On the stability and boundedness of a class of higher order delay differential equations

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Abstract

We establish some sufficient conditions which guarantee asymptotic stability of the null solution and boundedness of all the solutions of the following nonlinear differential equation of third order with the variable delay, r(t)

x(t)+g(x(tr(t)))x(t)+ψ(x(t))+f(x(tr(t)))+h(x(tr(t)))=p(t,x(t),x(t),x(tr(t)),x(tr(t)),x(t)),

when p(t, x(t), x′(t), x(tr(t)), x′(tr(t)), x′′(t))=0 and ≠0, respectively. By defining an appropriate Lyapunov functional, we prove two new theorems on the stability and boundedness of the solutions of the above equation. We also give an example to illustrate the theoretical analysis in this work. Our results improve a stability result in the literature, which was obtained for nonlinear differential equations of third order without delay, to the above differential equation with delay for stability and boundedness of the solutions.

Introduction

By a recent paper published in 2007, Zhang and Si [23] proved an asymptotic stability result for solutions to the following nonlinear third order scalar differential equation without delay:x(t)+g(x(t))x(t)+f(x(t),x(t))+h(x(t))=0.

At the same time, for some papers published on the qualitative behaviors of solutions of various nonlinear third order differential equations with delay or without delay, we refer the reader to the papers of Palusinski et al. [3], Tunç [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and the references thereof.

The object of this paper is to consider nonlinear third order differential equation with variable delay, r(t):x(t)+g(x(tr(t)))x(t)+ψ(x(t))+f(x(tr(t)))+h(x(tr(t)))=p(t,x(t),x(t),x(tr(t)),x(tr(t)),x(t)).

Section snippets

Preliminaries

Eq. (1) can be written as the following system:x(t)=y(t),y(t)=z(t),z(t)=g(y(t))z(t)ψ(y(t))f(y(t))h(x(t))+tr(t)tg(y(s))z(s)ds+tr(t)tf(y(s))z(s)ds+tr(t)th(x(s))y(s)ds+p(t,x(t),y(t),x(tr(t)),y(tr(t)),z(t)),where 0≤r(t)≤ρ, ρ is a positive constant to be determined later, and r′(t)≤β, 0<β<1; the primes in Eq. (1) denote differentiation with respect to t, t∈ℝ+, ℝ+=[0,∞); the functions g, ψ, f, h and p are continuous in their respective arguments on ℝ, ℝ, ℝ, ℝ and ℝ+×ℝ5,

Problem description

LetΩ:={(t,x,y,z)+×3:0t<,|x|<H1,|y|<H1,|z|<H1,H1H}.

Suppose there exist positive constants a, b, μ, δ, α, L and M such that the following assumptions hold for every t, x, y and z in Ω:

  • (a1)g(y)a+μand|g(y)|M.

  • (a2)ψ(y)sgnyα|y|.

  • (a3)f(y)sgny(b+δ)|y|and|f(y)|L.

  • (a4)0<h(x)<abandsgnh(x)=sgnx.

  • (a5)|p(t,x,y,x(tr(t)),y(tr(t)),z)|q(t),

where qL1(0,∞), L1 is space of Lebesgue integrable functions.

Let p(t,x,x,x(tr(t)),x(tr(t)),x)0 in (1).

Theorem 1

Consider Eq. (1), where the functions g, ψ, f and h

Proofs of main theorems

Proof of Theorem 1

By utilizing (7), it follows thatV(xt,yt,zt)D7(x2+y2+z2),where D7=min{D1, D2, D3}. The existence of a continuous function u(s)≥0 with u(|ϕ(0)|)≥0 such that u(|ϕ(0)|)≤V(ϕ) is now readily verified.

It can also be followed that the largest invariant set in Z is Q={0}, where

Z={ϕCH:V̇(ϕ)=0}. That is, the only solution of Eq. (1) for which ddtV(xt,yt,zt)=0 is the solution x≡0. This discussion guarantees that the null solution of Eq. (1) is asymptotically stable ([4, Lemma]).

This

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