非单特征值的广义分歧定理

讨论了抽象算子方程F(λ,u)=0的局部分歧问题,其中F:R×X→Y是一个C~2微分映射,λ是参数,X,Y为Banach空间.利用Lyapunov-Schmidt约化过程及偏导算子F_u(λ~*,O)的有界线性广义逆,在dim N(F_u(λ~*,0))≥codim R(F_u(λ~*,O))=1的条件下,证明了一个广义跨越式分歧定理.当参数空间的维数等于值域余维数时,应用同样的方法又得到了多参数方程的抽象分歧定理.

The Generalized Bifurcation Theorem from Multiple Eigenvalues

The authors discuss the local bifurcation problem of the abstract operator equation $F(\lambda,u)=0$, where $F:\mathbb{R} \times X \rightarrow Y$ is a $\mathcal{C}^{2}$ differential mapping, $\lambda$ is a parameter and $X,Y$ are Banach spaces. By the Lyapunov-Schmidt reduction and the bounded linear generalized inverse of $F_{u}(\lambda^{*},0)$, a generalized transcritical bifurcation theorem is obtained under the assumption that $\dim N(F_{u}(\lambda^{*},0)) \geq {\rm codim}\, R(F_{u}(\lambda^{*},0))=1$. When the dimension of parameter space is equal to the codimension of range space, the authors get an abstract bifurcation theorem of the equation with multiparameter by applying the same method.