A Variational Integral for Banach-Valued Functions

Abstract

It is shown that for Banach-space-valued functions the variational Henstock integral is equivalent to the Henstock integral if and only if the range space is of a finite dimension. The same is true for the equivalence of the variational McShane integral and the McShane integral.

Order-type Henstock and McShane integrals in banach lattice setting

“…The previous result can be used to show that the (H)-integrability in general does not imply the (oH)integrability: indeed, if X is any infinite-dimensional Banach space, there exists a McShane (norm)-integrable map f : [0, 1] → X that is not Bochner integrable (see [39]). In particular, when X is an L-space (of infinite dimension), such function f cannot be (oH)-integrable, in view of Theorem 15.…”

Denjoy-type Integrals in Locally Convex Topological Vector Space

“…We now show that for functions taking values in a Banach space X, the Denjoy-Bochner integral defined by Solodov in [12] is stronger than the Denjoy integral.…”

“…More specifically, they showed that a function f : [a, b] → R is HK-integrable on [a, b] if and only if there exists an ACG * -function on [a, b] such that F ′ (t) = f (t) almost everywhere. In 1995, following Lee's alternative definitions of AC * and ACG * , Canoy and Navarro [1] defined a Denjoy-type integral for Banach-valued functions and in 1998 Skvortsov and Solodov [12] adopted such definition and called that integral the Denjoy-Bochner integral. In this paper, following these earlier definitions, we introduce AC * and ACG * -type properties for LCTVS-valued functions and, subsequently, define two Denjoy-type integrals.…”

New extension of the variational McShane integral of vector-valued functions

“…For more information about the McShane integral we refer to [17], [2], [4], [5]- [7], [11], [10], [14] and [13].…”