Coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori via Poincaré operators

Topology and its Applications 255 (2019) 126–140

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Coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori via Poincaré operators Jan Andres 1 Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

a r t i c l e

i n f o

Article history: Received 14 May 2018 Received in revised form 21 January 2019 Accepted 27 January 2019 Available online 30 January 2019 MSC: primary 34B37, 34C28, 37E15 secondary 34A60, 34C40 Keywords: Coexistence of subharmonics on tori Impulsive differential equations and inclusions Poincaré translation operators Sharkovsky-type theorems Multivalued admissible maps

a b s t r a c t The coexistence of subharmonic periodic solutions of various orders is investigated to the first-order vector system of impulsive (upper-) Carathéodory differential equations and inclusions on tori. As the main tool, our recent Sharkovsky-type results for multivalued maps on tori are applied via the associated Poincaré translation operators along the trajectories of given systems. The solvability criteria are formulated, under natural bi-periodicity assumptions imposed on the right-hand sides, in terms of the Lefschetz numbers of admissible impulsive maps. Since the criteria become effective on the circle, the main general theorem can be improved and reformulated there in a more transparent way. The obtained results can be regarded in a certain sense as a nontrivial extension of those due to Poincaré [28], Denjoy [17] and van Kampen [24]. © 2019 Published by Elsevier B.V.

1. Introduction It is well known that the standard Sharkovsky cycle coexistence theorem for interval maps [29] (cf. [3, Chapter 2]) cannot be applied to Carathéodory differential equations x = f (t, x), where f (t, x) ≡ f (t +ω, x) for some ω > 0, because the associated Poincaré translation operators Tω along the trajectories of given scalar equations, satisfying a uniqueness condition, are strictly increasing homeomorphisms. Thus, the periodic orbits of Tω different from fixed point tuples are obviously excluded. For those who are not familiar with the notions of Carathéodory differential equations and the standard Poincaré translation operator Tω along their trajectories, we will recall both of them. E-mail address: [email protected]. The author was supported by the grant IGA_PrF_2017_ 019 “Mathematical Models” of the Internal Grant Agency of Palacký University in Olomouc. 1

https://doi.org/10.1016/j.topol.2019.01.008 0166-8641/© 2019 Published by Elsevier B.V.

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Hence, the scalar equation x = f (t, x), where f (t, x) ≡ f (t +ω, x) for some ω > 0, is called Carathéodory, provided (i) f (·, x) : [0, ω] → R is Lebesgue measurable, for every x ∈ R, (ii) f (t, ·) : R → R is continuous, for almost all t ∈ [0, ω], (iii) |f (t, x)| ≤ r(t)(1 + |x|), for all (t, x) ∈ [0, ω] × R, where r : [0, ω] → [0, ∞) is a Lebesgue integrable function. By its (Carathéodory) solution, we mean a locally absolutely continuous function x : R → R which satisfies this equation for almost all values of argument t ∈ R. Furthermore, Tω : R → R is the associated Poincaré translation operator if it takes the form Tω (x0 ) := {x(t0 + ω) : x(·) is a solution of x = f (t, x) such that x(t0 ) = x0 }. Let us note that, in the literature, the Poincaré translation operators are also called time-t maps, Levinson operators, Poincaré–Andronov operators or shortly just Poincaré operators. For some more details, see e.g. [25]. The same is impossible for the standard Sharkovsky-type theorems on the circle (see e.g. [13,14,19,30], and [3, Chapter 3]), where the Poincaré operators are orientation-preserving homeomorphisms (see e.g. [16, Chapter XVII]), by which all periodic orbits (if any) have the same period (see e.g. [21, Proposition 4.3.8]). Moreover, since the related Lefschetz number (see e.g. [15]) is there trivial, unlike for interval maps, one cannot assure in general the existence of fixed points, determining harmonic periodic solutions, i.e. periodic solutions whose period is the same as the one of the right-hand sides of given differential equations and inclusions. In the lack of uniqueness, the application becomes already feasible. Since the Poincaré operators are then multivalued, the appropriate multivalued versions of Sharkovsky-type theorems were successfully developed for such applications in [4–6,11]. Nevertheless, since the related Lefschetz number remains trivial on the circle (see e.g. [7, Chapter I.6]), our multivalued analogy [4] of the standard (single-valued) theorems, obtained independently by Efremova [19] and Block et al. [14] for circle maps whose Lefschetz number is exclusively nontrivial, cannot be applied to usual differential equations or inclusions (without impulses) again. On higher-dimensional tori Tn , n > 1, where the same obstruction occurs and especially the theorems like those in [5,6,11,13,29,30] have no higher-dimensional analogues, the situation is technically even more delicate; for n = 2, see [1,2], and for n = 3, see [23]. Let us note that our theorem in [4], obtained by means of the multivalued Nielsen theory in [8,10] (cf. also [7, Chapter I.10] and [22, Chapter VII.4]), had to be necessarily formulated in terms of irreducible periodic orbits of coincidences, but not of periodic orbits or periodic points like in [1,2,14,19,23]. In spite of these facts, there is a chance to apply all the mentioned Sharkovsky-type theorems to impulsive differential equations and inclusions (see e.g. [12,18,20,27], because the Lefschetz numbers of the associated Poincaré translation operators need not be any longer trivial. Moreover, these Lefschetz numbers, whose absolute values are equal to Nielsen numbers on tori, can be easily calculated from the assumptions imposed just on the impulsive maps. In this way, for instance, the theorem of Block et al. [14] and Efremova [19] (cf. [3, Chapter 3]) can be applied to impulsive scalar differential equation 

x = F (t, x), x(t+ j )

=

I(x(t− j )),

t = tj := t0 + jω, for some given ω > 0, j ∈ Z,

(1)

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where F : R2 → R is (for the sake of simplicity) continuous, satisfying a uniqueness condition, and such that F (t, x) ≡ F (t + ω, x) and F (t, x) ≡ F (t, x + 1),

(2)

and I : R → R is a continuous impulsive function such that I(x + 1) − I(x) = d ∈ Z, for all x ∈ R.

(3)

As we will see later (cf. Corollary 4.7 below), we will be easily able to formulate the related theorem by ourselves as follows. Theorem 1.1. If d > 1 or d < −2 in (3), then the impulsive equation (1), satisfying a uniqueness condition, admits, under (2), a kω-periodic (mod 1), (piece-wise) C 1 -solution x ∈ C 1 ((tj , tj+1 ), R), j ∈ Z, for every k ∈ N. If d = −2 in (3), then the same is true, for every k ∈ N \ {2}. If d = −1 or d = 0 in (3), then (1) admits, under (2), at least an ω-periodic (mod 1), piece-wise C 1 -solution x ∈ C 1 ((tj , tj+1 ), R), j ∈ Z. Our main aim in this paper is a significant generalization of Theorem 1.1, by virtue of our results in [4], to impulsive, vector, upper-Carathéodory differential inclusions, where the impulsive mappings can be also multivalued. For impulsive scalar differential equations and inclusions, the obtained criteria will be still reformulated in a more transparent way and supplied by further statements, obtained by means of our Sharkovsky-type theorem in [5]. Hence, our paper will be organized as follows. At first, we introduce the useful definitions and recall the Sharkovsky-type theorems which will be applied. Then the multivalued Poincaré translation operators along the trajectories of impulsive differential equations and inclusions will be examined in detail. On this basis, the main theorems about the coexistence of subharmonic periodic solutions of various orders will be carried out on tori and, in particular, on the circle. Finally, we add some concluding comments and remarks. 2. Preliminaries and auxiliary results By the n-torus Tn , n > 1, we will mean as usually either the factor space Rn /Zn = (R/Z)n or the Cartesian product S 1 × · · · × S 1 , where    n-times

S 1 := {x ∈ R2 : |x| = 1} or S 1 := {z = e2πs i : s ∈ [0, 1]}. If not explicitly specified, we will not distinguish between the additive and multiplicative notations, because the logarithm map e2πs i → s, s ∈ [0, 1], establishes an isomorphism between these two representations. Let us also note that the relation between the Euclidean space Rn and its factorization Rn /Zn can be realized by means of the natural projection, sometimes also called a canonical mapping, τ : Rn → Rn /Zn , x → [x], where the symbol [x] := {y ∈ Rn : (y − x) ∈ Zn } stands for the equivalence class of elements with x in Rn /Zn , i.e. Rn /Zn := {[x] : x ∈ Rn }, where [x] = x + Zn , x ∈ [0, 1)n . n In this way, for instance, the continuous maps f : Rn → Rn can be factorized into fˆ: Rn /Zn → Rn /Z , but  √  fˆ becomes a (single-valued) continuous mapping, in the respective metric dˆ: (R/Z)n × (Rn /Zn ) → 0, n , 2

where ˆ y) = |x − y| := min{|b − a| : a ∈ [x], b ∈ [y]}, d(x, provided only f (x) ≡ f (x + 1) (mod 1). For n = 1, we will also employ the reverse way characterized by the following proposition, involving the definition of the degree of fˆ.

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Proposition 2.1. (cf. e.g. [21, Proposition 4.3.1]) If fˆ: R/Z → R/Z is continuous, then there exists a continuous function f : R → R, called a lift of fˆ to R, such that fˆ ◦ τ = τ ◦ f , i.e. fˆ([x]) = [f (x)]. Such a lift is unique, up to an addition integer constant, and the degree deg(fˆ) := f (x + 1) − f (x) of fˆ is an integer independent of x ∈ R and the lift f . Moreover, if fˆ is a homeomorphism, then | deg(fˆ)| = 1. If fˆ is a homeomorphism such that deg(fˆ) = 1, then it is called an orientation preserving homeomorphism. As we have already pointed out, if fˆ: R/Z → R/Z is an orientation-preserving homeomorphism, then ˆ have the same (minimal) period (see e.g. [21, all periodic orbits of fˆ, determined by periodic points of f, Proposition 4.3.8]). Since the Poincaré operators, investigated in the next section, will be shown admissible in the sense of Górniewicz (see e.g. [7, Chapters I.4 and III.4]), let us consider this important class of multivalued maps. By a multivalued mapping ϕ : X  Y , where X, Y are metric spaces, we understand ϕ : X → 2Y \ {∅}. In the entire text, we will still assume that ϕ has closed values. Definition 2.2. A map ϕ : X  Y is said to be upper semicontinuous (u.s.c.) if, for every open U ⊂ Y , the set {x ∈ X : ϕ(x) ⊂ U } is open in X, or equivalently, if for every closed U ⊂ Y , the set {x ∈ X : ϕ(x) ∩ U = ∅} is closed in X. Obviously, in the single-valued case, if f : X → Y is u.s.c., then it is continuous. Furthermore, every u.s.c. map ϕ : X  Y has a closed graph Γϕ := {(x, y) ∈ X × Y | y ∈ ϕ(x)}, but not vice versa. Nevertheless, if the graph Γϕ of a compact map ϕ : X  Y (i.e. when the set ϕ(X) = x∈X ϕ(x) is contained in a compact subset of Y ) is closed, then ϕ is u.s.c. If ϕ : X  Y is u.s.c. with compact values and A ⊂ X is compact, then ϕ(A) ⊂ Y is compact, too. The composition ψ ◦ ϕ : X  Z of two u.s.c. maps with compact values, ϕ : X  Y and ψ : Y  Z, is again u.s.c. with compact values. For more details, see e.g. [7, Chapter I.3]. p

q

Definition 2.3. Assume that we have a diagram X ⇐= Z −→ Y (Z is a metric space), where p : Z ⇒ X is a (single-valued) Vietoris map, namely (i) p is onto, i.e. p(Z) = X, (ii) p is proper, i.e. p−1 (K) is compact, for every compact K ⊂ X, (iii) p−1 (x) is acyclic (i.e. homologically equivalent to a one point space), for every x ∈ X, where acyclicity is understood in the sense of Čech homology functor with compact carriers and coefficients in the field Q of rationals, and q : Z → Y is a continuous mapping. Then the map ϕ : X  Y is called admissible (in the sense of Górniewicz) if it is induced by ϕ(x) = q(p−1 (x)), for every x ∈ X. Thus, we identify the admissible map ϕ with the pair (p, q) called an admissible (selected) pair. One can readily check that every admissible map is u.s.c. with nonempty, compact, connected values, but not vice versa. The more general u.s.c. maps with nonempty, compact, connected values will be called M-maps. The class of admissible maps is closed w.r.t. finite compositions of admissible maps, i.e. a finite composition of admissible maps is also admissible. In fact, a map is admissible if and only if it is a finite composition of acyclic maps, i.e. u.s.c. maps with acyclic and compact values. Let us note that the composition of two acyclic maps need not be acyclic (see e.g. [4, Example 2.4], [7, Example I.4.11]). The class of admissible maps is very rich and important for applications of topological methods in nonlinear analysis. It contains u.s.c. maps with convex compact values, u.s.c. maps with contractible compact values, u.s.c. maps with Rδ -values, acyclic maps, and their compositions.

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If in particular X = Y = Tn then, in view of the above arguments, the upper semicontinuity of maps can be equivalently replaced by the closedness of their graphs. For X = Y = T1 = S 1 , the class of acyclic maps (i.e. ϕ : S 1  S 1 , where still ϕ(x) = S 1 , for all x ∈ S 1 ) reduces to the one of u.s.c. maps with compact contractible values (i.e. homotopically equivalent to a one point space) which are homeomorphic to compact intervals. On the other hand, although every admissible map on S 1 can be regarded as a finite composition of acyclic maps, it is not true that every M-map is admissible on S 1 , as illustrated by the following example. Example 2.4. Consider the M-mapping ϕ : R/Z  R/Z, where  ϕ(x) :=

1 x

(mod 1),

R/Z,

for x ∈ (0, 1), for x ∈ {0, 1}.

We will show that ϕ is not admissible. If it would be so, then it could be approximated on the graph Γϕ := {(x, y) | y ∈ ϕ(x)} with an arbitrary accuracy by a single-valued continuous function of the form τ ◦ f , and the degree d ∈ Z both ϕ (cf. its definition below) and τ ◦ f is constant, finite and identical, for all x ∈ [0, 1] (see [8, Lemma 3.4] and [7, Theorem I.10.24]), i.e. |d| = | deg(ϕ)| = | deg(τ ◦ f )| = |f (x + 1) − f (x)| < ∞. The function τ ◦ f , where ⎧ ⎪ ⎪ ⎨1 + f (x) := x1 , ⎪ ⎪ ⎩1,

1−ε ε2 ,

for x ∈ [0, ε), for x ∈ [ε, 1), for x = 1,

is even a single-valued continuous selection of the M-map ϕ, for every x ∈ [ε, 1] and an arbitrarily small ε > 0, but lim f (1) − f (ε) = 1 − lim

ε→0+

ε→0+

1 = −∞, ε

which is a contradiction. Moreover, f is no longer a (continuous) function at x = 0, for ε = 0. For the admissible maps on tori ϕ : Tn  Tn , we can define the global topological invariants, namely the Lefschetz number λ(ϕ) ∈ Z and the Nielsen number N (ϕ) ∈ N ∪ {0}, satisfying there (by the Anosov theorem) the equality N (ϕ) = |λ(ϕ)|. For more details, see (in the single-valued case) [15], and (in the multivalued case) [7, Chapters I.6 and I.10]. For the admissible maps on the circle ϕ : S 1  S 1 , we can also define their degree deg(ϕ) := 1 − λ(ϕ) which coincides, in the single-valued case, with the one defined in Proposition 2.1 (see [4]). Besides their existence property, when the existence of a fixed point of a (compact) ϕ, i.e. x ∈ Tn such that x ∈ ϕ(x), is implied by λ(ϕ) = 0, resp. by N (ϕ) > 0, or by deg(ϕ) = 1 (n = 1), all these numbers are invariant under an admissible homotopy ϕμ : Tn × [0, 1]  Tn , i.e. λ(ϕ0 ) = λ(ϕ1 ) = λ(ϕμ ), N (ϕ0 ) = N (ϕ1 ) = N (ϕμ ), and (n = 1) deg(ϕ0 ) = deg(ϕ1 ) = deg(ϕμ ), for all μ ∈ [0, 1]. The Nielsen number N (ϕ) (= |λ(ϕ)|), resp. N (ϕ) (= |1 − deg(ϕ)|, for n = 1), of an admissible mapping p q ϕ : Tn  Tn , induced by Tn ⇐= Z −→ Tn , gives still the lower estimate of the number of coincidences of (p, q), i.e. z ∈ Z such that p(z) = q(z), in the whole admissible homotopy class, but not of fixed points in general. Obviously, every fixed point of ϕ implies the existence of a coincidence of ϕ = q(p−1 ), but not vice versa.

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Definition 2.5. Let ϕ0 , ϕ1 : X  Y be two admissible maps. We say that ϕ0 is admissibly homotopic to ϕ1 −1 (written ϕ0 ∼ ϕ1 ) if there are selected pairs (p0 , q0 ), (p1 , q1 ), (p, q) such that ϕ0 = q0 (p−1 0 ), ϕ1 = q1 (p1 ), −1 ϕ = q(p ), and continuous (single-valued) maps γ0 , γ1 , γ such that the following diagram is commutative: X

p0

Z0

p

X × [0, 1]

Z γ1

i1

X

q0

γ0

i0

p1

q

Y q1

Z1

where i0 (x) = (x, 0), i1 (x) = (x, 1), and Z0 , Z1 , Z are metric spaces. On tori, admissible maps ϕ : Tn  Tn are admissibly homotopic to single-valued maps f : Tn → Tn , i.e. ϕ ∼ f . More precisely, every pair (p0 , q0 ) inducing ϕ is admissibly homotopic to a pair (p1 , q1 ) representing a single-valued map f (see e.g. [8, Lemma 4.3] and [7, Theorem I.10.24]). In [4], we have even proved that ϕk ∼ f k , k ∈ N, holds for the iterates of ϕ and f . Furthermore, since every single-valued continuous map f : Tn → Tn is homotopic to a map induced by a homology homomorphism on Rn , we obtain that (cf. [4]) N (ϕk ) = |λ(ϕk )| = N (f k ) = |λ(f k )| = | det(J − Ak )|, k ∈ N,

(4)

resp. in particular (n = 1) N (ϕk ) = |λ(ϕk )| = N (f k ) = |λ(f k )| = |1 − degk (f )| = |1 − dk |, k ∈ N,

(5)

where A, resp. d (n = 1), is a linearization matrix representing this homomorphism. The formulas (4), (5) allow us to calculate all these invariants for admissible maps and their iterates on tori. k Definition 2.6. Let ϕ : X  X be an M-mapping on a metric space X. A sequence {xi }k−1 i=0 ∈ X is called a k-periodic orbit (shortly, k-orbit) of ϕ, if xi+1 ∈ ϕ(xi ), for i < k − 1, x0 ∈ ϕ(xk−1 ), and there is no m < k k such that m|k and xsm+i = xi , for i < m, and s < m , where m|k means that m is a divisor of k.

Remark 2.7. We could also define a k-periodic point of ϕ : X  X as x0 ∈ X such that x0 ∈ ϕk (x0 ), and x0 ∈ / ϕj (x0 ), for all natural j < k, but one can easily check that this notion has not much meaning for multivalued maps. As a trivial example, we can consider a multivalued constant ϕ(x) = [0, 1], for every x ∈ [0, 1]. This constant ϕ has evidently infinitely many k-orbits, for any k ∈ N, but there is no k-periodic point of ϕ, for k > 1. Definition 2.8. Let ϕ : X  X be an admissible mapping on a metric space X, induced by a selected pair p q (p, q), i.e. ϕ = q(p−1 ), where X ⇐= Z −→ X and Z is a metric space. A sequence of points {z0 , . . . , zk−1 }, where zi ∈ Z, i = 0, . . . , k − 1, q(zi ) = p(zi+1 ), i = 0, . . . , k − 2, q(zk−1 ) = p(z0 ), is called a k-periodic orbit of coincidences (shortly, a k-orbit of coincidences) for the pair (p, q). Furthermore, we say that two orbits   {z0 , . . . , zk−1 } and {z0 , . . . , zk−1 } are cyclically equal if {z0 , . . . , zk−1 } = {zl , . . . , zk−1 ; z0 , . . . , zl−1 }, for all l = 0, . . . , k − 1. Otherwise, we call them cyclically different. At last, a k-orbit of coincidences (z0 , . . . , zk−1 ) is called irreducible if it is not a product orbit formed by going j-times around a shorter l-orbit, where jl = k.

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Obviously, every k-orbit of an admissible mapping ϕ : X  X implies the existence of an irreducible k-orbit of coincidences of ϕ = q(p−1 ), but not vice versa. Now, we are ready to formulate the Sharkovsky-type theorems for multivalued maps on tori and circles which will be applied to impulsive differential equations and inclusions. They will be formulated in the form of propositions. Proposition 2.9. (cf. [4, Theorem 5.3]) Let ϕ : Tn  Tn be an admissible map induced by a selected pair p q Tn ⇐= Z −→ Tn , where Z is a metric space. Assume that the sequence {N (ϕk )} of Nielsen numbers of ϕk = ϕ ◦ . . . ◦ ϕ,    k-times

where ϕ = q(p−1 ), is strictly increasing. Then ϕ has, for each k ∈ N, an irreducible k-orbit of coincidences. Remark 2.10. In the single-valued case, i.e. for a continuous mapping f : Tn → Tn , Proposition 2.9 was obtained in terms of periodic points in [1,2]. For n = 1, i.e. Tn = S 1 , Proposition 2.9 can be equivalently reformulated in a more explicit form as follows. Proposition 2.11. (cf. [4, Theorem 4.6]) Let ϕ : S 1  S 1 be an admissible map induced by a selected pair p q S 1 ⇐= Z −→ S 1 , where Z is a metric space, whose degree deg(ϕ) = d ∈ Z. If d > 1 or d < −2, then ϕ = q(p−1 ) admits an irreducible k-orbit of coincidences, for every k ∈ N. If d = −2 in (3), then the same is true for every k ∈ N \ {2}. In fact, for d = −2, either there exists or is absent an irreducible 2-orbit of coincidences, and both possibilities can occur. For d = −1 or d = 0, ϕ admits a fixed point. Remark 2.12. In the single-valued case, i.e. for a continuous mapping f : S 1 → S 1 , Proposition 2.11 was obtained in terms of periodic points, independently in [14] and [19]. Let us note that, for n = 2, 3, Proposition 2.9 can be also reformulated in a more transparent way by a classification made for single-valued maps in [1,2,23] (cf. also [22, Chapter VI.5]). For n = 2, this classification, characterising completely the minimal sets of periods, was done in terms of the determinants and traces of linearization 2 × 2 matrices A in formula (4). For n = 3, the classification was done in [23], in already a rather sophisticated way, in terms of the coefficients of the characteristic polynomials of the linearization 3 × 3 matrices A in formula (4). For more general circle M-maps, we can give the following proposition. Proposition 2.13. (cf. [5, Theorem 1]) Let ϕ : S 1  S 1 be an M -map which possesses a 3-orbit and a fixed point. Then either for every k ∈ N \ {2} or, but not at the same time, for every k ∈ N \ {4, 6}, ϕ admits a k-orbit. Remark 2.14. In the single-valued case, i.e. for a continuous mapping f : S 1 → S 1 , Proposition 2.13 was obtained in terms of periodic points, independently in [13] and [30]. In fact, unlike in Proposition 2.13, a possible absence of 4-orbits and 6-orbits can be eliminated there. Moreover, the main theorem in [13] is more complete, because the forced minimal periods were solved there for all assumed m-orbits, m ∈ N. An analogous extension of Proposition 2.13 would be also possible but technically rather demanding. If, for instance, m = 5, then k-orbits are implied, for every k ∈ N \ {2, 3, 4, 6}, provided ϕ has a fixed point. Concrete examples of absent orbits can be constructed.

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Remark 2.15. In view of Proposition 2.11, Proposition 2.13 has especially meaning for admissible maps of degrees d ∈ {−1, 0, 1}. For circle admissible maps of degrees d ∈ {−1, 0}, the assumption about the existence of a fixed point can be omitted. In order to apply Propositions 2.9, 2.11, 2.13 to impulsive differential equations and inclusions on tori and, in particular, on the circle, we will show that the associated Poincaré translation operators are admissible and that the calculations of the Nielsen numbers of their iterates, resp. of their degrees (n = 1), can be reduced to those of the given impulsive maps via admissible homotopy bridges. 3. Poincaré translation operators Hence, consider the impulsive vector differential inclusion  x ∈ F (t, x), x(t+ j )

∈

t = tj := t0 + jω, for some given ω > 0,

I(x(t− j )),

(6)

j ∈ Z,

where F : R × Rn  Rn is an upper-Carathéodory mapping such that F (t, x) ≡ F (t + ω, x) and F (t, x) ≡ F (t, x + 1),

(7)

and I : Rn  Rn is an admissible (in the sense of Definition 2.3) mapping such that I(x) ≡ I(x + 1) (mod 1),

(8)

+ − and x(t+ j ) := limt→t+ x(t), j ∈ Z, i.e. x(tj ) stands for the right limit, x(tj ) := limt→t− x(t), j ∈ Z, i.e. j

j

x(t− j ) stands for the left limit. Let us recall that a multivalued map F : R × Rn  Rn is, under (7), upper-Carathéodory, provided: (i) F (·, x) : [0, ω]  Rn is Lebesgue measurable, for every x ∈ [0, 1]n , i.e. {t ∈ [0, ω] : F (·, x) ⊂ U } is Lebesgue measurable, for each open U ⊂ [0, 1]n , (ii) F (t, ·) : [0, 1]n  R is u.s.c., for almost all (a.a.) t ∈ [0, ω], (cf. Definition 2.2), (iii) F has convex and compact values {F (t, x)}, for all (t, x) ∈ [0, ω] × [0, 1]n , (iv) |y| ≤ r(t)(1 + |x|), for all (t, x) ∈ [0, ω] × [0, 1]n , y ∈ F (t, x), where r : [0, ω] → [0, ∞) is a Lebesgue integrable function. By a solution of (6), we understand a (piece-wise) absolutely continuous vector function x ∈ AC((tj , tj+1 ), Rn ), j ∈ Z, satisfying (6), for a.a. t ∈ R \ {tj }j∈Z . Since we would like to study kω-periodic (mod 1) solutions of (6), for various k ∈ N, i.e. solutions x(·) of (6) such that x(t) ≡ x(t + kω) (mod 1) and x(t) ≡ x(t + mω) (mod 1), m < k, on R \ {tm }m∈Z , let us still prescribe the related boundary condition + x(t+ 0 ) = x(tk ) (mod 1), k ∈ N,

(9)

n where x(t+ k ) := xk denotes this time a vector xk ∈ R such that

xk ∈ I



 lim x(t) , k ∈ N.

t→t− k

For (6), we can associate the composition I ◦ Tμω : Rn × [0, 1]  Rn , where Tω : Rn  Rn is the Poincaré translation operator along the trajectories of the inclusion x ∈ F (t, x) without impulses (I = id), i.e.

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Tω (x0 ) := {x(tj + ω) ∈ Rn : x(·) ∈ AC(R, Rn ) is a solution of x ∈ F (t, x) such that x(tj ) = x0 }, j = 0, 1, . . . , k − 1.

(10)

Lemma 3.1. The composition I ◦ Tμω : Rn × [0, 1]  Rn of Tμω : Rn × [0, 1]  Rn , where Tω : Rn  Rn is the Poincaré translation operator defined in (10), and an admissible impulsive mapping I : Rn  Rn builds an admissible homotopy bridge between I ◦ Tω and I. Proof. The mapping I ◦ Tμω can be decomposed as ημ

eμ

I

Rn × [0, 1] − Z × [0, 1] −→ Rn − Rn , where Z := ημ (Rn × [0, 1]), ημ (x0 ) := {x ∈ AC([t0 , t0 + ω], Rn ) : x(·) is a solution of x ∈ F (t, x) such that x(t0 ) = x0 ∈ Rn }, μ ∈ [0, 1], and eμ := {x(t0 + μω) ∈ Rn : x ∈ Z × [0, 1]}. It is well known (see e.g. [7, Chapter III.4]) that ημ is an acyclic mapping, and so Z is compact. Furthermore, since the evaluation mapping eμ (·) : Z → Rn is obviously continuous, for every μ ∈ [0, 1], and eμ (x) : [0, 1] → Rn is continuous w.r.t. the parameter μ, for every x ∈ Z, the map eμ : Z × [0, 1] → Rn must be continuous as a whole. Thus, Tμω : Rn × [0, 1]  Rn is admissible, and since so is (by the hypotheses) I : Rn  Rn , their composition I ◦ Tμω is admissible, too.  At last, since I ◦ T1ω = I ◦ Tω (μ = 1) and I ◦ T0ω = I ◦ idRn = I (μ = 0), the proof is complete. 2 In order to check, under (7) and (8), the same for the respective operators directly in the factor space Rn /Zn , we could proceed quite analogously as in Lemma 3.1, but when using the appropriate metric ˆ y) := min{|b − a| : a ∈ [x], b ∈ [y]}. d(x, ˆ However, to prove there the acyclicity of the mapping ηˆμ : Rn /Zn ×[0, 1]  Z×[0, 1], where Zˆ := ηˆμ (Rn /Zn × n n [0, 1]), and ηˆμ , μ ∈ [0, 1], prescribes to every point [x0 ] ∈ R /Z the solutions of the given inclusion on [t0 , t0 + ω], would not be so straightforward. Therefore, we will preferably proceed in an alternative way, when applying Lemma 3.1. Hence, assuming (7) and (8), consider the composition Iˆ ◦ Tˆμω : Rn /Zn × [0, 1]  Rn /Zn , where Tˆμω : Rn /Zn × [0, 1]  Rn /Zn is defined by Tˆμω ([x0 ]) := {ˆ x(t0 + μω) ∈ Rn /Zn : x ˆ(·) ∈ AC(R, Rn ) is a solution of ˆ(t0 ) = [x0 ]}, (11) x ∈ F (t, x) considered on Rn /Zn , such that x ˆ and Iˆ: Rn /Zn  Rn /Zn is the factorization of the impulsive mapping I : Rn  Rn , i.e. [τ ◦ I(x)] = I([x]), n for every x ∈ R . ◦ Tμω := Iˆ ◦ Tˆμω , μ ∈ [0, 1], where Tˆμω is the factorized Poincaré Proposition 3.2. The above composition I translation operator defined in (11) and Iˆ is the (admissible) factorized impulsive mapping, builds an adˆ missible homotopy bridge between I ◦ Tω and I.

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Proof. The claim follows, by means of Lemma 3.1, directly from the commutative diagram: [0, 1]n × [0, 1]

ημ

◦Z

× [0, 1]

eμ

τ

i

Rn /Zn × [0, 1]

ηˆμ

ˆ ◦Z

× [0, 1]

Rn

I

◦ Rn

τ eˆμ

Rn /Zn

τ Iˆ

◦ Rn /Zn ,

where i : [x] → x is an isomorphism (⇒ i−1 (= τ ) : x → [x]), τ : Rn → Rn /Zn is a (continuous) natural projection, and Tˆμω := eˆμ ◦ ηˆμ , μ ∈ [0, 1], because ηˆμ ([x0 ]) := {ˆ x ∈ AC([t0 , t0 + ω], Rn /Zn ) : x ˆ(·) is a solution of x ∈ F (t, x) considered on Rn /Zn , such that x ˆ(t0 ) = [x0 ] ∈ Rn /Zn }, μ ∈ [0, 1], ˆ τ ◦ Z) := ηˆμ (Rn /Zn × [0, 1]), and satisfies ηˆμ ([x]) = [τ ◦ ημ (x)], for any x ∈ Rn , μ ∈ [0, 1], Z(= eˆμ (ˆ x) := {ˆ x(t0 + μω) ∈ Rn /Zn : x ˆ ∈ Zˆ × [0, 1]} satisfies eˆμ ([z]) = [τ ◦ eμ (z)], for every z ∈ Z.  ◦ Tω (μ = 1) and I ◦ T1ω = I ◦ T0ω = Iˆ ◦ idRn /Zn = Iˆ (μ = 0). Moreover, I

2

4. Main theorems Now, Propositions 2.9, 2.11, 2.13 will be applied, by means of Proposition 3.2 and formulas (4), (5), to impulsive inclusions (6), satisfying (7) and (8). For this application, the following correspondence is crucial. Lemma 4.1. There exists a selected pair (ˆ p, qˆ) which induces the (admissible) operator I ◦ Tω , treated in Proposition 2.9, such that there is a one-to-one correspondence, for every k ∈ N, between the irreducible k-orbits of coincidences of the mapping pˆ q Rn /Zn ⇐= Zˆ −→ Rn /Zn

(12)

and kω-periodic (mod 1) solutions of the impulsive inclusion (6), provided (7) and (8) holds. Proof. It is enough to take, for n = 1, pˆ in order pˆ−1 := ηˆ1 (i.e. pˆ(ˆ x(·)) = x ˆ(t0 )), and qˆ := Iˆ ◦ eˆ1 . Such a pˆ exists by the definition of a Vietoris mapping (cf. Definition 2.3). One can readily check that then the irreducible k-orbits of coincidences of (ˆ p, qˆ) can be even identified (cf. Definition 2.8) with kω-periodic n n solutions of (6), on the factor space R /Z . 2 Remark 4.2. On the other hand, there is no longer such a one-to-one correspondence for k-orbits of I ◦ Tω . They determine the existence of kω-periodic (mod 1) solutions of (6), satisfying (7) and (8), but reversely these periodic solutions only fulfil the boundary condition (9). Therefore, in order to employ a desired existence correspondence, one must still impose certain additional restrictions on some values of kω-periodic solutions on tori. Since the operator I ◦ Tω is, according to Proposition 3.2, admissible in the sense of Górniewicz (see Definition 2.5), and it is admissibly homotopic to the factorization Iˆ of the impulsive mapping I : Rn  Rn ,

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satisfying (8), we can use the formulas (4) and (5) for the calculation of the Nielsen numbers, Lefschetz ˆ ◦ Tω by those of I. numbers and (for n = 1) the degrees of I Theorem 4.3. Let A be an n × n linearization matrix (cf. (4)), representing the factorization Iˆ: Rn /Zn  ˆ = [I(x)], for Rn /Zn of the impulsive mapping I : Rn  Rn , satisfying (8), i.e. Iˆ ◦ τ = τ ◦ I, where I([x]) n k all x ∈ R . Assume that the sequence {| det(J − A )|} is strictly increasing, where J is the unit (identity) matrix. Then the impulsive inclusion (6) admits, under (7), a kω-periodic (mod 1) solution, for every k ∈ N. Equivalently, the impulsive inclusion (6), considered on the factor space Rn /Zn , possesses under the above assumptions a kω-periodic solution, for every k ∈ N. Proof. In view of Lemma 4.1, it is equivalent to show, for every k ∈ N, the existence of irreducible k-orbits of coincidences to the operator I p, qˆ) (cf. (12)). Since I ◦ Tω := (ˆ ◦ Tω : Rn /Zn  Rn /Zn is by Proposition 3.2 admissible, it has, according to Proposition 2.9, the irreducible k-orbits of coincidences, for all k ∈ N, provided the sequence {N ((I ◦ Tω )k )} of the Nielsen numbers is strictly increasing. Since it is still by ˆ the same is true (by means of the admissible homotopy property Proposition 3.2 admissibly homotopic to I, of the Nielsen number) for the sequence {N (Iˆk )}. But N (Iˆk ) = |λ(Iˆk )| = | det(J − Ak )|, k ∈ N, according to (4), which completes the proof. 2 By virtue of [1, Theorem 4.6], Theorem 4.3 has the following direct consequence which we state here in the form of corollary. Corollary 4.4. If A in Theorem 4.3 is a square integer matrix whose eigenvalues are real, positive, different from 1 and det A > 1, then the sequence {| det(J − Ak )|} is strictly increasing, and subsequently the conclusion of Theorem 4.3 holds. Remark 4.5. It follows from the existence and (admissible) homotopy properties of the Lefschetz number ˆ = 0 implies at least the existence of an ω-periodic (mod 1) solution to (6). that, in view of Lemma 4.1, λ(I) For n = 1, Theorem 4.3 can be equivalently reformulated in a much more explicit way as follows. ˆ ∈ Z. If d > 1 or Theorem 4.6. Let I : R  R be an admissible map satisfying (8) and such that d := 1 − λ(I) d < −2, then the impulsive inclusion (6) admits, under (7), a kω-periodic (mod 1) solution, for every k ∈ N. If d = −2, then the same is true, for every k ∈ N \ {2}. Equivalently, under the above assumptions, the impulsive inclusion (6), considered on the factor space Rn /Zn , possesses a kω-periodic solution, for every k ∈ N, resp. (when d = −2) for every k ∈ N \ {2}. Proof. Since det(J − Ak ) reduces, for n = 1, to 1 − dk , k ∈ N, and N (Iˆk ) = |λ(Iˆk )| = |1 − dk |, k ∈ N, according to (5), we can proceed in the same way as in the proof of Theorem 4.3, when applying Proposition 2.11, instead of Proposition 2.9. 2 One can readily check that Theorem 1.1 in the introduction is only a very special case of the following corollary, when taking into account the arguments in Remark 4.5. Corollary 4.7. Let I : R → R be a (single-valued) continuous function such that I(x + 1) − I(x) = d ∈ Z, for all x ∈ R. Then the same conclusions as those in Theorem 4.6 hold for (6). If d = −1 or d = 0, then, in view of Remark 4.5, inclusion (6) admits, under (7), at least an ω-periodic (mod 1) solution.

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Remark 4.8. In higher dimensions (n > 1), Theorem 4.3 could be also expressed more explicitly by means of the elegant classification made in [1,2] (n = 2), and [23] (n = 3), but because of a rather sophisticated description, we restricted ourselves just to Theorem 4.6, dealing with n = 1. For n = 1 and d = −1 or d = 0, in view of the (not necessarily one-to-one, cf. Remark 4.2) correspondence between kω-periodic solutions of (6), considered on R/Z, and k-orbits of I ◦ Tω , we can apply Propositions 2.11 and 2.13 as follows. ˆ = 1 or Theorem 4.9. Let I : R  R be an admissible mapping satisfying (8) and such that either λ(I) ˆ λ(I) = 2. If the impulsive inclusion (6), satisfying (7), admits a 3ω-periodic (mod 1) solution x(·) such + + + + + that x(t+ 0 ) = x(t1 ) (mod 1) or x(t1 ) = x(t2 ) (mod 1) or x(t0 ) = x(t2 ) (mod 1), then it also possesses a kω-periodic (mod 1) solution, for every k ∈ N \ {2} or (but not at the same time) every k ∈ N \ {4, 6}. + Proof. The existence of a 3ω-periodic (mod 1) solution x(·) such that x(t+ 0 ) = x(t1 ) (mod 1) or + + + x(t+ 1 ) = x(t2 ) (mod 1) or x(t0 ) = x(t2 ) (mod 1) implies, in view of (9), a 3-orbit of the admissible ◦ Tω := Iˆ ◦ Tˆω : R/Z  R/Z, where Tˆω is defined in (11). (cf. Proposition 3.2) operator I ˆ by ˆ i.e. I  ◦ Tω ∼ I, Furthermore, since I ◦ Tω is, according to Proposition 3.2, admissibly homotopic to I, the existence and homotopy properties of the Lefschetz number, there also exists a fixed point of I ◦ Tω .

Proposition 2.13 therefore yields, for every k ∈ N \{2} or k ∈ N \{4, 6}, the existence of k-orbits, determining irreducible k-orbits of coincidences, of I ◦ Tω . Thus, the condition follows by virtue of Lemma 4.1. 2 In the single-valued case, i.e. for the impulsive Carathéodory differential equations, satisfying a uniqueness condition, Theorem 4.9 can be slightly improved by means of the main theorem due to Block in [13] as follows. Theorem 4.10. Let I : R → R be a (single-valued) continuous mapping such that I(x + 1) − I(x) = −1 or I(x) = I(x + 1), for all x ∈ R. If the impulsive Carathéodory equation (1), satisfying (2) and a uniqueness condition, admits a 3ω-periodic (mod 1) solution, then it also possesses a kω-periodic (mod 1) solution, for every k ∈ N \ {2}. Proof. The only differences to the proof of Theorem 4.9 consist of the usage of the alternative definition of ˆ := I(x + 1) − I(x), for all x ∈ R, the additional assumption under which the conditions the degree deg(I) imposed on the solution values at the points tj , j = 0, 1, 2, are not necessary (they are directly implied), and the replacement of the applied Proposition 2.13 by the theorem of Block in [13] (see Remark 2.14), ◦ Tω . 2 which allowed us to eliminate possibly missing 4-orbits and 6-orbits of I Remark 4.11. The existence of kω-periodic (mod 1) solutions of (6), resp. (1), in Theorems 4.9 and 4.10 can be again equivalently expressed as the existence of kω-periodic solutions, considered on the factor space R/Z. Example 4.12. Consider the impulsive inclusion (6), satisfying (7), where I : R → R takes the particular form dx + p(x), d ∈ Z, and p(x) ≡ p(x + 1) is (single-valued) continuous. According to Corollary 4.7, inclusion (6) considered on R/Z admits a kω-periodic solution: • for every k ∈ N, when d > 1 or d < −2, • for every k ∈ N \ {2}, when d = −2, • for k = 1, when d = −1 or d = 0.

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Fig. 1. ω-periodic (mod 1) solution and 3ω-periodic (mod 1) solution from Example 4.12.

Like in the above theorems, under a natural bi-periodicity assumption (7), the coexistence of subharmonic periodic solutions is governed only by the degree of the given impulsive mapping on R/Z. This illustrates the efficiency of our approach (Fig. 1). On the other hand, the additional assumptions by means of the 3ω-periodic (mod 1) solutions in Theorems 4.9 and 4.10 are rather implicit and not so effective. 5. Concluding remarks Remark 5.1. The most difficult case occurs for d = 1, because the admissible circle maps need not have a fixed point. If it is so, then their degree d must be 1. In order to avoid it, Zhao [31] gave a nice theorem, saying that if a (single-valued) continuous map f : S 1 → S 1 admits a 3-orbit {x1 , x2 , x3 } such that f as the relative mapping, i.e. f : (S 1 , {x1 , x2 , x3 }) → (S 1 , {x1 , x2 , x3 }), is not relatively homotopically conjugated with a standard 120◦ rotation f0 : (S 1 , {v0 , v1 , v2 }) → (S 1 , {v0 , v1 , v2 }), where vj = e2jπ i/3 , j = 0, 1, 2, then f possesses a fixed point. We have established a multivalued version of Zhao’s theorem for admissible maps in [5], by means of the relative multivalued Nielsen theory developed in [10]. Thus, if the admissible ◦ Tω considered above satisfies the assumptions of our theorem in [10], then because of the operator I correspondence in Lemma 4.1, inclusion (6) would have guaranteed an ω-periodic (mod 1) solution, and ˆ = 1 or λ(I) ˆ = 2 in Theorem 4.9 can be replaced by those of our theorem subsequently the assumptions λ(I) in [10] related to I ◦ Tω , provided the same additional restrictions are imposed on the 3ω-periodic (mod 1) solution determined by {x1 , x2 , x3 }. Remark 5.2. In [9] (cf. also [7, Chapter II.6]), we have also defined another invariant for the admissible maps ϕ : Tn  Tn , denoted by Sk (ϕ), which gives the lower estimate of the number of irreducible, cyclically different, k-orbits of coincidences, k ∈ N, again in the whole (admissible) homotopy class of ϕ, but not of its k-orbits in general. Therefore, some authors speak with this respect about the families of homotopy minimal periods, resp. the homotopy classes of irreducible, cyclically different orbits of coincidences. On this basis, we can calculate, in view of Lemma 4.1, for each k ∈ N, the least number of kω-periodic (mod 1) solutions of (6), determined by the homotopy classes of irreducible, cyclically different orbits of coincidences ◦ Tω . to I Remark 5.3. Defining the impulsive differential equations (1) and inclusions (6), considered on tori, to be chaotic as far as the associated operators I ◦ Tω , resp. their single-valued continuous selections, are so, we

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can say that at least the Carathéodory equation (1) in Theorem 4.10 is chaotic. It follows e.g. from the theorem in [30], where the chaos in the sense of Devaney, resp. equivalently the positive topological entropy, and subsequently also the chaos in the sense of Li and Yorke, is implied by the existence of a 3-orbit and a fixed point of a given circle mapping. More generally, the same follows from the existence of a k-orbit, where k = 2m , m ∈ N, and a fixed point of a given circle map (see e.g. [3]). Finally, let us point out that, unlike e.g. [12,18,20], the multivalued impulsive maps were also considered e.g. in [27, Chapter 2]. For the practical applications, see e.g. [26] and the related references in [12,18,20]. On the other hand, as far as we know, the coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori (whence the title) has never been investigated before. References [1] L. Alsedà, S. Baldwin, J. Llibre, R. Swanson, W. Szlenk, Torus maps and Nielsen numbers, in: C. McCord (Ed.), Nielsen Theory and Dynamical Systems, in: Contemp. Math. Series, vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 1–7. [2] L. Alsedà, S. Baldwin, J. Llibre, R. Swanson, W. Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pac. J. Math. 169 (1) (1995) 1–32. [3] L. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2nd edition, World Scientific Publ., Singapore, 2000. [4] J. Andres, On the coexistence of irreducible orbits of coincidences for multivalued admissible maps on the circle via Nielsen theory, Topol. Appl. 221 (2017) 596–609. [5] J. Andres, J. Fišer, Sharkovsky-type theorems on S 1 applicable to differential equations, Int. J. Bifurc. Chaos 27 (2017) 1–21. [6] J. Andres, T. Fürst, K. Pastor, Period two implies all periods for a class of ODEs: a multivalued map approach, Proc. Am. Math. Soc. 135 (10) (2007) 3187–3191. [7] J. Andres, L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003. [8] J. Andres, L. Górniewicz, J. Jezierski, A generalized Nielsen number and multiplicity results for differential inclusions, Topol. Appl. 100 (2–3) (2000) 193–209. [9] J. Andres, L. Górniewicz, J. Jezierski, Relative versions of the multivalued Lefschetz and Nielsen theorems and their application to admissible semi-flows, Topol. Methods Nonlinear Anal. 16 (1) (2001) 73–92. [10] J. Andres, L. Górniewicz, J. Jezierski, Periodic points of multivalued mappings with applications to differential inclusions on tori, Topol. Appl. 127 (3) (2003) 447–472. [11] J. Andres, K. Pastor, P. Šnyrychová, A multivalued version of Sharkovski˘ı’s theorem holds with at most two exceptions, J. Fixed Point Theory Appl. 2 (1) (2007) 153–170. [12] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, vol. 2, Hindawi Publ. Corp., New York, 2006. [13] L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Am. Math. Soc. 82 (3) (1981) 481–486. [14] L. Block, J. Guckenheimer, M. Misiurewicz, L.-S. Young, Periodic points and topological entropy of one-dimensional maps, in: Z. Nitecki, C. Robinson (Eds.), Global Theory of Dynamical Systems, in: Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. [15] R.F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., Glenview, Ill., 1971. [16] E.A. Coddington, N. Levinson, Theory of Differential Equations, McGraw-Hill, New York, 1955. [17] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl. 11 (1932) 333–376. [18] S. Djebali, L. Górniewicz, A. Ouahab, Existence and Structure of Solution Sets for Impulsive Differential Inclusions: A Survey, Lecture Notes in Nonlinear Analysis, vol. 13, Juliusz Schauder Center for Nonlinear Studies, Nicolaus Copernicus University, Toruń, 2012. [19] L.S. Efremova, Periodic orbits and a degree of a continuous map of a circle, Differ. Integral. Equ. (Gor’kii) 2 (1978) 109–115 (in Russian). [20] J.R. Graef, J. Henderson, A. Ouahab, Impulsive Differential Inclusions: A Fixed Point Approach, De Gruyter Series in Nonlinear Analysis and Applications, vol. 20, De Gruyter, Berlin, 2013. [21] B. Hasselblatt, A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge Univ. Press, Cambridge, 2003. [22] J. Jezierski, W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Springer, Berlin, 2006. [23] B. Jiang, J. Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dyn. Syst. 4 (2) (1998) 301–320. [24] E.R. van Kampen, The topological transformations of a simple closed curve into itself, Am. J. Math. 57 (1935) 142–152. [25] M.A. Krasnosel’skii, The Operator of Translation Along the Trajectories of Differential Equations, Translations of Mathematical Monographs, vol. 19, Amer. Math. Soc., Providence, RI, 1968. [26] Z. Li, Y. Soh, C. Wen, Switched and Impulsive Systems: Analysis, Design, and Applications, Springer, Berlin, 2005. [27] N.A. Perestyuk, V.A. Plotnikov, A.M. Samoilenko, N.V. Skripnik, Differential Equations with Impulse Effects. Multivalued Right-hand Sides with Discontinuities, De Gruyter Studies in Mathematics, vol. 40, De Gruyter, Berlin, 2011.

140

J. Andres / Topology and its Applications 255 (2019) 126–140

[28] H. Poincaré, Sur les courbes définies par les équations différentielles (iii), J. Math. Pures Appl. 1 (1885) 167–244. [29] A.N. Sharkovsky, Coexistence of cycles of a continuous map of the line into itself, Int. J. Bifurc. Chaos Appl. Sci. Eng. 5 (5) (1995) 1263–1273, translated from the Russian by J. Tolosa Ukr. Mat. Zh. 16 (1) (1964) 61–71. [30] H.W. Siegberg, Chaotic mappings on S 1 , periods one, two, three imply chaos on S 1 , in: Proceedings of the Conference: Numerical Solutions of Nonlinear Equations, Bremen, 1980, in: Lecture Notes in Math., vol. 878, Springer, Berlin, 1981, pp. 351–370. [31] X. Zhao, Periodic orbits with least period three on the circle, Fixed Point Theory Appl. 2008 (1948) 194875.