Nonexpansive mappings and iterative methods for finding their fixed points are an ongoing topic that receives a lot of attention from various areas such that nonlin- ear analysis and optimization. In this paper we leverage the connections between nonexpansive mappings, resolvent of maximal monotone operators, and proximal mappings of proper closed convex functions to introduce a new fixed-point problem, propose some algorithms, based on Krasnoselski-Mann and Halpern’s iteration methods. Then, we provide some related convergence results and we translate the analysis developed in DC-optimization problems and some non-monotone inclusions.