## Taiwanese Journal of Mathematics

### Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line

#### Abstract

For the following class of partial neutral functional differential equations $\begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), \\ u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases}$ we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 4 (2019), 897-923.

Dates
Revised: 9 December 2018
Accepted: 17 December 2018
First available in Project Euclid: 18 July 2019

https://projecteuclid.org/euclid.twjm/1563436874

Digital Object Identifier
doi:10.11650/tjm/181209

Mathematical Reviews number (MathSciNet)
MR3982067

Zentralblatt MATH identifier
07088953

#### Citation

Nguyen, Thieu Huy; Trinh, Xuan Yen. Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line. Taiwanese J. Math. 23 (2019), no. 4, 897--923. doi:10.11650/tjm/181209. https://projecteuclid.org/euclid.twjm/1563436874

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