Indexed on: 01 Feb '96Published on: 01 Feb '96Published in: Aequationes mathematicae
Theorem S1. Assume that (i) V ∈ C[J × Sϱ, K], V(t, x) is locally Lipschitzian in x and for (t, θ(0)) ∈ J × Sϱ, D+ V(t, θ(0), θ) ≤k g(t, V(t, θ(0)), Vt), Vt = V(t + s, θ(s)) (ii) g ∈ C[R+ × Rn × K, Rn], g(t, u, ut) is quasimonotone in ut relative to K for each (t, u) ∈ R+ × Rn. (iii) f(t, 0, 0) = 0, g(t, 0, 0) = 0 and for some φ0 ∈ K0*, b(∥x(t0, θ0)(t)∥) ≤ (φ0, V(t, x(t0, θ0)(t))) ≤ a(t, ∥x(t0, θ0)(t)∥). Then the trivial solution x = 0 of (S1) is (a) stable (b) uniformly stable (c) asymptotically stable and (d) uniformly asymptotically stable if the trivial solution u = 0 of (S4) is respectively (e) φ0-stable (f) uniformly ø0-stable (g) asymptotically φ0-stable and (h) uniformly asymptotically ø0-stable.Theorem S2. Let cij(t, x) ≤ aij, dσij(t, x, s) ≤ dij, i, j = 1,..., n, C∥x(t0, θ0)(t)∥d ≤ (φ0, V(t, x(t0, θ0)(t))), V(t, u) is a cone-valued Lyapunov function for (S2). Then the steady state solution x = 0 of (S2) is exponentially stable if and only if there exist a nonnegative nonsingular n × n matrix B such that all the off-diagonal elements of B−1 AB> are nonnegative.