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

Krajščáková, Věra; Tomeček, Jan

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) ϵ int K, x'(t ) = -x'(t-) if x(t) ϵ ∂K, x(0) = A, x(T) = B, T > 0, K = [0, R], R > 0, f is a Carathéodory function on [0, T] × K × R, A, B ϵ 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 righthand side has at most sublinear growth in the last variable, there exist infinitely many solutions of the problem.

DIRICHLET problem; VELOCITY; PROBLEM solving; MATHEMATICAL variables; MATHEMATICAL formulas

Lietuvos Matematikos Rinkinys, 2024, Vol 65, Issue Series A, p25

0132-2818

Academic Journal

10.15388/lmd.2024.37775