Existence and global exponential stability conditions of almost periodic solution in DCNNs with variable coefficients (Q1879106)

For the almost-periodic (ap) nonlinear system \[ x'(t)= A(t) x(t)+ B(t, x(t), x(t-\tau))+ f(t), \] \(0\leq t\), \(x(t)= \varphi(t)\) for \(-\tau\leq t\leq 0\), it is shown that to each ap \(f\) and \(\varphi\) there exists exactly one ap solution \(x\), and this solution is globally exponentially stable. Here, the \(A\), \(B\) are \(t\)-ap (real valued) \(n\times n\)-matrices with \(A\) diagonal and the coefficients \(b_{i,j}\) sufficiently small with respect to the \(a_{i,i}\) and the Lipschitz constants of \(B(t,u,v)\) with respect to \(u\), \(v\). This is done with a contraction \(T(\varphi)= x\) defined by \(x'= Ax+ B(t,\varphi(t), \varphi(t-\tau))+ f\), \(\varphi\) ap.