Archivio Istituzionale della RicercaA local bifurcation theorem is proved under standard transversality assumptions by means of a reduction to finite dimensions. The Krasnosel'skii Bifurcation Theorem is generalized to C 1 - Fredholm maps. Let X and Y be Banach spaces, F: R x X —> Y be C 1 - Fredholm of index 1 and such that F(t, 0) = 0 . If I is a closed, bounded interval at whose endpoints DF(t, 0) is invertible, and the parity of the path is -1 , then I contains a bifurcation point of the equation F(t, x) = 0 . At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.
A local bifurcation theorem for C1-Fredholm maps / Fitzpatrick, P.; Pejsachowicz, Jacobo. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - STAMPA. - 109:4(1990), pp. 995-1002.
A local bifurcation theorem for C1-Fredholm maps
PEJSACHOWICZ, JACOBO
1990
Abstract
A local bifurcation theorem is proved under standard transversality assumptions by means of a reduction to finite dimensions. The Krasnosel'skii Bifurcation Theorem is generalized to C 1 - Fredholm maps. Let X and Y be Banach spaces, F: R x X —> Y be C 1 - Fredholm of index 1 and such that F(t, 0) = 0 . If I is a closed, bounded interval at whose endpoints DF(t, 0) is invertible, and the parity of the path is -1 , then I contains a bifurcation point of the equation F(t, x) = 0 . At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.
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https://hdl.handle.net/11583/2495707
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