Compact perturbations of linear differential equations in locally convex spaces

S. A. Shkarin. "Compact perturbations of linear differential equations in locally convex spaces." Studia Mathematica 172.3 (2006): 203-227. <http://eudml.org/doc/284773>.

@article{S2006,
abstract = { Herzog and Lemmert have proven that if E is a Fréchet space and T: E → E is a continuous linear operator, then solvability (in [0,1]) of the Cauchy problem ẋ = Tx, x(0) = x₀ for any x₀ ∈ E implies solvability of the problem ẋ(t) = Tx(t) + f(t,x(t)), x(0) = x₀ for any x₀ ∈ E and any continuous map f: [0,1] × E → E with relatively compact image. We prove the same theorem for a large class of locally convex spaces including: • DFS-spaces, i.e., strong duals of Fréchet-Schwartz spaces, in particular the spaces of Schwartz distributions 𝓢'(ℝⁿ), the spaces of distributions with compact support 𝓔'(Ω) and the spaces of germs of holomorphic functions H(K) over an arbitrary compact set K ⊂ ℂⁿ; • complete LFS-spaces, i.e., complete inductive limits of sequences of Fréchet-Schwartz spaces, in particular the spaces 𝓓(Ω) of test functions; • PLS-spaces, i.e., projective limits of sequences of DFS-spaces, in particular, the spaces 𝓓'(Ω) of distibutions and 𝓐(Ω) of real-analytic functions. Here Ω is an arbitrary open domain in ℝⁿ. We construct an example showing that the analogous statement for (smoothly) time-dependent linear operators is invalid already for Fréchet spaces. },
author = {S. A. Shkarin},
journal = {Studia Mathematica},
keywords = {ordinary differential equation in locally convex spaces; existence theorem; uniqueness theorem; linear differential equation; lifting},
language = {eng},
number = {3},
pages = {203-227},
title = {Compact perturbations of linear differential equations in locally convex spaces},
url = {http://eudml.org/doc/284773},
volume = {172},
year = {2006},
}

TY - JOUR
AU - S. A. Shkarin
TI - Compact perturbations of linear differential equations in locally convex spaces
JO - Studia Mathematica
PY - 2006
VL - 172
IS - 3
SP - 203
EP - 227
AB - Herzog and Lemmert have proven that if E is a Fréchet space and T: E → E is a continuous linear operator, then solvability (in [0,1]) of the Cauchy problem ẋ = Tx, x(0) = x₀ for any x₀ ∈ E implies solvability of the problem ẋ(t) = Tx(t) + f(t,x(t)), x(0) = x₀ for any x₀ ∈ E and any continuous map f: [0,1] × E → E with relatively compact image. We prove the same theorem for a large class of locally convex spaces including: • DFS-spaces, i.e., strong duals of Fréchet-Schwartz spaces, in particular the spaces of Schwartz distributions 𝓢'(ℝⁿ), the spaces of distributions with compact support 𝓔'(Ω) and the spaces of germs of holomorphic functions H(K) over an arbitrary compact set K ⊂ ℂⁿ; • complete LFS-spaces, i.e., complete inductive limits of sequences of Fréchet-Schwartz spaces, in particular the spaces 𝓓(Ω) of test functions; • PLS-spaces, i.e., projective limits of sequences of DFS-spaces, in particular, the spaces 𝓓'(Ω) of distibutions and 𝓐(Ω) of real-analytic functions. Here Ω is an arbitrary open domain in ℝⁿ. We construct an example showing that the analogous statement for (smoothly) time-dependent linear operators is invalid already for Fréchet spaces.
LA - eng
KW - ordinary differential equation in locally convex spaces; existence theorem; uniqueness theorem; linear differential equation; lifting
UR - http://eudml.org/doc/284773
ER -