Quotient space Henstock-Kurzweil integration on time scales

Abstract

We introduce Henstock-Kurzweil integrability for functions whose values lies in the quotient space
on time scales. We define the Henstock-Kurzweil integral with respect to the Δ-derivative and ∇-derivative
namely the quotient Henstock-Kurzweil Δ-integral and quotient Henstock-Kurzweil ∇-integral respectively.
Result establishing the criterion of integrability is observed, and few properties of the integrals are formulated.
Relations between quotient Henstock-Kurzweil integral and quotient Riemann integral, and quotient Henstock-
Kurzweil integral and Banach Henstock-Kurzweil integral are also presented.
In addition, as a linear combination of the Δ- and ∇-integrals we introduce the quotient Henstock-Kurzweil
♢α-integral and conclude with a theorem depicting the relation between the three integrals.