On the stability and boundedness of a class of higher order delay differential equations
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:
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):
Section snippets
Preliminaries
Eq. (1) can be written as the following system: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
Let
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:
Let 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 thatwhere 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
That is, the only solution of Eq. (1) for which 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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