Difference between revisions of "Delta Lyapunov inequality"
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==Theorem== | ==Theorem== | ||
Let $p \colon \mathbb{T} \rightarrow \mathbb{R}^+$ be positive-valued and [[continuity | rd-continuous]]. If the [[Sturm-Liouville dynamic equation]] | Let $p \colon \mathbb{T} \rightarrow \mathbb{R}^+$ be positive-valued and [[continuity | rd-continuous]]. If the [[Sturm-Liouville dynamic equation]] | ||
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has a nontrivial solution $x$ with $x(a)=x(b)=0$, then the Lyapunov inequality | has a nontrivial solution $x$ with $x(a)=x(b)=0$, then the Lyapunov inequality | ||
$$\displaystyle\int_a^b p(t) \Delta t \geq \dfrac{b-a}{f(d)}$$ | $$\displaystyle\int_a^b p(t) \Delta t \geq \dfrac{b-a}{f(d)}$$ | ||
| − | holds, where $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined by | + | holds, where $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined by $f(t) = (t-a)(b-t)$ and $d \in \mathbb{T}$ is such that |
| − | |||
| − | and $d \in \mathbb{T}$ is such that | ||
$$\left| \dfrac{a+b}{2} - d \right| = \min \left\{ \left| \dfrac{a+b}{2} -s \right| \colon s \in [a,b] \cap \mathbb{T} \right\}.$$ | $$\left| \dfrac{a+b}{2} - d \right| = \min \left\{ \left| \dfrac{a+b}{2} -s \right| \colon s \in [a,b] \cap \mathbb{T} \right\}.$$ | ||
Latest revision as of 14:29, 22 January 2023
Theorem
Let $p \colon \mathbb{T} \rightarrow \mathbb{R}^+$ be positive-valued and rd-continuous. If the Sturm-Liouville dynamic equation $$x^{\Delta^2} + p(t) x^{\sigma} = 0$$ has a nontrivial solution $x$ with $x(a)=x(b)=0$, then the Lyapunov inequality $$\displaystyle\int_a^b p(t) \Delta t \geq \dfrac{b-a}{f(d)}$$ holds, where $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined by $f(t) = (t-a)(b-t)$ and $d \in \mathbb{T}$ is such that $$\left| \dfrac{a+b}{2} - d \right| = \min \left\{ \left| \dfrac{a+b}{2} -s \right| \colon s \in [a,b] \cap \mathbb{T} \right\}.$$
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 |
