I am new on difference equations
I am strugling in the definition of the Cauchy function and how to use it to find the Green's function for a difference equations. For example: for the deference operator $$\Delta^2 y(t)=0$$ the cauchy function is given by y(t,s)=t-s
The problem is that in application they always work with $$-\Delta^2 y(t)=0$$ so how we can find this cauchy function if in self adjoint operator $$Ly(t)= \Delta(p(t-1)\Delta y(t-1))+q(t)y(t)$$p(t) is supposed to be positive.
definition Cauchy function : The Cauchy function $y(t,s)$ for difference equations, defined for $a\leq t\leq b+2,\; a+1\leq s\leq b+1,$ is defined as the function that, for each fixed $s\in [a+1,b+1]=\{a+1,a+2,..., b+1\}$, is the solution of the initial value problem $$Ly(t)=0$$ $$y(s,s)=0$$ $$y(s+1,s)=\frac{1}{p(s)}$$ Where $Ly(t)= \Delta(p(t-1)\Delta y(t-1))+q(t)y(t)$ is the self adjoint operator
