On oscillatory solutions of certain forced Emden–Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x″(t)+a(t)λ|x(t)|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” . 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].