\lambda \in [0,1]} r_\lambda (I(M)\cap K)$ is relatively compact.

Let $V$ be an open subset of $\partial K$, and $Fix\, g_\lambda \cap U_{V_{\lambda }}$ is compact.

Then the numbers $\hbox {ind}_K'(\pi _\lambda ,I,V)$ are well defined, and

$$\begin{aligned} \hbox {ind}_K'(\pi _0,I,V)=\hbox {ind}_K'(\pi _1,I,V). \end{aligned}$$

Proof

Notice that $U_{V_\lambda }=int_{\partial K}(I^{-1}(K)\cap V)=U_{V_0}$ is independent of the choice of $\lambda$. So, we have a homotopy $g_\lambda :U_{V_0}\rightarrow \partial K$ with assumptions allowing to apply the homotopy property of the index on ANRs. This finishes the proof. $\square$

One would think that the index $\hbox {ind}_K'$ is better because it simplifies considerations. However, it does not cover some important cases. Indeed, the set $V$ cannot be equal to $K^-$ since $K^-$ is closed and usually not open in $\partial K$. The following concrete example shows the difference between the results obtained by the use of $\hbox {ind}_K$ and $\hbox {ind}_K'$.

Example 5.2

Let $K$ be a rectangle in $\mathbb R^2$, for instance $K=[0,3]\times [0,2]$ and $\pi$ is generated by the equation $(\dot{x},\dot{y})=(g(t),0)$ with $g(t)>0$. Then, obviously, $K^-=\{3\}\times [0,2]$. Put $M=K^-$, see Figs. 4, 5, 6 and 7.

At first, assume that $I(x,y):=(1,y)$ and $V=M=K^-$ which is allowed in Sect. 2 (Fig. 4). Then $U_V=K^-$ and $\hbox {ind}_K(\pi ,I,V)=\hbox {ind}(id, K^-) = \chi (K^-)=1$, where $\chi (K^-)$ is the Euler characteristic of the set $K^-$ (equal to the Lefschetz number of $id$).

Notice that, for every open subset $V_0$ of $\partial K$, the set $U_{V_0}$ constructed in (5.1), even the largest one, is not equal to $K^-$. It implies that the index $\hbox {ind}_K'(\pi ,I,V_0)$ for no $V_0$ (see Fig. 5) is well defined, because $Fix\, id \cap U_{V_0}=U_{V_0}$ is not compact.

Now, let $M, V, V_0$ be as above, and $I(x,y):=(1,\root 3 \of {y-1}+1)$. Consider $V_1:=\{3\}\times (1/2,3/2)$ (see Figs. 6, 7). Since $U_{V_1}=V_1$ in both approaches [see (2.4) and (5.1)], and $g(3,y)=(3,\root 3 \of {y-1}+1)$, we easily obtain $\hbox {ind}_K(\pi ,I,V_1)=\hbox {ind}_K'(\pi ,I,V_1)=-1$ and $\hbox {ind}_K(\pi ,I,V)=1$, while $\hbox {ind}_K'(\pi ,I,V_0)=-1$.

Fig. 4

$\hbox {ind}_K(\pi ,I,V)=1$ for $V=M=K^-$

Fig. 5

$\hbox {ind}_K'(\pi ,I,V_0)$ is not defined

Fig. 6

$\hbox {ind}_K(\pi ,I,V_1)=\hbox {ind}_K'(\pi ,I,V_1)=-1$ and $\hbox {ind}_K(\pi ,I,V)=1$

Fig. 7

$\hbox {ind}_K(\pi ,I,V_0)=\hbox {ind}_K'(\pi ,I,V_0)=-1$

Our final remarks concerns possible generalizations of the impulsive index. One knows that a differential inclusion $\dot{x}\in F(x)$ taken instead of $\dot{x}=f(x)$ in (1.1), where $F$ is upper semicontinuous (usc) convex compact valued with a sublinear growth, generates a multivalued dynamical system. Moreover, the solution map ${\mathbb R}^n\ni x_0\multimap S_F(x_0)$, where $S_F(x_0)\in C([0,\infty ),{\mathbb R}^n)$ is a solution set for the Cauchy problem with $x(0)=x_0$, is usc with compact $R_\delta$-values. Under suitable assumptions on the set of constraints $K$ and an exit set, one can obtain results on the existence of viable and stationary trajectories (see, e.g., [12, 13]). If we suppose that there is no viable solution in $K$, it is sensible to assume that there exists a multivalued retraction $\Phi$ of $K$ onto $K_e(F)$, where

$$\begin{aligned} K_e(F) :=\{x_0\in \partial K \ | \ \exists x\in S_F(x_0): \ x \hbox { leaves}\, K\, \hbox {immediately}\} \end{aligned}$$

is a bigger exit set (see [14], or [12] where a discussion on two exit sets $K^-(F)$ and $K_e(F)$ is presented). Now, if $I:M\multimap {\mathbb R}^n$ is a multivalued impulse map, then the construction of the fixed point index of the composition $\Phi \circ I$ is still possible under suitable geometric assumptions on $K_e(F)$ and regularity assumptions on $I$. In particular, if $K_e(F)$ is a compact ANR and $I$ is usc with $R_\delta$-values, then the index proposed in [2] could be used (see also [15] for other versions of the index). Details of the construction and properties of the impulsive index for multivalued flows and jumps we leave for further considerations.