Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms

and Applied Analysis 3 In the light of (H1)–(H4), it is easy to see that problems (1)-(2) admit an equilibrium point u = 0. Definition 1. The equilibrium point u = 0 of problems (1)(2) is said to be globally exponentially stable if there exist constants κ > 0 andM ≥ 1 such that 󵄩󵄩󵄩󵄩u (t, x; t0, φ) 󵄩󵄩󵄩󵄩Ω ≤ M 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩Ω e −κ(t−t 0 ) , t ≥ t 0 , (10) where ‖φ‖ 2 Ω = sup t 0 −τ≤s≤t 0 ∑ n i=1 ∫ Ω φ i 2 (s, x)dx . Lemma 2 (see [30] (Gronwall-Bellman-type impulsive integral inequality)). Assume the following. (A1) The sequence {t k } satisfies 0 ≤ t 0 < t 1 < t 2 < ⋅ ⋅ ⋅, with lim k→∞ t k = ∞. (A2) q ∈ PC1[R + , R] and q(t) is left-continuous at t k , k = 1, 2, . . .. (A3) p ∈ C[R + , R + ] and for k = 1, 2, . . .,