Asymptotic Behavior of Solutions to Nonlinear Neutral Differential Equations

Abstract

In Chapter 2 of this thesis, in the first part, we deal with asymptotic behavior of nonoscillatory solutions to higher order nonlinear neutral differential equations of the form (x (t) + p (t) x (t − τ ))(n) + f (t, x (t) , x (ρ (t)) , x (t) , x (σ (t))) = 0,for n ≥ 2. We formulate sufficient conditions for all non-oscillatory solutions to behave like polynomial functions at infinity. For the higher order differential equation (x (t) + p (t) x (t − τ ))(n) + f (t, x (t) , x (ρ (t))) = 0, we provide necessary and sufficient conditions that guarantee existence of non-oscillatory solutions with polynomial-like behavior at infinity. In Chapter 3, we look into oscillation problem of second order nonlinear neutral differential equations r (t) ψ (x (t)) (x (t) + p (t) x (τ (t))) + q (t) f (x (t) , x (σ (t))) = 0 and r (t) (x (t) + p (t) x (τ (t))) + q (t) f (x (t) , x (σ (t))) = 0.