Existence of Chatterjea mappings with gamma in [1/3, 1/2) that are not generalized Chatterjea

Determine whether there exist mappings T: X → X on a metric space (X, d) that satisfy the Chatterjea condition d(Tx, Ty) ≤ γ [d(x, Ty) + d(y, Tx)] for all x, y ∈ X with a constant γ ∈ [1/3, 1/2), but do not satisfy the generalized three-point Chatterjea condition d(Tx, Ty) + d(Ty, Tz) + d(Tx, Tz) ≤ γ′ [d(x, Ty) + d(x, Tz) + d(y, Tx) + d(y, Tz) + d(z, Tx) + d(z, Ty)] for all pairwise distinct x, y, z ∈ X for any constant γ′ ∈ [0, 1/3).

Background

The paper introduces generalized Chatterjea type mappings as a three-point analogue of classical Chatterjea mappings. Proposition 2.1 shows that any Chatterjea type mapping with γ ∈ [0, 1/3) automatically satisfies the generalized three-point Chatterjea condition. Conversely, Example exa4 exhibits a mapping that satisfies the generalized Chatterjea condition but not the classical two-point Chatterjea condition, demonstrating that the two classes are distinct.

This raises a boundary question: while overlap is guaranteed for γ < 1/3 in the classical setting, it is unknown whether classical Chatterjea mappings with larger contraction parameter γ ∈ [1/3, 1/2) can fail to satisfy the generalized three-point condition. Resolving this would clarify the sharp relationship between the two classes across parameter ranges.