Convergence and Stability of the Split-Step θ-Milstein Method for Stochastic Delay Hopfield Neural Networks

and Applied Analysis 3 Now we give the definition of local and global errors. Definition 1. Let x(t) denote the exact solution of (1). The local approximate solution x̃(t k+1 ) starting from x(t k ) by SSTM (2) given by x̃ (t k+1 ) := x (t k ) + Λ (x (t k ) , ?̃? (t k ) , h, ΔW k ) , (9) where ?̃?(t k ) denotes the evaluation of (3) using the exact solution, yields the difference δ k+1 := x (t k+1 ) − x̃ (t k+1 ) . (10) Then the local error of SSTM is defined by ‖δ‖, whereas its global error means ‖ε‖ where ε := x(t k ) − X . Definition 2. If the global error satisfies E ( 󵄩󵄩󵄩󵄩 ε 󵄩󵄩󵄩󵄩 ) 2 ≤ Γh 2p ∀h ∈ (0, h 0 ) (11) with positive constants h 0 and Γ and a finite p, then we say that the order of mean-square convergence accuracy of the method is p. Here E is the expectation with respect to P. We then give the following lemmas that are useful in deriving the convergence results. Lemma 3 (see also [17]). Let the linear growth condition (7) hold, and the initial function ψ(t) is assumed to be F 0 -measurable and right continuous. And one puts E ψ := E(sup −τ≤t≤0 ‖ψ(t)‖ 2 ) < ∞. For any given positive T, there exist positive numbers Γ ψ and Γ 2 such that the solution of (1) satisfies E( sup −τ≤s≤T ‖x (s)‖2) ≤ Γψ, (12) where the constant Γ ψ is independent of step-size h but dependent on T. Moreover, for any 0 ≤ s < t ≤ T, t − s < 1, the estimation E(‖x (t) − x (s)‖)2 ≤ Γ2 (t − s) (13) holds. The Jensen inequality derives E( sup −τ≤s≤T ‖x (s)‖) ≤ √Γψ (14) from (12). Lemma 4. For s ∈ [t k , t k + h], one has E( 󵄩󵄩󵄩󵄩 z (s) − ?̃? (t k ) 󵄩󵄩󵄩󵄩 2 ) ≤ Γ τ h. (15) Here the constant Γ τ is independent of step-size h. Proof. If t k − τ j ≤ 0 and s − τ j ≤ 0, under Assumption 2 we have E[x j (s − τ j ) − ?̃? j (t k )] 2 = E[ψ j (s − τ j ) − ψ j (t k − τ j )] 2