Implicit operator theorems under group symmetry conditions (Q5917909)

The authors study a general problem of stationary branching for a nonlinear equation in Banach spaces, as well as the Poincaré-Andronov-Hopf bifurcation problem for a differential equation unsolved with respect to the derivative. The main subject of the paper is the investigation of applicability of the Lyapunov-Schmidt method in the case of equations with a continuous symmetry group (equations equivariant under the action of a Lie group of a positive dimension). Little-known cases of a noncompact group and cases of a general (non-simple) Jordan structure of the linear part of the equation in a generating stationary point are at the center of considerations. The verification of the equivariant implicit operator theorem for stationary and non-stationary problems of bifurcation theory (without the assumption of compactness for a permissible continuous group) is given. The heritability of the symmetry by the branching equation in the root-subspace is demonstrated and the description of the general analytical structure of the branching equation is presented. Possible applications are outlined.