Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 240450, 11 pages doi:10.1155/2010/240450 Research Article Strong and Weak Convergence of the Modified Proximal Point Algorithms in Hilbert Space Xinkuan Chai, 1 Bo Li, 2 and Yisheng Song 1 1 College of Mathematics and Information Science, Henan Normal University, XinXiang 453007, China 2 School of Mathematics and Statistics, AnYang Normal University, AnYang 455000, China Correspondence should be addressed to Yisheng Song, songyisheng123@yahoo.com.cn Received 26 October 2009; Revised 25 November 2009; Accepted 10 December 2009 Academic Editor: Tomonari Suzuki Copyright q 2010 Xinkuan Chai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a monotone operator T , we shall show weak convergence of Rockafellar’s proximal point algorithm to some zero of T and strong convergence of the perturbed version of Rockafellar’s to P Z u under some relaxed conditions, where P Z is the metric projection from H onto Z  T -1 0. Moreover, our proof techniques are simpler than some existed results. 1. Introduction Throughout this paper, let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖·‖, and let I be on identity operator in H. We shall denote by N the set of all positive integers, by Z the set of all zeros of T , that is, Z  T -1 0  {x ∈ DT ;0 ∈ Tx} and by FT  the set of all fixed points of T , that is, FT  {x ∈ E; Tx  x}. When {x n } is a sequence in E, then x n → x resp., x n ⇀x, x n ∗ ⇀x will denote strong resp., weak, weak ∗  convergence of the sequence {x n } to x. Let T be an operator with domain DT  and range RT  in H. Recall that T is said to be monotone if  x - y,x ′ - y ′  ≥ 0, ∀x, y ∈ DT ,x′∈ Tx, y′∈ Ty. 1.1 A monotone operator T is said to be maximal monotone if T is monotone and RI  rT  H for all r> 0.