Second–order discontinuous ODEs and billiard problems

J. Math. Anal. Appl. 536 (2024) 128237 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa Regular Articles Second–order discontinuous ODEs and billiard problems Jorge Rodríguez-López a,∗, Jan Tomeček b aCITMAga & Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago, Spain bDepartment of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Olomouc, Czechia a r t i c l e i n f o a b s t r a c t Article history: Received 30 October 2023 Available online 20 February 2024 Submitted by M. Quincampoix Keywords: Discontinuous differential equations Differential inclusions Impulsive differential equations Billiard problem Multiple solutions Dirichlet problem We present an existence principle for boundary value problems involving discontinuous ordinary differential equations of the second order using the Krasovskii regularization technique. Especially we obtain sufficient conditions of transversality type for Krasovskii solutions to be also Carathéodory solutions of the original problem. This result is applied on a certain billiard problem, which can be thought as an ordinary differential equation with state-dependent impulses that is equivalent to certain discontinuous differential equation. In particular, we obtain new existence and multiplicity results for Dirichlet problems in billiard spaces with time-varying boundaries. © 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/). 1. Introduction In this paper, we deal with second–order ordinary differential equations with discontinuous nonlinearities, i.e., discontinuous differential equations. It is well-known that if the right-hand side in the differential equation is discontinuous with respect to the spatial variable, then existence of Carathéodory solutions is not guaranteed. Here, in order to obtain existence results, we follow the line of the previous papers [4,11,13]: firstly, we look for solutions of a differential inclusion, which can be seen as a regularization of the former problem, and secondly, we provide conditions that ensure that solutions of the differential inclusion are also solutions of the differential equation. We will refer to such condition as a transversality condition and, in our case, it was inspired by that for first–order systems due to Bressan and Shen [3], later relaxed in [13]. To the best knowledge of the authors, this one is the first paper in which this type of transversality condition is adapted to second-order ordinary differential equations. *Corresponding author. E-mail addresses: jorgerodriguez.lop[email protected] (J. Rodríguez-López), [email protected] (J. Tomeček). https://doi.org/10.1016/j.jmaa.2024.128237 0022-247X/© 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/). 2J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 Discontinuous differential equations appear in a natural way in many physical processes (see, for instance, the recent paper [2]). As an application of our theory, here we focus on Dirichlet problems in one-dimensional billiard spaces with possible uneven surfaces, which will be reduced to second–order problems where the nonlinearity has jump discontinuities. Classical billiard spaces are some compact connected sets with sufficiently nice boundary and the structure of trajectories of a ball moving inside of the table in a uniform linear motion having absolutely elastic impact with the boundary is investigated, see e.g. [12]. Each such trajectory corresponds to a solution of the system of ODEs x =0. On the other hand, less is known when the ball does not move uniformly along the straight lines, i.e. the trajectories correspond to the more general differential equations x =f(t, x), where the ball is subjected to some force (caused by uneven surface or some external force). Some recent results in finite-dimensional billiard spaces can be found in [7–9,14]. Here we investigate Dirichlet problems for one-dimensional billiard table with time-changing boundary, which can be understood as boundary value problems for differential equations of the second order with state-dependent impulses in the form x =h(t, x),for a.a. t∈I:= [0,T],such that x(t)∈(α(t),β(t)),(1.1) x(s+) −γ(s)=γ(s)−x(s−),if s∈(0,T),x(s)=γ(s),for γ∈{α, β},(1.2) x(0) = A, x(T)=B, (1.3) where α, β:I→R, α, β∈W2,1(I; R), α<βon I, A ∈(α(0), β(0)), B∈(α(T), β(T)) and h ∈Car(Eα,β; R) where Eα,β := {(t, x) ∈I×R :α(t) ≤x ≤β(t)}. Problem (1.1)–(1.3)can be transformed into a Dirichlet second–order discontinuous problem and so using the results from the first part of this paper, we give sufficient conditions for the existence and multiplicity of solutions to the impulsive BVP. As it is shown, we generalize the results from [14]. The paper is organized as follows. First part is devoted to a general discontinuous differential equations with some boundary conditions. It is shown the Krasovskii and Filippov regularization, which convert the discontinuous differential equation into a regular differential inclusion. So called transversality conditions guarantee that solutions of this regularization are also solutions to the original problem – see Theorem 2.2 and 2.3 and Corollary 2.5. The main existence results are stated in Theorem 2.7 and 2.8. In the second part we are interested in impulsive problems of type (1.1)–(1.3), which are then converted into nonimpulsive (but discontinuous) problems. Depending on the way how the boundary of the billiard table is changing we investigate the linear and also nonlinear cases. Existence and multiplicity results are given in Theorem 3.11, 3.12, 3.18 and 3.19. 2. Second-order discontinuous ODEs Consider the second order problem x =f(t, x),for a.a. t∈I:= [0,T],x∈B,(2.4) where f:I×Rn→Rnmay be discontinuous with respect to both variables and the set B⊂C(I; Rn) denotes initial or boundary conditions. We will say that a function x :I→Rnis a Carathéodory solution of (2.4)if x ∈W2,1(I; Rn), x(t) =f(t, x(t)) for a.a. t ∈Iand it belongs to the set B. Let us note that W2,1(I; Rn) stands for all functions with values in Rnhaving absolutely continuous first derivatives and Lebesgue integrable second derivatives on I. When the right-hand side of the differential equation in (2.4)is discontinuous with respect to the state variable, the classical technique in the literature consists in replacing the differential equation by an inclusion and so considering a problem of the following type J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 3 x ∈F(t, x),for a.a. t∈I:= [0,T],x∈B,(2.5) where F:I×Rn→P(Rn)is a multivalued map which regularizes the discontinuous function f(i.e. P(Rn) is the set of all subsets of Rn). If Fis defined as F(t, x)= ε>0 co ft, Bε(x)=: K(f)(t, x),(t, x)∈I×Rn,(2.6) where co means closed convex hull and Bε(x)is the ball centered at xand radius ε, Carathéodory solutions of the differential inclusion (2.5)are called Krasovskii solutions of (2.4). On the other hand, in case that F is defined as F(t, x)= ε>0 m(N)=0 co ft, Bε(x)\N=: F(f)(t, x),(t, x)∈I×Rn,(2.7) where mdenotes the Lebesgue measure, Carathéodory solutions of the differential inclusion (2.5)are called Filippov solutions of (2.4). Following with the same terminology, the maps Fdefined in (2.6)and (2.7)are, respectively, the Krasovskii and Filippov envelopes of f. In the sequel, we assume that the following condition holds: (HF)if f(t, ·)is continuous at the point x, then F(t, x) ={f(t, x)}. Observe that both Krasovskii and Filippov envelopes satisfy condition (HF). Now, we shall discuss under what additional conditions on f, Carathéodory solutions of the differential inclusion (2.5)are also Carathéodory solutions of (2.4). To do that, we need the following technical result, see [1, Lemma 5.8.13]. Lemma 2.1. Let a, b ∈R, a <b. If ϕ :[a, b] −→ Ris almost everywhere differentiable on [a, b], then for each null measure set A ⊂Rthere exists a null measure set B⊂ϕ−1(A)such that ϕ(t)=0 for all t∈ϕ−1(A)\B. We are in a position to give sufficient conditions in order to ensure that the solutions of (2.5)are also solutions of (2.4). Theorem 2.2. Assume that condition (HF)holds and that there exist null measure sets Ak⊂R, k∈Cwith at most countable set C, and differentiable mappings τk:[ak, bk] ×Rn→R, [ak, bk] ⊂I, such that for a.a. t ∈I, f(t, ·)is continuous in Rn\ k∈C :t∈[ak,bk]{x∈Rn:τk(t, x)∈Ak} and for each k∈Cand each (t, x) ∈τ−1 k(Ak)we have ∇τk(t, x)·(1,z)=0 for all z∈K, (2.8) where K⊂Rnis a compact set such that x(t) ∈Kfor all t ∈Iand every solution xof (2.5). Then, if xis a Carathéodory solution of the inclusion (2.5), it is also a Carathéodory solution of the discontinuous problem (2.4). 4J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 Proof. Let xbe a Carathéodory solution of the differential inclusion (2.5). Let us show that m({t∈[ak,b k]:τk(t, x(t)) ∈Ak})=0 for all k∈C. For an arbitrarily fixed k∈C, we define ϕ(t) =τk(t, x(t)) for all t ∈[ak, bk]. Since m(Ak) =0, Lemma 2.1 guarantees the existence of a set B⊂ϕ−1(Ak), with m(B) =0, such that for every t ∈ ϕ−1(Ak) \Bwe have ϕ(t) =0, i.e. d dtτk(t, x(t)) = 0. By the chain rule, we can rewrite the previous expression as ∇τk(t, x(t)) ·(1,x (t)) = 0 for all t∈ϕ−1(Ak)\B. Since x(t) ∈Kfor all t ∈I, it follows from condition (2.8)that ϕ−1(Ak)is a null-measure set. Therefore, for a.a. t ∈Ithe function f(t, ·)is continuous at x(t), which due to hypothesis (HF) implies that F(t, x(t)) ={f(t, x(t))}for a.a. t ∈I. Since xis a solution of the differential inclusion (2.5), x(t) ∈ F(t, x(t)) ={f(t, x(t))}for a.a. t ∈I, thus xis a Carathéodory solution of (2.4).  Note that in order to apply Theorem 2.2 we need to construct the set Kand thus we need to have some a priori estimates on the derivatives of any solution of (2.5). This drawback is avoided by the following result. Theorem 2.3. Assume that condition (HF)holds and that there exist null measure sets Ak⊂R, k∈Cwith at most countable set C, and two times differentiable mappings τk:[ak, bk] ×Rn→R, [ak, bk] ⊂I, such that for a.a. t ∈I, f(t, ·)is continuous in Rn\ k∈C :t∈[ak,bk]{x∈Rn:τk(t, x)∈Ak} and for each k∈Cand each (t, x) ∈τ−1 k(Ak)we have (1,v)H(τk)(t, x)(1,v)T+∇xτk(t, x)·z=0 (2.9) for all v∈Rnsuch that ∇τk(t, x) ·(1, v) =0and all z∈F(t, x)(where H(τk)denotes the Hessian matrix of τk). Then, if xis a Carathéodory solution of the inclusion (2.5), it is also a Carathéodory solution of the discontinuous problem (2.4). Proof. Let xbe a Carathéodory solution of the differential inclusion (2.5). Again it suffices to see that for each k∈Cthe set Jk:= {t∈[ak,b k]:τk(t, x(t)) ∈Ak} has Lebesgue null measure. For a fixed k∈C, by Lemma 2.1 we have d dtτk(t, x(t)) = ∇τk(t, x(t)) ·(1,x (t)) = 0 for a.a. t∈Jk. J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 5 Applying again Lemma 2.1 we deduce that for a.a. t ∈Jk, d2 dt2τk(t, x(t)) = (1,x (t)) H(τk)(t, x(t)) (1,x (t))T+∇xτk(t, x(t)) ·x(t)=0. This fact joint with condition (2.9) implies that m(Jk) =0.  Remark 2.4. If τkis linear, i.e., τk:Rn→Ris defined as τk(x1,x 2,...,x n)=a1x1+a2x2+···+anxn, with ai∈R, i =1, 2, ..., n, then the transversality condition (2.9)reads simply as ∇τk(x)·z=0 forallz∈F(t, x). Equivalently, ν·z=0 forallz∈F(t, x), where νdenotes the vector (a1, a2, ..., an) ∈Rn, that is, νis a normal vector of the hyperplane τk(x) =c where fmay be discontinuous. Let us now focus on the scalar case of (2.4), i.e., with n =1. By the implicit function theorem, if τis regular enough, the discontinuity hypersurfaces of type τ(t, x)=c can be seen, at least locally, as the graphs of time-dependent curves of type x =γ(t). Corollary 2.5. Assume that n =1, condition (HF)holds and there exist null measure sets Ak⊂R, k∈C⊂ Z, and two times differentiable mappings γk:[ak, bk] ⊂I→Rsuch that for a.a. t ∈I, f(t, ·)is continuous in R\ k∈C :t∈[ak,bk] ck∈Ak{γk(t)+ck} and for each k∈C, the function γksatisfies either (i)γ k(t) /∈F(t, γk(t) +ck)for a.a. t ∈[ak, bk]and each ck∈Ak; or (ii)γ k(t) /∈Kfor a.a. t ∈[ak, bk], where K⊂Ris a compact set such that x(t) ∈Kfor all t ∈Iand every solution xof (2.5). Then, if xis a Carathéodory solution of the inclusion (2.5), it is also a Carathéodory solution of the discontinuous problem (2.4). Proof. It suffices to apply Theorems 2.2 and 2.3 with τk(t, x) =x −γk(t). Indeed, for (t, x) ∈[ak, bk] ×R we have ∇τk(t, x)=(−γ k(t),1) and 6J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 H(τk)(t, x)=−γ k(t)0 00 . Thus, condition (2.9)reads as follows: for each k∈Cand each x =γk(t) +ck, with ck∈Ak, we have −γ k(t)+z=0 forallz∈F(t, x). This is exactly condition (i) here. Note that alternative (ii) implies (2.8), so the conclusion follows.  Notice that there is a variety of existence results in the literature concerning differential inclusion (2.5) with different initial or boundary conditions, see for instance [5,6]. By means of them and the previous conditions which ensure that solutions of (2.5)are in fact Carathéodory solutions of (2.4), we are able to establish existence results for (2.4). Let us now deal with second order equations subject to Dirichlet boundary conditions x =f(t, x),for a.a. t∈I, x(0) = x0,x(T)=xT,(2.10) where x0, xT∈Rn, that is, problem (2.4)with B={x ∈C(I; Rn) :x(0) =x0, x(T) =xT}. Theorem 2.6. Assume that f:I×Rn→Rnsatisfies the following conditions: (C1)for all x ∈Rn, f(·, x)is measurable; (C2)there exists M∈L1(I)such that for a.a. t ∈Iand all x ∈Rn, we have f(t, x)≤M(t); (C3)there exist null measure sets Ak⊂R, k∈C⊂Z, and two times differentiable mappings τk:[ak, bk] × Rn→R, [ak, bk] ⊂I, such that for a.a. t ∈I, f(t, ·)is continuous in Rn\ k∈C :t∈[ak,bk]{x∈Rn:τk(t, x)∈Ak} and for each k∈Cand each (t, x) ∈τ−1 k(Ak)we have (1,v)H(τk)(t, x)(1,v)T+∇xτk(t, x)·z=0 for all v∈Rnsuch that ∇τk(t, x) ·(1, v) =0and all z∈K(f)(t, x). Then problem (2.10)has at least one Carathéodory solution. Proof. By [5, Theorem 12.2], conditions (C1)and (C2) ensure that the differential inclusion x ∈K(f)(t, x),for a.a. t∈I, x(0) = x0,x(T)=xT, has a solution. Then condition (C3) guarantees that it is also a Carathéodory type solution to the Dirichlet problem (2.10), as a consequence of Theorem 2.3. In the scalar case, we have the following existence result for the Dirichlet problem (2.10). Theorem 2.7. Assume that n =1and f:I×R →Rsatisfies conditions (C1), (C2)and J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 7 (C∗ 3)there exist null measure sets Ak⊂R, k∈C⊂Z, and two times differentiable mappings γk:[ak, bk] ⊂ I→Rsuch that for a.a. t ∈I, f(t, ·)is continuous in R\ k∈C :t∈[ak,bk] ck∈Ak{γk(t)+ck} and for each k∈C, the function γksatisfies either (i)γ k(t) /∈K(f)(t, γk(t) +ck)for a.a. t ∈[ak, bk]and each ck∈Ak; or (ii)γ k(t) /∈xT−x0 T−ML1,xT−x0 T+ML1for a.a. t ∈[ak, bk]. Then problem (2.10)has at least one Carathéodory solution. Proof. Note that again conditions (C1)and (C2) ensure that problem (2.10)has a Krasovskii solution. We need to see that it is a Carathéodory solution of (2.10). It follows as a direct application of Corollary 2.5 since the compact interval K=xT−x0 T−ML1,xT−x0 T+ML1 satisfies that x(t) ∈Kfor all t ∈Iand every Krasovskii solution of (2.10).  Observe that the previous result can be even improved allowing fto be discontinuous over the graphs of a countable number of solutions of the differential equation x =f(t, x). Theorem 2.8. Assume that n =1and f:I×R →Rsatisfies conditions (C1), (C2)and (˜ C3)there exist null measure sets Ak⊂R, k∈C, j∈D(with Cand Dat most countable sets) and two times differentiable mappings γk:[ak, bk] ⊂I→Rand ϕj:[˜aj, ˜ bj] ⊂I→Rsuch that for a.a. t ∈I, f(t, ·)is continuous in R\N(t), where N(t)=N1(t)∪N2(t),N 1(t)=  k∈C :t∈[ak,bk] ck∈Ak{γk(t)+ck},N 2(t)=  j∈D :t∈[˜aj,˜ bj]{ϕj(t)}, for each k∈C, the function γksatisfies either (i)γ k(t) /∈K(f)(t, γk(t) +ck)for a.a. t ∈[ak, bk]and each ck∈Ak; or (ii)γ k(t) /∈xT−x0 T−ML1,xT−x0 T+ML1for a.a. t ∈[ak, bk]; and for each j∈D, ϕ j(t) =f(t, ϕj(t)) for a.a. t ∈[˜aj, ˜ bj]. Then problem (2.10)has at least one Carathéodory solution. Proof. By conditions (C1)and (C2), problem (2.10)has a Krasovskii solution x, so we will show that xis also a Carathéodory solution of (2.10). It can be proven (just as above) that the set Jγ= k∈C {t∈[ak,b k]:x(t)−γk(t)∈Ak} 8J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 x(t)α(t)β(t) Fig. 1. A ball inside a one-dimensional billiard table with moving borders. has Lebesgue measure zero. Suppose that there exists j∈Dsuch that m(Jϕ j) >0, with Jϕ j=t∈[˜aj,˜ bj]:x(t)=ϕj(t)and Jϕ= j∈D Jϕ j. Then x(t) =ϕ j(t)for a.a. t ∈Jϕ j. By the definition of ϕj, we have that ϕ j(t) =f(t, ϕj(t)) for a.a. t ∈Jϕ j and thus x(t) =f(t, x(t)) for a.a. t ∈Jϕ j. Hence, x(t) =f(t, x(t)) for a.a. t ∈Jϕ. Finally, since f(t, ·)is continuous at x(t)for a.a. t ∈I\(Jγ∪Jϕ), we conclude that xis a Carathéodory solution of (2.10).  3. Applications to differential problems in time–dependent billiard spaces In this section, we apply the developed tools to the impulsive boundary value problem (1.1)–(1.3). We will use the following concept of solution for problem (1.1)-(1.2). Definition 3.1. We say that a function x ∈C(I;R)is a Carathéodory solution of the billiard problem (1.1)-(1.2)if • graph x ⊂Eα,β, •x ∈W2,1(J;R)and xsatisfies the differential equation in (1.1)for a.e. t ∈J, for every interval J⊂I for which x(t) ∈(α(t), β(t)) for every t ∈J, •if x(s) =α(s), for some s ∈(0, T), then there exist x(s+) and x(s−) satisfying x(s+) −α(s) = α(s) −x(s−), •if x(s) =β(s), for some s ∈(0, T), then there exist x(s+) and x(s−) satisfying x(s+) −β(s) = β(s) −x(s−). Remark 3.2. Note that the impulsive problem (1.1)–(1.2)is called “a billiard problem”, because it is a mathematical model of a ball moving in a line segment (=billiard table) between the “walls” changing their positions in time – see Fig. 1. Moreover, if the ball is inside of the segment, its motion is determined by the differential equation (1.1). And if the ball “hits the boundary”, the bounce is determined by (1.2), i.e. it is absolutely elastic. Without any loss of generality (see Lemma 3.3), we assume a special case of BVP for the lower barrier αidentically equal to zero, i.e. x =f(t, x),for a.a. t∈I:= [0,T],such that x(t)∈(0,γ(t)),(3.11) x(s+) = −x(s−),if s∈(0,T),x(s)=0,(3.12) x(s+) −γ(s)=γ(s)−x(s−),if s∈(0,T),x(s)=γ(s),(3.13) x(0) = A, x(T)=B, (3.14) where γ∈W2,1(I;R), γ>0on I, A ∈(0, γ(0)), B∈(0, γ(T)) and f∈Car(E0,γ ; R)with E0,γ := {(t, x) ∈ I×R :0 ≤x ≤γ(t)}. J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 9 Lemma 3.3. Let α, β∈W2,1(I; R), α<βon I, h ∈Car(Eα,β; R). If xis a Carathéodory solution of the billiard problem (3.11)–(3.13)with γ:= β−αand f(t, x):=h(t, x +α(t)) −α(t),(3.15) then the function y(t)=x(t)+α(t),t∈I, is a solution of (1.1)–(1.2)with the same number of impacts with the boundary. If, moreover, xsatisfies boundary conditions x(0) =A −α(0), x(T) =B−α(T)with A ∈(α(0), β(0)) and B∈(α(T), β(T)), then yis a solution of boundary value problem (1.1)–(1.3). Proof. Let xbe a solution of (3.11)–(3.13). First, let us note that x(t) ∈(0, γ(t)) iff y(t) ∈(α(t), β(t)), x(t) =0iff y(t) =α(t)and x(t) =γ(t)iff y(t) =β(t). Let J⊂Ibe an interval such that y(t) ∈(α(t), β(t)) for each t ∈J. Then x(t) ∈(0, γ(t)) for each t ∈J, x ∈W2,1(J;R)and y(t)=x(t)+α(t)=f(t, x(t)) + α(t)=h(t, x(t)+α(t)) = h(t, y(t)) for each t ∈J. Let y(s) =α(s)for some s ∈I. Then x(s) =0and y(s+) −α(s)=x(s+) + α(s)−α(s)=x(s+) = −x(s−)=−(y(s−)−α(s)) = α(s)−y(s−). Finally, if y(s) =β(s)for some s ∈I, then x(s) =γ(s)and similarly y(s+) −β(s) =β(s) −y(s−). The rest of the proof is trivial.  Remark 3.4. Let us note that if h ∈Car(Eα,β;R)and α, β∈W2,1(I;R), then for fand γdefined in (3.15) we have f∈Car(E0,γ ;R)and γ∈W2,1(I;R). Therefore, from Lemma 3.3 we can see that in order to investigate BVP (1.1)–(1.3), it is sufficient to give results for the problem (3.11)–(3.14). Also note that there is a one-to-one correspondence between solutions of problem (3.11)–(3.13)with (3.15)and problem (1.1), (1.2). Now, let us focus our attention on the billiard problem (3.11)–(3.13). Consider the map Δ :I×R →R defined as Δ(t, x)=x−2kγ(t),if x∈[2kγ(t),(2k+1)γ(t)),for some k∈Z, 2(1 + k)γ(t)−x, if x∈[(2k+1)γ(t),2(k+1)γ(t)),for some k∈Z. Remark 3.5. Note that Δis continuous and Δ(t, x) ∈[0, γ(t)] for all (t, x) ∈I×R. Moreover (a) Δ(t, x) ∈(0, γ(t)) if and only if x ∈(γ(t), ( +1)γ(t)) for some integer , (b) Δ(t, x) =0if and only if x =2kγ(t)for some integer kand (c) Δ(t, x) =γ(t)if and only if x =(2k+1)γ(t)for some integer k. For each t ∈I, the function x → Δ(t, x)is Lipschitz continuous with the Lipschitz constant equal to 1, i.e. for each (t, x), (t, y) ∈[0, T] ×Rwe have |Δ(t, x)−Δ(t, y)|≤|x−y|. 16 J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 γ k(t)/∈K(f)(t, γk(t)+ck) for a.a. t∈[ak,b k]andeachck∈Ak. Lemma 3.9 can be proven again as an application of Theorem 2.7. In the particular case in which γis constant, that is, γ(t) =Rfor all t ∈[0, T], with R>0, we obtain the following multiplicity result [14]. Corollary 3.14. Let A, B∈(0, R)and p ∈Nbe such that p≥T RML1+1. Then there exist at least two solutions of x =f(t, x),for a.a. t∈I:= [0,T],such that x(t)∈(0,R),(3.28) x(s+) = −x(s−),if s∈(0,T),x(s)∈{0,R},(3.29) x(0) = A, x(T)=B, (3.30) having exactly pimpacts. Remark 3.15. If the right-hand side fis identically equal to zero, the problem (3.28)-(3.30) becomes trivially solvable (it is a “classical” billiard table with uniform linear motion of the ball). In this case, there exists exactly one solution with no impact, and for each positive integer pthere exist exactly two solutions having exactly pimpacts with the boundary. In general, the existence of impact-free solution is not guaranteed. For example, let us put γ(t)=R, t ∈[0,T],f(t, x)=a>0,(t, x)∈[0,T]×(0,R),A,B=R/2.(3.31) Then the function x(t)=a 2t2−aT 2t+R 2,t∈[0,T] is the only function x ∈W2,1([0, T]) satisfying x(t) =afor a.e. t ∈[0, T]and x(0) =R/2, x(T) =R/2. Its values stay in the interval (0, R)if and only if xT 2=−a 2T 22 +R 2>0. Therefore if R≤a(T/2)2, then the problem (3.28)-(3.30)for (3.31)doesn’t have any impact-free solution. Let us note that if moreover (3 +2 √2)R≤a(T/2)2, then the problem has no solution with exactly one impact. 3.2. The case of a nonlinear γ Some existence result concerning (3.11)–(3.14)can be obtained even in the case in which γis nonlinear. Lemma 3.16. Let A, B∈Rand assume that there exists N∈Nsuch that M(t)≤Nγ(t)for a.a. t∈I. (3.32) Then the modified problem (3.16), (3.14)has at least one Carathéodory solution. J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 17 Proof. Let C:= max{|A|, |B|} +Nmaxt∈[0,T ]γ(t)and consider the auxiliary problem x =˜ f(t, x),t∈I, x(0) = A, x(T)=B, (3.33) where ˜ f(t, y)=⎧ ⎪ ⎨ ⎪ ⎩ f∗(t, C),if y>C, f∗(t, y),if |y|≤C, f∗(t, −C),if y<−C. Note that since fis a Carathéodory function, then •for all y∈R, ˜ f(·, y)is measurable on I; • there exists ˜ M∈L1(I)such that for a.a. t ∈Iand all y∈Rwe have ˜ f(t, y)≤˜ M(t); and thus ˜ fsatisfies conditions (C1)and (C2). Moreover, observe that for a.a. t ∈I, ˜ f(t, ·) is continuous in [−C, C]\ j∈Z{jγ(t)}. Clearly, ˜ f(t, ·)is continuous in R \[−C, C]. Hence, choose C=∅, D=Z, ϕj:I→Rthe function defined as ϕj(t) =jγ(t), j∈D, and without loss of generality the subintervals [˜aj, ˜ bj] =ϕ−1 j([−C, C]) (if jγ−1([−C, C]) is not connected just consider as many functions ϕjas components have). Since ϕ j(t)=jγ(t)=f∗(t, jγ(t)) = ˜ f(t, ϕj(t)),t∈[˜aj,˜ bj], we deduce that ˜ fsatisfies condition (C∗ 3). Therefore, Theorem 2.8 guarantees that (3.33)has a Carathéodory solution. Let us see that if yis a Carathéodory solution of the modified problem (3.33), then |y(t)|≤Cfor all t ∈I. To prove it, note that condition (3.32) implies that f∗(t, y) ≥0if y>Nγ(t)and f∗(t, y) ≤0if y<−Nγ(t). Then, since C≥Nmaxt∈[0,T ]γ(t), it follows that ˜ f(t, y)≥0ify>C and ˜ f(t, y)≤0ify<−C. Assume, to the contrary, that there exists t0∈(0, R)such that y(t0)= max t∈[0,T ]y(t)>C and y(t0) >y(t)for all t ∈(t0, R). By the continuity of y, we deduce that there exists r>0such that y(t) >Cfor all t ∈(t0−r, t0+r)and so y(t)= ˜ f(t, y(t)) >0 for a.a. t∈(t0−r, t0+r). By integration, taking into account that y(t0) =0, we have y(t)= t  t0 y(s)ds = t  t0 ˜ f(s, y(s)) ds > 0fort∈(t0,t 0+r). 18 J. Rodríguez-López, J. Tomeček / J. Math. Anal. Appl. 536 (2024) 128237 Hence, yis increasing in the interval (t0, t0+r), which contradicts the choice of t0. Analogously, we can show that y(t) >−Cfor all t ∈I. Therefore, if yis a Carathéodory solution of the modified problem (3.33), then it is also a Carathéodory solution of (3.16), (3.14).  Remark 3.17. Clearly, condition (3.32)is satisfied if fis a continuous function in E0,γ and γis a two times continuously differentiable function such that γ(t) >0for all t ∈I. Condition (3.32) provides “a priori” bounds for the solutions of the modified problem (3.16), (3.14). Our reasoning goes in the line of [10]and reminds the method of lower and upper solutions. Indeed, the constant functions α≡−Cand β≡Ccan be seen, respectively, as a lower and an upper solution for (3.16), (3.14). As a direct consequence of Lemmas 3.6 and 3.16, we derive the following existence result for the timedependent billiard problem (3.11)–(3.14). Theorem 3.18. Assume that there exists N∈Nsuch that condition (3.32)holds. Then the impulsive problem (3.11)–(3.14)has at least one solution. Theorem 3.19. Let M∈L1(I)be such that for a.a. t ∈Iand x ∈[α(t), β(t)] we have (3.25). Moreover, assume that there exists N∈Nsuch that M(t)≤N(β(t)−α(t)) for a.a. t∈I. Then the impulsive problem (1.1)–(1.3)has at least one solution. Proof. As in the proof of Theorem 3.12, we consider the problem (3.11)–(3.14)with (3.15)and A := A −α(0), B:= B−α(T)and use Lemma 3.3 together with Theorem 3.18. Acknowledgments Jorge Rodríguez–López was partially supported by Agencia Estatal de Investigación, Spain, and Feder, Project PID2020-113275GB-I00. Jan Tomeček was supported by Palacký University in Olomouc (grant no. IGA_PrF_2023_009). References [1] V.I. Bogachev, Measure Theory, vol. I, Springer, 2007. [2] D. Bothe, On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows, Nonlinearity 33 (2020) 5425–5456. [3] A. Bressan, W. Shen, On discontinuous differential equations, in: J. Andres, L. Górniewicz, P. Nistri (Eds.), Differential Inclusions and Optimal Control, in: Lect. Notes Nonlinear Anal., vol. 2, 1998, pp. 73–87. [4] J.Á. Cid, R. 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