On one approach to the study of the asymptotic behaviour of the Riccati equation with complex-valued coefficients

Summary The asymptotic behaviour of the solutions of the Riccati differential equation with complex-valued coefficients, especially the existence of bounded solutions and solutions satisfying the conditions of a form , is studied. The investigation is based on the reduction of the Riccati equation to the equation (E) z= =zg(t,z)+h(t, z).Here g, h are complex-valued functions, t and z being a real and a complex variable, respectively. The equation (E) is studied by means of various suitable techniques, such as Lyapunov functions, Wazewzki topological principle and by the use of results of M.Cecchi, M.Furi and M.Marini 3] on the solutions of a certain boundary value problem. Even though the obtained results are only of local character, they complete the previous results on the Riccati equation with complex-valued coefficients, which was intensively investigated by M.Ráb, and are effective in some cases not covered by known results.The paper was written during the author's stay at the Mathematical Institute U. Dini in Firenze and the author is very grateful for the kind hospitality.