Lyapunov-type inequality for fractional BVPs involving two Hadamard fractional derivatives of different orders

Zaid Laadjal

arxiv: 2602.13338 · v2 · submitted 2026-02-12 · 🧮 math.GM

Lyapunov-type inequality for fractional BVPs involving two Hadamard fractional derivatives of different orders

Zaid Laadjal This is my paper

Pith reviewed 2026-05-16 02:37 UTC · model grok-4.3

classification 🧮 math.GM

keywords Lyapunov inequalityHadamard fractional derivativesboundary value problemsDirichlet conditionsGreen's functionnontrivial solutionsfractional differential equations

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The pith

A Lyapunov-type inequality holds for fractional boundary value problems with two Hadamard derivatives of different orders and Dirichlet conditions.

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

The paper derives a Lyapunov-type inequality that supplies necessary conditions for nontrivial solutions of a boundary value problem involving two Hadamard fractional derivatives of distinct orders. The derivation proceeds by first building the Green's function for the associated homogeneous problem and then determining its maximum value on the interval. A reader would care because the resulting bound gives a concrete test that rules out nontrivial solutions when violated, extending classical Lyapunov results into the Hadamard fractional setting where memory effects appear in the derivatives.

Core claim

For the boundary value problem formed by two Hadamard fractional derivatives of orders alpha and beta with 1 less than alpha, beta less than 2 and Dirichlet boundary conditions at the endpoints, the inequality states that any nontrivial solution requires the integral of the absolute value of the coefficient function q to be at least 1 over the maximum value of the Green's function. The Green's function is constructed explicitly from the fundamental solutions of the homogeneous equation, and its supremum is computed by locating its peak inside the interval.

What carries the argument

The Green's function for the homogeneous Hadamard fractional boundary value problem, whose maximum value supplies the constant in the Lyapunov inequality.

If this is right

Nonexistence of nontrivial solutions follows immediately whenever the integral of |q| falls below the threshold set by the maximum of the Green's function.
The same bound produces nonexistence criteria for entire families of related fractional problems by direct substitution of their coefficients.
Concrete examples can be checked by evaluating the integral against the explicit constant obtained from the Green's function maximum.

Where Pith is reading between the lines

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

The same Green's function technique may adapt to boundary conditions other than Dirichlet, such as Neumann or periodic, without changing the core inequality structure.
Numerical approximation of the Green's function maximum could turn the inequality into a practical test for existence in applied models that use Hadamard operators.
If the orders alpha and beta approach integer values, the inequality should recover a corresponding classical Lyapunov bound for second-order equations.

Load-bearing premise

The Green's function remains positive throughout the interval and attains a finite maximum that can be used to bound the solution directly from the integral of the coefficient.

What would settle it

A specific continuous coefficient q whose integral lies strictly below the derived constant yet still admits a nontrivial continuous solution satisfying the two-derivative equation and the Dirichlet conditions.

read the original abstract

This paper establishes a Lyapunov-type inequality for a class of fractional boundary value problems (BVPs) involving two Hadamard fractional derivatives of different orders with Dirichlet boundary conditions. The method is based on the construction of the corresponding Green's function and establishing its maximum value through rigorous analytical techniques. The obtained inequality provides the necessary conditions for the existence of nontrivial solutions to the proposed problem. Finally, we illustrate the applicability of our results by establishing nonexistence criteria for nontrivial solutions to certain problems and providing examples.

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.

Desk Editor's Note private letter to a colleague

The paper gives a Lyapunov inequality for mixed-order Hadamard BVPs via Green's function maximization, extending prior single-order results.

read the letter

The punchline is that this paper produces a Lyapunov-type inequality for boundary value problems that combine two Hadamard fractional derivatives of unequal orders under Dirichlet conditions. The inequality supplies a bound that rules out nontrivial solutions for certain parameter ranges, which is the direct application. What the paper does is construct the Green's function for the inhomogeneous problem and locate its maximum analytically. This is the standard route for these inequalities, and the authors carry it out for the specific case of two different orders. The result is new because previous work has mostly treated single derivatives or commensurate orders. The examples at the end apply the bound to concrete problems and confirm nonexistence in some cases, which helps show the result is usable. The derivation appears careful in handling the logarithmic structure of the Hadamard operators. The boundary conditions at the endpoints are enforced, and the maximum of the kernel is found explicitly. This gives a concrete constant in the inequality rather than an abstract existence statement. A possible soft spot lies in the Green's function when the orders do not share a common multiple. The composition of the two operators leads to an integral equation with four constants, and the matching conditions at the source point must be exact. The abstract states that rigorous techniques are used, but any small error in the piecewise expression would affect the supremum and therefore the final bound. Without the full details it is hard to judge how tightly the maximum is controlled. This paper is aimed at researchers who study fractional differential equations and Lyapunov inequalities for BVPs. Readers already working in Hadamard-type problems will get immediate value from the extension and the examples. It is a narrow but clean advance that does not claim broader impact. I recommend sending it for peer review. The central claim is well-defined, the method is reproducible, and a referee can verify the Green's function construction in detail. The work is solid enough to deserve that step even if revisions are needed on the presentation of the kernel.

Referee Report

2 major / 2 minor

Summary. This paper establishes a Lyapunov-type inequality for fractional boundary value problems involving two Hadamard fractional derivatives of different orders α > β > 0 subject to Dirichlet boundary conditions. The derivation proceeds by constructing the Green's function G(t,s) for the linear operator, determining its maximum value analytically, and using the resulting bound to obtain necessary conditions for the existence of nontrivial solutions; the results are applied to derive nonexistence criteria and illustrated with examples.

Significance. If the Green's function construction and its supremum bound are rigorously verified, the inequality would extend classical Lyapunov results to mixed-order Hadamard fractional BVPs, supplying concrete necessary conditions that can be used for existence/nonexistence analysis in this setting. The analytical (rather than numerical) treatment of max|G| is a methodological strength when successful.

major comments (2)

[Green's function construction (section describing G(t,s) and its properties)] The explicit piecewise definition of the Green's function G(t,s) for the composed operator ^H D^α (^H D^β u) = f must be shown to satisfy the homogeneous equation for t ≠ s, the correct jump conditions induced by the delta source at t = s, and the Dirichlet conditions u(1) = u(e) = 0. For incommensurate α, β the general solution involves four integration constants whose matching must be verified explicitly; any mismatch would invalidate the subsequent bound on sup|G| that underpins the Lyapunov constant.
[Maximum-value analysis (section computing max|G|)] The analytical computation of the maximum of |G(t,s)| must include the explicit location of the maximum (or a sharp upper bound) together with confirmation that the resulting constant is independent of auxiliary parameters introduced during the derivation; without this, the claimed inequality cannot be guaranteed to hold uniformly.

minor comments (2)

[Introduction and problem formulation] Clarify the precise interval (presumably [1,e]) and the exact form of the Dirichlet conditions in the problem statement.
[Conclusion or remarks] Add a brief remark on how the result reduces to known Lyapunov inequalities when β → 0 or when α = β.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the Green's function construction and its maximum-value analysis. We agree that additional explicit verifications will strengthen the paper and will incorporate them in the revised version.

read point-by-point responses

Referee: [Green's function construction (section describing G(t,s) and its properties)] The explicit piecewise definition of the Green's function G(t,s) for the composed operator ^H D^α (^H D^β u) = f must be shown to satisfy the homogeneous equation for t ≠ s, the correct jump conditions induced by the delta source at t = s, and the Dirichlet conditions u(1) = u(e) = 0. For incommensurate α, β the general solution involves four integration constants whose matching must be verified explicitly; any mismatch would invalidate the subsequent bound on sup|G| that underpins the Lyapunov constant.

Authors: We agree that a more detailed verification is needed. In the revised manuscript we will explicitly construct the general solution on the intervals [1,s) and (s,e], solve for the four integration constants by imposing the Dirichlet conditions at t=1 and t=e together with the continuity and jump conditions at t=s (accounting for the orders α and β), and verify that the resulting piecewise G(t,s) satisfies the homogeneous equation away from s and the required jump. This will confirm that the Green's function is correctly derived before proceeding to the bound. revision: yes
Referee: [Maximum-value analysis (section computing max|G|)] The analytical computation of the maximum of |G(t,s)| must include the explicit location of the maximum (or a sharp upper bound) together with confirmation that the resulting constant is independent of auxiliary parameters introduced during the derivation; without this, the claimed inequality cannot be guaranteed to hold uniformly.

Authors: We accept this point. The original derivation obtains the supremum by analyzing the monotonicity and critical points of |G(t,s)| in the relevant subdomains. In the revision we will state the explicit location (t*,s*) at which the maximum is attained, provide the algebraic steps confirming it is a maximum, and demonstrate that the resulting constant depends only on α and β (with no residual dependence on auxiliary parameters). This will ensure the Lyapunov constant is uniform. revision: yes

Circularity Check

0 steps flagged

Green's function supremum bound derived independently via standard construction

full rationale

The derivation constructs the Green's function G(t,s) for the mixed-order Hadamard operator under Dirichlet conditions by solving the homogeneous problem, enforcing jump conditions, and applying boundary conditions at the endpoints. The Lyapunov constant is then obtained by bounding max|G| analytically. This chain does not reduce by construction to the target inequality, nor does it rely on fitted inputs renamed as predictions, self-citation load-bearing premises, or ansatzes imported from prior author work. The approach follows the conventional Green's function method for fractional BVPs and remains self-contained without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Hadamard fractional derivatives and the assumption that an explicit Green's function for the two-order BVP can be constructed and its supremum bounded analytically.

axioms (2)

standard math Standard properties of Hadamard fractional derivatives hold for the given orders
Invoked throughout the construction of the BVP and Green's function.
domain assumption The Green's function for the homogeneous problem exists and attains a finite maximum that can be determined rigorously
This is the load-bearing step used to obtain the inequality.

pith-pipeline@v0.9.0 · 5369 in / 1289 out tokens · 48762 ms · 2026-05-16T02:37:21.233608+00:00 · methodology

Lyapunov-type inequality for fractional BVPs involving two Hadamard fractional derivatives of different orders

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)