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Nabla Discrete fractional Calculus and Nabla Inequalities
George A. Anastassiou
TL;DR
The paper addresses the development of discrete nabla fractional calculus, including a Caputo-like fractional difference and fractional Taylor formulas. It derives remainder estimates and related Opial, Ostrowski, Poincare, and Sobolev-type inequalities under stated discrete-domain and boundary assumptions.
Problem
The paper addresses the need for discrete nabla fractional-calculus constructions, including a Caputo-like fractional difference and discrete fractional Taylor formulas.
Method
The paper defines a Caputo-like discrete nabla fractional difference and develops discrete fractional Taylor representations with remainder estimates, then derives related fractional inequalities.
Results
The paper presents discrete fractional Taylor representations and derives fractional Opial, Ostrowski, Poincare, and Sobolev-type inequalities.
Takeaways & Limitations
The resulting framework connects discrete nabla fractional differences and Taylor representations with several discrete fractional inequality types.
Takeaways & Limitations
The stated Taylor representations are valid only on a discrete interval with t ∈ [a + m, b] and require a + m < b.
Abstract
from arXiv · showhide
Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincare and Sobolev type inequalities.
1 Introduction
The paper develops the discrete nabla fractional-calculus framework by defining factorials, integer and fractional sums, and a Caputo-like fractional difference. It builds on existing discrete Taylor formulas and related fractional-difference results.
- The introduction situates the work alongside prior definitions, exponent laws, and discrete Taylor formulas for functions on integer domains.
- The paper defines the rising factorial n = t(t + 1) ... (t + n − 1) for n ∈ N.
- Higher-order nabla differences are defined inductively from the first-order nabla operator.
- The paper defines integer-order and fractional-order nabla sums, using ρ(s) = s − 1 in the integer-order construction.
- For µ > 0 with m − 1 < µ < m, the paper introduces a Caputo-like discrete nabla fractional difference with ν = m − µ.
- A related prior definition uses the m-th order forward difference operator.
2 Main Results
The paper develops discrete backward fractional Taylor representations and derives discrete nabla fractional Opial, Ostrowski, Poincaré, Sobolev, and average Sobolev inequalities under stated order, domain, and boundary assumptions.
- Fractional Taylor formulas: The paper presents a discrete backward fractional Taylor formula for µ > 0, with Theorem 10 specifying its representation for t ≥ a + m.The formulation uses m = ⌈µ⌉ and considers p ∈ N with µ > p.
- Fractional Taylor formulas: Theorem 4 establishes a discrete fractional Taylor representation for t ≥ a + m, with related corollaries under vanishing nabla differences at a.The paper also states an extended Taylor formula and notes that the representations are restricted to t ∈ [a + m, b] when the function is defined on a finite discrete interval.
- Fractional inequalities: A discrete fractional Opial inequality is derived for µ > 2 under vanishing initial nabla differences and conditions on auxiliary functions C and D.The result includes bounds involving g(t)g(t − 1) and the corresponding initial terms for t ≥ a + m.
- Fractional inequalities: The paper next derives discrete nabla fractional Ostrowski and Poincaré type inequalities for noninteger µ on finite discrete intervals with a + m < b.Theorem 18 and Theorem 19 impose vanishing nabla differences at a over specified index ranges.
- Fractional inequalities: Discrete nabla Sobolev and average Sobolev type fractional inequalities are obtained for noninteger orders, including multiple fractional orders µ1 < ... < µk.The average Sobolev result assumes vanishing nabla differences through order mk − 1 and a + mk < b.