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.