Nonlinear Analysis 71 (2009) 293–300 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Equivalent theorems of the convergence between proximal type algorithms ✩ Yisheng Song ∗ College of Mathematics and Information Science, Henan Normal University, XinXiang, 453007, PR China article info Article history: Received 2 January 2008 Accepted 20 October 2008 MSC: 47H06 47J05 47J25 47H10 47H17 Keywords: Rockafellar type proximal point algorithms Viscosity approximation methods Weak contractions Halpern type iterations Browder type iterations abstract The main aim of this work is to give the equivalence of the convergence between proximal type algorithms. Our convergence analysis covers Rockafellar type proximal point algorithm (with a weak contraction) for finding zeros of accretive operators as well as Browder and Halpern type iterations and viscosity approximation methods for finding fixed points of nonexpansive type mappings in Banach spaces. We also provide simple proofs of some results of Wong, Sahu and Yao [N.C. Wong, D.R. Sahu, J.C. Yao, Solving variational inequalities involving nonexpansive type mappings, Nonlinear Anal. 69 (12) (2008) 4732–4753]. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Throughout this paper, let E be a real Banach space with the norm ‖·‖ and the dual space E ∗ , and 〈·, ·〉 denote the generalized duality pairing, and K a nonempty closed convex subset of E . The fixed point set of an operator T is denoted by F (T ) := {x ∈ K ; Tx = x}. Let J denote the normalized duality mapping from E into 2 E ∗ given by J (x) ={f ∈ E ∗ , 〈x, f 〉=‖x‖‖f ‖, ‖x‖=‖f ‖}, ∀ x ∈ E . It is well known (see, for example, [35, Theorem 4.3.1, 4.3.2]) that E is smooth (equivalently, E ∗ is strict convex [35, Page 113, Problem 3]) if and only if J is single-valued. In the sequel, we shall denote the single-valued normalized duality map by j. A mapping A : D(A) ⊂ E → 2 E is called accretive if for all x, y ∈ D(A) there exists j(x − y) ∈ J (x − y) such that 〈u − v, j(x − y)〉≥ 0, for u ∈ Ax and v ∈ Ay; If E is a Hilbert space, accretive operators are also called monotone. Let A −1 0 ={x ∈ D(A); 0 ∈ Ax}. An operator A is called m-accretive if it is accretive and R(I + rA), the range of (I + rA), is E for all r > 0; and A is said to satisfy the range condition if D(A) ⊂ R(I + rA), ∀r > 0, where I is an identity operator of E and D(A) denotes the closure of the domain of A. Interest in ✩ This work is supported by the Chinese National Tianyuan Foundation (10726073). ∗ Tel.: +86 03733326148; fax: +86 03733326174. E-mail addresses: songyisheng123@yahoo.com.cn, songyisheng123@163.com. URLs: http://songys.mysinamail.com/, http://songys.16f.cn. 0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.067