Behavior of the solution of a nonlinear problem with a change in stability

If the function that determines the stable and unstable interval changes the stability conditions several times, then they are well studied. But under the condition the initial point and the point of change of stability do not coincide. Therefore, in this paper, we study solutions of nonlinear singularly perturbed differential equations with initial conditions. The peculiarity and novelty of this work lies in the fact that here the considered area changes the stability conditions several times. And also, the area has an infinitely long delay time. To prove the existence of solutions, the method of successive approximations is used. And also, for the convergence of solutions, we apply the majorant method. To prove the uniqueness of solutions, we use the contradiction method. The solution of the stated problem is considered in the real area.