A survey on the oscillation of differential equations with several deviating arguments

Ioannis P Stavroulakis

doi:10.1186/1029-242x-2014-399

Abstract

Consider the first-order linear differential equation with several retarded arguments x ′ (t)+ ∑ i = 1 m p i (t)x ( τ i ( t ) ) =0,t≥ t 0 , where the functions p i , τ i ∈C([ t 0 , ∞), R + ) for every i=1,2,…,m, τ i (t)≤t for t≥ t 0 and lim t → ∞ τ i (t)=∞. A survey on the oscillation of all solutions to this equation is presented in the case of several non-monotone arguments and especially when well-known oscillation conditions are not satisfied. Examples illustrating the results are given.MSC:34K11, 34K06.

Highlights

Consider the differential equation with several non-monotone retarded arguments m x (t) + pi(t)x τi(t) =, t ≥ t, ( . ) i=where the functions pi, τi ∈ C([t,∞), R+) for every i =, . . . , m), τi(t) ≤ t for t ≥ t and limt→∞ τi(t) = ∞.Let T ∈ [t, +∞), τ (t) = min{τi(t) : i =, . . . , m} and τ(– )(t) = inf{τ (s) : s ≥ t}
In this paper we present a survey on the oscillation of all solutions to these equations in the case of several non-monotone arguments and especially when the well-known oscillation conditions t t lim sup p(s) ds > and lim inf p(s) ds >
2 Oscillation criteria for equation (1.2) we study the delay equation x (t) + p(t)x τ (t) =, t ≥ t, where the functions p, τ ∈ C([t,∞), R+), τ (t) < t for t ≥ t and limt→∞ τ (t) = ∞

Summary

Introduction

In , Ladas et al [ ] proved that the same conclusion holds if in addition τ is a non-decreasing function and t (C ) A := lim sup p(s) ds >. In , Ladas [ ] established integral conditions for the oscillation of equation In , Erbe and Zhang [ ] developed new oscillation criteria by employing the upper bound of the ratio x(τ (t))/x(t) for possible non-oscillatory solutions x(t) of equation ) and assume that τ (t) is continuously differentiable and that there exists θ > such that p(τ (t))τ (t) ≥ θ p(t) eventually for all t Under this additional assumption, in , Kon et al [ ] and in , Sficas and Stavroulakis [ ] established the conditions.

In the case where

Note that the value of the lower bound on

We see that

We can take

Pdξ ds