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Global continuation for first order systems over the half‐line involving parameters
Author(s) -
Evéquoz Gilles
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810789
Subject(s) - mathematics , continuation , bounded function , projection (relational algebra) , fredholm operator , mathematical analysis , operator (biology) , nonlinear system , zero (linguistics) , order (exchange) , a priori and a posteriori , fredholm integral equation , differential operator , ordinary differential equation , line (geometry) , pure mathematics , differential equation , compact operator , integral equation , algorithm , geometry , philosophy , repressor , linguistics , chemistry , computer science , biochemistry , epistemology , quantum mechanics , transcription factor , programming language , physics , finance , economics , extension (predicate logic) , gene
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)