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On minimal nonlinearities which permit bifurcation from the continuous spectrum
Author(s) -
Küpper T.,
Roach G. F.
Publication year - 1979
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670010414
Subject(s) - mathematics , spectrum (functional analysis) , bifurcation , nonlinear system , mathematical analysis , continuous spectrum , bifurcation theory , monotone polygon , operator (biology) , convolution (computer science) , geometry , biochemistry , chemistry , physics , repressor , quantum mechanics , machine learning , artificial neural network , transcription factor , computer science , gene
Bifurcation from the continuous spectrum of a linearized operator is of interest in many physical problems. For example it occurs in the nonlinear Klein‐Gordon equation and in nonlinear integrodifferential equations as the Choquard problem; it further appears in nonlinear integral equations of the convolution type. A general theory enclosing all these problems is not yet known. To understand the basic phenomena, we therefore consider monotone differential operators whose linearisations have a purely continuous spectrum. It is shown that in fact the lowest point of the continuous spectrum is a bifurcation point, if the nonlinearity grows sufficiently strong.