Stability of solutions to systems of nonlinear differential equations with infinite distributed delay

The paper considers a certain class of systems of nonlinear differential equations with infinite distributed delay. It is assumed that the coefficients in the linear terms are $T$-periodic, the nonlinear term is a continuous Lipschitz-like vector function, whose the degree of smallness greater than one. Such systems of differential equations arise when modeling various processes occurring in biology, chemistry, physics, economics. The Lyapunov — Krasovskii functional is proposed, on the basis of which sufficient conditions for exponential stability of the zero solution of the class of systems under consideration are established, estimates for the set of attraction of the zero solution and estimates for the norm of the solution to the initial value problem characterizing exponential decrease at infinity are indicated. All parameters involved in the estimates are specified explicitly. The conditions of exponential stability of the zero solution established in the paper are expressed in terms of integral inequality. The conditions of global exponential stability of the zero solution are also obtained.