Nonlinear Analysis 72 (2010) 1257–1265 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Weak convergence theorems for a finite family of strict pseudocontractions C.E. Chidume a , Naseer Shahzad b,∗ a The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy b Department of Mathematics, King Abdul Aziz University, P.O.B. 80203, Jeddah 21589, Saudi Arabia article info Article history: Received 25 March 2009 Accepted 3 August 2009 MSC: 47H09 47H10 47J05 65J15 Keywords: Fixed points Strict pseudocontractions Weak convergence theorems Uniformly smooth Banach spaces Uniformly convex Banach spaces abstract Let E be a uniformly smooth real Banach space which is also uniformly convex and K be a nonempty closed convex subset of E. Let T : K → K be a λ-strict pseudocontraction for some 0 ≤ λ< 1 with x ∗ ∈ F (T ) := {x ∈ K : Tx = x} =∅. For a fixed x 0 ∈ K , define a sequence {x n } by x n+1 = (1−α n )x n +α n Tx n , where {α n } is a sequence in [0, 1] satisfying the following conditions: (i) ∑ ∞ n=0 α n =∞; (ii) ∑ ∞ n=0 α 2 n < ∞. Then, {x n } converges weakly to a fixed point of T . Furthermore, weak convergence theorems are proved for a common fixed point for a finite family of strict pseudocontractions. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Let E be a real Banach space and E ∗ the dual space of E . Let J denote the normalized duality mapping from E into 2 E ∗ defined by J (x) ={f ∈ E ∗ :〈x, f 〉=‖x‖ 2 =‖f ‖ 2 }, where 〈., .〉 denotes the duality pairing. A mapping T with domain D(T ) and the range R(T ) in E is called a λ-strict pseudocontraction [1] for some 0 ≤ λ< 1 if for all x, y ∈ D(T ), there exists j(x − y) ∈ J (x − y) such that 〈Tx − Ty, j(x − y)〉≤‖x − y‖ 2 − λ‖x − y − (Tx − Ty)‖ 2 . (1.1) If I is the identity operator, then (1.1) can be written in the form 〈(I − T )x − (I − T )y, j(x − y)〉≥ λ‖(I − T )x − (I − T )y‖ 2 . (1.2) In Hilbert spaces, (1.1) (and so (1.2)) is equivalent to the following inequality ‖Tx − Ty‖ 2 ≤‖x − y‖ 2 + κ ‖x − y − (Tx − Ty)‖ 2 (1.3) ∗ Corresponding author. E-mail addresses: chidume@ictp.it (C.E. Chidume), Naseer_shahzad@hotmail.com, nshahzad@kau.edu.sa (N. Shahzad). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.08.009