DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE Variations of Gronwall’s Lemma Gronwall’s lemma, which solves a certain kind of inequality for a function, is useful in the theory of differential equations. Here is one version of it 1, p, 283: 0. Gronwall’s inequality. Let y(t),f(t), and g(t) be nonnegative functions on 0,T. Then we give an explicit solution to the linear discrete fractional sum equation. This allows us to state and prove an analogue of Gronwall's inequality on discrete fractional calculus. We employ a nabla, or backward difference; we employ the Riemann-Liouville definition of the fractional difference.
Consider a sequence $(a_i)_{iinmathbb{N}}$ of non-negative real numbers and $(b_i)_{iinmathbb{N}}$ a positive sequence tending to zero as $itoinfty$. Furthermore, suppose that there is a $c>0$ such that $$ a_igeq b_i + ca_{i-1}.$$
I try to get a lower bound on $a_i$ for each $iinmathbb{N}$ and hoped to find a Gronwall type lemma but did not succeed. What quite obviously holds is $$a_igeq b_i+c(b_{i-1} +c(b_{i-2}+c(...)))$$
and what also follows directly is the lower bound $a_igeq c^i$ by omitting all the $b_i$. But I want to find a lower bound that depends on the $b_i$. How can I compute the extended right hand side or is there some lemma I could use?
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$begingroup$Consider a sequence $(a_i)_{iinmathbb{N}}$ of non-negative real numbers and $(b_i)_{iinmathbb{N}}$ a positive sequence tending to zero as $itoinfty$. Furthermore, suppose that there is a $c>0$ such that $$ a_igeq b_i + ca_{i-1}.$$
I try to get a lower bound on $a_i$ for each $iinmathbb{N}$ and hoped to find a Gronwall type lemma but did not succeed. What quite obviously holds is $$a_igeq b_i+c(b_{i-1} +c(b_{i-2}+c(...)))$$
and what also follows directly is the lower bound $a_igeq c^i$ by omitting all the $b_i$. But I want to find a lower bound that depends on the $b_i$. How can I compute the extended right hand side or is there some lemma I could use?