Generalizing the concept of bounded variation

Abstract

Let $$[a,b]\subseteq \mathbb {R}$$ [ a , b ] ⊆ R be a non-empty and non singleton closed interval and $$P=\{a=x_0<\cdots <x_n=b\}$$ P = { a = x 0 < ⋯ < x n = b } is a partition of it. Then $$f:I\rightarrow \mathbb {R}$$ f : I → R is said to be a function of r-bounded variation, if the expression $$\sum \nolimits ^{n}_{i=1}|f(x_i)-f(x_{i-1})|^{r}$$ ∑ i = 1 n | 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 $$r\in ]0,1]$$ r ∈ ] 0 , 1 ] , a function of r-bounded variation can be written as the difference of two monotone functions. While for $$r>1$$ 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 $$0<r_1<r_2$$ 0 < r 1 < r 2 , the function class of $$r_1$$ r 1 -bounded variation is contained in the class of functions satisfying $$r_2$$ r 2 -bounded variations. We go through approximately monotone functions and present a possible decomposition for $$f:I(\subseteq \mathbb {R}_+)\rightarrow \mathbb {R}$$ f : I ( ⊆ R + ) → R satisfying the functional inequality $$f(x)\le f(x)+(y-x)^{p}\quad (x,y\in I \text{ with } x<y \text{ and } p\in ]0,1[ ).$$ f ( x ) ≤ f ( x ) + ( y - x ) p ( x , y ∈ I with x < y and p ∈ ] 0 , 1 [ ) . A generalized structural study has also been done in that specific section. On the other hand, for $$\ell [a,b]\ge d$$ ℓ [ a , b ] ≥ d , a function satisfying the following monotonic condition under the given assumption will be termed as d-periodically increasing $$f(x)\le f(y)\quad \text{ for } \text{ all }\quad x,y\in I\quad \text{ with }\quad y-x\ge d.$$ f ( x ) ≤ f ( y ) for all x , y ∈ I with y - x ≥ 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.