Difference between revisions of "Delta Bihari inequality"
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| − | + | __NOTOC__ | |
| − | + | ==Theorem== | |
| − | $$w^{\Delta}=p(t)g(w) | + | Suppose that $g$ is continuous and nondecreasing, $p$ is [[continuity | rd-continuous]] and nonnegative, and $y$ is rd-continuous. Let $w$ be the solution of |
| + | $$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$$ | ||
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$$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]$$ | ||
for all $t \in \mathbb{T}$. | for all $t \in \mathbb{T}$. | ||
| − | + | ||
| − | + | ==Proof== | |
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==References== | ==References== | ||
Revision as of 00:02, 15 September 2016
Theorem
Suppose that $g$ is continuous and nondecreasing, $p$ is rd-continuous and nonnegative, and $y$ is rd-continuous. Let $w$ be the solution of $$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
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 |
