  We prove an analogue of the Korneichuk--Stechkin lemma for functions with
values in $L$-spaces. As applications, we obtain sharp Ostrowski type
inequalities and solve problems of optimal recovery of identity and
convexifying operators, as well as the problem of integral recovery, on the
classes of $L$-space valued functions with given majorant of modulus of
continuity. The recovery is done based on $n$ mean values of the functions over
some intervals. Moreover, on the classes of functions with given majorant of
modulus of continuity of their Hukuhara type derivative, we solve the problem
of optimal recovery of the function and the Hukuhara type derivative. The
recovery is done based on $n$ values of the function. We also obtain some sharp
Landau type inequalities and solve an analogue of the Stechkin problem about
approximation of unbounded operators by bounded ones and the problem of optimal
recovery of an unbounded operator on a class of elements, known with error.
Consideration of $L$-space valued functions gives a unified approach to
solution of the mentioned above extremal problems for the classes of multi- and
fuzzy-valued functions as well as for the classes of functions with values in
Banach spaces, in particular random processes, and many other classes of
functions.
