2-regularity and theorem on bifurcation (Q2777855)

This paper deals with application of general results about structure of level set of 2-regular non-linear mapping to the obtaining of bifurcation theorems. Here bifurcation theorems men theorems on local structure of solutions set of non-linear operator equation, depending on parameter, in the neighborhood of singular points. One of the presented results is the following. NEWLINENEWLINENEWLINELet \(X\) be a Banach space, let \(\Upsilon\) be a neighborhood of a point \(\sigma_{*}\) in \(R\), let \(V\) be a neighborhood of a point \(x_{*}\) in \(X\). Let a mapping \(F:\Upsilon\times V\to X\) be twice Fréchet differentiable at the point \((\sigma_{*},x_{*})\) and the following conditions hold true: NEWLINE\[NEWLINE\begin{aligned} \text{Im} \partial_2F(\sigma_{*},x_{*}) &=\{x\in X|\langle x^{*},x\rangle=0\;\forall x^{*}\in\text{Ker} (\partial_2F(\sigma_{*},x_{*}))^{*}\}; F(\sigma,x_{*})=0\;\forall\sigma\in \Upsilon;\\ \dim \partial_2F(\sigma_{*},x_{*}) &=\dim \text{Ker} (\partial_2F(\sigma_{*},x_{*}))^{*}=1;\;X=\text{Im} \partial_2F(\sigma_{*},x_{*})\oplus\text{Ker} \partial_2F(\sigma_{*},x_{*});\\ &\partial_{12}F(\sigma_{*},x_{*})\bar h\notin \text{Im} \partial_2F(\sigma_{*},x_{*})\;\text{for} \bar h\in\text{Ker} \partial_2F(\sigma_{*},x_{*})\setminus\{0\}. \end{aligned}NEWLINE\]NEWLINE Then there exists a unique number \(\chi\) such that \(T_{(\sigma_{*},x_{*})}M=\text{lin}\{(1,0)\}\cup\text{lin} \{(\chi,\bar h)\}\), and \(\chi\neq 0\) if and only if the mapping \(F(\sigma_{*},\cdot): V\to X\) is 2-regular at the point \(x_{*}\) on the element \(\bar h\). Here \(\partial_2F(\sigma_{*},x_{*})\) denotes partial derivative with respect to the second variable, \(\partial_{12}F(\sigma_{*},x_{*})\) denotes the second order partial derivative with respect to the first and the second variables at the point \((\sigma_{*},x_{*})\); \(T_{(\sigma_{*},x_{*})}M\) denotes the tangent cone to the set \(M\) at the point \((\sigma_{*},x_{*})\); \(M=F^{-1}(0)\subset \Upsilon\times V\) is a level set. Applications of the obtained results to the boundary value problems for ordinary differential equations are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00043].