Abstract
Employing a generalized Riccati transformation and integral averaging technique, we show that all solutions of the higher order nonlinear delay differential equation \t\t\ty(n+2)(t)+p(t)y(n)(t)+q(t)f(y(g(t)))=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ y^{(n+2)}(t)+p(t)y^{(n)}(t)+q(t)f\\bigl(y\\bigl(g(t)\\bigr)\\bigr)=0 $$\\end{document} will converge to zero or oscillate, under some conditions listed in the theorems of the present paper. Several examples are also given to illustrate the applications of these results.
Highlights
The objective of this paper is to investigate the oscillation and asymptotic behavior of solutions to the following higher order nonlinear delay differential equation: y(n+2)(t) + p(t)y(n)(t) + q(t)f (y g(t) = 0, t ∈ I, (1.1)where n is a positive integer, I = [a, +∞) ⊂ R (a ≥ 0)
Our attention is restricted to the solutions of equation (1.1) which exist on the interval I and satisfy supt≥T |y(t)| > 0 for any T ≥ a
3 Asymptotic dichotomy we present some sufficient conditions which guarantee that every solution to equation (1.1) oscillates or converges to zero as t approaches to infinity
Summary
In 2014, the oscillation and asymptotic behavior of solutions to the following nonlinear delay differential equation were studied in [15]: x(n+3)(t) + p(t)x(n)(t) + q(t)f x g(t) = 0. The goal of the present paper is to use a generalized method to study the oscillation and asymptotic behavior of solutions to the nonlinear delay differential equation (1.1). We can use the induction method to obtain limt→+∞ y(t) = –∞ This is a contradiction and we finish the proof of Theorem 3.1. Proof Suppose that y(t) is a non-oscillatory solution of equation (1.1) on the interval [T, +∞), where T ≥ a.
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