On interval Riemann double integration on time scales

Abstract

This article explores the theory of Riemann double integration for functions whose values are intervals in the framework of time scale calculus. We define the Riemann double ∆-integral and Riemann double ∇-integral for interval valued functions, namely interval Riemann ∆∆-integral and interval Riemann ∇∇-integral. Some key theorems in the article discuss the uniqueness of the integral, the equality of the interval Riemann double integral to the Riemann double integral when function is degenerate, necessary and sufficient conditions for integrability, proving integrability of a function without knowing the actual value of the integral. Additionally the relationship between the interval Riemann double integral and Riemann double integral for two interval-valued functions is estableshed via Hausdorff-Pompeiu distance. Elementary properties of the integral such as linearity property, subset property and others are established. Using the concept of generalized Hukuhara difference, alternate definitions of the interval Riemann ∆∆-integral and interval Riemann ∇∇-integral are formulated and theorems proving the equivalence of the integrals defined in both approaches are established. Theorems proving the equivalence of interval Riemann Δ - and ∇integrals previously defined in both approaches are also shown.