Boundedness for the general semilinear Duffing equations via the twist theorem

BOUNDEDNESS FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH JUMPING $p$-LAPLACIAN AND AN OSCILLATING TERM

“…The following results are similarity to [9],here for the convenience of readers, we still give the proof of these lemmas. where R satisfies…”

Boundedness of semilinear Duffing equations with singularity

“…They proved the existence of quasiperiodic solutions and the boundedness of all solutions. In the above quoted results, ψ(x, t) satisfies the growth condition lim |x|→+∞ x m D m x ψ(x, t) = 0 (5) for some finite m.…”

“…We point out that [9,16] do with V (x) is regular potential and ψ(x, t) satisfies the growth condition (5). But in this paper, we try to study the boundedness of (1) without condition (5).…”

“…But in this paper, we try to study the boundedness of (1) without condition (5). For example, ψ(x, t) = cos(x + t).…”

Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials

“…To cope with the nonsmoothness in φ and use Moser's small twist theorem to prove the Lagrangian stability, we will exchange the positions of variables (ρ, φ) and (H 3 , t) below. This trick has been used in [18,19,20].…”