On oscillatory solutions of certain forced Emden–Fowler like equations

Abstract

We give a constructive proof of existence to oscillatory solutions for the differential equations x(t) + a(t)|x(t)| λ sign[x(t)] = e(t), where t t0 1 and λ > 1, that decay to 0 when t → +∞ as O(t−μ) for μ > 0 as close as desired to the “critical quantity” μ = 2 λ−1 . For this class of equations, we have limt→+∞ E(t) = 0, where E(t) < 0 and E(t) = e(t) throughout [t0,+∞). We also establish that for any μ > μ and any negative-valued E(t) = o(t−μ) as t → +∞ the differential equation has a negative-valued solution decaying to 0 at +∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722–732]