A connection between Chidume iteration and the contraction-proximal point algorithm

Abstract

In this short note we comment on some recent results on the convergence of two type of iterative algorithms for finding a zero of a maximal monotone operator. The first one is based on a proximal approach and was investigated by many authors, see for example (Boikanyo and Morosanu in Nonlinear Anal TMA 74:544–555, ;2011 Kamimura and Takahashi in J Approx Theory 106:226–240, 2000, Marino and Xu in Comm Pure Appl Anal 3:791–808, 2004, Wang and Cui in J Global Optim 54:485–491, 2012, Yao and Noor in J Comput Appl Math 217:46–55, 2008) and the references therein. The second is based on a gradient approach and was essentially developed by Chidume et al. see for instance (Fixed Point Theory 2013, Chidume and Djitte in Abst Appl Anal doi:10.1155/2012/6813482012, Chidume et al. in J Nonlinear Conv Anal (to appear) 2013, Ofoedu and Zegeye in J Math Anal Appl 372:68–76, 2010) and the references therein. Then, relying on the Yosida approximate, we compare and make a link between these two approaches.