Global bifurcation for Fredholm operators. (English) Zbl 1454.47092
In Sections 2–6, the author practically repeats his (not cited) paper [J. López-Gómez, in: A mathematical tribute to Professor José María Montesinos Amilibia on the occasion of his seventieth birthday. Madrid: Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología. 437–451 (2016; Zbl 1380.47045)]. Therefore I cite below my review concerning that paper in [MR3525948]:
“The author considers parametrised (by an open interval of \(\mathbb{R}\)) families of Fredholm maps and the respective families of their linearisations. As an important tool, he uses the algebraic multiplicity of so-called \(k\)-transversal eigenvalue (being, in fact, a parameter admitting a bifurcation). He compares this notion with some others: the transversality condition of M. G. Crandall and P. H. Rabinowitz [J. Funct. Anal. 8, 321–340 (1971; Zbl 0219.46015)], the conditions implying changes of orientations [P. Benevieri and M. Furi, Topol. Methods Nonlinear Anal. 16, No. 2, 279–306 (2000; Zbl 1007.47026)] and the parity of P. M. Fitzpatrick and J. Pejsachowicz [Trans. Am. Math. Soc. 326, No. 1, 281–305 (1991; Zbl 0754.47009)]. The main results are the respective versions of bifurcation theorems concerning the existence of bifurcation points and properties of branches of nontrivial solutions.”
Section 7 contains a generalization of so-called Unilateral Bifurcation Theorem in the same context. Moreover, in the Introduction, the author presents a detailed history of the considered problems as well as their relations to other ones.
“The author considers parametrised (by an open interval of \(\mathbb{R}\)) families of Fredholm maps and the respective families of their linearisations. As an important tool, he uses the algebraic multiplicity of so-called \(k\)-transversal eigenvalue (being, in fact, a parameter admitting a bifurcation). He compares this notion with some others: the transversality condition of M. G. Crandall and P. H. Rabinowitz [J. Funct. Anal. 8, 321–340 (1971; Zbl 0219.46015)], the conditions implying changes of orientations [P. Benevieri and M. Furi, Topol. Methods Nonlinear Anal. 16, No. 2, 279–306 (2000; Zbl 1007.47026)] and the parity of P. M. Fitzpatrick and J. Pejsachowicz [Trans. Am. Math. Soc. 326, No. 1, 281–305 (1991; Zbl 0754.47009)]. The main results are the respective versions of bifurcation theorems concerning the existence of bifurcation points and properties of branches of nontrivial solutions.”
Section 7 contains a generalization of so-called Unilateral Bifurcation Theorem in the same context. Moreover, in the Introduction, the author presents a detailed history of the considered problems as well as their relations to other ones.
Reviewer: Dorota Gabor (Toruń)
MSC:
| 47J15 | Abstract bifurcation theory involving nonlinear operators |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
