A Simple proof for the algorithms of relaxed $(u,v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings

In this paper, a simple proof is presented for the convergence of the algorithms for the class of relaxed $(u,v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings. Based on $\alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u,v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.