Normalized Eigenvectors for Nonlinear Abstract and Elliptic Operators

We consider the eigenvalue problemAu−C(λ, u)=0for nonlinear operators A, C in a Banach space. The operator A is of type (S)+, or m-accretive, or maximal monotone, while the operator C is compact, or just continuous and bounded, whenever the resolvents of A are compact. The eigenvalues are not necessarily of multiplicative nature (as in the problem Au−λCu=0). Methods are introduced with applications to nonlinear eigenvalue problems involving the existence of eigenvectors with new normalized conditions. Most of the results are new even in the special case of Au−λCu=0.