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arxiv: 1907.08847 · v1 · pith:ASJ75Q4Enew · submitted 2019-07-20 · 🧮 math.CA

Lyapunov Inequalities for Nabla Caputo Boundary Value Problems

Areeba Ikram This is my paper

Pith reviewed 2026-05-24 18:33 UTC · model grok-4.3

classification 🧮 math.CA
keywords nabla Caputofractional differenceboundary value problemsGreen's functionLyapunov inequalitiesuniqueness of solutions
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The pith

Boundary value problems for nabla Caputo fractional differences have unique solutions under two-point conditions, with explicit Green's functions enabling Lyapunov inequalities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to prove that solutions to boundary value problems involving the nabla Caputo fractional difference are unique when subject to two-point boundary conditions. It derives explicit expressions for the Green's functions of these problems. These Green's functions are then applied to obtain Lyapunov inequalities for specific instances of the boundary value problems. A reader would care because such inequalities can help determine the existence of solutions or bound the behavior of fractional difference equations in discrete settings.

Core claim

The central claim is that uniqueness of solutions holds for the nabla Caputo fractional difference boundary value problems under two-point boundary conditions, and that explicit Green's functions can be constructed, which in turn allow the development of Lyapunov inequalities for these problems.

What carries the argument

The Green's function for the nabla Caputo fractional difference operator subject to two-point boundary conditions, which encodes the solution operator and enables the derivation of inequalities.

If this is right

  • Unique solutions exist for the considered nabla Caputo BVPs.
  • Explicit Green's functions provide a way to represent solutions to the inhomogeneous problems.
  • Lyapunov inequalities can be formulated for specific nabla Caputo boundary value problems.
  • These results extend classical Lyapunov theory to fractional difference equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar techniques might apply to other types of fractional difference operators or boundary conditions.
  • The Lyapunov inequalities could be used to study stability in discrete fractional systems.
  • Testing the inequalities on simple cases like integer order differences could verify consistency with known results.

Load-bearing premise

The nabla Caputo fractional difference operator admits an explicit Green's function representation under the two-point boundary conditions considered.

What would settle it

A counterexample consisting of a specific nabla Caputo BVP with two-point conditions where either multiple solutions exist or the constructed Green's function fails to satisfy the equation would disprove the claims.

read the original abstract

We will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper establishes uniqueness of solutions to boundary value problems for the nabla Caputo fractional difference operator subject to two-point boundary conditions. It derives explicit expressions for the Green's functions associated with these problems and subsequently uses them to obtain Lyapunov inequalities for specific cases of such boundary value problems.

Significance. Assuming the derivations are correct, this contributes to the literature on fractional difference equations by providing concrete tools for uniqueness and inequality estimates. The explicit Green's function approach strengthens the results by allowing direct application to Lyapunov-type problems. This is particularly useful in the field of discrete fractional calculus where such explicit forms are often sought after.

minor comments (1)
  1. [Abstract] The abstract is written in future tense ('We will establish', 'we will then develop'), which is atypical for published work; revise to present tense to reflect the completed nature of the research.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained

full rationale

The paper derives uniqueness of solutions and explicit Green's functions for nabla Caputo fractional difference BVPs under two-point boundary conditions directly from the operator definition, then extracts Lyapunov inequalities from those Green's functions. No steps reduce by construction to fitted parameters, self-citations, or ansatzes; the chain is a standard explicit construction in fractional difference theory and remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the established theory of the nabla Caputo operator and Green's function methods for boundary value problems; no new free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The nabla Caputo fractional difference operator satisfies the standard properties required to construct Green's functions for two-point boundary value problems.
    This property is invoked to establish uniqueness and to obtain the explicit Green's function expressions.

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Reference graph

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