Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary Conditions via Picard Operator

J. Vanterler da C. Sousa; Kishor D. Kucche; Pallavi U. Shikhare

REVIEW 3 major objections 4 minor 36 references

Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary Conditions via Picard Operator

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For second-order Volterra integrodifferential equations with a nonlocal condition and a slope boundary condition, a unique solution exists and depends Lipschitz-continuously on the data whenever a weighted contraction constant is below one.

desk verdict The result is likely salvageable, but the printed contraction condition H2 uses |Σ c_k| where Σ|c_k| is needed, and both numerical examples have arithmetic failures that make them invalid checks. read the letter →

arxiv 1908.08224 v1 pith:7I6AIUVO submitted 2019-08-22 math.CA

Pallavi U. Shikhare , Kishor D. Kucche , J. Vanterler da C. Sousa This is my paper · ORCID

classification math.CA MSC 45J0534G2047H1034B15

keywords VolterraintegrodifferentialequationnonlocalconditionboundaryPicardoperatordatadependenceweightednormcontractionmappingfixedpointtheorem

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 treats second-order Volterra integrodifferential equations whose data consist of a nonlocal condition at several interior times and a boundary condition relating the slopes at the two ends. It proves that, under global Lipschitz assumptions on the two nonlinearities and a contraction condition on a weighted norm, the problem has exactly one twice-differentiable solution. It further proves that the solution depends continuously on the nonlocal data and on the right-hand-side functions, with an explicit Lipschitz bound obtained from Picard operator theory. The practical point is that these are the two properties an applied user needs before trusting a numerical solution: existence, uniqueness, and controlled sensitivity to model inputs.

What carries the argument

The engine is the integral reformulation (3.1), which converts (1.1)-(1.3) into a fixed-point equation $w=P(w)$ for an operator $P$ on $C^1(J,\mathbb{R})$. The paper equips this space with the weighted norm $\|w\|_1=\max_{t\in J}(|w(t)|+|w'(t)|)e^{-\gamma t}$; the exponential weight lets the contraction constant absorb the interval length $T$, the boundary ratio $\beta$, and the nonlocal coefficients $c_k$ through a factor involving $e^{\gamma T}/(\beta-1)$. When $q<1$, the contraction mapping theorem yields the unique fixed point. For data dependence, the paper uses the notion of a Picard operator — a map with exactly one fixed point reached by iteration from every starting point — and the general fact, stated as Theorem 2.1, that if two operators are uniformly $\rho$-close and one is a contraction with constant $\alpha$, then their fixed points are at distance at most $\rho/(1-\alpha)$; applying this to the original and perturbed integral operators produces the bound (4.9).

What would settle it

For fixed data, compute the function $q(\gamma)$ defined in H2 and find its infimum over $\gamma>0$; if that infimum is at least 1, no weight makes the operator a contraction in the weighted space, so the hypotheses of Theorem 3.2 cannot be satisfied even though H1 may hold. A direct numerical check is also available because the proof's estimate (3.11) contains an extra term $e^{\gamma T}/(\beta-1)$ before the $e^{\gamma t}$ division, so the contraction constant read off from (3.11) need not match the one printed in H2.

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Extended reading notes

Core claim

The central claim is that the second-order Volterra integrodifferential problem (1.1)-(1.3) has a unique solution in $C^2(J,\mathbb{R})$ whenever the nonlinearities $F$ and $G$ are globally Lipschitz and a strictly positive weight $\gamma$ makes the contraction constant $q$ in assumption H2 smaller than 1. The companion data-dependence statement is Theorem 4.1: if the data $w_0$ and the functions $F,G$ are replaced by nearby data $\tilde{w}_0$ and $\tilde{F},\tilde{G}$, the distance between the two solutions in the weighted norm is bounded by an explicit multiple of $|w_0-\tilde{w}_0|$ plus the $L^1$ norm of the difference between the right-hand sides, divided by $1-q$. The paper therefore treats the nonlocal boundary-value problem as a fixed-point problem whose solution map is Lipschitz continuous in the problem data.

Load-bearing premise

The load-bearing premise is that a positive weight $\gamma$ exists making the displayed constant $q$ in H2 smaller than 1; the paper states 'Choose $\gamma>0$' but does not prove that such a choice is available for every problem satisfying the Lipschitz conditions H1.

Editorial extensions

If this is right

If the hypotheses hold, the boundary-value problem has exactly one solution in $C^2(J,\mathbb{R})$, and iterating the operator $P$ from any starting function converges to it.
Replacing $F,G$ by nearby functions and $w_0$ by $\tilde{w}_0$ moves the solution by an amount controlled by the data distance divided by $1-q$, so the solution map is Lipschitz continuous in the data.
Setting the forcing perturbation to zero isolates dependence on the nonlocal data; setting $w_0=\tilde{w}_0$ isolates dependence on $F$ and $G$; when both perturbations vanish, the bound reduces to uniqueness.
The contraction constant depends on the interval length $T$, the boundary ratio $\beta>1$, the coefficients $c_k$, and the two Lipschitz constants, so shortening the interval or weakening the nonlocal coupling makes $q<1$ easier to achieve.

Reading between the lines

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

The absence of a proof that some $\gamma>0$ realizes $q<1$ means the theorem should be read as a conditional existence test: for a concrete problem one can compute $q(\gamma)$ and check its minimum before invoking the result.
The bound (4.9) visibly amplifies by $1/|1+\sum c_k|$; when the nonlocal denominator is close to zero, small errors in the nonlocal data can produce large solution changes even if $q$ is comfortably below 1.
The same contraction scheme could in principle extend to fractional-order or delay analogues, provided the integral equivalent equation and the weighted-norm estimate are re-derived; the paper signals this direction but does not prove it.
Recomputing the contraction constant from the proof's own estimate (3.11) rather than from the printed H2 may give a different $q$, so a numerical check of the actual bound is the safer route in applications.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the existence and data-dependence results are probably fixable, but as written the contraction hypothesis is not derived correctly and the examples do not verify the theorem. I would send it out, but the authors need to change the constant in H2 and rework the examples.

The novel piece is the specific problem: a second-order Volterra integrodifferential equation with a nonlocal condition and a derivative boundary condition. The route they take—convert to an integral equation, apply Banach's contraction with a Bielecki norm in C^1, then use Rus's Picard operator comparison for data dependence—is standard but clearly laid out. Lemma 3.1 is fine, and the structure of the data-dependence bound in Theorem 4.1 is reasonable.

The soft spots are real. In the proof of Theorem 3.2, the nonlocal term is bounded by pulling out |Σ c_k/(1+Σ c_k)| from a weighted sum of nonnegative bracketed quantities. That is only legitimate when all c_k have the same sign. For mixed signs the triangle inequality gives (Σ|c_k|)/|1+Σ c_k|, which is strictly larger. So H2, which uses |Σ c_k|, does not actually guarantee q<1. Example 5.1 is exactly this case: c=(1,1,-1,0,1), and replacing 2/3 by 4/3 pushes q above 1, so the example does not demonstrate contraction. Both examples also have arithmetic slips: the proposed w(t) in Example 5.1 gives a nonlocal sum of about 3.125, not 3.10, and Example 5.2 gives 1.25, not 1.35. These are likely corrigible, but as printed the examples do not check the theorem.

A smaller gap: Theorem 4.1 asserts that the perturbed operator S is a contraction with constant \tilde q<1, but no hypothesis guarantees that the chosen γ (or any γ) yields \tilde q<1. One can probably add a condition, but it does not follow from what is assumed.

Core idea is sound and the result is probably true once H2 is corrected to use Σ|c_k| and once S's contraction is explicitly assumed. That is a substantial revision, not a desk reject. I'd send it to a referee, but the authors should fix the constant and re-check the examples.

Referee Report

3 major / 4 minor

Summary. The paper studies existence, uniqueness, and data dependence of solutions to a second-order Volterra integrodifferential equation with nonlocal and boundary conditions (1.1)–(1.3). The authors convert the problem to an equivalent integral equation (Lemma 3.1), define a fixed-point operator, and prove a contraction estimate in a Bielecki-type norm under global Lipschitz assumptions (H1) and a contraction condition (H2), yielding Theorem 3.2. Section 4 then uses Picard operator theory to derive an explicit bound (4.9) for the distance between solutions of the original problem and a nearby problem with perturbed right-hand side and initial data. Section 5 presents two numerical examples. The overall strategy is standard and self-contained, but the manuscript contains several load-bearing technical errors in the contraction estimate, in the assumptions of Theorem 4.1, and in the numerical examples.

Significance. If corrected, the paper would provide a straightforward but potentially useful application of Banach's contraction principle and Picard operator theory to a nonlocal boundary value problem for Volterra integrodifferential equations. The derivation is self-contained, the data-dependence bound (4.9) is explicit, and the authors do not fit constants to force the conclusion. However, the central contraction argument is flawed for sign-changing coefficients, the data-dependence theorem is missing a key hypothesis, and the numerical examples do not verify the stated results. These issues concern the main claims, so the current version cannot be accepted as is. With the corrections indicated below, the contribution would be a modest but publishable application of known techniques.

major comments (3)

[Section 3, proof of Theorem 3.2, estimates (3.9)–(3.11)] The passage from the nonlocal term |(1/(1+Σc_k)) Σ c_k [ t_k/(β−1)∫F + ∫(t_k−s)F ]| to |Σc_k/(1+Σc_k)| times the maximum of the bracketed expression is not justified. The triangle inequality gives (Σ|c_k|)/|1+Σc_k| times that maximum. Therefore the contraction constant in H2 should contain Σ|c_k| rather than |Σc_k|. As printed, H2 can hold while the actual contraction constant exceeds 1: in Example 5.1, Σc_k=2 but Σ|c_k|=4, and the printed value q=0.901 becomes approximately 1.13 when the factor 4/3 replaces 2/3. Hence the proof of Theorem 3.2 does not establish contraction under the stated H2. In addition, the placement of e^{γT}/(β−1) in the displayed H2 differs from the expression derived in the proof: the proof yields 1+[1+{Tβ+Tβ|A|}]e^{γT}/(β−1), not 1+[1+{Tβ+Tβ|A|}e^{γT}/(β−1)]. This bracketing should also be corrected.
[Section 4, Theorem 4.1 and its proof] The proof asserts that the operator S for the tilded problem is a contraction with constant q̃<1, but no assumption in Theorem 4.1 guarantees this. Assumption (H2)' only postulates Lipschitz constants L_{F̃} and L_{G̃}; there is no analogue of H2 for the tilded data, and the gamma that works for the original problem need not work for F̃,G̃. Thus S need not have a fixed point, and Theorem 2.1 cannot be applied. The statement of Theorem 4.1 must include a contraction hypothesis for the tilded problem, or another argument must be supplied that ensures S is a Picard operator.
[Section 5, Examples 5.1 and 5.2] The numerical examples do not verify the abstract results. In Example 5.1, equation (5.2) contains a typo ('-w(t2)' should presumably be '-w(t3)'), and with the claimed solution w(t)=e^{t/10} the left-hand side of the nonlocal condition equals approximately 3.104, not w0=3.10. In Example 5.2, the left-hand side of (5.5) for w(t)=(t+t^2)/10 equals 1.25, not 1.35. Moreover, substituting T=2, β=5, γ=2, L_F=1/100, L_G=1, and Σc_k=4 into the corrected contraction formula gives q≈1.95, not the reported 0.8395, so Example 5.2 does not satisfy H2. These failures undermine the only concrete illustrations of the main theorems.

minor comments (4)

[Section 3, Theorem 3.2] The definition of the norm is ambiguous as printed: ||w||_1 = max_{t∈J} { |w(t)| + |w'(t)| / e^{γt} } should read ||w||_1 = max_{t∈J} (|w(t)| + |w'(t)|) e^{-γt}.
[Section 3, proof of Lemma 3.1] The closing sentence 'Which is conditions (1.3)' should refer to the nonlocal condition (1.2); the boundary condition (1.3) was already verified in the preceding lines.
[Section 4, proof of Theorem 4.1] The text 'From (4.5) and (4.5)' should read 'From (4.5)–(4.8)', and the norm estimate that follows should cite equation (4.7) for the derivative term.
[Section 3, after estimate (3.11)] The phrase 'Choose γ>0 such that ... < 1' is misleading because H2 is an assumption, not a consequence of H1; in general, for fixed L_F, L_G, T and β, no such γ may exist, since the expression tends to infinity as γ→0 and as γ→∞.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a self-contained contraction-argument proof, and no conclusion is equivalent to its inputs by construction.

full rationale

The paper's central chain is Lemma 3.1 (equivalence of the boundary-value problem with the Volterra integral equation) followed by Banach/Picard fixed-point arguments. Lemma 3.1 constructs (3.1) by integrating (1.1) and substituting the nonlocal and boundary conditions, and conversely differentiates (3.1) to recover (1.1)-(1.3); this is a direct algebraic equivalence, not a renaming of the conclusion. Theorem 3.2 defines the operator P explicitly and estimates ||Pw-Pv||_1 using only the Lipschitz assumptions H1; the contraction constant q in H2 is the resulting bound, and uniqueness follows from Banach's contraction principle rather than from an assumed solution. Theorem 4.1 applies the standard Picard-operator comparison theorem, cited to external sources [25,31,32], and obtains (4.9) by triangle-inequality estimates; the dependence on |w0-w0tilde| and L_mu is the actual content of the estimate, not a relabeled input. The only self-citations in the paper are background stability papers and closing remarks on fractional calculus; none carries the existence, uniqueness, or data-dependence argument. The paper's demonstrated weakness is technical rather than circular: in the proof around (3.9) the absolute value of the weighted sum of c_k is bounded by |sum c_k/(1+sum c_k)|, whereas for sign-changing c_k the triangle inequality requires (sum |c_k|)/|1+sum c_k|, so H2 as printed may be insufficient and Example 5.1 may not verify the corrected contraction condition. This is a mathematical correctness issue, not a circularity: no fitted parameter is called a prediction, and no uniqueness theorem is imported from the authors' own prior work.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities. The main extra postulates are the Lipschitz and contraction conditions, plus the Bielecki weight gamma; these are technical assumptions rather than invented physical objects.

free parameters (1)

gamma (Bielecki norm weight) = gamma=1 in Example 5.1; gamma=2 in Example 5.2
The theorem requires existence of gamma > 0 such that the contraction constant q < 1. The paper chooses gamma by hand in examples and does not prove a general existence result.

assumptions (5)

domain assumption F and G satisfy global Lipschitz conditions with constants LF and LG (H1).
Used in the proof of Theorem 3.2 to bound differences of Pw and Pv. Global Lipschitz on the whole real spaces is strong and often fails in applications.
ad hoc to paper There exists gamma > 0 with q < 1, where q is the contraction constant (H2, corrected per the proof).
This condition restricts the admissible T, beta, and c_k; the paper does not show it is satisfiable for general data.
domain assumption The sum of the coefficients c_k is not equal to -1.
Needed to solve the nonlocal condition for w(0) in Lemma 3.1.
domain assumption H3': |F(t,u,v,w) - Ftilde(t,u,v,wtilde)| <= mu(t) uniformly in u,v,w,wtilde, with mu in L1 intersect C.
Used in Theorem 4.1 for data dependence; a very strong uniform closeness assumption.
standard math Banach contraction principle and Picard operator theorem (Theorem 2.1) are valid.
Standard fixed point results cited from [25,31,32].

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Pith. "Pith review of Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary Conditions via Picard Operator." pith.science (2026). https://pith.science/paper/7I6AIUVO

@misc{pith2026190808224,
  author       = {Pith},
  title        = {Pith review of: Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary Conditions via Picard Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7I6AIUVO}},
  note         = {Machine review of arXiv:1908.08224}
}

read the original abstract

This article investigates the existence and uniqueness of solutions to the second order Volterra integrodifferential equations with nonlocal and boundary conditions through its integral equivalent equations and fixed point of Banach. Further, utilising the Picard operator theory we obtain the dependency of solutions on the initial nonlocal data and on functions involved on the right hand side of the equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of Mathematical Analy sis and Applications 162(2) (1991), 494–505

work page 1991

[2] [2]

L. Byszewski, Existence and uniqueness of a classical solution to a functional-diﬀerential abstract nonlocal Cauchy problem, International Journal of St ochastic Analysis 12(1) (1999), 91–97

work page 1999

[3] [3]

Bednarz, L

A. Bednarz, L. Byszewski, On abstract nonlocal Cauchy proble m, Czasopismo Techniczne (2015), 11–17

work page 2015

[4] [4]

Byszewski, Continuous dependence of mild solutions, on initial n onlocal data, of the nonlocal semilinear evolution Cauchy problems, Czasopismo Technicz ne 9 (2017), 151–158

L. Byszewski, Continuous dependence of mild solutions, on initial n onlocal data, of the nonlocal semilinear evolution Cauchy problems, Czasopismo Technicz ne 9 (2017), 151–158

work page 2017

[5] [5]

Balachandran, J

K. Balachandran, J. Y. Park, Existence of solutions and contro llability of nonlinear in- tegrodiﬀerential systems in Banach spaces, Math. Problems in Eng in., 2003(2) (2003), 65–79

work page 2003

[6] [6]

Byszewski, An abstract nonlocal functional-diﬀe rential second order evolu- tion Cauchy problem

A Bednarz, L. Byszewski, An abstract nonlocal functional-diﬀe rential second order evolu- tion Cauchy problem. Czasopismo Techniczne, 2018(Volume 4), 139 –147

work page 2018

[7] [7]

Y. Lin, J. H. Liu, Semilinear integrodiﬀerential equations with nonlo cal Cauchy problem, Nonlinear Anal. 26 (1996), 1023–1033

work page 1996

[8] [8]

Muresan, A boundary value problem for some functional-diﬀer ential equations, via Picard operators, Carpathian Journal of Mathematics (2007), 1 41–148

V. Muresan, A boundary value problem for some functional-diﬀer ential equations, via Picard operators, Carpathian Journal of Mathematics (2007), 1 41–148

work page 2007