Global Stability of Almost Periodic Solution of a Class of Neutral-Type BAM Neural Networks

and Applied Analysis 3 where i 1, 2, . . . , n; j 1, 2, . . . , m. xi t , yj t are the states of the ith neuron of X layer and the jth neuron of Y layer, respectively; aij t , pji t and bji t , qij t are the delayed strengths of connectivity and the neutral delayed strengths of connectivity, respectively; f1j , f2j , g1i, g2i are activation functions; Ii t , Jj t stands for the external inputs; τij t , τij t , δji t , and δji t correspond to the delays, they are nonnegative; ci t , dj t > 0 represent the rate with which the ith neuron of X layer and the jth neuron of Y layer will reset its potential to the resting state in isolation when disconnected from the networks. Throughout this paper, we assume the following. H1 ci t , dj t , aij t , pji t , bji t , qij t , τij t , τij t , δji t , δji t , Ii t , and Jj t are continuous almost periodic functions. Moreover, we let c i sup t∈R {ci t }, c− i inf t∈R{ci t } > 0, d j sup t∈R { dj t } , d− j inf t∈R { dj t } > 0, aij sup t∈R ∣aij t ∣∣} < ∞, bji sup t∈R ∣bji t ∣∣} < ∞, pji sup t∈R ∣pji t ∣∣} < ∞, qij sup t∈R ∣qij t ∣∣} < ∞, Ii sup t∈R {|Ii t |} < ∞, Jj sup t∈R ∣Jj t ∣∣} < ∞. 2.2 H2 f1j , f2j , g1i, and g2i are Lipschitz continuous with the Lipschitz constants F1j , F2j , G1i, G2i, and f1j 0 f2j 0 g1i 0 g2i 0 0. H3 Consider α max ⎧ ⎨ ⎩ 1≤i≤n max { 1 c− i , 1 c i c− i }⎛ ⎝ m ∑ j 1 aijF1j n ∑ j 1 bjiF2j ⎞ ⎠, max 1≤j≤m max { 1 d− j , 1 d j d− j }( n ∑ i 1 pjiG1i m ∑ i 1 qijG2i )⎬ ⎭ < 1. 2.3 The initial conditions of system 2.1 are of the following form: xi t φi t , t ∈ −δ, 0 , δ sup t∈R max i,j max { δji t , δji t } , yj t φj t , t ∈ −τ, 0 , τ sup t∈R max i,j max { τij t , τ ij t } , 2.4 where i 1, 2, . . . , n; j 1, 2, . . . , m; φi t , φj t are continuous almost periodic functions. Let X {ψ|ψ φ1, φ2, . . . , φn, φ1, φ2, . . . , φm T , where φi, φj : R → R are continuously differentiable almost periodic functions. For any ψ ∈ X, ψ t φ1 t , φ2 t , . . . , φn t , φ1 t , φ2 t , . . . , φm t T . We define ‖ψ t ‖1 max{‖ψ t ‖0, ‖ψ̇ t ‖0}, where ‖ψ t ‖0 max{max1≤i≤n{|φi t |},max1≤j≤m{|φi t |}}, and ψ̇ t is the derivative of ψ at t. Let ‖ψ‖ supt∈R‖ψ t ‖1, then X is a Banach space. The following definitions and lemmas will be used in this paper. 4 Abstract and Applied Analysis Definition 2.1 see 11 . Let x t : R → R be continuous in t. x t is said to be almost periodic on R, if for any ε > 0, the set T x, ε {w|x t w − x t < ε, for all t ∈ R} is relatively dense, that is, for all ε > 0, it is possible to find a real number l l ε > 0, for any interval length l ε , there exists a number τ τ ε in this interval such that |x t τ −x t | < ε, for all t ∈ R. Definition 2.2 see 11 . Let x ∈ C R,R and Q t be n × n continuous matrix defined on R. The following linear system: