The Existence, Uniqueness, and Carathéodory’s Successive Approximation of Fractional Neutral Stochastic Differential Equation - Xiaolin Yuan; Yongguang Yu; Guojian Ren; Wei Chen; Weiyi Xu (Fractal and Fractional, 2022)
The existence, uniqueness, and Carath<semantics><mover accent="true">e´</mover></semantics>odory’s successive approximation of the fractional neutral stochastic differential equation (FNSDE) in Hilbert space are considered in this paper. First, we give the Carath<semantics><mover accent="true">e´</mover></semantics>odory’s approximation solution for the FNSDE with variable time delays. We then establish the boundedness and continuity of the mild solution and Carath<semantics><mover accent="true">e´</mover></semantics>odory’s approximation solution, respectively. We prove that the mean-square error between the exact solution and the approximation solution depends on the supremum of time delay. Next, we give the Carath<semantics><mover accent="true">e´</mover></semantics>odory’s approximation solution for the general FNSDE without delay. Under uniform Lipschitz condition and linear growth condition, we show that the proof of the convergence of the Carath<semantics><mover accent="true">e´</mover></semantics>odory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution. Furthermore, under the non-Lipschitz condition, which is weaker than Lipschitz one, we establish the existence and uniqueness theorem of the solution for the FNSDE based on the Carath<semantics><mover accent="true">e´</mover></semantics>odory’s successive approximation. Finally, a simulation is given to demonstrate the effectiveness of the proposed methods.