Thai Journal of Mathematics Volume 15 (2017) Number 1 : 193–206 http://thaijmath.in.cmu.ac.th ISSN 1686-0209 Generalized Projection Methods for Nonlinear Mappings Preedaporn Kanjanasamranwong, Sarapee Chairat and Siwaporn Saewan 1 Department of Mathematics and Statistics, Faculty of Science Thaksin University, Thailand e-mail : ypreedaporn@hotmail.com (P. Kanjanasamranwong) sarapee@tsu.ac.th (S. Chairat) siwaporn@scholar.tsu.ac.th (S. Saewan) Abstract : We present a new hybrid iterative process for finding an element in the solution of variational inequality problem and the fixed point set of relatively nonexpansive multi-valued mapping in Banach spaces. This theorem improve and extend some recent results. Keywords : multi-valued mapping; variational inequality; relatively nonexpan- sive. 2010 Mathematics Subject Classification : 47H05; 47H09: 47H10; 47J10. 1 Introduction Let C be a nonempty closed and convex subset of a Banach space E with dual E ∗ . A mapping A : C → E ∗ is said to be: (1) monotone if 〈x − y, Ax − Ay〉≥ 0 for all x, y ∈ C; (2) α-inverse-strongly monotone if there exists a constant α> 0 such that 〈x − y, Ax − Ay〉≥ α‖Ax − Ay‖ 2 1 Corresponding author. Copyright c  2017 by the Mathematical Association of Thailand. All rights reserved.