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.

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?Lemma. Let \(\displaystyle G(t)\) and \(\displaystyle F(t)\) be two differentiable functions defined on \(\displaystyle (0,âˆž)\) such that \(\displaystyle \lim_{tâ†’âˆž}G(t)=\lim_{tâ†’âˆž}F(t)=+âˆž\) then

\(\displaystyle \limsup_{tâ†’âˆž}\frac{G(t)}{F(t)}â‰¤\limsup_{tâ†’âˆž}\frac{Gâ€²(t)}{Fâ€²(t)}; \liminf_{tâ†’âˆž}\frac{G(t)}{F(t)}â‰¥\liminf_{tâ†’âˆž}\frac{Gâ€²(t)}{Fâ€²(t)}.\)

Proof.By the Cauchy theorem for differentiable functions, for any \(\displaystyle t_1,t_2>0\) there is a \(\displaystyle \thetaâˆˆ(t_1,t_2)\) such that

\(\displaystyle \frac{Gâ€²(\theta)}{Fâ€²(\theta)} =\frac{G(t_1)âˆ’G(t_2)}{F(t_1)âˆ’F(t_2)} =\frac{G(t_2)}{F(t_2)}\times\frac{1âˆ’\frac{G(t_1)}{G(t_2)}}{1âˆ’\frac{F(t_1)}{F(t_2)}}.\)

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.

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