On the Dirichlet problem in billiard spaces

Abstract

The constrained Dirichlet boundary value problemẍ = f (t, x), x(0) = x(T ), is studied in billiard spaces, where impacts occur in boundary points. Therefore we develop the research on impulsive Dirichlet problems with state-dependent impulses. Inspiring simple examples lead to an approach enabling to obtain both the existence and multiplicity results in one dimensional billiards. Several observations concerning the multidimensional case are also given.

Multiple solutions of a Dirichlet problem in one-dimensional billiard space

“…Note that a similar result in [4] (Theorem 3.6) cannot be applied in this case, because the right-hand side of the differential equation is not Lipschitz-continuous in the second variable. Moreover, Theorem 1 gives more detailed multiplicity results than the result in [4].…”

“…R, .t / D jt j for t 2 OE R; R and is 2R-periodic. Therefore, range of is OE0; R, it is a piecewise linear function, it is Lipschitz continuous on R. Moreover, Unlike the paper [4], here the existence result is obtained by using an appropriate "extension" of the right-hand side of the equation (1.1) onto the set OE0; T R. Note that this extension is not neccessarily a Carathéodory function on OE0; T R, but it has possible discontinuities in the state variable.…”

“…The existence of infinitely solutions are then guaranteed. The purpose of this paper was to give more general and more detailed information than in [4] for the one-dimensional case. Here, the Lipschitz condition is not needed.…”

“…Moreover, the exact number of impacts is given (see Theorem 1). The reason of the weakened assumptions lies in the fact that paper [4] relies on the shooting method and continuous dependence on initial values of solution. On the other hand, the idea in [4] can be quite easily generalized for higher dimensions.…”

“…The reason of the weakened assumptions lies in the fact that paper [4] relies on the shooting method and continuous dependence on initial values of solution. On the other hand, the idea in [4] can be quite easily generalized for higher dimensions. Here, the idea is to investigate an auxiliary problem without impulses.…”