Superlinear Hammerstein Operators and the Nonoccurrence of Secondary Bifurcation

Abstract

In the secondary bifurcation problem for a branch $x_\lambda ^{(j)} $ of eigensolutions of a superlinear Hammerstein operator with positive definite kernel, it is already known that if the primary bifurcation point $\lambda _j^B $ is an eigenvalue of unit multiplicity, there exists a nonvoid region of no secondary bifurcation $R_{nb}^{(j)} $. As long as $x_\lambda ^{(j)} \in R_{nb,}^{(j)} \lambda > \lambda _j^B $ , secondary bifurcation does not occur. $R_{nb}^{(j)} $ contains at least a spherical neighborhood of the origin. Conditions are herein given such that $R_{nb}^{(j)} $ also contain a corridor leading out of this central sphere. This affords an exit for bifurcation-free global extension of $x_\lambda ^{(j)} $. Then the question of having $x_\lambda ^{(j)} $ actually stay in this corridor is considered, with conditions given that are happily compatible with conditions for existence of the corridor. The requirements are met by an augmentation of a given kernel, which involves a suitable degenerate k...