Viable periodic solutions in state-dependent impulsive problems

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.


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 ⊂ R n of a state space. In other words, we are going to find a solution to the following periodic problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ẋ for someT x > 0, (1.1) where f : R n ⊃ → R n is sufficiently smooth to generate a semiflow, and I : M → 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) ⊂ K . Analysis of models in mathematical biology shows that it is worth considering a weaker assumption I (M) ⊂ R n , in practice at least I (M) ∩ K = ∅. 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.

Impulsive index
Let K ⊂ R n be an arbitrary closed subset, ⊃ K an open neighborhood in R n , and f : → R n be a sufficiently smooth vector field such that the equationẋ = f (x) generates a semiflow π on . We are interested in viable in K solutions so, without any loss of generality, we can assume in the sequel that = R n . Let us recall that by a solution of problem (1.1) we mean a left-continuous function x : [0, ∞) → R n , with a discrete set of discontinuity points, absolutely continuous between these points, and satisfyingẋ(t) = f (x(t)) in a.e. t ≥ 0, and x(t + ) = I (x(t)) in discontinuity points. The set of functions from [0, ∞) to 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, ∞), R n ).
In each point x ∈ K we can consider the Bouligand tangent cone Obviously, if the following tangency condition 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 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: All absolutely continuous solutions toẋ = f (x) leave the set K . Hence, in the whole paper we assume that Of course, (2.2) implies that K − is a closed subset of K . When we consider a family π λ of semiflows, we will assume, analogously, that deformations h λ are given, see below.
Let M ⊂ K − be an impulse set, i.e., there is an impulse function I : M → R n moving instantly each trajectory reaching the set M in a point p ∈ M to a point I ( p) ∈ R n . In practice, we will be interested in impulse functions satisfying I (M) ∩ K = ∅ which means that some trajectories are moving back to K . In our considerations we assume that where int K − A, cl K − A and, in the sequel, ∂ K − A denote the relative interior, closure and boundary of a set A ⊂ K − in K − , respectively. Note that, if M is compact, which is often the case, (A4) is implied by I (M) ∩ M = ∅. 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 We define M I := I −1 (K ) and Notice that, since every absolutely continuous solution toẋ = f (x) leaves the set K , there is a correspondence between elements of P and fixed points of iterations of the map g defined in (2.3). Indeed, let x ∈ P start from x 0 ∈ K . Then z = π(x 0 , τ K (x 0 )) ∈ M, and there is a minimal m ∈ N such that x 0 = π(I (g m (z)), s) for some s ∈ [0, τ K (I (g m (z)))]. It implies that z = π(x 0 , τ 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) := π(I (g k (z)), t − σ k (I (z))) for t ∈ (σ k (I (z)), σ k+1 (I (z))], k ≥ 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 and assume that We define an impulsive index of π on V as where 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 2.4 (Homotopy II)
Let π λ , λ ∈ [0, 1], be a continuous family of semiflows, and assume that, for every λ ∈ [0, 1], the exit set K − λ for π λ is a closed ANR, there are retractions and impulse functions I λ → R n . Moreover, assume that, for every pair Before proofs, let us comment some of assumptions in the above theorems. (2.5) and Theorem 2.1, then Fi x g ∩ 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 ) ⊂ K , K is closed in R n , and cl(I (M)) is compact. Hence, cl(I (M I )) ⊂ K is also compact. Now, cl(r (I (M I ))) ⊂ cl(r (cl(I (M I )))) = r (cl(I (M I ))) ⊂ K − , and r (cl(I (M I ))) is compact because r is continuous.
The assumption M ⊂ λ∈[0,1] K − λ in Theorem 2.3 is sensible. One knows that is a continuous family of maps. Then, for λ sufficiently The existence of a family of retractions r λ in Theorem 2.3 is also natural. Indeed, if K is bounded and π λ 0 does not have a viable trajectory in K for some λ 0 ∈ [0, 1], then π λ does not have a viable trajectory in K for λ sufficiently close to λ 0 . To check this, let us assume, by contrary, that there is a sequence λ n → λ 0 and a viable trajectory π λ n (·, x n ) in K for every n ≥ 1. By the compactness of K we can assume, without any loss of generality, that x n → x ∈ K . From the continuity of a family of semiflows it follows that π λ 0 (·, x) is viable in K ; a contradiction.
Observe that in Theorem 2.4 we have a family of maps g λ with possibly different domains and codomains, and, in fact, we consider an otopy introduced in [4] in the context of vector fields on manifolds.
This means that a solution of (1.1) starting from I ( p) is viable in K and periodic, and touches the barrier in p ∈ V .

Proof of Theorem 2.2. Notice that
We are going to check that Now, from the assumption it follows that Fi x g ∩ (U V 1 ∪U V 2 ) is a compact set. Moreover, By the additivity property of the fixed point index on ANRs we obtain Proof of Theorem 2.3. At first we prove that U V λ = U V λ 0 for every λ ∈ [0, 1]. To do this we take any open set where these balls are in spaces V, Consider the maps g λ as g λ : M I → X , i.e., with a codomain X . By the assumption, the set U V λ 0 is open in X . Therefore, we have a homotopy g λ : U V λ 0 → X . Now, we use the contraction and homotopy properties of the fixed point index on ANRs to obtain where, to distinguish, ind X stands for the fixed point index on the ANR X .

Remark 2.6
It is not obvious that U V λ 0 , which is open in every K − λ , is open in X , simultaneously. Some sufficient conditions for this property are presented in a separate section (see Sect. 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: Indeed, in a simple proof for every x ∈ U and i ∈ {1, . . . , m} we find r i > 0 such that U ⊃ B A i (x, r i ) = B X (x, r i ) ∩ A i . Taking r := min{r 1 , . . . , r m } we immediately obtain 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 λ ), λ ∈ [0, 1], is a continuous family of vector fields in R n generating, via equationsẋ = f λ (x), a continuous family of semiflows π λ , and K is a closed subset of R n with compact exit sets K
Proof Suppose that φ − is not lsc. Then there are λ 0 ∈ [0, 1] and ε > 0 such that for every δ > 0 we can find λ ∈ [0, 1] with |λ − λ 0 | < δ and K − λ 0 ⊂ N ε (K − λ ), where N ε (·) denotes an ε-neighborhood of a set. Taking δ = 1 m , m ≥ 1, we obtain, for every m ≥ 1, a number By the compactness of K λ 0 we obtain, up to subsequence, that x m → x 0 for some x 0 ∈ K − λ 0 . Consider a family of Cauchy problems with solutions y m defined on a common interval [0, T ], T > 0. From the convergence λ m → λ 0 it follows that y m → y, where y is a solution to We know that, for every η > 0, there is a positive time s ∈ (0, ρ) with y(s) / ∈ K . Take s such that |x 0 − y(t)| < ε 3 for every t ∈ [0, s]. By the convergence y m → y and closedness of the set K we get that y m (s) / ∈ K for sufficiently big m ≥ 1. It implies that y m (s m ) ∈ K − λ m for some s m ∈ [0, s).
We can assume that m is so big that |y m (t) − y(t)| < ε 3 for every t ∈ [0, s]. Then Under some regularity assumptions on the set K one obtains the following Proposition 3.3 Assume that K ⊂ R n is a closed C 1 -manifold with a boundary ∂ K , ( f λ ) is as in Proposition 3.2, and int ∂ K K − λ = K λ ⇒ for every λ ∈ [0, 1] (see Remark 2.5.4). Then the map φ + : λ m and |x m − y m | < 1 n , for every n ≥ 1. Then, obviously, x m → y. We shall show that y ∈ cl(∂ K \K − λ 0 ). Suppose it is not true. Then y ∈ int ∂ K K − λ 0 which implies, by the assumption, that n(y), f λ 0 (y) =: d > 0, where n(y) stands for the outer normal vector to K in y. Using the continuity of the family ( f λ ), the map n(·) and an inner product, we get, for m big enough, It implies that n(x m ), f λ m (x m ) > 0 for sufficiently big m. Hence, x m ∈ K − λ m ; a contradiction.
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
Proof Suppose that for some x ∈ U we have Take r = 1 m and δ = 1 m . Then, for every m ≥ 1, we can find 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
Proof It is easy to see that U is open in ∂ K . Moreover, U ⊂ X ⊂ ∂ K . This completes the proof.
Open problem. Is the following property true?
In other words, is Proposition 3.3 true without C 1 -regularity of the set and an additional assumption int ∂ K K − λ = K λ ⇒ on the exit set?

Computation of the index via Poincaré-type maps
In the section we briefly remind the technique of computation of the fixed point index ind(g λ , U ) and, consequently, the impulsive index ind K (π, 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 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. In what follows we assume that the vector field f : R n → R n and the impulse function I : M → R n are of class C 1 , at least. Let p ∈ M ∩ V be a fixed point of g = r I : M I → K − . Denote q := I ( p) ∈ K and the time T := τ K (q). Obviously, p = π(q, T ), where π is a flow generated by f . We shall also use the assumption that M is a level set {x ∈ R n | γ (x) = 0} for some smooth function γ : R n → R.
Finally assume that p ∈ ∂ K (M ∩ V ), the vector f ( p) is transversal to ∂ K , and f (q) is transversal to I (M). Then, for some N ⊂ M ∩ V with p ∈ int K − N there exists a δ > 0 such that for every y ∈ {π(x, t) | x ∈ N , |t| < δ} one can find a unique δ(y) ∈ (−δ, δ) with π(y, δ(y)) ∈ N . Such a neighborhood N is a Poincaré section for π.
Under the above assumptions we can check that the exit time function τ 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) = π(I (x), τ K (I (x))), we get where, e.g., g x (x) stands for the state variable derivative of g. Remind that (one easily checks it) t → f (π(q, t)) is a solution on [0, T ] to the linear differential equationẏ = f x (π(q, t))y with an initial condition y(0) = f (q). It is seen that y(T ) = f ( p). We know also that

Now we choose in
where T p I : T p M → T q I (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 Taking a sufficiently small neighborhood U of p we obtain where β is the sum of the multiplicities of the negative real eigenvalues of id T p M − B · T p I . To illustrate how the above technique works in practice we present an example which is a modification of Example 4.1 in [14].
This implies that ind(g, p) := ind(g, W ) = deg(i − g, W ) = 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 ind(g, = −1 = 0 which implies the existence of another periodic trajectory.

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) ∂ K is a closed ANR, and the exit set K − is closed, As before, we assume (2.1) with the retraction r : and assume that Fi x g ∩ U V is a compact set. Now we put ind K (π, I, V ) := ind(g, U V ), (5.2) where ind(g, U V ) is a fixed point index in an ANR ∂ 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 Proof Notice that U V λ = int ∂ K (I −1 (K ) ∩ V ) = U V 0 is independent of the choice of λ. So, we have a homotopy g λ : U V 0 → ∂ K with assumptions allowing to apply the homotopy property of the index on ANRs. This finishes the proof. One would think that the index 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 ∂ K . The following concrete example shows the difference between the results obtained by the use of ind K and ind K . 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 ind K (π, I, V ) = ind(id, K − ) = χ(K − ) = 1, where χ(K − ) is the Euler characteristic of the set K − (equal to the Lefschetz number of id).
Our final remarks concerns possible generalizations of the impulsive index. One knows that a differential inclusionẋ ∈ F(x) taken instead ofẋ = f (x) in (1.1), where F is upper semicontinuous (usc) convex compact valued with a sublinear growth, generates a 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 of K onto K e (F), where K e (F) := {x 0 ∈ ∂ K | ∃x ∈ S F (x 0 ) : x leaves K immediately} 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 R n is a multivalued impulse map, then the construction of the fixed point index of the composition • 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 δ -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.