Let $\mathbb{T}$ be a time scale. A second order dynamic equation of the form $$(p(t)y^{\Delta}(t))^{\Delta}+q(t)y(\sigma(t))=0$$ is called a self-adjoint second order dynamic equation.

Properties

Theorem: If $a,b \colon \mathbb{T} \rightarrow \mathbb{C}$ are rd continuous and $1-a(t)\mu(t)+b(t)\mu^2(t)\neq 0$ for all $t \in \mathbb{T}^{\kappa^2}$, then the second order dynamic equation $$y^{\Delta \Delta}(t) + a(t)y^{\Delta}(t) + b(t)y(t)=0$$ can be written in self-adjoint form if we choose $p(t)=e_{\alpha}(t,t_0)$ where $$\alpha(t)=\dfrac{a(t)-\mu(t)b(t)}{1-a(t)\mu(t)+b(t)\mu^2(t)}$$ and if we choose $q(t)=e_{\alpha}(\sigma(t),t_0)b(t)=(1+\mu(t)\alpha(t))e_{\alpha}(t,t_0)$.

Proof:

Theorem: If $a$ is a regressive function, then the second order dynamic equation $$y^{\Delta \Delta}(t)+a(t) y^{\Delta}(\sigma(t))+b(t)y(\sigma(t))=0$$ can be written in self-adjoint form if we choose $p(t)=e_a(t,t_0)$ and $q(t)=b(t)p(t)$.

Proof:

Theorem: (Variation of constants) Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$ be a rd continuous function. Let $\hat{y}(t,s)$ be the Cauchy function for $(py^{\Delta})+qy^{\sigma}=0$. Then $$y(t) = \displaystyle\int_a^t \hat{y}(t,s)f(s) \Delta s$$ is the solution of the initial value problem $$Ly=f(t);y(a)=0,y^{\Delta}(a)=0.$$

Proof:

Theorem: (Comparison theorem for IVP's) Assume the Cauchy function $\hat{y}$ for $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$ satisfies $\hat{y}(t,s) \geq 0$ for $t \geq \sigma(s)$. If $u,v$ are functions satisfying $L u(t) \geq Lv(t)$ for all $t \in [a,b] \cap \mathbb{T}$, $u(a)=v(a)$, and $u^{\Delta}(a)=v^{\Delta}(a)$, then $u(t) \geq v(t)$ for all $t \in [a,\sigma^2(b)]\cap \mathbb{T}$.

Proof:

Theorem (Wintner's theorem): Consider the self-adjoint equation $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. Assume that $\sup \mathbb{T}=\infty, a \in \mathbb{T}, \mu(t) \geq K > 0$ and $0 < p(t) \leq M$ for all $t \in [a,\infty)$, and $\displaystyle\int_a^{\infty} q(t) \Delta t = \infty.$ Then the self-adjoint equation is oscillatory on $[a, \infty)$.

Proof:

Theorem (Leighton-Wintner theorem): Consider the self-adjoint equation $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. Assume $a \in \mathbb{T}, p>0, \sup \mathbb{T}=\infty,$ and $$\displaystyle\int_a^{\infty} \dfrac{1}{p(t)} \Delta t = \displaystyle\int_a^{\infty} q(t) \Delta t = \infty.$$ Then the self-adjoint equation is oscillatory on $[a,\infty)$.

Proof: