Difference between revisions of "Induction on time scales"

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(Created page with "Let $\mathbb{T}$ be a time scale. <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $t_0 \in \mathbb{T}$ and assume t...")
 
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Let $\mathbb{T}$ be a [[time scale]].
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==Theorem==
 
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Let $\mathbb{T}$ be a [[time scale]]. Let $t_0 \in \mathbb{T}$ and assume that $\{S(t) \colon t \in [t_0,\infty)\}$ is a family of statements satisfying the following axioms:
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> Let $t_0 \in \mathbb{T}$ and assume that $\{S(t) \colon t \in [t_0,\infty)\}$ is a family of statements satisfying the following axioms:
 
 
* the statement $S(t_0)$ is true
 
* the statement $S(t_0)$ is true
 
* if $t \in [t_0,\infty)$ is right-scattered and $S(t)$ is true, then $S(\sigma(t))$ is true
 
* if $t \in [t_0,\infty)$ is right-scattered and $S(t)$ is true, then $S(\sigma(t))$ is true
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* if $t \in (t_0, \infty)$ is left-dense and $S(s)$ is true for all $s \in [t_0,t)$, then $S(t)$ is true.
 
* if $t \in (t_0, \infty)$ is left-dense and $S(s)$ is true for all $s \in [t_0,t)$, then $S(t)$ is true.
 
Then $S(t)$ is true for all $t \in [t_0,\infty)$.
 
Then $S(t)$ is true for all $t \in [t_0,\infty)$.
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
=References=
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==Proof==
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==References==
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* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Forward jump|next=}}: Theorem 1.7
 
[http://web.mst.edu/~bohner/sample.pdf]
 
[http://web.mst.edu/~bohner/sample.pdf]
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[[Category:Theorem]]
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[[Category:Unproven]]

Revision as of 05:09, 10 June 2016

Theorem

Let $\mathbb{T}$ be a time scale. Let $t_0 \in \mathbb{T}$ and assume that $\{S(t) \colon t \in [t_0,\infty)\}$ is a family of statements satisfying the following axioms:

  • the statement $S(t_0)$ is true
  • if $t \in [t_0,\infty)$ is right-scattered and $S(t)$ is true, then $S(\sigma(t))$ is true
  • if $t \in [t_0,\infty)$ is right-dense and $S(t)$ is true, then there is a neighborhood $U$ of $t$ such that $S(s)$ is true for all $s \in U \cap (t,\infty)$
  • if $t \in (t_0, \infty)$ is left-dense and $S(s)$ is true for all $s \in [t_0,t)$, then $S(t)$ is true.

Then $S(t)$ is true for all $t \in [t_0,\infty)$.

Proof

References

[1]