# On fuzzy Henstock-Stieltjes integral on time scales with respect to bounded variation function

## TL;DR

This paper introduces a new type of fuzzy integral on time scales, expanding the theory of fuzzy calculus.

## Contribution

The paper introduces the fuzzy Henstock-Stieltjes Δ-integral on time scales with respect to bounded variation functions.

## Key findings

- The paper defines the FHS-Δ-integral and outlines its basic properties.
- It provides necessary and sufficient conditions for integrable functions.
- A characterization theorem is presented using the embedding theorem of fuzzy number space.

## Abstract

In present paper we will investigate the basic theory of fuzzy Henstock-Stieltjes Δ-integral with respect to a bounded variation function on time scale. Firstly, we define the notion of fuzzy Henstock-Stieltjes Δ-integral (or briefly FHS-Δ-integral) on time scales, and propose some basic properties and several necessary and sufficient conditions for fuzzy Henstock-Stieltjes Δ-integrable functions. Secondly, we present a characterization theorem of fuzzy Henstock-Stieltjes Δ-integrable function by using the embedding theorem of fuzzy number space. Therefore, this paper complements and enriches the theory of fuzzy integral, and the results of this paper will contribute to establishing discontinuous fuzzy dynamic equations on time scales.