Nabla Dynamic Equations

If $$ \mathbb{T} $$ has a right-scattered minimum m, define $$ \mathbb{T}_\kappa : = \mathbb{T} - \{ m\} $$ ; otherwise, set $$ \mathbb{T}_\kappa = \mathbb{T} $$ . The backwards graininess $$ \nu :\mathbb{T}_\kappa \to \mathbb{R}_0^ + $$ is defined by $$ \nu (t) = t - \rho (t). $$ For $$ f:\mathbb{T} \to \mathbb{R} $$ and $$ t \in \mathbb{T}_\kappa $$ , define the nabla derivative [42] of f at t, denoted f ∇(t), to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t such that $$ |f(\rho (t)) - f(s) - f^\nabla (t)(\rho (t) - s)| \leqslant \varepsilon |\rho (t) - s) $$ for all s € U. For $$ \mathbb{T} = \mathbb{R} $$ , we have f ∇=f′, the usual derivative, and for $$ \mathbb{T} = \mathbb{Z} $$ we have the backward difference operator, f ∇(t)=∇f(t):=f(t)-f(t-1). Note that the nabla derivative is the alpha derivative when α = p. Many of the results in this chapter can be generalized to the alpha derivative case. Many of the results in this chapter can be found in [35, 37].