Volume 210, Nº 9 (2019)

A sufficient condition is proposed which ensures that a Dulac series that formally satisfies an algebraic ordinary differential equation (ODE) is convergent. Such formal solutions of algebraic ODEs are quite common: in particular, the Painleve III, V and VI equations have formal solutions given by Dulac series; they are convergent in view of the sufficient condition presented. Bibliography: 13 titles.

Sequences of complex numbers satisfying functional relations of bilinear type are investigated. The results obtained are used in describing all 1-periodic entire functions $f\colon \mathbb C\to\mathbb C$ such that the expansion ${f(x+y)f(x-y)}=\varphi_1(x)\psi_1(y)+…+\varphi_4(x)\psi_4(y)$ holds for some $\varphi_j,\psi_j\colon\mathbb C\to\mathbb C$.Bibliography: 38 titles.

A sufficient condition for the existence of an absolutely continuous solution for a sweeping process is given by the absolute continuity, in a definite sense, of the multivalued mapping which generates the process. This property is described in terms of the Hausdorff distance between values of the multivalued mapping. However, there exist multivalued mappings for which the Hausdorff distance between those values is infinite; for instance, mappings which take hyperplanes as values. For such mappings absolute continuity cannot be described in terms of the Hausdorff distance. In this paper we study conditions which provide local absolute continuity of a multivalued mapping. By using these conditions we prove an existence theorem for an absolutely continuous solution of a sweeping process. We apply the results obtained to the study of sweeping processes with nonconvex and with convexified perturbations. For such sweeping processes we prove an existence theorem for solutions and a relaxation theorem. Bibliography: 13 titles.