On the contraction-proximal point algorithms with multi-parameters

In this paper we consider the contraction-proximal point algorithm: xn+1 = αnu+λnxn+γnJβnxn, where Jβn denotes the resolvent of a monotone operator A. Under the assumption that limn αn = 0, ∑ n αn = ∞, lim infn βn > 0, and lim infn γn > 0, we prove the strong convergence of the iterates as well as its inexact version. As a result we improve and recover some recent results by Boikanyo and Morosanu (Four parameter proximal point algorithms, Nonlinear Anal., 74 (2011), 544–555, and Inexact Halpern-type proximal point algorithm, to appear in J. Global Optim.).