Difference between revisions of "Delta Bihari inequality"

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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 5.8
  
 
{{:Delta inequalities footer}}
 
{{:Delta inequalities footer}}

Revision as of 00:34, 15 September 2016

Theorem

Suppose that $g$ is continuous and nondecreasing, $p$ is rd-continuous and nonnegative, and $y$ is rd-continuous. Let $w$ be the solution of $$w^{\Delta}=p(t)g(w), \quad w(a)=\beta$$ and suppose there is a bijective function $G$ with $(G \circ w)^{\Delta} = p$. Then $$y(t) \leq \beta + \displaystyle\int_a^t p(\tau)g(y(\tau)) \Delta \tau$$ for all $t \in \mathbb{T}$ implies $$y(t) \leq G^{-1} \left[ G(\beta) + \displaystyle\int_a^t p(\tau) \Delta \tau \right]$$ for all $t \in \mathbb{T}$.

Proof

References

Ravi AgarwalMartin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 5.8

$\Delta$-Inequalities

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