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

Volume 116 / 2016

Nguyen Thieu Huy, Ngo Quy Dang Annales Polonici Mathematici 116 (2016), 173-195 MSC: Primary 34C25, 34D20; Secondary 35B10, 35B35. DOI: 10.4064/ap3677-10-2015 Published online: 29 February 2016


We prove the existence and conditional stability of periodic solutions to semilinear evolution equations of the form $\dot{u}=A(t)u+g(t,u(t)),$ where the operator-valued function $t\mapsto A(t)$ is $1$-periodic, and the operator $g(t,x)$ is $1$-periodic with respect to $t$ for each fixed $x$ and satisfies the $\varphi$-Lipschitz condition $ \|g(t,x_1)-g(t,x_2)\|\leq \varphi(t)\|x_1-x_2\|$ for $\varphi(t)$ 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 $(A(t))_{t\geq 0}$ 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.


  • Nguyen Thieu HuySchool of Applied Mathematics
    and Informatics
    Hanoi University of Science and Technology
    Vien Toan ung dung va Tin hoc
    Dai hoc Bach khoa Hanoi
    1 Dai Co Viet
    Hanoi, Vietnam
  • Ngo Quy DangThai Binh College of Education
    and Training
    Cao Dang Su Pham Thai Binh
    Chu Van An
    Quang Trung
    Thai Binh, Vietnam

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