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 $$\ddot{x}(t) = f(t,x(t))$$ x ¨ ( t ) = f ( t , x ( t ) ) together with Dirichlet boundary conditions $$x(0) = A$$ x ( 0 ) = A , $$x(T) = B$$ 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.