| Summary: | Abstract Stieltjes derivatives represent a new unification of discrete and continuous calculus consisting in a differentiation process with respect to a given non-decreasing function g. This notion infers a new class of differential equations which has shown to have many applications. Herein we explore the use of such derivatives in the study of multivalued equations, the so-called g-differential inclusions. Such multivalued differential problems simply consist in replacing the usual derivatives by Stieltjes derivatives (also known as g-derivatives). Using Baire category methods, we investigate extremal solutions for g-differential inclusions. It is shown that g-differential inclusions offer an alternative approach to measure-driven problems; therefore, the existence of extremal solutions for measure differential inclusions is obtained as a simple consequence of the results for this new type of inclusions.
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