In this paper, we consider a generalized forward-backward splitting (G-FBS) operator for solving the monotone inclusions, and analyze its nonexpansive properties in a context of arbitrary variable metric. Then, for the associated fixed-point iterations (i.e. the G-FBS algorithms), the global ergodic and pointwise convergence rates of metric distance are obtained from the nonexpansiveness. The convergence in terms of objective function value is also investigated, when the G-FBS operator is applied to a minimization problem. A main contribution of this paper is to show that the G-FBS operator provides a simplifying and unifying framework to model and analyze a great variety of operator splitting algorithms, where the convergence behaviours can be easily described by the fixed-point construction of this simple operator.
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