Abstract :
In this paper we consider the following nth-order neutral delay differential equation:
dn
dtn x(t) +cx(t −τ) +(−1)n+1f t,x(t −σ1), x(t −σ2), . . . , x(t − σk) = g(t), t t0,
where n is a positive integer, c ∈ R, τ > 0, σi > 0 for i = 1, . . . , k, f ∈ C([t0,∞) × Rk,R) and
g ∈ C([t0,∞),R+). By employing the contraction mapping principle, we obtain several existence results of
nonoscillatory solutions for the above equation, construct a fewMann-type iterative approximation schemes
for these nonoscillatory solutions and establish several error estimates between the approximate solutions
and the nonoscillatory solutions. In addition, we obtain some sufficient conditions for the existence of infinitely
many nonoscillatory solutions. These results presented in this paper extend, improve and unify many
known results due to Cheng and Annie [J.F. Cheng, Z. Annie, Existence of nonoscillatory solution to second
order linear neutral delay equation, J. Systems Sci.Math. Sci. 24 (2004) 389–397 (in Chinese)], Graef, Yang
and Zhang [J.R. Graef, B. Yang, B.G. Zhang, Existence of nonoscillatory and oscillatory solutions of neutraldifferential equations with positive and negative coefficients, Math. Bohem. 124 (1999) 87–102], Kulenovi
´c and Hadžiomerspahi´c [M.R.S. Kulenovi´c, S. Hadžiomerspahi´c, Existence of nonoscillatory solution
of second order linear neutral delay equation, J. Math. Anal. Appl. 228 (1998) 436–448; M.R.S. Kulenovi´c,
S. Hadžiomerspahi´c, Existence of nonoscillatory solution for linear neutral delay equation, Fasc. Math. 32
(2001) 61–72], Zhang and Yu [B.G. Zhang, J.S. Yu, On the existence of asymptotically decaying positive
solutions of second order neutral differential equations, J. Math. Anal. Appl. 166 (1992) 1–11], Zhang
[B.G. Zhang, On the positive solutions of a kind of neutral equations, Acta Math. Appl. Sinica 19 (1996)
222–230] and Zhou and Zhang [Y. Zhou, B.G. Zhang, Existence of nonoscillatory solutions of higher-order
neutral differential equations with positive and negative coefficients, Appl. Math. Lett. 15 (2002) 867–874]
and others. Some nontrivial examples are given to illustrate the advantages of our results.
© 2006 Elsevier Inc. All rights reserved
Keywords :
Error estimate , nth-order neutral delay differential equation , Nonoscillatory solution , Contraction mapping , Infinitely many nonoscillatorysolutions , Mann iterative sequence
