Bull. Korean Math. Soc. 51 (2014), No. 3, pp. 773–788 http://dx.doi.org/10.4134/BKMS.2014.51.3.773 APPROXIMATION METHODS FOR A COMMON MINIMUM-NORM POINT OF A SOLUTION OF VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS IN BANACH SPACES N. Shahzad and H. Zegeye Abstract. We introduce an iterative process which converges strongly to a common minimum-norm point of solutions of variational inequality problem for a monotone mapping and fixed points of a finite family of relatively nonexpansive mappings in Banach spaces. Our theorems im- prove most of the results that have been proved for this important class of nonlinear operators. 1. Introduction Let E be a real Banach space with dual E * . We denote by J the normalized duality mapping from E into 2 E * defined for each x ∈ E by Jx := {f * ∈ E * : 〈x, f * 〉 = ||x|| 2 = ||f * || 2 }, where 〈·, ·〉 denotes the generalized duality pairing between members of E and E * . It is well known that E is smooth if and only if J is single-valued and if E is uniformly smooth, then J is uniformly continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex real Banach space with a strictly convex dual, then J -1 is single valued, one-to-one, surjective, and it is the duality mapping from E * into E and thus JJ -1 = I E * and J -1 J = I E (see [17]). If E = H , a real Hilbert space, then the duality mapping becomes the identity map on H . Let E be a smooth real Banach space with dual E * . Let the Lyapunov functional φ : E × E → R, introduced by Alber [1], be defined by φ(y,x)= ||y|| 2 − 2〈y,Jx〉 + ||x|| 2 for x, y ∈ E, (1.1) Received April 7, 2013; Revised May 21, 2013. 2010 Mathematics Subject Classification. 47H05, 47H09, 47H10, 47J05, 47J25. Key words and phrases. monotone mappings, relatively nonexpansive mappings, strong convergence, variational inequality problems. c 2014 Korean Mathematical Society 773