The Fréchet Derivative and Critical Points of Extremum

Radu Precup
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Radu Precup: Babeş-Bolyai University, Department of Applied Mathematics

Chapter Chapter 7 in Methods in Nonlinear Integral Equations, 2002, pp 97-110 from Springer

Abstract: Abstract In this chapter we present the notion of Fréchet derivative of a functional and we illustrate it by some examples. Then we build a functional E : L 2 (Ω; R n ) → R whose Fréchet derivative is the operator $$I - {H^ * }{N_f}H:{L^2}\left( {\Omega ;{R^n}} \right) \to {L^2}\left( {\Omega ;{R^n}} \right)$$ associated to (6.8). We prove the infinite-dimensional version of the classical Fermat’s theorem about the connection between extremum points and critical points, and we give sufficient conditions for that a functional admits minimizers. The abstract results are then applied to establish the existence of L p solutions for Hammerstein integral equations in R n .

Keywords: Banach Space; Bounded Linear Operator; Reflexive Banach Space; Unique Critical Point; Frechet Derivative (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-015-9986-3_8

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DOI: 10.1007/978-94-015-9986-3_8

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