Boundedness in asymmetric oscillations under the non-resonant case

In this paper, we are concerned with the boundedness of all solutions of the asymmetric oscillation x '' + ax(+) - bx(-) = p(t), where x(+) = max{x, 0}, x(-) = max{-x, 0}, p(t) is a real analytic 2 pi periodic function, a and b are two different positive constants satisfying omega(0) := 1/2(1/root a + 1/root b) is an element of R\Q and the condition vertical bar k omega(0) - l vertical bar >= c(0)/Omega(vertical bar k vertical bar), k is an element of Z\{0}, l is an element of Z, where Omega is an approximation function and c(0) is a small positive constant. In particular, when a = 5, b = 1, p(t) = cos 4t, the boundedness of all solutions will be proved. (C) 2020 Elsevier Inc. All rights reserved.