Cauchy function

Let $\mathbb{T}$ be a time scale. Consider the self-adjoint operator $Ly=(py^{\Delta})^{\Delta}+qy^{\sigma}$. 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 $$L\hat{y}(\cdot,s)=0; \hat{y}(\sigma(s),s)=0, \hat{y}^{\Delta}(\sigma(s),s)=\dfrac{1}{p(\sigma(s))}.$$