Limes superior and inferior inequalities with functions and its derivatives

Sep 2019
4
0
Poland
Hi, I have got some troubles with Lemma 2. from N.H. Duu, On The Existence of Bounded Solutions for Lotka- Volterra Equations, Acta Mathematica Vietnamica 25(2) (2000), 145-159.

Lemma. Let \(\displaystyle G(t)\) and \(\displaystyle F(t)\) be two differentiable functions defined on \(\displaystyle (0,∞)\) such that \(\displaystyle \lim_{t→∞}G(t)=\lim_{t→∞}F(t)=+∞\) then
\(\displaystyle \limsup_{t→∞}\frac{G(t)}{F(t)}≤\limsup_{t→∞}\frac{G′(t)}{F′(t)}; \liminf_{t→∞}\frac{G(t)}{F(t)}≥\liminf_{t→∞}\frac{G′(t)}{F′(t)}.\)

Proof. By the Cauchy theorem for differentiable functions, for any \(\displaystyle t_1,t_2>0\) there is a \(\displaystyle \theta∈(t_1,t_2)\) such that
\(\displaystyle \frac{G′(\theta)}{F′(\theta)} =\frac{G(t_1)−G(t_2)}{F(t_1)−F(t_2)} =\frac{G(t_2)}{F(t_2)}\times\frac{1−\frac{G(t_1)}{G(t_2)}}{1−\frac{F(t_1)}{F(t_2)}}.\)
Letting \(\displaystyle t_1\) and \(\displaystyle t_2\to\infty\) such that \(\displaystyle \lim\frac{G(t_1)}{G(t_2)}=\lim\frac{F(t_1)}{F(t_2)}=0\) we get the result.
Can anybody help me understand the last part of this proof? Exactly, how this transformation using Cauchy theorem with included convergences leads to inequalities from lemma?
 
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mathman

Forum Staff
May 2007
6,932
774
It looks like $t_2 \to \infty$ first.