Nabla Dynamic Equations

Douglas Anderson (), John Bullock (), Lynn Erbe (), Allan Peterson () and HoaiNam Tran ()
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Douglas Anderson: Concordia College, Department of Mathematics and Computer Science
John Bullock: Concordia College, Department of Mathematics and Computer Science
Lynn Erbe: University of Nebraska-Lincoln, Department of Mathematics and Statistics
Allan Peterson: University of Nebraska-Lincoln, Department of Mathematics and Statistics
HoaiNam Tran: University of Nebraska-Lincoln, Department of Mathematics and Statistics

Chapter Chapter 3 in Advances in Dynamic Equations on Time Scales, 2003, pp 47-83 from Springer

Abstract: Abstract 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].

Keywords: Prove Theorem; Adjoint Equation; Semigroup Property; Cauchy Function; Constant Formula (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-0-8176-8230-9_3

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