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

Document Type : Research Paper


1 Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran

2 Department of Mathematics Texas and University-Kingsville 700 University Blvd., MSC 172 Kingsville, Texas 78363-8202, USA

3 School of Mathematics, Statistics, National University of Ireland, Galway, Ireland


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.