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A continuation principle for Fredholm maps II: application to homoclinic solutions

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  • Christian Pötzsche
  • Robert Skiba

Abstract

When studying the behavior of autonomous ordinary differential equations under time‐dependent perturbations vanishing for t→±∞, their equilibria generically persist locally as homoclinic solutions. Using an abstract and flexible continuation theorem, we find even global branches of such homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach is based on degree‐theoretical arguments. In particular, Landesman–Lazer conditions are proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.

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  • Christian Pötzsche & Robert Skiba, 2020. "A continuation principle for Fredholm maps II: application to homoclinic solutions," Mathematische Nachrichten, Wiley Blackwell, vol. 293(6), pages 1174-1199, June.
    • Handle: RePEc:bla:mathna:v:293:y:2020:i:6:p:1174-1199
    DOI: 10.1002/mana.201800451
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    1. Christian Pötzsche & Robert Skiba, 2020. "A continuation principle for Fredholm maps I: theory and basics," Mathematische Nachrichten, Wiley Blackwell, vol. 293(5), pages 983-1003, May.
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