Approximating solutions of variational inequalities for asymptotically nonexpansive mappings

By using viscosity approximation methods for asymptotically nonexpansive mappings in Banach spaces, some sufficient and necessary conditions for a new type of iterative sequences to converging to a fixed point which is also the unique solution of some variational inequalities are obtained. The results presented in the paper extend and improve some recent results in [C.E. Chidume, Jinlu Li, A. Udomene, Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 138 (2) (2005) 473-480; N. Shahzad, A. Udomene, Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 64 (2006) 558-567; T.C. Lim, H.K. Xu, Fixed point theories for asymptotically nonexpansive mappings, Nonlinear Anal. TMA, 22 (1994) 1345-1355; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004) 279-291].