Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side - Krajščáková, Vĕra; Tomeček, Jan (Lietuvos matematikos rinkinys, 2024)
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.