Set Differential Equations with Causal Operators

Differential equations involving causal operators have gained much attention of late and some results are assembled in a recent monograph [1]. The term causal is adopted from the engineering literature. Basically, a causal operator is a nonanticipative operator. The theory of these operators has the powerful quality of unifying ordinary differential equations, integrodifferential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral functional equations, to name a few. The study of set differential equations (SDE) in a metric space is interesting due to its applicability to multivalued differential inclusions and fuzzy differential equations and its inclusion of ordinary differential systems as a special case [2, 4]. A combination of these two concepts leads to set differential equations with causal operators. In this paper, using this setup, we obtain some basic results on existence, uniqueness, and continuous dependence of solutions with respect to initial values.