Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W 1, p ((0, ∞), ℝ N ) or C 0 1 ([0, ∞), ℝ N ), we study the global continuation in λ for solutions ( λ , u , ξ ) ∈ ℝ × X × ℝ k of the following system of ordinary differential equations:where ℝ N = X 1 ⊕ X 2 is a given decomposition, with associated projection P : ℝ N → X 1 . Under appropriate conditions upon the given functions F and φ , this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ℝ × X × ℝ k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)