Positive eigenvectors and simple nonlinear maps

For linear operators L , T and nonlinear maps P we describe classes of simple F = I ? between Banach Hilbert spaces, for which no point has more than two preimages. The encompass known examples (homeomorphisms, global folds) the weaker, geometric, hypotheses suggest new ones. operator may be Laplacian with various boundary conditions, as in original Ambrosetti-Prodi theorem, or associated quantum harmonic oscillator, hydrogen atom, a spectral fractional Laplacian, elliptic non-divergent form. include Nemitskii map ( u ) f but non-local, even non-variational. self-adjoint employ familiar results on nondegeneracy ground state. On use variation Krein-Rutman theorem.