ON HENSTOCK-STIELTJES INTEGRALS OF INTERVAL-VALUED FUNCTIONS ON TIME SCALES

Abstract

In this paper we introduce the interval-valued Henstock- Stieltjes integral on time scales and investigate some properties of these integrals. 1. Introduction and preliminaries The Henstock integral for real functions was first defined by Henstock (2) in 1963. The Henstock integral is more powerful and simpler than the Lebesgue, Wiener and Feynman integrals. The Henstock delta integral on time scales was introduced by Allan Peterson and Bevan Thompson (5) in 2006. In 2000, Congxin Wu and Zengtai Gong introduced the concept of the Henstock integral of interval-valued functions (6). In this paper we introduce the concept of the Henstock-Stieltjes delta integral of interval-valued functions on time scales and investigate some properties of the integral. A time scale T is a nonempty closed subset of real number R with the subspace topology inherited from the standard topology of R. For t ∈ T we define the forward jump operator σ(t) = inf{s ∈ T : s > t} where inf ϕ = sup T , while the backward jump operator ρ(t) = sup{s ∈ T : s t, we say that t is right-scattered, while if ρ(t) 0 on (a, b)T , δR(t) > 0 on (a, b)T , δL(a) ≥ 0, δR(b) ≥ 0 and δR(t) ≥ µ(t) for each t ∈ (a, b)T.