Persistence of Singular Solutions in State-Dependent Delay Equations

Definitions and mathematical statement

Consider the supplied family of state-dependent delay equations \[ \varepsilon\dot x(t)=g\left(x(t-r_1(x(t))),\ldots,x(t-r_m(x(t)))\right),\qquad \varepsilon>0, \] where $m$ is the number of delays, $x$ is the unknown state, $g$ is the prescribed response function, and each $r_i$ is a prescribed function of the current state. The equation is to hold at times for which the derivative and all delayed evaluations are defined. The source does not specify the state space, the time interval, the domains or regularity of these functions, or conditions ensuring that the delays are nonnegative and the delayed evaluations are available.

Formally setting $\varepsilon=0$ gives the outer constraint \[ 0=g\left(x(t-r_1(x(t))),\ldots,x(t-r_m(x(t)))\right). \] This formal constraint alone does not define the inner solutions or their matching to outer solutions.

The supplied persistence question is the following: given a singular solution $x_0$, composed of inner and outer solutions at $\varepsilon=0$, must there exist $\varepsilon_0>0$ such that, for every $\varepsilon\in(0,\varepsilon_0)$, there is a genuine solution $x_\varepsilon$ of the displayed delay equation corresponding to $x_0$? Here the requested correspondence is the persistence of the prescribed singular solution, rather than bare existence of an unrelated solution. However, the source supplies no definition of the singular composite, the inner equations, the matching conditions, or the approximation or convergence relation between $x_\varepsilon$ and $x_0$. It also supplies no initial history, boundary conditions, periodicity requirement, or intended solution class. Consequently this is a schematic persistence question requiring those specifications before it becomes a precise mathematical assertion.

Short English statement

Does a singular solution assembled from inner and outer pieces persist as a corresponding actual solution for every sufficiently small positive $\varepsilon$ in the stated state-dependent delay equation? The supplied record leaves the hypotheses, matching conditions, and meaning of persistence unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001124$, source problem code catalogue entry, titled "Persistence of Singular Solutions in State-Dependent Delay Equations," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2008.
• Supplied source citation: Dynamics of Patterns (2009), Oberwolfach Reports, pp. 3201--3271, DOI 10.4171/owr/2008/57. Original report link: https://doi.org/10.4171/owr/2008/57. This is the authoritative source identified by the record; it has not been independently consulted for this draft.
• Supplied literature reference, labeled "Mallet-Paret--Nussbaum delay analysis": https://doi.org/10.1016/S0022-0396(02)00088-8. The record supplies no full bibliographic citation or specific theorem identifying its relevance to this formulation.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports rigorous boundary-layer and transition-layer existence results under substantial hypotheses, while stating that general persistence of every singular inner--outer solution requested here is not proved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The record labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This is a supplied source-verification claim, not verification performed for this draft.