Singular Lyapunov operator equations: applications to $$C^*-$$algebras, Fréchet derivatives and abstract Cauchy problems

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Bogdan D. Djordjević
Keyword(s):  
Cauchy Problems ◽  
Operator Equations ◽  
C Algebras ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Lyapunov Operator ◽  
Abstract Cauchy Problems

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.

scholarly journals The Super-Diffusive Singular Perturbation Problem

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 403
Author(s):  
Edgardo Alvarez ◽  
Carlos Lizama
Keyword(s):  
Singular Perturbation ◽  
Unique Solution ◽  
Analytic Semigroup ◽  
Singular Perturbation Problem ◽  
Singularly Perturbed ◽  
Regularity Conditions ◽  
Cauchy Problems ◽  
Perturbation Problem ◽  
Bounded Analytic Semigroup ◽  
Abstract Cauchy Problems

In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] , 1 < α < 2 , ϵ > 0 , for the parabolic equation ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem ( P ϵ ) has a unique solution u ϵ ( t ) for each small ϵ > 0 . Moreover u ϵ ( t ) converges to u 0 ( t ) as ϵ → 0 + , the unique solution of equation ( P ) .

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