Numerical Functional Analysis and Optimization, 29(9–10):987–1033, 2008 Copyright © Taylor & Francis Group, LLC ISSN: 0163-0563 print/1532-2467 online DOI: 10.1080/01630560802418391 MANN-TYPE STEEPEST-DESCENT AND MODIFIED HYBRID STEEPEST-DESCENT METHODS FOR VARIATIONAL INEQUALITIES IN BANACH SPACES Lu-Chuan Ceng 1 , Qamrul Hasan Ansari 2 , and Jen-Chih Yao 3 1 Department of Mathematics, Shanghai Normal University, Shanghai; and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China 2 Department of Mathematics & Statistics, College of Science, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia; and Department of Mathematics, Aligarh Muslim University, Aligarh, India 3 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan  In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality. Keywords Convergence analysis; Mann-type steepest-descent method; Modified hybrid steepest-descent method; Nonexpansive maps; Resolvent operators; Variational inequalities. AMS Subject Classification 49J40; 47J20; 47H10; 47H06; 65K10. Address correspondence to Qamrul Hasan Ansari, Department of Mathematics & Statistics, College of Science, P.O. Box 1169, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia; E-mail: qhansari@kfupm.edu.sa 987 Downloaded By: [Ansari, Qamrul Hasan] At: 09:01 13 November 2008