Existence Theorems for Nonlinear Functional Differential Equations of Neutral Type

Conditions are found upon satisfaction of which the differential equation x(n)(t)− λx(n)(t− σ) + f(t, x(g(t))) = 0 has solutions which are asymptotically equivalent to the solutions of the equation x(n)(t)− λx(n)(t− σ) = 0. § 0. Introduction We consider the neutral functional differential equation x(n)(t)− λx(n)(t− σ) + f(t, x(g(t))) = 0 (A) under the assumptions that (i) n ≥ 1 is an integer; λ and σ are positive constants with λ ≤ 1; (ii) g : [t0,∞) → R is continuous, t0 > 0, and lim t→∞ g(t) = ∞; (iii) f : [t0,∞)× R→ R is continuous and satisfies |f(t, x)| ≤ F (t, |x|), (t, x) ∈ [t0,∞)× R, for some continuous function F (t, u) on [t0,∞)× [0,∞) which is nondecreasing in u for each fixed t ≥ t0. (0.1) 1991 Mathematics Subject Classification. 34K15.