International Journal of Analysis and Applications ISSN 2291-8639 Volume 1, Number 1 (2013), 79-99 http://www.etamaths.com CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS FOR NON-LIPSCHITZIAN ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPINGS G. S. SALUJA Abstract. The aim of this paper is to study an implicit iterative process with errors for two finite families of non-Lipschitzian asymptotically quasi- nonexpansive type mappings in the framework of real Banach spaces. In this paper, we have obtained a necessary and sufficient condition to converge to common fixed points for proposed scheme and mappings and also obtained strong convergence theorems by using semi-compactness and Condition (B′). 1. Introduction Let E be a real Banach space and UE = {x ∈ E : ‖x‖ = 1}. E is said to be uniformly convex if for any ε ∈ (0, 2] there exists δ > 0 such that for any x,y ∈UE, ‖x−y‖≥ ε implies ∥∥∥∥x−y2 ∥∥∥∥ ≤ 1 − δ. In 1973, Petryshyn and Williamson [13] established a necessary and sufficient condition for a Mann [12] iterative sequence to converge strongly to a fixed point of quasi-nonexpansive mapping. Subsequently, Ghosh and Debnath [5] extended the results of [13] and obtained some necessary and sufficient conditions for an Ishikawa-type iterative sequence to converge to a fixed point of quasi-nonexpansive mapping. In 2001, Liu in [10, 11] extended the results of Ghosh and Debnath [5] to a more general asymptotically quasi-nonexpansive mappings. In 2003, Sahu and Jung [15] studied Ishikawa and Mann iteration process in Banach spaces and they proved some weak and strong convergence theorems for asymptotically quasi- nonexpansive type mapping. In 2006, Shahzad and Udomene [17] gave the necessary and sufficient condition for convergence of common fixed point of two-step modified Ishikawa iterative sequence for two asymptotically quasi-nonexpansive mappings in real Banach space. Recently, Qin et al. [14] studied a general implicit iterative process for a finite family of generalized asymptotically quasi-nonexpansive mappings and established 2010 Mathematics Subject Classification. 47H09, 47H10, 47J25. Key words and phrases. Asymptotically quasi-nonexpansive type mapping, general implicit iterative process with errors, common fixed point, strong convergence, uniformly convex Banach space. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 79 80 SALUJA strong convergence theorem of the proposed iterative process in the framework of real Banach space. The main goal of this paper is to establish the strong convergence of general implicit iterative process studied by Qin et al. [14] which includes Schu’s explicit iterative processes and Sun’s implicit iterative processes as special cases for two fi- nite families of non-Lipschitzian asymptotically quasi-nonexpansive type mappings on a closed convex unbounded set in a real uniformly convex Banach spaces. Our results unify, improve and generalize many known results given in the existing cur- rent literature. 2. Preliminaries and lemmas Let C be a nonempty subset of a normed space E and T : C → C be a given mapping. The set of fixed points of T is denoted by F(T), that is, F(T) = {x ∈ C : T(x) = x}. The mapping T is said to be (1) nonexpansive if ‖Tx−Ty‖ ≤ ‖x−y‖(2.1) for all x,y ∈ C. (2) quasi-nonexpansive [2] if ‖Tx−p‖ ≤ ‖x−p‖(2.2) for all x ∈ C, p ∈ F(T). (3) asymptotically nonexpansive [6] if there exists a sequence {un} in [0,∞) with limn→∞un = 0 such that ‖Tnx−Tny‖ ≤ (1 + un)‖x−y‖(2.3) for all x,y ∈ C and n ≥ 1. (4) asymptotically quasi-nonexpansive if F(T) 6= ∅ and there exists a sequence {un} in [0,∞) with limn→∞un = 0 such that ‖Tnx−p‖ ≤ (1 + un)‖x−p‖(2.4) for all x ∈ C, p ∈ F(T) and n ≥ 1. (5) uniformly L-Lipschitzian if there exists a positive constant L such that ‖Tnx−Tny‖ ≤ L‖x−y‖(2.5) for all x,y ∈ C and n ≥ 1. (6) asymptotically nonexpansive type [8], if lim sup n→∞ { sup x,y∈C ( ‖Tnx−Tny‖−‖x−y‖ )} ≤ 0.(2.6) CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 81 (7) asymptotically quasi-nonexpansive type [15], if F(T) 6= ∅ and lim sup n→∞ { sup x∈C, p∈F (T ) ( ‖Tnx−p‖−‖x−p‖ )} ≤ 0.(2.7) Remark 2.1. It is easy to see that if F(T) is nonempty, then asymptotically nonexpansive mapping, asymptotically quasi-nonexpansive mapping and asymp- totically nonexpansive type mapping are the special cases of asymptotically quasi- nonexpansive type mappings. The Mann and Ishikawa iteration processes have been used by a number of authors to approximate the fixed points of nonexpansive, asymptotically nonex- pansive mappings, and quasi-nonexpansive mappings on Banach spaces (see, e.g., [5, 7, 9, 10, 11, 13, 18, 23]). Recall that the modified Mann iteration which was introduced by Schu [16] generates a sequence {xn} in the following manner: x1 ∈ C, xn+1 = (1 −αn)xn + αnTnxn, n ≥ 1,(2.8) where {αn} is a real sequence in the interval (0, 1) and T : C → C is an asymptot- ically nonexpansive mapping. In 2001, Xu and Ori [24] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a Hilbert space H. Let C be a nonempty subset of H. Let T1,T2, . . . ,TN be self-mappings of C and suppose that F = ∩Ni=1F(Ti) 6= ∅, the set of common fixed points of Ti, i = 1, 2, . . . ,N. An implicit iteration process for a finite family of nonexpansive mappings is defined as follows, with {tn} a real sequence in (0, 1), x0 ∈ C: x1 = t1x0 + (1 − t1)T1x1, x2 = t2x1 + (1 − t2)T2x2, ... xN = tNxN−1 + (1 − tN )TNxN, xN+1 = tN+1xN + (1 − tN+1)T1xN+1, ... which can be written in the following compact form: xn = tnxn−1 + (1 − tn)Tnxn, n ≥ 1(2.9) where Tk = Tk mod N . (Here the mod N function takes values in N). And they proved the weak convergence of the process (2.4). In 2003, Sun [19] extend the process (2.9) to a process for a finite family of asymptotically quasi-nonexpansive mappings, with {αn} a real sequence in (0, 1) 82 SALUJA and an initial point x0 ∈ C, which is defined as follows: x1 = α1x0 + (1 −α1)T1x1, ... xN = αNxN−1 + (1 −αN )TNxN, xN+1 = αN+1xN + (1 −αN+1)T21 xN+1, ... x2N = α2Nx2N−1 + (1 −α2N )T2Nx2N, x2N+1 = α2N+1x2N + (1 −α2N+1)T31 x2N+1, ... which can be written in the following compact form: xn = αnxn−1 + (1 −αn)Tki xn, n ≥ 1(2.10) where n = (k − 1)N + i, i ∈N . Sun [19] proved the strong convergence of the process (2.10) to a common fixed point, requiring only one member T in the family {Ti : i ∈N} to be semi-compact. The result of Sun [19] generalized and extended the corresponding main results of Wittmann [21] and Xu and Ori [24]. In 2010, Qin et al. [14] studied the following general implicit iteration process for two finite families of generalized asymptotically quasi-nonexpansive mappings {S1,S2, . . . ,SN} and {T1,T2, . . . ,TN}: x1 = α1x0 + β1S1x0 + γ1T1x1 + δ1u1, x2 = α2x1 + β2S2x1 + γ2T2x2 + δ2u2, ... xN = αNxN−1 + βNSNxN−1 + γNTNxN + δNuN, xN+1 = αN+1xN + βN+1S 2 1xN + γN+1T 2 1 xN+1 + δN+1uN+1,(2.11) ... x2N = α2Nx2N−1 + β2NS 2 Nx2N−1 + γ2NT 2 Nx2N + δ2Nu2N, x2N+1 = α2N+1x2N + β2N+1S 3 1x2N + γ2N+1T 3 1 x2N+1 + δ2N+1u2N+1, ... which can be written in the following compact form: xn = αnxn−1 + βnS h(n) i(n) xn−1 + γnT h(n) i(n) xn + δnun, n ≥ 1(2.12) where x0 is the initial value, {un} is a bounded sequence in C, and {αn}, {βn}, {γn} and {δn} are sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Since for each n ≥ 1, it can be written as n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . ,N} = N , h = h(n) ≥ 1 is a positive integer and h(n) →∞ as n →∞. CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 83 In this paper, motivated by [14], we study general implicit iterative process (2.12) for two finite families of non-Lipschitzian asymptotically quasi-nonexpansive type mappings {S1,S2, . . . ,SN} and {T1,T2, . . . ,TN} in Banach spaces. Also we estab- lish some strong convergence theorems for said scheme and mappings. We remark that implicit iterative process (2.12) which includes the explicit itera- tive process (2.8) and the implicit iterative process (2.10) as a special case in general. If Si = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then the implicit iterative process (2.12) is reduced to the following implicit iterative process: xn = (αn + βn)xn−1 + γnT h(n) i(n) xn + δnun, n ≥ 1.(2.13) If Ti = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N} = N , then the implicit iterative process (2.12) is reduced to the following explicit iterative process: xn = αn 1 −γn xn−1 + βn 1 −γn S h(n) i(n) xn + δn 1 −γn un, n ≥ 1.(2.14) Denote the indexing set {1, 2, . . . ,N} by N . Let {Ti : i ∈ N} be N uniformly Lt,i-Lipschitzian asymptotically quasi-nonexpansive type self-mappings of C and {Si : i ∈ N} be N uniformly Ls,i-Lipschitzian asymptotically quasi-nonexpansive type self-mappings of C. We show that (2.12) exists. Let x0 ∈ C and x1 = α1x0 + β1S1x0+γ1T1x1+δ1u1. Define W : C → C by: Wx = α1x0+β1S1x0+γ1T1x1+δ1u1 for all x ∈ C. The existence of x1 is guaranteed if W has a fixed point. For any x,y ∈ C, we have ‖Wx−Wy‖ ≤ γ1 ‖T1x−T1y‖≤ γ1Lt,1 ‖x−y‖ ≤ γ1Lt ‖x−y‖ ,(2.15) where Lt = max{Lt,i : 1 ≤ i ≤ N}. Now, W is a contraction if γ1Lt < 1 or Lt < 1/γ1. As γ1 ∈ (0, 1), therefore W is a contraction even if Lt > 1. By the Banach contraction principle, W has a unique fixed point. Thus, the existence of x1 is established. Similarly, we can establish the existence of x2,x3,x4, . . . . Thus, the implicit algorithm (2.12) is well defined. The distance between a point x and a set C and closed ball with center zero and radius r in E are, respectively, defined by d(x,C) = inf y∈C ‖x−y‖ , Br(0) = {x ∈ E : ‖x‖≤ r}.(2.16) In order to prove our main results, we need the following definition and lemmas. Definition 2.1.(see [19]) Let C be a closed subset of a normed space E and let T : C → C be a mapping. Then T is said to be semi-compact if for any bounded sequence {xn} in C with ‖xn −Txn‖→ 0 as n →∞, there is a subsequence {xnk} of {xn} such that xnk → x ∗ ∈ C as nk →∞. 84 SALUJA Lemma 2.1.(see [20]) Let {an} and {bn} be sequences of nonnegative real num- bers satisfying the inequality an+1 ≤ an + bn, n ≥ 1.(2.17) If ∑∞ n=1 bn < ∞, then limn→∞an exists. In particular, if {an} has a subsequence converging to zero, then limn→∞an = 0. Lemma 2.2.(see [14]) Let E be a uniformly convex Banach space, s > 0 a positive number, and Bs(0) a closed ball of E. Then there exists a continuous, strictly increasing and convex function g : [0,∞) → [0,∞) with g(0) = 0 such that ‖ax + by + cz + dw‖2 ≤ a‖x‖2 + b‖y‖2 + c‖z‖2 + d‖w‖2 −abg(‖x−y‖)(2.18) for all x,y,z,w ∈ Bs(0) = {x ∈ E : ‖x‖ ≤ s} and a,b,c,d ∈ [0, 1] such that a + b + c + d = 1. 3. Main Results We begin with a necessary and sufficient condition for convergence of {xn} gen- erated by the general implicit iterative process (2.12) to prove the following result. Theorem 3.1. Let C be a nonempty closed convex subset of a Banach s- pace E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymptotically quasi- nonexpansive type mapping and let Si : C → C be a uniformly Ls,i-Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) ⋂ ∩Ni=1F(Si) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn +βn +γn +δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.12). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N}(3.1) and Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N} (3.2) such that ∑∞ n=1 An < ∞ and ∑∞ n=1 Bn < ∞. Assume that the following restric- tions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. Then {xn} converges strongly to some point in F if and only if lim inf n→∞ d(xn,F) = 0. CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 85 Proof. The necessity is obvious and so it is omitted. Now, we prove the suffi- ciency. For any p ∈ F, from (2.12), (3.1) and (3.2), we have ‖xn −p‖ = ∥∥∥αnxn−1 + βnSh(n)i(n) xn−1 + γnTh(n)i(n) xn + δnun −p∥∥∥ = ∥∥∥αn(xn−1 −p) + βn(Sh(n)i(n) xn−1 −p) + γn(Th(n)i(n) xn −p) + δn(un −p)∥∥∥ ≤ αn ‖xn−1 −p‖ + βn ∥∥∥Sh(n)i(n) xn−1 −p∥∥∥ + γn ∥∥∥Th(n)i(n) xn −p∥∥∥ +δn ‖un −p‖ ≤ αn ‖xn−1 −p‖ + βn(‖xn−1 −p‖ + Bn) + γn(‖xn −p‖ + An) +δn ‖un −p‖ = (αn + βn)‖xn−1 −p‖ + γn ‖xn −p‖ + βnBn + γnAn +δn ‖un −p‖ = (1 −γn −δn)‖xn−1 −p‖ + γn ‖xn −p‖ + βnBn + γnAn +δn ‖un −p‖ ≤ (1 −γn)‖xn−1 −p‖ + γn ‖xn −p‖ + An + Bn +δn ‖un −p‖ .(3.3) Since from restriction (a) γn ≤ d it follows from (3.3) that ‖xn −p‖ ≤ ‖xn−1 −p‖ + 1 1 −γn (An + Bn) + δn 1 −γn ‖un −p‖ ≤ ‖xn−1 −p‖ + 1 1 −d (An + Bn) + δn 1 −d ‖un −p‖ ≤ ‖xn−1 −p‖ + 1 1 −d (An + Bn) + M 1 −d δn,(3.4) where M = supn≥1{‖un −p‖}, since {un} is a bounded sequence in C. This implies that d(xn,F) ≤ d(xn−1,F) + mn,(3.5) where mn = 1 1−d (An + Bn) + M 1−dδn. Since by assumptions of the theorem,∑∞ n=1 An < ∞, ∑∞ n=1 Bn < ∞ and ∑∞ n=1 δn < ∞, it follows that ∑∞ n=1 mn < ∞. Therefore, from Lemma 2.1, we know that limn→∞d(xn,F) exists. Since by hy- pothesis lim infn→∞d(xn,F) = 0, so by Lemma 2.1, we have lim n→∞ d(xn,F) = 0.(3.6) 86 SALUJA Next we prove that {xn} is a Cauchy sequence in C. It follows from (3.4) that for any m ≥ 1, for all n ≥ n0 and for any p ∈ F, we have ‖xn+m −p‖ ≤ ‖xn+m−1 −p‖ + 1 1 −d (An+m + Bn+m) + M 1 −d δn+m ≤ ‖xn+m−2 −p‖ + 1 1 −d (An+m−1 + Bn+m−1) + M 1 −d δn+m−1 + 1 1 −d (An+m + Bn+m) + M 1 −d δn+m ≤ ‖xn+m−2 −p‖ + 1 1 −d [(An+m + An+m−1) + (Bn+m + Bn+m−1)] + M 1 −d [δn+m + δn+m−1] ≤ . . . ≤ . . . ≤ ‖xn −p‖ + 1 1 −d n+m∑ k=n+1 (Ak + Bk) + M 1 −d n+m∑ k=n+1 δk.(3.7) So, we have ‖xn+m −xn‖ ≤ ‖xn+m −p‖ + ‖xn −p‖ ≤ ‖xn −p‖ + 1 1 −d n+m∑ k=n+1 (Ak + Bk) + M 1 −d n+m∑ k=n+1 δk +‖xn −p‖ = 2‖xn −p‖ + 1 1 −d n+m∑ k=n+1 (Ak + Bk) + M 1 −d n+m∑ k=n+1 δk. (3.8) Then, we have ‖xn+m −xn‖ ≤ 2d(xn,F) + 1 1 −d n+m∑ k=n+1 (Ak + Bk) + M 1 −d n+m∑ k=n+1 δk, ∀n ≥ n0.(3.9) For any given ε > 0, there exists a positive integer n1 ≥ n0 such that for any n ≥ n1, d(xn,F) < ε 6 , n+m∑ k=n+1 (Ak + Bk) < (1 −d)ε 3 ,(3.10) and n+m∑ k=n+1 δk < (1 −d)ε 3M .(3.11) CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 87 Thus, from (3.9)-(3.11) and n ≥ n1, we have ‖xn+m −xn‖ < 2. ε 6 + 1 1 −d . (1 −d)ε 3 + M (1 −d) . (1 −d)ε 3M = ε.(3.12) This implies that {xn} is a Cauchy sequence in C. Thus, the completeness of E implies that {xn} must be convergent. Assume that limn→∞xn = p. Now, we have to show that {xn} converges to some common fixed point in F . Indeed, we know that the set F = ∩Ni=1F(Ti) ⋂ ∩Ni=1F(Si) is closed. From the continuity of d(x,F) = 0 with limn→∞d(xn,F) = 0 and limn→∞xn = p, we get d(p,F) = 0,(3.13) and so p ∈ F, that is, {xn} converges to some common fixed point in F. This completes the proof. If Si = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then Theorem 3.1 is reduced to the following result: Corollary 3.1. Let C be a nonempty closed convex subset of a Banach space E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.13). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N}, such that ∑∞ n=1 An < ∞. Assume that the following restrictions are satisfied: (a) there exist constants a,b,c ∈ (0, 1) such that a ≤ αn + βn and b ≤ γn ≤ c < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. Then {xn} converges strongly to some point in F if and only if lim inf n→∞ d(xn,F) = 0. If Ti = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then Theorem 3.1 is reduced to the following result: Corollary 3.2. Let C be a nonempty closed convex subset of a Banach s- pace E. Let Si : C → C be a uniformly Ls,i-Lipschitz and asymptotically quasi- nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Si) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} 88 SALUJA be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.14). Put Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N}, such that ∑∞ n=1 Bn < ∞. Assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn and c ≤ γn, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. Then {xn} converges strongly to some point in F if and only if lim inf n→∞ d(xn,F) = 0. We prove a lemma which plays an important role in establishing strong conver- gence of the general implicit iterative process (2.12) in a uniformly convex Banach space. Lemma 3.1. Let C be a nonempty closed convex subset of a real uniformly convex Banach space E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymp- totically quasi-nonexpansive type mapping and let Si : C → C be a uniformly Ls,i- Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) ⋂ ∩Ni=1F(Si) is nonempty. Let {un} be a bounded se- quence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.12). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N} and Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N}, such that ∑∞ n=1 An < ∞ and ∑∞ n=1 Bn < ∞. Assume that the following restric- tions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. Then lim n→∞ ‖xn −Trxn‖ = lim n→∞ ‖xn −Srxn‖ = 0, ∀ r ∈{1, 2, . . . ,N}. CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 89 Proof. As in the proof of Theorem 3.1, limn→∞‖xn −q‖ exists for all q ∈ F . It follows that the sequence {xn} is bounded. In view of Lemma 2.2, we see that ‖xn −q‖ 2 ≤ αn ‖xn−1 −q‖ 2 + βn ∥∥∥Sh(n)i(n) xn−1 −q∥∥∥2 + γn ∥∥∥Th(n)i(n) xn −q∥∥∥2 +δn ‖un −q‖ 2 −αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ + Bn]2 + γn[‖xn −q‖ + An]2 +δn ‖un −q‖ 2 −αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ 2 + K′n] + γn[‖xn −q‖ 2 + K′′n] +M1δn −αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) = (αn + βn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnK ′ n + γnK ′′ n +M1δn −αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) = (1 −γn − δn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnK ′ n + γnK ′′ n +M1δn −αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ (1 −γn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + (K′n + K ′′ n) +M1δn −αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥),(3.14) where M1 is a appropriate constant such that M1 = supn≥1{‖un −q‖ 2} and K′n = B2n + 2‖xn −q‖Bn and K′′n = A2n + 2‖xn −q‖An, since ∑∞ n=1 An < ∞ and∑∞ n=1 Bn < ∞, it follows that ∑∞ n=1 K ′ n < ∞ and ∑∞ n=1 K ′′ n < ∞. This implies that αnβng (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ (1 −γn)[‖xn−1 −q‖2 −‖xn −q‖2 ] +(K′n + K ′′ n) + M1δn.(3.15) In view of restrictions (a), (b), ∑∞ n=1 K ′ n < ∞ and ∑∞ n=1 K ′′ n < ∞, we obtain that lim n→∞ g (∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥) = 0.(3.16) Since g : [0,∞) → [0,∞) is a continuous, strictly increasing, and convex function with g(0) = 0, we obtain that lim n→∞ ∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥ = 0.(3.17) Next, we show that lim n→∞ ∥∥∥Th(n)i(n) xn −xn−1∥∥∥ = 0.(3.18) 90 SALUJA From Lemma 2.2, we also see that ‖xn −q‖ 2 ≤ αn ‖xn−1 −q‖ 2 + βn ∥∥∥Sh(n)i(n) xn−1 −q∥∥∥2 + γn ∥∥∥Th(n)i(n) xn −q∥∥∥2 +δn ‖un −q‖ 2 −αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ + Bn]2 + γn[‖xn −q‖ + An]2 +δn ‖un −q‖ 2 −αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ 2 + K′n] + γn[‖xn −q‖ 2 + K′′n] +M1δn −αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) = (αn + βn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnK ′ n + γnK ′′ n +M1δn −αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) = (1 −γn − δn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnK ′ n + γnK ′′ n +M1δn −αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) ≤ (1 −γn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + (K′n + K ′′ n) +M1δn −αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥),(3.19) This implies that αnγng (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) ≤ (1 −γn)[‖xn−1 −q‖2 −‖xn −q‖2 ] +(K′n + K ′′ n) + M1δn.(3.20) In view of restrictions (a), (b), ∑∞ n=1 K ′ n < ∞ and ∑∞ n=1 K ′′ n < ∞, we obtain that lim n→∞ g (∥∥∥Th(n)i(n) xn −xn−1∥∥∥) = 0.(3.21) Since g : [0,∞) → [0,∞) is a continuous, strictly increasing, and convex function with g(0) = 0, we obtain that (3.18) holds. Notice that ‖xn −xn−1‖ ≤ βn ∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥ + γn ∥∥∥Th(n)i(n) xn −xn−1∥∥∥ +δn ‖un −xn−1‖ .(3.22) In view of (3.17) and (3.18), we see from the restriction (b) that lim n→∞ ‖xn −xn−1‖ = 0,(3.23) which implies that lim n→∞ ‖xn −xn+j‖ = 0, ∀ j ∈{1, 2, . . . ,N}.(3.24) CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 91 Since for any positive integer n > N, it can be written as n = (h(n) − 1)N + i(n), where i(n) ∈{1, 2, . . . ,N} = I, observe that ‖xn−1 −Tnxn‖ ≤ ∥∥∥xn−1 −Th(n)i(n) xn∥∥∥ + ∥∥∥Th(n)i(n) xn −Tnxn∥∥∥ ≤ ∥∥∥xn−1 −Th(n)i(n) xn∥∥∥ + Lt ∥∥∥Th(n)−1i(n) xn −xn∥∥∥ ≤ ∥∥∥xn−1 −Th(n)i(n) xn∥∥∥ +Lt (∥∥∥Th(n)−1i(n) xn −Th(n)−1i(n−N)xn−N∥∥∥ + ∥∥∥Th(n)−1i(n−N)xn−N −x(n−N)−1∥∥∥ + ∥∥x(n−N)−1 −xn∥∥).(3.25) Since for each n > N, n = (n−N)(mod N), on the other hand, we obtain from n = (h(n)−1)N +i(n) that n−N = ((h(n)−1)−1)N +i(n) = (h(n−N)−1)N +i(n−N). That is, h(n−N) = h(n) − 1, i(n−N) = i(n).(3.26) Notice that∥∥∥Th(n)−1i(n) xn −Th(n)−1i(n−N)xn−N∥∥∥ = ∥∥∥Th(n)−1i(n) xn −Th(n)−1i(n) xn−N∥∥∥ ≤ Lt ‖xn −xn−N‖ ,(3.27) and ∥∥∥Th(n)−1i(n−N)xn−N −x(n−N)−1∥∥∥ = ∥∥∥Th(n−N)i(n−N) xn−N −x(n−N)−1∥∥∥ .(3.28) Substituting (3.27) and (3.28) into (3.25), we obtain that ‖xn−1 −Tnxn‖ ≤ ∥∥∥xn−1 −Th(n)i(n) xn∥∥∥ + Lt(Lt ‖xn −xn−N‖ + ∥∥∥Th(n−N)i(n−N) xn−N −x(n−N)−1∥∥∥ + ∥∥x(n−N)−1 −xn∥∥). (3.29) In view of (3.18) and (3.24), we obtain that lim n→∞ ‖xn−1 −Tnxn‖ = 0.(3.30) Notice that ‖xn −Tnxn‖ ≤ ‖xn −xn−1‖ + ‖xn−1 −Tnxn‖ .(3.31) It follows from (3.23) and (3.30) that lim n→∞ ‖xn −Tnxn‖ = 0.(3.32) Notice that ‖xn −Tn+jxn‖ ≤ ‖xn −xn+j‖ + ‖xn+j −Tn+jxn+j‖ +‖Tn+jxn+j −Tn+jxn‖ ≤ (1 + Lt)‖xn −xn+j‖ + ‖xn+j −Tn+jxn+j‖(3.33) for all j ∈{1, 2, . . . ,N}. 92 SALUJA From (3.24) and (3.32), we obtain that lim n→∞ ‖xn −Tn+jxn‖ = 0, ∀ j ∈{1, 2, . . . ,N}.(3.34) Note that any subsequence of a convergent sequence converges to the same limit, it follows that lim n→∞ ‖xn −Trxn‖ = 0, ∀ r ∈{1, 2, . . . ,N}.(3.35) Letting Ls = max{Ls,i : 1 ≤ i ≤ N}, we have∥∥∥Sh(n)i(n) xn −xn−1∥∥∥ ≤ ∥∥∥Sh(n)i(n) xn −Sh(n)i(n) xn−1∥∥∥ + ∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥ ≤ Ls ‖xn −xn−1‖ + ∥∥∥Sh(n)i(n) xn−1 −xn−1∥∥∥ .(3.36) In view of (3.17) and (3.23), we see that∥∥∥Sh(n)i(n) xn −xn−1∥∥∥ = 0.(3.37) Observe that ‖xn−1 −Snxn−1‖ ≤ ∥∥∥xn−1 −Sh(n)i(n) xn−1∥∥∥ + ∥∥∥Sh(n)i(n) xn−1 −Snxn−1∥∥∥ ≤ ∥∥∥xn−1 −Sh(n)i(n) xn−1∥∥∥ + Ls ∥∥∥Sh(n)−1i(n) xn−1 −xn−1∥∥∥ ≤ ∥∥∥xn−1 −Sh(n)i(n) xn−1∥∥∥ + Ls(∥∥∥Sh(n)−1i(n) xn−1 −Sh(n)−1i(n−N)xn−N∥∥∥ + ∥∥∥Sh(n)−1i(n−N)xn−N −x(n−N)−1∥∥∥ + ∥∥x(n−N)−1 −xn−1∥∥). (3.38) In view of∥∥∥Sh(n)−1i(n) xn−1 −Sh(n)−1i(n−N)xn−N∥∥∥ = ∥∥∥Sh(n)−1i(n) xn−1 −Sh(n)−1i(n) xn−N∥∥∥ ≤ Ls ‖xn−1 −xn−N‖ ,(3.39) and ∥∥∥Sh(n)−1i(n−N)xn−N −x(n−N)−1∥∥∥ = ∥∥∥Sh(n−N)i(n−N) xn−N −x(n−N)−1∥∥∥ ,(3.40) we obtain that ‖xn−1 −Snxn−1‖ ≤ ∥∥∥xn−1 −Sh(n)i(n) xn−1∥∥∥ + Ls(Ls ‖xn−1 −xn−N‖ + ∥∥∥Sh(n)−1i(n−N)xn−N −x(n−N)−1∥∥∥ + ∥∥x(n−N)−1 −xn−1∥∥). (3.41) In view of (3.17), (3.24) and (3.37), we obtain that lim n→∞ ‖xn−1 −Snxn−1‖ = 0.(3.42) Notice that ‖xn −Snxn‖ ≤ ‖xn −xn−1‖ + ‖xn−1 −Snxn−1‖ + ‖Snxn−1 −Snxn‖ ≤ (1 + Ls)‖xn −xn−1‖ + ‖xn−1 −Snxn−1‖ .(3.43) From (3.23) and (3.42), we see that lim n→∞ ‖xn −Snxn‖ = 0.(3.44) CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 93 On the other hand, we have ‖xn −Sn+jxn‖ ≤ ‖xn −xn+j‖ + ‖xn+j −Sn+jxn+j‖ + ‖Sn+jxn+j −Sn+jxn‖ ≤ (1 + Ls)‖xn −xn+j‖ + ‖xn+j −Sn+jxn+j‖ , ∀ j ∈{1, 2, . . . ,N}. (3.45) It follows from (3.24) and (3.45) that lim n→∞ ‖xn −Sn+jxn‖ = 0, ∀ j ∈{1, 2, . . . ,N}.(3.46) Note that any subsequence of a convergent sequence converges to the same limit, it follows that lim n→∞ ‖xn −Srxn‖ = 0, ∀ r ∈{1, 2, . . . ,N}.(3.47) This completes the proof. Now, we are in a position to prove our strong convergence theorems. Theorem 3.2. Let C be a nonempty closed convex subset of a real uniformly convex Banach space E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymp- totically quasi-nonexpansive type mapping and let Si : C → C be a uniformly Ls,i- Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) ⋂ ∩Ni=1F(Si) is nonempty. Let {un} be a bounded se- quence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.12). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N} and Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N}, such that ∑∞ n=1 An < ∞ and ∑∞ n=1 Bn < ∞. Assume that the following restric- tions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. If one of {S1,S2, . . . ,SN} or one of {T1,T2, . . . ,TN} is semicompact, then the sequence {xn} converges strongly to some point in F . Proof. By Lemma 3.1, it follows that lim n→∞ ‖xn −Trxn‖ = lim n→∞ ‖xn −Srxn‖ = 0, ∀ r ∈{1, 2, . . . ,N}.(3.48) Without any loss of generality, we may assume that S1 is semi-compact. Therefore, by (3.48), it follows that limn→∞‖xn −S1xn‖ = 0. Since S1 is semi-compact, 94 SALUJA therefore there exists a subsequence {xnj} of {xn} such that xnj → x∗ ∈ C. For each r ∈{1, 2, . . . ,N}, we get that ‖x∗ −Srx∗‖ ≤ ∥∥x∗ −xnj∥∥ + ∥∥xnj −Srxnj∥∥ + ∥∥Srxnj −Srx∗∥∥ . (3.49) Since Sr is Lipschitz continuous, we obtain from (3.48) that x ∗ ∈ ∩Nr=1F(Sr). Notice that ‖x∗ −Trx∗‖ ≤ ∥∥x∗ −xnj∥∥ + ∥∥xnj −Trxnj∥∥ + ∥∥Trxnj −Trx∗∥∥ . (3.50) Since Tr is Lipschitz continuous, we obtain from (3.48) that x ∗ ∈∩Nr=1F(Tr). This means that x∗ ∈ F . In view of Theorem 3.1, we obtain that limn→∞ ‖xn −q‖ exists for all q ∈ F, therefore {xn} converges to x∗ ∈ F, and hence the result. This completes the proof. If Si = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then Theorem 3.2 is reduced to the following result: Corollary 3.3. Let C be a nonempty closed convex subset of a Banach space E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.13). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N}, such that ∑∞ n=1 An < ∞. Assume that the following restrictions are satisfied: (a) there exist constants a,b,c ∈ (0, 1) such that a ≤ αn + βn and b ≤ γn ≤ c < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. If one of {T1,T2, . . . ,TN} is semicompact, then the sequence {xn} converges strongly to some point in F . If Ti = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then Theorem 3.2 is reduced to the following result: Corollary 3.4. Let C be a nonempty closed convex subset of a Banach s- pace E. Let Si : C → C be a uniformly Ls,i-Lipschitz and asymptotically quasi- nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Si) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.14). Put Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N}, CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 95 such that ∑∞ n=1 Bn < ∞. Assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn and c ≤ γn, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. If one of {S1,S2, . . . ,SN} is semicompact, then the sequence {xn} converges strongly to some point in F. Remark 3.1. Theorem 3.2 extends and improves Theorem 3.3 due to Sun [19] to the case of more general class of asymptotically quasi-nonexpansive mapping and general implicit iterative process and without the boundedness of C which in turn generalizes Theorem 2 by Wittmann [21] from Hilbert spaces to uniformly convex Banach spaces. In 2005, Chidume and Shahzad [1]) introduced the following conception. Recall that a family {Ti}Ni=1 : C → C with F = ∩ N i=1F(Ti) 6= ∅ is said to satisfy Condition (B) on C if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that for all x ∈ C max 1≤i≤N {‖x−Tix‖}≥ f(d(x,F)).(3.51) Based on Condition (B), Qin et al. [14] introduced the following conception for two finite families of mappings. Recall that two families {Si}Ni=1 : C → C and {Ti}Ni=1 : C → C with F = ∩ N i=1F(Si) ⋂ ∩Ni=1F(Ti) 6= ∅ are said to satisfy Condition (B′) if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that for all x ∈ C max 1≤i≤N {‖x−Six‖ + ‖x−Tix‖}≥ f(d(x,F)).(3.52) Note that Condition (B′) defined above reduces to the Condition (B) [1] if we choose Si = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}. Finally, an application of the convergence criteria established in Theorem 3.1 is given below to obtain yet another strong convergence result in our setting. 4. Application Theorem 4.1. Let C be a nonempty closed convex subset of a real uniformly convex Banach space E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymp- totically quasi-nonexpansive type mapping and let Si : C → C be a uniformly Ls,i- Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) ⋂ ∩Ni=1F(Si) is nonempty. Let {un} be a bounded se- quence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.12). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N} 96 SALUJA and Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N}, such that ∑∞ n=1 An < ∞ and ∑∞ n=1 Bn < ∞. Assume that the following restric- tions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. If {S1,S2, . . . ,SN} and {T1,T2, . . . ,TN} satisfy Condition (B′), then the itera- tive sequence {xn} converges strongly to some point in F. Proof. As in the proof of Theorem 3.2, (3.48) holds. Taking lim inf on both sides of Condition (B′) and using (3.48), we have that lim infn→∞ f(d(xn,F)) = 0. Since f is a nondecreasing function with f(0) = 0 and f(r) > 0 for all r ∈ (0.∞), it follows that lim infn→∞d(xn,F) = 0. Now by Theorem 3.1, xn → p ∈ F, that is, {xn} converges strongly to a point in F. This completes the proof. If Si = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then Theorem 4.1 is reduced to the following result: Corollary 4.1. Let C be a nonempty closed convex subset of a Banach space E. Let Ti : C → C be a uniformly Lt,i-Lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Ti) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.13). Put An = max { 0, sup p∈F, n≥1 (∥∥∥Th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈N}, such that ∑∞ n=1 An < ∞. Assume that the following restrictions are satisfied: (a) there exist constants a,b,c ∈ (0, 1) such that a ≤ αn + βn and b ≤ γn ≤ c < 1/Lt, where Lt = max{Lt,i : 1 ≤ i ≤ N}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. If {T1,T2, . . . ,TN} satisfies Condition (B), then the iterative sequence {xn} con- verges strongly to some point in F . If Ti = I, where I denotes the identity mapping, for each i ∈{1, 2, . . . ,N}, then Theorem 4.1 is reduced to the following result: CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH ERRORS 97 Corollary 4.2. Let C be a nonempty closed convex subset of a Banach s- pace E. Let Si : C → C be a uniformly Ls,i-Lipschitz and asymptotically quasi- nonexpansive type mapping for each 1 ≤ i ≤ N. Assume that F = ∩Ni=1F(Si) is nonempty. Let {un} be a bounded sequence in C. Let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. Let {xn} be a iterative sequence generated in (2.14). Put Bn = max { 0, sup p∈F, n≥1 (∥∥∥Sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈N}, such that ∑∞ n=1 Bn < ∞. Assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn and c ≤ γn, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. If {S1,S2, . . . ,SN} satisfies Condition (B), then the iterative sequence {xn} con- verges strongly to some point in F. Remark 4.1. (i) Our results extend the corresponding results of Ud-din and Khan [4] to the case of more general class of asymptotically quasi-nonexpansive mappings considered in this paper. (ii) Our results also generalize and improve the corresponding results of Sun [19], Wittmann [21] and Xu and Ori [24] to the case of more general class of nonex- pansive, asymptotically quasi-nonexpansive mappings and general implicit iterative process for two finite families of mappings considered in this paper. (iii) Our results also extend the corresponding results of [1, 3, 7, 15] and many others. Example 4.1. Let E be the real line with the usual norm |.| and K = [0, 1]. Define T : K → K by T(x) = sinx, x ∈ [0, 1], for x ∈ K. Obviously T(0) = 0, that is, 0 is a fixed point of T, that is, F(T) = {0}. Now we check that T asymptotically quasi-nonexpansive type mapping. In fact, if x ∈ [0, 1] and p = 0 ∈ [0, 1], then |T(x) −p| = |T(x) − 0| = |sinx− 0| = |sinx| ≤ |x| = |x− 0| = |x−p|, that is, |T(x) −p| ≤ |x−p|. Thus, T is quasi-nonexpansive. 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