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 delaysxi̇(t)=-ai(t)xi(hi(t))+∑j=1mFijt,xj(gij(t)),i=1,…,mwhich 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 using the system’s parameters, is an M-matrix. Comparison with the tests obtained by Gopalsamy (2007) and Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper.