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Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 258))

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Abstract

Many problems in mathematics, and its applications to theoretical physics, chemistry, and biology, lead to a problem of the form

$$ f(\lambda ,\,x)\, = \,0, $$
(13.1)

where f is an operator on R x B1 into B2, with B1 B2 Banach spaces. For example, (13.1) could represent a system of differential or integral equations, depending on a parameter λ. We are interested in the structure of the solution set; namely, the set

$$ {f^{ - 1}}(0)\, = \,\{ (\lambda ,\,x)\, \in \,R\, \times \,{B_1}\,:\,f(\lambda ,\,x)\, = \,0\} .$$
(13.2)

In particular, we seek conditions on f in order that we can determine when a solution (λ, ) of (13.1) lies on a “curve” of solutions (λ, x(λ)), at least locally; i.e., for |λλ| < ε. We may also inquire as to when (λ̄, ) lies on several solution curves, (λ, x1(λ)), (λ, x2(λ)),….

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© 1983 Springer-Verlag New York Inc.

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Smoller, J. (1983). Bifurcation Theory. In: Shock Waves and Reaction—Diffusion Equations. Grundlehren der mathematischen Wissenschaften, vol 258. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0152-3_13

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  • DOI: https://doi.org/10.1007/978-1-4684-0152-3_13

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4684-0154-7

  • Online ISBN: 978-1-4684-0152-3

  • eBook Packages: Springer Book Archive

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