Positive Solutions for a Higher‐Order Nonlinear Neutral Delay Differential Equation

Abstract

This paper deals with the higher‐order nonlinear neutral delay differential equation (dn/dtn)[x(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, where n, 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), and limt→+∞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.