Difference between revisions of "Induction on time scales"
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| − | Let $\mathbb{T}$ be a [[time scale]]. | + | ==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: | |
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* 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 $t \in [t_0,\infty)$. | ||
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| − | =References= | + | ==Proof== |
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| + | ==References== | ||
| + | * {{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]] | ||
| + | [[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
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous): Theorem 1.7
