Abstract
The aim of this paper is to study the solvability of a third-order nonlinear neutral delay differential equation of the form{α(t)[β(t)(x(t)+p(t)x(t−τ))′]′}′+f(t,x(σ1(t)),x(σ2(t)),…,x(σn(t)))=0,t≥t0. By using the Krasnoselskii's fixed point theorem and the Schauder's fixed point theorem, we demonstrate the existence of uncountably many bounded nonoscillatory solutions for the above differential equation. Several nontrivial examples are given to illustrate our results.
Highlights
Introduction and PreliminariesIn recent years, the study of the oscillation, nonoscillation, asymptotic behaviors and existence of solutions for various kinds of first- and second-order neutral delay differential equations and systems of differential equations have attracted much attention, for example, see 1–12 and the references therein
Motivated by the papers mentioned above, in this paper, we investigate the following third-order nonlinear neutral delay differential equation αtβtxtptxt−τ f t, x σ1 t, x σ2 t, . . . , x σn t 0, t ≥ t0, 1.7 where n ≥ 1 is an integer, τ > 0, α, β ∈ C t0, ∞, R \ {0}, p ∈ C t0, ∞, R, and f ∈ C t0, ∞ × Rn, R
We study those conditions under which 1.7 possesses uncountably many bounded nonoscillatory solutions
Summary
The study of the oscillation, nonoscillation, asymptotic behaviors and existence of solutions for various kinds of first- and second-order neutral delay differential equations and systems of differential equations have attracted much attention, for example, see 1–12 and the references therein. Dorociakovaand Olach 2 discussed the existence of nonoscillatory solutions and asymptotic behaviors for the first-order delay differential equation xtptxtqtxτt 0, t ≥ 0. Tang and Liu 10 studied the existence of bounded oscillation for the second-order linear delay differential equation of unstable type xtptxt − τ , t ≥ t0, 1.3 where τ > 0, p ∈ C t0, ∞ , R and p t /≡ 0 on any interval of length τ. In view of the Banach fixed point theorem, Kulenovicand Hadziomerspahic 7 deduced the existence of a nonoscillatory solution for the second-order linear neutral delay differential equation with positive and negative coefficients x t cx t − τ. By utilizing the Krasnoselskii’s fixed point theorem, Zhou 12 discussed the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation m rtxtptxt−τ. S has at least one fixed point in Ω
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