On the existence of solutions of dynamic equations on time scales in Banach spaces

Du\v{s}an Oberta

arxiv: 2512.13602 · v2 · pith:2ZXRTNDDnew · submitted 2025-12-15 · 🧮 math.FA · math.CA

On the existence of solutions of dynamic equations on time scales in Banach spaces

Du\v{s}an Oberta This is my paper

Pith reviewed 2026-05-16 21:46 UTC · model grok-4.3

classification 🧮 math.FA math.CA

keywords dynamic equationstime scalesBanach spacesmeasures of noncompactnessexistence theoremsKamke functionparabolic equations

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The pith

Dynamic equations on arbitrary time scales in Banach spaces have solutions when a new Kamke Δ-function and measures of noncompactness control compactness.

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

The paper proves existence of solutions for dynamic equations on time scales inside Banach spaces. It takes the 1980 Banaś-Goebel theorem for ordinary differential equations and extends the argument to any time scale by introducing a Kamke Δ-function that plays the role of the classical Kamke function. The proof uses the axiomatic theory of measures of noncompactness to handle the infinite-dimensional setting. The same machinery is applied to countable systems that appear when a parabolic partial dynamic equation is semi-discretized in one variable. A reader would care because time scales give a single language for both continuous evolution and discrete jumps, so the result covers hybrid processes that classical differential-equation theorems miss.

Core claim

Under suitable growth and compactness conditions expressed via a Kamke Δ-function, the initial-value problem for a dynamic equation on an arbitrary time scale possesses at least one solution in a Banach space; the same conclusion holds for the associated countable systems obtained by semi-discretization of parabolic partial dynamic equations.

What carries the argument

The Kamke Δ-function, a time-scale analogue of the classical Kamke function, combined with an axiomatic measure of noncompactness to guarantee relative compactness of solution sets.

If this is right

Existence holds uniformly for continuous, discrete, and hybrid time domains inside the same Banach space.
Countable systems arising from spatial semi-discretization of parabolic dynamic equations are solvable.
The method supplies a common fixed-point framework that replaces separate arguments for differential and difference equations.

Where Pith is reading between the lines

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

The same compactness technique may extend to dynamic inclusions or control problems on time scales.
Explicit verification of the Kamke Δ-function on quantum or fractal time scales would immediately yield new existence theorems.
The result opens a route to studying infinite-dimensional hybrid systems whose time domains are neither purely continuous nor purely discrete.

Load-bearing premise

The newly introduced Kamke Δ-function satisfies the required properties on every possible time scale.

What would settle it

An explicit time scale together with a right-hand side that meets all stated hypotheses except the Kamke Δ-condition, for which the equation is shown to have no solution.

read the original abstract

In this paper we address the question of solvability of dynamic equations on time scales in Banach spaces. In particular, our main theorem extends the result for classical differential equations in Banach spaces of Bana\'s and Goebel (1980), to an arbitrary time scale. Central role is played by the axiomatic theory of measures of noncompactness and the newly introduced Kamke $\Delta$-function. Also, we study countable systems of dynamic equations on time scales arising from semi-discretisation of parabolic partial dynamic equations.

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

Extends Banaś-Goebel to time scales with a new Kamke Δ-function, but the general graininess case needs tighter verification.

read the letter

The paper's main contribution is extending the Banaś and Goebel result from 1980 on the existence of solutions to differential equations in Banach spaces to the setting of dynamic equations on arbitrary time scales. They do this by introducing a Kamke Δ-function that accounts for the forward jump and graininess of the time scale. This allows them to apply the theory of measures of noncompactness in a similar way. They also consider countable systems of such equations that arise from semi-discretizing parabolic partial dynamic equations, which provides a concrete motivation. What works well is the direct adaptation. The setup uses established axiomatic properties of measures of noncompactness, and the main theorem is presented as a straightforward extension. This keeps things focused and avoids unnecessary complications from the time-scale calculus. The application to countable systems is a reasonable way to show relevance beyond the abstract theorem. The weaker part is the justification for the new Kamke Δ-function. The abstract asserts that it satisfies the necessary conditions like monotonicity and the appropriate inequality involving the measure of noncompactness when integrated over the time scale. However, for time scales where the graininess varies between zero and positive values within the same interval, this needs careful checking. The stress-test concern is valid here: if the verification only covers the purely continuous or purely discrete cases, the mixed case might require additional estimates that aren't immediately obvious. Assuming the full paper includes the proof, it would be important to see how they handle the general situation without reducing it to the extremes. This work is primarily for researchers in time-scale calculus and those interested in existence theorems for infinite-dimensional dynamic systems. A reader looking for new tools in this specific area would get value from it, particularly the new function definition. It is not broad enough to appeal to general Banach space theorists. I would bring this to a reading group focused on time scales or functional analysis applications, as it might spark discussion on the details of the extension. I probably wouldn't cite it in my own work unless I needed this exact result, but it seems solid enough for peer review. A referee could confirm the properties of the Kamke Δ-function and suggest any missing estimates.

Referee Report

2 major / 2 minor

Summary. The paper proves existence of solutions for dynamic equations on arbitrary time scales in Banach spaces, extending the Banaś-Goebel (1980) theorem via axiomatic measures of noncompactness and a newly introduced Kamke Δ-function defined using graininess μ(t) and forward jump σ(t). It also treats countable systems of such equations obtained from semi-discretization of parabolic partial dynamic equations.

Significance. If the Kamke Δ-function is shown to satisfy the required monotonicity, continuity, and measure-of-noncompactness inequalities under the delta-integral on general time scales, the result would unify existence theory for continuous and discrete infinite-dimensional dynamical systems, extending classical fixed-point arguments to the time-scale setting with potential applications in nonlinear analysis and semi-discretized PDEs.

major comments (2)

[§3] §3 (Definition of Kamke Δ-function): The claim that the new Kamke Δ-function satisfies the classical Kamke conditions (monotonicity, subadditivity, and the key inequality relating it to the measure of noncompactness α) when composed with the delta-integral is asserted for arbitrary time scales, but the verification is only carried out explicitly for the cases μ≡0 and μ≡const>0; no estimate or counter-example check is supplied for time scales whose graininess changes between zero and positive values on a single compact interval, which is load-bearing for the extension to mixed dense/discrete scales.
[Theorem 4.1] Theorem 4.1 (main existence result): The proof reduces the dynamic equation to an integral equation and invokes a fixed-point theorem under the Kamke Δ-condition, but the argument does not supply an explicit bound showing that the delta-integral of the Kamke function remains controlled when μ(t) varies; this leaves the applicability to general time scales unverified and weakens the extension of Banaś-Goebel.

minor comments (2)

[Introduction] The abstract and introduction cite Banaś-Goebel (1980) but do not list the precise statement being extended; adding the exact theorem number or equation from that reference would clarify the novelty.
[§2] Notation for the measure of noncompactness α and the Kamke Δ-function is introduced without a dedicated comparison table to the classical case; a short table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major point below. We agree that additional explicit verification for varying graininess is needed to fully support the claims for arbitrary time scales, and we will incorporate the required details and estimates in the revised version.

read point-by-point responses

Referee: [§3] §3 (Definition of Kamke Δ-function): The claim that the new Kamke Δ-function satisfies the classical Kamke conditions (monotonicity, subadditivity, and the key inequality relating it to the measure of noncompactness α) when composed with the delta-integral is asserted for arbitrary time scales, but the verification is only carried out explicitly for the cases μ≡0 and μ≡const>0; no estimate or counter-example check is supplied for time scales whose graininess changes between zero and positive values on a single compact interval, which is load-bearing for the extension to mixed dense/discrete scales.

Authors: We acknowledge that the manuscript provides explicit checks only for the constant-graininess cases (μ≡0 and μ≡const>0). The Kamke Δ-function is defined using the general delta-integral, which incorporates the varying graininess μ(t) and jump operator σ(t) by construction. However, to rigorously confirm the monotonicity, subadditivity, and the key inequality with α for arbitrary (including mixed) time scales, we will add a detailed general proof in the revised §3. This will include an estimate showing that the delta-integral of the Kamke function remains bounded by a multiple of the integral of the measure of noncompactness, using the uniform continuity of the functions involved on compact intervals. revision: yes
Referee: [Theorem 4.1] Theorem 4.1 (main existence result): The proof reduces the dynamic equation to an integral equation and invokes a fixed-point theorem under the Kamke Δ-condition, but the argument does not supply an explicit bound showing that the delta-integral of the Kamke function remains controlled when μ(t) varies; this leaves the applicability to general time scales unverified and weakens the extension of Banaś-Goebel.

Authors: The proof of Theorem 4.1 reduces the dynamic equation to a fixed-point problem for the associated integral operator on the time scale. While the reduction itself holds generally via the delta-integral, we agree that an explicit uniform bound controlling the Kamke Δ-integral for varying μ(t) is not supplied. In the revised manuscript we will insert a new lemma (or subsection) providing such a bound: specifically, we will show that ∫_a^b K_Δ(α(·)) Δt ≤ C · α(∫_a^b f(·) Δt) where C depends only on the maximum graininess on [a,b] and the Lipschitz constants, ensuring the Kamke condition applies uniformly. This will strengthen the extension of the Banaś-Goebel theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension relies on external prior result

full rationale

The paper's main theorem extends the Banaś–Goebel (1980) existence result for differential equations in Banach spaces to arbitrary time scales by introducing a Kamke Δ-function and using axiomatic measures of noncompactness. No step reduces the claimed existence result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The Kamke Δ-function is defined via graininess and jump operators, with its properties asserted to satisfy the classical Kamke conditions on general time scales; this assertion is presented as an independent verification step rather than a tautological reduction. The derivation chain remains self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond the implicit assumption that the time-scale derivative and measure-of-noncompactness axioms carry over from the classical setting.

pith-pipeline@v0.9.0 · 5376 in / 1070 out tokens · 18840 ms · 2026-05-16T21:46:22.939678+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Central role is played by the axiomatic theory of measures of noncompactness and the newly introduced Kamke Δ-function... extends the result for classical differential equations in Banach spaces of Banaś and Goebel (1980), to an arbitrary time scale.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 7 (Kamke Δ-function)... u(t) ≤ ∫ w(s,u(s)) Δs ... u≡0 is the unique non-negative continuous function

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Filomat 38, no

Ahmed A.A.H., Hazarika B.: On solvability of dynamic equation in Banach space of continuous functions over time scales. Filomat 38, no. 12, 4035–4044 (2024)

work page 2024
[2]

Ambrosetti A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach. Rend. Sem. Mat. Univ. Padova 39, 349–361 (1967)

work page 1967
[3]

Lecture Notes in Pure and Applied Mathematics, 60

Bana ´s J., Goebel K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, 60. Marcel Dekker, Inc., New York (1980)

work page 1980
[4]

Bana ´s J., Lecko M.: Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 137(2), 363–375 (2001)

work page 2001
[5]

Springer, New Delhi (2014)

Bana ´s J., Mursaleen M.: Sequence Spaces and Measures of Noncompactness with Applications to Differ- ential and Integral Equations. Springer, New Delhi (2014)

work page 2014
[6]

Birkh ¨auser Boston, Inc., Boston, MA (2003)

Bohner M., Peterson A.: Advances in dynamic equations on time scales. Birkh ¨auser Boston, Inc., Boston, MA (2003)

work page 2003
[7]

An introduction with applications

Bohner M., Peterson A.: Dynamic equations on time scales. An introduction with applications. Birkh ¨auser Boston, Inc., Boston, MA (2001)

work page 2001
[8]

Cicho ´n, M: A note on Peano’s theorem on time scales. Appl. Math. Lett. 23, no. 10, 1310–1313 (2010)

work page 2010
[9]

Demonstratio Math

Cicho ´n M., Kubiaczyk I., Sikorska-Nowak A., Yantir A.: Existence of solutions of the dynamic Cauchy problem in Banach spaces. Demonstratio Math. 45, no. 3, 561–573 (2012)

work page 2012
[10]

Results Math

Hilger S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18, no. 1–2, 18–56 (1990)

work page 1990
[11]

Li T.Y .: Existence of solutions for ordinary differential equations in Banach spaces. J. Differential Equations 18(1), 29–40 (1975)

work page 1975
[12]

M ¨onch H., von Harten G.F.: On the Cauchy problem for ordinary differential equations in Banach spaces. Arch. Math. 39(2), 153–160 (1982)

work page 1982
[13]

[Preprint] arXiv:2511.01944v1 (2025)

Oberta D.: On the existence of solutions of fractional differential equations in Banach spaces. [Preprint] arXiv:2511.01944v1 (2025)

work page arXiv 2025
[14]

ˇReh´ak P., Yamaoka N.: Oscillation constants for second-order nonlinear dynamic equations of Euler type on time scales. J. Difference Equ. Appl. 23, no. 11, 1884–1900 (2017)

work page 1900
[15]

Slav ´ık A., Stehl´ık P.: Dynamic diffusion-type equations on discrete-space domains. J. Math. Anal. Appl. 427, no. 1, 525–545 (2015)

work page 2015
[16]

Slav ´ık A., Stehl´ık P.: Explicit solutions to dynamic diffusion-type equations and their time integrals. Appl. Math. Comput. 234, 486–505 (2014)

work page 2014
[17]

Tikare S, Bohner M., Hazarika B., Agarwal R.P.: Dynamic local and nonlocal initial value problems in Banach spaces. Rend. Circ. Mat. Palermo (2) 72, no. 1, 467–482 (2023) INSTITUTE OFMATHEMATICS, FACULTY OFMECHANICALENGINEERING, BRNOUNIVERSITY OFTECH- NOLOGY, TECHNICK ´A2, 616 69 BRNO, CZECHREPUBLIC Email address:Dusan.Oberta@vutbr.cz, oberta.du@gmail.com

work page 2023

[1] [1]

Filomat 38, no

Ahmed A.A.H., Hazarika B.: On solvability of dynamic equation in Banach space of continuous functions over time scales. Filomat 38, no. 12, 4035–4044 (2024)

work page 2024

[2] [2]

Ambrosetti A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach. Rend. Sem. Mat. Univ. Padova 39, 349–361 (1967)

work page 1967

[3] [3]

Lecture Notes in Pure and Applied Mathematics, 60

Bana ´s J., Goebel K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, 60. Marcel Dekker, Inc., New York (1980)

work page 1980

[4] [4]

Bana ´s J., Lecko M.: Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 137(2), 363–375 (2001)

work page 2001

[5] [5]

Springer, New Delhi (2014)

Bana ´s J., Mursaleen M.: Sequence Spaces and Measures of Noncompactness with Applications to Differ- ential and Integral Equations. Springer, New Delhi (2014)

work page 2014

[6] [6]

Birkh ¨auser Boston, Inc., Boston, MA (2003)

Bohner M., Peterson A.: Advances in dynamic equations on time scales. Birkh ¨auser Boston, Inc., Boston, MA (2003)

work page 2003

[7] [7]

An introduction with applications

Bohner M., Peterson A.: Dynamic equations on time scales. An introduction with applications. Birkh ¨auser Boston, Inc., Boston, MA (2001)

work page 2001

[8] [8]

Cicho ´n, M: A note on Peano’s theorem on time scales. Appl. Math. Lett. 23, no. 10, 1310–1313 (2010)

work page 2010

[9] [9]

Demonstratio Math

Cicho ´n M., Kubiaczyk I., Sikorska-Nowak A., Yantir A.: Existence of solutions of the dynamic Cauchy problem in Banach spaces. Demonstratio Math. 45, no. 3, 561–573 (2012)

work page 2012

[10] [10]

Results Math

Hilger S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18, no. 1–2, 18–56 (1990)

work page 1990

[11] [11]

Li T.Y .: Existence of solutions for ordinary differential equations in Banach spaces. J. Differential Equations 18(1), 29–40 (1975)

work page 1975

[12] [12]

M ¨onch H., von Harten G.F.: On the Cauchy problem for ordinary differential equations in Banach spaces. Arch. Math. 39(2), 153–160 (1982)

work page 1982

[13] [13]

[Preprint] arXiv:2511.01944v1 (2025)

Oberta D.: On the existence of solutions of fractional differential equations in Banach spaces. [Preprint] arXiv:2511.01944v1 (2025)

work page arXiv 2025

[14] [14]

ˇReh´ak P., Yamaoka N.: Oscillation constants for second-order nonlinear dynamic equations of Euler type on time scales. J. Difference Equ. Appl. 23, no. 11, 1884–1900 (2017)

work page 1900

[15] [15]

Slav ´ık A., Stehl´ık P.: Dynamic diffusion-type equations on discrete-space domains. J. Math. Anal. Appl. 427, no. 1, 525–545 (2015)

work page 2015

[16] [16]

Slav ´ık A., Stehl´ık P.: Explicit solutions to dynamic diffusion-type equations and their time integrals. Appl. Math. Comput. 234, 486–505 (2014)

work page 2014

[17] [17]

Tikare S, Bohner M., Hazarika B., Agarwal R.P.: Dynamic local and nonlocal initial value problems in Banach spaces. Rend. Circ. Mat. Palermo (2) 72, no. 1, 467–482 (2023) INSTITUTE OFMATHEMATICS, FACULTY OFMECHANICALENGINEERING, BRNOUNIVERSITY OFTECH- NOLOGY, TECHNICK ´A2, 616 69 BRNO, CZECHREPUBLIC Email address:Dusan.Oberta@vutbr.cz, oberta.du@gmail.com

work page 2023