Difference between revisions of "Delta Bihari 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> Suppose that $g$ is continuous and nondecreasing, $p$ is [[continuity | rd-continuous]] and nonnegative, and $y$ is rd-continuous. Let $w$ be the solution of  
+
==Theorem==
$$w^{\Delta}=p(t)g(w); w(a)=\beta$$
+
Suppose that $g$ is continuous and nondecreasing, $p$ is [[continuity | 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  
 
and suppose there is a bijective function $G$ with $(G \circ w)^{\Delta} = p$. Then  
 
$$y(t) \leq \beta + \displaystyle\int_a^t p(\tau)g(y(\tau)) \Delta \tau$$
 
$$y(t) \leq \beta + \displaystyle\int_a^t p(\tau)g(y(\tau)) \Delta \tau$$
Line 7: Line 8:
 
$$y(t) \leq G^{-1} \left[ G(\beta) + \displaystyle\int_a^t p(\tau) \Delta \tau \right]$$
 
$$y(t) \leq G^{-1} \left[ G(\beta) + \displaystyle\int_a^t p(\tau) \Delta \tau \right]$$
 
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 5.8
 +
 
 +
{{:Delta inequalities footer}}
 +
 
 +
[[Category:Theorem]]
 +
[[Category:Unproven]]

Latest revision as of 00:36, 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