Oscillatory Behavior of Second-Order Nonlinear Neutral Differential Equations

Tongxing Li,Yuriy V Rogovchenko

doi:10.1155/2014/143614

Abstract

We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.

Highlights

This paper is concerned with the oscillation of a class of second-order nonlinear neutral functional differential equations (r (t) ((x + p x (η (t)))󸀠 γ󸀠 )) f (t, x (g (t))) = (1)where t ≥ t0 > 0
We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments
The increasing interest in problems of the existence of oscillatory solutions to second-order neutral differential equations is motivated by their applications in the engineering and natural sciences

Summary

Introduction

This paper is concerned with the oscillation of a class of second-order nonlinear neutral functional differential equations (r (t). ≥ t0 such that x + p ⋅ continuously differentiable x∘ and η x satisfies (1) for all t ≥ Tx. In what follows, we assume that solutions of (1) exist and can be continued indefinitely to the right. Baculıkovaand Dzurina [6] studied oscillation of a second-order neutral functional differential equation Obtaining sufficient conditions for all solutions of (1) either to be oscillatory or to satisfy limt → ∞x(t) = 0; see [21, Theorem 2.3]. Sufficient conditions for all solutions of (1) either to be oscillatory or to satisfy limt → ∞x(t) = 0 were obtained under the assumptions that (17) holds and 0 ≤ p(t) ≤ p1 < 1; see [13, Theorem 3.8]. Our principal goal in this paper is to analyze the oscillatory behavior of solutions to (1) in the case where (17) holds and without assumptions (H3), (19)–(23), and γ ≥ 1

Oscillation Criteria

Examples and Discussion