Monotone Hybrid Projection Algorithms for an Infinitely Countable Family of Lipschitz Generalized Asymptotically Quasi-Nonexpansive Mappings

and Applied Analysis 3 for all x, y ∈ C, where cn max{0, supx,y∈C ‖Tnx − Tny‖ − ‖x − y‖ } so that limn→∞cn 0. Hence, T is a generalized asymptotically nonexpansive mapping. The mapping T : C → C is said to be demiclosed at 0 if for each sequence {xn} in C converging weakly to x and {Txn} converging strongly to 0, we have Tx 0. A Banach space E is said to satisfy Opial’s property, see 4 , if for each x ∈ E and each sequence {xn}weakly convergent to x, the following condition holds for all x / y : lim inf n→∞ ‖xn − x‖ < lim inf n→∞ ∥ xn − y ∥ ∥. 1.9 Let τ be a Hausdorff linear topology and let E satisfy the uniform τ-Opial property. In 1993, Bruck, Kuczumow, and Reich proved that {Tnx} is τ-convergent if and only if {Tnx} is τ-asymptotically regular, that is, T 1x − Tx τ − → 0. 1.10 Moreover, they also proved that the τ-limit of {Tnx} is a fixed point of T . In 1953, Mann 5 introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H: xn 1 αnxn 1 − αn Txn, ∀n ∈ N, 1.11 where the initial point x0 is taken in C arbitrarily and {αn} is a sequence in 0, 1 . However, we note that Mann’s iteration process 1.11 has only weak convergence, in general; for instance, see 6–8 . In 2003, Nakajo and Takahashi 9 proposed the following modification of the Mann iteration for a single nonexpansive mapping T in a Hilbert space. They proved the following theorem. Theorem 1.1. Let C be a closed and convex subset of a Hilbert space H and let T : C → C be a nonexpansive mapping such that F T / ∅. Assume that {αn}n 0 is a sequence in 0, 1 such that αn ≤ 1 − δ for some δ ∈ 0, 1 . Define a sequence {xn}n 0 in C by the following algorithm: x0 ∈ C chosen arbitrarily, yn αnxn 1 − αn Txn, Cn { z ∈ C : ∥∥yn − z ∥ ∥ ≤ ‖xn − z‖ } , Qn {z ∈ C : 〈x0 − xn, xn − z〉 ≥ 0}, xn 1 PCn∩Qnx0. 1.12 Then {xn} defined by 1.12 converges strongly to PF T x0. Recently, Kim and Xu 10 extended the result of Nakajo and Takahashi 9 from nonexpansive mappings to asymptotically nonexpansive mappings. They proved the following theorem. 4 Abstract and Applied Analysis Theorem 1.2. Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H and let T : C → C be an asymptotically nonexpansive mapping with a sequence {kn} such that kn → 1 as n → ∞. Assume that {αn}n 0 is a sequence in 0, 1 such that lim supn→∞αn < 1. Define a sequence {xn} in C by the following algorithm: x0 ∈ C chosen arbitrarily, yn αnxn 1 − αn Txn, Cn { z ∈ C : ∥∥yn − z ∥ ∥ 2 ≤ ‖xn − z‖ θn } , Qn {z ∈ C : 〈x0 − xn, xn − z〉 ≥ 0}, xn 1 PCn∩Qnx0, 1.13 where θn 1 − αn k2 n − 1 diamC 2 → 0, as n → ∞. Then {xn} defined by 1.13 converges strongly to PF T x0. Since 2003, the strong convergence problems of the CQmethod for fixed point iteration processes in a Hilbert space or a Banach space have been studied bymany authors; see 9–20 . Let {Ti}i 1 be an infinitely family of uniformly Li-Lipschitzian and generalized asymptotically quasi-nonexpansivemappings and let F : ⋂∞ i 1 F Ti . In this paper, motivated by Kim and Xu 10 andNakajo and Takahashi 9 , we introduce two kinds of new algorithms for finding a common fixed point of a countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings which are modifications of the normal Mann iterative scheme. Our iterative schemes are defined as follows. Algorithm 1.3. For an initial point x0 ∈ C, compute the sequence {xn} by the iterative process: yi,n αi,nxn 1 − αi,n T i xn, Ci,n { z ∈ C : ∥∥yi,n − z ∥ ∥ 2 ≤ ‖xn − z‖ − αi,n 1 − αi,n ∥ ∥T i xn − xn ∥ ∥ 2 1 − αi,n θi,n } ,