Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients

Qi Li,Rui Wang,Feng Chen,Tongxing Li

doi:10.1186/s13662-015-0377-y

Abstract

We present several oscillation criteria for a second-order nonlinear delay differential equation with a nonpositive neutral coefficient. Two examples are given to illustrate the main results.

Highlights

1 Introduction In this work, we study the oscillation of a nonlinear second-order neutral delay differential equation r(t) z (t) α + q(t)f x σ (t) =, t ≥ t >, ( . )
Throughout, we assume that the following hypotheses are satisfied: (H ) r, p, q ∈ C([t, ∞), R), r(t) >, ≤ p(t) ≤ p , σ (t) ≤ t, and limt→∞ σ (t) = ∞; (H ) f ∈ C(R, R), uf (u) > for all u =, and there exists a positive constant k such that f (u) uα
We provide some background details regarding the study of oscillation of second-order differential equations which motivated our study

Summary

Introduction

Throughout, we assume that the following hypotheses are satisfied: (H ) r, p, q ∈ C([t , ∞), R), r(t) > , ≤ p(t) ≤ p < , q(t) ≥ , and q is not identically zero for large t; (H ) τ ∈ C([t , ∞), R), τ (t) ≤ t, and limt→∞ τ (t) = ∞; (H ) σ ∈ C ([t , ∞), R), σ (t) > , σ (t) ≤ t, and limt→∞ σ (t) = ∞; (H ) f ∈ C(R, R), uf (u) > for all u = , and there exists a positive constant k such that f (u) uα ) is termed oscillatory if all its solutions oscillate. There has been increasing interest in studying oscillation of solutions to different classes of differential equations due to the fact that they have numerous applications in natural sciences and engineering; see, e.g., Hale [ ] and Wong [ ].

Results

Conclusion