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Periodicity and stability of solutions to neutral differential equations with $\varphi $-Lipschitz and infinite delays

Volume 125 / 2020

Thieu Huy Nguyen, Thi Loan Nguyen, Thi Ngoc Ha Vu Annales Polonici Mathematici 125 (2020), 155-181 MSC: Primary 34K13, 34K19; Secondary 35B10. DOI: 10.4064/ap190917-18-5 Published online: 19 October 2020

Abstract

We prove the existence and uniqueness of a periodic solution to neutral function partial differential equations with infinite delay of the form $$ \frac {\partial Fu_t}{\partial t}= A(t)Fu_t +g(t,u_t)\quad \mbox {for } t \gt 0,\quad u(t)=\phi (t)\quad \hbox {for } t\le 0, $$ where the family $(A(t))_{t\ge 0}$ of linear partial differential operators is such that the mapping $t\mapsto A(t)$ is 1-periodic, and the nonlinear delay operator $g(t,u)$ is 1-periodic with respect to $t$ and $\varphi $-Lipschitz with respect to $u$ (i.e. $\|g(t,u)-g(t,v)\|\le \varphi (t)\|u-v\|$ for some function $\varphi $ in an admissible function space). Then we apply the results to study the existence, uniqueness and conditional stability of a periodic solution to the above equation when $(A(t))_{t\ge 0}$ generates an evolution family which has an exponential dichotomy. Moreover, we prove the existence of a local stable manifold around such a periodic solution.

Authors

  • Thieu Huy NguyenSchool 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
    e-mail
  • Thi Loan NguyenDepartment of Basic Science
    Hung Yen University
    of Technology and Education
    Khoai Chau, Hung Yen, Vietnam
    e-mail
  • Thi Ngoc Ha VuSchool 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
    e-mail

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