A Combination Method for Local Bifurcation from Characteristic Values with Multiplicity

✍ Scribed by Nguyen Xuan Tan

• Publisher: John Wiley and Sons
• Year: 1991
• Tongue: English
• Weight: 665 KB
• Volume: 154
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19911540116

✦ Synopsis

A combination of the LIAPUNOV-SCHMIDT procedure, the implicit function theorems and the topological degree theory is used to investigate bifurcation points of equations of the form

where A is an open subset in a normed space and for every fixed 1 E A, 7; L(1, .) and M(1, .) are mappings from the closure d of a neighborhood D of the origin in a BANACH space X into another BANACH space Y with T(0) = L(1,O) = M ( I , 0) = 0. Let A be a characteristic value of the pair (7; L) such that T-L(1, .) is a FREDHOLM mapping with nullity p and index s, p > s 2 0. Under suitable hypotheses on 7; L and M, (A, 0) is a bifurcation point of the above equations. This generalizes the results of [4], [6], [8], [13] and [14] etc. An application of the obtained results to the axisymmetric buckling problem of a thin spherical shell will be given. *) This research was supported by the ALEXANDER VON HUMBOLDT Foundation of the Federal Republic of Germany at the Mathematical Institute of the University of Cologne.