Solution branches of nonlinear eigenvalue problems on restricted domains

We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through the bifurcation point either is unbounded, admits an accumulation point on the boundary, or contains an even number of odd multiplicity points. In the simple multiplicity case, we show that branches of solutions in the directions of corresponding eigenvectors satisfy similar conditions on our domains.