Quasi-periodic solutions for the general semilinear Duffing equations with asymmetric nonlinearity and oscillating potential

Abstract

In this paper, we prove the existence of quasi-periodic solutions and boundeness of all solutions of the general semilinear quasi-periodic differential equation x″ + ax+ − bx− = Gx(x, t) + f(t), where x+ = max{x, 0}, x− = max{ − x, 0}, a and b are two different positive constants, f(t) is $${{\cal C}^{39}}$$ smooth in t, G(x, t) is $${{\cal C}^{35}}$$ smooth in x and t, f (t) and G(x, t) are quasi-periodic in t with the Diophantine frequency ω = (ω1, ω2), and $$D_x^iD_t^jG\left({x,\;t} \right)$$ is bounded for 0 ⩽ i + j ⩽ 35.