Abstract:
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.
$$
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