*(English)*Zbl 1056.35082

The paper deals with the initial value problem

where $x\in {\mathbb{R}}^{N}$, $t>0$, here $\alpha $ is a bounded uniformly continuous function and $r$ is bounded, measurable but generally a non-continuous one. The problem is motivated by the model of best response dynamics arising in game theory [see the first author, Ann. Oper Res. 89, 233–251 (1999; Zbl 0942.91018)]. The theorem proved by the authors claims that (1), (2) has a generalized (in the sense derived in the paper) solution. The proof is based on solving the problem (1), (2) with $f$ replaced by ${f}_{n}$, where $\left({f}_{n}\right)$ is a sequence of ${C}^{\infty}$ functions approximating $f$. Then the Arzela-Ascoli theorem is applied to the corresponding sequence $({u}_{n}$ of solutions and the uniform limit of a subsequence of $\left({u}_{n}\right)$ is the desired solution of (1), (2).