A finite difference scheme for finding viscosity solutions of specifically stated problems for Hamilton–Jacobi equations is proposed. Solutions of such problems satisfy two differential inequalities involved into the definition of viscosity solutions. The specifics is that the inequalities must hold only in the region where solutions are less than a given function or greater than another given function. From the point of view of optimal control theory and the theory of differential games, these results can be applied to finding value functions in problems with nonterminal payoff functionals and state constraints.
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