In the paper, some random fixed point theorems are proved for random operators in separable Banach spaces with so-called PPF dependence (i.e., for the operators which may depend on the past, present or future) with different domain and range. This type of fixed points was first examined by A. Špaček [Czech. Math. J. 5(80), 462–466 (1955; Zbl 0068.32701)] and O. Hanš [in: Trans. 1st Prague Conf. Information Theory, statist. Decision functions, Random Processes, Liblice Nov. 28–30, 1956, 105–125 (1957; Zbl 0087.33104)].
These results are applied to some nonlinear random functional differential equations to get the existence and uniqueness of PPF dependent random solutions of the initial value problems of this type of differential equations.
The methods of proof of the results presented in the paper are similar to the methods known for ordinary differential equations and utilize the method of “successive approximations”.