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

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ (0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. 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 ? × X × ?^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)