# RELATIONSHIP BETWEEN INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

### Abstract

In this paper, we describe the relationship between the solutions of the equation below with

under the conditions that the family of linear operators (B( ) t )t J ∈ defned on a Banach space X generates the evolution family (U t s ( , ))t s ≥ having an exponential dichotomy or trichotomy on the J, the diﬀerence operator F X : → is bounded and linear, and the nonlinear delay operator Φ satisfes the ϕ -Lipschitz condition, i.e.,‖ ‖ ‖ ‖ Φ - Φ ≤ - ( , ) ( , ) t t φ ψ ϕ φ ψ for φ ψ , ∈ , where ϕ belongs to an admissible function space defned on J, where J is a subinterval of the real line . 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.

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*UTEHY Journal of Science and Technology*,

*38*, 92-97. Retrieved from http://tapchi.utehy.edu.vn/index.php/jst/article/view/618