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

Nguyen Thieu Huy¹ ; Ngo Quy Dang²

¹School 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
²Thai Binh College of Education and Training Cao Dang Su Pham Thai Binh Chu Van An Quang Trung Thai Binh, Vietnam

Annales Polonici Mathematici, Tome 116 (2016) no. 2, pp. 173-195

Résumé

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.

Détail

DOI : 10.4064/ap3677-10-2015

Keywords: prove existence conditional stability periodic solutions semilinear evolution equations form dot g where operator valued function mapsto periodic operator periodic respect each fixed satisfies varphi lipschitz condition g leq varphi x varphi being real positive function which belongs admissible function space apply results study existence uniqueness conditional stability periodic solutions above semilinear equation the family geq generates evolution family having exponential dichotomy prove existence local stable manifold near periodic solution