Some bifurcation results including Banach space valued parameters (Q1332677)

Summary: A contribution to the so-called limit point bifurcation is given. In the paper [J. Math. Anal. Appl. 75, 417-430 (1980; Zbl 0452.47075)] by \textit{D. W. Decker} and \textit{H. B. Keller} a bifurcation or branching phenomenon which they call multiple limit point bifurcation has been shown for equations \(T(\lambda,x)= 0\) with real parameter \(\lambda\). If one has a solution \((\lambda^ 0,x^ 0)\) of equation \(T(\lambda,x)= 0\), then one speaks from a limit point if the Fréchet derivative \(T_ x(\lambda^ 0,x^ 0)\) is singular and \(T_ \lambda(\lambda^ 0,x^ 0)\) is not in the range of \(T_ x(\lambda^ 0,x^ 0)\). Here, a method will be given, which generalizes the notion of limit point to that what is called \((\alpha,n)\)-limit point. This makes it possible to handle equations of the form \(T(u,x)= 0\) which may have \(u\) as a Banach space valued parameter. This equation with singular operator \(T_ x(u^ 0,x^ 0)\) may be ``embedded'' in a larger system with a linearization, which is non-singular and hence to which an implicit function theorem can apply. An estimation for the number of branching solutions of this new system is given.