Nonlinear eigenvalue problems and spherical fibrations of Banach spaces

Eigenvalue problems of the form g′(v) = λh′(v) are considered, with the normalizations g(v) = r or h(v) = r, where g and h are real-valued C¹ functions on a real Banach space which are invariant under a periodic linear isometry. Theorems are proved on the existence of solutions λ(r), v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the linearized problem g″(0)v = λh″(0)v is discussed. The theorems are applied to elliptic problems for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces.