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 + eB(u) =cu, where A is a linear Fredholm operator of index zero (with nontrivial kernel KerA) acting in a real Banach space X, and B from X to 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 u0 in S \ KerA (thus satisfying (*) for e = c = 0) which either can be continued as solutions of (*) for e different from 0 or - more generally - are bifurcation points for solutions of that kind.