Differential equations with discontinuous right-hand sides☆

Many phenomena from physics, control theory, and economics have been successfully described by differential equations with discontinuous functions. This in turn leads to a great need to develope a theory for such differential equations. However, as far as existence and qualitative behavior are concerned, such a theory is far from being complete. Even existence results are available only for very special discontinuous function f (as the discontinuous part in equation), e.g., see [S] for piecewise continuous f; [9] for f having one discontinuous point, [S, 121 for f of bounded variation. In this paper, we are going to establish some existence results for differential equations with discontinuous right-hand sides with great generality. Some of the results in this paper can be stated and proved for high order ordinary and partial differential equations. For example, we may consider the following problem