Optimal multiplicity in local bifurcation theory. II: General case
The author considers an equation of the form \[ (*)\quad L(\lambda)u+F(\lambda,u)=0, \] where \(\lambda\) is a small parameter, L(\(\lambda)\) is an uniparametric family of linear bounded operators between two Banach spaces U and V, \(u\in U\). Let \[ L(\lambda)=L_ 0+\lambda L_ 1+...+\lambda^ kL_ k+0(\lambda^{k+1}) \] with \[ (+)\quad R(L_ 0)\oplus L_ 1N(L_ 0)\oplus L_ 2[N(L_ 0)\cap N(L_ 1)]\oplus...\oplus L_ k[N(L_ 0)\cap...\cap N(L_{k-1})]=V, \] then the multiplicity \(\chi\) at \(\lambda =0\) is defined by \[ (**)\quad \chi:=\sum^{k}_{i=1}i\cdot \dim L_ i[N(L_ 0)\cap...\cap N(L_{i- 1})]. \] The author and \textit{J. López-Gómez} proved in an earlier paper [part I, J. Differential Equations 71, 72-92 (1988; Zbl 0648.34027)]: The following conditions are equivalent: (a) \(\chi\) is odd; (b) for all F(\(\lambda\),u) of second order in u, \((\lambda,u)=(0,0)\) is a bifurcation point of (*). In this paper the concept of multiplicity is generalized eliminating the hypothesis \((+)\). Furthermore it is shown that the previous types of multiplicity considered in the literature all coincide and are equivalent to the one defined by (**).
scientific article; zbMATH DE number 4094228