Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 187–207 | Cite as

Unstable Manifolds for Partial Neutral Differential Equations and Admissibility of Function Spaces

  • Thieu Huy NguyenEmail author
  • Van Bang Pham


We prove the existence and attraction property of an unstable manifold for solutions to the partial neutral functional differential equation of the form

\(\left\{\begin{array}{ll} \frac{\partial}{\partial t}Fu_{t}= B(t)Fu_{t} +\varPhi(t,u_{t}),\quad t\ge s;~ t,s\in \mathbb{R},\\ u_{s}=\phi\in \mathcal{C}:=C([-r, 0], X) \end{array}\right.\)

under the conditions that the family of linear operators \((B(t))_{t\in \mathbb {R}}\) defined on a Banach space X generates the evolution family (U(t, s)) ts having an exponential dichotomy on the whole line \(\mathbb {R}\), the difference operator \(F:\mathcal {C}\to X\) is bounded and linear, and the nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition, i.e., \(\| \Phi (t,\phi ) -\Phi (t,\psi )\| \le \phi (t)\|\phi -\psi \|_{\mathcal {C}}\) for \(\phi ,~ \psi \in \mathcal {C}\), where ϕ(⋅) belongs to an admissible function space defined on \(\mathbb {R}\). Our main method is based on Lyapunov-Perron’s equations combined with the admissibility of function spaces and the technique of choosing F-induced trajectories.


Exponential dichotomy Partial neutral functional differential equations Unstable manifolds Attractiveness Admissibility of function spaces 

Mathematics Subject Classification (2010)

34K19 35R10 



This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant No. 101.02-2014.02.


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam
  2. 2.Department of Basic SciencesUniversity of Economic and Technical IndustriesHanoiVietnam

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