Difference between revisions of "Cauchy function"
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Let $\mathbb{T}$ be a [[time scale]]. Consider the [[self-adjoint]] equation $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. We say that a function $\hat{y} \colon \mathbb{T} \times \mathbb{T}^{\kappa^2} \rightarrow \mathbb{C}$ is a Cauchy function for the self-adjoint equation if for each fixed $s \in \mathbb{T}^{\kappa^2}$ the function $\hat{y}(\cdot,s)$ is a solution of the initial value problem | Let $\mathbb{T}$ be a [[time scale]]. Consider the [[self-adjoint]] equation $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. We say that a function $\hat{y} \colon \mathbb{T} \times \mathbb{T}^{\kappa^2} \rightarrow \mathbb{C}$ is a Cauchy function for the self-adjoint equation if for each fixed $s \in \mathbb{T}^{\kappa^2}$ the function $\hat{y}(\cdot,s)$ is a solution of the initial value problem | ||
| − | $$L\hat{y}(\cdot,s)=0; \hat{y}(\sigma(s),s), \hat{y}^{\Delta}(\sigma(s),s)=\dfrac{1}{p(\sigma(s)}.$$ | + | $$L\hat{y}(\cdot,s)=0; \hat{y}(\sigma(s),s), \hat{y}^{\Delta}(\sigma(s),s)=\dfrac{1}{p(\sigma(s))}.$$ |
Revision as of 21:01, 27 June 2015
Let $\mathbb{T}$ be a time scale. Consider the self-adjoint equation $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. We say that a function $\hat{y} \colon \mathbb{T} \times \mathbb{T}^{\kappa^2} \rightarrow \mathbb{C}$ is a Cauchy function for the self-adjoint equation if for each fixed $s \in \mathbb{T}^{\kappa^2}$ the function $\hat{y}(\cdot,s)$ is a solution of the initial value problem $$L\hat{y}(\cdot,s)=0; \hat{y}(\sigma(s),s), \hat{y}^{\Delta}(\sigma(s),s)=\dfrac{1}{p(\sigma(s))}.$$
