The paper deals with the initial value problem
where , , here is a bounded uniformly continuous function and 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 replaced by , where is a sequence of functions approximating . Then the Arzela-Ascoli theorem is applied to the corresponding sequence of solutions and the uniform limit of a subsequence of is the desired solution of (1), (2).