International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 3, Pages 285-296
doi:10.1155/S0161171278000319

A representation theorem for operators on a space of interval functions

Abstract

Suppose N is a Banach space of norm |•| and R is the set of real numbers. All integrals used are of the subdivision-refinement type. The main theorem [Theorem 3] gives a representation of TH where H is a function from R×R to N such that H(p+,p+), H(p,p+), H(p−,p−), and H(p−,p) each exist for each p and T is a bounded linear operator on the space of all such functions H. In particular we show that TH=(I)∫abfHdα+∑i=1∞[H(xi−1,xi−1+)−H(xi−1+,xi−1+)]β(xi−1)+∑i=1∞[H(xi−,xi)−H(xi−,xi−)]Θ(xi−1,xi)where each of α, β, and Θ depend only on T, α is of bounded variation, β and Θ are 0 except at a countable number of points, fH is a function from R to N depending on H and {xi}i=1∞ denotes the points P in [a,b]. for which [H(p,p+)−H(p+,p+)]≠0 or [H(p−,p)−H(p−,p−)]≠0. We also define an interior interval function integral and give a relationship between it and the standard interval function integral.