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Analysis of Impulsive $\varphi$--Hilfer Fractional Differential Equations

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Pith's one-line read Stability and data dependence proven for impulsive fractional equations

desk verdict The order-dependence theorem is broken; the rest is a routine extension of the authors' own prior work. read the letter →

arxiv 1908.07785 v1 pith:4ON6XCFY submitted 2019-08-21 math.DS math.AP

Kishor D. Kucche , Jyoti P. Kharade This is my paper · ORCID
classification math.DSmath.AP MSC 34A0834A1245M1034A37
keywords impulsivefractionaldifferentialequationsphi-HilferderivativeweightedpiecewisecontinuousfunctionsUlam-HyersstabilitydependenceondataordergeneralizedGronwallinequalityMittag-Lefflerfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that a broad class of impulsive fractional differential equations, those built from the $\varphi$-Hilfer derivative, has the qualitative properties a model needs: existence of solutions, continuous dependence on the initial condition, on the nonlinearities, and on the order of the derivative, and Ulam-Hyers type stability. The $\varphi$-Hilfer derivative contains Caputo, Riemann-Liouville, Hilfer, Hadamard, and Katugampola derivatives as special cases. The proofs work in the weighted space of piecewise continuous functions with norm $\sup_t |(\varphi(t)-\varphi(a))^{1-\sigma} u(t)|$, converting the differential problem into a fixed-point equation and then applying a generalized Gronwall inequality. A sympathetic reader would take the paper's thesis to be that a single Gronwall estimate, expressed through Mittag-Leffler functions, controls all of these properties uniformly across the entire $\varphi$-Hilfer family.

What carries the argument

The load-bearing object is the equivalent fractional integral representation of the solution (Lemma 2.4): a function $u$ solves the impulsive $\varphi$-Hilfer problem exactly when $u(t)=(\varphi(t)-\varphi(a))^{\sigma-1}\Gamma(\sigma)^{-1}(u_a+\sum_{a<t_k<t}J_k(u(t_k^-)))+I^{\rho;\varphi}_{a+}f(t,u(t))$. This identity turns the differential problem into a fixed-point equation for an operator on the weighted piecewise-continuous space, and it is the formula that every later estimate differentiates. The argument is carried by the generalized Gronwall inequality (Lemma 2.3), which takes an inequality of the form $U(t)\le V(t)+g(t)\int_a^t \varphi'(s)(\varphi(t)-\varphi(s))^{\rho-1}U(s)\,ds+\sum_{a<t_k<t}\beta_k U(t_k^-)$ and returns an explicit bound by products and single Mittag-Leffler functions $E_\rho(g(t)\Gamma(\rho)(\varphi(t)-\varphi(a))^\rho)$; existence, dependence, and stability results are all applications of that one estimate.

What would settle it

Check the admissibility of the Gronwall step in Theorem 4.5 with $a=0$, $\varphi(t)=t$, $\nu=0$, $\rho=0.8$, $\delta=0.5$, $f\equiv 0$, $J_k\equiv 0$, $u_a=v_a=1$. Then $u(t)=t^{\sigma-1}/\Gamma(\sigma)$ and $v(t)=t^{\sigma-\delta-1}/\Gamma(\sigma-\delta)$, so $U(t)=t^{1-\sigma}|u-v|$ behaves like $t^{-\delta}=t^{-0.5}$, while the lemma's hypothesis $U\in\mathrm{PC}_{1-\sigma;\varphi}$ requires $t^{1-\sigma}U(t)$ to be bounded; here $t^{1-\sigma}U(t)\sim t^{1-\sigma-\delta}=t^{-0.3}$, which diverges. If this admissibility fails, the proof of Theorem 4.5 is incomplete for $\nu<1$; if it can be repaired, the order-dependence claim survives.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the impulsive $\varphi$-Hilfer Cauchy problem (1.2) with initial value $I^{1-\sigma;\varphi}_{a+}u(a)=u_a$ is well behaved in the weighted space $\mathrm{PC}_{1-\sigma;\varphi}$: under Lipschitz hypotheses on $f$ and the impulse maps $J_k$, Schaefer's fixed point theorem gives at least one solution (Theorem 3.1); the generalized Gronwall lemma gives explicit bounds showing that the solution depends continuously on the initial condition (Theorem 4.1), on the right-hand-side functions (Theorem 4.3), and on the order of the derivative (Theorem 4.5); and the same Gronwall machinery yields Ulam-Hyers and Ulam-Hyers-Rassias stability (Theorems 5.2 and 5.4). The inequality (4.7) for order-dependence is the most detailed of these claims: it bounds $(\varphi(t)-\varphi(a))^{1-\sigma}|u(t)-v(t)|$ by the Mittag-Leffler product times a coefficient $B(t)$ that records the mismatch in initial data, impulse strengths, and the change $\delta$ in the derivative order.

Load-bearing premise

Everything about dependence on the derivative order rests on the assumption that the weighted difference $(\varphi(t)-\varphi(a))^{1-\sigma}|u(t)-v(t)|$ is finite and that the coefficient $B(t)$ in (4.8) is admissible in the Gronwall lemma, which is not established because $B(t)$ can contain factors $(\varphi(t)-\varphi(a))^{\delta(\nu-1)}$ that blow up as $t\to a$ when $\nu<1$.

Editorial extensions

If this is right

  • Small perturbations of the initial value $u_a$ move the entire weighted solution curve by a controlled amount, with the bound $|u_a-v_a|\Gamma(\sigma)^{-1}$ times the Mittag-Leffler product in (4.2).
  • Small perturbations of the nonlinearity $f$ and impulse maps $J_k$ produce linearly controlled changes, with separate $\varepsilon_f$ and $\varepsilon_J$ terms in (4.5).
  • Changing the derivative order $\rho$ to $\rho-\delta$ changes the solution by the amount bounded in (4.7); the bound records exactly how the singularity at the left endpoint $t=a$ depends on $\delta$ and the type parameter $\nu$.
  • Every solution covered by the theorem is Ulam-Hyers stable, with the explicit constant $C_{m,\rho}=A_{m,\rho}(m/\Gamma(\sigma)+(\varphi(T)-\varphi(a))^{1-\sigma+\rho}/\Gamma(\rho+1))$, and Ulam-Hyers-Rassias stable under the stated $\theta$-condition.
  • Because $\varphi$ is arbitrary increasing, the same results specialize to the Caputo, Riemann-Liouville, Hadamard, Hilfer, and Katugampola fractional derivatives, so the paper's claims cover a large existing literature at once.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The order-dependence theorem is the member of the package most sensitive to the weights: when $\nu<1$, the weight $(\varphi(t)-\varphi(a))^{1-\sigma}$ is more singular than the natural weight of the perturbed solution, so the coefficient $B(t)$ contains factors like $(\varphi(t)-\varphi(a))^{\delta(\nu-1)}$ that blow up as $t\to a$. A testable consequence is that order-continuity is likely to be m
  • The same machinery would support a theorem the paper does not state: continuous dependence on the defining function $\varphi$ itself, obtained by comparing the integral kernels $(\varphi(t)-\varphi(s))^{\rho-1}$ for two different $\varphi$'s; the Gronwall lemma is already tailored to that kernel.
  • The stability constant grows with the number of impulses $m$ and with the Mittag-Leffler factor $E_\rho((\varphi(T)-\varphi(a))^{1-\sigma+\rho})$, suggesting that the guaranteed stability margin degrades as the time horizon or the number of impulses grows. Whether that degradation is intrinsic or an artifact of the Gronwall bound could be checked numerically on the example in Section 6.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper studies an initial-value problem for a nonlinear impulsive φ-Hilfer fractional differential equation (1.2) in the weighted space PC_{1-σ;φ}. The authors prove an existence result via Schaefer's fixed point theorem, establish continuous dependence of solutions on the initial data, on the functions f and Jk, and on the order of the fractional derivative, and prove Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias stability. The proofs are based on an equivalent integral representation (Lemma 2.4) and a generalized Gronwall inequality (Lemma 2.3), both imported from earlier works, together with standard fixed point arguments.

Significance. If correct, the results would generalize several known existence and stability theorems to impulsive φ-Hilfer equations in a unified framework, and the order-dependence estimate would be a novel contribution. The paper is clearly organized and many estimates follow familiar patterns. However, the order-dependence theorem, which is one of the advertised main objectives, is not established: the proof compares solutions living in different weighted spaces and produces a bound that is singular near the initial point. In addition, the existence proof has gaps in the continuity and equicontinuity steps. The stability results may be correct, but their proofs depend on unproved imported lemmas. Given these issues, the contribution in its current form does not meet the standard required for publication.

major comments (4)
  1. [Section 4.3, Theorem 4.5] Theorem 4.5 is not established and, as stated, cannot hold for generic admissible data. The solution u of (1.2) lies in PC_{1-σ;φ}, while the solution v of (4.6) lies in PC_{1-σ*;φ} with σ* = σ + δ(ν-1) ≤ σ. Since (φ(t)-φ(a))^{1-σ}|v(t)| = (φ(t)-φ(a))^{σ*-σ}(φ(t)-φ(a))^{1-σ*}|v(t)| and σ*-σ = δ(ν-1) ≤ 0, the left side of (4.7) is not finite for generic admissible v when ν<1. The proof then applies Lemma 2.3 to U(t) = (φ(t)-φ(a))^{1-σ}|u(t)-v(t)|, but U is not known to belong to PC_{1-σ;φ}. Moreover, the quantity B(t) in (4.8) contains the factor (φ(t)-φ(a))^{δ(ν-1)} (written as δ(β-1)), which diverges as t→a^+, so the bound (4.7) cannot provide a finite estimate in the claimed norm. Thus the advertised continuous dependence on the order of the derivative is unsupported.
  2. [Section 3, Step 3 of Theorem 3.1] The equicontinuity step is not valid in the space PC_{1-σ;φ}. The proof estimates |(Fu)(t2)-(Fu)(t1)|, whereas the PC_{1-σ;φ} Arzelà-Ascoli theorem (Lemma 2.1) requires equicontinuity of the weighted functions (φ(t)-φ(a))^{1-σ}(Fu)(t). The displayed sum over a<tk<t2-t1 is not meaningful, because t2-t1 is not an endpoint of the partition, and the intended estimate over impulses lying between t1 and t2 is not supplied. Consequently the complete continuity of F is not established as written.
  3. [Section 3, Step 1 of Theorem 3.1] The continuity argument for F is incomplete: it invokes only pointwise continuity of f and Jk and does not use the Lipschitz hypothesis (H1)(ii). Pointwise convergence of |f(s,u_n(s))-f(s,u(s))| does not by itself imply convergence of the weighted fractional integral term (φ(t)-φ(a))^{1-σ} I^{ρ;φ}_{a+}(f(·,u_n)-f(·,u)) uniformly in t; a dominated-convergence estimate based on (H1)(ii) is needed. This gap is likely repairable, but the proof as written is not complete.
  4. [Section 2, Lemmas 2.3 and 2.4] The two central tools of the paper, the integral representation Lemma 2.4 and the impulsive Gronwall inequality Lemma 2.3, are imported from References [36] and [41], respectively, without proofs. Since [36] is an unpublished companion work by the same authors and the representation formula is known to be delicate for impulsive fractional problems (see the discussion of References [1,2]), the paper should either prove these lemmas or state them with the exact hypotheses needed for the present setting.
minor comments (5)
  1. [Equation (4.8) and proof of Theorem 4.5] The symbol β is used where ν is intended: σ* is defined as σ + δ(ν-1) in (4.6), but (4.8) and the surrounding proof repeatedly write δ(β-1) and Γ(σ+δ(β-1)). Please make the notation consistent.
  2. [Proofs of Theorems 4.1 and 4.3] The phrase 'as given in the proof of Theorem 5.2' is a forward reference; the Gronwall data U(t), g(t), and βk are first defined in the proof of Theorem 3.1, so that theorem should be cited instead.
  3. [Example 6.1] The displayed inequality for J1 contains the exponent '1−γ' which should be '1−σ'. Also, Theorem 3.1 establishes existence only, so the statement that problem (6.1) 'has a unique solution' should be justified by Theorem 4.1 or Remark 4.2, or rephrased as existence.
  4. [Lemma 2.3] The statement of Lemma 2.3 is somewhat imprecise: the product runs over i=1,...,k with t∈(t_k,t_{k+1}], but the summatory condition a<tk<t is written with the same symbol k; the index and the dependence of the product on the interval should be clarified, and the implicit nonnegativity assumption on U should be stated explicitly.
  5. [Throughout] There are numerous typos and grammatical errors, e.g., 'it is have to research' in the Introduction and 'Erdlyi-Kober' for 'Erdélyi-Kober'; the manuscript would benefit from a careful language revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's derivations are carried out with fixed-point and Gronwall arguments from imported parameter-free lemmas; the flagged Theorem 4.5 concern is a correctness gap, not a self-referential reduction.

full rationale

The paper's claimed results—existence via Schaefer's theorem, dependence on initial data, on the right-hand-side functions, on the order, and the Ulam-type stabilities—are all obtained by substituting the integral representations and applying the cited Gronwall inequality (Lemma 2.3) and fixed-point arguments. The most heavily used imported tool, Lemma 2.4, is a parameter-free equivalence statement from the authors' own prior work [36]; its assumptions do not include the estimates or stability conclusions of the present paper, so citing it is ordinary reliance on a prior theorem, not circularity. Lemmas 2.1 and 2.3 are likewise used as standard tools rather than as restatements of the paper's conclusions. None of the output inequalities (4.2), (4.5), (4.7), or the stability constants are fitted parameters or definitions disguised as predictions; each bound follows from the preceding integral estimates. The skeptical criticism of Theorem 4.5—that v is only shown to lie in PC_{1−σ*;φ} with σ* ≤ σ while the Gronwall lemma is applied to U(t)=(φ(t)−φ(a))^{1−σ}|u(t)−v(t)|, and that B(t) may contain singular or non-admissible factors—is a legitimate mathematical-rigor concern about an unproved regularity premise and the applicability of the Gronwall lemma, but it is not a circular step in the defined sense: the theorem's conclusion is not assumed as a hypothesis, and no equation is reused as its own input. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on imported analytical tools: a fixed point theorem, a weighted Arzela-Ascoli theorem, a generalized Gronwall inequality, and the integral representation formula. The first is standard; the last three come from prior papers, one of which is by the same authors ([36]). No new entities or fitted parameters are introduced.

assumptions (4)
  • standard math Schaefer fixed point theorem (Theorem 2.2)
    Used in Theorem 3.1 to prove existence of a fixed point for the operator F.
  • domain assumption PC_{1-sigma;phi} Arzela-Ascoli theorem (Lemma 2.1)
    Cited from [36]; used to prove the operator F is completely continuous.
  • domain assumption Generalized Gronwall inequality with impulses (Lemma 2.3)
    Cited from [41]; used for a priori bounds and stability estimates.
  • domain assumption Equivalent fractional integral representation for the impulsive phi-Hilfer Cauchy problem (Lemma 2.4)
    Cited from [36]; the whole analysis reduces the differential equation to this integral equation.

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Pith. "Pith review of Analysis of Impulsive $\varphi$--Hilfer Fractional Differential Equations." pith.science (2026). https://pith.science/paper/4ON6XCFY

@misc{pith2026190807785,
  author       = {Pith},
  title        = {Pith review of: Analysis of Impulsive $\varphi$--Hilfer Fractional Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ON6XCFY}},
  note         = {Machine review of arXiv:1908.07785}
}
abstract

This paper is concerned with the existence and uniqueness, and Ulam--Hyers stabilities of solutions of nonlinear impulsive $\varphi$--Hilfer fractional differential equations. Further, we investigate the dependence of the solution on the initial conditions, order of derivative and the functions involved in the equations. The outcomes are acquired in the space of weighted piecewise continuous functions by means of fixed point theorems and the generalized version of Gronwall inequality.

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