Set Valued Riemann-Liouville integral and some Regular Selections

TL;DR

This paper introduces a fractional integral for set-valued functions, explores its properties, and demonstrates how regularity is preserved, with implications for differential inclusions and future research directions.

Contribution

It defines a new set-valued Riemann-Liouville fractional integral and studies its fundamental properties and regularity preservation, extending classical concepts to set-valued analysis.

Findings

Fractional integral preserves convexity, boundedness, and continuity.

Bounded variation and Lipschitz continuity are inherited by the fractional integral.

Extremal selections are identified with the same regularity properties.

Abstract

In this article, we introduce the notion of the Riemann-Liouville fractional integral of set-valued mappings via integrable selections. We establish fundamental properties of this fractional integral, including convexity, boundedness, and continuity with respect to the Hausdorff metric. The investigation of preservation of regularity under fractional integration with respect to the Hausdorff metric is given. We show that bounded variation and Lipschitz continuity of a set-valued mapping are inherited by its Riemann-Liouville fractional integral. We discuss the existence of regular selections for the fractional integral under the corresponding regularity assumptions on the original mapping. In the scalar case, we further identify extremal selections given by the pointwise minimum and maximum of the fractional integral and show that they possess the same regularity properties. Finally, we discuss possible applications in differential inclusion and directions for future research.