Justification of the existence of a group of asymptotics of the general fifth Painlevé transcendent

There are several existing ways in developing the asymptotics of the Painlevé transcendents. But it is always a hard task to justify the existence of these asymptotics. In this note, we apply the successive approximation to the general fifth Painlevé equation and rigorously prove the existence of a group of asymptotics of its solutions. 1. Introduction. The mathematical and physical significance of the six Painlevé tran-scendents has been well established. Their mathematical importance originates from the work by Painlevé [7, 8] and Garnier [2]. Their physical significance follows their applicability to a wide range of important physical problems, such as nonlinear waves in quantum field theory and statistical mechanics [6]. There have been many results on the asymptotics of the Painlevé transcendents. In 1980, Hastings and McLeod [4] developed a method and applied it to the second Painlevé equation. In their paper, they rigorously proved the existence of a group of asymptotics to the second Painlevé equation and obtained a connection formula of the asymptotics. In 1997, Abdullayev [1] further developed the ideas used by Hastings and McLeod, " linearized " a special case of the fourth Painlevé equation and proved the existence of a group of its asymptotics. In [5], we studied the general fifth Painlevé equation (PV)