On the asymptotics of the real solutions to the general sixth Painlevé equation.
We apply the averaging method to ordinary differential inclusions with maxima perturbed by a small parameter and illustrate the method by some examples.
We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.
Autonomous linear neutral delay and, especially, (non-neutral) delay difference equations with continuous variable are considered, and some new results on the behavior of the solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation.
The ideas of the present paper have originated from the observation that all solutions of the linear homogeneous differential equation (DE) satisfy the non-trivial linear homogeneous boundary conditions (BCs) , . Such a BC is referred to as a natural BC (NBC) with respect to the given DE, considered on the interval . This observation suggests the following queries : (i) Will each second-order linear homogeneous DE possess a natural BC ? (ii) How many linearly independent natural BCs can a...