Asymptotics of Solutions to a Differential Equation with Delay and Nonlinearity Having Simple Behaviour at Infinity

In this paper, we study nonlocal dynamics of a nonlinear delay differential equation. This equation with different types nonlinearities appears in medical, physical, biological, and ecological applications. The type nonlinearity paper is generalization two important for applications nonlinearities: piecewise constant compactly supported functions. We asymptotics solutions under the condition that multiplied by large parameter. construct all initial conditions from wide subset phase space find on parameters equations having periodic solutions.