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 ⩾ t 0 ⩾ 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 lim t → + ∞ E ( t ) = 0 , where E ( t ) < 0 and E ″ ( t ) = e ( t ) throughout [ t 0 , + ∞ ) . 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].