The resolvent is a fundamental concept in studying various operator splitting algorithms.\nIn this paper, we investigate the problem of computing the resolvent of compositions of operators\nwith bounded linear operators. First, we discuss several explicit solutions of this resolvent operator\nby taking into account additional constraints on the linear operator. Second, we propose a fixed point\napproach for computing this resolvent operator in a general case. Based on the Krasnoselskiiâ??Mann\nalgorithm for finding fixed points of non-expansive operators, we prove the strong convergence of\nthe sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative\nalgorithm for solving the scaled proximity operator of a convex function composed by a linear\noperator, which has wide applications in image restoration and image reconstruction problems.\nFurthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite\nsum of maximally monotone operators as well as the proximal operator of a finite sum of proper,\nlower semi-continuous convex functions.
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