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.
References
N.T. Hanh, D.T. Thu, “Đa tạp tâm không ổn định đối với phương trình vi phân trung tính”, Conference GNH2022 Hung Yen University of Technology and Education, 2022, pp. 18-26.
N.T. Huy, P.V. Bang, “Invariant stable manifolds in admissible spaces on a half-line”, Discrete and Continnous Dynamical Systems – Series B, 2015, 20(9), pp. 2993-3011.
N. T. Huy, T. V. Duoc, “Integral manifolds for partial functional differential equations in admissible spaces on a half-line”. J.Math.Anal.Appl., 2014, 411, pp. 816-828.
N.T. Huy, D.X. Khanh, “Local stable manifolds of admissible classes for parabolic functional equations and applications to Hutchinson models”. International Journal of Evolution Equations, 2017, 10, pp. 391-406.
R. Nagel, G. Nickel, “Well-posedness for non-autonomous abstract Cauchy problems”. Prog. Nonl. Diff.Eq.Appl, 2002, 50, pp. 279-293.
A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
