Metric constructions and fixed point theorems in product spaces
Doan Huu Hieu, Vo Minh Tam, Nguyen Duy Cuong
TL;DR
The paper develops a general scheme to construct metrics on finite product spaces using a family of continuous convex functions on the standard simplex, unifying and extending the conventional $p$-metrics and proving topological equivalence to standard product metrics. It then derives fixed-point and approximate fixed-point results for nonexpansive maps on product spaces under broad, interpretable conditions, showing that many existing results are special cases of this framework. The framework is further extended to products of length and geodesic spaces, providing explicit formulas for lengths of product curves and establishing that the product inherits the length or geodesic structure exactly when each factor has it. Collectively, the work offers a versatile toolkit for fixed-point analysis and metric-geometry on product spaces, with implications for variational analysis and geometric reasoning in nonlinear analysis.
Abstract
The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are topologically equivalent to the conventional ones. As an application, we study fixed point and approximate fixed point properties for nonexpansive maps on a product space equipped with the constructed metric. We show that existing fixed point results of this type are consequences of our framework. Examples are provided to illustrate the established results. The construction machinery is also used to study products of length and geodesic spaces. The obtained results encompass existing ones and provide a background for potential studies of fixed point properties on these product spaces.