Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation

Abstract: We consider a singularly perturbed boundary value problem for a second-order ordinarydifferential equation with nonlinear right-hand side containing functions of delayed argument. Weprove the existence of a solution with a transition layer that has a more sophisticated structurethan the ones studied before and construct a uniform asymptotic approximation to this solutionwith respect to a small parameter. Vasil’eva’s method is used when constructing the asymptoticapproximation, while the existence theorem is proved by combining the matching method and theasymptotic differential inequality method. Conditions for the existence of a solution withmonotone internal transition and boundary layers are stated. An example illustrating the class ofproblems studied here is given.