Generalized Jordan Chains and Two Bifurcation Theorems of Krasnoselskii

摘要

Given two Banach spaces X and Y over K = R or C and a parameterized family A (mu) an element of L(X, Y) with mu an element of K, partial and algebraic multiplicities of any value mu sub 0 an element of K such that A (mu sub 0) if Fredholm with index zero are defined by the means of generalized Jordan chains. These notions are developed in close connection with bifurcation problems and we show that partial and algebraic multiplicities are not affected by Lyapunov-Schmidt reduction. Properties of invariance under equivalence are also established. These general results are used to give a proof of Magnus' generalization of the classical bifurcation theorem by Krasnoselskii through a somewhat more natural approach than his. But the convincing evidence of the usefulness of the notions developed here has to be found in a new and wide extension of the Boehme-Marino-Rabinowitz theorem on bifurcation for gradient operators, the ancestor of which is also due to Krasnoselskii.