Generalized Aizerman–Myshkis problem for abstract differential-delay equations Original Research Article

We consider a class of semilinear functional-differential equations in a Hilbert space. Conditions for the absolute stability are established. Moreover, it is shown that these conditions separate equations that satisfy the generalized Aizerman–Myshkis hypothesis. The suggested approach is based on a combined usage of the properties of operators on tensor products of Hilbert spaces and recent estimates for the norm of the resolvent. Our results are new even in the finite-dimensional case. We also discuss applications of the mentioned results to coupled systems of parabolic equations with delay and integro-differential equations with delay.