Bifurcation from simple eigenvalues for several parameter families☆

In this paper, we investigate the structure of the nontrivial solution set for the following nonlinear operator equation where m is a positive integer, X is a Banach space, $L(\cdot ):X\to X$ is a (positively) homogeneous compact operator and $H:{\mathbb{R}}^m\times X\to X$ is compact with $H=o(\Vert u\Vert )$ near $u=0$ uniformly on bounded λ sets. We obtain some results involving (unilateral) global bifurcation. Two examples of p-Laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied.

The general solutions of the homogeneous matrix equation AXC^T − BXD^T = 0 and the system of the matrix equations AX + BY = 0, XC^T + YD^T = 0 are described in terms of Kronecker canonical forms, i.e., in terms of Kronecker invariants and Kronecker bases, for pairs of matrices (A, B) and (C, D). A canonical form for a pair of commuting matrices (E, F) such that E² = F² = EF = 0 is discussed. These results are applied to construct a canonical basis for the second root subspace of a two-parameter eigenvalue problem. The corresponding relations for canonical invariants are given.

Let X and Y be real Banach spaces, let f: $\text{R}$^p × X → Y be a C^l mapping, l ≥ 2, and let L(λ) = Df(λ, 0), for λ ∈ $\text{R}$^p. Suppose also that f(λ, 0) = 0, for all λ, and at some point λ⁰, dim N(L(λ⁰)) = n > 0, codim R(L(λ⁰)) = m > 0. We examine the structure of the zero set of f in a neighbourhood of (λ⁰, 0), and give conditions under which this zero set is diffeomorphic to the zero set of the lower order terms in the Taylor expansion of f near (λ⁰, 0). These results tell us about the structure of the set of non-trivial solutions bifurcating from zero when there are several parameters and the dimension of the null space of the linearization is greater than 1. Finally, we consider the genericity of the conditions used.

In this paper we give a result in multiparameter local bifurcation theory. This result is a generalization of the Hopf bifurcation theorem and of a previous result by J. K. Hale. Our result can be applied to Fredholm operators with arbitrary index.