Halanay Inequality on Time Scales with Unbounded Coefficients and Its Applications
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DOI: 10.1007/s13226-020-0447-z
HALANAY INEQUALITY ON TIME SCALES WITH UNBOUNDED COEFFICIENTS AND ITS APPLICATIONS Boqun Ou Department of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, Guangdong, People’s Republic of China e-mail: [email protected] (Received 2 September 2018; accepted 21 May 2019) In the present paper, we obtain a Halanay inequality on time scales with unbounded coefficient for a dynamic problem, which extends a result of Wen et al. (J. Math. Anal. Appl., 347 (2008), 169178.) to the inequality of integral type on time scales. Moreover, we list two dynamic problems to which the Halanay inequality obtained above can be applied and prove the zero solution of two delay dynamic problems are asymptotically stable. Moreover, it is worth mentioning that the Halanay inequality obtained in the present paper is more precise than the results in [3, 14, 17]. Key words : Halanay inequality of integral type; time scales; unbounded coefficients; asymptotically stable. 2010 Mathematics Subject Classification : 34A34, 34C11, 34D05.
1. I NTRODUCTION In 1966, Halanay [7] proved that: Lemma 1.1 — [7]. Let x(t) be any non-negative solutions of x0 (t) ≤ −αx(t) + β
sup x(x), t ≥ t0
s∈[t−τ,t]
and α > β > 0, then there exist two positive constants γ > 0 and K > 0 such that x(t) ≤ Ke−γ(t−t0 ) for t ≥ t0 . In 1996, Baker and Tang [3] obtained that:
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Lemma 1.2 — [3]. Let x(t) be any non-negative solution of x0 (t) ≤ −a(t)x(t) + b(t) sup x(s), t > t0 ,
s∈[q(t),t]
x(t) = |φ(t)| for t ≤ t0 ,
where φ(t) : (−∞, t0 ] → R is a bounded and continuous function, a(t) ≥ 0, b(t) ≥ 0 for any t ≥ t0 , q(t) ≤ t, and q(t) → ∞ as t → ∞. If there exists σ > 0 such that −a(t) + b(t) ≤ −σ < 0 for t ≤ t0 , then
(
x(t) ≤ kφk(−∞,t0 ] , t ≥ t0 , x(t) → 0, as t → ∞,
where kφk(−∞,t0 ] = sup |φ(t)|. t≤t0
Wen et al. [17] proved the following Halanay inequality with unbounded coefficients. Lemma 1.3 — [17]. Let x(t) be any non-negative solutions of x0 (t) ≤ γ(t) + α(t)x(t) + β(t)
sup ξ∈[t−τ (t),t]
x(ξ), t ≥ t0 ,
x(t) = |ψ(t)| for t ≤ t0 ,
where ψ(t) : (−∞, t0 ] → R is a bounded and continuous function, continuous functions γ(t) ≥ 0, α(t) ≤ 0, β(t) ≥ 0 for t ∈ [t0 , ∞), τ (t) ≥ 0 and t − τ (t) → ∞ as t → ∞, and if there exists σ > 0 such that α(t) + β(t) ≤ −σ < 0 for t ≤ t0 , then, we have (i) u(t) ≤
γ∗ + G, t ≥ t0 . σ
If we assume further that there exists 0 < δ < 1 such that δα(t) + β(t) < 0 for t ≥ t0 , then, we have
HALANAY INEQUALITY ON TIME SCALES
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(ii) for any given ² > 0, there exists tˆ = G > t0 such that u(t) ≤
γ∗ + ², t ≥ tˆ, σ
where G=
|ψ(t)| and γ ∗ =
sup t∈(−∞,t0 ]
sup
γ(t).
t∈[t0 ,+∞)
Recently, Jia et al. [14] proved the following Halanay inequality of integral type on times scales. Lemma 1.4 — [14]. Let x(t) be any non-negative solution of R x∆ (t) ≤ −a(t)x(t) + b(t) sup x(s) + c(t) + d(t) 0∞ K(t, s)x(t − s)∆s, t ≥ t0 ,
s∈[t−τ (t),t]
x(t) = |ψ(t)| for t ≤ t0 ,
where ψ(t) : (−∞, t0 ]T → R is a bounded and rd-continuous function with M = sup |φ(t)|,
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