ON HENSTROCK INTEGRALS OF INTERVAL-VALUED FUNCTIONS ON TIME SCALES

Abstract

In this paper we introduce the interval-valued Hen- stock integral on time scales and investigate some properties of these integrals. 1. Introduction and preliminaries The Henstock integral for real functions was flrst deflned 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 delta integral of interval-valued function on time scales and investigate some properties of the integral. A time scale T is a nonempty closed subset of real numberR with the subspace topology inherited from the standard topology ofR. For t 2 T we deflne the forward jump operator ae(t) = inffs 2 T : s > tg where inf ` = supfTg, while the backward jump operator ‰(t) = supfs 2 T : s t, we say that t is right-scattered, while if ‰(t) < t, we say that t is left-scattered. If ae(t) = t, we say that t is right-dense, while if ‰(t) = t, we say that t is left-dense. The forward graininess function (t) of t 2 T is deflned by (t) = ae(t) i t, whlie the backward graininess function (t) of t 2 T is deflned by (t) = t i ‰(t). For a;b 2 T we denote the closed interval (a;b)T = ft 2 T : atbg.