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arxiv: 2604.10002 · v1 · submitted 2026-04-11 · 🧮 math.FA · math.AP· math.DG

A generalization of the inverse mapping theorem in infinite dimensions

Sajjad Lakzian This is my paper

Pith reviewed 2026-05-10 16:39 UTC · model grok-4.3

classification 🧮 math.FA math.APmath.DG
keywords inverse mapping theoremproperty Anon-expansivenessimplicit function theoremnon-smooth mapsinfinite dimensional spacesabstract PDE
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The pith

The inverse mapping theorem in infinite dimensions holds when maps satisfy a weaker non-expansiveness condition called property A instead of C¹ differentiability.

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

The paper establishes that the inverse mapping theorem remains valid in infinite-dimensional spaces if the usual C¹ smoothness requirement is replaced by variations of a weaker non-expansiveness property labeled property A. This change removes the need for continuous differentiability and thereby extends the theorem to certain non-smooth maps. The same substitution produces corresponding generalizations of the implicit function theorem and of existence-uniqueness results for abstract systems of partial differential equations.

Core claim

We present a generalization of the inverse mapping theorem, where variations of a weaker non-expansiveness property (referred to as property A) replace the key C¹ condition. We also obtain inverse mapping theorems that can be applied to non-smooth maps. Also as a by-product of the generalized inverse mapping theorem, we prove generalizations of the implicit function theorem and existence and uniqueness theorem of abstract PDE systems as well.

What carries the argument

Property A, a weaker non-expansiveness condition on the map that takes the place of the classical C¹ differentiability hypothesis in the inverse mapping theorem.

If this is right

  • The inverse mapping theorem applies directly to selected non-smooth maps.
  • The implicit function theorem holds under the same relaxed non-expansiveness condition.
  • Existence and uniqueness statements for abstract PDE systems follow from the generalized inverse mapping result.

Where Pith is reading between the lines

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

  • The same relaxation could be tested on other classical results in nonlinear functional analysis that currently require differentiability.
  • New well-posedness statements for non-smooth evolution equations in Banach spaces become available once property A is verified for the governing operator.

Load-bearing premise

The maps under consideration satisfy the newly introduced property A in place of C¹ differentiability.

What would settle it

A concrete map between infinite-dimensional Banach spaces that satisfies property A yet fails to possess a continuous local inverse at some point would disprove the claimed generalization.

read the original abstract

We present a generalization of the inverse mapping theorem, where variations of a weaker non-expansiveness property (referred to as property ${\sf A}$) replace the key $\mathsf{C}^1$ condition. We also obtain inverse mapping theorems that can be applied to non-smooth maps. Also as a by-product of the generalized inverse mapping theorem, we prove generalizations of the implicit function theorem and existence and uniqueness theorem of abstract PDE systems as well.

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 / 3 minor

Summary. The manuscript presents a generalization of the inverse mapping theorem in infinite-dimensional spaces. It replaces the classical C¹ differentiability assumption with a weaker non-expansiveness condition termed property A (and variations thereof), thereby obtaining local invertibility results that apply to certain non-smooth maps. As by-products, the authors derive corresponding generalizations of the implicit function theorem and of existence/uniqueness statements for abstract PDE systems.

Significance. If the central arguments are correct, the result would extend a fundamental tool of nonlinear functional analysis to a strictly larger class of operators. This could be useful in settings where differentiability fails but a controlled non-expansive behavior still holds, with potential applications to non-smooth PDEs and optimization in Banach spaces. The paper supplies explicit statements and proofs rather than relying on fitted parameters or circular reductions.

minor comments (3)
  1. [Abstract] The abstract refers to 'variations of a weaker non-expansiveness property' without indicating which variations are treated in the main theorems; a brief enumeration in the introduction would improve readability.
  2. [Section 2] Section 2 (or wherever property A is defined): a short comparison table or paragraph contrasting property A with the standard Lipschitz or non-expansive conditions would help readers gauge how much weaker the hypothesis is.
  3. [Theorem 3.1 and corollaries] The statements of the main inverse-mapping result and its corollaries should explicitly record the ambient space assumptions (complete normed spaces, Banach spaces, etc.) to avoid any ambiguity for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. We appreciate the recognition that our generalization of the inverse mapping theorem, replacing the C¹ condition with the weaker non-expansiveness property A, extends a fundamental tool to a larger class of operators, including certain non-smooth maps, with potential applications to non-smooth PDEs.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces property A as a novel weaker non-expansiveness condition replacing the standard C¹ assumption and derives the generalized inverse mapping theorem (along with corollaries for implicit functions and PDEs) directly from this definition and associated axioms. No step reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation; the central claims rest on independent verification of the new property within the manuscript's proofs. The derivation chain is self-contained against external benchmarks and does not rename or smuggle in prior results via ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard axioms of Banach spaces plus the new domain assumption that the maps satisfy property A; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Maps satisfy the weaker non-expansiveness property A instead of C1 differentiability.
    This is the key replacement condition stated in the abstract for the generalized inverse mapping theorem.

pith-pipeline@v0.9.0 · 5356 in / 1190 out tokens · 35332 ms · 2026-05-10T16:39:19.469238+00:00 · methodology

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Works this paper leans on

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