Weakly convergent fixed point iterations for weakly sequentially non-expansive mappings

Thomas P. Wihler

arxiv: 2604.20471 · v2 · submitted 2026-04-22 · 🧮 math.NA · cs.NA· math.FA

Weakly convergent fixed point iterations for weakly sequentially non-expansive mappings

Thomas P. Wihler This is my paper

Pith reviewed 2026-05-09 23:58 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.FA

keywords fixed point iterationsweak convergencenon-expansive mappingsOpial spacesreflexive Banach spacesnumerical analysis

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The pith

Fixed point iterations converge weakly when mappings satisfy only a weak sequential non-expansiveness property in reflexive Opial spaces.

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

The paper develops a framework showing that fixed point iterations converge weakly for mappings obeying an asymptotic non-expansiveness condition along weakly convergent sequences. This condition is weaker than the usual global Lipschitz or contraction requirements, so it covers more nonlinear problems that lack monotonicity or convexity. The result holds in reflexive Opial spaces and removes the need for uniform convexity or other strong geometric assumptions on the space. A sympathetic reader would care because many practical equations in scientific computing fall outside the scope of classical Banach fixed-point arguments, and this broadens the class of problems where iterative methods can be guaranteed to work.

Core claim

The authors establish that if a mapping is weakly sequentially non-expansive, then fixed point iterations converge weakly in reflexive Opial spaces. Weak sequential non-expansiveness means that distances between iterates do not increase in an asymptotic sense along weakly convergent sequences, which is a significantly weaker requirement than global Lipschitz continuity. This framework extends classical convergence results to a broader class of mappings without additional geometric assumptions such as uniform convexity.

What carries the argument

The weak sequential non-expansiveness property, which requires that distances between points do not increase along weakly convergent sequences in an asymptotic sense, serves as the central condition that replaces stronger Lipschitz assumptions and enables the weak convergence proof.

If this is right

Classical fixed-point convergence theorems extend to mappings that are not globally Lipschitz.
Iterative methods become applicable in reflexive Opial spaces without requiring uniform convexity.
Asymptotic bounds on distances suffice for proving weak convergence instead of stronger contraction properties.

Where Pith is reading between the lines

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

The framework may apply directly to certain nonlinear integral equations or optimization problems whose mappings satisfy the weaker asymptotic condition but not standard non-expansiveness.
Numerical implementations could test the property on sequences generated by the iteration to verify applicability before running the full computation.

Load-bearing premise

The mapping must satisfy the weak sequential non-expansiveness property and the underlying space must be reflexive and Opial.

What would settle it

An explicit mapping on a reflexive Opial space that meets the weak sequential non-expansiveness condition yet whose fixed point iteration fails to converge weakly would falsify the result.

read the original abstract

Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that a suitable contraction property can be verified. However, such conditions are typically too restrictive for more complex nonlinear equations that lack key structural features such as monotonicity or convexity. In this paper, we develop a general framework for the weak convergence of fixed point iterations based on asymptotic bounds. In particular, we introduce and exploit a weak sequential non-expansiveness property, which is significantly weaker than the global Lipschitz assumptions commonly employed in this context. This approach permits to extend classical convergence results to a broader class of mappings in general (reflexive) Opial spaces, without relying on additional geometric assumptions such as uniform convexity.

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

This paper introduces weak sequential non-expansiveness plus asymptotic bounds to prove weak convergence of fixed-point iterations in reflexive Opial spaces without uniform convexity or global Lipschitz assumptions.

read the letter

The one or two things your colleague should know: the paper defines a weak sequential non-expansiveness property that is weaker than the usual global Lipschitz condition and combines it with an asymptotic-bound framework to obtain weak convergence of iterations like x_{n+1} = T(x_n) in reflexive Opial spaces, dropping the need for uniform convexity. This is a direct extension of classical results in a setting where those stronger assumptions often fail for nonlinear problems. The work does a clean job of motivating the restrictions of the Banach theorem and then laying out the new property and the convergence argument without adding extra geometric structure. It rests on standard functional-analytic background and shows no sign of circularity or self-referential definitions. The abstract and stress-test material give no indication of load-bearing gaps in the logic. Soft spots are modest and mostly practical. The abstract stays high-level, so the real test is whether the full proofs go through cleanly and whether the paper supplies concrete mappings where the new property holds but older theorems do not; without those examples the broadening of the class remains somewhat abstract. Opial spaces are a reasonable minimal setting, but readers will still need to check how often the property can be verified in applications. Overall this is a technical but coherent contribution in numerical analysis. It is aimed at researchers working on iterative methods for nonlinear equations in Banach spaces who care about weak convergence. A reader already comfortable with Opial spaces and fixed-point theory will get the most out of it. The paper deserves a serious referee because the claimed relaxation is meaningful and the framework is grounded in reproducible functional-analytic arguments rather than fitting or ad-hoc constructions. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper develops a general framework for proving weak convergence of fixed point iterations by introducing a weak sequential non-expansiveness property on the mapping. This property is claimed to be strictly weaker than global Lipschitz or contraction conditions and is used to obtain weak convergence results in reflexive Opial spaces without requiring uniform convexity or other strong geometric assumptions on the space.

Significance. If the derivations hold, the work would meaningfully broaden the scope of weak-convergence theory for fixed-point iterations, allowing application to a larger class of nonlinear mappings that lack monotonicity or convexity. The reliance on asymptotic bounds and the minimal ambient assumptions (reflexivity + Opial property) aligns with standard functional-analytic techniques while relaxing a common restrictive hypothesis.

minor comments (2)

[Abstract / Introduction] The abstract and introduction should explicitly state the precise form of the fixed-point iteration (e.g., the recurrence relation) and the exact mathematical definition of weak sequential non-expansiveness, including the role of the asymptotic bounds, before any convergence theorem is stated.
All notation (e.g., the sequence {x_n}, the mapping T, the Opial modulus) must be introduced with clear definitions prior to its first use in any theorem or proposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary and recommendation of minor revision. We are pleased that the significance of introducing weak sequential non-expansiveness to broaden weak-convergence results in reflexive Opial spaces, without uniform convexity or similar assumptions, has been recognized.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the weak sequential non-expansiveness property as a new, weaker assumption than global Lipschitz continuity and then derives weak convergence results for fixed-point iterations in reflexive Opial spaces. This development rests on standard functional-analytic background (reflexivity, Opial property) together with the newly stated definition; no step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The abstract and context show an independent framework extension rather than renaming or smuggling prior results. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the central claim rests on the new non-expansiveness property and standard properties of reflexive Opial spaces. No free parameters or invented entities are visible.

axioms (1)

domain assumption The space is reflexive and satisfies Opial's condition.
Invoked to guarantee weak convergence of the iteration sequence.

pith-pipeline@v0.9.0 · 5431 in / 1132 out tokens · 28943 ms · 2026-05-09T23:58:38.977383+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 6 canonical work pages

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work page doi:10.2307/2042038 1976
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and Keeler, E

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Baillon, J.-B. and Bruck, R. E. and Reich, S. , title =. Houston Journal of Mathematics , volume =
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Bruck, R. E. and Reich, S. , title =. Houston Journal of Mathematics , volume =
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and Shafrir, I

Reich, S. and Shafrir, I. , title =. Proceedings of the American Mathematical Society , volume =. 1987 , doi =

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, title =

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[1] [1]

, TITLE =

Opial, Z. , TITLE =. Bull. Amer. Math. Soc. , FJOURNAL =. 1967 , PAGES =. doi:10.1090/S0002-9904-1967-11761-0 , URL =

work page doi:10.1090/s0002-9904-1967-11761-0 1967

[2] [2]

Bruck, Jr., R. E. , TITLE =. Pacific J. Math. , FJOURNAL =. 1973 , PAGES =

1973

[3] [3]

, TITLE =

Edelstein, M. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1974 , PAGES =. doi:10.2307/2040439 , URL =

work page doi:10.2307/2040439 1974

[4] [4]

and Tintarev, C

Solimini, S. and Tintarev, C. , TITLE =. Commun. Contemp. Math. , FJOURNAL =. 2016 , NUMBER =. doi:10.1142/S0219199715500388 , URL =

work page doi:10.1142/s0219199715500388 2016

[5] [5]

, TITLE =

Zeidler, E. , TITLE =. 1986 , PAGES =. doi:10.1007/978-1-4612-4838-5 , URL =

work page doi:10.1007/978-1-4612-4838-5 1986

[6] [6]

, pages =

Sims, B. , pages =. A support map characterization of the. Proc. Centre Math. Anal. Austral. Nat. Univ. , volume =

[7] [7]

P. D. Lax , title =. 2002 , month = apr, pages =

2002

[8] [8]

Bruck, R. E. , title =. Journal of Mathematical Analysis and Applications , year =

[9] [9]

, TITLE =

van Dulst, D. , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 1982 , NUMBER =. doi:10.1112/jlms/s2-25.1.139 , URL =

work page doi:10.1112/jlms/s2-25.1.139 1982

[10] [10]

, journal =

Schaefer, H. , journal =

[11] [11]

Fixed points and iteration of a nonexpansive mapping in a Banach space

Ishikawa, S. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1976 , NUMBER =. doi:10.2307/2042038 , URL =

work page doi:10.2307/2042038 1976

[12] [12]

, title =

Edelstein, M. , title =. Journal of the London Mathematical Society , volume =. 1962 , doi =

1962

[13] [13]

Boyd, D. W. and Wong, J. S. W. , title =. Proceedings of the American Mathematical Society , volume =. 1969 , doi =

1969

[14] [14]

and Keeler, E

Meir, A. and Keeler, E. , title =. Journal of Mathematical Analysis and Applications , volume =. 1969 , doi =

1969

[15] [15]

and Bruck, R

Baillon, J.-B. and Bruck, R. E. and Reich, S. , title =. Houston Journal of Mathematics , volume =

[16] [16]

Bruck, R. E. and Reich, S. , title =. Houston Journal of Mathematics , volume =

[17] [17]

and Shafrir, I

Reich, S. and Shafrir, I. , title =. Proceedings of the American Mathematical Society , volume =. 1987 , doi =

1987

[18] [18]

, title =

Pata, V. , title =. 2010 , series =

2010