Functions of bounded variation arising from a generalized two-normed space
Osmin Ferrer, Kandy Ferrer, Jaffeth Cure
TL;DR
This work extends the classical theory of functions of bounded variation from normed and 2-normed spaces to generalized two-normed spaces. It develops a theory of bounded generalized $2$-variation, proves that such functions are $(2,k)$-bounded, and shows that the collection $BV([a,b], abla^k)$ forms a vector space equipped with a generalized symmetric two-norm $||\,ullet\,||_{2G}$, thereby unifying BV concepts within a broader framework. The paper also establishes key inequalities, presents a counterexample to the converse implication, and provides new applications that align with and extend prior results (Chistyakov; Cure–Ferrer), highlighting the practical potential of BV analysis in generalized two-normed contexts. Overall, the results offer a rigorous, generalized foundation for BV in spaces beyond the classical normed and 2-normed settings, with implications for functional analysis and related applications.
Abstract
The first formal development of functions with bounded variation in normed spaces is attributed to Chistyakov [5], and was later extended to the context of 2-normed spaces by Cure, Ferrer S., and Ferrer V. [6]. In this paper, we elaborate on this extension (Definition 4.5), exploring its fundamental properties. The foundational characteristics of functions of bounded 2-variation within the context of generalized two-normed spaces are explored in detail (see Theorems 4.4, 4.12, 4.13). Additionally, it is shown that a function of bounded variation defined on a semi-normed space can generate a function of bounded 2-variation in the context of generalized two-normed spaces (Proposition 4.10).