Difference between revisions of "Delta Gronwall inequality"
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| − | + | __NOTOC__ | |
| − | + | ==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 $t \in \mathbb{T}$. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | |||
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==References== | ==References== | ||
Revision as of 00:01, 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
R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
