# Delta Gronwall inequality

From timescalewiki

## 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 Agarwal, Martin 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 |