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The structure of Rabinowitz' global bifurcating continua for problems with weak nonlinearities
Author(s) -
Bari Rehana,
Rynne Bryan P.
Publication year - 1997
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300012717
Subject(s) - mathematics , eigenvalues and eigenvectors , differentiable function , multiplicity (mathematics) , nonlinear system , bifurcation , mathematical analysis , simple (philosophy) , pure mathematics , philosophy , physics , epistemology , quantum mechanics
Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum ( i.e ., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1‐dimensional curves and that there are no non‐trivial solutions of the equation other than those lying on the bifurcating continua.