  Let the sets of functions $Z$ and $\Omega$ be given on the time interval $T$,
let there also be a multifunction (m/f) $\alpha$ acting from $\Omega$ to $Z$
and a finite set of moments $\Delta$ from $T$. The work deals with two
questions: the first one is the connection between the possibility of stepwise
construction (specified by $\Delta$) of a value $z$ of $\alpha(\omega)$ for an
unknown step-by-step implemented argument $\omega\in\Omega$ and the existence
of a multiselector $\beta$ of the m/f $\alpha$ with a non-anticipatory property
of special kind defined by $\Delta$; and the second question is how to build
the above $\beta$ for a given pair $(\alpha,\Delta)$. The consideration of
these questions is motivated by the presence of similar step-by-step procedures
in the differential game theory, for example, in the alternating integral
method, in pursuit-evasion problems posed with use of counter-strategies, and
in the method of guide control. It is shown that the step-by-step construction
of the value $z\in\alpha(\omega)$ can be carried out for any in steps
implemented argument $\omega$ if and only if the multiselector $\beta$ is
non-empty-valued. In this case, the desired value $z$ can be selected from
$\beta(\omega)$ in step-by-step procedure for any unknown in advance argument
$\omega$. The key point of the work is the procedure for calculation the
multiselector $\beta$, for which a constructive and finite-step description is
given. Illustrative examples are considered that include, in particular,
problems of a guaranteed result optimization under functional constraints on
control and/or disturbance implementations.
