Monotone convergence theorems for pointwise Henstock-Kurzweil integrable operator-valued functions and applications

Abstract

In this paper we prove convergence theorems for pointwise Henstock-Kurzweil integrable functions from a compact real interval to the space $L(H)$ of bounded linear operators of a separable complex Hilbert space $H$, ordered by the cone of positive operators. Using the so obtained theorems and fixed point theorems in partially ordered sets we prove fixed point theorems in spaces of $L(H)$- valued functions, and apply them to prove existence and comparison results for extremal solutions of discontinuous functional integral equations containing pointwise Henstock-Kurzweil integrable operator-valued functions.