On refining the asymptotics of a singular perturbed problem solution as a result of separation of the roots of a degenerate equation

We carry the constructing and founding of solution asymptotics for initial singular perturbed problem when the roots of degenerate equation are cross. The problem is characterized by a presence of inner transition layer near which the solution has a change of own behavior, it means, passes from one branch of complex stable root to another one. It turns out, the roots of degenerate equation could be separated by means of defined their representation. The same representation is true for research function also. It lets to reduce the problem to new one which estimate of solution is easily found. In the first place, the order of the terms of right part of equation is set up into and out of small neighborhood of bifurcation point; secondly, the solution asymptotics of reference problem is improved out of this small neighborhood. The last one is made by means of a certain boundary function, which aim of introduce is to go of asymptotics to regime given left and right of bifurcation point. The proof of existence and uniqueness theorem having the point out asymptotics we carry by differential inequalities method.