Abstract: In this paper, we aim to find eventually vanished solutions, a special class of bounded solutions which tend to 0 as t→±∞, to a Li´enard system with a time-dependent force. Since it is not a Hamiltonian system with small perturbations, the well-known Melnikov method is not applicable to the determination of the existence of eventually vanished solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. Difficulties caused by the non-Hamiltonian form are overcome by applying the Schauder’s fixed point theorem. We show that the sequence of the periodic solutions has an accumulation giving an eventually vanished solution of the forced Li´enard system.

Key words: eventually vanished, bounded solution, non-Hamiltonian, accumulation