Generalizing the Concept of Bounded Variation

Abstract

Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression $\overset{n}{\underset{i=1}{\sum}}|f(x_i)-f(x_{i-1})|^{r}$ is bounded for all possible partitions like $P$. One of the main result of the paper deals with the generalization of Classical Jordan decomposition theorem. We have shown 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 functions can be treated as approximately monotone function which can be closely approximated by a nondecreasing majorant. We also proved 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 $f:I(\subseteq \mathbb{R_+})\to\mathbb{R}$ satisfying the functional inequality $$f(x)\leq f(x)+(y-x)^{p}\quad (x,y\in I\mbox{ with $x<y$ and $ p\in]0,1[ $}).$$ A generalized structural study has also be done in that specific section. On the other hand for $\ell[a,b]\geq d$; a function satisfying the following monotonic condition under the given assumption will be termed as $d$-periodically increasing $$f(x)\leq f(y)\quad \mbox{for all}\quad x,y\in I\quad\mbox{with}\quad y-x\geq d.$$ we establish that in a compact interval any bounded function can be decomposed as the difference of a monotone and a $d$-periodically increasing function.