Geodesic
On a class of nonlinear equations in a space of measurable functions
Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 241-251 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     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/}
}
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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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