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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The bifurcation of solutions in Banach spaces

Author: William S. Hall
Journal: Trans. Amer. Math. Soc. 161 (1971), 207-218
MSC: Primary 47.80
MathSciNet review: 0282267
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Abstract: Let $ L:D \subset X \to D \subset {X^ \ast }$ be a densely defined linear map of a reflexive Banach space X to its conjugate $ {X^\ast}$. Define M and $ {M^\ast}$ to be the respective null spaces of L and its formal adjoint $ {L^\ast}$. Let $ f:X \to {X^\ast}$ be continuous. Under certain conditions on $ {L^\ast}$ and f there exist weak solutions to $ Lu = f(u)$ provided for each $ w \in X,v(w) \in M$ can be found such that $ f(v(w) + w)$ annihilates $ {M^ \ast }$. Neither M and $ {M^\ast}$ nor their annihilators need be the ranges of continuous linear projections. The results have applications to periodic solutions of partial differential equations.

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Keywords: Bifurcation of solution, periodic solutions, partial differential equations
Article copyright: © Copyright 1971 American Mathematical Society