Questions tagged [gronwall-type-inequality]

Questions on inequalities similar to the classical Gronwall's lemma. Typically, a function's derivative is bounded by (some variation of) itself. This may also be given in the corresponding integral version instead. From there an estimate for the original function can be derived.

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Gronwall inequality for backward (linear) differential equation

In the differential form of the Gronwall's Lemma, we have the following: $$ \frac{d}{dt} \phi(t) \leq \psi(t) \phi(t), $$ for all $t\geq t_0$. Then, we get $$ \phi(t) \leq \phi(t_0) exp\left(\int_{t_0}...
hanava331's user avatar
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About Gronwall's inequality

I knew the following Gronwall's inequality (Integral form) If $\alpha$ is non-negative and $H(t)$ satisfies the integral inequality \begin{align*} H(t) \leq c+ \int_0^t \alpha(s) H(s)ds \quad \text{(c:...
bluejyellow's user avatar
2 votes
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Using Gronwall to prove bi-Lipschitz

I am working through a proof which, for fixed $x \in \mathbb{R}^k$ considers an initial value problem of the form: $$\frac{d}{d t}u_t(x)=v_t(u_t(x)), \quad u_t(0)=x$$ where $u_t:\mathbb{R}^k \...
JDoe2's user avatar
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Gronwall's Inequality Application

Let say we have $$ \frac{\partial |F|}{\partial t} \le K |F| + K \alpha^{2},\:\:\:\:\: t \in [t_{n}, t_{n+1}]$$ with $F(x, t_{n})=0$ and $t_{n+1}-t_{n}=\triangle t$. The above is the same thing as $$ \...
Redsbefall's user avatar
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8 votes
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Gronwall lemma with highly oscillatory kernel

As a toy model for a larger problem, I want to show that if $A,B\geq 0$ and $u(t)$ satisfies $$|u(t)|\leq A+\left|\int_0^t B\cos(s^2)u(s)ds\right|$$ then $u$ satisfies a bound like $$|u(t)|\leq AC$$ ...
kieransquared's user avatar
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A possible upper bound for a function that satisfies a singular integral inequality

I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality: $$ |v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left( |...
Taki Zeg's user avatar
1 vote
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Generalized Gronwall Inequality covering many different applications

Recently, I needed some generalized version of Gronwall's Lemma, which I couldn't find in a quick search. However, I discovered that MSE is full of questions differing only in details on this very ...
junjios's user avatar
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2 votes
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"Nonlinear" Gronwall-type inequality

Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$...
Dal's user avatar
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Gronwall inequality

This question concerns a proof of a theorem involving Gronwall type inequality. We have the following: The question is: how did we apply Gronwall type inequality to get estimate (3.6)?
Masr's user avatar
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Nonlinear Gronwall inequality

Suppose a (continuous, non-negative) function $f$ satisfies $$ f'(t) \leq f(t)^2 $$ for $t \in [0,1]$. Set $$ g(t) = \exp\left(-\frac{1}{f(t)}\right). $$ Then $$ g'(t) = \frac{f'(t)}{f(t)^2} g(t) \...
onamoonlessnight's user avatar
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1 answer

Gronwall-type inequality in higher dimension

I would like to know if there is any kind of Gronwall inequality for a smooth function $u \colon \mathbb{R}^n \to \mathbb{R}$ satisfying $$ |\nabla u | \le K u, $$ where $K$ is a constant.
Onil90's user avatar
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1 vote
2 answers

gronwall lemma involves gradient

Could anyone see how the following two lines follow from Gronwall lemma? I use the usual differential form gronwall lemma from in Evans book. I do not know how to deal with the term involves the ...
math101's user avatar
  • 3,065
2 votes
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Kind of Gronwall Inequality

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$ f(t) \leq A+\int_0^{2t} g(s)f(s) ds $$. Where $f$ and $g$ are as smooth as ...
Birkhofff's user avatar
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37 votes
2 answers

Intuition of Gronwall lemma

The Gronwall lemma is a well known and very useful statement which is used in many situations, in particular in the theory of differential equations. I have seen it so many times and even the proof is ...
asd's user avatar
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5 votes
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Gronwall type inequality

Is there a Gronwall-type inequality for bounding $u(t)$ such that $$\vert \partial_t u(t)\vert\leq C \{ u(t)+u(t)^\alpha\}$$ with $\alpha>1$ ?
max's user avatar
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About the Gronwall inequality

If I have that $$||\eta_u(t)||\leq 1+C_1\int_0^t \frac{1}{||\eta(s)||}||\eta_u(s)||ds$$ and $$\sqrt{1-\frac{2\varepsilon}{C}}||u||\leq ||\eta(s)||\leq 2||u||$$ how to obtain using the Gronwall ...
Vrouvrou's user avatar
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1 answer

Reference Request: Weak ODEs and weak Gronwall inequality

During my research I came across a weak gronwall-type inequality of the following type: $$-\int_0^T f'(t)(u(t)-u_0) \leq \int_0^T f(t)u(t)$$ for non-negative $f\in C_c^\infty(0,T)$, $u\in L^1(0,T)$ ...
Quickbeam2k1's user avatar
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2 votes
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A Gronwall-type inequality

I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that. $$ f^2(t) \leqslant g^2(t) +...
Ann's user avatar
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