  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.
