Gauge transforms of metrics in fixed point theory: Contractive and nonexpansive mappings

Let $ (Q, \delta) $ be a metric space and let $ \psi:[0, \infty)\to[0, \infty) $ be an admissible gauge. We study the transform

$ \delta_\psi(u, v) = \psi(\delta(u, v)) $

without assuming that $ \delta_\psi $ is a metric. The class $ \mathcal G $ preserves convergence to zero and Cauchy sequences, while the subclasses $ \mathcal G(q) $ provide a local power lower estimate that is sufficient for summability of Picard increments.

We prove fixed point theorems for linear gauge contractions and nonlinear gauge comparison contractions in the original complete metric space. We also obtain a range-zero-separating transform theorem, which requires neither monotonicity nor differentiability of the scalar function and contains both the linear gauge theorem and the recent $ k $-admissible transform theorem of Jleli and Samet. The global two-point inequalities give the fixed point property at the limit, so no continuity assumption on the mapping is needed. Every $ \theta $-hyperbolic sine distance is exactly a gauge transform with $ \psi = \theta\circ\sinh $. For the regular subclass in which this function belongs to $ \mathcal G(q) $, the corresponding regular-subclass versions of the hyperbolic-sine fixed point results follow directly; under these additional regularity assumptions, the sequential condition in the original formulation need not be imposed separately.

We also study the chain metric generated by a gauge transform. A local linear lower estimate guarantees that the chain distance is a metric; a root normalization gives a canonical equivalent-metric interpretation of every linear $ \mathcal G(q) $-gauge contraction. An integral operator is given, which is not a Banach contraction in the original metric. Finally, we construct a nonlinear Hutchinson operator, which has no uniform linear constant in the original Hausdorff metric or in the associated nonmetric gauge distance, and is not Banach contractive in the canonical chain metric generated by that gauge.

For admissible gauges that are continuous on $ [0, \infty) $, we also treat nonexpansive mappings using proofs that do not use the contraction results or the chain-metric construction. Following the normal-structure method of Takahashi, we work directly with gauge balls of the possibly nonmetric function $ D_\psi = \psi\circ\delta $. Gauge versions of convexity, $ \psi $-property (C), normal structure, and a compact radius-reduction property are formulated for $ D_\psi $. Fixed point results are obtained for $ \psi $-nonexpansive mappings, families with the invariant property, left amenable semigroups, commuting families, and one-parameter semigroups. The transformed distance is not assumed to satisfy the triangle inequality.