Abstract
We establish a criterion for the global exponential stability of the zero solution of the scalar retarded functional differential equation whose linear part generates a monotone semiflow on the phase space with respect to the exponential ordering, and the nonlinearity has at most linear growth.
Highlights
Given r ∈ R+ = [0, ∞), let C = C([−r, 0], R) denote the Banach space of continuous functions mapping [−r,0] into R equipped with the supremum norm φ = sup φ(θ), φ ∈ C
We are concerned with the stability properties of the zero solution of the scalar retarded functional differential equation x (t) = L xt + g t, xt, (1.2)
In [9, Chapter 6, Theorem 1.1], it is shown that the above semiflow is monotone with respect to the ordering ≤μ, that is, φ ≤μ ψ implies yt 0, φ ≤μ yt(0, ψ), ∀t ∈ R+, (2.4)
Summary
Recall that the zero solution of (1.2) is (globally uniformly) exponentially stable if there exist constants M > 1 and λ < 0 such that if x is a noncontinuable solution of (1.2) with initial value (1.4) for some σ ∈ R+ and φ ∈ C, x is defined on [σ − r,∞) and satisfies the inequality x(t) ≤ M φ eλ(t−σ), t ≥ σ. The proof of Theorem 1.1 by Gyori [6] is based on the variation-of-constants formula and the fact that under condition (1.7), the fundamental solution of the linear part of (1.5), y (t) = −d y(t − τ),. Equation (2.1) generates a continuous global semiflow on C by t −→ yt(0, φ), t ∈ R+, φ ∈ C
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