Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

and Applied Analysis 3 (3) φ i (s) = φ i (s) on s ∈ [−m∗, 0], (4) eφ i (t) → 0 as t → ∞, where α = const and 0 < α < min i∈N{ai}, where t k and φ i (s) are defined as shown in Section 2. AlsoH is a complete metric space when it is equipped with a metric defined by d (q (t) , h (t)) = n ∑ i=1 sup t≥−m ∗ 󵄨󵄨󵄨󵄨qi (t) − hi (t) 󵄨󵄨󵄨󵄨 , (10) where q(t) = (q 1 (t), . . . , q n (t)) ∈ H and h(t) = (h 1 (t), . . . , h n (t)) ∈ H. Theorem 3. Assume that conditions (A1)–(A4) hold provided that (i) there exists a constant μ such that inf k=1,2,... {t k −t k−1 } ≥ μ, (ii) there exist constants p i such that p ik ≤ p i μ for i ∈ N and k = 1, 2, . . ., (iii) ∑n i=1 {(1/a i )max j∈N|bijlj| + (1/ai)maxj∈N|cijkj| + (ρ/a i )max j∈N|ωjdij|} +maxi∈N{pi(μ + (1/ai))} ≜ χ < 1, and then the trivial equilibrium x = 0 is globally exponentially stable. Proof. Multiplying both sides of (1) with ei gives, for t > 0 and t ̸ = t k , deaitx i (t) = eaitdx i (t) + a i x i (t) eaitdt = eait { { { n ∑ j=1 b ij f j (x j (t)) + n ∑ j=1 c ij g j (x j (t − τ ij (t)))