Symmetry and potentiality in the general problem of the branching theory (Q1022195)

In real Banach functional spaces \(E_1\) and \(E_2\), the authors study the general problem of the branching theory: \[ F(x,\varepsilon)=0,\;F(x_0,0)=0,\;B_{x_0}=-F_x'(x_0,0), \] where \(B_{x_0}\) is a Fredholm operator, \(N(B_{x_0})=\text{span} \{\varphi_i\}^n_{i-1}\) is the subspace of zeros (the kernel) of the operator \(B_{x_0}\). It is supposed that the operator \(F(\cdot,\varepsilon)\) is continuously differentiable with respect to \(x\) and sufficiently smooth with respect to \(\varepsilon\) in a neighbourhood of the branch point \((x_0,0)\) and admits a group \(G\), and the Fredholm operator \(B_{x_0}\) has a symmetry only with respect to the stationary subgroup of \(x_0\). The authors prove a theorem on the inheritance of symmetry, \(BEs\) of potential type, a cosymmetric identity and a theorem on reduction of \(BE\). It is given an applications to symmetry breaking problems.