Periodic solutions to evolution equations: existence, conditional stability and admissibility of function spaces

Nguyen Thieu Huy, and Ngo Quy Dang. "Periodic solutions to evolution equations: existence, conditional stability and admissibility of function spaces." Annales Polonici Mathematici 116.2 (2016): 173-195. <http://eudml.org/doc/280349>.

@article{NguyenThieuHuy2016,
abstract = {We prove the existence and conditional stability of periodic solutions to semilinear evolution equations of the form u̇ = A(t)u + g(t,u(t)), where the operator-valued function t ↦ A(t) is 1-periodic, and the operator g(t,x) is 1-periodic with respect to t for each fixed x and satisfies the φ-Lipschitz condition ||g(t,x₁) - g(t,x₂)|| ≤ φ(t)||x₁-x₂|| for φ(t) being a real and positive function which belongs to an admissible function space. We then apply the results to study the existence, uniqueness and conditional stability of periodic solutions to the above semilinear equation in the case that the family $(A(t))_\{t≥0\}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.},
author = {Nguyen Thieu Huy, Ngo Quy Dang},
journal = {Annales Polonici Mathematici},
keywords = {semilinear evolution equations; periodic solutions; admissibility of function spaces; conditional stability; local stable manifolds},
language = {eng},
number = {2},
pages = {173-195},
title = {Periodic solutions to evolution equations: existence, conditional stability and admissibility of function spaces},
url = {http://eudml.org/doc/280349},
volume = {116},
year = {2016},
}

TY - JOUR
AU - Nguyen Thieu Huy
AU - Ngo Quy Dang
TI - Periodic solutions to evolution equations: existence, conditional stability and admissibility of function spaces
JO - Annales Polonici Mathematici
PY - 2016
VL - 116
IS - 2
SP - 173
EP - 195
AB - We prove the existence and conditional stability of periodic solutions to semilinear evolution equations of the form u̇ = A(t)u + g(t,u(t)), where the operator-valued function t ↦ A(t) is 1-periodic, and the operator g(t,x) is 1-periodic with respect to t for each fixed x and satisfies the φ-Lipschitz condition ||g(t,x₁) - g(t,x₂)|| ≤ φ(t)||x₁-x₂|| for φ(t) being a real and positive function which belongs to an admissible function space. We then apply the results to study the existence, uniqueness and conditional stability of periodic solutions to the above semilinear equation in the case that the family $(A(t))_{t≥0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.
LA - eng
KW - semilinear evolution equations; periodic solutions; admissibility of function spaces; conditional stability; local stable manifolds
UR - http://eudml.org/doc/280349
ER -