Difference between revisions of "Delta Gronwall inequality"

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__NOTOC__
<strong>Theorem:</strong> Let $y$ and $f$ be [[continuity | rd-continuous]] and $p$ be [[regressive_function | positively regressive]] and $p \geq 0$. If for all $t \in \mathbb{T}$
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==Theorem==
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Let $y$ and $f$ be [[continuity | rd-continuous]] and $p$ be [[regressive_function | positively regressive]] and $p \geq 0$. If for all $t \in \mathbb{T}$
 
$$y(t) \leq f(t) + \displaystyle\int_a^t y(\tau) p(\tau) \Delta \tau,$$
 
$$y(t) \leq f(t) + \displaystyle\int_a^t y(\tau) p(\tau) \Delta \tau,$$
 
then
 
then
 
$$y(t) \leq f(t) + \displaystyle\int_a^t e_p(t,\sigma(\tau))f(\tau)p(\tau)\Delta \tau$$
 
$$y(t) \leq f(t) + \displaystyle\int_a^t e_p(t,\sigma(\tau))f(\tau)p(\tau)\Delta \tau$$
 
for all $t \in \mathbb{T}$.
 
for all $t \in \mathbb{T}$.
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<strong>Proof:</strong> █
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==Proof==
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==References==
 
==References==
[http://www.math.unl.edu/~apeterson1/pub/ineq.pdf R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey]
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{{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=findme|next=findme}}: Theorem 5.6
  
 
{{:Delta inequalities footer}}
 
{{:Delta inequalities footer}}
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 00:36, 15 September 2016

Theorem

Let $y$ and $f$ be rd-continuous and $p$ be positively regressive and $p \geq 0$. If for all $t \in \mathbb{T}$ $$y(t) \leq f(t) + \displaystyle\int_a^t y(\tau) p(\tau) \Delta \tau,$$ then $$y(t) \leq f(t) + \displaystyle\int_a^t e_p(t,\sigma(\tau))f(\tau)p(\tau)\Delta \tau$$ for all $t \in \mathbb{T}$.

Proof

References

Ravi AgarwalMartin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 5.6

$\Delta$-Inequalities

Bernoulli Bihari Cauchy-Schwarz Gronwall Hölder Jensen Lyapunov Markov Minkowski Opial Tschebycheff Wirtinger