Stability loss protraction for singularly perturbed equations with continuous right-hand sides

Since the middle of the last century, mathematicians' attention was attracted by differential equations with small parameters at highest derivatives. Such equations are called singularly perturbed. The central problem in the theory of singularly perturbed equations is solving the problem about the asymptotic proximity of solutions of singularly perturbed equations and degenerate equation. First, this problem was solved by А. N. Tikhonov. In works by А. N. Tikhonov, one of the conditions is the stability of the rest point. It is proved that under these conditions (restrictions), there is a limit transition. The limit transition is not uniform over the whole interval. In a vicinity of the point, there appears so-called boundary layer. In works by A. B. Vasilyeva, solutions' asymptotic expansions by the small parameter of ordinary differential equations were constructed. М. I. Imanaliev developed a method for expanding solutions of singularly perturbed and integro-differential equations. The first work in which the stability condition is violated and nevertheless there exists the transition limit is M. A. Shishkova’s work. The phenomenon that is described in the works of M. A. Shishkova was called the loss stability protraction. In those works, the posed problems were solved in the space of analytic functions, i.e., right-hand sides of the equations were supposed to be analytic in a certain region of the complex plane.
In this paper, we consider singularly perturbed equations with continuous right-hand sides such that the stability condition for the rest point of the adjoint equation is not satisfied on the considered interval. We prove the existence and uniqueness of the solution. The existence of the solution is proved using the method of successive approximations. An example is presented.