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adjoint analysis Arctan Arctan(v assumptions auxiliary equation auxiliary map Banach bifurcation equation bifurcation map boundary value problem bounded linear operator coefficients compact set consider the equation constant continuous map contraction mapping defined differential operator Dirichlet boundary conditions domain easily seen eigenfunctions eigenvalue eK(I elliptic example finite follows form Lu Fredholm indices function on G G+ G_ G hence homeomorphic homogeneous Dirichlet boundary homotopic homotopy Hr+m hypotheses independent variable inner product lemma lim h(s linear operator M(Arctan(v multiple resonance N(ve non-selfadjoint null space orthogonal projection P N(v partial derivatives positive and bounded preceeding Proof prove Range L,B regions of positivity RK(I satisfied Schauder self-adjoint share regions Simple Resonance Sobolev embedding theorem Sobolev space solvable subset of H sufficiently small takes the form theory topological degree topological theorem uniformly Lipschitzian unique solution zero Dirichlet