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16 views
Lemma 3.3 from “Positive solutions for third order semipositone boundary value problems”
I have this lemma :
Assume that : $w(t)$ is nondecreasing and $w(t)>0$ on $(q,1]$ , and let $M(t)$ such that $M\in L(0,1)$; $M(t)>0 $ on $(0,1)$ and $f(t,x+\gamma(t))\geq -M(t)$ for $(t,x)\in ...
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0answers
18 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 ...
1
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1answer
44 views
Show $\left| \int_a^b f d\alpha \right| \le \int_a^b|f|dV$ if $V$ is variation of $\alpha$ on $[a, b]$
Let $\alpha$ be of bounded variation on $[a, b]$ and assume that $V(x)$ be the total variation of $\alpha$ on $[a, x]$, $a<x\le b$ and $V(a) = 0$. Let $f$ be defined and bounded on $[a, b]$. If ...
0
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1answer
135 views
Proving a function has a bounded variation
$f$ is continuous on $[a,b]$ and $\vert f \vert$ has a bounded variation. I would like to show $f$ has bounded variation.
Using the intermediate value theorem we can take a partition such that (1) ...
1
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1answer
71 views
variation of $\vert f \vert^{1.5}$
$f$ has a bounded variation. $\vert f\vert ^{1.5}$ also has a bounded variation. $\vert f \vert $ is a bounded variation function, as well as integer powers of $\vert f \vert$. $\vert f \vert ^{1.5}$ ...
1
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0answers
198 views
Show that $f$ is of Bounded Variation by $f'$' being integrable.
For $\alpha$, $\beta$ > 0, define the function $f$ on $[0, 1]$ by $f(x) = x^{\alpha}\sin(1/x^{\beta})$ for $0 < x \leq 1$ and $f(x) = 0$ for $x=0$. Show that if $\alpha > \beta$, then $f$ is of ...
