Abstract
This paper deals with the higher-order nonlinear neutral delay differential equation(dn/dtn)[x(t)+∑i=1mpi(t)x(Ti(t))]+(dn−1/dtn−1)f(t,x(α1(t)),…,x(αk(t)))+h(t,x(β1(t)),…,x(βk(t)))=g(t),t≥to, wheren,m,k∈ℕ,pi,τi,βj,g∈C([to,+∞),ℝ),αj∈Cn−1([to,+∞),ℝ),f∈Cn−1([to,+∞)×ℝk,ℝ),h∈C([to,+∞)×ℝk,ℝ), andlimt→+∞τi(t)=limt→+∞αj(t)=limt→+∞βj(t)=+∞,i∈{1,2,…,m},j∈{1,2,…,k}. By making use of the Leray-Schauder nonlinear alterative theorem, we establish the existence of uncountably many bounded positive solutions for the above equation. Our results improve and generalize some corresponding results in the field. Three examples are given which illustrate the advantages of the results presented in this paper.
Highlights
Zhou et al 7 used the Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation: dn dtn xt p t x t−τ m qi t f x t − σi i1 g t, t ≥ t0, 1.10 where τ, σi ∈ R, p, qi, g ∈ C t0, ∞, R for i ∈ {1, 2, . . . , m} and f ∈ C R, R
It follows from Theorem 2.3 that 3.5 has uncountably many bounded positive solutions in U M
Summary
Introduction and PreliminariesThis paper is concerned with the higher-order nonlinear neutral delay differential equation: dn dtn xt m pi t x τi t i1 dn−1 dtn−1 f t, x α1 t , . . . , x αk t1.1 h t, x β1 t , . . . , x βk t g t , t ≥ t0, where n, m, k ∈ N, pi, τi, βj , g ∈ C t0, ∞ , R , αj ∈ Cn−1 t0, ∞ , R , f ∈ Cn−1 t0, ∞ × Rk, R , h ∈ C t0, ∞ × Rk, R , and t−l→im∞τi t t−l→im∞αj t t−l→im∞βj t∞, i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , k}.Abstract and Applied AnalysisTheory of neutral delay differential equations has undergone a rapid development in the last over thirty years. Zhou and Zhang 8 extended the results in 1 to the higher-order neutral functional differential equation with positive and negative coefficients: dn dtn x t px t − τ Zhou et al 7 used the Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation: dn dtn xt p t x t−τ m qi t f x t − σi i1 g t , t ≥ t0, 1.10 where τ, σi ∈ R , p, qi, g ∈ C t0, ∞ , R for i ∈ {1, 2, .
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