An analytic bifurcation principle for Fredholm operators

Smooth Equations of the form G[z]=0 are investigated in Banach spaces with the aim of continuing the basic solution G[0]=0 to a solution curve of G[z]=0 with the implicit function theorem. If the linearization is surjective, then the transversality condition of the implicit function theorem can be satisfied in a straightforward way, yielding a regular solution curve, whereas otherwise the equation G[z]=0 has to be extended appropriately for reaching a surjective linearization accessible to the implicit function theorem. This extension process, implying in the first step the standard bifurcation theorem of simple bifurcation points, is continued arbitrarily, yielding a sequence of bifurcation results presumably being applicable to bifurcation points with finite degeneracy.