On a Class of Nonlinear Equations in a Space of Measurable Functions

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 followingTheorem. Let be a -space of measurable functions, a -space with weakly compact sphere for some norm, a subspace of that is dense in , and .Let (, ) be a family of operators defined on with positive resolvents ( for ), and let (, ) be a family of functions such that for all and .Then (under certain additional assumptions on , , , ) the equation has a unique solution in for , . This solution has the form Bibliography 6 items.