Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

scholarly journals Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Kairi Kasemets ◽  
Jaan Janno
Keyword(s):  
Inverse Problems ◽  
Sobolev Spaces ◽  
Integrodifferential Equation ◽  
Weak Form ◽  
Weak Formulation ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Cost Functionals ◽  
Derivatives Of

We deduce formulas for the Fréchet derivatives of cost functionals of several inverse problems for a parabolic integrodifferential equation in a weak formulation. The method consists in the application of an integrated convolutional form of the weak problem and all computations are implemented in regular Sobolev spaces.

A generalization of Krasnosel’skii fixed point theorem for sums of compact and contractible maps with application

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Bogdan Przeradzki
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Point Theorem ◽  
Generalized Contraction ◽  
Nonlinear Integral Equations ◽  
Bounded Set ◽  
Compact Map

AbstractThe existence of a fixed point for the sum of a generalized contraction and a compact map on a closed convex bounded set is proved. The result is applied to a kind of nonlinear integral equations.

scholarly journals AN ALTERNATIVE APPROACH TO FRÉCHET DERIVATIVES

2020 ◽  
pp. 1-19
Author(s):  
SHANE ARORA ◽  
HAZEL BROWNE ◽  
DANIEL DANERS
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Banach Spaces ◽  
Short Proof ◽  
Banach Fixed Point Theorem ◽  
Alternative Approach ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
The Schwarz Lemma

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

scholarly journals Existence of Tripled Fixed Points for a Class of Condensing Operators in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Vatan Karakaya ◽  
Nour El Houda Bouzara ◽  
Kadri Doğan ◽  
Yunus Atalan
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Integral Equations ◽  
Existence Of Solutions ◽  
General System ◽  
Nonlinear Integral Equations ◽  
Condensing Operators

We give some results concerning the existence of tripled fixed points for a class of condensing operators in Banach spaces. Further, as an application, we study the existence of solutions for a general system of nonlinear integral equations.

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