Nonlinear Analysis: Hybrid Systems 1 (2007) 398–413 www.elsevier.com/locate/nahs A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces Liwei Li a,b , W. Song b,∗ a Department of Mathematics, Northeast Normal University, Changchun 130024, China b Department of Mathematics, Harbin Normal University, Harbin 150080, China Received 14 July 2006; accepted 31 August 2006 Abstract In the paper, we introduce two iterative sequences for finding a point in the intersection of the zero set of a inverse strongly monotone or inverse-monotone operator and the zero set of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. We prove weak convergence theorems under appropriate conditions, respectively. c  2006 Elsevier Ltd. All rights reserved. Keywords: Extragradient method; Proximal point algorithm; Uniformly convex and uniformly smooth Banach space; Weakly sequentially continuous; Nonexpansive operators; Inverse-monotone operator; Maximal monotone operators; Weak convergence 1. Introduction Let X be a Banach space with the dual space X ∗ and let T : X → X ∗ be an operator. The problem of finding v ∈ X satisfying 0 ∈ T v is connected with the convex minimization problems and variational inequalities. A set-valued mapping T : X → X ∗ with domain D(T ) ={x ∈ X | Tx =∅} and range R(T ) ={x ∗ ∈ X ∗ | x ∗ ∈ Tx , x ∈ D(T )} is said to be monotone if 〈x − y , x ∗ − y ∗ 〉≥ 0 for all x ∗ ∈ Tx , y ∗ ∈ Ty . T is said to be maximal monotone if the graph G(T ) of T is not properly contained in the graph of any other monotone operator. When T is maximal monotone, a well-known method for solving the equation 0 ∈ T v in a Hilbert space H is the proximal point algorithm (see [13]): x 1 = x ∈ H and x n+1 = J r n x n , n = 1, 2,..., (1.1) where {r n }⊂ (0, ∞) and J r = ( I + rT ) −1 for all r > 0 is the resolvent operator for T . Rockafellar (see [13]) proved the weak convergence of the algorithm (1.1). ∗ Corresponding author. E-mail address: wsong218@yahoo.com.cn (W. Song). 1751-570X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2006.08.003