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On differentiability and bifurcation.
- Source :
- Advances in Mathematical Economics (9784431308980); 2006, p155-184, 30p
- Publication Year :
- 2006
-
Abstract
- For a function acting between Banach spaces, we recall the notions of Hadamard and w-Hadamard differentiability and their relation to the common notions of Gâteaux and Fréchet differentiability. We observe that even for a function F: H → H that is both Hadamard and w-Hadamard differentiable but not Fréchet differentiable at 0 on a real Hilbert space H, there may be bifurcation for the equation F(u) = λu at points λ which do not belong to the spectrum of F′(0). We establish some necessary conditions for λ to be a bifurcation point in such cases and we show how this result can be used in the context of partial differential equations such as $$ - \Delta u\left( x \right) + q\left( x \right)u\left( x \right) = \lambda \left( {e^{\left<INNOPIPE> x \right<INNOPIPE>} u\left( x \right)} \right) for u \in H^2 \left( {\mathbb{R}^N } \right) $$ where this situation occurs. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISBNs :
- 9784431308980
- Database :
- Supplemental Index
- Journal :
- Advances in Mathematical Economics (9784431308980)
- Publication Type :
- Book
- Accession number :
- 26350803
- Full Text :
- https://doi.org/10.1007/4-431-30899-76