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A local bifurcation theorem for C1-Fredholm maps
| Content Provider | Semantic Scholar |
|---|---|
| Author | Fitzpatrick, Peter B. Pejsachowicz, Jacobo |
| Copyright Year | 1990 |
| Abstract | The Krasnosel'skii Bifurcation Theorem is generalized to C - Fredholm maps. Let X and Y be Banach spaces, F: R x X —> Y be C - Fredholm of index 1 and F(X, 0) = 0 . If / C R is a closed, bounded interval at whose endpoints |£ |j (X, 0) is invertible, and the parity of |£ (1, 0) on / is -1 , then / contains a bifurcation point of the equation F(X, x) = 0 . At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary. The celebrated Krasnosel'skii Local Bifurcation Theorem ((Kr)) asserts that if A' is a Banach space and C : X —► X is compact and differentiable at 0, with C(0) = 0, then each characteristic value of C'(0) of odd algebraic multiplicity is a bifurcation point of x - XC(x) = 0. Our purpose in this note is to present a generalization, based on the concept of parity introduced in (F - P1 ), of the Krasnosel'skii Theorem to one-parameter families of C -Fredholm maps. The proof is short, simple and uses only the classical change of degree argument. Moreover, at isolated potential bifurcation points, our sufficient condition for bifurcation is also necessary. A number of extensions of the Krasnosel'skii Theorem have been given in the context of generalized multiplicities (see (M, S, II, 12, L-M, C-H, Ki, E-L, E, Ra), among others). Certain technical aspects of the definitions of these multiplicities impose material restrictions on the class of maps for which bifurcation theorems can be proved. As is standard, given Banach spaces X and Y, by L(X, Y), Q>n(X, Y), GL(.Y, Y) and AA(X, Y) we denote the space of all bounded linear operators, the space of all linear Fredholm operators of Fredholm index « , the space of all linear isomorphisms, and the space of linear compact operators, respectively, from X to Y , endowed with the norm topology. |
| Starting Page | 995 |
| Ending Page | 1002 |
| Page Count | 8 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1990-1009988-5 |
| Volume Number | 109 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1990-109-04/S0002-9939-1990-1009988-5/S0002-9939-1990-1009988-5.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1990-1009988-5 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
