In the present paper, by means of the successive approximations method, the local or global existence and uniqueness theorems for a stochastic functional differential equation of the Ito type are proved.
We establish the existence of mild solutions and periodic mild solutions for a class of abstract first-order non-autonomous
neutral functional differential equations with infinite delay in a Banach space.
The existence of an algebraic functional-differential equation P (y′(x), y′(x + log 2), …, y′(x + 5 log 2)) = 0 is proved such that the real-analytic solutions are dense in the space of continuous functions on every compact interval. A similar result holds for an algebraic functional-differential equation P(y′(x − 4πi), y′(x − 2πi), …, y′(x + 4πi)) = 0 (with i2 = −1), which is explicitly given: There are real-analytic solutions on the real line such that every continuous function defined on a compact interval can be approximated by these solutions with arbitrary accuracy.