Some Applications Related to Differential Inclusions Based on the Use of a Weighted Space

In this paper, we present an existence theorem for the problem of discontinuous dynamical system related to ordinary differential inclusion, based on the use of the concepts related to weighted spaces introduced by Gorka and Rybka, without using any fixed point theorem. The solution concept in this theorem is considered to belong to the weighted space. For comparison with the classical case and as an application of the theorem, we give an example problem that has such a solution but no continuously differentiable solution.