Measure Functional Differential Equations in the Space of Functions of Bounded Variation

Abstract

We establish general conditions for the unique solvability of nonlinear measure functional differential equations in terms of properties of suitable linear majorants.

Closure properties for integral problems driven by regulated functions via convergence results

“…There is a wide literature treating this subject (we refer to [1], [9], [13], [14] in the single-valued case and to [8], [11], [31], [37] in the set-valued setting). The motivation comes from the fact that one can thus cover the framework of usual differential problems (when g is absolutely continuous), of discrete problems (when g is a sum of step functions), of impulsive equations (for g being the sum between an absolutely continuous function and a sum of step functions), as well as retarded problems (see [1]). As proven in [13], dynamic equations on time scales and generalized differential equations can also be seen as measure differential equations.…”

Well-posedness results for abstract generalized differential equations and measure functional differential equations

“…These equations were introduced only relatively recently in [10], and were subsequently studied in [3,11,12,26,27,37]. Fairly general results concerning the well-posedness for linear equations with finite delay have been obtained in [27], while the nonlinear case with finite delay was considered in [10].…”