Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

Abstract

Dirichlet problem in an n-dimensional billiard space is investigated. In particular, the system of ODEs ddot{x}(t) = f(t,x(t)) together with Dirichlet boundary conditions x(0) = A, x(T) = B in an n-dimensional interval K with elastic impact on the boundary of K is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulating it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.