  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.
