What question did this study set out to answer?

This work aims to redefine the Riemann ∆ Banach-valued integral and establish key properties related to its integrability.

May 30, 2026Open Access

Riemann integration of Banach space-valued functions on time scale

Key Points

This work aims to redefine the Riemann ∆ Banach-valued integral and establish key properties related to its integrability.
Defined an alternate approach to Riemann integration for Banach space-valued functions on time scale.
Formulated criteria for integrability, uniqueness, linearity, and additivity properties.
Proofs established for the Fundamental Theorem of Calculus regarding the integral.
Proved necessary and sufficient conditions for the integrability of Banach space-valued functions.
Demonstrated the uniqueness of the integral under defined properties.
Established results on boundedness and continuity related to the newly defined integral.

Abstract

In this article, we explore the Riemann integration of Banach-valued functions on time scale. We attempt to give an alternate definition of the previously defined Riemann ∆ Banach-valued integral. Basic properties and results such as the criterion of integrability of the function, uniqueness of integral, linearity property, additivity property and results related to boundedness and continuity of functions are formulated and established. Finally, we state and proof the Fundamental Theorem of Calculus for the integral.

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