Difference between revisions of "Delta Gronwall inequality"

From timescalewiki
Jump to: navigation, search
 
(4 intermediate revisions by the same user not shown)
Line 1: Line 1:
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
__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 $t \in \mathbb{T}$
+
==Theorem==
 +
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 $t \in \mathbb{T}$.
<div class="mw-collapsible-content">
+
 
<strong>Proof:</strong> █
+
==Proof==
</div>
 
</div>
 
  
 
==References==
 
==References==
[http://www.math.unl.edu/~apeterson1/pub/ineq.pdf R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey]
+
{{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}}
 +
 
 +
[[Category:Theorem]]
 +
[[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