Internal Layer for a System of Singularly Perturbed Equations with Discontinuous Right-Hand Side

We study a system of two singularly perturbed first-order equations on an interval. The equations have discontinuous right-hand sides and equal powers of the small parameter multiplying the derivatives. We consider a new class of problems with discontinuous right-hand side, prove the existence of a solution with an internal transition layer, and construct its asymptotic approximation of arbitrary order. The asymptotic approximations are constructed by the Vasil’eva method, and the existence theorems are proved by the matching method.