Dynamical Behaviors of Stochastic Hopfield Neural Networks with Both Time-Varying and Continuously Distributed Delays

and Applied Analysis 3 where ∗ denotes the corresponding symmetric terms, Σ 11 = Q 1 + τQ 2 − PC − CP − 2L 1 U 1 − 2M 1 U 4 , Σ 22 = − (1 − μ)Q 1 − 2L 1 U 2 , Σ 33 = Q 3 + τQ 4 − 2U 1 , Σ 44 = − (1 − μ)Q 3 − 2U 2 , Σ 55 = U 3 K (]) − 2U 4 , L 1 = diag (l− 1 l + 1 , . . . , l − n l + n ) , L 2 = diag (l− 1 + l + 1 , . . . , l − n + l + n ) , M 1 = diag (m− 1 m + 1 , . . . , m − n m + n ) , M 2 = diag (m− 1 + m + 1 , . . . , m − n + m + n ) . (7) Proof. The key of the proof is to prove that there exists a positive constantC ∗ , which is independent of the initial data, such that lim sup t→∞ E‖x(t)‖ 2 ≤ C ∗ . (8) If (8) holds, then it follows from Chebyshev’s inequality that for any ε > 0 and C = √C ∗ /ε, lim sup t→∞ P {‖x (t)‖ > C} ≤ lim sup t→∞ E‖x (t)‖ 2 C = ε, (9) which implies that (4) holds. Now, we begin to prove that (8) holds. From Σ < 0 and Lemma 2, one may obtain