2
votes
1answer
174 views

About functions of bounded variation

I got the following the following idea in one of the articles that I'm reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions ...
0
votes
0answers
26 views

Lemma 3.2 from “Positive solutions for third order semipositone boundary value problems”

How de prove this lemma please : Assume that: $w(t)$ is nondercreasing and $w(t)>0$ on $(q,1]$ , $\frac12<p<q<1$ hods . Let $z\in C^2[0,1]\cap C^3(0,1)$ satisfy $z'''(t)\geq 0$ 0n ...
5
votes
1answer
1k views

Prove the normed space of bounded variation functions is complete

Let $\Vert f \Vert = |f(0)| + \mathrm{Var}f$ for all $f \in BV([0,1])$; we are given that it is a norm. Show that $BV([0,1])$ is a complete normed space with this norm. I have shown that any ...