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REVIEW 3 major objections 5 minor 28 references

On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes unique mild solutions for a nonlinear impulsive Volterra-Fredholm integrodifferential equation and derives data-dependence bounds from a new mixed Gronwall-type integral inequality.

desk verdict Theorem 5.1, the paper's new mixed Volterra-Fredholm integral inequality, is false; the existence-uniqueness and Picard sections are sound, but the advertised removal of smallness restrictions is unsupported. read the letter →

arxiv 1908.09545 v1 pith:X7TTL2BW submitted 2019-08-26 math.CA

Pallavi U. Shikhare , Kishor D. Kucche , J. Vanterler da C. Sousa This is my paper · ORCID
classification math.CA MSC 26D1034K3034A1247D6034K45
keywords Volterra-Fredholmintegrodifferentialequationsimpulsivedifferentialintegralinequalitypiecewisecontinuousfunctionsepsilon-approximatesolutionsdatadependencefixed-pointoperatorsBanachspaces
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

The paper's central claim is Theorem 5.1, a Gronwall-type integral inequality for piecewise continuous functions that includes both Volterra and Fredholm kernels in the same estimate. Around that inequality the paper builds existence and uniqueness results for nonlinear impulsive Volterra-Fredholm integrodifferential equations in Banach spaces, together with two data-dependence routes: fixed-point operators and the new inequality. The inequality route is intended to deliver continuous dependence and uniqueness without the smallness condition that the contraction argument requires.

What carries the argument

The load-bearing object is the auxiliary function $V(\tau)$ introduced in the proof of Theorem 5.1: the right-hand side of (5.1), which by construction dominates $u$ and is nondecreasing. The proof replaces $u$ by $V$ under all integrals, uses monotonicity of $V$ and of the kernels to estimate the inner integrals, and then applies a known piecewise Gronwall-type comparison supplied by Lemma 2.2 of [25] to the resulting Volterra inequality for $V$. This conversion from an inequality with the unknown $u$ on both sides to a closed exponential bound for $u$ is the mechanism that carries every later data-dependence estimate.

What would settle it

Take the interval end $b=1$, set the kernel $b(\cdot,\cdot)=0$, $k_1=0$, $\beta_k=0$, and $k_2(\tau,\sigma,\varsigma)=c>0$. Define $u(\tau)=1+c\tau\int_0^1 u(s)\,ds$; then (5.1) holds as an equality and $u(\tau)=1+\frac{c\tau}{1-c/2}$. At $\tau=1$, the claimed bound (5.2) is $e^c$, while $u(1)=\frac{1+c/2}{1-c/2}$. For $c=0.1$, $u(1)\approx1.10526$ exceeds $e^{0.1}\approx1.10517$, which would settle the theorem's conclusion by a direct calculation.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 5.1: if a nonnegative piecewise continuous function $u$ on $[0,b]$ satisfies $$u(\tau)\le a(\tau)+\int_0^\tau b(\tau,\$\sigma$)u(\$\sigma$)\,d\$\sigma$+\int_0^\tau\int_0^\$\sigma$ k_1(\tau,\$\sigma$,\varsigma)u(\varsigma)\,d\varsigma\,d\$\sigma$+\int_0^\tau\int_0^b k_2(\tau,\$\sigma$,\varsigma)u(\varsigma)\,d\varsigma\,d\$\sigma$+\sum_{0<\tau_k<\tau}\beta_k(\tau)u(\tau_k),$$ with $a$ and $\beta_k$ nondecreasing and the kernels nonnegative and nondecreasing in $\tau$, then $$u(\tau)\le a(\tau)\prod_{0<\tau_k<\tau}(1+\beta_k(\tau))\exp\left(\int_0^\tau b(\tau,\$\sigma$)\,d\$\sigma$+\int_0^\tau\int_0^\$\sigma$ k_1(\tau,\$\sigma$,\varsigma)\,d\varsigma\,d\$\sigma$+\int_0^\tau\int_0^b k_2(\tau,\$\sigma$,\varsigma)\,d\varsigma\,d\$\sigma$\right).$$ The authors apply this inequality to the semigroup solution formula and obtain the data-dependence estimate (6.1) and the epsilon-approximate comparison (6.3), with uniqueness and continuous dependence as corollaries.

Load-bearing premise

The load-bearing premise in the proof of Theorem 5.1 is that the function $V(\tau)$ (the whole right-hand side of the starting inequality) satisfies $V(\varsigma)\le V(\sigma)$ for every $\varsigma$ in $[0,b]$; the stated hypotheses only make $V$ nondecreasing, so this comparison is unavailable for $\varsigma>\sigma$.

Editorial extensions

If this is right

  • If Theorem 5.1 holds, continuous dependence on the initial condition and on the nonlinearities follows without any smallness condition on the Lipschitz constants, unlike the fixed-point route where a contraction constant below 1 is required.
  • Uniqueness follows by setting the perturbation sizes to zero in the epsilon-approximate comparison: two approximate solutions with the same initial data must coincide.
  • The explicit bounds show exponential growth in the interval length $b$, the Lipschitz constants, and the number of impulses, giving a quantitative picture of sensitivity.
  • The mixed inequality is a template for stability and boundedness arguments for other impulsive mixed integrodifferential equations.
  • The same argument extends, as the authors indicate, to fractional-order impulsive integrodifferential equations.

Reading between the lines

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

  • A reader who wants to preserve the proof of Theorem 5.1 can add the kernel support condition $k_2(\tau,\sigma,\varsigma)=0$ for $\varsigma>\sigma$; under that hypothesis the monotonicity comparison in the proof is valid and the remaining argument is unchanged.
  • The mixed inequality, once repaired, should apply to nonlocal parabolic equations by taking the Banach space to be a Hilbert space and the evolution operator to be a heat or analytic semigroup, yielding a priori error bounds for semidiscrete approximations.
  • Setting all kernels constant turns the inequality into a closed-form integral equation; comparing that exact solution with the exponential bound is a fast way to screen which kernel classes an admissible mixed inequality can cover.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies nonlinear impulsive Volterra-Fredholm integrodifferential equations (VFIIDEs) in a Banach space. It proves existence and uniqueness of mild solutions by a Bielecki-norm contraction argument (Theorem 3.1), establishes data dependence of solutions via Picard operator theory (Theorem 4.1), then extends a Gronwall-type inequality for piecewise continuous functions to a mixed Volterra-Fredholm setting (Theorem 5.1). This mixed inequality is subsequently used to derive data-dependence and epsilon-approximate-solution estimates without the smallness restrictions required by the Picard-operator approach (Theorems 6.1 and 6.2). The advertised novelty is the mixed integral inequality and the resulting removal of smallness conditions.

Significance. If the mixed integral inequality were valid, the paper would provide a useful contribution by eliminating the smallness condition in data-dependence estimates for impulsive Volterra-Fredholm equations. The contraction argument in Theorem 3.1 is standard and appears correct, and the Picard-operator estimate in Theorem 4.1 is plausible. However, the central new tool, Theorem 5.1, is false as stated, and the subsequent data-dependence theorems all rely on it. The claimed removal of restrictions is therefore unsupported. The paper also has extensive presentation issues, but the decisive problem is mathematical.

major comments (3)
  1. [Section 5, proof of Theorem 5.1, immediately before Eq. (5.3)] The step replacing V(ς) by V(σ) in the Fredholm double integral ∫_0^τ ∫_0^b k2(τ,σ,ς)V(ς) dς dσ is invalid. The proof has just established that V is nondecreasing, so for ς > σ one has V(ς) ≥ V(σ), not V(ς) ≤ V(σ). Because ς ranges over [0,b] and can exceed σ, the comparison used to pass from the first displayed inequality to Eq. (5.3) is false. This is the load-bearing step of Theorem 5.1.
  2. [Theorem 5.1, Eq. (5.2)] The claimed inequality is not merely unproved; it is false. Take b=1, a(τ)=1, b(τ,σ)=k1(τ,σ,ς)=0, k2(τ,σ,ς)=3/2, β_k=0, and u(τ)=1+6τ. Then ∫_0^1 u(ς) dς = 4, so the right-hand side of (5.1) equals 1 + (3/2)·4τ = 1+6τ = u(τ); the hypothesis holds with equality for all τ∈[0,1]. The conclusion (5.2) at τ=1 would assert 7 ≤ exp(3/2) ≈ 4.48, which is false. Theorem 5.1 is therefore not a valid extension of Theorem 2.3 to the mixed case.
  3. [Theorems 6.1 and 6.2, Eqs. (6.1) and (6.3)] The main advertised results in Section 6 inherit the gap in Theorem 5.1. In the proof of Theorem 6.1, the term ∫_0^τ ∫_0^b M L_G L_{F2} ||w(ς)-v(ς)|| dςdσ is exactly of the Fredholm type for which Theorem 5.1 fails, and the same is true of the corresponding term in Eq. (6.9) in the proof of Theorem 6.2. Since the mixed integral inequality is the only tool used to remove the smallness condition, the conclusions of Theorems 6.1 and 6.2 are unsupported as stated.
minor comments (5)
  1. [Statement of Theorem 5.1] The theorem statement uses 'τ ∈ [0, τ]' both in the hypothesis and in the conclusion; this is a variable confusion and should be 'τ ∈ [0, b]'.
  2. [End of proof of Theorem 5.1] The last sentence refers to 'the desired inequality (5.3)', but the conclusion of the theorem is Eq. (5.2).
  3. [Proof of Theorem 6.1] The passage '∏_{0<τ_k<0}(1+M L_{I_k})' is an empty product over the wrong index set; the intended product is over 0<τ_k<τ and is later bounded by the product over k=1,...,n.
  4. [Proof of Theorem 6.2] There are several typographical errors in the displayed inequalities, e.g., 'F2(σ,ς,w1(ς))dσ' should presumably be 'F2(σ,ς,w1(ς))dς', and the mixed notation in the first double integral in Eq. (6.8) is garbled.
  5. [Abstract and Section 7] The phrase 'integral inequity' appears in the abstract and the claim of 'less restrictions' is not quantified by comparing the assumptions of Theorem 4.1 and Theorem 6.1; a precise comparison of the two sets of hypotheses would be needed even if the core inequality were correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are self-contained and rely on standard external theorems; the known defect in Theorem 5.1 is a monotonicity-direction error, not a circular reduction.

full rationale

The central results are Theorem 3.1 (Banach fixed point for mild solutions), Theorem 4.1 (data dependence via the Picard-operator comparison theorem), and Theorems 6.1–6.2 (data dependence via the extended integral inequality). Each is derived from explicit Lipschitz hypotheses and standard external tools (Pazy's semigroup growth bound, Bainov–Hristova's Theorem 2.3/Lemma 2.2, and Rus's fixed-point comparison theorem). There are no fitted parameters that are later renamed as predictions, and no quantity is defined in terms of the quantity it is supposed to determine. The authors' prior works appear only as background citations in the introduction and the reference list; none is invoked to justify an existence, uniqueness, or dependence claim. The proof of Theorem 5.1 does contain a serious mathematical error: after defining V as the right-hand side of (5.1), it replaces V(ς) by V(σ) in the Fredholm term using monotonicity of V; this is valid for ς≤σ but not for ς∈[0,b] with ς>σ, and the resulting inequality can fail (as a one-dimensional counterexample shows). This is a correctness flaw in the derivation chain, not a circularity: the claimed inequality is neither an identity with its hypotheses nor a renamed fit. Accordingly, the circularity score is 0.

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

The results rest on standard semigroup theory and a known integral inequality. The only questionable ingredient is the monotonicity comparison inside the Fredholm term, which is where the proof fails.

assumptions (4)
  • domain assumption Hypotheses (H1)-(H3): G, F1, F2, Ik are continuous and Lipschitz in state variables
    These are stated explicitly in Section 3 and are required for the contraction estimate and for applying the integral inequality.
  • standard math C0-semigroup growth bound from Pazy's Theorem (Theorem 2.4)
    Used to bound the semigroup norm by M in the contraction and dependence estimates.
  • standard math Bainov-Hristova integral inequality (Lemma 2.2 / Theorem 2.3)
    Cited external result for piecewise continuous functions, used as the base of the extended inequality.
  • domain assumption V(t) is nondecreasing on [0,b]
    Relies on monotonicity of a, b, k1, k2, beta and nonnegativity of u; used in the invalid comparison step.

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Pith. "Pith review of On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations." pith.science (2026). https://pith.science/paper/X7TTL2BW

@misc{pith2026190809545,
  author       = {Pith},
  title        = {Pith review of: On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7TTL2BW}},
  note         = {Machine review of arXiv:1908.09545}
}
abstract

In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $\epsilon$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.

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Works this paper leans on

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