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Bifurcation without Fréchet differentiability at the trivial solution
Author(s) -
Stuart C.A.
Publication year - 2015
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.3409
Subject(s) - mathematics , differentiable function , bifurcation , hadamard transform , fréchet derivative , derivative (finance) , nonlinear system , pure mathematics , line (geometry) , mathematical analysis , sequence (biology) , class (philosophy) , elliptic curve , geometry , banach space , computer science , physics , quantum mechanics , financial economics , economics , artificial intelligence , biology , genetics
Criteria for the bifurcation of small solutions of an equation F ( λ , u ) = 0 from a line { ( λ , 0 ) : λ ∈ R } of trivial solutions are usually based on properties of the D u F ( λ ,0) at the trivial solutions, where the partial derivative is taken in the sense of Fréchet. When this derivative only exists in some weaker sense, the situation charges considerably and much remains to be carried out to understand the conditions under which bifurcation takes place. This paper summarizes and extends one direction of research in this direction, where only Hadamard differentiability is required. In addition to presenting the abstract results, their application to the study of bound states of a class of nonlinear elliptic equations onR Nis also examined. Copyright © 2015 John Wiley & Sons, Ltd.