New Global Exponential Stability Criteria for Nonlinear Delay Differential Systems with Applications to BAM Neural Networks

We consider a nonlinear non-autonomous system with time-varying delays $$\dot{x_i}(t)=-a_i(t)x_i(h_i(t))+\sum _{j=1}^m{F}_{ij}(t,x_j(g_{ij}(t)))$$ which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed by the system’s parameters, is an $M$-matrix. Comparison with the tests obtained by K. Gopalsamy (2007) and B. Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper.