Difference between revisions of "Forward regressive"

Revision as of 22:09, 25 May 2014

Let $\mathbb{T}$ be a time scale. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$. We say that $p$ is regressive if for all $t \in \mathbb{T}^{\kappa}$ $$1+\mu(t)p(t)\neq 0.$$ We let $\mathcal{R}(X,Y)$ denote the set of regressive functions $p \colon X \rightarrow Y$. Let $p,q \in \mathcal{R}$ and define the "circle plus" operation $\oplus \colon \mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R}$ by the formula, for $t \in \mathbb{T}^{\kappa}$, $$(p \oplus q)(t) = p(t)+q(t)+\mu(t)p(t)q(t).$$ We define the inverse operation of $\oplus$ by the formula $(\ominus p)(t) = -\dfrac{p(t)}{1+\mu(t)p(t)}$. The ordered pair $(\mathcal{R},\oplus)$ is an Abelian group with subtraction $(p \ominus q)(t) = (p \oplus (\ominus q))(t) = \dfrac{p(t)-q(t)}{1+\mu(t)q(t)}$.

Related definitions

• The set of positively regressive functions is

$$\mathcal{R}^+(\mathbb{T},X)=\{p \in \mathcal{R} \colon \forall t \in \mathbb{T}, 1+\mu(t)p(t)>0 \}.$$

• The set of negatively regressive functions is

$$\mathcal{R}^-(\mathbb{T},X)=\{p \in \mathcal{R} \colon \forall t \in \mathbb{T}, 1+\mu(t)p(t)<0 \}.$$

• Consider the dynamic equation $y^{\Delta \Delta}(t)+p(t)y^{\Delta}(t)+q(t)y(t)=f(t)$. We say this equation is regressive if $,p,q,f \in C_{rd}$ and for all $t \in \mathbb{T}$

$$1-\mu(t)p(t)+\mu^2(t)q(t)\neq 0.$$