Generalizing the concept of bounded variation

Abstract

Let \([a,b]\subseteq \mathbb {R}\) be a non-empty and non singleton closed interval and \(P=\{a=x_0<\cdots <x_n=b\}\) is a partition of it. Then \(f:I\rightarrow \mathbb {R}\) is said to be a function of r-bounded variation, if the expression \(\sum \nolimits ^{n}_{i=1}|f(x_i)-f(x_{i-1})|^{r}\) is bounded for all possible partitions like P. One of the main results of the paper deals with the generalization of the classical Jordan decomposition theorem. We establish that for \(r\in ]0,1]\), a function of r-bounded variation can be written as the difference of two monotone functions. While for \(r>1\), under minimal assumptions such a function can be treated as an approximately monotone function which can be closely approximated by a nondecreasing majorant. We also prove that for \(0<r_1<r_2\), the function class of \(r_1\)-bounded variation is contained in the class of functions satisfying \(r_2\)-bounded variations. We go through approximately monotone functions and present a possible decomposition for \(f:I(\subseteq \mathbb {R}_+)\rightarrow \mathbb {R}\) satisfying the functional inequality

$$f(x)\le f(x)+(y-x)^{p}\quad (x,y\in I \text{ with } x<y \text{ and } p\in ]0,1[ ).$$

A generalized structural study has also been done in that specific section. On the other hand, for \(\ell [a,b]\ge d\), a function satisfying the following monotonic condition under the given assumption will be termed as d-periodically increasing

$$f(x)\le f(y)\quad \text{ for } \text{ all }\quad x,y\in I\quad \text{ with }\quad y-x\ge d.$$

We establish that in a compact interval any function satisfying d-bounded variation can be decomposed as the difference of a monotone and a d-periodically increasing function. The core details related to past results, motivation, structure of each and every section are thoroughly discussed below.