In this note, we study the nonexpansive properties based on arbitrary variable metric and explore the connections between firm nonexpansiveness, cocoerciveness and averagedness. A convergence rate analysis for the associated fixed-point iterations is presented by developing the global ergodic and non-ergodic iteration-complexity bounds in terms of metric distances. The obtained results are finally exemplified with the metric resolvent, which provides a unified framework for many existing first-order operator splitting algorithms.
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