Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side

Abstract

The paper brings multiplicity results for a Dirichlet problem in one-dimensional billiard space with right-hand side depending on the velocity of the ball, i.e. a problem in the form
x'' = f(t, x, x')    if x(t) ∈ int K,       x'(t+) = -x'(t-)    if x(t) ∈ ∂K,
                                     x(0) = A,    x(T) = B,
where T > 0, K = [0, R], R > 0, f is a Carathéodory function on [0, T] × K  × ℝ, A, B ∈ int K. Sufficient conditions ensuring the existence of at least two solutions having prescribed number of impacts with the boundary of the billiard table K are obtained. In particular, if the right-hand side has at most sublinear growth in the last variable, there exist infinitely many solutions of the problem.