Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 413468, 15 pages doi:10.1155/2012/413468 Research Article Existence and Algorithm for Solving the System of Mixed Variational Inequalities in Banach Spaces Siwaporn Saewan and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 22 December 2011; Accepted 29 January 2012 Academic Editor: Hong-Kun Xu Copyright q 2012 S. Saewan and P. Kumam. 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. The purpose of this paper is to study the existence and convergence analysis of the solutions of the system of mixed variational inequalities in Banach spaces by using the generalized f projection operator. The results presented in this paper improve and extend important recent results of Zhang et al. 2011 and Wu and Huang 2007 and some recent results. 1. Introduction Let E be a real Banach space with norm ‖·‖, let C be a nonempty closed and convex subset of E, and let E ∗ denote the dual of E. Let 〈·, ·〉 denote the duality pairing of E ∗ and E. If E is a Hilbert space, 〈·, ·〉 denotes an inner product on E. It is well known that the metric projection operator P C : E → C plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problems, and so forth see, e.g., 1, 2 and the references therein. In 1993, Alber 3 introduced and studied the generalized projections π C : E ∗ → C and Π E : E → C from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber 1 presented some applications of the generalized projections to approximately solving variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li 2 extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang 4 introduced a new generalized f -projection operator in Banach spaces which extended the definition of the generalized projection operators introduced by Abler 3 and proved some properties of the generalized f -projection operator.