Implicit operator theorems under group symmetry conditions (Q5894419)

The present article is a short presentation of the author's work [``Implicit operator theorems in stationary and dynamical problems of branching theory under group symmetry conditions'', Izv. Irkutsk Univ., Math. 4, No. 1, 31--43 (2011)]. Its aim is to give proofs of \(G\)-invariant implicit operator theorems in stationary and dynamical problems of branching theory, based on general group symmetry inheritance theorems for nonlinear equations in Banach spaces by relevant Lyapounov and Schmidt branching equations in the root subspaces, taking into account the movement of branching point \(x_0\) along its orbit under the action of the group symmetry [\textit{B. V. Loginov}, ``General problem of branching theory under group symmetry conditions,'' Uzbek. Math. J. 1, 39--45 (1991); \textit{B. V. Loginov}, ``Branching equations in the root subspaces'', Nonlinear Anal., Theory Methods Appl. 32, No. 3, 439--445 (1998; Zbl 0949.47051)]. The formulation of the relevant theorem for Andronov-Hopf bifurcation contains an inaccuracy: the implicit operator theorem here should be understood with respect to the nonlinear operator \(\mathfrak{F}\) in complexified Banach space arising after application of A.\,Poincaré substitution.