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 reformulation it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.

10 pages, 1 figure

References in corpus (2)

• Almost periodic evolution systems with impulse action at state-dependent moments
• On the Dirichlet problem in billiard spaces