arxiv: 2604.10265 · v1 · submitted 2026-04-11 · 🧮 math.CA · math.DS

Recognition: unknown

On the non-uniqueness of continuous solutions to differential equations with a discrete state-dependent delay

Alexander Rezounenko

Pith reviewed 2026-05-10 15:18 UTC · model grok-4.3

classification 🧮 math.CA math.DS

keywords state-dependent delaynon-uniquenesscontinuous solutionsdelay differential equationssolution classificationdiscrete delay

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The pith

Differential equations with discrete state-dependent delays admit multiple continuous solutions from continuous initial data.

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

The paper shows that delay differential equations can have more than one continuous solution even when the initial function is continuous, provided the delay itself is discrete and depends on the current state of the solution. It supplies explicit examples where this occurs and introduces a partial classification scheme that labels the different solutions by combinations of three colors. A reader would care because many applied models in biology, control, or physics use exactly this form of equation, and non-uniqueness changes what it means to predict future behavior from known starting conditions. The classification gives a simple way to organize the alternatives without claiming to exhaust all possibilities.

Core claim

The central claim is that for differential equations containing a discrete state-dependent delay, continuous initial functions do not guarantee a unique continuous solution; explicit examples demonstrate this non-uniqueness, and the distinct solutions can be partially classified by means of three colors and their simple combinations.

What carries the argument

The discrete state-dependent delay that selects past times according to the present value of the solution, thereby opening multiple continuous extensions of the same initial function.

If this is right

Non-uniqueness occurs in simple, verifiable examples without requiring discontinuous data.
The number of distinct continuous solutions can be finite and organized by color combinations.
The classification distinguishes solution types by intuitive color groupings rather than exhaustive case analysis.
Further research is required to determine whether the three-color scheme covers all instances of non-uniqueness.

Where Pith is reading between the lines

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

Standard uniqueness results for delay equations may require additional conditions on the delay function when dependence on the state is discrete.
In applied models the existence of multiple paths could produce qualitatively different long-term outcomes from identical starts.
The color-labeling approach might be checked against numerical integrations of specific state-dependent delay equations to test its utility.

Load-bearing premise

The delay must remain discrete and state-dependent while the initial function stays continuous, without further restrictions that would force uniqueness.

What would settle it

A concrete differential equation with discrete state-dependent delay together with a proof that exactly one continuous solution exists for every continuous initial function on the initial interval.

Figures

Figures reproduced from arXiv: 2604.10265 by Alexander Rezounenko.

**Figure 1.** Figure 1: Example 1/1963 The meaning of supscripts will be explained below. The same idea (an example) was used in [11, p.395] (1970). The analysis of this elegant Drivers‘s example leads to different conclusions why the non-uniqueness appears. Naturally, the initial function is not Lipschitz and the SDD does not satisfy the additional condition [7]. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Example 2/2010 Since questions (A) and (B) are very general, we restrict ourself to the simplest case of one discrete state-dependent delay and continuous initial functions with only one point of non-Lipschitz. It is natural that several SDDs and several points of nonLipschitz may multiply the number of solutions. To answer question (A) we construct the following example, which is a natural generalizati… view at source ↗

**Figure 3.** Figure 3: Example 3 One may verify that we have two families of solutions (𝜏 ≥ 0 is a parameter). The right-side family 𝑥 𝜏 (𝑡) = {︃ 1 + 𝑡, 𝑡 ∈ [0, 𝜏 ], 1 + 𝑡 − 𝐶(𝑡 − 𝜏 ) 1 1−𝛼 , 𝑡 ∈ (𝜏, 𝜏 + 𝜎], (14) with 𝐶 ≡ 𝐴 1 1−𝛼 (1 − 𝛼) 1 1−𝛼 and the left-side family 𝑦 𝜏 (𝑡) = {︃ 1 + 𝑡, 𝑡 ∈ [0, 𝜏 ], 1 + 𝑡 + 𝐷(𝑡 − 𝜏 ) 1 1−𝛽 , 𝑡 ∈ (𝜏, 𝜏 + 𝜎], (15) with 𝐷 ≡ 𝐵 1 1−𝛽 (1 − 𝛽) 1 1−𝛽 . Here 𝜎 > 0 is fixed. The above example gives an an… view at source ↗

read the original abstract

We discuss the non-uniqueness of continuous solutions to differential equations with a {\it discrete } state-dependent delay and continuous initial functions. We are interested not only in the fact (conditions) of non-uniqueness, but in additional information on the number of non-unique solutions and discuss an approach to classify them. We provide a few explicit (easy to verify) examples of the non-uniqueness of continuous solutions and propose an approach to their classification. This partial classification may be illustrated by using just three collors and their simple and intuitive combination. We recognize that this initial classification is not exhaustive, and further study is necessary to build a complete picture.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript discusses non-uniqueness of continuous solutions to differential equations with discrete state-dependent delays under continuous initial functions. It supplies a few explicit examples to demonstrate the phenomenon, provides additional information on the number of such solutions, and proposes a partial classification approach illustrated via combinations of three colors, while explicitly noting that the classification is not exhaustive and requires further study.

Significance. If the claimed explicit examples can be directly verified against the differential equations as stated, the work offers concrete illustrations of non-uniqueness in a class of equations where uniqueness is not guaranteed in general. This is useful for the theory of functional differential equations, particularly state-dependent delays, and the intuitive three-color classification provides a starting framework for organizing solution multiplicity, even if partial. The focus on verifiable instances rather than a general theorem aligns with the paper's scope and could inform numerical or modeling work, though broader impact depends on the rigor and completeness of the derivations in the full text.

major comments (1)

The abstract asserts that the examples are 'explicit (easy to verify)' and support the non-uniqueness claim, but the provided text contains no specific equations, derivations, or verification steps. This makes it impossible to confirm that the solutions satisfy the equation and initial conditions without additional material, which is load-bearing for the central claim.

minor comments (1)

The classification is described as 'partial' and 'not exhaustive'; the manuscript should clarify in the relevant section what criteria (e.g., number of delay switches or solution branches) the three-color scheme captures and what cases remain unclassified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for recognizing the utility of explicit examples and the three-color classification framework in our work on non-uniqueness for state-dependent delay equations. We address the single major comment below and will revise the manuscript to strengthen the verifiability of the central examples.

read point-by-point responses

Referee: The abstract asserts that the examples are 'explicit (easy to verify)' and support the non-uniqueness claim, but the provided text contains no specific equations, derivations, or verification steps. This makes it impossible to confirm that the solutions satisfy the equation and initial conditions without additional material, which is load-bearing for the central claim.

Authors: We agree that the presentation of the examples can be made more immediately verifiable. The manuscript does contain specific functional forms for the right-hand side, the state-dependent delay, continuous initial functions, and candidate continuous solutions in the body text (following the abstract), but we acknowledge that explicit equation statements, initial conditions, and substitution verifications are not laid out in a single, self-contained block. In the revised version we will add a dedicated subsection that states each example in full (equation, initial data, proposed solutions) and performs the direct substitution checks step by step, confirming that each solution satisfies both the differential equation and the initial condition. This will make the non-uniqueness claim transparent without requiring the reader to assemble the pieces. revision: yes

Circularity Check

0 steps flagged

No significant circularity: claims rest on explicit verifiable examples

full rationale

The paper centers on concrete, directly verifiable examples of non-uniqueness for continuous solutions of state-dependent delay equations, together with an explicitly partial and non-exhaustive classification scheme illustrated by color combinations. No equations, parameters, or uniqueness theorems are fitted or derived in a way that reduces the stated non-uniqueness back to the inputs by construction. Any self-citations to prior work by the same author are not load-bearing for the central claims, which are supported by the supplied instances that can be checked independently against the differential equation and initial data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, invented entities, or non-standard axioms; the work relies on the standard framework of delay differential equations.

axioms (1)

domain assumption Continuous initial functions and discrete state-dependent delay are admissible for the equation class under study
Stated directly in the abstract as the setting for non-uniqueness.

pith-pipeline@v0.9.0 · 5399 in / 1145 out tokens · 45164 ms · 2026-05-10T15:18:39.681915+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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