We consider a nonlinear eigenvalue problem in the form F(x, λ) = Ax – T(λ)x – R(x, λ) = 0 with F:X × ℝ →Y where X, Y are Banach spaces. We assume that F(0, λ) = 0 for all λ ∈ ℝ and seek bifurcation points; that is, values λ0 ∈ ℝ for which there are solutions to F(x, λ) = 0 with x ≠ 0 in any neighbourhood of (0, λ0). Corresponding to these bifurcation points we obtain global properties of the maximal connected subset of solutions to F(x, λ) = 0 containing (0, λ0).
Generalised topological degree techniques are employed in the proofs of our results without requiring a transversality condition. The operators involved belong to the general class of A-proper mappings which include compact and k-ball contractive perturbations of the identity operator, accretive mappings, and many more.