The spaces of regulated functions and differential equations with generalized functions in coefficients

A function defined on an open (finite, semi-finite, infinite) interval is called regulated if it has finite one-sided limits at each point of its domain. In the present paper we study spaces of regulated functions, in particular, their dense subsets. Our motivation is applications to differential equations. Namely, we consider the Cauchy problem for a scalar linear differential equation with coefficients, which are derivatives of regulated functions. We immerse the Cauchy problem into the space of the Colombeau generalized functions. If the coefficients are derivatives of step functions, we find explicit solution $R(\varphi_\mu,t)$ of the Cauchy problem (in terms of representatives); its limit as $\mu\to+0$ is defined to be the solution of the original problem. In this way, we obtain a densely defined (on the space of regulated functions) operator $\mathbf T$, which associates the solution to a Cauchy problem with this problem. Next, using a well-known topological result on a continuous extension, we extend the operator $\mathbf T$ to the operator $\widehat{\mathbf T}$ defined on the entire space of regulated functions. We have given the explicit representation of solution of the Cauchy problem for the inhomogeneous differential equation. Illustrative examples are also offered.