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arxiv: 1907.00017 · v1 · pith:T6YKOQ7Jnew · submitted 2019-06-28 · 🧮 math.AP

On a multivalued differential equation with nonlocality in time

Andr\'e Eikmeier , Etienne Emmrich This is my paper

Pith reviewed 2026-05-25 13:33 UTC · model grok-4.3

classification 🧮 math.AP
keywords multivalued differential equationVolterra operatormonotone operatorKakutani fixed-point theoremexistence of solutionsnonlocal in timeBanach spaces
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The pith

Existence of solutions is proved for a multivalued differential equation combining a monotone operator and a Volterra integral operator on non-embedded Banach spaces.

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

The paper establishes existence of solutions to an initial value problem for a differential inclusion. The right-hand side is the sum of a multivalued map and the action of a monotone hemicontinuous coercive operator plus a convolution Volterra operator with exponential decay. These operators act in different Banach spaces without an embedding relation between them. The proof adapts Kakutani's fixed-point theorem to this setting under suitable measurability, continuity and growth assumptions. If correct, the result extends existence theory to nonlocal problems in mismatched function spaces.

Core claim

The central claim is that the initial value problem for the multivalued differential equation has a solution. This is shown by verifying that the associated set-valued operator satisfies the hypotheses of a generalized Kakutani fixed-point theorem, given the properties of the monotone operator, the Volterra operator, and the set-valued right-hand side.

What carries the argument

A generalization of the Kakutani fixed-point theorem applied to the sum of the monotone operator and the Volterra integral operator acting on the set-valued map.

If this is right

  • The result guarantees existence without requiring one Banach space to embed into the other.
  • Exponential decay in the Volterra kernel helps control the nonlocal term in the proof.
  • The growth conditions on all terms ensure the fixed-point set is nonempty and the solution is integrable.
  • Measurability of the set-valued map allows the use of measurable selections in the argument.

Where Pith is reading between the lines

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

  • This framework could be adapted to other convolution kernels beyond exponential decay.
  • Applications in viscoelasticity or heat conduction with memory might benefit from this non-embedded space setting.
  • Future work could investigate uniqueness or regularity of the solutions obtained this way.

Load-bearing premise

The two operators act on different Banach spaces where one is not embedded in the other, with the set-valued right-hand side measurable and satisfying continuity and growth conditions.

What would settle it

Constructing an explicit counterexample satisfying all the operator conditions but lacking any solution to the initial-value problem would disprove the claim.

read the original abstract

The initial value problem for a multivalued differential equation is studied, which is governed by the sum of a monotone, hemicontinuous, coercive operator fulfilling a certain growth condition and a Volterra integral operator in time of convolution type with exponential decay. The two operators act on different Banach spaces where one is not embedded in the other. The set-valued right-hand side is measurable and satisfies certain continuity and growth conditions. Existence of a solution is shown via a generalisation of the Kakutani fixed-point theorem.

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

0 major / 2 minor

Summary. The manuscript studies the initial value problem for a multivalued differential equation governed by the sum of a monotone hemicontinuous coercive operator (with growth condition) and a convolution-type Volterra integral operator with exponential decay. The operators act on distinct Banach spaces with no embedding between them. The set-valued right-hand side is assumed measurable and to satisfy continuity and growth conditions. Existence of a solution is established by applying a generalization of the Kakutani fixed-point theorem.

Significance. If the result holds, it extends existence theory for differential inclusions to settings without the usual embedding between the spaces on which the monotone and nonlocal operators act. This is potentially useful for evolution problems with time-nonlocal terms where standard Sobolev-type embeddings fail. The direct application of a generalized fixed-point theorem to an operator constructed from the given terms is a standard and appropriate technique in this area.

minor comments (2)
  1. [Abstract] The abstract states that the two operators act on different Banach spaces with no embedding, but the precise functional setting (e.g., the space in which the sum is taken or the domain of the solution) is not indicated; adding this would improve readability without affecting the argument.
  2. [Abstract] The growth and continuity hypotheses on the multivalued term and the coercivity/growth hypotheses on the monotone term are listed but not cross-referenced to the precise statement of the generalized Kakutani theorem used; a short remark on compatibility would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments or requested changes appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes existence of solutions to a multivalued evolution equation by applying a generalized Kakutani fixed-point theorem to an operator assembled directly from the given monotone hemicontinuous coercive term and the convolution-type Volterra operator. The hypotheses (measurability and growth on the set-valued map, hemicontinuity/coercivity/growth on the monotone operator, exponential decay on the kernel) are stated as external assumptions; the argument does not reduce any prediction or central claim to a fitted parameter, a self-referential definition, or a load-bearing self-citation chain. The two operators acting on non-embedded spaces is an explicit modeling choice, not a derived result. The derivation is therefore self-contained within standard fixed-point theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard functional-analytic assumptions listed in the abstract (monotonicity, hemicontinuity, coercivity, measurability, growth, exponential decay) together with the applicability of a generalized Kakutani theorem; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption A generalization of the Kakutani fixed-point theorem applies to the multivalued operator formed by the sum of the monotone operator and the Volterra term
    Invoked to conclude existence from the listed operator properties.

pith-pipeline@v0.9.0 · 5605 in / 1312 out tokens · 33030 ms · 2026-05-25T13:33:12.046427+00:00 · methodology

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