A new theme in nonlinear analysis: continuation and bifurcation of the unit eigenvectors of a perturbed linear operator

We review some recent results concerning nonlinear eigenvalue problems of the form (∗) Au + epsilon B(u) = delta u, where A is a linear Fredholm operator of index zero (with nontrivial kernel KerA) acting in a real Banach space X, and B : X → X is a (possibly) nonlinear perturbation term. We seek solutions u of (∗) in the unit sphere S of X, and the emphasis is put on the existence - under appropriate conditions on B - of points u_0 ∈ S ∩ KerA (thus satisfying (∗) for epsilon = delta = 0) which either can be continued as solutions of (∗) for epsilon eq 0 or - more generally - are bifurcation points for solutions of that kind