In this paper, we study the nonexpansive properties of metric resolvent, and present a convergence rate analysis for the associated fixed-point iterations (Banach-Picard and Krasnoselskii-Mann types). Equipped with a variable metric, we develop the global ergodic and non-ergodic iteration-complexity bounds in terms of both solution distance and objective value. A byproduct of our expositions also extends the proximity operator and Moreaus decomposition identity to arbitrary variable metric. It is further shown that many classes of the first-order operator splitting algorithms, including alternating direction methods of multipliers, primal-dual hybrid gradient and Bregman iterations, can be expressed by the fixed-point iterations of a simple metric resolvent, and thus, the convergence can be analyzed within this unified framework.
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