Nonlinear operator-valued Reich contractions via cone b-metric spaces
Year: 2026
Type: article
Abstract: Maximum-type nonlinear contractions in real-valued b -metric spaces depend essentially on the total order of \([0,\infty )\) . Such conditions cannot be transferred directly to cone-valued distances, since a general ordered Banach space is only partially ordered and finite maxima of cone elements need not exist. This paper develops a cone b -metric fixed point framework that avoids this obstruction by separating intrinsic cone-order arguments from scalar-gauge reductions. The main theorem is a nonlinear comparison principle governed by an order-preserving map \(\Psi :P\to P\) whose iterates satisfy an s -Picard summability condition. It yields existence, uniqueness, constructive Picard convergence, and a posteriori cone-order error estimates without assuming linearity, additivity, or homogeneity of the comparison mechanism. As structured consequences, we derive positive operator-valued contractions and an operator-valued Reich-type theorem involving the resolvent \(\mathcal{R}=(I-B)^{-1}(A+B)\) under the spectral condition \(r(\mathcal{R})<1/s\) . Maximum-type nonlinear contractions are then recovered only after nonlinear scalarization, where real maxima are well defined. The framework also gives Ulam–Hyers stability, data-dependence estimates, examples over the positive cone of \(C[0,1]\) , and an application to a Caputo fractional Volterra-type initial value problem. In the application, the positive fractional Volterra majorant has spectral radius zero, so uniqueness follows without the customary scalar smallness condition on the Lipschitz bound, fractional horizon, or interval length.
Author: Thokchom Chhatrajit Singh
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,790
OpenAlex ID: W7203704875
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