We present two iterative schemes with errors which are proved to be strongly convergent to a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using the result we consider strong convergence theorems for variational inequalities and equilibrium problems in a real Hilbert space and strong convergence theorems for maximal monotone operators in a real uniformly smooth and uniformly convex Banach space. 1. Introduction Let be a real Banach space, and the dual space of . The function is denoted by for all , where is the normalized duality mapping from to . Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of [1] if contains a sequence which converges weakly to such that the strong equals 0. The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive if for all and nonexpansive with respect to the Lyapunov functional [2] if for all and it is called relatively nonexpansive [3–6] if and for all and . The asymptotic behavior of relatively nonexpansive mapping was studied in [3–6]. There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [7] introduced the iteration as follows: a sequence is defined by where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [1]. In an infinite-dimensional Hilbert space, Mann iteration can yield only weak convergence (see [8, 9]). Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [10] proposed the following modification of Mann iteration method (1.2) for nonexpansive mapping in a Hilbert space: in particular, they studied the strong convergence of the sequence generated by where and is the metric projection from onto . Recently, Takahashi et al. [11] extended iteration (1.6) to obtain strong convergence to a common fixed point of a countable family of nonexpansive mappings; let be a nonempty closed convex subset of a Hilbert space . Let and be families of nonexpansive mappings of into itself such that and let . Suppose that satisfies the NST-condition (I) with ; that is, for each bounded
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