TY - JOUR

T1 - Existence of positive solutions of higher order nonlinear neutral differential equations

AU - Tanaka, Satoshi

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2000

Y1 - 2000

N2 - The neutral differential equation (1.1) dn/dtn [x(t) + h(t)x(t – τ)] + σf (t, x(g(t))) = 0 is considered under the following conditions: n ≥ 2; σ = ±1; τ > 0; h ∈ C[t0, ∞), lim t → ∞ g(t)=∞ ; f ∈ C([t0, ∞), × (0, ∞)), f(t, u) ≥ 0 for (t, u) ∈ [t0, ∞)× (0, ∞), and f(t, u) is nondecreasing in u ∈ (0, ∞) for each fixed t ∈ [t0, ∞). It is shown that, for the case where h(t) > –1 and h(t) = h(t – τ) on [t0, ∞), equation (1.1) has a positive solution x(t) satisfying x(t) = [c/1+h(t)+o(1)]tk as t → ≈ for some c > 0 if and only if ∫∞ tn-k-1 (t, a[g(t)]k) dt < ≈ for some a > 0. Here k is an integer with 0 ≤ k ≤ n – 1.

AB - The neutral differential equation (1.1) dn/dtn [x(t) + h(t)x(t – τ)] + σf (t, x(g(t))) = 0 is considered under the following conditions: n ≥ 2; σ = ±1; τ > 0; h ∈ C[t0, ∞), lim t → ∞ g(t)=∞ ; f ∈ C([t0, ∞), × (0, ∞)), f(t, u) ≥ 0 for (t, u) ∈ [t0, ∞)× (0, ∞), and f(t, u) is nondecreasing in u ∈ (0, ∞) for each fixed t ∈ [t0, ∞). It is shown that, for the case where h(t) > –1 and h(t) = h(t – τ) on [t0, ∞), equation (1.1) has a positive solution x(t) satisfying x(t) = [c/1+h(t)+o(1)]tk as t → ≈ for some c > 0 if and only if ∫∞ tn-k-1 (t, a[g(t)]k) dt < ≈ for some a > 0. Here k is an integer with 0 ≤ k ≤ n – 1.

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U2 - 10.1216/rmjm/1021477264

DO - 10.1216/rmjm/1021477264

M3 - Article

AN - SCOPUS:0034259816

VL - 30

SP - 1139

EP - 1149

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 3

ER -