Absolute continuity for Banach space valued mappings (Q997549)

The authors generalize the concept of variation, namely \(p,\lambda,\delta\)-variation, of Banach-space valued functions defined in an open set of \(\mathbb R^n\), by means of gage-fine partitions. The idea of considering gage-fine partitions for defining the variation of a function is not new. In a previous paper by the same authors [J. Math. Anal. Appl. 299, No. 1, 227--234 (2004; Zbl 1124.26009)], a variation-type concept using gage-fine partitions is considered for real valued functions. In the present paper, the authors characterize the class of functions of bounded \(p,\lambda,\delta\)-variation (Theorems 5 and 6) and prove that bounded \(p,\lambda,\delta\)-variation implies Frechét differentiability almost everywhere. They also specialize some results for \(p,\lambda,\delta\)-absolute continuous functions.