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arxiv: 2605.21549 · v1 · pith:AXFXTKBUnew · submitted 2026-05-20 · 🧮 math.FA

Another Perspective on Chatterjea Contraction

Shallu Sharma , Irfan Ahmed , Sahil Billawria This is my paper

Pith reviewed 2026-05-22 00:53 UTC · model grok-4.3

classification 🧮 math.FA
keywords fixed pointChatterjea contractioncomplete metric spacek-continuitym-Chatterjea contractiondiscontinuous mappingiterative sequence
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The pith

An m-Chatterjea contraction on a complete metric space has a unique fixed point when the map is k-continuous.

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

The paper begins with the known result that if an iterate of a map is a Banach contraction then the map itself has a unique fixed point. It adapts this idea to Chatterjea contractions while keeping the original mapping on the left side of the defining inequality. The resulting m-Chatterjea contractions form a new class that properly contains the standard Chatterjea mappings. Adding the assumption of k-continuity yields existence and uniqueness of a fixed point on complete metric spaces. This matters for extending fixed-point methods to maps that may be discontinuous at their fixed points.

Core claim

The authors define m-Chatterjea contractions by retaining the original mapping on the left-hand side of a Chatterjea-type inequality. They prove that any such mapping that is also k-continuous possesses a unique fixed point in a complete metric space. Examples confirm that the new class strictly contains the Chatterjea mappings and that the fixed point need not be a point of continuity.

What carries the argument

The m-Chatterjea contraction inequality retained on the original mapping, combined with k-continuity to guarantee convergence of iterates to a unique fixed point.

If this is right

  • Every k-continuous m-Chatterjea contraction has exactly one fixed point.
  • The sequence of iterates starting from any point converges to that fixed point.
  • The mapping may be discontinuous precisely at the fixed point.
  • The new class properly contains all standard Chatterjea contractions.

Where Pith is reading between the lines

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

  • Similar retention of the original map on the left side could be tried for other known contraction classes to obtain wider fixed-point results.
  • The approach may help construct fixed-point theorems for operators arising in differential or integral equations that are discontinuous at their solutions.

Load-bearing premise

The mapping must satisfy the m-Chatterjea contraction inequality and also be k-continuous; failure of either condition removes the guarantee of a unique fixed point.

What would settle it

Exhibit a complete metric space together with a mapping that meets the m-Chatterjea inequality and is k-continuous yet possesses no fixed point, or a mapping that meets the inequality without k-continuity and has more than one fixed point.

read the original abstract

Inspired by the well-known result stating that if any iterate of a mapping is a Banach contraction on a complete metric space, then the mapping itself possesses a unique fixed point, we investigate that claim for a Chatterjea contraction but by retaining the left-hand side of the inequality as per the mapping itself. With an additional assumption of k- continuity, the existence and uniqueness of a fixed point is obtained for a new class of contractions, m-Chatterjea contraction, on a complete metric space. Several examples are given in order to substantiate many theoretical claims such as discontinuity at the unique limit point of the iterative sequence, as well as examples demonstrating that this new class strictly contains the class of Chatterjea mappings.

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 / 2 minor

Summary. The manuscript defines a new class of m-Chatterjea contractions on a complete metric space by the inequality d(Tx, Ty) ≤ k [d(x, T^m y) + d(y, T^m x)] (retaining the original mapping on the left-hand side) for positive integer m. Under the additional hypothesis of k-continuity, it proves existence and uniqueness of a fixed point by showing that the Picard iterates form a Cauchy sequence via iterated application of the inequality, passing to the limit with k-continuity, and obtaining uniqueness by a direct contradiction argument. Several examples are supplied to illustrate that the class properly contains ordinary Chatterjea mappings and that the fixed point may be a point of discontinuity for T.

Significance. If the central claim holds, the work supplies a modest but concrete extension of the Chatterjea fixed-point theorem by shifting iterates to the right-hand side while preserving the original mapping on the left. The resulting class is strictly larger and admits mappings discontinuous at their unique fixed point, which is of interest in metric fixed-point theory. Explicit examples verifying both strict containment and discontinuity constitute a positive feature, as does the internally consistent use of a telescoping Cauchy estimate followed by k-continuity.

major comments (1)
  1. [Theorem 3.1] Main theorem (existence): the Cauchy estimate obtained by iterating the m-Chatterjea inequality must be shown to produce a geometric ratio strictly less than 1; the manuscript should state explicitly the admissible range for k (independent of m) that guarantees convergence of the telescoping sum, since this step is load-bearing for the existence claim.
minor comments (2)
  1. [Preliminaries] The definition of k-continuity should be stated in a numbered display equation rather than inline, to facilitate later reference in the limit passage.
  2. [Theorem 3.2] In the uniqueness argument, the same inequality is applied to a pair of putative fixed points; a brief remark clarifying that the argument is independent of the particular value of m would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the significance of the work, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Theorem 3.1] Main theorem (existence): the Cauchy estimate obtained by iterating the m-Chatterjea inequality must be shown to produce a geometric ratio strictly less than 1; the manuscript should state explicitly the admissible range for k (independent of m) that guarantees convergence of the telescoping sum, since this step is load-bearing for the existence claim.

    Authors: We thank the referee for highlighting this point. In the proof of Theorem 3.1 the m-Chatterjea inequality is iterated along the Picard sequence to produce a telescoping bound on d(x_n, x_{n+p}). The resulting estimate is a geometric series whose common ratio is 2k. Consequently the series converges whenever 0 ≤ k < 1/2. This admissible interval for k is independent of the positive integer m and guarantees that the geometric ratio is strictly less than 1, so the sequence is Cauchy. We will add an explicit statement of the range k ∈ [0, 1/2) both in the hypotheses of Theorem 3.1 and in the detailed iteration step of the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the m-Chatterjea contraction as d(Tx, Ty) ≤ k [d(x, T^m y) + d(y, T^m x)] and proves existence/uniqueness of a fixed point on a complete metric space by showing that the Picard iterates form a Cauchy sequence (via direct iteration and telescoping of the given inequality), invoking completeness to obtain a limit, and using the added k-continuity hypothesis to verify that the limit is a fixed point. Uniqueness is obtained by a standard contradiction argument reusing the same inequality. These steps rely only on the metric axioms, completeness, and the explicit contraction condition; no equation reduces to a fitted parameter, self-referential definition, or load-bearing self-citation. The result is therefore independent of its own inputs and follows from the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard axioms of metric spaces plus the newly introduced m-Chatterjea inequality and k-continuity; no free parameters are fitted to data.

axioms (2)
  • domain assumption The underlying space is a complete metric space
    Invoked as the setting for the fixed-point theorem, standard in the field.
  • ad hoc to paper The mapping is k-continuous
    Additional assumption required for the existence-uniqueness result; introduced specifically for this class.
invented entities (1)
  • m-Chatterjea contraction no independent evidence
    purpose: Define a new broader class of mappings for which a fixed-point theorem holds while retaining the original mapping on the left-hand side of the inequality
    Postulated in the paper as a generalization that strictly contains Chatterjea mappings; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.0 · 5643 in / 1388 out tokens · 51241 ms · 2026-05-22T00:53:15.155193+00:00 · methodology

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