ALGEBRAIC-DELAY DIFFERENTIAL SYSTEMS: C-0-EXTENDABLE SUBMANIFOLDS AND LINEARIZATION

作者：Kosovalic N^*; Chen Y; Wu J

来源：Transactions of the American Mathematical Society, 2017, 369(5): 3387-3419.

DOI：10.1090/tran/6760

• Linearization
• nonlinear transport equation
• state dependent delay
• structured population dynamics
• Banach manifold
• interpolation space
• functional differential equation

Consider the abstract algebraic-delay differential system,
x'(t) = Ax(t) + F(x(t), alpha(t)),
alpha(t) = H(x(t), alpha(t)).
Here A is a linear operator on D(A) subset of X satisfying the Hille-Yosida conditions, x(t) is an element of <(D(A))over bar> subset of X, alpha(t) is an element of R-n, and X is a real Banach space. Let C-0 subset of <(D(A))over bar> be closed and convex, and K subset of R-n be a compact set contained in the ball of radius h > 0 centered at 0. Under suitable Lipschitz conditions on the nonlinearities F and H and a subtangential condition, the system generates a continuous semiflow on a subset of the space of continuous functions C([-h, 0], C-0 x R-n), which is induced by the algebraic constraint. The object of this paper is to find conditions under which this semiflow is also differentiable with respect to initial data. In the motivating example coming from modelling the dynamics of an age structured population, the nonlinearities F and H are not Fr ' echet differentiable on the sets C-0 x K and C([-h, 0], C-0 x K), respectively. The main challenge of obtaining the differentiability of the semiflow is to determine the right type of differentiability and the right phase space. We develop a novel approach to address this problem which also shows how the spaces on which the derivatives of solution operators act reflect the model structure.

• 出版日期2017-5