Existence results for some classes of differential systems with "maxima"

Eduardo Pascali; Elisabetta Mangino

REVIEW 2 major objections 4 minor 4 references

Existence results for some classes of differential systems with "maxima"

T0 review · 2 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read Local existence is proved for differential systems whose equations depend on past maxima.

desk verdict Solid general local-existence theorem for systems with maxima, but two of the three worked examples have flawed proofs as printed, so the paper needs correction before it can be trusted. read the letter →

arxiv 2009.13327 v1 pith:A2WLEEMF submitted 2020-09-28 math.CA

Elisabetta Mangino , Eduardo Pascali This is my paper · ORCID

classification math.CA MSC 34K07

keywords differentialequationswithmaximainitialvalueproblemslocalexistencecontractionmappingtheoremPicarditerationmodifiedlogisticequationfunctionalrunningmaximum

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 proves that broad classes of initial value problems for differential systems with 'maxima'—where the derivative at time $t$ depends on the maximum, over the whole past interval $[0,t]$, of functions of each component—have local $C^1$ solutions. The general theorem treats the $m$-dimensional system (1.5) under continuity and local Lipschitz conditions and obtains existence from the Banach fixed-point theorem on a ball of continuous functions. For the scalar modified logistic equation $\dot x = x - \max_{[0,t]} x^2$ and for two particular two-component systems involving $\max y$, $\max x$, $\max y^2$, and $\max x^2$, the authors construct solutions by Peano–Picard approximation. These equations matter because they model situations in which future behaviour depends on the whole past history, and where classical logistic or population models require parameters or delays that cannot be fixed experimentally.

What carries the argument

The load-bearing object is the running-maximum functional $M_t(g)=\max_{s\in[0,t]}g(s)$ together with the elementary inequality $|\max_{[a,b]}g-\max_{[a,b]}h|\le \max_{[a,b]}|g-h|$, which makes $M_t$ non-expansive in the uniform norm. In Theorem 2.3 this turns the integral operator $F$ defined by (2.1) into a contraction on the ball $\{x:\|x-x_0\|_\infty\le\alpha\}$, yielding existence by Banach's fixed-point theorem. In Section 3 the same functional is used to build Peano–Picard sequences whose maxima can be evaluated explicitly—by monotonicity of the iterates, $\max_{[0,s]}\eta$ becomes $\eta(s)$ or the initial value—so that the iterates are shown to converge uniformly to solutions.

What would settle it

Run the Picard recursion of Theorem 3.3 with concrete positive unequal initial data, for instance $x_0=2$, $y_0=1$: the proof requires $x_n(t)\ge y_0$ and $y_n(t)\le x_n(t)$ for every $n,t$, and it requires the increment $x_{n+1}(t)-y_{n+1}(t)=x_0-y_0+\int_0^t(x_n(s)-y_n(s))\,ds$ to be nonnegative; checking those inequalities for $n=1,2$ settles whether the ordering premise used to evaluate the maxima is valid.

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Extended reading notes

Core claim

On its own terms, the paper establishes that local existence for systems with maxima is not an exotic problem: the non-expansiveness of the running-maximum functional lets a standard contraction argument work for a general system, and for special systems the Picard iterates can be controlled so that their uniform limits satisfy the integral equations. Theorem 2.3 gives a $C^1$ solution of (1.5) on $[0,T]$ for sufficiently small $T$, with the time horizon bounded by an expression involving $\alpha/M_{\alpha,T}$ and $1/(L_{\alpha,T}(1+L_\alpha\sqrt m))$, where the constants are defined from the data. Theorem 3.1 handles the modified logistic equation for $x_0\neq 1$, Theorem 3.3 claims solutions on every finite interval for the two-component system (3.3) with positive unequal initial values, and Theorem 3.6 gives a local solution for the squared-max system (3.5) under explicit inequalities relating $T$ and $c_0$.

Load-bearing premise

In the general theorem the load-bearing premise is the local Lipschitz continuity of $f$ and $g_i$ on the compact sets used to define $\alpha$, $M_{\alpha,T}$, and the Lipschitz constants; in the two-component theorem the load-bearing premise is the ordering of the Picard iterates that identifies each running maximum with a known endpoint, and the printed induction step verifying that ordering contains a sign error (the displayed $x_{n+1}(t)-y_{n+1}(t)$ should be $\ge0$, not $\le0$).

Editorial extensions

If this is right

The general system (1.5) has a unique local $C^1$ solution on an explicit time interval whenever $f$ and the $g_i$ satisfy the stated continuity and Lipschitz assumptions.
The scalar modified logistic equation $\dot x=x-\max_{[0,t]}x^2$ has a local solution for every $x_0\neq1$, with a computable existence time $T^*$.
The two-component system $\dot x=x-\max y$, $\dot y=y-\max x$ is claimed to have $C^1$ solutions on every finite interval when $x_0,y_0>0$ and $x_0\neq y_0$.
The squared-max system $\dot x=x-\max y^2$, $\dot y=y-\max x^2$ has a local solution whenever $T$ and $c_0$ satisfy the inequalities in Theorem 3.6.
Because the fixed-point argument gives a contraction, the local solution of the general system is not merely existent but unique inside the ball where the constants are checked.

Reading between the lines

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

A natural extension would allow moving or weighted past windows in place of $[0,t]$, provided the history functional remains Lipschitz in the uniform norm; the paper does not treat that case.
The contraction argument indicates that the local solution should depend continuously on the initial data, a property the paper does not investigate.
For the two-component system (3.3), a repaired proof will likely need an a priori bound or a comparison argument if the ordering of the iterates is ever violated; numerical iteration with $x_0>y_0$ would show whether that repair is necessary.
Connecting the maxima formulation to comparison principles could give global existence or blow-up criteria for the scalar logistic-type equation, since the sign of $\dot x$ is governed by whether $x(t)$ lies above the running maximum of $x^2$.

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Formalized claims in Lean

Claim #1: On its own terms, the paper establishes that local existence for systems with maxima is not an exotic problem: the non-expansiveness of the running-maximum functional lets a standard contraction argument work for a general system, and for special systems the Picard iterates can be controlled so that their uniform limits satisfy the integral equations. Theorem 2.3 gives a $C^1$ solution of (1.5) on

/-- @claim 1 On its own terms, the paper establishes that local existence for systems with maxima is not an exotic problem: the non-expansiveness of the running-maximum functional lets a standard contraction argument work for a general system, and for special systems the Picard iterates can be controlled so that their uniform limits satisfy the integral equations. Theorem 2.3 gives a $C^1$ solution of (1.5) on -/
def central_claim : Prop :=

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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

Dear colleague,

Quick take: this paper gives a clean local existence result for systems of differential equations with memory of maxima, then illustrates it with three examples. The general theorem is worth having. But two of the three example theorems are not actually proven as printed, because the Picard iterations are written incorrectly. One is a sign mistake in the ordering argument; the other is a recurrence that solves the wrong system. Both look fixable, but they are not typos you can overlook.

What is new: the general system (1.5) with multiple maxima terms is a genuine extension of the scalar case from Bainov–Hristova, and Theorem 2.3 is a straightforward but solid contraction argument. The proof checks out modulo minor misprints; the contraction bound is fine. Remarks 2.1 and 2.2 are correct. The paper is honest about its scope in the body, though the abstract promises 'initial boundary value problems' that do not appear.

The soft spots. Theorem 3.3 claims x_{n+1} − y_{n+1} ≤ 0, but given x0 > y0 and the induction hypothesis x_n ≥ y_n, the expression is nonnegative, not nonpositive. That breaks the monotonicity argument used to identify the maxima. The theorem may be true, but this proof does not establish it. Theorem 3.6 has a worse issue: the y-sequence is defined as y_{n+1} = x0 + ∫ x_n − ∫ max x_n^2, which starts at x0 instead of y0 and integrates x_n instead of y_n. The limit then satisfies the wrong second equation. The stress-test note is on the mark: this is not a sign slip, it is the wrong functional iteration. Replacing x0 with y0 and x_n with y_n in the first integral of the y-recurrence fixes it, but as written the theorem is unproven.

Overall: the main result is correct and the examples are plausible, but the paper needs a careful revision of the two example proofs. It is a modest contribution, not a breakthrough. A serious referee should look at it, and the authors should be asked to correct the recurrences and re-check the monotonicity argument. If the errors are fixed as described, I would accept it.

Referee Report

2 major / 4 minor

Summary. The paper investigates local existence of solutions for initial value problems for systems of differential equations with 'maxima', i.e., equations whose right-hand side depends on the maximum over a past interval of functions of the solution components. Theorem 2.3 gives a general local existence result for system (1.5) by a Banach fixed point argument under continuity and local Lipschitz conditions on f and g_i. The remainder of the paper studies three concrete examples by Peano-Picard iteration: the scalar equation (3.1), the two-component system (3.3), and the two-component system with squared maxima (3.5). The paper claims a C^1 solution on a small interval for (3.1) and on every finite interval for (3.3) and (3.5) under suitable sign and boundedness conditions.

Significance. Local existence for functional-differential equations with maxima is a legitimate but specialized topic, and the contraction argument in Theorem 2.3 is a clean and potentially correct result that extends standard treatments. The concrete Picard-iteration results, however, are not established in the present manuscript: the proof of Theorem 3.3 contains a sign error in the key monotonicity induction, and the recurrence in Theorem 3.6 is set up for the wrong functional equation. Since these two theorems are among the paper's advertised contributions, the manuscript needs substantial repair. No numerical verification or machine-checked proofs are supplied; the positive aspects are the explicit fixed-point framework and the transparent statement of assumptions.

major comments (2)

[Theorem 3.3, proof] The induction step contains a sign error. After assuming x_n(t) ≥ y0 and y_n(t) ≤ x_n(t), the paper claims x_{n+1}(t)-y_{n+1}(t)=x0-y0+∫_0^t(x_n(s)-y_n(s))ds ≤ 0. With x0>y0 and x_n-y_n≥0, the right-hand side is nonnegative, not nonpositive. Moreover, using the recurrences as printed (x_{n+1}=x0+∫(x_n-y0)ds and y_{n+1}=y0+∫(y_n-x_n)ds), the difference is actually x0-y0+∫(2x_n-y_n-y0)ds, so the displayed formula is also algebraically incorrect. Consequently the identities max_{[0,s]}x_n=x_n(s) and max_{[0,s]}y_n=y0 are not proved, and the identification of the simplified iteration with the integral system (3.4) is unjustified. The existence proof for Theorem 3.3 is therefore incomplete as written.
[Theorem 3.6, proof] The Picard sequence for the second component is defined incorrectly. The printed recurrence y_{n+1}=x0+∫_0^t x_n(s)ds-∫_0^t max_{[0,s]}x_n^2 ds should read y_{n+1}=y0+∫_0^t y_n(s)ds-∫_0^t max_{[0,s]}x_n^2 ds. As printed, every y_n starts at x0 and its dynamics are driven by x_n rather than y_n, so the uniform limit (x_∞,y_∞) satisfies y_∞(0)=x0 and y_∞'=x_∞-max x_∞^2, not the second equation of (3.5). The subsequent boundedness and convergence estimates are derived for this incorrect sequence and therefore do not establish Theorem 3.6. Replacing x0 by y0 and x_n by y_n in the first integral of the y-recurrence would repair the proof, but as it stands the theorem is not proved.

minor comments (4)

[Theorems 2.3 (statement and proof)] The condition on T is misstated: '0<T<min{α/M_{α,T}, 1/(L_{α,T}(1+L_α√m)), T}' contains a redundant and impossible final T; the proof's contraction condition should be T<1/(L_{α,T}(1+√m L_α)), not T<1/(L_{α,T}L_α√m). In addition, the displayed Lipschitz condition on f should read ||f(t,u_1,v_1)-f(t,u_2,v_2)||≤L_{α,T}(||u_1-u_2||+||v_1-v_2||).
[Theorem 3.1, proof] For x0=0, which is allowed by the hypothesis x0≠1, the claimed induction bound |g_n(t)|≤α is false because the recurrence for g_n gives g_n(t)=∑_{k=0}^n t^k/k! when x0=0. Since x≡0 is an obvious solution, the case can be handled separately, but the proof as written does not cover it.
[Remark 2.2] The continuity proof contains a typo: '|g(τ)-g(s)|>ε if |τ-s|<δ' should be '<ε'.
[Throughout] There are numerous typographical errors (e.g., 'RESUL TS', 'probl ems', 'om [0,T]', 'with respect the the distance') and the proof of Theorem 3.3 writes max_{[0,s]}x(τ)=x(s) where x_n is intended. A careful proofreading is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proofs rest on standard contraction and Picard-iteration arguments with no self-citations, fitted parameters, or definitional reductions.

full rationale

The paper contains no circular reasoning. Theorem 2.3 is a standard Banach fixed-point argument: existence of a C^1 solution is reduced to the equivalent integral equation (2.1), and the map F is shown to map a complete ball into itself and to be a contraction under the stated hypotheses on f and g_i; the smallness conditions on T are derived from the contraction constant and from the bound M_{α,T}, not assumed from the conclusion. Theorems 3.1, 3.3, and 3.6 use explicit Peano–Picard iterations, and their convergence claims, whether or not fully correct, are meant to be proved by direct estimates from the defining recurrences. There are no fitted parameters, no data subsets, no normalizations that force the result, and no load-bearing self-citations: the only references are the Bainov–Hristova monograph and classical works by D’Ancona and Volterra. The manuscript does contain genuine mathematical gaps—notably, in Theorem 3.3 the displayed inequality x_{n+1}(t)-y_{n+1}(t)=x0-y0+∫(x_n-y_n)ds≤0 has the wrong sign under the stated induction hypothesis, and in Theorem 3.6 the printed y_{n+1} recursion starts at x0 and integrates x_n instead of y0 and y_n, so the limit solves a different system—but these are proof errors, not circularity. No step reduces by construction to its own input, so the circularity score is 0.

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

The paper introduces no free parameters, no fitted values, and no new postulated entities. All assumptions are standard mathematical hypotheses about continuity and Lipschitz behavior, plus the standard fixed-point theorem and Picard iteration. The only non-standard aspect is the specific form of the system, which is a modeling choice rather than an additional axiom.

assumptions (4)

standard math Banach fixed-point theorem (contraction mapping theorem)
Used in Theorem 2.3 to prove existence and uniqueness of the fixed point of the integral operator F.
domain assumption Locally Lipschitz continuity of f and g_i
Assumed in Theorem 2.3 and the specific examples; required for the contraction estimate and for the Picard iteration bounds.
standard math Continuity of the running maximum function t → max_{[0,t]} g(τ) for continuous g
Proved in Remark 2.2 (with a typo in the proof) and used to ensure the integral operator maps C([0,T]) into itself.
standard math Peano-Picard successive approximations converge for the considered systems
Used in Theorems 3.1, 3.3, 3.6; convergence is established by explicit bounds, but the method is a standard technique.

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Cite this review

Pith. "Pith review of Existence results for some classes of differential systems with "maxima"." pith.science (2026). https://pith.science/paper/A2WLEEMF

@misc{pith2026200913327,
  author       = {Pith},
  title        = {Pith review of: Existence results for some classes of differential systems with "maxima"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2WLEEMF}},
  note         = {Machine review of arXiv:2009.13327}
}

read the original abstract

Local existence properties of initial boundary value problems associated with a new type of systems of differential equations with "maxima" are investigated.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Bainov and S.G

D.D. Bainov and S.G. Hristova, Diﬀerential Equations with Maxima , Chapman & Hall, CRC Press, Boca Raton- London - New York, 2011

work page 2011
[2]

D’Ancona La lotta per l’esistenza Giulio Einaudi Editore, Torino, 1942

U. D’Ancona La lotta per l’esistenza Giulio Einaudi Editore, Torino, 1942

work page 1942
[3]

Volterra, Sui tentativi di applicazione delle matema tiche alle scienze biologiche e sociali; Arch

V. Volterra, Sui tentativi di applicazione delle matema tiche alle scienze biologiche e sociali; Arch. Fisiologica, vol. III, p. 175, 1906

work page 1906
[4]

Ennio De Giorgi

V. Volterra, Variazioni e ﬂuttuazioni del numero d’indi vidui in specie animali conviventi; Mem. R. Accad. Lincei, ser. VI, vol. II fasc. III, 1926. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Unive rsit`a del Salento, Via per Arnesano, I-73100 LECCE, Italy E-mail address : elisabetta.mangino@unisalento.it E-mail address : eduardo.pascali@unisalento.it

work page 1926

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Bainov and S.G

D.D. Bainov and S.G. Hristova, Diﬀerential Equations with Maxima , Chapman & Hall, CRC Press, Boca Raton- London - New York, 2011

work page 2011

[2] [2]

D’Ancona La lotta per l’esistenza Giulio Einaudi Editore, Torino, 1942

U. D’Ancona La lotta per l’esistenza Giulio Einaudi Editore, Torino, 1942

work page 1942

[3] [3]

Volterra, Sui tentativi di applicazione delle matema tiche alle scienze biologiche e sociali; Arch

V. Volterra, Sui tentativi di applicazione delle matema tiche alle scienze biologiche e sociali; Arch. Fisiologica, vol. III, p. 175, 1906

work page 1906

[4] [4]

Ennio De Giorgi

V. Volterra, Variazioni e ﬂuttuazioni del numero d’indi vidui in specie animali conviventi; Mem. R. Accad. Lincei, ser. VI, vol. II fasc. III, 1926. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Unive rsit`a del Salento, Via per Arnesano, I-73100 LECCE, Italy E-mail address : elisabetta.mangino@unisalento.it E-mail address : eduardo.pascali@unisalento.it

work page 1926