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)vertical bar x(t)vertical bar lambda sign[x(t)] = e(t), where t >= t(0) >= 1 and lambda > 1, that decay to 0 when t -> infinity as 0(t(-mu)) for mu > 0 as close as desired to the "critical quantity" mu* = 2/lambda-1 For this class of equations, we have lim(t ->+infinity) E(t) = 0, where E(t) 0 and E ''(t) e(t) throughout [t(0) + infinity). We also establish that for any mu > mu* and any negative-valued E(t) = 0(t(-mu)) as t ->+infinity the differential equation has a negative-valued solution decaying to 0 at +infinity as o(t(-mu)). 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]. (C) 2008 Elsevier Inc. All rights reserved.