We consider a class of equations in a space of measurable functions that contains a large number of equations involving the value of a game in the theory of optimal control by stochastic processes. We prove the following Theorem. {\it Let $L$ be a $B$-space of measurable functions, $W\subset L$ a $B$-space with weakly compact sphere for some norm, $V_0$ a subspace of $W$ that is dense in $L,$$v_0\in W$ and $V=V_0+v_0$. Let $L^{\alpha\beta}$$(\alpha\in\mathfrak U,$$\beta\in\mathfrak B(\alpha))$ be a family of operators defined on $W$ with positive resolvents $R_\lambda^{\alpha\beta}$$(R_\lambda^{\alpha\beta}f\in V_0$ for $f\in L),$ and let $f^{\alpha\beta}$$(\alpha\in\mathfrak U,$$\beta\in\mathfrak B(\alpha))$ be a family of functions such that $|f^{\alpha\beta}|\leqslant g\in L$ for all $\alpha$ and $\beta$. Then $($under certain additional assumptions on $L,$$W,$$L^{\alpha\beta},$$R_\lambda^{\alpha\beta})$ the equation $\lambda u-\inf_{\alpha\in\mathfrak U}\sup_{\beta\in\mathfrak B(\alpha)}(L^{\alpha\beta}u+f^{\alpha\beta})=f $ has a unique solution in $V$ for $\lambda\geqslant0$, $f\in L$. This solution has the form} $$ u=\inf_{\alpha\in\mathfrak U}\sup_{\beta\in\mathfrak B(\alpha)}R_\lambda^{\alpha\beta}(f^{\alpha\beta}+f+\lambda v_0-L^{\alpha\beta}v_0)+v_0. $$ Bibliography: 6 titles.
@article{SM_1969_9_2_a7,
author = {N. V. Krylov},
title = {On a~class of nonlinear equations in a~space of measurable functions},
journal = {Sbornik. Mathematics},
pages = {241--251},
year = {1969},
volume = {9},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1969_9_2_a7/}
}
TY - JOUR
AU - N. V. Krylov
TI - On a class of nonlinear equations in a space of measurable functions
JO - Sbornik. Mathematics
PY - 1969
SP - 241
EP - 251
VL - 9
IS - 2
UR - http://geodesic.mathdoc.fr/item/SM_1969_9_2_a7/
LA - en
ID - SM_1969_9_2_a7
ER -
%0 Journal Article
%A N. V. Krylov
%T On a class of nonlinear equations in a space of measurable functions
%J Sbornik. Mathematics
%D 1969
%P 241-251
%V 9
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1969_9_2_a7/
%G en
%F SM_1969_9_2_a7
N. V. Krylov. On a class of nonlinear equations in a space of measurable functions. Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 241-251. http://geodesic.mathdoc.fr/item/SM_1969_9_2_a7/
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