Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

Abstract

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form ( E, δ) L n x( t) + δq( t) f( x[ g 1( t)],…, x[ g m ( t)]) = 0, where δ = ± 1 and L 0 x( t) = x( t), L k x( t) = a k ( t)( L k − 1 x( t)) ., k = 1, 2,…, n ( . = d dt ) , is examined. A classification of solutions of ( E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of ( E, 1) and ( E, −1) with first and second order equations of the form y .( t) + c 1( t) f( y[ g 1( t)],…, y[ g m ( t)]) = 0 and ( a n − 1 ( t) z .( t)) . − c 2( t) f( z[ g 1( t)],…, z[ g m ( t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.