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

Abstract

The paper gives multiplicity results for the impulsive boundary value problemx 00 D f .t; x/; for a.e. t 2 OE0; T ; such that x.t/ 2 int K;x 0 .sC/ D x 0 .s /; if s 2 .0; T /; x.s/ 2 @K;x.0/ D A; x.T / D B; where K R is a compact interal, f is Carathéodory function on OE0; T K and A; B 2 int K. This problem can be understood as a problem in one-dimensional billiard space and it is also a generalization of oscillator with obstacles from below and from above and absolutely elastic impacts. A simple condition for the existence of solution with exact number of impacts is given, as well as the multiplicity result. The results are obtained by a transformation into problem without impulses (without impacts) and using Schauder fixed point theorem.

Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

“…First, we leave the impulsive conditions at the boundary of K at the cost of losing the regularity of the right-hand side of the differential equation and obtain (possibly singular) equation ( 8). The possible singularity lies in the state variable, and we overcome this obstacle by constructing the sequence of regular problems (9). By means of a priori bound technique and Arzelà-Ascoli Theorem we obtain the existence of multiple solutions of the two-point boundary value problem for singular equation (8).…”

“…Since the right-hand side f * can have (and it really does have) discontinuity points with respect to the second variable, it is not a Carathéodory map. Hence, inspired by [9], we are going to regularize f * . To do this, we define…”

“…. , n, then there exists a monotone solution z of problem (8) with z(0) = A and z(T (9) with y m (0) = A, y m (T ) = B, and satisfying (10) and (11). Of course, since…”

“…In [2] the authors gave the numerical treatment to the problem. In [9] the author transformed a one dimensional billiard problem to the problem without impulses and proved multiplicity results by the use of the Schauder Fixed Point Theorem. It was possible because the real line could be viewed as a mosaic built of copies of an interval K = [0, R], i.e., R = i∈Z [iR, (i + 1)R].…”